Eleni Karamouza
email: eleni.karamouza@gustaveroussy.fr

Rosalind M Glasspool

Caroline Kelly

Liz-Anne Lewsley

Karen Carty

Gunnar B Kristensen

Josee-Lyne Ethier

Tatsuo Kagimura

Nozomu Yanaihara

Sabrina Chiara Cecere

Gunnar B Kristens

Josee-Lyn Ethie

Sabri Chiara Cecere

Benoit You

Ingrid A Boere

Eric Pujade-Lauraine

Isabelle Ray-Coquard

Cécile Proust-Lima

Xavier Paoletti

Early Dynamics to Predict Overall Survival in Women with Newly Diagnosed Advanced Ovarian Cancer Based on Meta-Analysis Data

Keywords: ovarian cancer, CA-125, biomarker, linear mixed model, meta-analysis

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

Introduction

Ovarian cancer (OC) is the seventh most common cause of cancer mortality in women worldwide, with a survival rate of 46% at 5 years after diagnosis [START_REF] Doherty | Challenges and Opportunities in Studying the Epidemiology of Ovarian Cancer Subtypes[END_REF]. First-line treatment consists of primary or interval debulking surgery and platinum-and taxane-based chemotherapy, which may be combined with maintenance treatments including bevacizumab or poly (ADP-ribose) polymerase (PARP) inhibitors, if applicable. Recently, PARP inhibitors have demonstrated very promising gains in progression free survival (PFS) and overall survival (OS), especially for patients with BRCA-mutated ovarian cancer [START_REF] Ray-Coquard | Olaparib plus Bevacizumab as First-Line Maintenance in Ovarian Cancer[END_REF][START_REF] Banerjee | Maintenance olaparib for patients with newly diagnosed advanced ovarian cancer and a BRCA mutation (SOLO1/GOG 3004): 5-year follow-up of a randomised, double-blind, placebo-controlled, phase 3 trial[END_REF][START_REF] Grimm | Maintenance olaparib plus bevacizumab (bev) after platinum-based chemotherapy plus bev in patients (pts) with newly diagnosed advanced high-grade ovarian cancer (HGOC): Efficacy by timing of surgery and residual tumor status in the Phase III PAOLA-1 trial[END_REF][START_REF] González-Martín | Niraparib in Patients with Newly Diagnosed Advanced Ovarian Cancer[END_REF]. Cancer antigen 125 (CA-125) is a marker mainly related to ovarian cancer, but it may also be elevated in other conditions. According to the Gynecologic Cancer Intergroup (GCIG) criteria, it is used to monitor the patient's response to treatment in cases of recurrent disease or to define progression after first-line therapy [START_REF] Taylor | New guidelines to evaluate the response to treatment in solid tumors (ovarian cancer) (multiple letters)[END_REF][START_REF] Rustin | Early versus delayed treatment of relapsed ovarian cancer (MRC OV05/EORTC 55955): A randomised trial[END_REF]. Although in 2010, the routine measurement of CA-125 was discouraged [START_REF] Rustin | Early versus delayed treatment of relapsed ovarian cancer (MRC OV05/EORTC 55955): A randomised trial[END_REF][START_REF] Rustin | Follow-up with CA125 after primary therapy of advanced ovarian cancer has major implications for treatment outcome and trial performances and should not be routinely performed[END_REF], two strategies for the follow-up of women after primary treatment are now used: (i) clinical follow-up with CA-125 and imaging on clinical indication or (ii) CT-scans throughout follow-up and optional CA-125 testing [START_REF] Esselen | Use of CA-125 tests and computed tomographic scans for surveillance in ovarian cancer[END_REF]. However, the contribution of CA-125 is debated, as clinical trials implement systematic CA-125 and CT-scan surveillance, especially in maintenance treatment settings. The best ways to analyze CA-125 decrease, its timeframe and reproducibility across trials, and the risk of error are insufficiently documented.

Assessing the prognostic value of the early evolution of CA-125 for time-to-event endpoints raises statistical challenges. The CA-125 trajectory can be summarized in multiple ways (e.g., CA-125 value at baseline, KELIM at 3 months, CA-125 value, or decrease at the prediction time (e.g., 3 and 6 months)), and its prognostic role has been studied in many isolated studies [START_REF] Markman | Significance of early changes in the serum CA-125 antigen level on overall survival in advanced ovarian cancer[END_REF][START_REF] Skaznik-Wikiel | Possible use of CA-125 level normalization after the third chemotherapy cycle in deciding on chemotherapy regimen in patients with epithelial ovarian cancer[END_REF][START_REF] You | The strong prognostic value of KELIM, a model-based parameter from CA 125 kinetics in ovarian cancer: Data from CALYPSO trial (a GINECO-GCIG study)[END_REF][START_REF] Colloca | CA125-related tumor cell kinetics variables after chemotherapy in advanced ovarian cancer: A systematic review[END_REF][START_REF] Liao | CA125 normalization within 60 days as an independent prognostic factor for patients with advanced epithelial ovarian cancer[END_REF]. Nevertheless, the performance of these methods has never been evaluated in an individual patient data meta-analysis. CA-125 is a biomarker prone to measurement error and biological variation (i.e., some values may appear higher or lower than anticipated from the overall trajectory), so that raw summaries of observed CA-125 may be suboptimal for assessing CA-125 prognostic value. Furthermore, CA-125 may not be available at a particular timepoint, and an imputation method may be necessary [START_REF] Overall | Last-observation-carried-forward (LOCF) and tests for difference in mean rates of change in controlled repeated measurements designs with dropouts[END_REF]. Therefore, some authors have proposed reliance on modeling techniques that handle noisy and sparsely measured biomarkers [START_REF] Prentice | Covariate measurement errors and parameter estimation in a failure time regression model[END_REF][START_REF] Proust-Lima | Development and validation of a dynamic prognostic tool for prostate cancer recurrence using repeated measures of posttreatment PSA: A joint modeling approach[END_REF] to better assess the evolution of CA-125. Recently, the use of the CA-125 ELIMination Rate Constant K (KELIM) [START_REF] Colomban | Early Modeled Longitudinal CA-125 Kinetics and Survival of Ovarian Cancer Patients: A GINECO AGO MRC CTU Study[END_REF], which is based on longitudinal CA-125 measures as well as pharmacokinetic and pharmacodynamic parameters of treatment, has gained important interest in multiple settings (in recurrent disease, for neo-adjuvant treatments, and for initial treatments). In the adjuvant and neoadjuvant settings, the prognostic role of the KELIM at 100 days (i.e., 3 months) has been tested in several trials [START_REF] You | The role of the tumor primary chemosensitivity relative to the success of the medical-surgical management in patients with advanced ovarian carcinomas[END_REF][START_REF] Lauby | The Increasing Prognostic and Predictive Roles of the Tumor Primary Chemosensitivity Assessed by CA-125 Elimination Rate Constant K (KELIM) in Ovarian Cancer: A Narrative Review[END_REF]. Alternatively, statistical techniques such as mixed-effect modeling have been successfully developed in the case of prostate cancer to model the prostate-specific antigen dynamic over time [START_REF] Proust-Lima | Development and validation of a dynamic prognostic tool for prostate cancer recurrence using repeated measures of posttreatment PSA: A joint modeling approach[END_REF][START_REF] Carrot | Modeled Early Longitudinal PSA Kinetics Prognostic Value in Rising PSA Prostate Cancer Patients after Local Therapy Treated with ADT +/-Docetaxel[END_REF]. A recent study proposed a CA-125 rate estimate that is easy to quantify in order to aid in decision making regarding second-line treatment for patients with recurrent high-grade serous ovarian cancer [START_REF] Bartl | Tumor growth rate estimates are independently predictive of therapy response and survival in recurrent high-grade serous ovarian cancer patients[END_REF]. These statistical methods for longitudinal data are thus central to the assessment of the prognostic value of the trajectory estimated over various timeframes. Considering timepoints of less than three months could be challenging due to the scarcity of CA-125 measurements in this timeframe. Conversely, longer periods of monitoring of CA-125, such as six months, may improve the model's accuracy and the discriminatory value of the CA-125 decline. This may entail a CA-125 collection burden, which may lead to an elevated number of follow-up visits.

When studying the prediction of a time-to-event endpoint based on a prone-to-error biomarker, where the timeframe differs between the longitudinal process (a few months) and the survival endpoints (a few years), a landmark approach may be an appropriate approach. Briefly, its principle is to set up a landmark time from which the prediction is to be performed and restrict the population to event-free subjects at this timepoint. Summaries of the dynamics of the biomarker before the landmark time serve to predict progression or death after the landmark time [START_REF] Van Houwelingen | Dynamic prediction by landmarking in event history analysis[END_REF][START_REF] Ferrer | Individual dynamic predictions using landmarking and joint modelling: Validation of estimators and robustness assessment[END_REF].

