Guillaume Bouzillé

Camille Morival

Richard Westerlynck

Pierre Lemordant

Emmanuel Chazard

Pascal Lecorre

Yann Busnel

Marc Cuggia

An Automated Detection System of Drug-Drug Interactions from Electronic Patient Records Using Big Data Analytics

Keywords: Computing Methodologies, Drug Interaction, Machine Learning

The aim of the study was to build a proof-of-concept demonstratrating that big data technology could improve drug safety monitoring in a hospital and could help pharmacovigilance professionals to make data-driven targeted hypotheses on adverse drug events (ADEs) due to drug-drug interactions (DDI). We developed a DDI automatic detection system based on treatment data and laboratory tests from the electronic health records stored in the clinical data warehouse of Rennes academic hospital. We also used OrientDb, a graph database to store informations from five drug knowledge databases and Spark to perform analysis of potential interactions betweens drugs taken by hospitalized patients. Then, we developed a machine learning model to identify the patients in whom an ADE might have occurred because of a DDI. The DDI detection system worked efficiently and computation time was manageable. The system could be routinely employed for monitoring.

Introduction

Drug-drug interactions (DDIs) are a critical issue in patient care because they can lead to adverse events and ultimately increase care costs and patient mortality. Therefore, these events must be identified and prevented as early as possible [START_REF] Meier | Adverse drug events in patients admitted to an emergency department: an analysis of direct costs[END_REF]. However, many new drugs are released each year, and therefore, it is very difficult for healthcare professionals to be informed and to consider all DDIs. Moreover, the alarm functionalities of drug computerized physician order entry (CPOE) systems are frequently not used because they do not focus on clinically relevant DDIs and lead users to alarm fatigue. Although focused on specific interactions, studies on DDI prevalence show the existence of risks for polymedicated patients and highlight the importance of pharmacovigilance programmes [START_REF] Bonnet | Potential drug-drug interactions with abiraterone in metastatic castration-resistant prostate cancer patients: a prevalence study in France[END_REF][START_REF] Ramirez | A Pharmacovigilance Program From Laboratory Signals for the Detection and Reporting of Serious Adverse Drug Reactions in Hospitalized Patients[END_REF].

With the unprecedented development of digital health and hospital clinical data warehouses (CDW), data produced during the healthcare process are now easily reusable [START_REF] Price | What Can Big Data Offer the Pharmacovigilance of Orphan Drugs?[END_REF]. Electronic health records (EHR) contain real-time information on drug prescription/regimens during hospitalization as well as all clinical information. Such data could be analysed to estimate DDI prevalence, to facilitate health professionals' practice assessment and to detect the occurrence of DDI-linked adverse drug events (ADE). In France, pharmacovigilance currently relies mainly on the spontaneous reporting by physicians or/and detection of diagnoses that could be related to ADE from the hospital billing system (diagnosis related group, DRG, database). New data sources, such as national claim databases, are also leveraged to improve DDI and ADE detection [START_REF] Martin-Latry | Pharmacoepidemiological research using French reimbursement databases: yes we can![END_REF][START_REF] Yeh | Potential drug-drug interactions in pediatric outpatient prescriptions for newborns and infants[END_REF]. EHR data-mining also could help pharmacovigilance professionals to improve drug safety assessment.

All these health-related databases fit perfectly with the big data paradigm because they contain voluminous, highly complex and heterogeneous information that is produced in real time [START_REF] Bouzillé | Sharing health big data for research -A design by use cases: the INSHARE platform approach[END_REF]. In the last few years, many big data technologies have been developed. However, their implementation in a hospital information system for processing healthcare big data in realworld condition of use is still largely uncharted.

Here, we describe a method, which propose to use big data technology to improve drug safety monitoring in a hospital and could help pharmacovigilance professionals to make datadriven targeted hypothesis on ADEs.

Methods

Figure 1 presents the overall approach of the study and the big data technologies used in each step.

Patient Data

We used the Rennes academic hospital EHRs that are stored in a CDW called eHOP (entrepot HOPital). This CDW includes both structured data (e.g., laboratory results, drug prescriptions and regimens) and unstructured data (e.g., operative reports, discharge summaries), and is dedicated to data reuse for clinical research [START_REF] Bouzillé | An Integrated Workflow For Secondary Use of Patient Data for Clinical Research[END_REF]. The eHOP's star schema architecture and graphic user interface allows researchers, even without any database language knowledge, to quickly access and efficiently search information within millions of patient records.

For this study, we used information about drug administrations (used drug(s) and regimens) and laboratory results (date, nature of the test and results: normal, abnormally high, or abnormally low). To identify and collect information on potential DDIs, and also to compare information from different sources, we selected five drug knowledge databases: Thesaurus, Vidal, Theriaque, Micromedex and Drugs.com [START_REF] Ansm | Agence nationale de sécurité du médicament et des produits de santé[END_REF][START_REF]VIDAL -La base de données en ligne des prescripteurs libéraux[END_REF][START_REF] Thériaque | [END_REF][12][START_REF][END_REF]. These databases are commonly used by health professionals, but are not specifically targeted to DDI detection. They are available via a web application programming interface (API) that requires a specific procedure because each database stores data with its own structure. To avoid this, we extracted the relevant information from these databases and stored it in OrientDB, a graph-oriented model database [START_REF]OrientDB -Distributed Graph/Document Multi-Model Database[END_REF] that fits well with our objective because a DDI can be modelled as an edge between two drugs. Thus, once the information is stored in a single OrientDB database, no more computation is required to access such information.

DDI Automatic Detection System from Patient Records (Step 2)

For DDI identification, we collected drug data from the patient EHRs stored in eHOP and computed the active interval (i.e., the period during which a drug was effective) for all drugs taken by a patient during the hospital stay. If two active intervals overlapped (fully or partially), then analysis of the data collected in the OrientDB database allowed determining whether the two drugs interacted. In this case, the potential DDI event was stored in eHOP. As these are independent processes (each drug pair is checked independently), the Spark clustercomputing framework was used to perform distributed computing [START_REF] Zaharia | Spark: Cluster computing with working sets[END_REF][START_REF] Zaharia | Resilient distributed datasets: A fault-tolerant abstraction for inmemory cluster computing[END_REF]. As all the potential DDI events can be stored in eHOP, then we could compute the prevalence of a DDI for any specific drug, molecule, or population.

Creation of a Machine Learning Model (Step 3a)

The data stored in the CDW eHOP do not allow direct confirmation of whether a patient reported a DDI-linked ADE or not. Indeed, this needs to be validated by the pharmacovigilance experts who do not have the proper means to check all the patient records. Therefore, we wanted to create a system to report to drug safety professionals only the most interesting cases among all DDIs detected by the DDI automatic detection system (i.e., patients in whom an ADE might have occurred because of a DDI).

We assume that laboratory results will change if an ADE occurs. So, we can train a machine learning model with two populations: those who experienced an ADE and those who did not. Unfortunately, we cannot identify manually who experienced an ADE. For this reason, we performed one of the research design presented by Hennessy et al. [START_REF] Hennessy | Pharmacoepidemiologic Methods for Studying the Health Effects of Drug-Drug Interactions (DDIs)[END_REF]: we choose to compare the population exposed to a DDI with another population non-exposed to this DDI and who did not experience an ADE, by design. There are likely many patients who do not experience an ADE in the exposed population, but the model will present only the most suspected cases and this problem will be solved with the gradual feedback of drug safety professionals: the system will adjust weights of patients in the model, giving a greater weight to the well-predicted patients.

We developed an artificial neural network system that allows us to predict an output. This system has a single hidden layer and the number of perceptrons was decided during crossvalidation. Our machine learning model works in two phases. First, it uses all data available for patients who experienced a specific DDI and those who did not (exposed and non-exposed populations) to classify them as having reported an ADE or not. Then, the model is reinforced with information coming from drug safety professionals who infirm or confirm the previous classification (Fig. 2).

We then had to form the non-exposed population. Within a DDI, we called "Object" the drug under study, and "Precipitant" the other drug. Moreover, we called "Controlprecipitant" any drug that has the same therapeutic use as the Precipitant, but that does not interact with the Object. For a given Object, we compared the exposed population, found with the DDI automatic detection system, to the non-exposed population. The non-exposed population included all patients, who were not in the exposed population and who had an overlap (fully or partial) between the action interval of the Object and of the Control-precipitant. We created this non-exposed population using the same process as for the exposed population.

Figure 2 -Creation and use of the artifical neural network

Data processing was performed with Java 8, Spark 2.10 and OrientDB 2.2.4 on Intel(R) Xeon(R) CPU E5-2609 1,90GHz computer with 32,0 Go of RAM.

