Géraldine Dahlqvist

Farid Gaouar

Fabrice Carrat

Sofia Meurisse

Olivier Chazouillères

Raoul Poupon

Catherine Johanet

M.D. Christophe Corpechot
email: christophe.corpechot@aphp.fr

Large-scale Characterization Study of Patients with Antimitochondrial Antibodies but Nonestablished Primary Biliary Cholangitis

Large-Scale Characterization Study of Patients With Antimitochondrial Antibodies but Nonestablished Primary Biliary Cholangitis

G eraldine Dahlqvist

AQ8

, [START_REF] Kaplan | Primary biliary cirrhosis[END_REF]2 Farid Gaouar, [START_REF] Kaplan | Primary biliary cirrhosis[END_REF] Fabrice Carrat, [START_REF] Lindor | Primary biliary cirrhosis[END_REF][START_REF] Walker | Serological tests in diagnosis of primary biliary cirrhosis[END_REF] Sofia Meurisse, [START_REF] Lindor | Primary biliary cirrhosis[END_REF][START_REF] Walker | Serological tests in diagnosis of primary biliary cirrhosis[END_REF] Olivier Chazouille `res, [START_REF] Kaplan | Primary biliary cirrhosis[END_REF][START_REF] Munoz | Is mitochondrial antibody diagnostic of primary biliary cirrhosis?[END_REF] Raoul Poupon, [START_REF] Kaplan | Primary biliary cirrhosis[END_REF][START_REF] Munoz | Is mitochondrial antibody diagnostic of primary biliary cirrhosis?[END_REF] Catherine Johanet, [START_REF] Grimbert | Antimitochondrial antibodies in patients with chronic hepatitis C[END_REF] and Christophe Corpechot [START_REF] Kaplan | Primary biliary cirrhosis[END_REF][START_REF] Munoz | Is mitochondrial antibody diagnostic of primary biliary cirrhosis?[END_REF] ; the French network of Immunology Laboratories

The prevalence, clinical characteristics, and outcomes of patients with antimitochondrial antibodies (AMAs), but no clinical evidence of primary biliary cholangitis (PBC), are largely unknown. A prospective study of AMA incidence was conducted through a nation-wide network of 63 French immunology laboratories. Clinical data from 720 of 1,318 AMApositive patients identified in 1 year were collected. Patients were categorized as either newly diagnosed with PBC (n 5 275), previously diagnosed with PBC (n 5 216), or with nonestablished diagnosis of PBC (n 5 229). The latter group was specifically evaluated. Follow-up data were collected for up to 7 years after detection of AMAs. Prevalence of AMApositive patients without evidence of PBC was 16.1 per 100,000. These patients had the following characteristics: 78% female; median age 58 years; median AMA titer 1:160; extrahepatic autoimmune disorders 46%; normal serum alkaline phosphatases (ALP) 74%; ALP above 1.5 times the upper limit of normal 13%; and cirrhosis 6%. Compared to those newly diagnosed with PBC, the patients were slightly younger, had lower AMA titers, and lower sex-ratio imbalance. Among the patients with normal ALP and no evidence of cirrhosis, the 5-year incidence rate of PBC was 16%. Whereas no patients died from PBC, the 5-year survival rate was 75%, as compared to 90% in a control, standardized population matched for age and sex (P < 0.05). Conclusion: Nearly half of the newly detected AMAs in clinical practice does not lead to a diagnosis of PBC. PBC is unrecognized in 13% of those cases. Only 1 in 6 patients with AMAs and normal ALP will develop PBC after 5 years. The mortality of AMA-positive patients without PBC is increased irrespective of the risk of PBC development. (HEPATOLOGY 2016; 00:000-000). P rimary biliary cholangitis (PBC) is a chronic cholestatic liver disease of unknown, but presumably autoimmune, origin characterized by elevated alkaline phosphatases (ALP) activity and presence of antimitochondrial antibodies (AMAs) in serum and highly suggestive histological lesions, namely, granulomatous or lymphocytic nonsuppurative destructive cholangitis of interlobular bile ducts, on liver biopsy. [START_REF] Kaplan | Primary biliary cirrhosis[END_REF] The standardization in clinical practice of AMA detection has led to earlier diagnosis and management of the disease, avoiding, in most patients, an evolution to end-stage liver disease, at least in part, because of the prescription of ursodeoxycholic acid (UDCA), which is the only first-line pharmacological treatment currently approved by drug agencies and international medical societies. (2,[START_REF] Lindor | Primary biliary cirrhosis[END_REF] Abbreviations: AID, autoimmune disease; AIH, autoimmune hepatitis; ALP, alkaline phosphatase; ALT, alanine aminotransferase; AMAs, antimitochondrial antibodies type 2; ANAs, antinuclear antibodies; CI, confidence interval; GGT, gamma-glutamyl transpeptidase; IF, immunofluorescence; IgM, immunoglobulin M; PBC, primary biliary cholangitis; UDCA, ursodeoxycholic acid; ULN, upper limit of normal.

AMAs are regarded as highly specific (95%) of PBC in people with abnormal ALP, so that a diagnosis of PBC can be made with confidence without the use of a biopsy in patients with unexplained elevation of ALP and presence of AMAs. (2,[START_REF] Walker | Serological tests in diagnosis of primary biliary cirrhosis[END_REF][START_REF] Munoz | Is mitochondrial antibody diagnostic of primary biliary cirrhosis?[END_REF] However, their detection in the absence of any other symptoms or signs of the disease has been widely reported, notably in patients with extrahepatic autoimmune disorders, such as scleroderma, Sj€ ogren's syndrome, or autoimmune thyroid disease, as well as in patients with hepatic or extrahepatic nonautoimmune disorders such as chronic hepatitis C or hematological malignancies. [START_REF] Grimbert | Antimitochondrial antibodies in patients with chronic hepatitis C[END_REF][START_REF] Zamfir | Anti-pyruvate dehydrogenase autoantibodies in extrahepatic disorders[END_REF] In such a situation, longitudinal studies seem to indicate AMAs to be inevitably associated with PBC development, but these studies are rather old, all originated from a single UK center and limited area, and involve no more than 29 patients in total. [START_REF] James | Primary biliary cirrhosis-a revised clinical spectrum[END_REF][START_REF] Mitchison | Positive antimitochondrial antibody but normal alkaline phosphatase: is this primary biliary cirrhosis?[END_REF][START_REF] Metcalf | Natural history of early primary biliary cirrhosis[END_REF] So far, no large, prospective study has been conducted to assess specifically the significance of AMAs in the absence of clinical indications of PBC. The aim of the present study was therefore to evaluate, prospectively and at a large-scale level, the prevalence, clinical characteristics, and outcomes of patients with AMAs, but nonestablished diagnosis of PBC.

Patients and Methods

STUDY DESIGN AND POPULATION

This was a prospective, nation-wide, observational study conducted in France during May 2006-September 2013 in which an extensive network of clinical immunology laboratories, including 63 laboratories covering approximately 90% of the French metropolitan territory, was actively involved. The geographical location of the laboratories is shown in Fig. F1 1. The primary objective of the study was to assess the incidence of newly identified AMA-positive patients in France based on the tests routinely prescribed in clinical practice. For this purpose, every positive AMA tests identified between May 2006 and June 2007 were prospectively recorded in a secure web database through which the patients were registered in an anonymized format together with the name and address of the prescribing physician. This constituted the 1-year census phase of the study.

All laboratories used the same first-line detection methods and diagnostic algorithms. In line with the international recommendations, serum AMAs were first detected by indirect immunofluorescence (IF) on rat liver, kidney, and stomach tissue sections. Every antibody titer equal to or higher than 1:40 was considered positive and subsequently tested by immunodotting or immunoblotting on mitochondrial recombinant or native antigens for confirmation. For the latter purpose, most laboratories used ready-for-use immunodotting test commercial kits. The reference lab of Saint-Antoine hospital, Paris, used its own immunoblotting method, as described. [START_REF] Dubel | Kinetics of anti-M2 antibodies after liver transplantation for primary biliary cirrhosis[END_REF] PBC-specific antinuclear antibodies (ANAs), that is, ANA with either rim-like/membranous or multiple nuclear dot IF patterns, were detected on Hep-2 cells. A titer equal to or greater than 1:80 was considered significant.

