Nabil Mr

Pierre Badel

Pr

Flavien Balbo

Jean-François Bassereau

PR Batton-Hubert Mireille

Michel Beigbeder

Informatique Ma

Fayol

Appliquees M Mathematiques

Informatique O Batton-Hubert

Professeur Boissier

J C Formes

Professeur Genie Pinoli

N Industriel

Maitre Absi

Jules Le

Jules Le Lay
email: jules.le-lay@emse.fr

Julia Neveu

Benjamin Dalmas
email: benjamin.dalmas@emse.fr

Vincent Augusto
email: augusto@emse.fr

Edgar Alfonso-Lizarazo

Bienvenu Bongue

Malek Masmoudi

Xiaolan Xie

Baptiste Gramont

Thomas Célarier

ABSI

Keywords: Process Mining, Automation, Process Discovery

Lay

28/03/2022

Je tiens à remercier avant toute choses Vincent et Xiaolan pour les conseils avisés prodigués tout au long de cette thèse. Vous m'avez offert la chance de remettre les sciences au coeur de mon quotidien. Je tiens à remercier particulièrement Vincent pour m'avoir guidé à travers toutes les étapes de cette aventure. Merci à Edgar et Malek pour leur soutien et leurs conseils, qui se sont avérés précieux pour surmonter les difficultés techniques que nous avons rencontrés. Merci Bienvenu, Thomas et Baptiste d'avoir enrichi ce travail en y apportant votre expertise. Les différentes rencontres et collaborations ayant eu lieu durant ces 3 années ont grandement enrichies le travail présenté ici et m'ont fait grandir.

Merci à tous mes collègues du Centre Ingénierie Santé avec qui j'ai pu apprendre, travaillé et rire. Mes premières pensées vont à Asma avec qui j'ai la chance de partager le quotidien au CIS avec moi, c'est un plaisir de venir travailler à tes cotés. Merci aux doctorants, ceux déjà partis et ceux qui sont arrivés, pour les repas, pauses cafés et soirées que l'on a pu partagé. Je remercie également tous les collègues qui m'ont aidés pour les problèmes scientifiques ou pratiques auquel j'ai été confronté. Merci également à tous les coureurs du midi pour m'avoir attendus tant de fois.

Merci à mes amis, qui ont été d'un soutien précieux et constant tout au long de cette thèse, malgré les circonstances et kilomètres qui ont espacés nos rencontres, notamment les nantais, toujours présents avec bonne humeur, ne serait-ce que pour une soirée ou un week-end. Merci aux amis présents depuis l'école, pour tous ces messages qui donnent le sourire, en particulier à Hugo et Bastien, qui m'ont ré-accueilli à Saint-Étienne, et aux autres aurasiens, Victor et Bassel, pour les péripéties lyonnaises et stéphanoises. Je pense également aux colocataires qui se sont succédés rue Gambetta, Émilie, Laurie, Élise, Alice, Lucie, Antonin, Ahmed, Ivan et Xavier, les bons moments passés ensemble sont des souvenirs précieux, j'ai hâte d'en créer de nouveaux.

Je souhaite exprimer toute ma reconnaissance à ma famille, en particulier à mes parents, Béatrice et Laurent, ainsi qu'à mes grandes soeurs, Faustine et Charlotte. Vous m'avez fait grandir et je ne serai pas arrivé où je suis sans votre aide et vos conseils.

Merci enfin à Élodie, pour tes encouragements, ta bonne humeur et ton soutien sans faille, dans les bons moments comme dans les moins bons. Je ne pourrai imaginer meilleure compagne de voyage pour avancer dans la vie.

Abstract

The care for multimorbid patients (patients having multiple chronic diseases) in health care centers is a growing concern for health professional. Multiple research on care pathways in Europe aim to ease the simultaneous treatment of multiple diseases, both inside and outside the hospital. This has to be dealt with in a complex environment, where multiple factors interfere with the management of multidisciplinary pathways. The constant increase of hospital admissions through the emergency department delays care and support from the pertinent units of the hospital. The expected increase of the elderly population in the Loire department, in France, and the established link between age and multimorbidity raises concerns of an increase in multimorbid patient admissions.

The hospital of Saint-Étienne CHUSE is a university hospital at the head of the Loire Hospital Group (GHT, Groupement Hospitalier de Territoire in French), a large network of health care centers. CHUSE has 1,802 beds dispatched in more than 60 medical units. In 2020, CHUSE admitted more than 90 000 patients for hospitalization and 78 400 patients were seen in its Emergency Department.

We developed a methodology to generate a population with realistic clinical pathways using Process Mining techniques. A digital twin of the hospital was built using AnyLogic®and tested using the case-study of the COVID-19 pandemic management. We evaluated the ability of multimorbidity indexes to predict hospital readmission. Those elements allowed to investigate the creation of a Polyvalent Unit for the care of multimorbid patients in the CHUSE through simulation. An optimization algorithm was implemented to size this unit using the simulation model.

Résumé

La prise en charge des patients atteints de multimorbidité (ayant plusieurs pathologies chroniques simultanément) pose de nombreuses interrogations chez les professionnels de santé. Des dizaines de protocoles ont étés développés en France et en Europe pour faciliter la coordination des soins entre les différentes spécialités, à l'hôpital ainsi que pour les soins quotidiens à domicile. A cela s'ajoute un contexte hospitalier difficile, où de nombreux facteurs peuvent complexifier la prise en charge, comme la constante augmentation du nombre de visites aux urgences en France depuis 20 ans, et le vieillissement de la population, les patients âgés étant plus susceptibles d'être atteints de multimorbidité. Ce projet de recherche vise à construire un modèle de simulation généralisable du Centre Hospitalier Universitaire de Saint-Étienne (CHUSE) pour l'évaluation des parcours de soins. Composé de plus de 60 unités médicales pour un total de 1802 lits, le CHUSE a enregistré plus de 90 000 séjours en hospitalisation et 78 400 visites aux urgences en 2020. Nous présentons une méthode de génération de population pour la simulation en utilisant les techniques d'exploration de processus (Process Mining). Un jumeau numérique de l'hôpital a été développé sous la forme d'une simulation en utilisant le logiciel Any-Logic®, testé sur la COVID-19. Nous avons évalué la performance de score de multimorbidité pour prédire les réadmissions à l'hôpital. L'ensemble de ces éléments ont permis d'étudier la création d'une unité de soins polyvalent dédié notamment aux seau sein du CHUSE, et un algorithme d'optimisation a été ajouté pour dimensionner cette unité.

" Chercher est le propre de tous, trouver n'est pas chose commune. "

Proverbe breton.

List of publications

Bibliography

Note to the reader

This manuscript presents a thesis organized as a collection of research articles (Chapters 2-5). To improve the coherence of the manuscript, each article is preceded by an introduction and followed by a brief summary and conclusion, presenting the motivation behind this research and its position in relation to the rest of the thesis. Each article contains an overview of the relevant literature for the specific problematic of the chapter. The literature review presented in chapter 1 presents the global context of the study.