Big Data Technologies: Convenient Tools for Complex Data Processing

Here, we proposed a complete automated data treatment system, from the collection of heterogeneous data to their enhancement in a machine learning model. This system can monitor DDI prevalence and try to identify patients with a possible DDI-linked ADE, without the intervention of drug safety professionals. To achieve this, we used several convenient tools:

OrientDB is an easy-to-use tool to store pre-computed data. The OrientDB database model includes two main classes: vertices and edges that connect two vertices. In our study, the "vertex" interface represented the class "Drug" and included drug name, ID-code and half-life. The "edge" interface represented the class "Interactions" and included DDI severity level. We also specified from which drug database the information on the DDI came. Thus, via OrientDB, each drug knowledge database can be interrogated separately. The "edge" interface is also used to represent the class "Control-precipitant".

Figure 3 presents the database model through an example: Drug1 has an interaction with Drug3 according two different databases (two edges of class "Interaction"). Let consider the Object-Precipitant couple Drug1-Drug3, then Drug4 is a control-precipitant of Drug3 (one oriented edge of class "Control-precipitant"). An example of query would be: "give all the drugs that have an interaction with Drug1 according Micromedex and where the severity level is 1".

Figure 3 -OrientDb database model

The query language was very close to the structured query language (SQL) and allowed searching a vertex that walks along edges to another vertex, according to the chosen conditions. Data uploading is fast and based on a convenient Java Graph API. We manipulated a graph Java object that is automatically committed at the end of the process. Moreover, if access to a part of the graph is required (e.g., all the drugs that interact with pravastatin according to a severity level of 2), we used this object as a temporary store before processing.

Switching between different knowledge databases, stored in the same OrientDB database, involves only a variable on an edge. Ultimately, the little amount of time spent for the precalculation facilitates the storage and the access to multiple data sources. Only one kind of request is needed for all five databases. We could easily add information from other data sources (for example, composition of a drug and half-life of the active substances), or more precise information about DDIlinked ADEs (such as the relevant laboratory tests). Nevertheless, this task demands a manual work for each group of drugs [START_REF] Neubert | Epharmacovigilance: development and implementation of a computable knowledge base to identify adverse drug reactions[END_REF]. Spark allows parallel processing easily. As many processes are independent from each other, their parallel treatment with Spark leads to a big time saving [START_REF] Shoro | Big data analysis: Apache spark perspective[END_REF].

Evaluation (Step 3B)

To evaluate our DDI detection system (step 2 in Fig 1 ), we focused on a class of drugs called statins that are prescribed (long-term treatment) to patients with cardiovascular diseases, and particularly to elderly patients who are usually polymedicated and consequently prone to DDIs. We selected the study population (i.e., all patients taking statins) from all patients included in eHOP from January, 1 2015 to July, 8 2016. It included 10,506 hospitalized patients with a median hospitalization of 7 days, and a median age of 72 years (range: 19 to 98 years).

We defined statins as the "Object" and all the drugs that interact with them were considered as candidate "Precipitants". We selected as Control-precipitants (symbolized by * in fig 3) all the drugs that are in the same fifth level (i.e., chemical substance) as the Precipitant in the Anatomical Therapeutic Chemical (ATC) classification [START_REF]WHOCC -Structure and principles[END_REF], but do not interact with the Object. Thus, Control-precipitants have the same (or a similar) therapeutic usage as the Precipitant. We stored all these data in OrientDB because each DDI is a link (i.e., edge) between drugs (i.e., two vertices).

Concerning the action intervals, we chose a period of seven half-lives for each statin molecule and arbitrarily selected one day for the Precipitant, because this information could not always be extracted automatically from the five drug knowledge databases.

To determine how well the machine learning model can identify patients who may have a DDI-linked ADE (step 3B in Fig 1 ), we evaluated the model prediction error using the out-of-bag (OOB) error method: several models are built with a bootstrapped dataset, the OOB error is the mean of the errors computed with non-used data in each model.

The neural network gives the probability to belong to a class. We used cross-validatin resampling to optimize the threshold separating the two class. We chose to study a specific DDI in which atorvastatin was the Object and cyclosporine the Precipitant (i.e., exposed population). The non-exposed population consisted of patients who took atorvastatin and a Control-precipitant (Fig 3). The used variables were: demographic data, pathologies (ICD-10 codes) and laboratory test results. We used all the laboratory test results included between the beginning of the event and 3 days later. If a laboratory test appeared more than once, we took the mode of the results.

The reinforcement phase was not evaluated because it is currently under construction in collaboration with drug safety specialists.

Results

DDI identification with the automatic detection system was very fast due to the use of a graph-oriented model. For instance, for the simple query "is there an interaction between these two drugs?", or the more complex query "select all drugs that interact with this specific drug", the OrientDB database was always faster (less than 20ms) than the Theriaque SQL database (several seconds). Moreover, switching to another drug knowledge database was very easy with OrientDB because it only needed to change a condition in the query (which database = 'Theriaque').

The time required to create these graph databases was reasonable: for instance, the information coming from the Theriaque database, which is equivalent to 18,800 vertices and 23 million edges, was integrated in one hour. Afterwards, data access was immediate.

Once the OrientDB database was ready, from the eHOP CDW, we checked the DDI occurrence for all drug couples in the study population. To this aim, we computed all the fully or partially overlapping action intervals for all drug couples involving a statin. For each patient, we visualized all the detected DDIs: between 22.5% and 52.2% (depending on the drug knowledge database) of the 10,506 patients who were taking statins presented at least one DDI involving a statin.

Computation time was reduced with the use of the Spark framework: the processing time of 800,000 rows of patient records decreased from 60 minutes initially to only 12 minutes with Spark.

To test the ADE prediction performance of the machine learning model, we then focused only on one specific DDI (atorvastatin-cyclosporine) to create the training sample. We could identify 102 patients with atorvastatin-cyclosporine DDIs (i.e., the exposed population) and 150 patients without this DDI (i.e., the non-exposed population) (Table 1). For the optimal threshold, the neural network out-of-bag error was 17.06%, sensitivity and specificity were 90.20% and 78% respectively, and the AUC was 0.757. The processing time was short (less than 30 seconds) and could be easily performed again during the reinforcement phase.

Discussion

DDI Automatic Detection System: A New Source of Refined Data for Drug Safety Professionals

With this DDI detection system and the CDW, we can compute the overall DDI prevalence for any drug pairs, and also according to a chosen interaction severity level, or for a specific population subset. These data are useful for drug safety monitoring/research and have been already used in a study on the use of statins [START_REF] Stausberg | International prevalence of adverse drug events in hospitals: an analysis of routine data from England, Germany, and the USA[END_REF][START_REF] Morival | Prevalence and nature of statin drug-drug interactions in a university hospital by electronic health record mining[END_REF]. Moreover, currently, pharmacovigilance studies use different case report databases [START_REF] Kaguelidou | Neonatal adverse drug reactions: an analysis of reports to the French pharmacovigilance database[END_REF]. We find DDIs directly in the patient EHRs. Therefore, after DDI detection, we can link this information to other data included in the EHR (e.g., demographic data, laboratory test, etc.) to contextualize the case.

However, our DDI detection system cannot identify all DDIs. This could be due to several reasons. First, the choice of the drug knowledge database is important, and we actually observed heterogeneity between these databases that might lead to variability in DDI detection [START_REF] Morival | Prevalence and nature of statin drug-drug interactions in a university hospital by electronic health record mining[END_REF]. Moreover, with more information concerning the changes in the blood concentration (and half-life) of the involved drugs, we could compute more precise action intervals, thus improving the identification of overlapping treatment periods. However, this would require extensive manual search of literature data. Finally, our system cannot detect a DDI caused by a drug prescribed/administered outside the hospital. For instance, the regular treatment is usually stopped when a patient is hospitalized in the emergency service and is recorded in the emergency report. Accessing this information requires a specific treatment of unstructured text. Another option could be to link data on the drugs prescribed in primary care settings (i.e., the national health insurance database) to the hospital data (e.g., eHOP). Despite the linkage problems and the issues due to the national health insurance database features (data only on refundable drugs and only on the drug purchase but not the regimen), the analysis of the entire patient path could bring useful information on treatment ruptures, which could suggest DDIs.

A machine learning model for search reinforcement

The automatic way used to create the non-exposed population works and selects a population similar to the exposed group in terms of demographics and pathology. If the sample is big enough, we can ask the system to select the most similar patients.

Although the study of the temporal correlations between laboratory test changes and drug administration is relevant for ADE detection [START_REF] Newe | Towards a Computable Data Corpus of Temporal Correlations between Drug Administration and Lab Value Changes[END_REF][START_REF] Newe | Dramatyping: a generic algorithm for detecting reasonable temporal correlations between drug administration and lab value alterations[END_REF], we chose a robust prediction-oriented machine learning model that can work without requiring too many adjustments. Indeed, we expect that clinical variables in the exposed population will change in the presence of a DDI. However, we do not know whether the detection of a DDI implies automatically an ADE, and accessing the information to confirm the ADE involves a considerable work for drug safety professionals that we want to avoid. Therefore, to automate the monitoring of DDI-linked ADEs, we took the data immediately available from eHOP.