After centralization and unanonymization of the data, the prescribing physicians were contacted and invited to complete a questionnaire on the past history, clinical, biochemical, and (when appropriate) histological features of their patient(s), and to state whether a

ARTICLE INFORMATION:

From the 1 Service d'H epatologie, Centre de r ef erence des Maladies Inflammatoires des Voies Biliaires (MIVB), Filie `re de Sant e Maladies Rares du Foie de l'Adulte et de l'Enfant (FILFOIE), Hôpital Saint-Antoine, Assistance Publique-Hôpitaux de Paris (APHP), Paris, France; 2 Service d'H epato-Gastroent erologie, Cliniques Universitaires Saint-Luc, Brussels, Belgium; [START_REF] Lindor | Primary biliary cirrhosis[END_REF] diagnosis of PBC was officially retained (diagnosis was based on physician opinion in real-life conditions). The items constitutive of this questionnaire are available in Supporting Table S1. All patients were delivered an information form by their physician indicating that unless they notify their objection, some of their medical data will be recorded for research purposes in the strict and specific field of autoimmunity.

According to the medical information thus received, the patients were categorized as being in one of the three following situations: (1) patients newly diagnosed with PBC (i.e., incident case of PBC); (2) patients previously diagnosed with PBC (i.e., preexisting case of PBC); and (3) patients nondiagnosed with PBC (i.e., nonestablished case of PBC).

For the purpose of the study, the focus was made on the latter group of patients. Prospective follow-up data were collected by asking the prescribing physicians to complete a second questionnaire 2, 4, 5, and 7 years, respectively, after the beginning of the census period and to state about the health and diagnosis status of their patient(s) over time. This constituted the follow-up phase of the study. The items constitutive of this second questionnaire are available in Supporting Table S2. Survival and PBC incidence rates were directly estimated from these medical reports.

STATISTICAL ANALYSIS

Descriptive statistics were expressed as median (range) or number (%). The incidence rates of AMAs and PBC were calculated by dividing the number of incident cases registered during the 1-year census period by the total number of people in metropolitan France during the same period, taking into account the territory coverage of AMA screening, the participation rate of the prescribing physicians, and a 10% expected ratio of AMA-negative PBC. [START_REF] Galula | Quelle la fr equence r eelle des cirrhoses biliaires primitives s eron egatives?[END_REF] Rates were calculated by sex and 10-year-age categories. Assuming that these rates were invariable in time, point prevalence was estimated using the following formula: prevalence 5 incidence 3 life expectancy. Life expectancy was evaluated from a cohort of 378 patients with PBC of similar age structure followed-up in Saint-Antoine hospital, Paris, between 1985 and 2005. Patient groups were compared using the Student t test or the Mann-Whitney U test, when appropriate, for continuous variables and the chi-square test, or the Fisher's exact test, when appropriate, for categorical variables. Because survival data were determined between fixed intervals of time rather than using the exact date of death (a variable that was frequently missing), survival rates were calculated using an actuarial method (i.e., life table analysis) with 1-year constant intervals. A standardized population matched for age, sex, and follow-up period served as a control. Estimates of survival rates of this population were obtained from the French official census tables. (13) Observed and expected survival rates were compared using the log-rank test. Prognostic variables were studied using a Cox regression model. Cumulative incidence function for PBC was estimated using a nonparametric method dealing with competing risk of death. (14) Excluded from this analysis were patients who had serum ALP activity higher than 1 times the upper limit of normal (ULN), pruritus, cirrhosis, or PBC-compatible lesions on histology (i.e., those with possible or presumed PBC).

INSTITUTIONAL REVIEW BOARD APPROVAL

The study was approved by the French Advisory Committee for Data Processing in Health Research and the National Commission on Informatics and Liberty before data collection.

Results

INCIDENCE AND PREVALENCE OF AMA-POSITIVE PATIENTS WITH OR WITHOUT ESTABLISHED PBC

A total of 1,367 positive AMA tests were registered in 1,318 patients during the 1-year census period of the study (Fig. F2 2). The total number of AMA tests performed over this period was highly variable according to the laboratories (range, 64-18,500; median, 563), but the percentages of positive results were similar between them, with an average of 2.5% of the whole tests (95% confidence interval [CI], 1.6-3.5; Supporting Table S3). The prescribing physician was solicited for 1,149 (87%) of the AMA-positive patients. Medical information could be collected in 772 (67%) patients, but was really exploitable in 720 (63%). Among the latter were counted 275 (38%) incident cases of PBC, 216 (30%) prevalent cases of PBC, and 229 (32%) nonestablished cases of PBC (Fig. 2).

Incidence rates per 100,000 inhabitant-years of AMAs and PBC were 1.7 and 1.0, respectively. These rates varied as a function of age and sex (Fig. F3 3). They were 4 (all AMA-positive patients) to 6 (PBC patients) times higher in females than in males. In both sexes, incidence increased linearly with age from 20 years old to reach a peak during the eighth decade of life (Fig. 3). The estimated prevalence rates per 100,000 inhabitants of AMAs and PBC were 40.4 and 24.3, respectively. The prevalence rate of AMApositive patients with nonestablished PBC was 16.1 per 100,000 inhabitants.

CHARACTERISTICS OF AMA-POSITIVE PATIENTS WITH NONESTABLISHED PBC

Clinical characteristics of the 229 patients with a positive AMA test, but no established diagnosis of PBC, are shown in Table T1 1. These patients were mainly females (78%) aged over 40 years (median age, 58). The median titer of AMA was 1:160. Immunodotting/blotting tests, available in 150 (66%) patients, were positive in 91% of the tested individuals. Patients not tested with these second-line methods had similar AMA titers and demographic characteristics than those tested with (data not shown). The proportion of patients with PBC-specific ANAs was 6%. In nearly half of the cases (46%), these antibodies were revealed in the evaluation of an autoimmune disorder. The most frequent autoimmune diseases (AIDs) were systemic lupus erythematosus (n 5 18; 8%), Sj€ ogren's syndrome (n 5 14; 6%), and autoimmune hepatitis (AIH; n 5 10; 4%). In 12% of the cases, AMAs were found in the background of a nonautoimmune liver disease. The most frequent of these diseases were chronic hepatitis C (n 5 8; 3%) and alcoholic liver disease (n 5 8; 3%). All the clinical settings in which AMAs were evidenced are available in Supporting Table S4. Regretfully, because of too few longitudinal data, persistence of AMA positivity was not assessable.

Complete or partial biochemical data were available in 130 (57%) patients. Median serum levels of total bilirubin, ALP, alanine aminotransferase (ALT), and immunoglobulin M (IgM) were all in the normal range, whereas gamma-glutamyl transpeptidase (GGT) was slightly above the ULN (Table 1). Serum ALP and whole biochemical liver tests (i.e., bilirubin, ALP, GGT, and ALT) were normal in 74% and 44% of patients, respectively. An ALP level above 1.53 ULN without an alternative explanation (i.e., liver tumor, drug-induced liver injury, or any other wellidentified liver diseases) to PBC was observed in 13% of cases. A liver biopsy was performed in 28 (19%) of the 148 patients for which data were available. None of the histological reports were suggestive of PBC given that no bile duct lesions or granulomas were reported.

Histological reports could include normal histology, mild portal inflammation, steatohepatitis, histological features compatible with AIH, or cirrhosis. Cirrhosis, whether diagnosed on histology or on clinical and ultrasound findings, was reported in 13 (6%) patients. In these patients, alcohol abuse was reported in half of the cases (n 5 7), whereas no etiology was reported in 4 (30%). The patients with a negative immunodot/blot test (n 5 13) had similar characteristics than those with a positive one (n5137; Supporting Table S5). Likewise, the patients with a low IF AMA titer S6).