Glossary

Introduction

Multimorbidity refers to the co-occurrence of multiple chronic diseases [1], which complicates the management of hospitalized patients. Choosing which care plan is the best for multimorbid patients is a question being more and more investigated. Indeed, multimorbidity is often cited as a cause for patients' prolonged length of stay [2], readmission or in-hospital mortality [3], [4]. According to Institut National de la Statistique et des Études Économiques (INSEE) predictions, the number of elderly persons in France is expected to grow. In 2070, people aged 65 years or more should represent 28.9% of the population while they represent 20.5% in 2021 [5]. Advanced age is often linked to higher multimorbidity prevalence, foreshadowing that the care for multimorbid persons in the hospital will become more and more common.

The different chronic diseases affecting multimorbid patients can cause complications of the provided care. The worsening of a secondary chronic condition might force the care management team to transfer patients toward another medical unit, and discontinue of care for the primary condition. This extension of patients' length of stay is a problem for the hospital, as it increases the consumption of medical resources such as beds and associated human resources. From the patient's perspective, complications related to multimorbidity impact the satisfaction of care for obvious reasons (multiple transfers from one medical unit to another, extended stay, etc.).

This thesis aims to improve the clinical pathway of patients with multimorbidity through simulation and optimization. It is articulated around a generic Discrete Event Simulation (DES) model of an health care center, able to replay the pathways of patients using process mining. A pre-processing module using Process Mining (PM) techniques to generate fictitious patients pathways was developed. The choice of which multimorbid patients to act on have been studied using advanced statistics and Machine Learning (ML) techniques.

Introduction

Using DES to evaluate what-if scenarios and improve the care pathways of patients is a possible complement to traditional medico-economical studies. Comparing care management model could be achieved using a generic hospitalization model altering the care management of some patients and measuring the impact on the health care center.

The scientific challenges posed by this thesis can be formulated as follows:

• How to identify among patients with multimorbidity those at risk of complications ? We used during this work the medico-administrative data of multimorbid patients who were hospitalized. It is necessary to manage differently stable patients, whose clinical pathway was appropriate, to the at-risk patients whose pathway is to be modified.

• How to accurately model and simulate the normal course of operations of a health care center? accurately simulate the course of hospital operations is not a simple task. We identified two main challenges in developing such a model:

-Extract and format the clinical pathway of patients from the medico-administrative database;

-Develop a model that could accurately replay those pathways in a faithful representation of the health care center.

• How to size a separate part of the hospital dedicated to a particular category of patients ? Implementing a novel and physically separated care pathway for a precise category of patients implies to size this new section according to the targeted population and to the means allocated to this purpose.

Contributions

In this thesis, different contributions have been developed to tackle the challenges listed here above. Those contributions can be roughly sorted in two different categories: technical contributions, where pre-existing techniques were applied and assembled and automatized for our purposes and scientific contributions when we developed innovative methods.

Technical contributions

The first technical contribution of this thesis is the development of a PM framework to generate the input population of a simulation model from historical data. PM is a set of techniques dedicated to the study of business process [6]. It is a bridge between the fields Data Mining and Business Process Management. Applications of PM are generally classified into three categories : process discovery to map processes where no models have been established, conformance checking that compares the theoretical process and the model as it is executed and enhancement, when modifications are tested before implementation.

Here, we apply process discovery and used the output process model to generate the population's pathways and length of stay.

The second significant technical contribution consists in a generic optimization module for AnyLogic® using a tabu search. This module was used to size and compose the polyvalent unit for the care of multimorbid patients, but can be modified easily for other health care related problems.

Scientific contributions

As part of the generation of pathways for simulation, we thoroughly studied the generated pathways to ensure that they were sound. This contribution directly follows the first technical contribution listed here-above.

In medical environment, the use of scores to understand the profile of patients and their level of risk is common. Recently, ML techniques, like Tree-based algorithms or neural networks, have been extensively used for outcome prediction models. Between 2015 and 2019 alone, Huang et al. identified 43 articles using ML techniques for prediction of 30-days hospital readmission on US-based patient populations [7]. During the course of this thesis, we assessed the performance of different ML techniques to predict all-cause readmission. Other statistical procedures were used to predict abnormally complex pathways.

The development of a complete and generic simulation model, that could act as a digital twin of the existing health care center, was one of the main objectives of the present thesis. Digital twins are a concept introduced in [8] for the industry and that can formally described as " a mirror image of a physical process that is articulated alongside the process in question " [9]. We developed a DES model that could fit this purpose. DES is a technique that belongs to Operations Research (OR). It models time as the collection of occurrence dates of events which allows to simulate system operations as a sequence of distinct events in time. Interest entities (patients in our case) are passed through a network of queues and activities until completion of the process [10]. Its use in health care research have constantly increased over the past years [11]. In addition, when implemented with a sufficient level of details, the ability to account for uncertainty make DES fit for micro-costing evaluation.

Manuscript plan

The present manuscript is organized as follows:

• Chapter 1 explores the relevant literature on multimorbidity, its identification, its measurement and the development of specific care pathways to improve the care of multimorbid patients inside and outside health care centers.

• In Chapter 2, we present the development of a framework analyzing the pathway of multimorbid patients from electronic health records from a Process-mining perspective. This framework pursue two objectives: (i) automatically identify the clinical pathway of patients and (ii) generate a patient population following the identified characteristics (in both pathway and length of stay) to use in a simulation model.

• Chapter 3 describes the development and use of a macroscopic digital twin of a health care center, using DES. Its potential to evaluate the modification of existing process and use as a strategic decision-aid tool is explored by studying the addition of an Intensive Care Unit (ICU) for the care of patients infected with SARS-COV2 virus.

• We describe from an epidemiological perspective the population of patients with multimorbidity that were hospitalized in the Centre Hospitalier Universitaire de Introduction Saint-Étienne (CHUSE) in Chapter 4. Prediction of readmission for those patients using machine learning techniques is investigated as well.

• Chapter 5 presents the application of our macroscopic simulation model to the care of multimorbid patients in a newly created polyvalent unit. An optimization model for the sizing of this unit is proposed and solved using a Tabu Search module incorporated to the simulation model.

The articulation of each chapters and their associated contributions are summarized in

Résumé du chapitre

Nous présentons dans ce chapitre un aperçu de la littérature scientifique dans les différents domaines abordés dans ce manuscrit. La prise en charge des patients multimorbides à l'hôpital pose de nombreux problèmes du point de vue de l'organisation des soins. En particulier, définir et identifier les patients multimorbides représente un challenge conséquent. Nous explorons ici la littérature médicale autour du concept de multimorbidité dans le but de définir ce concept central de notre travail. Une attention particulière a été portée au choix d'un indice de multimorbidité. Enfin nous résumerons les principaux points qui ont pu motiver et nourrir le travail de recherche présenté dans cette thèse, et le positionnement global de cette dernière vis-à-vis de la littérature scientifique existante.