As they have very similar demographic characteristics, comparing exposed and non-exposed populations seemed to be an effective way to initialize the system. An improvement would be to take into account also the information included, for example, in ADE report databases. However, this system can be easily improved even without more data. Indeed, the model predicts candidate ADE cases that are likely to have been caused by DDIs and proposes them to drug safety professionals. If these cases are confirmed by drug safety professionals, they are included in the training sample to automatically enhance the model.

On the other hand, and like for any automatic detection model, our neural network model does not allow understanding which anomaly led to the prediction of an ADE and for this the analysis of the patient record is required. A machine learning model requires a lot of work, especially the choice of the model and the features engineering. In particular, a larger sample could ollow other resampling strategies to be used, that do not require the out of bag error, which is prone to overestimation of the true prediction error [START_REF] Janitza | On the overestimation of random forest's out-of-bag error[END_REF]. These questions need a suitable study including a better evaluation with drug safety specialists.

Conclusions

This study shows how to employ healthcare data for automated DDI monitoring and ADE prediction. It involves the complete data processing chain: data collection, processing and enrichment as well as the creation of a machine learning model. The developed statistical model is the first step for a simple and convenient use of data, and could be enriched with additional information from other databases that must be integrated (more specific drug knowledge databases, ADE report databases …).

Although no drug safety professional is required during the monitoring, their expertise is essential to properly understand the data and put them into context. Their recommendations were also important to build the monitoring system and to improve the model.

MEDINFO 2019 :Figure 1 -

 20191 Figure 1 -Overall approach of the study

Table 1 -

 1 Demographic data of the exposed and non-exposed population samples

		Exposed	Non-exposed
		population	population
		(n=102)	(n=150)
	Age (mean ± Sd)	72.1 ± 11.6	72.9 ± 10.9
	Sex (% of men)	79.8	83.5
	Cardiac pathology (%)	38.2	37.8

G. Bouzillé et al. / An Automated Detection System of Drug-Drug Interactions from Electronic Patient Records

Acknowledgements

We would like to thank the French National Research Agency (ANR), for funding this work inside the INSHARE (INtegrating and Sharing Health dAta for Research) project (grant no. ANR-15-CE19-0024).

Address for correspondence

Bouzillé Guillaume: guillaume.bouzille@univ-rennes1.fr

G. Bouzillé et al. / An Automated Detection System of Drug-Drug Interactions from Electronic Patient Records


Antoine Feuillet
email: antoine.feuillet@outlook.fr

Mickael Terrien

Nicolas Scelles

Christophe Durand

Determinants of coopetition and contingency of strategic choices: the

Keywords: coopetition, strategy, contingency, strategic group, football

come

Introduction

In his seminal article on the peculiar economics of professional sports, [START_REF] Neale | The peculiar economics of professional sports[END_REF] used the example of the New York Yankees to illustrate the peculiarity of the production in professional sports. Being the wealthiest team in North American professional baseball, the Yankees could have monopolised the league by owning all of the teams and players. Nevertheless, in order to produce a game and a championship, competition is needed. The firm as understood in economic theory is, as argued by Neale, not the team but the league as a whole. Therefore, a minimum level of cooperation is required between teams. The Yankees example provided by [START_REF] Neale | The peculiar economics of professional sports[END_REF] illustrates the concept of coopetition in professional team sports. Coopetition is also relevant in sport in general such as in individual sports [START_REF] Mignot | Strategic behavior in road cycling competitions[END_REF], inter-organizational dynamics in sport [START_REF] Vernhet | Dynamics between the French Federation of Rugby (FFR) and the National Rugby League (NRL): Competition, co-operation or co-opetition[END_REF] or the sports goods and services industry [START_REF] Rodrigues | Strategic coopetition of global brands: A game theory approach to 'Nike + iPod Sport Kit' co-branding[END_REF]. In professional football, revenue sharing is one of the regulatory rules that operationalises coopetition. For example, in European football, the different revenue sharing schemes existing in the different leagues illustrate the duality of the relationships between firms and the difficulty to determine the right balance between cooperation and competition, which make professional sports peculiar. These peculiarities may constitute a "theoretical justification and empirical evidence" for coopetition, still required in the mainstream literature according to Le Roy et al. (2018, p. 2).

The notion of coopetition has first emerged with [START_REF] Cherington | Advertising as a business force[END_REF]. After having been neglected for a long time, it has reappeared in the literature thanks to [START_REF] Brandenburger | The right game: Use game theory to shape strategy[END_REF]. The concept is commonly understood as "the dyadic and paradoxical relationship that emerges when two firms cooperate in some activities (…) and at the same time compete with each other in other activities" (Bengtsson & Kock, 2000, p. 412). Coopetition should not be confused with collusion (cooperation for mutual benefit), although both present some similarities [START_REF] Rusko | Exploring the concept of coopetition: A typology for the strategic moves of the Finnish forest industry[END_REF][START_REF] Walley | Coopetition: An introduction to the subject and an agenda for research[END_REF]. Substantial work has already developed the strategic field of coopetition [START_REF] Bengtsson | Cooperation and competition in relationships between competitors in business networks[END_REF]Brandenburger & Nalebuff, 1996;Lecoq & Yami, 2004).

By contrast, professional sport has been subject to relatively little research on the concept of coopetition, despite the latter being described as a new paradigm [START_REF] Roy | From strategizing coopetition to managing coopetition[END_REF]. Our research tackles coopetition in the context of the French men's football Ligue 1 and Ligue 2 over the 2006-2017 period. Therefore, it follows the exploratory studies by Le [START_REF] Roy | Coopétition et performances : Le cas du football professionnel français [Coopetition and performance: The case of French professional football[END_REF] and [START_REF] Robert | Coopetition between SMEs: An empirical study of French professional football[END_REF]. These authors do not find any links between coopetition and economic or sporting performance but identify different types of strategies across clubs and a link between coopetition and financial efficiency regarding player transactions.

In our paper, we develop a different approach based on the decomposition of player transactions and the use of strategic groups. We aim to investigate the determinants of coopetition and the contingency of club strategic choices. To do so, we examine player transactions with an emphasis on coopetitive strategies. Player transactions are fundamental in football economics. This is particularly the case in French professional leagues where the business models developed by most Ligue 1 and Ligue 2 clubs depend on (young) player trading. In addition, we cannot consider all clubs as part of a single strategic group due to the disparity of their resources from one division to another but also within the same division. The idea that multiple strategic groups exist within a league is essential in this paper. It corresponds to a segmentation in different groups based on economic criteria (income, payroll) between clubs made by the financial governing body itself (DNCG 1 ). Consequently, our approach allows us to verify the contingency of strategic choices, using a taxonomic approach based on the typology of strategic groups produced by the practitioners. This enables us to study two important concepts of strategic management in the context of professional football: competition and cooperation. To do so, we use the concept of coopetition to better describe some of the characteristics in football. We consider such characteristics through an extended number of variables mainly related to player transactions (decomposition of transfers and loans) and financial features (fixed assets and gross operating income). This article is structured as follows. In the next section, the literature on coopetition and strategic groups is reviewed, first generically then focusing on sport, with a specific attention on football. Second, the methodology is described with the case study and variables used. Third, results are provided through an exploratory statistical analysis.

Then, we conclude and provide some perspectives for further research.

Theoretical background

Determinants of coopetition

Why do some companies develop coopetitive strategies? [START_REF] Bengtsson | Coopetition" in business networks: To co-operate and compete simultaneously[END_REF] explain this as the result of the complementarity and the interdependence of heterogeneous resources between those companies. Nevertheless, it can also be explained by the interdependence of homogeneous resources (Dagnino & Gnyawali, 2009), such as broadcasting rights in football. In the football sector, coopetition also relates to match tickets, the quality of football fields, fan experience and corporate hospitality. This is in line with the resource dependence theory [START_REF] Pfeffer | The external control of organizations: A resource dependence perspective[END_REF] that underlines the interdependence between firms, in particular for the access to critical resources.

One of the advantages of coopetition is the mutualisation of different types of resources (Czakon, 2018), enabling organisations to save costs (Le [START_REF] Roy | Coopétition et performances : Le cas du football professionnel français [Coopetition and performance: The case of French professional football[END_REF]. The rewards coming from coopetition relate to the duality of the concept: cooperation brings an access to key complementary resources and technology; competition drives differentiation between partners [START_REF] Robert | Better, faster, stronger, the impact of market-oriented coopetition on product commercial performance[END_REF]. However, coopetition does not suit all cases. Indeed, some companies do not have the need, the ability or the resources to commit themselves to this type of relationship. This highlights the need to distinguish between types of organisations when dealing with coopetition.