Patients were compared to those (n 5 275; analyzable 5 247) concomitantly diagnosed with PBC during the same period (Table T2 2). Compared to the latter group, they tended to be slightly younger and to have fewer symptoms. The proportion of patients with fatigue did not differ between groups, but pruritus and jaundice were less frequently observed. Patients had a significantly lower titer of AMAs, a lower proportion of PBC-specific ANAs, a lesser sex ratio imbalance, and significantly lower serum levels of total bilirubin, ALP, GGT, ALT, and IgM (Table 2). Also, they were less frequently subjected to liver biopsy and were less likely to have cirrhosis. Finally, they had comparable personal and familial past histories of autoimmune disorders.

CLINICAL OUTCOMES OF AMA-POSITIVE PATIENTS WITH NONESTABLISHED PBC

Follow-up data were available in 92 (41%) of the 222 patients still alive at the end of the census period. Mean duration of follow-up was 4.0 6 1.8 years (range, 0.5-7.3). Taking all deaths into account (including those recorded during the census period), a total of 20 (9%) patients died during the whole study period. Considering the follow-up phase specifically, 17 (18%) of the 92 followed-up patients died. Median age at death was 73.6 years (range, 42.4-90.0). Causes of death were the following: non-liver-related cancers, 5 (bronchial squamous-cell carcinoma, 1; pulmonary adenocarcinoma, 1; and metastatic adenocarcinoma of undetermined origin 3); hematological malignancies 4 (non-Hodgkin's lymphoma, 2; acute myeloid leukaemia, 1; and multiple myeloma, 1); infectious diseases, 3 (lung infection, 2; infective endocarditis, 1); cardiovascular diseases, 2 (multifactorial heart failure, 1; chronic cor pulmonale, 1); digestive diseases, 3 (alcoholic chronic liver failure, 2; severe acute pancreatitis, 1); and undetermined cause, 3. No patients died officially from PBC. Actuarial and Kaplan-Meier survival curves are shown in Fig. F4 4. The 1-, 3-, and 5-year rates of survival were 95% (95% CI, 92-98), 90% (95% CI, 85-96), and 75% (95% CI, 63-87), respectively. These rates were significantly lower than those expected in the control population, specifically, 99% (95% CI, 97-100), 96% (95% CI, 92-100), and 90% (95% CI, 83-98), respectively (P 5 0.023).

Development of PBC was reported in 9 (10%) of the 92 followed-up patients, among whom 8 were female (89%). Median age at PBC diagnosis was 62.1 years (range, 35.9-69.8). Incidence rates were calculated after excluding patients with elevated ALP, pruritus, or cirrhosis at baseline (i.e., from 66 of 92 patients) and by taking care of the competing risk of death. The incidence curve is shown in Fig.

F5 5. Cumulative incidence rates of PBC at 1, 3, and 5 years were 2% (95% CI, 0-7), 7% (95% CI, 2-15), and 16% (95% CI, 6-29), respectively. Neither age, sex, AMA titer, PBC-specific ANAs, personal or familial history of AID, or baseline serum levels of bilirubin, ALP, GGT, ALT, or IgM were predictive of PBC development (Table T3 3). The incidence rate of PBC was not altered by exclusion of patients with a negative dotting/blotting test (Supporting Fig. S5). In addition, no statistical difference was found between the low (51:40) and high (1:80) IF AMA titer groups, although no case of incident PBC was reported in the former group (Supporting Fig. S6).

Discussion

In this study, we showed that nearly half of the prospectively detected AMAs in clinical practice was not related to a diagnosis of PBC. This observation raises again the question of the true significance of AMAs in human pathology. [START_REF] Onji | Significance of study on prevalence of AMA in healthy subjects[END_REF][START_REF] Dahlqvist | Are antimitochondrial antibodies the invariable hallmark of primary biliary cirrhosis?[END_REF] Using classical diagnostic criteria, however, we showed that 13% of these patients had definite PBC, suggesting that the disease may be underdiagnosed by physicians, more likely internists and nonspecialists of the liver given that a large proportion of AMAs are revealed in the settings of nonhepatic AIDs. Notwithstanding, taking care to exclude these false-negative cases at inclusion, our follow-up data clearly showed that only a few patients eventually developed PBC over a mean follow-up of 4 years, whereas, in the same time, this population was shown to have an increased risk of mortality as compared to a matched control population. PBC is asymptomatic for years and increasingly diagnosed incidentally on routine blood tests revealing mild chronic cholestasis. In this common situation, a positive AMA test is of major significance because of its high sensitivity and specificity for diagnosis of PBC. AMAs are one of the earlier hallmarks of the disease. They may antedate histological and biochemical manifestations by several years and persist thereafter throughout the course of the disease. This makes the AMA test particularly attractive for assessing the extent of the PBC spectrum on a population scale. Our study was designed on such an AMA screening-based strategy. However, it was not comparable to a seroprevalence study because AMA tests were substantiated by clinical indications, and thus the study was not able to capture all AMA-positive individuals. Consequently, our incidence and prevalence data should be regarded as low-end estimates of what reality is. Notwithstanding, it is emphasized that these estimates, which are the first epidemiological data available in France on both PBC and AMAs, are quite comparable to those previously reported in Western Europe countries. [START_REF] Metcalf | Incidence and prevalence of primary biliary cirrhosis in the city of Newcastle upon Tyne, England[END_REF][START_REF] Boberg | Incidence and prevalence of primary biliary cirrhosis, primary sclerosing cholangitis, and autoimmune hepatitis in a Norwegian population[END_REF][START_REF] Lofgren | Incidence and prevalence of primary biliary cirrhosis in a defined population in Sweden[END_REF] Systematic screening of blood donors and healthy subjects shows AMA-positive results in 0.07%-9.9% of individuals, depending on the techniques and diagnostic thresholds used, as well as on the type, age, and sex structures of the populations screened. [START_REF] Mattalia | Characterization of antimitochondrial antibodies in health adults[END_REF][START_REF] Omagari | Autoantibodies to M2 mitochondrial autoantigens in normal human sera by immunofluorescence and novel assays[END_REF][START_REF] Turchany | A study of antimitochondrial antibodies in a random population in Estonia[END_REF][START_REF] Shibata | Prevalence of antimitochondrial antibody in Japanese corporate workers in Kanagawa prefecture[END_REF][START_REF] Liu | Frequencies of autoantibodies specific for primary biliary cirrhosis in a general adult population group[END_REF] The relatively high rate (2.5% of all tests) of AMA positivity observed in our study may be related to the equivocal nature of low titers of AMAs detected by IF, but is more likely to be driven by the population concerned. The proportion of positive results is typically bigger in the female population above 40 years. The question is to know whether AMAs, in such circumstances, are associated with underlying indolent, but nevertheless slowly progressive, PBC that may necessitate UDCA treatment. Considering both PBC and AMA prevalences in the Japanese population, Shibata et al. inferred that only 0.73% of the AMA carriers in Japan should suffer from symptomatic PBC. [START_REF] Shibata | Prevalence of antimitochondrial antibody in Japanese corporate workers in Kanagawa prefecture[END_REF] These data, which are in keeping with ours, point out that AMArelated conditions are a wide, poorly known area from which PBC, as the tip of the iceberg, would just be the known side.

In 1996, Metcalf et al. described a cohort of 29 asymptomatic patients who were positive for AMAs without any other signs of the liver disease at first detection. [START_REF] Metcalf | Natural history of early primary biliary cirrhosis[END_REF] All the 29 patients were previously screened in a workup for another AID. [START_REF] Mitchison | Positive antimitochondrial antibody but normal alkaline phosphatase: is this primary biliary cirrhosis?[END_REF] Liver biopsies were performed in the majority of them, and at the screening, 24 patients had histological lesions compatible with or diagnostic of PBC, suggesting that, before the advent of any clinical or biochemical manifestations, those patients did have PBC. Median follow-up was 17.8 years. During that period, 76% developed symptoms of PBC and 83% had persistently abnormal liver tests showing cholestasis after a median time of only 5.6 years. No patients, however, developed portal hypertension or cirrhosis, and no patients died from PBC, pointing out that the progression of the disease in these patients was very slow.