When developing prediction tools, the assessment of the predictive performance needs to be carried out carefully. The value of the marker for predicting the risk of death can be measured using the Brier score and time-dependent area under the ROC curve (AUC) with estimators adapted to the time-to-event context [START_REF] Heagerty | Time-dependent ROC curves for censored survival data and a diagnostic marker[END_REF][START_REF] Gerds | Consistent estimation of the expected brier score in general survival models with right-censored event times[END_REF][START_REF] Blanche | Quantifying and comparing dynamic predictive accuracy of joint models for longitudinal marker and time-to-event in presence of censoring and competing risks[END_REF]. The predictive performance may be overoptimistic when assessed on the same data used for training the model. To correct this bias, external validation or cross-validation techniques can be used [START_REF] Gerds | The performance of risk prediction models[END_REF].

This work aims to assess the performance of CA-125 early dynamics and determine the most appropriate landmark timepoint (i.e., the timeframe required to enrich the statistical prediction models) so as to predict OS at the best future timepoint in newly diagnosed patients with advanced ovarian cancer treated with taxane-and platinum-based chemotherapy. We used individual patient data (IPD) meta-analysis from the GCIG meta-analysis group study, which has a large and diverse sample size, permitting a cross-validation of the results between patients and studies as well as between subgroups based on the patients' characteristics.

Materials and Methods

This report follows the Preferred Reporting Items for Systematic Reviews and Metaanalyses (PRISMA)-IPD guidelines and the Transparent reporting of a multivariable prediction model for individual prognosis or diagnosis (TRIPOD) statement for the registration of the protocol, trial identification, data collection and integrity, assessment of bias, and sensitivity analyses [START_REF] Debray | Individual Participant Data (IPD) Meta-analyses of Diagnostic and Prognostic Modeling Studies: Guidance on Their Use[END_REF][START_REF] Stock | Development and validation of a risk prediction model of preterm birth for women with preterm labour symptoms (the QUIDS study): A prospective cohort study and individual participant data meta-analysis[END_REF]. This meta-analysis was registered with PROSPERO (CRD42017068135). The Ethics Committee of Gustave Roussy Cancer Center, Villejuif, France, approved this study, and the French Data Protection Authority waived the need for informed consent due to the use of deidentified data.

Study Population

To develop and validate a statistical predictive model of the dynamics of CA-125 in relation to OS, we used the IPD of the GCIG meta-analysis. This included patients with newly diagnosed ovarian cancer of the International Federation of Gynecology and Obstetrics FIGO stages IC to IV whose data were collected from randomized controlled trials published from January 2001 to September 2016 (GCIG meta-analysis of ovarian cancer [START_REF] Paoletti | Assessment of Progression-Free Survival as a Surrogate End Point of Overall Survival in First-Line Treatment of Ovarian Cancer: A Systematic Review and Meta-analysis[END_REF]). From the initial set of 17 trials, we selected those that investigated initial systemic treatments after surgery (no maintenance treatments).

For the present analysis, only trials that collected serum CA-125 levels at baseline (before the start of the treatment) and repeatedly during follow-up were selected. Data collection followed the original research protocol. To be eligible, every patient had to have at least two CA-125 measures, including one at baseline, and complete information regarding overall survival and progression.

Data checking was performed to ensure the data quality. Levels of the biomarker greater than 15,000 were considered as outliers, and peri-operative CA-125 measures were excluded, as they are affected by the surgical intervention.

Outcomes

The primary endpoint was OS, defined as the time from randomization to death of any cause. Patients alive at the cut-off date were censored at the last follow-up date.

The prognostic value of the levels of CA-125 at 3 and 6 months from randomization and the rate of CA-125 decline were evaluated.

The model was developed based on patients treated with either the 'standard chemotherapy' (Paclitaxel + Carboplatin and Epirubicin/Doxorubicin) or the same chemotherapy with an investigational treatment, as none of the included trials showed significant differences in treatment effect between the two arms. A sensitivity analysis, after excluding the patients from the investigational arms, was carried out.

The following patient characteristics were available at baseline: age, performance status, FIGO stage, histological subtype and grade, and residual disease after surgery (≥1 cm, <1 cm, unknown).

Statistical Methods

In order to study the biomarker-OS association over time, a landmark analysis was performed, and different post-baseline times, noted as s, were considered [START_REF] You | The role of the tumor primary chemosensitivity relative to the success of the medical-surgical management in patients with advanced ovarian carcinomas[END_REF]. All the available data up to s were modeled to predict the risk of death from time s to time s + t, with t denoting the horizon. The investigated prognostic factors based on the early CA-125 trajectory included (i) the observed CA-125 value at the landmark time, where if the former was missing, the closest previous measure was retained (OVLT), and (ii) the observed relative decline (ORD) in the biomarker between baseline and the last CA-125 measure before the landmark. In addition, three CA-125 summaries estimated using a hierarchical linear mixed-effect model were considered: (iii) the estimated value at the landmark time (EVLT) and (iv-v) the estimated slopes at baseline (ESB) and at the landmark time (ESLT).

For these three summaries, repeated measures of the log-transformed CA-125 were analyzed using a hierarchical linear mixed model (HLMM) [START_REF] Verbeke | Linear Mixed Models for Longitudinal Data[END_REF], which handles the hierarchical structure of the data, the inherent measurement error, and missing values according to the missing-at-random mechanism. Three hierarchical levels were distinguished: (i) the between-trial variability, using random effects at the trial level, (ii) the within-patient correlation, using patient-specific random effects, and (iii) the variability due to observation-specific measurement errors.

To account for the nonlinear dynamics of log-CA-125 over time, we considered a basis of natural cubic splines over time with 3 or 4 internal knots for the landmark time at 3 or 6 months, respectively [START_REF] Perperoglou | A review of spline function procedures in R[END_REF]. The knots were placed at the beginning of the follow-up (i.e., 0.5, 1, and 2 or 3 months for 3 and 4 knots, respectively) in order to capture possible rapid declines in the biomarker, and external knots were placed at the 2%-and 98%-percentiles of the measurement times. Knot selection was based on the Akaike criteria (AIC) of the model. For each patient, we included random effects on each natural cubic spline function to capture the patient-specific CA-125 deviation to the mean trajectory. For each study, a random intercept was further used to capture the trial-specific deviation.

The summaries were derived from the HLMM using the Best Linear Unbiased Predictor for the trial-and patient-specific random effects.

Evaluation of the Predictive Performance of the CA-125 Summaries

The summaries were included as fixed effects in a Cox proportional hazards model considering study-specific baseline hazards. Heterogeneity across studies of the associations between the summaries and survival was tested with a likelihood ratio test that compared the partial likelihood of the stratified model with the partial likelihood of the stratified model using trial specific summary effects [START_REF] Smith | Investigating heterogeneity in an individual patient data meta-analysis of time to event outcomes[END_REF][START_REF] Smith | An overview of methods and empirical comparison of aggregate data and individual patient data results for investigating heterogeneity in meta-analysis of time-to-event outcomes[END_REF].

As shown in Figure 1, the probability of death was predicted from the landmark time s at the horizon time s + t according to the CA-125 summary computed at time s, with s = 3 and 6 months and t = 24, 36, 48, and 60 months in a subset of studies with a sufficient follow-up. The predictive performance of these predictions was assessed using the AUC and Brier score in order to determine the most appropriate landmark time and to explore the robustness for various prediction horizon times. To account for censored times-to-event between the time s and the time s + t, estimators of these two quantities were weighted by the inverse probability of censoring (IPCW) [START_REF] Blanche | Review and comparison of ROC curve estimators for a time-dependent outcome with marker-dependent censoring[END_REF]. Both measures were used to assess the quality of the prediction tool. The AUC can be seen as a concordance measure between patients with high and low risks of death [START_REF] Gerds | Absolute risk regression for competing risks: Interpretation, link functions, and prediction[END_REF], while the Brier score measures calibration and discrimination. Both measures range between 0 and 1. An AUC below 0.7 suggests a moderate discriminatory performance.

The calibration was assessed graphically: patients were split into groups defined by quantiles of the predicted event probabilities, and the results were plotted against the observed risk of death.