Multimorbidity

Multimorbidity refers to the co-occurrence of multiple health conditions within one patient. The fact that several diseases and affections are simultaneous complicates the care for one particular conditions. Interactions between diseases and treatments may interfere with the usual care plan, encouraging health practitioners to investigate new care pathways for the concerned patients. This section focuses on the formal definition of the multimorbidity concept and explores the methods used in the literature to identify multimorbid patients. Then the attempts to measure multimorbidity for predictive purposes is presented.

Multimorbidity concept and definition

The coexistence of different diseases within one patient and how to treat people in this situation is a growing concern for scientific community. The term multimorbidity derives from the term comorbidity, which was defined by Feinstein et al. [12] as "any distinct additional entity that existed or may occur during the clinical course of a patient who has the index diseases under study". The two words were used indistinctly for quite some time until a review by Van den Akker [1] clarified the differences between the two words. The term comorbidity kept its initial definition and multimorbidity was formally defined as the "co-occurrence of multiple chronic or acute conditions within one person". This review also added a layer of classification to these definitions, to characterize comorbidity and multimorbidity with the following elements:

• Simple: the co-occurrence of diseases whether coincidental or not;

• Associative: not known to be causal;

• Causal: implying a causal relation among co-occurring diseases.

In summary, we can define multimorbidity by the combination of the two following aspects (i) the co-occurrence of several diseases and (ii) the chronicity of those conditions, acute diseases being often excluded from the different analysis we found. For instance, Wei et al. [13] included only chronic conditions in its index of multimorbidity -a notion we define more precisely in 1.2 -and [14] explicitly excludes conditions based on their acute nature.

Multimorbidity is quite present in European elderly population, between 24 and 36 % in Europe according to Nielsen et al. [15]. As detailed in 1.1.2, those estimates vary with the chosen definition. In France, estimations of multimorbidity in elderly population vary from 15% in [16] to 93% [17] based on INSEE Measurements. Coste et al. [18] calculated a 30.4% to 39.0% prevalence of multimorbidity in general population1 .

Identification of multimorbidity in general population

Although this definition is widely agreed upon, the actual formalization of multimorbidity turns out to be more complex. A systematic review carried out in [19] describes three main definitions for multimorbidity identification:

• Using a pre-established list of diseases, multimorbidity can be defined as the detection of more than two (or three or more) conditions in one individual.

• Another way to define multimorbidity is to use a weighted count of conditions, assessing the severity of the condition regarding a particular outcome. Numerous versions of such counts have been built, with the score developed in [20] probably being the most widely used.

• Finally, one can measure multimorbidity through the simultaneous presence of symptoms, without relying on diagnosis per say.

The study by Sasseville et al. [21] explores the differences between two multimorbidity definitions in the prediction of mortality and poorer self-related health outcomes (namely health-related quality of life, mental health and physical functioning). It uses two definitions for multimorbidity : (i) a count of chronic conditions and (ii) a count of medication classes prescribed. Neither presented significant advantage on the other in sensitivity for mortality, however, with a better specificity, the medical condition count proved a better ability to pinpoint multimorbid patients at risk of presenting poorer health outcomes during the two years follow-up compared to classical condition counts.

Multimorbidity as a count

All those techniques rely on establishing a list of conditions contributing to multimorbidity beforehand. The lists from a study to another are variable, with included conditions depending greatly on data availability. [22] notes that conditions included in indexes are also variables because they consider different degrees of abstraction for the conditions. It namely uses cancer as an example, as it is included in some studies under the generic appellation "cancer", while other indexes distinguish different categories of cancer. This study ultimately recommends that any new index include at least 11 diagnoses chosen based on chronic diseases prevalence in Germany and their frequent appearance in mortality reports. However, these conditions have also been contested: Harrison et al. highlight the fact that this list fails to capture multimorbid population when considering patients having more than 2 or 3 chronic conditions [23], and although other lists based Chapter 1. Contextualization on prevalence of chronic conditions in the general population are considered, like Fortin et al. [24], the authors recommend to include all possible chronic conditions.

Barnett et al. [25] uses a list of 40 morbidities, including the 11 conditions identified in [22], to define multimorbidity as the presence of 2 or more conditions from the list. The affections considered for multimorbidity are variable from a study to another and depend greatly on the context of the study and availability of data. Contrary to the idea that multimorbidity primarily concerns elderly patients, Barnett et al. found that there was more multimorbid patients younger than 65 years in absolute numbers (210 500 and 194 996 respectively). This is confirmed by Taylor et al. [26] in a study on 3206 adults in Australia. The absolute number of multimorbid elders was higher (200) than the number of multimorbid persons younger than 60 years old (178), but the two figures are close and confirm that focusing on elderly patients might ignore a significant proportion of patients.

Other operationalization of multimorbidity have been built, especially with the emergence of large and standardized medico-administrative databases using the International Classification of Disease (ICD-9) or ICD-10 coding. The classical definition of multimorbidity in such cases is to vague and inadequate. Efforts to transpose existing indexes to medico-administrative databases have been made, [27] conducted a study to translate the list of diseases include for the calculation of Charlson's index [20] into groups of ICD-9 codes, a similar work used the same methodology to build groups of ICD-10 codes [START_REF] Sundararajan | New ICD-10 version of the Charlson comorbidity index predicted in-hospital mortality[END_REF]. Both studies report satisfying results for the prediction of in-hospital mortality, with ROC-AUC score of 0.86 to 0.87 and 0.85 to 0.86 respectively.

As explained here-above, the identification of multimorbidity in a population is highly dependent of the chronic conditions considered. [19] reports difficulties to compare the studies identified in the review because of the differences between the construction of multimorbidity measures. The identification of multimorbidity from a list of conditions or symptoms implies the equal participation of each condition to the phenomenon of multimorbidity and to the outcomes of interest.

Complex multimorbidity [START_REF] Singer | Social determinants of multimorbidity and multiple functional limitations among the ageing population of England, 2002-2015[END_REF] compares three different definitions of multimorbidity, the "basic" definition being the count of diseases as defined by [19]. Among other proposed definitions are "complex" multimorbidity introduced in [23], "the co-occurrence of two or more chronic conditions affecting two or more body systems within one person without an index chronic condition", and finally as the presence of multiple function limitations. Both Singer and Harrison come to the conclusion that complex multimorbidity is more useful as it results in the identification of a narrower portion of the population and shows a greater differentiation of multimorbidity among elderly patient.

[30] compares the 8 following definitions of multimorbidities and their predictive ability on 30-days readmission.

1. 2 or more distinct body system categories and number of health condition, 2. 2 or more distinct body system categories and number of chronic health conditions, 3. number of distinct body system categories, 4. number of CCS (Clinical Classification software [START_REF] For | Healthcare Research and Quality, Hcup ccs. healthcare cost and utilization project (hcup)[END_REF]) categories, 5. number of health conditions, 6. number of chronic health conditions, 7. Deyo-Charlson Comorbidity Index [27], [START_REF] Charlson | Validation of a combined multimorbidity index[END_REF], 8. Elixhauser-van-Walraven Comorbidity Index [START_REF] Elixhauser | Comorbidity Measures for Use with Administrative Data[END_REF], [START_REF] Van Walraven | A Modification of the Elixhauser Comorbidity Measures Into a Point System for Hospital Death Using Administrative Data[END_REF].