Unlike the UK cohort, only a minority (19%) of our patients had a liver biopsy at AMA detection. The absolute number of biopsied patients, however, was similar (n 5 28). In contrast with the UK experience, none of those patients displayed histological lesions suggestive of PBC. However, 13% of all patients met biochemical criteria for the disease. In addition, only 44% of the patients had their whole biochemical liver tests strictly normal at the time of AMA detection, thus suggesting that many of them did have a mild, but unrecognized, form of the disease. The fact that the patients were statistically younger and had lower AMA titers than those diagnosed with PBC at the same time supports the hypothesis of a very early, indolent phase of the disease. In contrast to the UK study, however, occurrence of PBC manifestations was reported in a minority (16%) of our patients after 5 years of follow-up. These inconsistent findings may result from inherent discrepancies between the populations studied and the methods used (single-center retrospective selection vs. prospective nation-wide screening). Notwithstanding, our data suggest that the breakdown of immune tolerance to PBC-specific mitochondrial epitopes does not necessarily result in disease emergence.

We sought to describe the mortality of this specific population. Our data showed that AMA carriers with no manifestations of PBC experienced an increased risk of mortality regardless of PBC risk. Patients mainly died from nonhepatic primary cancers or hematological malignancies, whereas there is classically no such increased risk of extrahepatic malignant conditions in PBC. [START_REF] Howel | Cancer risk in primary biliary cirrhosis: a study in northern England[END_REF][START_REF] Jackson | Influence of ursodeoxycholic acid on the mortality and malignancy associated with primary biliary cirrhosis: a population-based cohort study[END_REF][START_REF] Floreani | Extrahepatic autoimmune conditions associated with primary biliary cirrhosis[END_REF] Quite rightly, it may be argued that such a link between AMAs and non-liver-related mortality likely results from complications of concomitant diseases and/or treatments (immunosuppressive drugs) rather than from the consequences of AMAs. Unfortunately, we could not obtain data from the patients who were tested AMA negative during the same census period, which precludes any conclusions. However, whether AMAs could directly or indirectly give rise to an increased risk of death is a hypothesis that cannot totally be excluded. This hypothesis is indeed supported by the increase in non-liver-related mortality reported in some population-based studies [START_REF] Jackson | Influence of ursodeoxycholic acid on the mortality and malignancy associated with primary biliary cirrhosis: a population-based cohort study[END_REF][START_REF] Prince | Survival and symptom progression in a geographically based cohort of paients with primary biliary cirrhosis: follow-up for up to 28 years[END_REF] and the growing evidence that patients with PBC may suffer from significant systemic dysfunctions. [START_REF] Jones | Impaired cardiovascular function in primary biliary cirrhosis[END_REF][START_REF] Mcdonald | Central nervous system dysfunction in primary biliary cirrhosis and its relationship to symptoms[END_REF] The limitations of our study are inherent to its large-scale design mainly based on the goodwill of voluntary physicians and their declarative data. These include low rate of participation, incomplete exhaustiveness, frequent missing data, and residual uncertainties from unaudited data. The persistence of AMA positivity in time, for instance, could not be evaluated. However, the participation rate (67%) of physicians was quite satisfactory for such a large-scale study, just as was the mean exhaustiveness of data (64%). In addition, these intrinsic weaknesses are counterbalanced by the large amount of data collected and their representativeness in terms of population.

In conclusion, the present study highlights the relatively high proportion of AMA-positive patients with nonestablished PBC. The younger age and lower autoantibody titer of these patients, together with the frequent mild abnormalities of their biochemical liver tests, supports a very early, presymptomatic precholestatic stage of the disease. The incidence of clinical manifestations of PBC seems, however, much lower than previously reported. On the other hand, these patients may display an increased mortality risk whose link with AMAs remains uncertain. AQ1: A disclosure statement reporting no conflict of interest has been inserted in the footnotes section. Please verify whether statement is accurate. AQ2: Please provide English translation for each author affiliation. AQ3: A telephone number has been included for corresponding author; please confirm as correct. Please also provide an English for contact information.

AQ4: For all listed citations with two or more authors who share first authorship, HEPATOLOGY stipulates those authors' names to be in bold type. It is the authors' responsibility to ensure that these names appear in bold in the reference section when submitting a manuscript. This allows giving due credit to joint first authors. Also, please include the phrase, 'Author names in bold designate shared co-first authorship' at the end of the references section if you have citations that have joint first authors. Please see author guidelines for further details. AQ5: For reference 12, please provide an English translation for title. AQ6: For reference 13, please provide an English translation for title. Please also provide location of publisher. AQ7: In footnote section to Table 3, please define "inf." AQ8: Please confirm that given names (red) and surnames/family names (green) have been identified correctly.

  Service de Sant e Publique, Hôpital Saint-Antoine, APHP, Paris, France; 4 Sorbonne Universit es, INSERM, UPMC Universit e Paris 06, Institut Pierre Louis d' epid emiologie et de Sant e Publique (IPLESP UMRS 1136), Paris, France; 5 UMR_S938, Facult e de M edecine Pierre et Marie Curie-Site Saint-Antoine, Universit e Pierre et Marie Curie-Paris 6, Paris, France; and 6 Laboratoire d'Immunologie, Hôpital Saint-Antoine, APHP, Paris, France AQ2 .

FIG. 1 .

 1 FIG. 1. Map of the French immunology laboratories network. Sites of laboratories are represented as dark circles.

J_ID:

  HEP Customer A_ID: HEP28859 Cadmus Art: HEP28859 Ed. Ref. No.: 16-0887.R2 Date: 15-October-16 Stage: Page: 4 ID: pachiyappanm Time: 19:23 I Path: //chenas03/Cenpro/ApplicationFiles/Journals/Wiley/HEP#/Vol00000/160417/Comp/APPFile/JW-HEP#160417

FIG. 2 .

 2 FIG. 2. Flow chart of the study.

(

  51:40; n 5 36) did not differ significantly from those with a higher one (1:80; n 5 193; Supplementary Table

FIG. 3 .

 3 FIG. 3. Distribution by age and sex of the incidence rate per 100,000 person-years of the patients with newly detected AMAs (top) and of those with newly diagnosed PBC (down).

FIG. 4 .

 4 FIG. 4. Actuarial (thick line) and Kaplan-Meier (thin line) survival curves of patients with newly detected AMAs and nonestablished PBC compared to the actuarial survival curve of a standardized control population matched for age, sex, and time period (dotted line).

J_ID:

  HEP Customer A_ID: HEP28859 Cadmus Art: HEP28859 Ed. Ref. No.: 16-0887.R2 Date: 15-October-16 Stage: Page: 12 ID: pachiyappanm Time: 19:24 I Path: //chenas03/Cenpro/ApplicationFiles/Journals/Wiley/HEP#/Vol00000/160417/Comp/APPFile/JW-HEP#160417

TABLE 1 .