To correct for the over-optimistic performance obtained on the training datasets, we applied a 10-fold cross-validation technique on the patient level. The original dataset was partitioned into 10 sub-samples of the same size. For each sub-sample, the predictions were computed using the model trained on the 9 remaining sub-samples. The predictions from the 10 sub-samples were then pooled for the performance assessment. The average and standard deviation of the Brier score and the AUC over 50 replicates of the crossvalidation technique were finally reported to account for fluctuations in the partitions. The corresponding 95% interval of the bootstrap distribution was calculated.

To further investigate the added value of a summary of CA-125 dynamics, we finally compared the Cox model stratified by study and adjusted for the evaluated summary to either (i) the null model, i.e., a survival model stratified by study that served as a reference, or (ii) the same Cox model with the summary of the dynamics and with or without CA-125 level at baseline.

In a secondary analysis, the prognostic value of CA-125 was evaluated based on subgroups of patients defined by their baseline characteristics, such as the FIGO stage and residual disease.

Finally, for the known cutoff of 35 for the CA-125 value, we evaluated the positive predictive value (PPV) and the negative predictive value (NPV) at the analyzed time horizons.

The statistical analyses were performed using R (version 4.1.2, R foundation for Statistical Computing, Vienna, Austria), with R package RiskRegression for the predictive performances assessment and the lme4 package for the HLMM.

Results

Study Selection and Characteristics

Among the 17 trials in the GCIG meta-analysis, repeated CA-125 measures were available for 13 trials. Fours trials that investigated maintenance treatments were further excluded. A total of nine trials were then selected, with four and five trials, respectively, investigating new initial treatments and the intensification of an initial treatment, leading to a total sample of 5573 patients.

More than 72,967 CA-125 measures were collected (see the flowchart in Scheme 1). The median number of CA-125 measures per patient was 11 (IQR [START_REF] Rustin | Early versus delayed treatment of relapsed ovarian cancer (MRC OV05/EORTC 55955): A randomised trial[END_REF][START_REF] Rustin | Follow-up with CA125 after primary therapy of advanced ovarian cancer has major implications for treatment outcome and trial performances and should not be routinely performed[END_REF][START_REF] Esselen | Use of CA-125 tests and computed tomographic scans for surveillance in ovarian cancer[END_REF][START_REF] Markman | Significance of early changes in the serum CA-125 antigen level on overall survival in advanced ovarian cancer[END_REF][START_REF] Skaznik-Wikiel | Possible use of CA-125 level normalization after the third chemotherapy cycle in deciding on chemotherapy regimen in patients with epithelial ovarian cancer[END_REF][START_REF] You | The strong prognostic value of KELIM, a model-based parameter from CA 125 kinetics in ovarian cancer: Data from CALYPSO trial (a GINECO-GCIG study)[END_REF][START_REF] Colloca | CA125-related tumor cell kinetics variables after chemotherapy in advanced ovarian cancer: A systematic review[END_REF][START_REF] Liao | CA125 normalization within 60 days as an independent prognostic factor for patients with advanced epithelial ovarian cancer[END_REF][START_REF] Overall | Last-observation-carried-forward (LOCF) and tests for difference in mean rates of change in controlled repeated measurements designs with dropouts[END_REF][START_REF] Prentice | Covariate measurement errors and parameter estimation in a failure time regression model[END_REF][START_REF] Proust-Lima | Development and validation of a dynamic prognostic tool for prostate cancer recurrence using repeated measures of posttreatment PSA: A joint modeling approach[END_REF]), with a range between 2 and 100 values, as described in the Supplementary Materials, Table S1. The trials and patient characteristics are described in Tables 1 and2, respectively and in the Supplementary Materials, Table S1. A total of 2570 deaths were reported, and the median OS

Endpoint and Landmark Timeframe

We computed survival at the time horizons of 24, 36, 48, and 60 months for two landmark times, s = 3 and 6 months. A total of 5209 and 4946 patients were alive at these two landmark times, respectively, and included in the analysis set.

Performance of CA-125 Summaries

The cross-validated performances of the various CA-125 summaries are reported in Figures 2 and3, and further results are described in the Supplementary Materials, Tables S2 andS3. Regardless of the summary in question, the AUC were relatively similar at the 24-, 36-, and 48-month horizon times, with a drop in performance at 60 months. In contrast, the shorter the prediction horizon was, the smaller the error in the prediction was, as measured by the Brier score. The model had a good calibration, as shown in Supplementary Materials Figures S1-S5.

Landmark at 3 Months

The estimated value at 3 months and the last observed value before or at 3 months provided the best predictive performances at all the horizon times with, for instance, AUCs of 0.745 IC 95% [0.699, 0.796] and 0.749 [0.702, 0.796], respectively, and statistically significant heterogeneity across trials for both summaries. These AUCs can be compared with the AUC of the null model (only accounting for the trial effect), which was 0.648 [0.593; 0.702] at the 24-month horizon time (Figure 2, Supplementary Materials Table S2).

The estimated slope showed a modest predictive accuracy. For instance, at 24 months, the estimated slope at baseline, without adding the CA-125 value at baseline, provided an AUC of 0.685 [0.637; 0.739], with significant heterogeneity across trials (p = 0.0009). The estimated slope and ORD at 3 months showed AUCs of 0.668 [0.616; 0.722] (with heterogeneity (p = 0.005)) and 0.646 [0.590, 0.697] (with non-significant heterogeneity across trials (p = 0.102)), respectively.

At the horizon time of 24 months, the prognostic impact of CA-125 at baseline (with a single variable included in the model stratified by trial) was 0.703 [0.659, 0.752]. After adding the CA-125 value at baseline, the predictive performances of the estimated and observed CA-125 values at the landmark time remained virtually the same as those without the CA-125 value at baseline (0.745 [0.701, 0.794] and 0.748 [0.705, 0.796], respectively). However, the estimated slope and ORD at 3 months, together with CA-125 at baseline, showed improved predictive performances at all times with, for instance, AUCs of 0.703 [0.658-0.752] and 0.738 [0.693, 0.789] at 24 months.

Of note, the AUC of 0.75 indicates that for two random patients, there is a 75% probability that the patient with the lowest CA-125 value at 3 months will have the longest survival when restricting to the window between 3 and 24 months.

Landmark at 6 Months

Six-month landmark analyses performed in a manner similar to the 3-month analyses, with a lower performance for a longer prediction horizon time. The estimated value at 6 months and the last observed value before 6 months had AUCs of 0.731 [0.678; 0.778] and 0.735 [0.683; 0.783], respectively, at 24 months and AUCs of 0.656 [0.586; 0.734] and 0.66 [0.59; 0.734] at 60 months (Figure 3, Supplementary Materials Table S3).

CA-125 Summary Performance for Subgroups of Patients Based on Characteristics

In the case of the FIGO stage III patients, the estimated value at 3 months provided an AUC value of 0.709 [0.640; 0.772] and a Brier score of 0.190 [0.167, 0.214] vs. 0.684 [0.446, 0.878] and a Brier score of 0.09 [0.062; 0.123] for FIGO IC and II at 24 months. For residual disease ≥1 cm vs. <1 cm at 24 months, the AUCs, without adding CA-125 at baseline, were 0.707 [0.634; 0.768] and 0.687 [0.589; 0.790], respectively. Note that the model performed equivalently after adding CA-125 baseline value (see Supplementary Materials Tables S4 andS5 and Figures S6 andS7).

Sensitivity Analysis and Predictive Values of the CA-125 Normal Range

The sensitivity analysis of the patients from the standard treatment arm showed equivalent results for both landmark times. (Supplementary Materials, Tables S6 andS7 and Figures S8 andS9).

Finally, we examined the sensitivity and specificity of the CA-125 value at 3 months. A patient with a CA-125 value < 35 had a 20% probability of death at 24 months, and 30%, 38%, and 66% at 36, 48, and 60 months, respectively. On the contrary, a patient with a CA-125 value > 35 had a 51% probability of death at 24 months, and 66%, 75%, and 79% at 36, 48, and 60 months, respectively.

Discussion

In patients with newly diagnosed FIGO stage II to IV ovarian cancer, the serum CA-125 level at 3 months after the end of primary treatment showed the best prediction of the overall survival probability at 24, 36, and 48 months compared to various measures of the CA-125 decrease rate. The last observed CA-125 value before or at 3 months and the estimate based on the repeated-measures model were associated with AUCs at 24 months of 0.749 and 0.745, respectively. A landmark time of 6 months did not improve the predictive performances, suggesting that early assessment provides the best trade-off between clinical application and statistical performance. The predictive capacity of both CA-125 summary measures remained equivalent after adding the CA-125 baseline value. Important improvement in the rate measures was observed after adding the CA-125 baseline value; for example, the observed rate of decline reached 0.738 at 24 months vs. 0.646 without the addition of the CA-125 baseline value. This reflects the importance of considering the CA-125 value at baseline in order to better interpret CA-125 kinetics. Despite the presence of heterogeneity, probably due to different inclusion criteria or healthcare standards between studies, the IPD meta-analytic context and the cross-validation allowed for robust results to be drawn.