Each multimorbidity definitions was used to classify patients in three risk categories of resource utilization, and to predict 30-days readmission and prolonged length of stays. Although the results for each outcome were fair, no definition was performing better over all 3 outcomes.

Beyond counting diseases or affected body systems, many studies have attempted to build more complex measurement of multimorbidity, able to discriminate the present conditions based on their severity or their relations can be established.

Measurement of multimorbidity

Building exhaustive measure of multimorbidity allow practitioners to assess the severity of multimorbidity from the present conditions better than a binary classification between multimorbid and non multimorbid patients. For the rest of this manuscript we will refer to a quantification of multimorbidity as a multimorbidity index. Literature often discriminates indexes in two categories: counts of conditions and weighted indexes [START_REF] Huntley | Measures of Multimorbidity and Morbidity Burden for Use in Primary Care and Community Settings: A Systematic Review and Guide[END_REF]. A review performed in [START_REF] Stirland | Measuring multimorbidity beyond counting diseases: Systematic review of community and population studies and guide to index choice[END_REF] also focus on more complex multimorbidity indexes and identify 4 main categories: drug counts, weighted condition counts, clusters of conditions and additional variables, with some indexes belonging in several categories.

Measuring Multimorbidity in the Literature

Simple counts of diseases

This first and more common measure of multimorbidity consists in counting the number of diseases, which is the natural continuation of the identification of multimorbidity from a list of condition. From the binary variable indicating if the patient has more than a defined number of conditions it is possible to build an index giving directly the number of conditions the patients' have. Counts are mainly used in epidemiological studies. [25], [26] both report the proportion of population suffering from multimorbidity using different cutoff points for the definition (two, three or four diseases).

The main criticism that is made against counts of diseases is the fact that they do not reflect the patients' experience of the disease or its severity. As already mentioned, the definition of multimorbidity as a simple addition of chronic diseases does not have a great discriminating ability for multimorbidity in general population [23], [START_REF] Singer | Social determinants of multimorbidity and multiple functional limitations among the ageing population of England, 2002-2015[END_REF].

Weighted counts of diseases

Weighted indexes follow the same baseline than the count of conditions, it sums the number of condition present in a patient, but the count is weighted by the estimated severity of the condition. The development of this type of index is not new, the Cumulative Illness Rating Scale (CIRS) was developed in 1968 and weight condition on a 0 to 4 scale of severity (from 0 when the condition is absent to 4 for extremely severe conditions) [START_REF] Linn | Cumulative Illness Rating Scale[END_REF]. The conditions can be rated according to the perceived severity of the diseases or by calculating the association between the disease and a particular outcome, such as mortality or Health Related Quality of Life (HRQoL). This is the case for instance with Chapter 1. Contextualization indexes developed by Charlson et al. and Stanley et al. [20], [START_REF] Stanley | The new measuring multimorbidity index predicted mortality better than Charlson and Elixhauser indices among the general population[END_REF]. Weights are assigned the conditions according to their predictive power on mortality.

Weighted counts are not exempt of problems and one of the main critics about them is their lack of portability in several ways:

• Regarding outcomes, the indexes are built to predict a precise outcome, their predictive power need to be validated before being used for other outcomes;

• Geographically, indexes must be validated on population of interest in order to account for potential differences with the development population;

• Temporally, with treatments always evolving, the evaluation of a condition's severity can evolve, as it was the case for Charlson et al. [20] and it evaluation of AIDS.

For these reasons, it is common to see popular indexes being updated over time. The calculation of the different weights also varies from an index to the other. The most common method is to use some sort of regression model to compute the part each condition has on one particular outcome. This is the case for Bannay et al. [START_REF] Bannay | The Best Use of the Charlson Comorbidity Index With Electronic Health Care Database to Predict Mortality[END_REF], which uses the methodology of the original Charlson's index. The hazard ratio generated from a Cox model are rounded to the nearest integer and used as weights. Outcomes used are varied, we can cite HRQoL (Millá-Perseguer et al. [START_REF] Millá-Perseguer | Measurement of health-related quality by multimorbidity groups in primary health care[END_REF]), physical functioning (Wei et al. [START_REF] Wei | Multimorbidity and Physical and Cognitive Function: Performance of a New Multimorbidity-Weighted Index[END_REF]) or hospital admissions (Byles et al. [START_REF] Byles | Single index of multimorbidity did not predict multiple outcomes[END_REF]).

Clusters of conditions

Another attempt to measure multimorbidity is to form groups of pathologies affecting the body in similar manners. Singer et al. [START_REF] Singer | Social determinants of multimorbidity and multiple functional limitations among the ageing population of England, 2002-2015[END_REF] defines 8 body systems and defines a complex multimorbidity score corresponding to the number of affected body system. Calderón-Larrañaga et al. [14] gathered a group of health care professionals to systematically cluster chronic conditions, identified using ICD-10 codes, into 60 categories. Calderón-Larrañaga et al's define the patient's score as the number of categories containing the patient diagnoses. It includes more the 900 ICD-10 codes at the 3-digit level, making it a viable option for measuring multimorbidity in medico-administrative databases.

Considering other information than just diagnosis

The last noticeable method to enrich a multimorbidity index is the addition of other information to chronic condition data. A systematic review identified several variables that could be added to multimorbidity indexes like physiological measures or drug prescriptions Stirland et al. [START_REF] Stirland | Measuring multimorbidity beyond counting diseases: Systematic review of community and population studies and guide to index choice[END_REF]. Newman et al. [START_REF] Newman | A physiologic index of comorbidity: Relationship to mortality and disability[END_REF] performed several measures including pulmonary function, carotid wall thickness, cystatin C, white matter grade and fasting glucose. Only the latter was coded as a binary variable, the others being evaluated on a scale from 0 (best) to 2 (worse). The authors report a strong predictive power for mortality and disability, even for people presenting low-risks.

Similarly, the study performed by Meems et al. [START_REF] Meems | Low levels of vitamin D are associated with multimorbidity: Results from the LifeLines Cohort Study[END_REF] built a multimorbidity index that can be classified as complex according to Harrison et al. [23] and compared its predictive ability to plasma 25-hydroxyvitamin D3 levels. Classically associated with calcium absorption and bone health, recent research give an important role to vitamin-D levels in immunoregulation [START_REF] Rosa | Vitamin D3: A helpful immuno-modulator[END_REF]. It can be noticed that low vitamin-D levels were associated with increasing morbidity prevalence and its inclusion in an index might be of interest, although Meems et al. reports that associations of low vitamin-D levels with certain diseases are still under debate and might be a redundant information. This information is important when we consider that recent research associate low vitamin-D levels to poorer outcomes in case of SARS-COV-2 infection. However, no conclusive evidence of a causal link has been published yet according to this recent review by Ghelani et al. [START_REF] Ghelani | Vitamin D and COVID-19: An Overview of Recent Evidence[END_REF].