 1 Characteristics of AMA-Positive Patients With Nonestablished PBC (n 5 229) HEP Customer A_ID: HEP28859 Cadmus Art: HEP28859 Ed. Ref. No.: 16-0887.R2 Date: 15-October-16

						Stage:	Page: 7
	HEPATOLOGY, Vol. 00, No. 00, 2016				DAHLQVIST ET AL.
			Median (Range)/
	Variable	Available Data	No. (%)
	Age (years)	229	58 (15-90)
	Female sex (%)	229	179 (78)
	BMI (kg/m -2 )	92	24 (17-46)
	Past history of AID (%)	130	31 (24)
	Familial history of AID (%)	102	11 (11)
	Clinical settings of AID (%)	225	104 (46)
	AMA titer	229	1:160 (1:40-1:640
			or higher)
	Positive dotting/blotting	150	137 (91)
	test (%)
	PBC-specific ANAs (%)	143	9 (6)
	Fatigue (%)	148	82 (55)
	Pruritus (%)	145	5 (3)
	Jaundice (%)	145	6 (4)
	Ascites (%)	144	3 (2)
	Total bilirubin (lmol/L)	110	10 (2-149)
	ALP (3ULN)	119	0.7 (0.2-8.6)
	GGT (3ULN)	127	1.1 (0.2-66.5)
	ALT (3ULN)	130	0.6 (0.2-31.3)
	IgM (3ULN)	41	0.6 (0.3-7.8)
	Platelet (310 9 /L)	119	247 (11-769)
	Prothrombin index (%)	99	97 (20-110)
	Liver biopsy (%)	148	28 (19)
	Cirrhosis (%)	229	13 (6)
	Abbreviation: BMI, body mass index.
	TABLE 2. Comparison of Established Versus Nonestablished PBC Groups
			Established PBC	Nonestablished PBC
			Group (n 5 247)	Group (n 5 229)
	Variable	Available Data	Results	Available Data	Results	P Value
	Age (years)	247	60 (20-91)	229	58 (15-90)	0.0567
	Female sex (%)	247	220 (89)	229	179 (78)	0.0012
	BMI (kg/m -2 )	147	24 (17-45)	92	24 (17-46)	0.4440
	History of AID (%)	173	41 (24)	130	31 (24)	0.9168
	Familial history of AID (%)	144	13 (9)	102	11 (11)	0.8612
	AMA titer*	247	5 (0-6)	229	3 (0-6)	<0.0001
	Positive dotting/blotting test (%)	139	134 (96)	150	137 (91)	0.0551
	PBC-specific ANAs (%)	142	18 (13)	143	9 (6)	0.0196
	Fatigue (%)	174	81 (47)	148	66 (45)	0.7253
	Pruritus (%)	172	41 (24)	145	5 (3)	<0.0001
	Jaundice (%)	174	23 (13)	145	6 (4)	0.0050
	Ascites (%)	171	9 (5)	144	3 (2)	0.2365
	Total bilirubin (lmol/L)	142	12.0 (2.0-357)	110	9.9 (2.0-149.0)	0.0358
	ALP (3ULN)	161	1.7 (0.2-16.5)	119	0.7 (0.2-8.6)	<0.0001
	GGT (3ULN)	169	5.8 (0.3-71.9)	127	1.1 (0.2-66.5)	<0.0001
	ALT (3ULN)	166	1.4 (0.2-46.7)	130	0.6 (0.2-31.3)	0.2809
	IgM (3ULN)	84	1.4 (0.2-5.5)	41	0.6 (0.3-7.8)	0.0069
	Platelet (310 9 /L)	157	249 (43-500)	119	247 (11-769)	0.9781
	Prothrombin index (%)	152	100 (38-111)	99	97 (20-110)	0.1479
	Liver biopsy (%)	102	87 (85)	148	28 (19)	<0.0001
	Cirrhosis (%)	168	21 (13)	229	13 (6)	0.0096

Variables are expressed as median (range) or number (% of available data). *AMA titer was evaluated according to the following semiquantitative score: 1:40 5 1; 1:80 5 2; 1:160 5 3; 320 5 4; 1:640 5 5; and >1:640 5 6. Abbreviation: BMI: body mass index. J_ID: ID: pachiyappanm Time: 19:23 I Path: //chenas03/Cenpro/ApplicationFiles/Journals/Wiley/HEP#/Vol00000/160417/Comp/APPFile/JW-HEP#160417

TABLE 3 .

 3 Univariate Analysis of Factors Associated With PBC Development in AMA-Positive Patients With Normal ALP and Available Follow-up (n 5 66)

		Stage:	Page: 8
	Variable	Hazard Ratio (95% CI)	P Value
	Age (years)	0.98 (0.93-1.04)	0.4726
	Male sex	0.44 (0.06-3.65)	0.4529
	Past history of AID	0.54 (0.05-5.29)	0.5930
	Familial history of AID	2.52 (0.26-24.46)	0.4245
	AMA titer	1.19 (0.84-1.69)	0.3180
	PBC-specific ANAs	0.00 (0.00-inf.)	0.9913	AQ7
	Total bilirubin (lmol/L)	0.92 (0.72-1.17)	0.4993
	ALP (3ULN)	5.53 (0.00-40769)	0.7066
	GGT (3ULN)	1.37 (0.79-2.40)	0.2631
	ALT (3ULN)	0.91 (0.58-1.43)	0.6781
	IgM (3ULN)	0.77 (0.19-3.06)	0.7093

J_ID: HEP Customer A_ID: HEP28859 Cadmus Art: HEP28859 Ed. Ref. No.: 16-0887.R2 Date: 15-October-16 ID: pachiyappanm Time: 19:24 I Path: //chenas03/Cenpro/ApplicationFiles/Journals/Wiley/HEP#/Vol00000/160417/Comp/APPFile/JW-HEP#160417

  HEP Customer A_ID: HEP28859 Cadmus Art: HEP28859 Ed. Ref. No.: 16-0887.R2 Date: 15-October-16

		Stage:	Page: 9
	HEPATOLOGY, Vol. 00, No. 00, 2016	DAHLQVIST ET AL.

J_ID:

ID: pachiyappanm Time: 19:24 I Path: //chenas03/Cenpro/ApplicationFiles/Journals/Wiley/HEP#/Vol00000/160417/Comp/APPFile/JW-HEP#160417

Acknowledgments: We thank Benjamin Diemert, Anne-Violaine Salle, Fr ed eric Chau, and Gr egory Pannetier for their important contribution to the development and management of the web database. We thank all the physicians (around 700), general practitioners and specialists, who kindly and voluntarily accepted to participate in this study. Without them, the study would not have been possible.

23 I Path: //chenas03/Cenpro/ApplicationFiles/Journals/Wiley/HEP#/Vol00000/160417/Comp/APPFile/JW-HEP#160417 23 I Path: //chenas03/Cenpro/ApplicationFiles/Journals/Wiley/HEP#/Vol00000/


Philippe Beltrame
email: philippe.beltrame@univ-avignon.fr

A T E X template Drop train flow in a microtube

Keywords: wetting, microflow, annular flow, bifurcation, localized state

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

Drop train flow in a microtube

Introduction

The paper studies a 3D-axisymmetric annular flow in a microtube driven by a force in the tube axis. This type of problem is found in many contexts, whether industrial such as oil recovery by water injection [START_REF] Olbricht | Pore-scale prototypes of multiphase flow in porous media[END_REF] or physiological such as flow in human pulmonary capillaries [START_REF] Grotberg | Respiratory fluid mechanics[END_REF] or agronomic such as water transfer in earthworm galleries [START_REF]Identifying the functional macropore network related to preferential flow in structured soils[END_REF]. A lot of work was devoted to the analogous problem of a film flowing down a vertical fiber [START_REF] Duprat | Spatial evolution of a film flowing down a fiber[END_REF][START_REF] Frenkel | Annular flows can keep unstable films from breakup: Nonlinear saturation of capillary instability[END_REF][START_REF] Ruyer-Quil | Modelling film flows down a fibre[END_REF]. Due to azimuthal free-surface curvature, the Plateau-Rayleigh (PR) instability destabilizes an annular film. Contrary to the plane case, the surface instability does not necessary lead to film break-up but instead to an unduloid [START_REF] Kalliadasis | Drop formation during coating of vertical fibres[END_REF] or to an annular ridge separated by a thick film [START_REF] Duprat | Spatial evolution of a film flowing down a fiber[END_REF]. Moreover, for tubes with a sufficient large radius and thick enough films, the inertia terms play a relevant role in the dynamics and a rich scenario of traveling waves occurs [START_REF] Kerchman | Strongly nonlinear interfacial dynamics in core annular flows[END_REF]. In the context of inner flow in a microtubre, most of recent literature have studied the resulting flow, focussing especially on the formation of liquid bridges (plugs) [START_REF] Camassa | Viscous film flow coating the interior of a vertical tube. part 1. gravity-driven flow[END_REF][START_REF] Dietze | Films in narrow tubes[END_REF][START_REF] Jensen | Draining collars and lenses in liquid-lined vertical tubes[END_REF]. Therefore, the wettability plays a negligible role in the framework.