A value of CA-125 above 35 at 3 months led to a higher risk of death, such as 51% at 24 months or up to 79% at 60 months.

In the case of 3-month surveillance, CA-125 kinetics could help in decision making regarding the intensification of the follow-up or the anticipation of a treatment change. More precisely, a patient with a CA-125 level concentration above 35 at 3 months would benefit from closer monitoring, with the measurement of the CA-125 concentration and CT scans in order to anticipate a potential need for treatment change. These patients may be good candidates for clinical trials upon progression.

Predictions at over 60 months post-landmark should be interpreted carefully, since all the measures performed poorly compared to the earlier prediction times. A lack of follow-up could explain such a performance. The main limitation is that recently approved drugs such as PARP-inhibitors were not included in this meta-analysis, since the trials were published between 2001 and 2016. Although the standard is still the chemotherapy backbone, maintenance treatment including PARP-inhibitors is now the standard of care. This may lead to different associations between CA-125 and OS. An update including trials with newly approved drugs, such as PARP-inhibitors, would be of interest to further quantify the prognostic ability of CA-125 in this context. Furthermore, we relied on linear mixed models to overcome the problems of missing and sparse data. Although we considered flexible functions of time and carefully assessed the goodness-of-fit, this approach is still based on parametric assumptions.

Recently, the subgroup of patients with HRD-or BRCA-positive status have gained great interest as a separate subtype and may lead us to question the robustness of our results for this subgroup [START_REF] Kim | Effect of BRCA mutational status on survival outcome in advanced-stage high-grade serous ovarian cancer[END_REF]. This matter could be investigated through an update of the meta-analysis with the most recent agents. Finally, this meta-analysis is limited to serum CA-125, although other biomarkers with potential prognostic capacities, such as human epididymis protein type 4 (HE4), osteopontin, mesothelin (MSLN), and folate receptor α (FOLR1), can also be measured depending on the circumstances and state of the patients [START_REF] Giampaolino | Role of biomarkers for early detection of ovarian cancer recurrence[END_REF]. As a result, we concentrated our research on serum CA-125, which is routinely used in practice, and tried to quantify its prognostic capacity. Nevertheless, the same type of work could be extended, combining other markers in specific contexts to assess the risk of progression or death.

The assessment of KELIM in patients treated with maintenance veliparib and those treated with maintenance bevacizumab showed promising results that suggested a similar performance between KELIM and the reported chemotherapy agents. Nevertheless, further assessment of the predictive value of CA-125 at 3 months in the setting of PARP maintenance is required.

The development of model-based approaches has been motivated by the impact of the variability between different assays used in routine practice on the interpretation of outcomes of the one-and two-timepoint strategies. Interestingly, the 3-month time window found in the present study is consistent with the 100-first-treatment-day period used for KELIM calculation.

Conclusions

To conclude, the surveillance of CA-125 at 3 months after the initiation of treatment could help to provide individual information for patients based on the initial CA-125 trajectory. More precisely, this could help to provide an informative tool so as to guide clinicians in decision making regarding overall survival up to 48 months.

Supplementary Materials:

The following supporting information can be downloaded at: https:// www.mdpi.com/article/10.3390/cancers15061823/s1, Table S1: Description of patients' characteristics (N = 5573). Table S2: 10-fold cross-validated AUC (A,B) and Brier score (C,D) for the prediction of overall survival at 24, 36, 48 and 60 months from CA125 history up to 3 months when considering different CA125 summaries. Table S3: 10-fold cross-validated AUC (A,B) and Brier score (C,D) for the prediction of overall survival at 24, 36, 48 and 60 months from CA125 history up to 6 months when considering different CA125 summaries (Landmark time 6 months). Table S4: Predictive performances by subgroups for estimated CA-125 value at 3 months. Table S5: Predictive performances by subgroups for estimated CA-125 value at 6 months. Table S6: 5-fold cross-validated AUC (A,B) and Brier score (C,D) for the prediction of overall survival at 24, 36, 48 and 60 months from CA125 history up to 3 months when considering different CA125 summaries. Half population including only Standard regimen arm. Table S7: Institutional Review Board Statement: This study was conducted according to the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) [START_REF] Stewart | Preferred reporting items for a systematic review and meta-analysis of individual participant data: The PRISMA-IPD statement[END_REF]-IPD guidelines for the registration of the protocol, trial identification, data collection and integrity, assessment of bias, and sensitivity analyses. This meta-analysis was registered with PROSPERO (CRD42017068135). The Ethics Committee of Gustave Roussy Cancer Center, Villejuif, France, approved this study, and the French Data Protection Authority waived the need for informed consent for the use of deidentified data.

Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.

Figure 1 .

 1 Figure1. Illustration of the landmark framework. CA-125 trajectories of observed values up to the landmark time of s = 6 months for two patients at risk of death at the landmark time (#1 alive after 56.9 months, and #2 deceased after 24.5 months). Their overall survival was assessed at t = 24 months. Summaries of the early CA-125 trajectory were obtained from values up to the landmark time (the blue dashed lines represent the observed slope, and the red dashed lines represent the estimated slope).

  Scheme 1. Flowchart: study selection.

Figure 2 .

 2 Figure 2. The 10-fold cross-validated AUCs (A,B) and Brier scores (C,D) for the prediction of overall survival at 24, 36, 48, and 60 months from CA-125 history up to 3 months, considering different CA-125 summaries (ORD, OVLT, EVLT, ESB, ESLT) adjusted (B,D) or not (A,C) based on the CA-125 value at baseline. Null model: survival model stratified by study, ORD: observed relative decline, OVLT: observed value at landmark time, EVLT: estimated value at landmark time, ESB: estimated slope at baseline, ESLT: estimated slope at landmark time. * models adjusted for CA-125 at baseline. Note: The SCOTROC-1 trial was excluded for the landmark time at 6 months and prediction time at 36 months and for the landmark time at 3 months and prediction time at 48 months. The SCOTROC-4 and MITO-7 trials were excluded for both landmark times and for the prediction time of 60 months.

Figure 3 .

 3 Figure 3. The 10-fold cross-validated AUCs (A,B) and Brier scores (C,D) for the prediction of overall survival at 24, 36, 48, and 60 months from CA-125 history up to 6 months, considering different CA-125 summaries (ORD, OVLT, EVLT, ESB, ESLT) adjusted (B,D) or not (A,C) based on the CA-125 value at baseline. Null model: survival model stratified on study, ORD: observed relative decline, OVLT: observed value at landmark time. Null model: survival model stratified by study, ORD: observed relative decline, OVLT: observed value at landmark time, EVLT: estimated value at landmark time, ESB: estimated slope at baseline, ESLT: estimated slope at landmark time. * models adjusted for CA-125 at baseline. Note: The SCOTROC-1 trial was excluded for the landmark time at 6 months and prediction time at 36 months and for the landmark time at 3 months and prediction time at 48 months. The SCOTROC-4 and MITO-7 trials were excluded for both landmark times and for the prediction time of 60 months.

  5-fold cross-validated AUC (A,B) and Brier score (C,D) for the prediction of overall survival at 24, 36, 48 and 60 months from CA125 history up to 6 months when considering different CA125 summaries. Half population including only Standard regimen arm.

Figure S1 :

 S1 Calibration plot for the observed rate of decline (ORD)-3 months.

  Figure S2: Calibration plot for the observed value at landmark time (OVLT)-3 months.

  Figure S3: Calibration plot for the estimated value at landmark time (EVLT)-3 months.

  Figure S4: Calibration plot for the estimated slope at baseline (ESB) (for landmark 3 months).

  Figure S5: Calibration plot for the estimated slope at landmark time (ESLT)-3 months.

  Figure S6: Predictive performances measured by AUC (A) and Brier Score (B) by subgroups for estimated CA-125 value at 3 months.