Coopetition and strategic groups

The objectives targeted by organisations depend on their strategic groups understood as groups of homogeneous organisations within an industry [START_REF] Lassalle | Management des organisations sportives [Management of Sports Organisations[END_REF][START_REF] Porter | Competitive strategy: Techniques for analyzing industries[END_REF]. Using strategic groups helps to understand the competition structure and dynamics in an industry (Frynas & Mellahi, 2015;[START_REF] Mas-Ruiz | Asymmetric rivalry within and between strategic groups[END_REF]). An industry has mobility barriers likely to prevent an organisation from moving from a group to another [START_REF] Caves | From entry barriers to mobility barriers: Conjectural decisions and contrived deterrence to new competition[END_REF][START_REF] Guedri | Multimarket competition, mobility barriers, and firm performance[END_REF]. Therefore, mobility barriers are what materialises the frontiers between strategic groups. They can be defined "as structural forces impeding firms from freely changing their competitive position" (Cool & Schendel, 1988, p. 3). Firm strategies are considered as depending on their own capabilities and their environment [START_REF] Aragón-Correa | A contingent resource-based view of proactive corporate environmental strategy[END_REF][START_REF] Mintzberg | The structuring of organization: A synthesis of the research[END_REF], as highlighted in the contingency theory [START_REF] Lawrence | Differentiation and integration in complex organizations[END_REF].

Coopetition and sport

Coopetition has been researched in a few studies in the context of sport. For example, [START_REF] Lorgnier | Considering coopetition strategies in sport tourism networks: A look at the nonprofit nautical sports clubs on the northern coast of France[END_REF] analyse the complexity and diversity of coopetition between non-profit nautical sports clubs. [START_REF] Wemmer | The impact of coopetition-based open innovation on performance in nonprofit sports clubs[END_REF] extend [START_REF] Lorgnier | Considering coopetition strategies in sport tourism networks: A look at the nonprofit nautical sports clubs on the northern coast of France[END_REF] by looking at the performance of non-profit sports organisations and, in particular, the impact of coopetition-based open innovation. Both articles show the importance of coopetition strategies for non-profit sports organisations. Coopetition strategies can also be considered through sports clusters and networks. Thus, [START_REF] Gerke | Cluster concept: Lessons for the sport sector? Toward a two-step model of sport cluster development based on socioeconomic proximity[END_REF] examine sports clusters in the sailing and surfing industry, focusing on the choices of location and socio-economic proximity. In addition, Crick and Crick (2016) investigate the different forms of cooperation and competition between taekwondo clubs. [START_REF] Vernhet | Dynamics between the French Federation of Rugby (FFR) and the National Rugby League (NRL): Competition, co-operation or co-opetition[END_REF] consider coopetition in relation to the structure of professional rugby in France.

Recently, [START_REF] Barden | Horizontal competition and interorganizational exchange partner selection: An analysis of Major League Baseball player trades[END_REF] deal with coopetition while looking at player trades in Major League Baseball (MLB), whereas [START_REF] Scelles | Temporary organizational forms and coopetition in cycling: What makes a breakaway successful in the Tour de France? Team Performance Management[END_REF] explore coopetition patterns of temporary organisations in cycling through breakaways in the Tour de France.

Indices of cooperation in football

Revenue sharing may be considered as one example of the existing coopetition level between clubs. In France, TV revenue sharing is decided by member clubs, with the league proceeding to the distribution through its board of administrators (Feuillet et al., 2018). [START_REF] Lardo | Co-opetition models for governing professional football[END_REF] use coopetition models to indicate that the clubs in professional football are also part of the decision-making bodies when dealing with the election of their representatives within those power systems. In particular, the most important clubs use their power to influence to the decision-making process. This relates to negotiation and the political decisions made within the leading bodies of the league, but also the different levels of power that different strategic groups may have.

Strategic groups in football

The typology of strategic groups used for French professional football comes from practitioners (DNCG reports from 2006(DNCG reports from -2007(DNCG reports from to 2016(DNCG reports from -2017)). Similar typologies have already been used previously in the academic literature [START_REF] Barros | Efficiency measurement of the English football Premier League with a random frontier model[END_REF][START_REF] Barros | Analysing the technical efficiency of the Spanish Football League First Division with a random frontier model[END_REF]Sener & Karapolatgil, 2015;Terrien & Durand, 2017). Strategic groups from DNCG reports are based on income and payroll. The rationale behind this choice is the strong causality identified in the literature between payroll and sporting performance in football [START_REF] Hall | Testing causality between team performance and payroll: The cases of Major League Baseball and English soccer[END_REF]. Based on this, teams from different strategic groups are not supposed to be in sporting competition in the league table. Consequently, two teams from two different groups can have a mutual interest about a player. For example, when a player needs to gain experience and/or playing time, these two teams can benefit from a loan. On one side, a team reduces its payroll [START_REF] Robert | Coopetition between SMEs: An empirical study of French professional football[END_REF] and can take advantage of the development (skills and / or experience) of its player after his return. On the other side, the team that welcomes the player attract talent without paying a transfer fee. From an economic perspective, teams in a league are not exclusively in partnership with each other. Indeed, they are also in economic competition since, for example, a better sporting ranking means more money through TV rights. If we assume that a team cannot move from one group to another (in a given season), teams are in economic competition with the other teams in their group but not with those in the other groups. However, a group of teams can try to get more money from TV rights relatively to another group of teams which was the case in England regarding international TV rights distribution (Conn, 2018). In this case, teams in the same group are in economic partnership between them and in economic competition with teams in the other groups.

Eventually, there is a potential for coopetition with both cooperation and competition between teams. Nonetheless, the extent of cooperation and competition between teams vary from one case to another. Besides, there is a need to confirm empirically these assumptions developed above. To do so, the research investigates the determinants of strategic choices across the generic strategies of competition and cooperation [START_REF] Porter | Competitive strategy: Techniques for analyzing industries[END_REF], as well as coopetition. Moreover, this research also examines the contingency of strategic choices to strategic groups.

Methodology

Strategic groups

In order to fit with the specificities of the football industry, we assume that player transactions (purchases, sales and loans) between clubs depend on the strategic group of each club taking part in the deal.

In our model, differences are considered between clubs in Ligue 1 and Ligue 2, but also within a league since we use different strategic groups within a single division.

Thereby, there is no competition between clubs from different strategic groups regarding purchases and sales of players. Indeed, the different strategic groups are based on income and payroll differences between clubs as determined by the DNCG over the years. The DNCG strategic groups used to determine the decomposition of player transaction variables are provided in Annex 1 (the annex is in a separated document).

Variables

The variables used to study coopetitive strategies are primarily financial in nature [START_REF] Walley | Coopetition: An introduction to the subject and an agenda for research[END_REF], but some non-economic exchange factors are also included [START_REF] Bengtsson | Cooperation and competition in relationships between competitors in business networks[END_REF][START_REF] Kotzab | Value-adding partnerships and co-opetition models in the grocery industry[END_REF]. The variables chosen and the corresponding strategies are introduced in Table 1.

[Insert here: Table 1]

Non-player transaction variables

Our economic variables are more precise than Le [START_REF] Roy | Coopétition et performances : Le cas du football professionnel français [Coopetition and performance: The case of French professional football[END_REF], based on the idea that economic performance cannot be restrain to assets and profits. Therefore, gross operating income and fixed assets are included to better describe the economic and financial health of clubs. Operating income is a positive determinant of a firm value [START_REF] Scelles | Professional sports firm values: Bringing new determinants to the foreground? A study of European soccer, 2005-2013[END_REF]. Assets and more particularly fixed assets inform on a firm's capacity to cover its debts. Some of the variables used in our study are the same as Le [START_REF] Roy | Coopétition et performances : Le cas du football professionnel français [Coopetition and performance: The case of French professional football[END_REF] (TV rights, sponsoring and advertising revenues, payroll), while the merchandising revenue variable is omitted since it has not been provided in the club financial reports after 2006-2007. On the non-economic side, we use the presence in the board of administrators (BA). Belonging to the G142 is no longer a criterion to consider (contrary to Le [START_REF] Roy | Coopétition et performances : Le cas du football professionnel français [Coopetition and performance: The case of French professional football[END_REF] since this organisation was disbanded in 2008. Sporting performance is measured by the sports performance index (SPI3 ) and the UEFA coefficient (Le [START_REF] Roy | Coopétition et performances : Le cas du football professionnel français [Coopetition and performance: The case of French professional football[END_REF].