Singer et al. [START_REF] Singer | Social determinants of multimorbidity and multiple functional limitations among the ageing population of England, 2002-2015[END_REF] also introduced a count of functional limitations for comparison with multimorbidity scores. Stirland et al. [START_REF] Stirland | Measuring multimorbidity beyond counting diseases: Systematic review of community and population studies and guide to index choice[END_REF] identified 4 indexes using drugs counts for identifying multimorbidity, like Korff et al. [START_REF] Korff | A chronic disease score from automated pharmacy data[END_REF] which established a chronic diseases count based on pharmacy data.

Examples of Multimorbidity scores on fictitious patients

For illustrative purposes we present the case of fictitious patient and the different scores of multimorbidity using different definitions. Figure 1.1 presents two patients and their chronic conditions. We use in this example two adaptations of the Charlson's score using ICD-10 developed by Bannay et al. and Quan et al. [START_REF] Bannay | The Best Use of the Charlson Comorbidity Index With Electronic Health Care Database to Predict Mortality[END_REF], [START_REF] Quan | Coding Algorithms for Defining Comorbidities in ICD-9-CM and ICD-10 Administrative Data[END_REF]. We also present the method of calculation developed by Calderón-Larrañaga et al's score [14], For patient 2, we have only one active category: Congestive heart failure. Indeed, contrary to Bannay et al's classification, the two codes fall in the same category.

Using Calderón-Larrañaga et al's score, Patient 1 has 3 active categories on the 60 existing, and thus has a score of 3. The active categories are: "Diabetes", "COPD, Emphysema, chronic bronchitis" and "Chronic Kidney Disease". Patient 2 has a score of 2 with the following categories active: "Heart Failure" with code I50 and "Ischemic heart disease" with code I25.

Concluding remark on indexes calculation methods

It is to be noticed that the index calculations methods presented here are not mutually exclusive. For instance Korff et al. [START_REF] Korff | A chronic disease score from automated pharmacy data[END_REF] uses a count of drugs to establish a chronic disease score but the count is also weighted, and drugs are grouped in several classes to avoid redundancy. This CDS score belongs in three of the index categories we presented here.

Multimorbidity indexes are subject to several criticisms. Counts of diseases especially are controversial for their vague definition and lack of specificity [18], [START_REF] Stirland | Measuring multimorbidity beyond counting diseases: Systematic review of community and population studies and guide to index choice[END_REF], [START_REF] Bastian | Measuring Multimorbidity: A Risky Business[END_REF], [START_REF] Diederichs | THE IMPORTANCE OF A STAN-DARDIZED INSTRUMENT TO ASSESS THE BURDEN OF MULTIMORBIDITY[END_REF]. Indexes in general are criticized by Boeckxstaens et al. for their poor use as predictors in several studies [START_REF] Boeckxstaens | Multimorbidity measures were poor predictors of adverse events in patients aged ≤80 years: A prospective cohort study[END_REF], [START_REF] Boeckxstaens | Should we keep on measuring multimorbidity?[END_REF].

We can also see that the majority of indexes are built in a precise context, and that using an index based on a study of mortality in a study about HRQoL is likely to yield mixed results. Data availability may also an issue. The CDC developed by Korff et al. [START_REF] Korff | A chronic disease score from automated pharmacy data[END_REF] or the index built in Newman et al. [START_REF] Newman | A physiologic index of comorbidity: Relationship to mortality and disability[END_REF] are based on physiological measures that might not be accessible in other studies.

Another limitation of multimorbidity index lies in the fact that it needs reliable data for calculations. Some study use prospective methods, with doctors performing individual examinations of patients to identify conditions, and reproducing the exact same methodology would mobilize costly resources. The use of self-reports or medicoadministrative data in other cases also poses the question of the quality of the information. It has been pointed out before that ICD coding in the French electronic health medical record database the national hospital discharge database or "Programme de Médicalisation des Systèmes d'Information" (PMSI) was of unequal quality [START_REF] De Léotoing | French hospital discharge database (pmsi) and bacterial resistance: Is coding adapted to hospital epidemiology?[END_REF].

Generic Formulation for Multimorbidity Indexes

We explored in this section the academic literature relative to the concept of multimorbidity. The widely used definition of multimorbidity "the co-occurrence of multiple chronic conditions within one person" underwent multiple adaptations, and the methods to identify multimorbid patients enclose the identification of diagnosis as well as drugs, and researchers now tend to focus on the co-occurrence of groups of chronic conditions affecting similar body groups rather than on conditions alone. Indexes have been developed to assess the severity of multimorbidity by weighting conditions according to the association of specific outcomes.

Despite the numerous differences between indexes, we can identify several points that are common to each study and standardize the formulation of multimorbidity by using the two following elements. Definition 1 (Category). A category i is a list of conditions, clusters of conditions, drugs or additional variables. We note variables x i the binary variable associated to category i.

A category is said active (x i = 1) if the patient is affected by at least one of the element of the category inactive otherwise (x i = 0). Definition 2 (Weight). The weight a i quantifies the contribution of a category i to predicting a certain outcome, or is a factor figuring the perceived severity of category i. It is equal to 1 when no weights are defined.

Those two elements allow us to define a multimorbidity index as follows: Definition 3 (Multimorbidity index). A multimorbidity index is defined by a set of N categories of multimorbidity assigned each a corresponding weight. The calculation giving the score s on the index is equal to the weighted sum of the category variables x i with their corresponding weights a i which can be summed up by the following formula:

s = n i=1 a i * x i (1.1)
To perform the research presented in this manuscript, we needed a recent index, that was easy to implement on medico-administrative data and compatible with ICD-10 coding of diagnosis. We conducted a systematic review of multimorbidity indexes in order to identify the index that best suited our needs. The following section presents this systematic review and the results.

Choice of an index: Systematic Review

The choice of an index, for all the reasons listed here above, is primordial when conducting a study on multimorbidity. We focused this review on the most recent indexes developed and analyzed the changes and trends in the area on multimorbidity. We present in the following subsection 1.3.1 the method we used to retrieve and analyze the research articles developing new indexes of multimorbidity. The articles extracted are presented in subsection 1.3.2, before we analyze the results and conclude in 1.3.3. The index we chose to use and the arguments supporting this choice are detailed in subsection 1.3.3.

Method

A structured systematic review search was performed in three databases with extensive coverage of the public health literature, PubMed, SCOPUS and Web Science to identify medical research articles developing novel indexes of multimorbidity. The period of interest was defined between 2009 and November 2019 (date of the search). For articles published before 2009, we decided to include the articles present in Diederichs et al. (2011) [22], who conducted a thorough analysis of the literature. We only included articles written in French, English or Spanish.