  Figure S7: Predictive performances measured by AUC (A) and Brier Score (B) by subgroups for estimated CA-125 value at 6 months. Figure S8: 5-fold cross-validated AUC (A,B) and Brier score (C,D) for the prediction of overall survival at 24, 36, 48 and 60 months from CA125 history up to 3 months in Standard regimen arm when considering different CA125 summaries (ORD, OVLT, EVLT, ESB, ESLT) adjusted (B,D) or not (A,C) on the CA125 value at baseline. Figure S9: 5-fold cross-validated AUC (A,B) and Brier score (C,D) for the prediction of overall survival at 24, 36, 48 and 60 months from CA125 history up to 6 months in Standard regimen arm when considering different CA125 summaries (ORD, OVLT, EVLT, ESB, ESLT) adjusted (B,D) or not (A,C) on the CA125 value at baseline. Author Contributions: Conceptualization, E.K., C.P.-L. and X.P.; data curation, E.K.; formal analysis, E.K., C.P.-L. and X.P.; funding acquisition, X.P.; investigation, R.M.G., C.K., L.-A.L., K.C., G.B.K., J.-L.E., T.K., N.Y., S.C.C., I.A.B., E.P.-L. and I.R.-C.; methodology, E.K., C.P.-L. and X.P.; project administration, C.P.-L. and X.P.; resources, E.K., R.M.G., C.K., L.-A.L., K.C., G.B.K., J.-L.E., T.K., N.Y., S.C.C., I.A.B., E.P.-L., I.R.-C. and X.P.; software, E.K., C.P.-L. and X.P.; supervision, C.P.-L. and X.P.; validation, E.K., C.P.-L. and X.P.; visualization, E.K., C.P.-L. and X.P.; writing-original draft, E.K., C.P.-L. and X.P.; writing-review and editing, R.M.G., C.K., L.-A.L., K.C., G.B.K., J.-L.E., T.K., N.Y., S.C.C., B.Y., I.A.B., E.P.-L., I.R.-C., C.P.-L. and X.P. All authors have read and agreed to the published version of the manuscript. Funding: This study was partly supported by grant PHRC-K 2017 'PHRC-K 17-187' from the Programme Hospitalier de Recherche Clinique en Cancérologie, funded by the French Ministry of Health and the French Ligue Against Cancer.

Table 1 .

 1 Description of included studies.

		Trial Name	Number of pts	Number of CA-125 Measures	Median Survival
		total	2377	21,978	3.87 (3.6; 4.11)
		MITO-7	529	1429	NR
	Intensification therapy	GINECO-2007 TURBO-2014	109 253	634 3421	2.63 (2.25; 3.39) 3.74 (3.23; 4.30)
		JGOG-3016	557	3475	6.72 (5.30; NA)
		SCOTROC-4	929	14,019	2.75 (2.54; 2.97)
		total	3196	49,989	4.33 (4.08; 4.66)
		JGOG-3017	632	9403	NR
	No maintenance	NSGO-2012	774	9904	3.60 (3.21; 3.98)
		SCOTROC-1	1052	12,582	2.86 (2.73; NA)
		CCTG-OV.16	737	18,100	3.84 (3.49; 4.28)
	Total		5573	72,967	4.11 (3.94; 4.33)

Abbreviations: NR, not reached.

Table 2 .

 2 Patients' characteristics by trial (N = 5209).

				FIGO				Residual Disease
	Trials	I	II	III	IV	.	<1 cm	≥1 cm	NA	.
	MITO-7	41 (8.9%)	42 (9.1%)	273 (59.2%)	105 (22.8%)		NA	NA
	GINECO-2007	0	0	78 (80.4%)	19 (19.6%)	1	14 (14.4%)	83 (85.6%)		1
	TURBO	0	19 (7.9%)	156 (65%)	65 (27.1%)		173 (72.7%	63 (26.5%))	2 (0.8%)	2
	JGOG-3016	0	102 (18.6%)	364 (66.5%)	81 (14.8%)		258 (47.2%)	275 (50.3%)	14 (2.6%)
	SCOTROC-4	118 (13.9%)	75 (8.8%)	549 (64.6%)	108 (12.7%)		300 (35.3%)	550 (64.7%)
	JGOG-3017	395 (66.1%)	69 (11.5%)	112 (18.7%)	22 (3.7%)		548 (91.6%)	48 (8%)	2 (0.3%)
	NSGO-2012	0	94 (12.7%)	533 (71.8%)	115 (15.5%)		317 (42.7%)	425 (57.3%)
	SCOTROC-1	70 (7.2%)	127 (13%)	645 (65.9%)	137 (14%)		332 (33.9%)	647 (66.1%)
	CCTG-OV.16	0	65 (9.4%)	468 (67.4%)	161 (23.2%)		325 (57%)	238 (41.8%)	7 (1.2%)	124

Abbreviations: '.': Missing information, NA: Not available.

Data Availability Statement: Individual patient data (IPD) were requested for each eligible trial for all randomized patients. All data were checked with a standard procedure which follows the recommendations of the Cochrane working group on meta-analysis using individual patient data. Each trial was analyzed individually, and the resulting survival analyses and data description were sent to the trialists for review.

Conflicts of Interest: J.-L.E. reported participating as Speaker and being a member of Advisory board for Merck, AZ and GSK. B.Y. reported consulting for MSD, Astra-Zeneca, GSK-TESARO, BAYER, Roche-Genentech, ECS Progastrine, Novartis, LEK, Amgen, Clovis Oncology, Merck Serono, BMS, SEAGEN, Myriad, Menarini, and Gilead. No other conflict of interest were declared.


Antonio Chahine

Julien Sebilleau

Romain Mathis

Dominique Legendre

Sliding droplets in a laminar or turbulent boundary layer

Keywords: Droplet, Shear flow, Boundary layer, Critical velocity

I. INTRODUCTION

Sliding droplets on a solid surface is a common phenomenon that can be observed on windows and windshields during rainy days. This situation, that occurs when an external force (such as gravity and aerodynamic forces) overcomes the capillary force, is of fundamental interest because of the subtle hydrodynamics near the contact line. The case of droplets sliding along an inclined wall under partial wetting conditions has been studied since the 1950's [START_REF] Bikerman | Sliding of drops from surfaces of different roughnesses[END_REF][START_REF] Furmidge | Studies at phase interfaces. i. the sliding of liquid drops on solid surfaces and a theory for spray retention[END_REF][START_REF] Dussan | On the ability of drops or bubbles to stick to non-horizontal surfaces of solids[END_REF][START_REF] Dussan | On the ability of drops or bubbles to stick to non-horizontal surfaces of solids. part 2. small drops or bubbles having contact angles of arbitrary size[END_REF][START_REF] Podgorski | Corners, cusps, and pearls in running drops[END_REF][START_REF] Le Grand | Shape and motion of drops sliding down an inclined plane[END_REF][START_REF] Inmaculada | Effect of initial droplet shape on the tangential force required for spreading and sliding along a solid surface[END_REF]. In this situation, the onset of sliding is predicted thanks to a balance between gravity and capillary forces leading to critical Bond number that depends on the substrate inclination and the contact angle hysteresis: Bo c = ρ d gsin(α)V 2/3 d /σ ∝ cos θ rcos θ a where ρ d , V d and σ are respectively the droplet density, volume and surface tension, α is the inclination of the substrate and θ r and θ a are the receding and advancing contact angles. Then depending on its velocity, the droplet can adopt three different shapes: oval, corner, or cusp shape.

The case of droplets sliding on an horizontal substrate under the effect of an airflow has also received some attention in the past decades [START_REF] Dussan | On the ability of drops to stick to surfaces of solids. part 3. the influences of the motion of the surrounding fluid on dislodging drops[END_REF][START_REF] Dimitrakopoulos | Displacement of fluid droplets from solid surfaces in low-reynolds-number shear flows[END_REF][START_REF] Milne | Drop shedding by shear flow for hydrophilic to superhydrophobic surfaces[END_REF][START_REF] Fan | Displacement of liquid droplets on a surface by a shearing air flow[END_REF][START_REF] Moghtadernejad | Shear Driven Droplet Shedding on Surfaces with Various Wettabilities[END_REF][START_REF] Fu | Detachment of droplets in a fully developed turbulent channel flow[END_REF][START_REF] Roisman | Dislodging a sessile drop by a high-reynolds-number shear flow at subfreezing temperatures[END_REF][START_REF] Seiler | Aerodynamically driven motion of a wall-bounded drop on a smooth solid substrate[END_REF][START_REF] Barwari | Experimental study of the onset of downstream motion of adhering droplets in turbulent shear flows[END_REF][START_REF] Barwari | Hydrodynamic Instabilities of Adhering Droplets Due to a Shear Flow in a Rectangular Channel[END_REF][START_REF] Ma | An experimental investigation on wind driven droplet moving on surfaces with different wettabilities[END_REF][START_REF] White | Wind-and gravity-forced drop depinning[END_REF][START_REF] Zhang | Interaction of water droplets residing on a solid surface with wall-bounded shear flows[END_REF]. Experiments have been performed