Player transaction variables

Player transaction variables are decomposed in an attempt to better describe their complexity in football. Indeed, we argue that they are not limited to transfers (purchases and sales) for competitive behaviour and loans (incoming and outgoing) for cooperative behaviour, as used by Le [START_REF] Roy | Coopétition et performances : Le cas du football professionnel français [Coopetition and performance: The case of French professional football[END_REF]. Some variables present dual characteristics of cooperation and competition. For example, purchases intra (i.e. player transactions between clubs within the same strategic group) are sporting competition (Le [START_REF] Roy | Coopétition et performances : Le cas du football professionnel français [Coopetition and performance: The case of French professional football[END_REF]. At the same time, they are also economic cooperation as they soften the budget constraint for the seller. A DNCG (2015) report stipulates that for most French clubs, their business model depends on player trading, similar to a recent report by CIES 4 (2018). This is supported by [START_REF] Paché | Les stratégies de création et d'appropriation de la valeur dans un contexte d'incertitude majeure: Le cas du football professionnel [Strategies for creating and appropriating value in a context of major uncertainty: The case of professional football[END_REF] who argue (based on the 2009 DNCG report) that without such trading strategies, French clubs would suffer from larger deficits, a reason being that they generate less revenues from sponsoring and matchday compared to the Big 4 leagues (England, Germany, Italy and Spain). Intra group sales could be a mean to compete economically with its rivals but also sporting cooperation to retain talent in the domestic league, which can enhance its attractiveness [START_REF] Madden | Game theoretic analysis of basic team sports leagues[END_REF][START_REF] Terrien | Organisational efficiency of national football leagues in Europe[END_REF][START_REF] Terrien | French 75% tax rate: An opportunity to optimize the attractiveness of the French soccer league[END_REF], especially if the player is a superstar [START_REF] Lucifora | Superstar effects in sport: Evidence from Italian soccer[END_REF]. Purchases extra (i.e. player transactions between clubs from different strategic groups) are sporting competition (they can be viewed as a way to consolidate mobility barriers) and at the same time economic cooperation (they soften the budget constraint for the seller). Extra group sales are economic competition with other sellers and sporting cooperation (to retain talent in the domestic league). Loans (incoming and outgoing) are sporting cooperation.

On the international side, purchases in foreign leagues are a way to develop league attractiveness. They represent sporting competition since they enable a club to strengthen its squad compared to its domestic competitors. For the sales, we analyse them as economic competition since international markets increase the number of potential buyers 4 International Centre for Sports Studies.

that will pay more than a domestic club. Contrary to Le [START_REF] Roy | Coopétition et performances : Le cas du football professionnel français [Coopetition and performance: The case of French professional football[END_REF], we consider international loans (incoming and outgoing) as sporting competition (Olson & Schwarb, 2000). Indeed, loaning with foreign teams can mean a refusal to interact with clubs within the domestic league. This can be the consequence of buying a domestic stock of talent (players) then loaning the surplus to foreign leagues to avoid the domestic sporting competition those players would have represented. A further distinction is made at the international level between the Big 4 and others.

Methods

We proceeded with an exploratory statistical analysis. Second, we identified a taxonomy of behaviours among clubs and their contingency to strategic groups with a K-means classification aiming to form clusters. This is a classification method that generates groups following an iterative process. The advantage of this method is the possibility to choose the number of categories to display. The first axis (F1) of the PCA biplot represents most of the total variance (24.036 % out of 31.615 %). On this axis, observations on the right are characterised by high levels of sporting performance (SPI), resources (assets, TV rights, sponsoring and advertising), payroll, all sorts of purchases (intra, extra, intl), international and intra-group sales, outgoing loans and incoming international loans. Observations on the left are characterised by high levels of economic performance (profits, gross operating income); intra and extra-group incoming loans, free agent arrivals. The second axis (F2) provides a lower amount of the total variance (7.58% out of 31.615%). This explains why most of the variables and observations are closer to F1 than F2. Observations at the top are characterised by high levels of purchases in the Big 4 leagues or at the international level and sponsoring and advertising revenue. Observations at the bottom are characterised by high levels of incoming loans from international and all types of sales.

Results

Pearson

The projections for the PCA biplot variables show the positive correlation between SPI and some competitive variables (e.g. international purchases, payroll, sponsoring & advertising). These variables are negatively correlated with economic performance (especially profits and gross operating income), which may indicate the cost of performance, consistent with the idea of win maximisation [START_REF] Sloane | The economics of professional football: The football club as a utility maximiser[END_REF]. These economic variables are positively correlated with extra incoming loans (especially from higher division), extra group sales to higher division and free agent arrivals. This can be associated with clubs that do not invest in the transfer market and depend on other clubs (using loans) and the market for free agents. This could indicate a 'free rider' strategy. Its aim is to take advantage of the broadcasting rights without investing in players, based on the awareness that the risk of being relegated is high even when investing in players.

Although this player loan strategy is negatively correlated with sporting performance, it enables higher financial gains (profits). In most cases, this strategy is used by clubs that do not have the financial ability to recruit on the transfer market. A mix of variables are also correlated, some being competitive (international sales), some other more cooperative (presence in the BA) and some presenting both characteristics (intra group sales, extra group purchases).

The clubs are projected in Figure 1 along axes 1 and 2 in relation to the variables.

Olympique Lyonnais (OL) and Paris Saint-Germain (PSG) seem to have adopted competitive strategies based on payroll, sponsorship and advertising, and purchases intra across different seasons. However, the strategies are not the same from one year to another. Indeed, the position of the clubs varies across seasons. For example, PSG has encountered a massive strategic shift since the State of Qatar bought the club in 2011-2012 according to variables. AS Monaco (ASM) also seems to have completely switched its strategy from a more competitive to a more 'coopetitive' one since 2013-2014.

Olympique de Marseille (OM) also presents many different strategies. Thus, from 2009-2010 to 2011-2012, the club seems to follow a competitive strategy resulting in a championship title in 2009-2010, while in 2015-2016 and 2016-2017 it seems to have adopted a more coopetitive behaviour.

The K-means classification

We chose the number of classes (four) that minimise intra-group variance and maximise inter-group variance (Annexes 7 and 8). Each category or cluster is linked to a centre (centroid) which corresponds to an observation, and each observation will be associated to the nearest centre thereby forming a cluster. Table 2 shows the centroid classification and allows us to interpret the results of the K-means grouping method.

[Insert here: Table 2] The first category (C1) includes 250 observations. These are clubs with few resources that have therefore fewer opportunities in terms of competitive strategies (low levels of sponsorship and advertising, TV rights, payroll, purchases, sales, outgoing loans, assets, SPI). They cooperate with other clubs through a high level of extra group incoming loans. Since these clubs have few resources and a need for players, they use the loan system (and free agents) to strengthen their recruitment at a lower cost. This leads to slightly negative profits and gross operating income. These financial results seem to provide evidence of a 'safety' strategy. This refers to the sporting vulnerability of these teams that prevents them from investing additional resources by fear of bankruptcy resulting from the inability to recover considerable debts with even fewer resources in case of relegation (particularly TV rights). This risk has become real for certain clubs belonging to this category (e.g. Bastia, Arles Avignon, Sedan, Evian or Vannes). Despite their (assumed) cautious behaviour, these clubs could have suffered from demand shocks in relation to their attendance [START_REF] Scelles | Insolvency in French soccer: The case of payment failure[END_REF].

The second class (C2) includes 102 observations. It also presents some features of a revenue model depending on the transfer system that keeps the French clubs alive, given the uncertain nature of their own resources. C2 shows quite important investments in other international purchases (€2.473 million), compared to very low purchases from the Big 4 (€0.378 million). It also unveils relatively important sales to the Big 4 (€3.443 million) compared to sales to other international leagues (€1.998 million). This is consistent with the idea that the French league serves as intermediary 'springboard' for players before joining the Big 4 leagues, especially the English Premier League [START_REF] Feuillet | A winner's curse in the bidding process for broadcasting rights in football? The cases of the French and UK markets[END_REF][START_REF] Poli | Migrations de footballeurs et mondialisation : Du système-monde à une approche relationnelle [Footballer migration and globalization: From the world system to a relational approach[END_REF].

The third class (C3) includes 54 observations that have strong competition characteristics with high intra group purchases. These clubs also use loans from international leagues to get players. They send 3.59 players on outgoing loans, mainly in lower divisions with 2.40 players sent in Ligue 2, semi-professional or amateur divisions.

Their economic and financial results are negative, especially profits (€ -4.349 millions).

This could be the consequence of their attempt to acquire players that come from their own strategic groups (intra group purchases). It is worth noting that only two clubs from the strategic group 3 in Ligue 1 (Monaco in 2006-2007and Auxerre in 2007[START_REF] Barros | Efficiency measurement of the English football Premier League with a random frontier model[END_REF] The fourth class (C4) includes 22 observations. C4 has massive international purchases (€46.57 million) and also sent 2.54 players on outgoing loans to international clubs. It may explain their strongly negative gross operating income ( €-31.202 million), although their negative profits are not strongly superior in absolute value to C3 (€-4.901 million).

C4 is consistent with a win maximiser behaviour. Only one club from the strategic group 2 in Ligue 1 is in C4, the other clubs being all from the strategic group 1 in Ligue 1. This observation is Nice in 2016-2017. It can be explained by a specific context. Indeed, Nice was purchased in the 2016 summer by a duo of Sino-American investors. During that year, they tripled their fixed assets from €4.2 million in 2015-2016 to €13 million in 2016-2017. Then, the club qualified for the Europa League in 2016-2017, performing very well in the Ligue 1. This resulted in additional TV rights revenue. For these reasons, the club spent more money in the transfer market but also in payroll, jumping from €31.6 million to €49.9 million from 2015-2016 to 2016-2017.