Recently, we focused on the wettability effect by studying microtube with thin film in inertialess limit [START_REF] Beltrame | Partial and complete wetting in micro-tube[END_REF]. We have highlighted two specificities of the annular flow compared to the one-dimensional ridge flow on a plane. First, for tube radius small enough a periodic emission of pearls in the tail drop may occur. This dynamics dynamics does not occur for a ridge on a plane but it is reminiscent of pearls in the tail of a 3D drop [START_REF] Podgorski | Corners, Cusps, and Pearls in Running Drops[END_REF]. The second specificity of annular flow is a dynamical complete wetting regime for large enough driving force and for small tube radius: A sliding ridge coexists with a thick film (compared to the precursor film). This regime is referred to coating complete wetting in [START_REF] Beltrame | Partial and complete wetting in micro-tube[END_REF] and the thick film is the analogue of the Landau-Levich film when a plate is removed from a liquid bath [START_REF] Galvagno | Continuous and discontinuous dynamic unbinding transitions in drawn film flow[END_REF]. The latter regime becomes unstable if the mean water height is small. Then other attractors appear as train of annular drops. The objective of this article is the study of drop train in the state of progressive waves: their emergences and the different branches of solution in the parameter space. In the next section, we present our model of a thin film governed by a lubrication equation that includes wettability. The dynamics of the traveling waves are explored using the continuation method by varying the geometric parameters and the driving force while the wettability properties remain fixed.

Modeling

Governing equations

We consider an axisymmetric thin film flow inside a cylindrical tube of radius R driven by the force µ, e.g. the weight µ = ρg e x (Fig. 1). We note h(x, t) the film height and we impose a -periodic boundary condition of the inlet and outlet flow.

The effective interaction energy combines destabilizing short-range and stabilizing long-range interactions [START_REF] Pismen | Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics[END_REF][START_REF] Thiele | Film rupture in the diffuse interface model coupled to hydrodynamics[END_REF]. Results from the literature use many forms of the potential. The results suggest that the specific form of the potential has little effect on the qualitative results [START_REF]Thin films of soft matter[END_REF][START_REF] Thiele | On the importance of nucleation solutions for the rupture of thin liquid films[END_REF]. We have chosen an expression with exponentials first proposed by Pismen and Pomeau [START_REF] Pismen | Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics[END_REF]. As in [START_REF] Beltrame | Partial and complete wetting in micro-tube[END_REF], we expressed it with powers of two:

Π(h) = Π 0 -2 -h hp + 2 • 2 -2h hp , (1)
where h p is the precursor film height for a planar geometry, i.e. the ultra-thin film where short-and long-range interactions annihilate, and Π 0 is pressure term chosen positive in order to obtain a stable precursor film. The model is based on the inertialess lubrication approximation [START_REF] Beltrame | Rayleigh and depinning instabilities of forced liquid ridges on heterogeneous substrates[END_REF] applied to the cylindrical geometry [START_REF] Beltrame | Partial and complete wetting in micro-tube[END_REF][START_REF] Kalliadasis | Drop formation during coating of vertical fibres[END_REF]:

∂ t h = -∂ x (m(h)∂ x [γ (κ + κ a ) + Π(h) + µx]) , (2)
where γ is the surface tension. The mobility m(h) for a free surface Stokes flow with viscosity η and no-slip boundary condition is given by m(h) = h 3 3η [START_REF] Kalliadasis | Drop formation during coating of vertical fibres[END_REF]. The terms κ = ∂xxh √ 1+(∂xh) 2 and κ a = 1 R-h are the longitudinal and azimuthal free surface curvatures and then γ (κ + κ a ) is the Laplace pressure. In the context of the longwave approximation, the longitudinal curvature is approximated by κ ∂ xx h. Moreover, we assume that the film height h is small compared to the radius R, then κ a 1 R + h R 2 . Note that we can skip the term 1 R in the κ a expression, since it vanishes when applying the differential operator ∂ x in Eq. (2). We scale the x axis by λ, the height by h 0 = 20h p and the pressure by γh0 λ 2 . The dimensionless parameters are: the azimuthal curvature of the tube κ = λ 2 R 2 and the Bond number representing the adimensional driving force μ = µ λ 3 γh0 . The typical length of the problem is related to the instability of the Nusselt flow. In the limiting case κ = 0 (planar geometry) λ is scaled by the dewetting instability wavelength [START_REF] Beltrame | Partial and complete wetting in micro-tube[END_REF] while for narrow tubes, λ is scaled approximatively by R [START_REF] Kalliadasis | Drop formation during coating of vertical fibres[END_REF]. Dropping the tilde over the variables and parameters, the dimensionless evolution equation then reads

∂ t h = -∂ x h 3 ∂ x ∂ 2 xx h + κh + Π(h) + µx = F (h, µ, κ) (3)
Thus, if wettability is fixed, i.e. Π 0 and h p , the dimensionless pressure Π(h) is unchanged. The value Π 0 = 20 is fixed therefore thorough the paper. Note that the order of magnitude of the height h is about 20h p and then we have h R and h λ which justifies the longwave approximation and the linearized expression κh of the azimuthal curvature. In the context of the gravity driven flow in a vertical flow of a narrow tube, µ = ρg R 3 γh0 . Thus, considering the density and surface tension of water and h p 10nm, the adimensional driving force µ varies between 0.0015 and 1.5 if R varies between 30 and 300µm.

Instability of the Nusselt flow

The uniform film flow h ≡ h m , called Nusselt flow, can be unstable because of dewetting instability but also because of the Plateau-Rayleigh instability. The latter results from the competition between the longitudinal (stabilizing) and the azimuthal (destabilitzing) Laplace pressure. On a planar substrate (κ = 0), only the dewetting instability occurs and the Nusselt flow is unstable if and only if dΠ dh is positive. For the chosen disjoining pressure given by Eq. ( 1), this condition is realized for dimensionless thickness h m superior to h c = 0.1. According to [START_REF] Beltrame | Partial and complete wetting in micro-tube[END_REF] this condition is slightly modified in a narrow tube with a curvature κ > 0: the instability occurs for thickness superior to h c (κ) 0.1(1 -0.01κ). When the film is unstable, i.e. h m > h c , all harmonic perturbations with wavenumbers k inferior to k c = κ + Π (h m ) are unstable. Thus, all wavelengths superior to 2π/k c are unstable. A well-known result shows that the typical wavelength λ 0 is about 2π √ 2/k c . Throughout the paper, we fix h m = 0.5 and κ = 0.8, then k c 0.91 and the typical pattern wavelength is about of λ 0 = 9.78. Note that all these results are independent of the driving force µ. The instability gives rise to drop patterns that we calculate numerically.

Numerical method

If µ = 0 there exists an equilibrium which minimises the free energy of the system. This solution consists in a single ridge, noted R 1 . This solution is an attractor since it is the energy minimum. If µ > 0, i.e. a driving force is present, the drops slide and may travel at a constant, then speed because of the translation invariance. Then, traveling waves are expected. Due to the lateral force µ, there is no longer any minimal energy, which implies that R 1 is no longer necessarily an attractor. In this paper, we study the different traveling waves solutions in the parameter space and their stability. Using periodic boundary conditions, the traveling waves are stationary states in the comoving frame and solutions of

G(h, µ, κ, c) = F (h, µ, κ) + c ∂ x h = 0, ( 4
)
where c is the wave velocity. The branches of stationary states are computed using a continuation algorithm similar to [START_REF] Beltrame | Time integration and steady-state continuation for 2d lubrication equations[END_REF]. In the bifurcation diagram, we represent the norm δh as the L 2 -norm of (h -h m ), where h m is the mean height. The stability of the traveling waves is studied by computing the eigenvalues of the linear operator DG Dh for a stationary state. Hereafter, the eigenmodes refers to the eigenvectors of DG Dh . The numerical method to compute the spectrum of DG Dh is detailed in [START_REF] Beltrame | Time integration and steady-state continuation for 2d lubrication equations[END_REF]. With periodic boundary the mean volume remain invariant, then the mean height h m is constant during the dynamics. For this study, the mean height is fixed to h m = 0.5.