for laminar [START_REF] Milne | Drop shedding by shear flow for hydrophilic to superhydrophobic surfaces[END_REF][START_REF] Moghtadernejad | Shear Driven Droplet Shedding on Surfaces with Various Wettabilities[END_REF][START_REF] Roisman | Dislodging a sessile drop by a high-reynolds-number shear flow at subfreezing temperatures[END_REF][START_REF] Barwari | Experimental study of the onset of downstream motion of adhering droplets in turbulent shear flows[END_REF][START_REF] Ma | An experimental investigation on wind driven droplet moving on surfaces with different wettabilities[END_REF][START_REF] White | Wind-and gravity-forced drop depinning[END_REF][START_REF] Zhang | Interaction of water droplets residing on a solid surface with wall-bounded shear flows[END_REF] or turbulent [START_REF] Fu | Detachment of droplets in a fully developed turbulent channel flow[END_REF][START_REF] Seiler | Aerodynamically driven motion of a wall-bounded drop on a smooth solid substrate[END_REF][START_REF] Barwari | Experimental study of the onset of downstream motion of adhering droplets in turbulent shear flows[END_REF] boundary layer while theoretical and numerical studies have addressed the case of a linear shear flow [START_REF] Dussan | On the ability of drops to stick to surfaces of solids. part 3. the influences of the motion of the surrounding fluid on dislodging drops[END_REF][START_REF] Dimitrakopoulos | Displacement of fluid droplets from solid surfaces in low-reynolds-number shear flows[END_REF]. In these studies, the onset of sliding is generally predicted thanks to a balance between the drag force and the capillary force leading to a critical Weber number that depends on the contact angle hysteresis. Then, depending on its velocity, the droplet can adopt four different shapes: oval, corner, cusp, and rivulet shape. The latter is characterized by a long tail [START_REF] Barwari | Hydrodynamic Instabilities of Adhering Droplets Due to a Shear Flow in a Rectangular Channel[END_REF] that does not appear for droplets sliding on an inclined wall. The onset on sliding under the effect of an external airflow is usually reported using a critical Weber number W e = ρ a U 2 c H/σ, where ρ a is the air density, H is the height of the droplet and U c is a characteristic velocity either defined as the gas velocity U ∞ far from the substrate [START_REF] Moghtadernejad | Shear Driven Droplet Shedding on Surfaces with Various Wettabilities[END_REF][START_REF] Barwari | Experimental study of the onset of downstream motion of adhering droplets in turbulent shear flows[END_REF][START_REF] Barwari | Hydrodynamic Instabilities of Adhering Droplets Due to a Shear Flow in a Rectangular Channel[END_REF][START_REF] Ma | An experimental investigation on wind driven droplet moving on surfaces with different wettabilities[END_REF][START_REF] Hooshanginejad | Dynamics of a partially wetting droplet under wind and gravity[END_REF] or constructed using the shear rate ∂U ∂z [START_REF] Dussan | On the ability of drops to stick to surfaces of solids. part 3. the influences of the motion of the surrounding fluid on dislodging drops[END_REF][START_REF] Dimitrakopoulos | Displacement of fluid droplets from solid surfaces in low-reynolds-number shear flows[END_REF][START_REF] Roisman | Dislodging a sessile drop by a high-reynolds-number shear flow at subfreezing temperatures[END_REF][START_REF] Ma | An experimental investigation on wind driven droplet moving on surfaces with different wettabilities[END_REF]. The resulting values of W e and operating conditions are reported in Table I. It results in critical Weber numbers that differ about one order of magnitude for similar wetting conditions as illustrated in Fig. 1 where only measurements for water droplets are presented. It is also noticeable that, in these experiments, the droplets height and its ratio with boundary layer thickness δ estimated here with the scaling of Blasius theory for laminar boundary layer (δ = √ νax U∞ , where ν a is the air viscosity and x the distance to the leading edge) can change significantly as shown in Table I. Thus, some experiments are performed for droplets inside the linear part of the boundary layer ( H δ < 5) while in the rest the gas flow is no longer a linear sheared flow ( H δ > 5). (▸), Barwari et al. [START_REF] Barwari | Experimental study of the onset of downstream motion of adhering droplets in turbulent shear flows[END_REF]; (◇), Hooshanginejad and Lee [START_REF] Hooshanginejad | Droplet depinning in a wake[END_REF]; (▲), White and Schmucker [START_REF] White | Wind-and gravity-forced drop depinning[END_REF];

(▽), Zhang [START_REF] Zhang | Flow development over isolated droplet-inspired shapes[END_REF].

The aim of the present work is to clarify the onset of sliding prediction and to characterize the different shapes adopted by the sliding droplet thanks to two synchronized views (from the side and the bottom) with two different liquids (water and glycerin). In the following, we first introduce the experimental apparatus and the procedures employed for these measurements (Sec. II), then we present our experimental results obtained and compare them with the theory (Sec. III) and finally concludes this work (Sec. IV).

II. EXPERIMENTAL CONFIGURATION

The experiments were performed inside a closed-loop wind tunnel depicted in Fig. 2 (a). This facility has a working test section of 0.8 m × 0.5 m × 2.0 m with a maximum freestream velocity of 30 m/s and a free-stream turbulence intensity less than 0.1%. A 1.2 m long aluminum plate with a 3D print leading edge (to ensure a laminar boundary layer) is placed at the mid-height of the test section. This plate was designed to host a glass plate of 200 mm × 100 mm at a distance of 600 mm from the leading edge (see Fig. 2 (a), (b), and (c)), where the droplets are deposited with a micro-pipette. Prior to experiments, the glass plate was treated with a commercial rainproof product named Rain-X. This product consists of a hydrophobic silicone polymer that renders the glass hydrophobic (with typical contact angles about 90 ○ . According to the manufacturer's recommendation, the treatment was applied as follows:

1. the surface is cleaned with water, 2. the Rain-X is applied on the glass by rubbing in a circular motion a cloth, 3. the substrate is allowed to dry. Any excess powder on the glass is cleaned until a fully transparent surface is achieved.

The wetting conditions were tested by measuring the advancing and receding contact angles on several locations of the treated glass substrate thanks to a KRUSS, EASYDROP goniometer. The mean advancing and receding contact angles (θ a,sθ a,r ) were observed to be quite homogeneous on the treated glass surface with values (87 ○ -83 ○ ) and (92 ○ -86 ○ ) for water and glycerol droplets, respectively. The droplet deformation and sliding were recorded by shadowgraphy using two Imager SCMOS cameras from Lavision (see Fig. 2 (d)), with a resolution of 2560 pix × 2160 pix, at a frame rate of 100 fps. The side view field is approximately equal to 6.2 cm × 1.3 cm, and the bottom view is about 3.6 cm × 1.2 cm. Both cameras were synchronized to record at the same time. The recording begins when the wind tunnel is started. Each group of experiments was repeated at least two times under the same conditions. A summary of the experimental conditions is in Table III. The properties of the water and glycerol are presented in Table II.

III. RESULTS

A. Shapes of sliding droplets

The use of two cameras allows to record simultaneous side and bottom views of the droplet during its motion. Typical experimental visualisations of glycerin and water droplets are presented in Fig. 4 As one can see, when submitted to a shear flow, the droplet initially axisymmetric deforms along the flow direction and can adopt various shapes during its sliding. For the glycerin droplet presented in Fig. 4, the initially axisymmetric droplet is firstly tilted in the airflow direction. Prior to the onset of sliding the downstream (receding) contact moves first leading to an oval shape that begins to move with the airflow. Then the shape evolves to a corner shape that can easily be recognized on the bottom view thanks to the formation of a wedge on the downstream contact line. This wedge eventually evolves toward a rivulet shape that can exhibit waves on its surface if sufficiently long. These waves appear to have a strong influence on the droplet sliding but appear after the other sliding regimes (oval, corner and rivulet) and thus do not affect the onset of sliding. For water droplets, similar shapes are observed but additional oscillations can appear before the onset of sliding and remain for oval shapes. The appearance of these shapes depends on both the airflow conditions and the droplet volume. To further characterize these regimes, the latest droplet shapes observed (in the field of view of the cameras) are summarized for the studied volume and airflow conditions in Fig. 6 (a) and (b) for water droplets for laminar and turbulent boundary layer and in the Fig. 7 for glycerin droplets for laminar boundary layers. A similar trend is observed in the three figures. When moving along the diagonal with increasing both the droplets volume and the inlet flow velocity, the droplets shape varies from oval, to corner, then to rivulet. However, we can notice that immobile water droplets were observed at low velocity but not observed for glycerin droplets, a non-expected result because of droplets with a much larger fluid viscosity. The rivulet shape that only appears for sufficiently large droplets and velocities is not observed in laminar conditions for the range of water droplets considered here. The condition of sliding is discussed in the next section.