The four classes resulting from the K-means classification are analysed further to verify their contingency with the strategic groups formulated by the DNCG (Table 3).

[Insert here: Table 3] Results show an important contingency: C1 is composed by 78% of observations from Ligue 2; C2 is composed by 89% of observations of Ligue 1; C3 is composed mostly by observations from Ligue 1, in particular from its second strategic group but also for It appears that belonging to the highest strategic group (mostly group 1 in Ligue 1) permits different strategic choices. Indeed, we have observations for the clubs from the first strategic group, both in Ligue 1 or Ligue 2, in all four classes. This may indicate the range of choices available for this type of clubs. By contrast, a lower strategic group means a more limited number of classes with at least one observation. For the clubs corresponding to the lower strategic groups, this may illustrate the idea of partially constrained strategy, "between determinism and free choice" (Mintzberg & Waters, 1985, p. 268).

Discussion and conclusion

This research aimed to measure the determinants of strategic choices in French football, with an emphasis on coopetitive strategies in addition to cooperation and competition. Using strategic groups, this research goes further compared to previous research on coopetition in general since, to our knowledge, it is the first study to combine the study of coopetition and strategic groups. The taxonomy identified with the K-means classification demonstrates a variety of strategies related to cooperative, 'coopetitive' and competitive behaviours. The K-means classification indicates that clubs with low revenues tend to use incoming loans extra (mostly from higher division) as a recruitment strategy. This enables to constitute a team with low or zero transfer investment and, therefore, to limit the extent of the financial losses. Nevertheless, it does present a risk since it is highly negatively correlated with sporting performance (SPI). These teams are probably not able to choose another strategy without accepting a higher financial risk given their limited resources, consistent with the idea of the contingency of strategic choices.

Our research contributes to an understanding of the different strategies and behaviours adopted by the different clubs, instead of assuming a single strategy and behaviour for all of them. For the sake of modelling, the leagues are often considered as entities with clubs sharing similar behaviours. However, [START_REF] Zimbalist | Sport as business[END_REF] identified that clubs follow different objectives. It is in line with a direction first explored by [START_REF] Rascher | A model of professional sports leagues[END_REF] who modelled multiple objectives within a league. Since then, few researches have attempted to further develop this direction, some exceptions being Terrien and Durand (2017) who also modelled the multiple objectives. Terrien et al. (2017) also identified them empirically and found that objectives can be different from one club to another but also for a given club from one season to another. Our taxonomy also brings evidence of different club strategies and behaviours, while providing a more nuanced understanding through the use of the coopetition concept that enables to identify the relationships between clubs and discuss the contingency of strategic choices.

In a context where player transactions are more and more scrutinised by the regulatory authorities (UEFA, FIFA), this research may inform the potential consequences that some teams could face. FIFA has recently released a project that would limit the number of outgoing loans if they are conducted for 'commercial exploitation'5 .

Such reforms could put more pressure on teams that use or depend on loans (e.g. Chelsea FC has 40 players on loans in 2018-2019). Our research may help identify which clubs would be the most likely to suffer from that regulation.

Our research can also inform the recurrent debate about the creation of a European Superleague with the richest clubs in football [START_REF] Scelles | Star quality and competitive balance? Television audience demand for English Premier League football reconsidered[END_REF]Scelles, Durand et al., 2016).

The potential relationships between the richest clubs taking part in the Superleague and the other clubs in their respective domestic leagues may be enlightened by our results on the different types of strategy, especially regarding player transactions. On a broader perspective, the approach combining coopetition and strategic groups could be extended to other sports and other sectors to analyse strategic choices. CIES (2018). Financial analysis of the transfer market in the big-5 leagues (2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017)(2018)

Database

  Data analysis is based on longitudinal data for each club having belonged at least one season to Ligue 1 or Ligue 2 over the 2006-2017 period (53 different clubs). In theory, the number of observations should be 440 corresponding to 11 seasons multiplied by 40 clubs (20 in Ligue 1 and 20 in Ligue 2). However, our dataset does not contain the data for 12 club seasons that might have biased the results. Indeed, these observations are either incomplete due to bankruptcy or hardly comparable due to changes in the presentation of their financial rubrics. Hence, the number of observations used is 428. The data collection has been carried out based on the DNCG reports (LFP, n.d.a), as well as the websites of Transfermarkt (n.d.), Soccer Association (n.d.), (LFP, n.d.b) and UEFA (n.d.).

  First, a Pearson's principal component analysis (PCA) was conducted to summarise the structure of the observations in relation to the variables and help conduct the taxonomic analysis. The PCA summarises the information from variables to represent it through synthetic factors. This allows us to understand the different strategies or behaviours of each team and their evolution over time. Factors 1 and 2 (F1 and F2) represent the two synthetic dimensions of the whole data (all variables) that best represent the data. The length of the variables' vectors represents how well they are represented by the two factors of the PCA (F1 and F2).

  's principal component analysis (PCA) Correlations between variables are presented in Annex 2 (correlation matrix between non-player transaction variables) and Annex 3 (correlation between player transaction variables and non-transaction variables). The number of factors to study is decided based on the tests used. The adequacy of the sample is assessed by the KMO (Kaiser-Meyer-Ohlin) index (Annex 4) that provides an acceptable result (0.655). The Bartlett's test of sphericity (Annex 5) is used (excluding the player transaction variables decomposed by division extra groups and by types of international player transactions due to multicollinearity) to determine the quality of the PCA. This test provides a high level of significance (p-value<0.001). The criterion of explained variance (Annex 6) brings out two explanatory factors which, when cumulated, explain 31.615 % of the total variance. In the figure resulting from the PCA (Figure 1), sharp angles indicate a strong positive correlation between indicators, right angles no link and obtuse angles a strong negative correlation. [Insert here: Figure 1]

  appear in C3. This may be explained by these two clubs having historically qualified in European competitions before the season under consideration. Therefore, Monaco had three participations in a row in the Champions League before the 2006-2007 season, while Auxerre had five participations in a row in European competitions before the 2007-2008 season. During these seasons, both clubs have a positive transfer market balance but still manage to invest to purchase players.

  41% by observations from its first strategic group; and C4 is almost exclusively composed by observations from the first Ligue 1 strategic group. The only two exceptions (OGC Nice 2016-2017 and AS Monaco 2012-2013) can be explained by the arrival of new owners, as developed previously for OGC Nice. For AS Monaco, Dmitry Rybolovlev bought the club in the middle of the 2011-2012 season when it was at the very last position in the Ligue 2 and began his investments at the same time, i.e. during the 2011-2012 winter transfer window. His first full season as owner was in 2012-2013, corresponding to the only presence of an observation from Ligue 2 for C4.

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Jurisport

, 183, 39-44.

Table 2 K

 2 -Means centroid The presence of a representative is simply counted as 1 and the absence of representative as 0.

	Class	Unit	C1	C2	C3	C4
	Purchases intra group	Thousands €	19	389.95	3064.81	2590.90
	Purchases extra group	Thousands €	271.15	2629.41	7822.12	14931.81
	Purchases extra group same division	Thousands €	89.52	1454.41	6212.03	13250
	Purchases extra group lower division	Thousands €	139.62	1170.09	1591.57	1400
	Purchases extra group higher division	Thousands €	42.00	4.90	18.51	281.81
	Purchases international	Thousands €	189.03	2852.34	6267.59	46570.90
	Purchases big 4 leagues	Thousands €	11.80	378.43	2220.37	30606.81
	Purchases international other than big 4 leagues	Thousands €	177.23	2473.91	4047.22	15964.09
	Sales intra group	Thousands €	24.6	498.52	1244.90	3211.36
	Sales extra group	Thousands €	1150.12	4073.52	5420.37	2209.09
	Sales.extra group same division	Thousands €	282.52	3805.88	4925	2050
	Sales extra group lower division	Thousands €	23.4	98.03	115.74	90.90
	Sales group extra higher division	Thousands €	844.2	169.60	379.62	68.18
	Sales international	Thousands €	562.06	5442.51	13908.05	28506.59
	Sales big 4 leagues	Thousands €	256.72	3443.77	10762.22	25690
	Sales international other than big 4 leagues	Thousands €	300.94	1998.74	3145.83	2816.59
	Incoming loans intra group	Player(s)	0.04	0.06	0.05	8.0743E-17
	Incoming loans extra group	Player(s)	1.74	0.62	0.31	0.18
	Incoming loans extra group same division	Player(s)	0.42	0.49	0.16	0.13
	Incoming loans extra group lower division	Player(s)	0.06	0.05	0.05	0.04
	Incoming loans extra group higher division	Player(s)	1.28	0.08	0.09	-1.413E-16
	Incoming loans international	Player(s)	0.50	0.71	1.07	0.68
	Incoming loans big 4 leagues	Player(s)	0.15	0.26	0.62	0.27
	Incoming loans other than big 4 leagues	Player(s)	0.35	0.46	0.46	0.40
	Free agent arrivals	Player(s)	7.64	2.91	1.87	0.72
	Outgoing loans intra group	Player(s)	0.03	0.07	0.01	0.09
	Outgoing loans extra group	Player(s)	1.26	2.45	3.59	2.68
	Outgoing loans extra group same division	Player(s)	0.18	0.28	1.14	1.45
	Outgoing loans extra group lower division	Player(s)	1.00	2.16	2.40	1.18

Table 3

 3 Contingency table of Ligue 1 (L1) and Ligue 2 (L2) strategic groups from DNCG report (G1to G5) compared to

		Ligue 1				Ligue 2				Observations
		G1 G2	G3	G4	G5	G1	G2	G3 G4	G5
	C1	1	4	23	22	4	39	51	69	30	7	250
	C2	2	23	52	11	4	5	3	1	1	0	102
	C3	22	26	2	0	0	3	0	1	0	0	54
	C4	20	1	0	0	0	1	0	0	0	0	22
	Total 44	54	77	33	8	48	54	70	31	7	428

National Direction for Management Control as translated from French by Dermit-Richard et al. (2019).