To identify relevant studies we looked in the different databases for the combination of one of the terms "index", "measure", "measurement" with the word "multimorbidity". The results were exported in an Excel spreadsheet. Articles were selected in a two-step screening process. Firstly the articles were sorted based on their titles and abstracts. Remaining articles went through a full text review. We decided not to consider articles describing simple counts of conditions or limiting the analysis on the measure to binary values (multimorbidity or not). We focused on research developing new indexes and excluded the ones adapting an existing index.

Results

The search algorithm is presented in the flowchart Figure 1.2 with the inclusion count. The initial search raised 1584 articles across the three databases searched, the 39 articles described by Diederichs et al. [22] and one additional article was included based on authors' personal knowledge. It came down to 769 records after excluding duplicates. Title and abstract screening excluded 709 additional records. Only 25 records remained in the qualitative analysis after full-text assessment. On those 25 publications, 12 are present in Diederichs et al. [22] and 13 were published between 2010 and 2019. Of the 25 articles finally included in the search,it can be noted that eight of them build several variations of indexes, mainly for comparing a simple count of disease with a weighted count [START_REF] Byles | Single index of multimorbidity did not predict multiple outcomes[END_REF], [START_REF] Fung | The Relationship Between Multimorbidity and Patients' Ratings of Communication[END_REF]- [START_REF] Wang | Development and internal validation of a multimorbidity index that predicts healthcare utilisation using the Canadian Longitudinal Study on Aging[END_REF]. Among those, Byles et al. [START_REF] Byles | Single index of multimorbidity did not predict multiple outcomes[END_REF] computed 4 weighted measures using different weighting methods, Payne et al. [START_REF] Payne | Development and validation of the Cambridge Multimorbidity Score[END_REF] computed 3 indexes using regressions on 3 different outcomes and 1 on general outcomes obtained by averaging the 3 outcome-specific weights. Tooth et al. [START_REF] Tooth | Weighted multimorbidity indexes predicted mortality, health service use, and health-related quality of life in older women[END_REF] computed indexes for 13 different outcomes: mortality, GP or specialist visits, hospitalizations, activities of daily living and HRQoL Jules Le Lay (through the seven Short Form in 36 questions (SF-36) sub-scales).

Table 1.1 displays the key characteristics of the articles: the population studied, origin of the data used and design of the study. Two other important features of are considered: (i) the outcomes studied and (ii) the use of weights. In Review [START_REF] Stirland | Measuring multimorbidity beyond counting diseases: Systematic review of community and population studies and guide to index choice[END_REF] Weights Rozzini [START_REF] Rozzini | Geriatric Index of Comorbidity: Validation and comparison with other measures of comorbidity[END_REF] 2002

Weighted indexes

19 studies developed weighted counts of multimorbidity, calculating weights as the perceived severity of disease [START_REF] Bayliss | Subjective assessments of comorbidity correlate with quality of life health outcomes: Initial validation of a comorbidity assessment instrument[END_REF], [START_REF] Ubalde-Lopez | Beyond Return to Work: The Effect of Multimorbidity on Work Functioning Trajectories After Sick Leave due to Common Mental Disorders[END_REF] or using coefficients or Odds Ratio from regressions [13], [20], [START_REF] Stanley | The new measuring multimorbidity index predicted mortality better than Charlson and Elixhauser indices among the general population[END_REF], [START_REF] Mcgee | Patterns of comorbidity and mortality risk in blacks and whites[END_REF], [START_REF] Diederichs | How to weight chronic diseases in multimorbidity indices? Development of a new method on the basis of individual data from five population-based studies[END_REF].

The most studied outcome is mortality (in-hospital or during an interest period), with 9 studies investigating the predictive ability of multimorbidity scores on this measure. HRQoL follows with 7 studies. As displayed in Table1.1 several studies are interested in multiple outcomes. Although most indexes are related to an outcome, it is not the case for Singer et al. [START_REF] Singer | Social determinants of multimorbidity and multiple functional limitations among the ageing population of England, 2002-2015[END_REF], Ubalde-Lopez et al. (2016) [START_REF] Ubalde-Lopez | Measuring multimorbidity in a working population: The effect on incident sickness absence[END_REF] and Calderón-Larrañaga et al. [14].

Indexes interested in a particular outcome can be classified in two categories: (i) outcomes related to the patient's status, obtained prospectively (HRQoL, vitamin-D levels. . . ) and (ii) general outcomes evaluated retrospectively (mortality, readmission). The former seem to focus on the quality of life of the multimorbid patient, while the latter are more interested in the public health impacts through quality of care indicators (rehospitalization, costs of care and so on). Outcomes like mortality or re-hospitalization are also more accessible in retrospective studies on patients hospitalization records, whereas HRQoL need to be specifically measured and reported when doing the study.

Additional Factors to Consider

We note that outcomes such as HRQoL or physical functioning are closely related to features outside of diagnosis themselves, and that multimorbidity scores often disregard. The treatment burden -the additional effort required from the patient to look after their health -caused by the multiple chronic diseases must be taken into account. A crosssection study on 835 elderly patients conducted by Morris et al. [START_REF] Morris | Treatment burden for patients with multimorbidity: Cross-sectional study with exploration of a single-item measure[END_REF] reports that 18% multimorbid participants included declare having a high burden, using [START_REF] Duncan | Development and validation of the Multimorbidity Treatment Burden Questionnaire (MTBQ)[END_REF]'s multimorbidity treatment burden questionnaire. However due to the low response rate of the study (42%), the rate of patient with multimorbidity might be underestimated as patients with higher treatment burden could be less likely to participate in such a study.

Socio-economic level of patients can also be an important factor as Stringhini et al. [START_REF] Stringhini | Socioeconomic status, non-communicable disease risk factors, and walking speed in older adults: Multi-cohort population based study[END_REF] reports a 4.8 years loss in physical functioning for patients from low socio-economic status. Lower functional ability was also noticed in [START_REF] Marventano | Multimorbidity and functional status in institutionalized older adults[END_REF] for people with lower education levels and Cho et al. [START_REF] Cho | Effects of health literacy on health status and health service utilization amongst the elderly[END_REF] reports a direct correlation between higher health literacy and good health status and there is evidence in the literature that multimorbity is less prevalent for higher education and higher household income groups [15]. In particular in France, [18] shows that people with lower education and socio-economic status tend to present multimorbity earlier in life, the difference disappearing for older age groups. Low health literacy is strongly associated with high treatment burden according to Morris et al. [START_REF] Morris | Treatment burden for patients with multimorbidity: Cross-sectional study with exploration of a single-item measure[END_REF] as well, and people presenting high treatment burden reporting more financial difficulties with health care.

Conclusion

This systematic review of literature updated the work done by Diederichs et al. [22] with the identification of 13 additional indexes of multimorbidity, which shows that multimorbidity is increasingly investigated. On administrative database, new approaches allow the formalization of the calculation, and allow to account for most of the available information in the patient digital records, like biological tests.