B. Onset of sliding

The onset of sliding is defined by the simultaneous motion of the downstream (receding) and upstream (advancing contact line) of the droplet. This onset of sliding has been studied theoretically and numerically for droplets in a linear shear flow [START_REF] Dussan | On the ability of drops to stick to surfaces of solids. part 3. the influences of the motion of the surrounding fluid on dislodging drops[END_REF][START_REF] Dimitrakopoulos | Displacement of fluid droplets from solid surfaces in low-reynolds-number shear flows[END_REF]. Both these studies predict the onset of sliding through a critical shear rate that's scales with the contact angle hysteresis and the droplet volume as:

( ∂U ∂z ) c ∝ σ µ a (cos(θ r ) -cos(θ a )) V 1/3 d (1)
In our experiments with a laminar boundary layer, the droplet height is sufficiently small to have the entire droplet inside the linear sheared region of the boundary layer. Thus, using the velocity profile measurement depicted in Fig 3, the critical shear rate for this onset of sliding can be measured. with the theoretical scaling law (Eq. 1) for a water and a glycerin droplets, respectively. As one can see, the measured critical shear rate appears to evolve linearly with this theoretical prediction. Nevertheless, the each airflow velocity U ∞ appears to have its own evolution meaning that the theoretical scaling law does not catch all the effects induced by the airflow when considering Eq. 1. Considering now our experiments in turbulent boundary layers, the droplet still remains inside the boundary layer but is no longer submitted to a linear shear.

We thus propose to define an adapted shear rate with the ratio of the air velocity at the droplet height U H determined using the turbulent boundary layer profile reported in Fig. 3 and the droplet height H as (∂U /∂z) c ≈ U H /H. The corresponding evolution for (∂U /∂z) c , reported in Fig. 8 (c), appears very similar to those obtained for laminar boundary layers and still follows a linear evolution with the theoretical scaling law (Eq. 1), again with different lines dependant on the air velocity U ∞ far from the substrate. Another approach to predict the onset of sliding resides on a critical Weber number obtained through a force balance approach [START_REF] Milne | Drop shedding by shear flow for hydrophilic to superhydrophobic surfaces[END_REF][START_REF] Moghtadernejad | Shear Driven Droplet Shedding on Surfaces with Various Wettabilities[END_REF][START_REF] Fu | Detachment of droplets in a fully developed turbulent channel flow[END_REF][START_REF] Roisman | Dislodging a sessile drop by a high-reynolds-number shear flow at subfreezing temperatures[END_REF][START_REF] Seiler | Aerodynamically driven motion of a wall-bounded drop on a smooth solid substrate[END_REF][START_REF] Barwari | Experimental study of the onset of downstream motion of adhering droplets in turbulent shear flows[END_REF][START_REF] Ma | An experimental investigation on wind driven droplet moving on surfaces with different wettabilities[END_REF][START_REF] White | Wind-and gravity-forced drop depinning[END_REF][START_REF] Zhang | Interaction of water droplets residing on a solid surface with wall-bounded shear flows[END_REF]. At the onset of sliding, the aerodynamic force overcomes the capillary force. Thus, assuming that the droplet shape remains close to a spherical cap with a width W and a height H, this force balance reads:

1 2 ρ a U 2 m C d πHW ≃ σW (cos(θ r ) -cos(θ a )) (2)
where ρ a is the air density, C d is the droplet drag coefficient, and U m is the mean air velocity impacting the droplet. For experiments in laminar boundary layer, the mean velocity is given by U m = U H 2 as the portion of the boundary layer profile impacting the droplet is linear (i.e. H δ < 5). For experiments in turbulent boundary layer, the previous comparison with the theoretical scaling law (Eq. 1) suggests that taking U m = U H 2 is acceptable even if the portion of the boundary layer profile impacting the droplet is no longer linear. Thus, the force balance (Eq. 2) can be converted into a critical Weber number as

W e c = ρ a U 2 H H σ ≃ 8(cos(θ r ) -cos(θ a )) πC d (3)
It is noticeable that the drag coefficient in Eq. 3 depends on the Reynolds number Re H = 2U H H νa , that varies between 50 and 450 in our experiments. The drag coefficient for an hemispherical bubble set fixed on wall in a linear shear flow has been proposed by Legendre et al. [START_REF] Legendre | Lift, drag and added mass of a hemispherical bubble sliding and growing on a wall in a viscous linear shear flow[END_REF]. The proposed relation connects the analytical solution under Stokes flow condition (Re H ≪ 1) and the asymptotic limit at large Reynolds number (Re H ≫ 1) obtained using direct numerical simulation leading to

C d = 16 Re H + C ∞ d = 16
Re H + 0.137. The extension of the Stokes solution in the limit Re H ≪ 1 from the bubble to the solid hemisphere is straightforward and gives C d = 24/Re H . The limit at large Reynolds number is reported in Nardone et al. [START_REF] Nardone | Velocity field and drag force measurements of a cube and a hemisphere mounted on an artificial bed surface roughness. e3s web conf[END_REF] and in Saal et al. [START_REF] Saal | Shuffling gait motion of an aerodynamically driven wall-bound drop[END_REF] where C ∞ d = 0.40, so we propose the following relation to provide the description of the drag coefficient of a solid hemisphere in a linear shear flow as:

C d = 24 Re H + 0.40 (4)
Figure 9 presents the measured critical Weber number with the prediction (Eq. 3 combined with Eq. 4) for a water droplet in a laminar and turbulent boundary layer (open and close symbols respectively) and for a glycerin droplet in a laminar boundary layer. As one can see, linear trends independent of the airflow velocity U ∞ are now observed but the slopes appear to be dependent on the fluid and on the type of boundary layer (which is not surprising in regards of the "crude" approximation made for the mean air velocity for the turbulent boundary layer). The observed dependence on the fluid, suggests that the droplet viscosity (other physical properties being similar) has also an effect on the onset of sliding.

This effect may be linked to the development of a recirculation inside the droplet prior to its sliding. Such recirculation has been observed numerically [START_REF] Dimitrakopoulos | Displacement of fluid droplets from solid surfaces in low-reynolds-number shear flows[END_REF] and an effect of the viscosity ratio on the onset of sliding has already been suggested [START_REF] Dussan | On the ability of drops to stick to surfaces of solids. part 3. the influences of the motion of the surrounding fluid on dislodging drops[END_REF][START_REF] Dimitrakopoulos | Displacement of fluid droplets from solid surfaces in low-reynolds-number shear flows[END_REF]. Indeed, the recirculation inside the droplet is induced by the airflow through the tangential viscous shear stress at the interface. The magnitude of the recirculation is then connected to the external air flow with the viscosity ratio µ a /µ. To confront our prediction with the results of the literature, we select in Table I the experiments performed for water droplets in laminar boundary layer (for which it is possible to compute the air velocity at the top of the droplet U H ) and plot them together with our experiments with water droplet in laminar boundary layer in Fig. 10. As one can see, both our data and the ones from the literature appear to follow the same linear trend (except for the data of Roisman et al. [START_REF] Roisman | Dislodging a sessile drop by a high-reynolds-number shear flow at subfreezing temperatures[END_REF]). This confirms that our prediction captures well the physics behind the onset of sliding. Note that the Reynolds number has a strong influence on the droplet onset of sliding due to the dependency of the drag coefficient with the Reynolds number for the system considered here.

C. Transition between shapes

After the onset of sliding, the droplet can adopt various shapes depending on experimental conditions (droplet volume and airflow velocity). The transition from oval to corner shape is difficult to characterize with only the side view as the wedge formation is only accessible from the bottom view. Nevertheless, as the droplet length strongly evolves during the transition from corner to rivulet shape, it may be possible to characterize the transition from oval to corner shape thanks to the evolution of the droplet length. Figure 11 presents the length evolution with time for two different air velocities and for a water droplet of volume V d = 40 µL in dimensionless form (L = L/L 0 , L 0 being the initial droplet length, and t = t/t f , t f being the time when the droplet reaches the end of the field of view). As one can see, when the droplet adopts an oval shape, its length varies by less than 12%, while corner shape and rivulet shape are characterized by a larger variation allowing to characterize the transition between oval and corner shape on the evolution of the length. It is interesting to notice that the contact angle evolution with the Capillary number differs from the expected Cox-Voinov evolution and that the droplet length evolution appears to be significant when the contact angles have reached almost constant values. Thus one can expect a strong coupling between the droplet shape and its velocity. To confirm these results, phase diagrams of droplet shape in terms of capillary and Bond (Bo = ∆ρ g L 2 0 /σ) numbers are presented in Fig. 14 for glycerin and water droplets. These phase diagrams clearly show the link between the droplet shapes and the capillary number. It also appears that the rivulet shape only exists for a sufficiently large Bond number of order of 6.8 for water droplets and 3.2 for glycerin droplets. One can also remark that the capillary numbers for transition between shapes differ from water droplets to glycerin droplets. This confirm that the air-droplet viscosity ratio plays an important role in the transition between the shapes. A similar effect was observed for droplets sliding along an inclined wall [START_REF] Le Grand | Shape and motion of drops sliding down an inclined plane[END_REF].