Former organisation reuniting 14 (then 18) European football clubs.

We use a simple average of points per game (total number of points/total number of games).

"Development of the regulation of loans of players for the purpose of youth development as opposed to commercial exploitation. The number of loans per season and between each club shall be limited and bridge transfers and sub-loans shall be prohibited"(FIFA, 2018).

Tables


Victoriya Kashtanova
email: victoriya.kashtanova@inria.fr

Ibrahim Ayed

Andony Arrieula

Mark Potse

Patrick Gallinari

Maxime Sermesant
email: maxime.sermesant@inria.fr

Deep Learning for Model Correction in Cardiac Electrophysiological Imaging

Keywords: Physics-based learning, Electrophysiology, Deep learning, Simulations

Imaging the electrical activity of the heart can be achieved with invasive catheterisation. However, the resulting data are sparse and noisy. Mathematical modelling of cardiac electrophysiology can help the analysis but solving the associated mathematical systems can become unfeasible. It is often computationally demanding, for instance when solving for different patient conditions. We present a new framework to model the dynamics of cardiac electrophysiology at lower cost. It is based on the integration of a low-fidelity physical model and a learning component implemented here via neural networks. The latter acts as a complement to the physical part, and handles all quantities and dynamics that the simplified physical model neglects. We demonstrate that this framework allows us to reproduce the complex dynamics of the transmembrane potential and to correctly identify the relevant physical parameters, even when only partial measurements are available. This combined model-based and data-driven approach could improve cardiac electrophysiological imaging and provide predictive tools.

Introduction

Mathematical modelling of the heart has been an active research area for decades, and it is now more and more coupled with artificial intelligence approaches [START_REF] Mansi | Artificial Intelligence for Computational Modeling of the Heart[END_REF][START_REF] Giffard-Roisin | Non-invasive personalization of a cardiac electrophysiology model from body surface potential mapping[END_REF][START_REF] Karoui | Cardiac activation maps reconstruction: a comparative study between data-driven and physics-based methods[END_REF][START_REF] Trayanova | Machine learning in arrhythmia and electrophysiology[END_REF]. Among the multiphysics phenomena involved in cardiac function, cardiac electrophysiology (EP) models can accurately reproduce electrical behaviour of cardiac cells.

In order to describe the dynamics of transmembrane voltage, current, and different ionic concentrations in the cardiac cell, biophysically detailed models such as the Ten Tusscher-Panfilov model [START_REF] Ten Tusscher | A model for human ventricular tissue[END_REF][START_REF] Ten Tusscher | Alternans and spiral breakup in a human ventricular tissue model[END_REF] have been proposed. However, these models are complex and computationally expensive, and have many hidden variables which are impossible to measure, making model parameters difficult to personalise. Another type of model, phenomenological models, are simplified models derived from biophysical models. Examples include the FitzHugh-Nagumo, Aliev-Panfilov, and Mitchell-Schaeffer models (FitzHugh, 1961;[START_REF] Nagumo | An active pulse transmission line simulating nerve axon[END_REF][START_REF] Aliev | A simple two-variable model of cardiac excitation[END_REF][START_REF] Nash | Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias[END_REF][START_REF] Colleen | A two-current model for the dynamics of cardiac membrane[END_REF]. These models employ fewer variables, they have fewer parameters and are therefore especially useful for rapid computational modelling of wave propagation at the organ level. However, they are less realistic and therefore need a complementary mechanism to fit them to the measured data. Machine learning and in particular deep learning (DL) approaches could help providing such a correction mechanism. The combination of rapid phenomenological models and machine learning components could then allow the development of rapid and accurate models of transmembrane dynamics.

In the last few years, DL neural networks have been increasingly used in order to learn dynamical models from data motivating a large number of publications. For example, [START_REF] Long | PDE-net: Learning PDEs from data[END_REF][START_REF] Long | PDE-net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network[END_REF] endowed neural layers with additional structure, useful for learning PDEs. Chen et al. ( 2018) used the adjoint method to learn differential equations parametrised with neural networks. Ayed et al. (2019b) propose a framework for learning models using a purely data driven approach in partially observable settings. [START_REF] Willard | Integrating physics-based modeling with machine learning: A survey[END_REF] propose a broad survey of ML for physics-based modeling.

In spite of achieving good progress in cardiac electrophysiology simulations (Ayed et al., 2019a;[START_REF] Kashtanova | EP-Net 2.0: Out-of-domain generalisation for deep learning models of cardiac electrophysiology[END_REF] data-driven models alone could not reproduce the complex unseen dynamics like the repolarisation [START_REF] Kashtanova | EP-Net 2.0: Out-of-domain generalisation for deep learning models of cardiac electrophysiology[END_REF], and the maximum forecasting horizon is still limited. For this reason, researchers have begun to use coupled physico-statistical approaches for cardiac electrophysiology simulations with a high precision and at low cost. For example, [START_REF] Court | Design of the monodomain model by artificial neural networks[END_REF] designed a neural network that approximates the FitzHugh- Nagumo model, Sahli Costabal et al. (2020) used a physics-informed neural network for cardiac activation mapping accounting for underlying wave propagation dynamics, [START_REF] Fresca | POD-enhanced deep learning-based reduced order models for the real-time simulation of cardiac electrophysiology in the left atrium[END_REF] proposed an approach to create a nonlinear reduced order model with the help of deep learning algorithms (DL-ROM) for cardiac electrophysiology simulations, and Herrero Martin et al. ( 2022) present a physics-informed neural network for accurate action potential simulation and EP model parameter estimation. However, the majority of these coupled approaches bases on high-fidelity physical models and fits them to the data. This could be computationally expensive and cannot manage large discrepancies between simulated and real data.

To alleviate this limitation, we propose a framework to Augment incomplete PHYsical models with a deep learning component for ideNtifying comlex cardiac ElectroPhysiology dynamics (APHYN-EP) from data, based on a fast low-fidelity (or incomplete) physical model. This framework has two components which decompose the dynamics into a physical and a data-driven term. The data-driven deep learning component is designed so as to capture only the information that cannot be modeled by the incomplete physical model. The proposed model closely follows the approach of [START_REF] Yuan Yin | Augmenting physical models with deep networks for complex dynamics forecasting[END_REF]. But in contrast to this work, that considers fully-observable dynamics and simple test use cases, cardiac electrophysiology dynamics have a high complexity and represent simultaneously multiple underlying processes. Furthermore, most cardiac electrophysiology models lack measurements for some variables, which makes them partially-observable and requires inferring the dynamics from incomplete observations only.

Learning Framework

To learn cardiac electrophysiology dynamics (X t ) we solve an optimization problem via a physics-based data-driven framework APHYN-EP. This framework combines a physical model (F p ) representing an incomplete description of the underlying phenomenon and a neural network (F a ) which will complement the physical model by capturing the information that cannot be modeled by the physics component:

min Fp∈Fp,Fa∈Fa F a subject to ∀X ∈ D, ∀t, dX t dt = F (X t ) = (F p + F a )(X t ). (1)
Our incomplete physical model is the two-variable (v, h) model by [START_REF] Colleen | A two-current model for the dynamics of cardiac membrane[END_REF] for cardiac electrophysiology simulation (2). The variable v represents normalised (v ∈ [0, 1]) dimensionless transmembrane potential while the "gating" variable h controls the repolarisation (return to the initial state):

∂ t v = div (σI∇v) + hv 2 (1 -v) τ in - v τ out + J stim ∂ t h = 1-h τopen if v < v gate -h τ close if v > v gate (2)
where J stim is a transmembrane potential activation function, which is equal to 1 in the stimulated area during stimulation time (t stim ). This physical model has been successfully used in patient-specific modelling [START_REF] Relan | Coupled personalization of cardiac electrophysiology models for prediction of ischaemic ventricular tachycardia[END_REF], it covers general electrophysiology dynamics and, in contrast to more detailed electrophysiology models, it is flexible in terms of spatial and temporal steps. Assuming that we can obtain the coordinates of an applied electrical stimulation from the data and using v(t = 0) ≡ 0 and h(t = 0) ≡ 1 we can calculate an approximation of h for any time point t with the help of a simple integration scheme.