Drop train 3.1 Single sliding drop

The R 1 solution, corresponding to a single sliding drop, is not necessary stable when µ > 0 contrary to µ = 0. It may be unstable via Hopf bifurcation leading to the emission of pearls in the tail of the drop or it can ends via a saddle-node (see [START_REF] Beltrame | Partial and complete wetting in micro-tube[END_REF]). The continuation of the ridge R 1 exemplifies these different transitions in the bifurcation diagram of Fig. 2. The R 1 branch is stable till a saddle node. The profile of the solution shows a drop shape coexisting with a thin film. Beyond the first saddle-node, the solution profiles changes qualitatively. After five saddle-nodes the drop profile disappears and instead a wavy profile takes place. The branch ends at µ = µ 2 = 0.2164 via a reverse symmetry-breaking bifurcation with a branch invariant by the /2 translation. The latter branch is called R 2 branch in the next section. The presence of saddle-nodes depends on the forced -periodicity. If we study R 1 branch for the same parameters but with a larger we find out that for large enough µ the profile is almost flat except a small peak as shows Fig. 3 for = 30. We found that the solution is unstable. Subsequently, we show that drop trains can be a stable stationary solution for large domain lengths.

Emergence of Drop train

Coarsening modes

Depending on the periodicity , i.e. on the domain size, there may exist states of n-drops that are periodically spaced, noted R n . Note τ n , the translation of length /n acting on h such as

(τ n .h)(x) = h(x -/n). (5)
Then, the R n solution branch is invariant with respect to the translations kτ n with k an integer. If µ = 0, the R n solution is unstable and the n drops merge into a single drop R 1 solution through the so-called coarsening dynamics [START_REF] Beltrame | Time integration and steady-state continuation for 2d lubrication equations[END_REF][START_REF]Thin films of soft matter[END_REF]. The coarsening dynamics involve two processes. On one hand, any perturbation of drop volume implies that the smaller one is 'swallowed' by the larger one. On the other hand, if the distance between drops is not exactly the same, the nearest drop moves slowly towards one another until they collapse in a larger drop (see coarsening dynamics in e.g. [START_REF] Beltrame | Time integration and steady-state continuation for 2d lubrication equations[END_REF]). Each processus can be associated to a eigenvector responsible of the instability, called translation coarsening eigenmode if the drops gets closer or volume coarsening eigenmode otherwise [START_REF]Thin films of soft matter[END_REF]. Both eigenmodes are associated to a positive eigenvalue since the R n stationary state is unstable. Note that if the n drops of the R n solution are fairly separated, the interaction between drops is weak and then the eigenvalues are close to zero. If µ > 0, the coarsening modes still exist for the n sliding drops of R n stationary state . To study these eigenmodes, let us consider the simple case of two identical drops separated by the length /2, i.e. the R 2 stationary state. The profile of R 2 branch has the τ 2 invariance implying that their eigenmodes ξ(x) are symmetric (6a) or anti-symmetric (6b):

τ 2 .ξ(x) = ξ(x -/2) = ξ(x) (6a) or τ 2 .ξ(x) = ξ(x -/2) = -ξ(x). (6b)
The coarsening modes are anti-symmetric. Figure 4 depicts the profile of the R 2 solution at µ = µ t = 0.001915 and the associated coarsening eigenmodes.

The other physical parameters are the same as for R 1 in the bifurcation diagram Fig. 2, i.e. = 9, κ = 0.8, h m = 0.5. The translation coarsening eigemode ξ t has exactly a zero mean value in contrast to the volume coarsening mode ξ v .

If we perturb the R 2 stationary profile by the eigenvector ξ t , the left drop is shifted to right while the rightmost drop is shifted to the left. Then, the coarsening via translation appears. In the same way, if we perturb the R 2 profile by adding the eigenvector ξ v , the volume of the leftmost drop increases while the rightmost drop volume decreases. The coarsening occurs by mass transfer. In contrast to the case µ = 0, the eigenvalues of the coarsening eigenmodes are not necessary positive. Indeed µ = µ t in Fig. 4 corresponds to a zero eigenvalue of the translation coarsening eigenmode. Likewise, the volume coarsening eigenvalue can change its sign. We show Section 3.2.2 that the bifurcated branches arising from these critical points are responsible of the drop train structure emergence.

Stabilization of the coarsening modes

To study the bifurcated branch, we employ the continuation of the R 2 stationary branch by increasing the driving force µ. The R 2 branch and bifurcated branches are shown in bifurcation diagram Fig. 5. The R 2 branch ends at a saddle-node µ r2 sn 1.0386. The solution profile of the branch with the largest norm is depicted in Fig. 4: two identical drops translated by an spatial halfperiod. The returning branch after the saddle node is linked to profiles similar to a wavy film with two holes. On the R 2 branch, for the small critical value µ t , the translation coarsening mode becomes negative and it implies the emergence of a new branch, noted D 2 . The solution profiles near the bifurcation show that the two drops are closer together and the front drop is larger than the back drop. By increasing µ, the distance between the drop decreases while the height difference increases (Fig. 6). Thus, the T 2 branch is referred to as a duck train. This equilibrium results from the competition between the translation coarsening modes which tend to bring the drops closer together and the drop weight difference which tends to draw away the downstream drop which is bigger and therefore faster. The branch D 2 becomes stable at a Hopf bifurcation at µ h 0.3553 which is responsible to pearl emission as explained in [START_REF] Beltrame | Partial and complete wetting in micro-tube[END_REF]. When the branch is stable, the two drops coalesced as shows the right panel of Fig. 6. The single drop spreads over the domain by increasing µ till a saddle-node. For this case the

Drop train branches

If the domain size is larger, T n branches may exist for n > 2. We study the drop train branches by increasing the domain size with the same parameters as in Fig. 5 and we fix the driving force to µ = 0.45. To display the norm of the drop train branches, we use a norm related to their translation symmetry. We define the τ n -norm h τ n such as

h τ n = h -τ n .h . (7)
Therefore, the τ n -norm . emerge from the abscissa line showing that they result from a symmetry breaking of the R n branches as explained in the previous section.

All the branches T n emerge supercritically from R n and are stable near the bifurcation. The stability ends at a Hopf bifurcation leading to a time dependent dynamics that we do not study in this section. In the following, we analyze each of the solution branches in more detail.

Different profiles of the T 2 branch are depicted in Fig. 9. The length of the thin flat film separating the train drop T 2 increases when increasing till the first saddle-node at 13.13. For this profile, the relative length of the flat film is maximum and is about one third of the domain length. On the turning branch, the upstream drop has a long tail and the flat film vanishes (see the profile for = 12.5 in Fig. 9). After two additional saddle-node bifurcations, the solution profile is an almost flat film except two localized pics (see the profile for = 59 in Fig. 9). This branch seems to exist for large value of length but it is unstable. Indeed the stability of the T 2 occurs only near the bifurcation.

Regarding the T 3 branch, two branches emerge from periodic branches R 3 . Their bifurcated profiles are slightly different for = 17. For the branch called T a 3 the drop train is constituted of three drops almost identical with separated by an equal distance (Fig. 9). The profile of the second branch T b 3 differs only by the downstream drop that is slightly detached from the other two , of the T a 4 branch when = 30 are displayed Fig. 10. All drop train profiles are constituted of four drops almost identical separated by an equal distance (Fig. 10). The main quantitative difference is the length of the thin flat film that connects the consecutive drop trains. This distance decreases when the norm h τ n of solution decreases. For T a1 4 there is a net separation about /4 between the periodic drop trains. For T a2 4 , this distance is smaller and for T a3 4 , the different trains are no longer separated. This solution branch is close to the 4-periodic solution R 4 . According to the bifurcation diagram Fig. 8, all the drop train branches are unstable via a Hopf bifurcation. In [START_REF] Beltrame | Partial and complete wetting in micro-tube[END_REF], it is shown that the Hopf bifurcation of the R 1 leads to a two frequency dynamics with a quasi-periodic pearls (i.e. very small drops) emission in the tail of the drop. In the present study, the Hopf instability of the drop train branches leads to a time dependent and complex dynamics. Starting from one of these branches for = 30 , the numerical simulation shows a complex and time dependent dynamics. There is not only a pearl emission, but the drop location in the drop train is moving, inducing a variable number of drops in an aperiodic way. Unfortunately, this complex dynamics always converges to a stable stationary state after a long duration. These new stationary states are 4-drops trains that we did not detect by the numerical bifurcation analysis. The time integration provides two stationary 4drops train, denoted T b 4 and T c 4 . Their profile is different from the T a 4 branches. For the T b 4 solution, we recognize the concatenation of the two 2-drops trains (Fig. 10)a nd for for the T c 4 solution, we recognize the concatenation of a 3drops train and a single drop. Since we did not find their emergence from the R 4 branch, we believe that they emerge from T n as secondary bifurcations.