To chose the appropriate index, we suggest to first carry out a careful assessment of the objective, population and available data. To perform the research presented in the following chapters of this manuscript, we needed an easily implementable index, that was not built for a specific outcome. Calderón-Larrañaga's index responds to this need. It was built by a large panel of medical specialist and takes into account a large number of ICD-10 codes (over 900), it is therefore suitable for medico-administrative database. Its categories design make it more susceptible to capture complex multimorbidity and it is non outcome-specific.

We recommend that researchers willing to use a multimorbidity index for a medical study refer to the systematic review and analysis performed in Stirland et al. [START_REF] Stirland | Measuring multimorbidity beyond counting diseases: Systematic review of community and population studies and guide to index choice[END_REF], where a flowchart guiding the choice of such index was built 1.3. This flowchart recapitulates the most robust multimorbidity indexes identified during their studies and classify them according to the data and outcome used to build each index, making the selection of an index easier. Although performed with the same motivation, most of the studies identified in our research were not presented in [START_REF] Stirland | Measuring multimorbidity beyond counting diseases: Systematic review of community and population studies and guide to index choice[END_REF] (17/25)3 .

Care of Multimorbid patients

Numerous attempts to improve interventions on patients with multimorbidity have been made [START_REF] Rijken | How to improve care for people with multimorbidity in Europe[END_REF]. Developments of integrated care protocols have occurred across Europe to enforce patient-centered care pathways [START_REF] Van Der Heide | How to strengthen patient-centredness in caring for people with multimorbidity in Europe[END_REF]. Context of care in the region, with demographic and organizational issues to resolve, is to take into account in the development Chapter 1. Contextualization of such pathways. This section describes the Integrated Care Pathways (ICP) principles and their applications to multimorbid patients, as well as the present context of care in Saint-Étienne (France).

Integrated Care Pathways

ICP can be defined as "structured multidisciplinary care plans" [START_REF] Campbell | Integrated care pathways[END_REF], standardized protocols describing, for a specific medical problem, the care plan to apply, and gives the expected progress of the patient other time. Campbell et al. notes many advantages in using ICP, including improvement of the patient care, cost-effectiveness and gains in time for the health care center staff. Olsson et al. [START_REF] Olsson | The integrated care pathway reduced the number of hospital days by half: A prospective comparative study of patients with acute hip fracture[END_REF] reports significant reduction of the length of stay and complications occurrence when ICP was used for elderly patients with hip fractures. Later publications by the same authors confirmed improvement both in recovery of the patient and cost-effectiveness of the treatment [START_REF] Olsson | A cost-effectiveness study of a patient-centred integrated care pathway[END_REF]. ICP for elderly patients with heart failure can also lower the risk of mortality and unplanned readmissions [START_REF] Kul | Modeling of in-hospital treatment outcomes for elderly patients with heart failure: Care pathway versus usual care[END_REF], although cost was similar for ICP and control groups. However, the implementation of ICP was not identical in all health care centers involved due to organizational differences.

ICP for multimorbid patients

Implementations of care practices for multimorbidity have been observed by Rijken et al. [START_REF] Rijken | Managing multimorbidity: Profiles of integrated care approaches targeting people with multiple chronic conditions in Europe[END_REF] in Europe. Most of the 112 care practices favored patient-centered approach for multimorbid patients without specifying combination of chronic conditions (58%), with a global interest in elderly (55% of the time) and frail patients (29%). This study highlights the high number of care practices that focus on involving the patients into the management of care. In Australia, Shakib et al. [START_REF] Shakib | Effect of a Multidisciplinary Outpatient Model of Care on Health Outcomes in Older Patients with Multimorbidity: A Retrospective Case Control Study[END_REF] developed a global outpatient care model with a 30% reduction of mortality for patients with multimorbidity.

However, such organization of care don't always result in improving readmissions rates, length of stay or mortality rates in general population [START_REF] Pannick | Effects of Interdisciplinary Team Care Interventions on General Medical Wards: A Systematic Review[END_REF]. For instance, [START_REF] Lanzeta | Cost-utility analysis of an integrated care model for multimorbid patients based on a clinical trial[END_REF] developed an ICP aiming at better communication between primary care and hospital professionals and didn't find significant improvements in cost-effectiveness nor in efficiency of interventions.

Aging population in AURA

Saint-Étienne is at the heart of the Loire department, located in the Auvergne-Rhône-Alpes region, in the south-east of France. Nationwide, demographic predictions at 50 years show that the population wouldn't vary much in number (68, 1 millions inhabitants in 2070 compared to 67, 4 millions in 2021), but a clear increase of the elderly population (+5.7 millions for people aged 75 years old and more) [5]. As a consequence, a report by Charpin and Tlili predicts a rapid growth of health care expenditures for dependent patients [START_REF] Charpin | Perspectives démographique et financières de la dépendance[END_REF] .

Concerning the Loire department, [START_REF] Desgouttes | 9.5 millions d'habitants à l'horizon 2050 (AURA)[END_REF] predict a moderate increase of elderly people compared to national standards and Thouilleux et Bianco [START_REF] Thouilleux | Des seniors dépendants de plus en plus nombreux d'ici[END_REF] predict that the increase in Loire's dependent patients population should be moderate until 2050, mainly due to the current age structure in the department. This may suggest a smaller expenditure increase in the department than anticipated nationwide by Charpin and Tlili's report [START_REF] Charpin | Perspectives démographique et financières de la dépendance[END_REF], although they also predict that those patients are likely to become more severely dependent than they are now.

Context of care in health care centers

Health care centers in France have encountered several difficulties in the past decades that can cause trouble in the management of care.

Firstly, we can observe a constant increase in emergency department visits of 3.6% per year between 1996 and 2018, making the annual number of visits increase from 10.1 million in 1996 to 21.1 million in 2018 [START_REF] Sous La Direction De Fabien Toutlemonde | Les établissements de santé -édition 2021 -Ministère des Solidarités et de la Santé[END_REF], as shown in Figure 1.4a. This evolution puts the hospital under tension to admit Emergency Department (ED) patients into short stay units. Indeed, [START_REF] Boisguérin | Urgences : La moitié des patients restent moins de deux heures, hormis ceux maintenus en observation[END_REF] shows that a high number of patients go to the emergency department because it represents the easiest access of care, although a high proportion of ED patients report going for a justified medical reason. This increasing importance of the emergency department occurs in spite of the development of systems to ease the direct admission of patients, like the geriatric hotline in place in the CHUSE since 2014 [START_REF] Bailly | Intérêt d'une ligne téléphonique directe (hotline) destinée aux médecins pour limiter le passage aux urgences des personnes âgées : Analyse de 198 appels[END_REF] and recently evaluated in the thesis of Thomas Franck from MINES Saint-Étienne [START_REF] Franck | Modélisation, analyse et pilotage des parcours des personnes âgées en soins aigus et sur le long terme[END_REF].