For the data-driven component we use a Residual Network (ResNet) [START_REF] He | Deep residual learning for image recognition[END_REF], because it can accurately reproduce transmembrane potential dynamics (Ayed et al., 2019a;[START_REF] Kashtanova | EP-Net 2.0: Out-of-domain generalisation for deep learning models of cardiac electrophysiology[END_REF]. The choice of data-driven component is not limited by the ResNet architecture. We performed preliminary experiments with other types of convolutional networks (out of scope for this paper), but overall the ResNet model was the more stable along the different simulations.

Instead of solving the ODE in Eq. ( 1), we use an integral trajectory-based approach which is robust and less sensitive to the time resolution [START_REF] Yuan Yin | Augmenting physical models with deep networks for complex dynamics forecasting[END_REF]. We compute the next state X(i) h∆t from the initial state X (i) 0 as an approximate solution of the integral Algorithm 1: APHYN-EP Initialization: θ 0 , λ 0 ≥ 0, τ > 0;

X (i) 0 +h∆t X (i) 0 (F θp p + F θa a )(X s
for epoch = 1 : N epochs do for batch in 1 : B do L traj (θ j ) = N i=1 T /∆t h=1 ||X (i) h∆t - X(i) h∆t || θ j+1 = θ j -∇ [λ j L traj (θ j ) + F a ] end λ j+1 = λ j + τ L traj (θ j+1 ) end
Additionally, in order to train simultaneously the physical and the data-driven components of APHYN-EP via automatic differentiation tools (provided by the Pytorch library [START_REF] Paszke | Pytorch: An imperative style, high-performance deep learning library[END_REF]) we implemented the Laplace operator in (2) with a simple finitedifference scheme. To avoid difficulties with high time resolution required in this numerical scheme we used two different time steps in the integration schemes for the physical component computations and for the computations of the final forecast given by the framework.

Experimental settings

Data collection

To evaluate our method, we used a dataset of transmembrane potential activation simulated with a monodomain reaction-diffusion equation and the Ten Tusscher -Noble -Noble -Panfilov ionic model [START_REF] Ten Tusscher | A model for human ventricular tissue[END_REF], which represents 12 different transmembrane ionic currents. The simulations were performed with a recent version of the propag-5 software [START_REF] Krause | Hybrid parallelization of a large-scale heart model[END_REF][START_REF] Potse | Scalable and accurate ECG simulation for reaction-diffusion models of the human heart[END_REF] with a spatial step of 0.2 mm and a time step of 1 ms, the same as used by Ten Tusscher et al. (2004). The computational domain represents a slab of 2D cardiac tissue of size 24 × 24 elements. For one data sample, one stimulation was applied for 1 ms in the selected area for transmembrane potential activation. Since the grid was symmetric under 90-degree rotations, the stimulations were done only on each grid point of the first quarter of the cardiac slab. Then, we applied data-augmentation techniques to translate the simulations on the three remaining quarters. Overall we had a database of 500 training samples and 100 validation samples. Each simulation represented 500 ms of propagation, which represents 40 seconds of computation on a 12-core Intel Xeon W-2133 CPU.

The data simulated via the Ten Tusscher model are considered here as the ground truth. The objective is then to learn the complex dynamics generated with this model with the APHYN-EP model combining a simplified physics description with a deep learning component. This will result in a low computational cost surrogate model of the computationally intensive Ten Tusscher model.

Training settings

The physical model (F p ) of Eq. 2 was implemented with a standard finite-difference scheme for the Laplace operator with a spatial resolution of 1 mm 2 pixels and inner time resolution of 0.01 ms. We estimated only σ and τ in as unknown parameters in (2), because they represent the major part of early dynamics (velocity and upstroke) and therefore the main difference between the Mitchell-Schaeffer and the Ten Tusscher -Noble -Noble -Panfilov models in our simulations. The initial Mitchell-Schaeffer model parameters are taken as in the original paper [START_REF] Colleen | A two-current model for the dynamics of cardiac membrane[END_REF]: τ out = 6, τ open = 120, τ close = 150, v gate = 0.13 and t stim = 1. The data-driven model (F a ) is ResNet with 8 filters at the initial stage, one down-sampling initial layers and three intermediary blocks, and starts with a re-weighted orthogonal initialisation for its parameters.

We used a time resolution of 0.1 ms to compute the forecast given by APHYN-EP. Training was performed using a horizon of 6 time frames. Training was performed by learning a 4 frames horizon for the first 10 epochs, and then 6 frames. This leads to more stable results. We trained APHYN-EP until full model convergence (about 100 epochs) using an ADAM optimiser [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF] with initial learning rate of 10 -3 . The algorithm hyper-parameters λ 0 and τ were set to 1 and 10 3 respectively.

The code and data used for APHYN-EP training are freely available on the official github project page.

Results

We present here qualitative results on the forecast over 9 ms after assimilating only one first frame of dynamics (see Fig. 2). These first 9 ms represent an important part of the cardiac dynamics from early depolarisation to full depolarisation. We can observe a very good agreement between the ground truth and the forecast transmembrane potentials generated by APHYN-EP in Fig. 3. The correction term brought by F a is clearly visible.

Table 1 shows the mean squared error (MSE) results for our framework for different forecasting horizons on validation data samples. To calculate this error, for each data sample, we fed the model with only one initial test measurement, then let it predict several steps forward without any additional information. We also added for comparison two baseline models corresponding to the two components of our model, each used alone: only the "incomplete" physical model and only the data-driven model (ResNet) trained on the same dataset as APHYN-EP, described in 3.1. As we can see, APHYN-EP captured the observed dynamics with good precision for different time horizons, even if for training we used only the first 6 ms. In the same time, the pure physical and the data-driven models when used alone struggle to learn the proper dynamics. Figures 3 and4 represent the performance of different components of APHYN-EP and their contribution to the final result. We can see which part of the generated transmembrane potential was created by the physical component of the framework (see Fig. 4 (second row)). The data-driven component was used only to correct the difference between the groundtruth dynamics and the physical part (see Fig. 4 (third and fourth rows)).

Conclusion

We have presented the APHYN-EP framework for modeling complex cardiac electrophysiology dynamics via a surrogate model combining simplified physics and deep neural networks. We demonstrated that this framework is able to reproduce with good precision the dynamics simulated by the Ten Tusscher -Noble -Noble -Panfilov ionic model, even using a simplified electrophysiology model as a physical component of the framework.

Such framework opens up possibilities in order to introduce prior knowledge in deep learning approaches through explicit equations and to correct model errors from data.

Future work will evaluate this framework on more challenging settings: presence of scars, multiple onsets and various conduction velocities in the cardiac tissue slab. We also left for the future the adaptation of the APHYN-EP application on the surface of real 3D heart. the reference number ANR-19-P3IA-0002, through the "Research and Teaching chairs in artificial intelligence (AI Chairs)" funding for DL4Clim project, and through ANR grant reference ANR-10-IAHU04-LIRYC. The research leading to these results has also received European funding from the ERC starting grant ECSTATIC (715093). The authors are grateful to the OPAL infrastructure from Université Côte d'Azur and to HPC resources of CINES under GENCI allocation 2020-A0090307379 for providing resources and support.

+Figure 1 :

 1 Figure 1: General APHYN-EP framework scheme. During the training phase twocomponent framework learn the parameters for the physical (F p ) and the datadriven (F a ) components from data. Then via an ODE solver the framework can forecast further the learned dynamics.

  ) dX s obtained by a differentiable ODE solver[START_REF] Ricky | Neural ordinary differential equations[END_REF][START_REF] Ricky | Learning neural event functions for ordinary differential equations[END_REF]. The APHYN-EP training uses an algorithm adapted from[START_REF] Yuan Yin | Augmenting physical models with deep networks for complex dynamics forecasting[END_REF].

Figure 2 :Figure 3 :

 23 Figure 2: APHYN-EP predicted dynamics for the transmembrane potential diffusion. The figure shows a 9 ms of forecast).

Figure 4 :

 4 Figure 4: Predicted dynamics for the transmembrane potential diffusion by APHYN-EP physical component (second row), error with ground-truth diffusion for this physical component of APHYN-EP (third row), and trained APHYN-EP data-driven component contribution (bottom row).

Table 1 :

 1 Mean-squared error (MSE) of normalised transmembrane potential forecasting per time-step for different forecasting horizons.

		MSE (6 ms) MSE (12 ms) MSE (24 ms) MSE (50 ms)
	APHYN-EP	0.0057	0.0037	0.0029	0.002
	Physical model only 0.0093	0.0111	0.0096	0.0085
	Resnet model only	0.0195	0.0220	0.1593	9.9212

Acknowledgments

This work has been supported by the French government, through the 3IA Côte d'Azur Investments in the Future project managed by the National Research Agency (ANR) with