(i) (ii) (iii) (iv) (i) (ii) (iii) (iv

IV. CONCLUSION

In this work, droplet sliding induced by an airflow on a horizontal surface has been studied experimentally. The onset of sliding is described thanks to a critical Weber number based on a force balance approach. This Weber number, based on the mean airflow velocity impacting the droplet, is dependent on the contact angle hysteresis, and the droplet drag coefficient itself depends on the Reynolds number. A relation for the dependence of the drag coefficient with the Reynolds number for an hemispherical droplet in a linear shear flow is proposed and used for the critical number prediction. This proposed Weber number appears to be in good agreement with our experiments and those from literature but appears to depend on the droplet viscosity. This dependence upon the droplet viscosity appears as an interesting perspective but needs more experiments with other viscosities. The various shape of a sliding droplet (oval, corner, and rivulet) has been determined and their transitions appear to be directly linked to the capillary number. It is interesting to notice that the rivulet shape only appears for sufficiently large droplets.

FIG. 2 :

 2 FIG. 2: Experimental Set-up: (a): The Wind Tunnel showing the bottom view setup. (b): Bottom View Set-up. (c): Side and Bottom View Set-up. (d): Examples of droplet photos taken by the cameras from side and bottom view.

  FIG. 3: Measurement of flow profiles for (a): Laminar Flow and (b): Turbulent Flow.

  and 5, respectively, for droplet volume V d = 40 µL and airflow with U ∞ = 22 m/s and U ∞ = 20 m/s respectively.

V

  FIG. 4: Side and bottom views of a glycerol droplet submitted to a laminar shear flow on a glass surface, V d = 40 µL -U ∞ = 22 m/s.

FIG. 6 :

 6 FIG. 6: Cartography of shape and behaviour for water droplets. (a): Laminar Flow. (b): Turbulent Flow. (×), Oval Immobile; (+), Oval Oscillation & immobile; (◻), Oval Oscillation & moving; (○), Oval moving; (△), Corner; (▷), Rivulet; (◇), Long Rivulet.

FIG. 7 :

 7 FIG. 7: Cartography of shape and behaviour for glycerol droplets (laminar flow only). (○), Oval moving; (▷), Rivulet; (▽), Rivulet Wave; ( ), Rivulet Break-up.

FIG. 8 :

 8 FIG. 8: Evolution of the critical shear rate (∂U /∂z) as a function of Eq. (1): (a): water droplet in laminar flow. (b): glycerol droplet in laminar flow (c): water droplet in turbulent flow.

Fig. 8 (

 8 Fig. 8 (a) and (b) shows the evolution of the critical shear rate in a laminar boundary layer

FIG. 9 :

 9 FIG. 9: Evolution of the critical Weber number, W e c for water and glycerol, as a function of 8[cos(θ r )cos(θ a )]/(πC d ). (○), Water droplets; (◻), Glycerol droplets. Droplets immersed in a laminar boundary layer are represented in open symbols and in solid symbols for a turbulent boundary layer.

FIG. 10 :

 10 FIG. 10: The distribution of critical Weber numbers determined using the shear stress (∂U /∂z) c as defined in the text for both our data and the literature on water droplets in Laminar flow, function of 8[cos(θ r )cos(θ a )]/[πC d ]. (○), Water droplet -Our data; (•), Milne and Amirfazli -2009, (∎), Roisman et al. -2015; (◇), Hooshanginejad and Lee -2017; (▲), White and Schmucker -2021; (▽), Zhang -2021. The symbols represent different droplet sizes with respect to the boundary layer: open symbols for H/δ < 5 and solid symbols for H/δ > 5

  FIG. 11: Dimensionless Length L of the droplet function of dimensionless time t to show the condition of transition between oval and corner for two different velocities U ∞ = 12 and 20 m/s.

FIG. 12 :FIG. 13 :

 1213 FIG. 12: Advancing and Receding contact angles function of the Capillary number. (a): water Droplet. (b): Glycerol Droplet. (△), Advancing contact angle θ a ; (▽), Receding contact angle θ r .

FIG. 14 :

 14 FIG. 14: Capillary number Ca function of Bond number Bo. Droplets immersed in laminar boundary layer are represented in open symbols and in solid symbols for turbulent boundary layer. (a): Water Droplet. (b): Glycerol Droplet.

TABLE I :

 I Critical depinning conditions for water droplets reported in literature W e and calculated W e c ; θ m is the mean contact angle at depining; studies marked with ✠ are experiments conducted in turbulent channel flows, and otherwise in laminar boundary layers formed over flat plates.

		8
		6
		W e
		4
		2
		0	0	0.2	0.4 cos(θr) -cos(θa) 0.6	0.8	1
	Substrate Weber number W e constructed using the shear rate V d (µL) Uc (m/s) θm (deg) H (mm) PDMS 50 3.8 79 2.6 100 3.5 79 3.3 PMMA 50 3.2 80 2.6 100 3.2 80 3.3 FIG. 1: Critical Weber number W e as a function of the hysteresis at the contact line H/δ W e W ec Roisman et al. [14] 7.316 0.6 0.630 8.846 0.7 1.259 6.779 0.4 0.460 8.541 0.6 0.574 cos(θ r ) -cos(θ a ) as reported in literature for water droplet. The symbols represent different droplet sizes with respect to the boundary layer: open symbols for H/δ < 5 and solid symbols for H/δ > 5. (•), Milne and Amirfazli [10]; (∎), Roisman et al. [14]; (◂), Seiler et al. [15];
	(∎)	Teflon	50 100	2.6 2.5	113 113	3.5 4.4	8.034 9.926	0.3 0.4	0.441 0.813
		SHS		50 100	2.2 2.2	150 150	4.3 5.4	9.160 11.541	0.3 0.4	0.612 1.224
				75	17.8	82	2.5	2.660	6.9	1.926
	Zhang [20] (▽)	PMMA	90 105	17.2 18.2	81.2 83	2.6 2.7	2.766 2.872	7.0 8.3	2.001 2.425
				120	16.8	83.5	2.8	2.979	7.7	2.309
		Aluminum	35	3.4	63	2.0	7.242	1.7	-
	Seiler et al. ✠ [15]	Aluminum (varnished)	35	3.3	60	1.9	6.885	1.6	-
	(◂)	PMMA	35	3.2	59	1.9	6.698	1.5	-
		Steel (varnished)	35	3.5	50	1.7	6.223	1.8	-
		Weber number W e constructed using the gas velocity
	Milne and Amirfazli [10] (•)	PMMA	58 100	4.8 3.5	66 66	2.2 2.4	8.388 7.814	1.8 1.6	0.860 0.500
	Hooshanginejad and Lee [22] (◇)	Aluminum (rough)	130	20.8	49	2.0	4.933	8.5	0.674
	White and Schmucker [19] (▲)	Aluminum (rough)	75 100 150	14.8 14.5 13.5	51 51 51	2.1 2.5 2.8	4.481 5.281 5.707	7.6 7.8 7.6	0.301 0.481 0.539
	Barwari et al. ✠ [16]	PMMA	39.9	10	51	1.8	10.356	3.6	-
	(▸)	Silicon (coated)	39.9	5.2	90	2.7	11.944	1.3	-

TABLE II :

 II Liquid properties: Dynamic viscosity µ, density ρ, surface tension σ and the test section temperature T .

	Water	0.001	997	75.64 * 10 -3	20
	Glycerol	1.1438	1260	63.40 * 10 -3	20

Fluid

µ (Pa.s)

ρ (Kg/m 3 ) σ (N/m) T ( ○ C)

TABLE III :

 III Experimental conditions: droplet volume V d , surface wetting properties and inlet air flow velocity U ∞ and boundary layer (laminar vs turbulent).

	Water	5-100	92 ± 2	75 ± 2	4-22	×	×
	Glycerol	10-100	95 ± 2	80 ± 2	8-22	×	-

Viscosity V d range (µL) θ a,s (deg) θ r,s (deg) U ∞ range (m/s) Laminar Turbulent