In particular, we argue that the T b 4 branch defined on a domain of length 2 results from a coarsening bifurcation involving two identical T 2 patterns defined on subdomains of length . As describing in Section 3.2, this transition corresponds to the breaking of the translation symmetry. It is noteworthy that is the less symmetric pattern that is stable.

Concluding remarks

We have highlighted a flow regime by train of drops that consists of a clustering of drops without coalescence. This regime is in contrast to 1D flows on a flat substrate where the flow takes place as a single very spread out drop, called pancake. [START_REF] Thiele | Sliding drops on an inclined plane[END_REF]. The drop train solution branch emerges from a periodic distribution of drops via the symmetry-breaking of the translation invariance. The coarsening modes are responsible of the clustering of drops. Contrary to the case without driving force, coalescence does not necessarily take place. We focused on drop trains where the constituent drops have similar heights because they are generally stable for large enough driving force µ. This result differs from [START_REF] Beltrame | Partial and complete wetting in micro-tube[END_REF] where the stable solution for a sufficiently large driving force was the single drop coexisting with a thick film. In the latter reference, the mean height is double that in the present study. This shows that drop trains are expected solutions only for low water content. There is a rich variety of solutions: the drops of the train may have identical shapes or in contrary not. For sufficiently large domain lengths, a large number of arrangements of these drops is possible giving rise to a large number of solution branches. Moreover, the droplet trains group into new clusters giving rise to hierarchical structures breaking all translational symmetries. Therefore, for thin films in a long microtube, we do not expect a single coarse drop dynamic that would cause a liquid bridge, but rather an aperiodic spatial distribution of drops of equivalent height. Apart from the quasi-periodic dynamics with pearl emission in the tail of the drop as presented in [START_REF] Beltrame | Partial and complete wetting in micro-tube[END_REF], the dynamics converge to stable stationary states, i.e. traveling waves. However, the transient dynamics reveal quite complex dynamics. We believe that this convergence may be related to the periodic boundary conditions. Further simulations with different boundary conditions closer to the experimental setup, such as an imposed flux at the inlet, might reveal complex dynamics.

Fig. 1

 1 Fig. 1 Sketch of the geometry of the problem. Trains of annular ridges slide along the inner tube driven by µ. The problem is -periodic, R is the tube radius and h(x, t) the film height.

Fig. 2 [Fig. 3

 23 Fig. 2 [left] Bifurcation diagram of the R 1 branch w.r.t. µ for hm = 0.5, κ = 0.8, = 9. The dots indicate the stationary solutions for which their profile is shown in the right panel. The end dot corresponds to a bifurcation with the R 2 branch (See Fig. 5). [right] Profiles of the stationary states of the bifurcation diagram for different values of the driving force µ.

Fig. 4 [

 4 Fig. 4 [above] Profile of the R 2 branch solution for µ = µt = 0.001915 at the first coarsening bifurcation. [Below] Volume and translation coarsening modes associated to the R 2 solution.

Fig. 5

 5 Fig. 5 Continuation of the (black) D 2 branch and (red) T 2 bifurcated from the (dashed black) R 2 branch. The bold lines indicate stability while the thin line instability. Circles indicate bifurcation points. The dot R 2 branch indicates the bifurcation with the R 1 branch at µ = µ 2 (see Fig.2)

Fig. 6 [

 6 Fig. 6 [left] Profiles of the D 2 branch of Fig. 5 for different values of µ before the saddlenodes bifurcation. [right] Profiles of the D 2 branch for different values of µ after the saddlenodes bifurcation.

Fig. 8 3 Fig. 9

 839 Fig. 8 Continuation of T 2 , T 3 and T 4 branches w.r.t. the domain size . The code color is as fllows: red line for T 2 , resp. black, green, blue lines for T a,b,c 3 and magenta line for T 4 . The bold lines indicate stability while the thin lines indicate instability.

Fig. 10

 10 Fig. 10 Profiles of the branches (black dashed) T 3 and T 4 branches of the bifurcation diagram of Fig. 8 for = 30 and µ = 0.45.

Data Availability Statement. Data will be made available on reasonable request.

drop width reaches the limits of the domain. This scenario is reminiscent of the one presented for drops sliding on an inclined plane. This study shows that the elongated shape of the single drop called pancake in [START_REF] Thiele | Sliding drops on an inclined plane[END_REF] results from the coalescence of two drops via the translation coarsening mode. The stable branch ends at a saddle-node at µ = µ sn1 0.9889. The a second saddle-node at µ sn2 0.5605 leads to a stable solution. The profile is then a wavy film, that is the unique stable solution for µ > µ sn1 .

On the R 2 branch, the eigenvalue of the volume coarsening mode becomes negative at µ v 0.3542. The bifurcated branch, denoted T 2 , breaks the τ 2 translation symmetry and as for the D 2 branch, the drops get closer when the force increases. Nevertheless, the drop height difference remains small and the back drop is the larger contrary to the D 2 branch. Thus, the T 2 branch is referred to as a drop train. The equilibrium is possible by a slightly difference of the drop shape as showed in Fig. 7. The branch T 2 is always unstable. The branch ends by a reverse bifurcation from the R 2 'hole' branch at µ v2 = 1.026. The bifurcation is also due to the volume mode ξ v associated to the bifurcation. The T 2 branch is always unstable for = 9. however, in the next section, we show that it can be stable.

We showed that two-periodic drops solutions branches may break the translation invariance τ 2 via a symmetry breaking bifurcation. This bifurcation can be interpreted as a pitchfork bifurcation in the comoving frame or a perioddoubling in the laboratory frame. We distinguish two types of branches according to the eigenmode responsible of the bifurcation. For the volume coarsening eigenmode, the resulting branch is the so-called drop train or T 2 and for the translation coarsening mode, the resulting branch is the so-called duck branch or D 2 . The duck branch leads to a coarsening drops by increasing the driving force. Then, in the following, we focus mainly on the drop train branches T n . drops (Fig. 9). A secondary bifurcation occurs at c 29.15 from T a 3 leading to a third branch T c 3 for > c . All these branches continue to exist for > c . Stability changes occur at Hopf bifurcations. Thus, there are different ranges of stability. The stability ranges of the three branches differ. Thus, there are ranges where two drop trains are stable and ranges where all three solutions are unstable as in the vicinity of = 30. The continuation for > 45 suggests that all these three branches become unstable for sufficiently large . We depict the three branches for = 30 in Fig. 10. The three drops are no longer identical for the branch T a 3 . The two downstream drops are larger than the upstream drop and look like the so-called pancake drop. For the new branch T c 3 only the drop downstream is larger. For the T b 3 , it is the middle drop which is the pancake drop. In addition, the profile is always characterized by the downstream drop slightly detached from the rest of the upstream train.

Finally, according to the bifurcation analysis we found one branch 4-drop train branch T a 4 emerging for the R 4 branch. The branch emerges supercritically and it is stable near bifurcation as the other branches. However, in contrast to other branches, its existing domain is finite. The branch reconnects at 33.99 the R 4 branch after 5 saddle-node bifurcations. This scenario is similar as the one for the T 2 branch when = 9 shown in the bifurcation diagram of Fig. 5. The profiles of the three solutions, noted T a1,2,3