In addition, a recent report by Boisguérin et al. shows that the number of open beds in complete hospitalization in France has decreased in the past 7 years, although the number of critical care has increased over the same period 1.4b [START_REF] Boisguérin | Entre fin 2019 et fin 2020, la capacité d'accueil hospitalière a progressé de 3,6 % en soins critiques et de 10,8 % en hospitalisation à domicile | Direction de la recherche, des études, de l'évaluation et des statistiques[END_REF]. This is particularly important given that information show that the likelihood of an adverse outcome occurring in the hospital can be related to overall hospital occupancy [START_REF] Boyle | Probability of severe adverse events as a function of hospital occupancy[END_REF].

Positioning of the present research

This research was motivated by the interrogations surrounding the care for multimorbid elderly patients that emerged from previous collaborations between the department of clinical Gerontology of the CHUSE and the Ingénierie des Systèmes de Soins et des Services de Santé (I4S) department of MINES Saint-Étienne. The identification of such patients in medico-administrative patients had not been attempted in previous research in the department, leading to the review of literature that was presented in this chapter. A first model of multimorbid patient's care was developed using preliminary incomplete data. The privileged solution at the time was to organize a shared care management in the Groupement Hospitalier de Territoire (GHT) Loire, by coordinating short stay and rehabilitation units to pull the flow of patient. Early results involving our first conclusions on multimorbidity indexes and this first simulation model were presented in the 2019 Operation Research Applied to health care Services conference.

Based on this first experience, definitive access to CHUSE patients with multimorbidity data was obtained. We developed the epidemiological description of the 16 000 patients with multimorbidity in the area of Saint-Étienne that had been hospitalized and discharged in the year 2017, in parallel to the preparation of a generic model of simulation of the CHUSE. The purpose of this generic simulation model was to obtain a DES that could serve as a base for our future work. The objective was twofold: 1 to develop a system allowing for all pathway to go through the hospital model an 2 prepare the addition of a secondary pathway that would become the altered multimorbid pathway. The emergence of SARS-COV-2 gave us a good case study for testing this layout and this work resulted in a participation in the Winter Simulation Conference 2020. [START_REF] Lay | Impact of covid-19 epidemics on bed requirements in a healthcare center using data-driven discrete-event simulation[END_REF].

Moreover, the formatting of patient's pathway information required for the simulation in [START_REF] Lay | Impact of covid-19 epidemics on bed requirements in a healthcare center using data-driven discrete-event simulation[END_REF] motivated the Process Mining analysis of patients pathway that is presented in chapter 2. Such work allow a standard representation of the population and ensures the anonymity of patients. This was of paramount importance to facilitate implementa- Integrated care for multimorbid patients, in the form of a new medical unit providing polyvalent care, is studied in chapter 5. The hospital generic layout from the previous chapter is adapted to this unit addition, and we further investigate the sizing of this unit using an optimization algorithm.

Chapter 2

Automated Generation of Patient

Population for Discrete Event Simulation using Process Mining

Motivation

Simulating patients admission and hospitalization of patients is a powerful way to assess the efficiency of a healthcare process. Stochastic simulation models, like DES, introduce randomness in timed procedures and allow for a more realistic representation of reality, where hazards can occur and interfere with the expected course of events. However, large amounts of data are needed to accurately set up the model. For this purpose, we need data about patients hospitalization: the list of units visited and the order in which they were visited, the length of stay at each step and so on. Medico-administrative databases, like PMSI, contain standardized data on patients stays in medical establishments and can provide such information.

Using personal information like these require authorizations of use by competent authorities, and requests take a long time to process. However this population will also include some unique patients that don't fit any predefined care pathways for medical or organizational reasons, and that we can't consider. Using aggregated data requires a less Chapter 2. Automated Generation of Patient Population for Discrete Event Simulation using Process Mining complicated procedure but does not allow access to detailed pathway information.

To propose a generic simulation pathway, easy to use with new healthcare center, we need to establish a pre-processing framework that can generate fictitious pathway information and length of stay. This pre-processing module could be applied inside the healthcare center and provide relevant information without disclosing patients personal information. We aim to filter the database to exclude unique patients without affecting the most representative pathways.

Summary

This chapter presents the development of a pre-processing unit that generates a synthetic patient population from historical stay data to populate a generic DES model of healthcare center. We used Process Mining techniques to map hospitalization pathways. Different filters have been tested to remove unusual pathways from the dataset. We enriched the process mining model with length of stay information. The resulting business process model was used to generate patient pathways and the length of stay at each step. The population was the formatted and used in a DES simulation.

Automated Generation of Patient Population for Discrete

Event Simulation using Process Mining

• Jules Le Lay, Julia Neveu, Benjamin Dalmas, Vincent Augusto, Automated Generation of Patient Population for Discrete Event Simulation using process mining (submitted to the 2022 Annual Modeling and Simulation Conference).

INTRODUCTION 1.Context

This research addresses the need for accessing a large quantity of real patient pathways to establish accurate simulation models. Medico-administrative databases record all pathways occurring in healthcare centers. However, some instances, like unique pathways caused by exceptional circumstances, need to be removed so that the model can focus on representative patients.

Since 2016, the General data protection regulation (GDPR), relating to the protection of individuals with regard to the processing of personal data and the free movement of such data, has changed the organization of health systems. Health data are described by GDPR as, "personal data related to the physical or mental health of a natural person, including the provision of health care services, which reveal information about his or her health status" (European Commission 2016), (article 4).

Macroscopic simulation models allow us to review the stays of patients in independent units and help to improve organization, identify bottlenecks and plan changes in a hospital. This research describes the automatic generation of population information for these types of simulation models.

Related Literature

Current research has investigated the creation of a complete framework to automate the use of medicoadministrative data in simulations. Here, we explore the related literature from different areas of the framework.

Process discovery algorithms map business processes using the data generated by their execution [START_REF] Aalst | Process Mining: Discovery, Conformance and Enhancement of Business Processes[END_REF]. All the steps, or events, of an instance form a trace. Discrete-event simulation (DES) is an operation research technique that relies on a stochastic modeling approach. In a recent review, (Vázquez-Serrano, Peimbert-García, and Cárdenas-Barrón 2021) identified 231 papers using DES in healthcare and highlighted a continued rise of publications in the last decade. Papers attempting to model healthcare operations tended to focus on one particular department, such as the Emergency department (ED) (Ben-Tovim et al. 2016) or Intensive care unit (ICU) (Busby and Carter 2017), and its interactions with the rest of the hospital. Here, we are interested in setting-specific simulation models, as described by (Fletcher and Worthington 2009), in which a generic simulation layout that uses input data to ensure the representation of a specific healthcare center is built. DES is particularly suitable to evaluate the economic aspects of processes. (Soto-Gordoa et al. 2017) built a model to quantify the cost of care of multimorbid patients following an integrated care pathway using Arena®Rockwell software.

(Maruster and van Beest 2009) proposed a simple 3-step approach to combine PM and DES for process improvement: (i) identify performance issues, (ii) map, modify and assess the existing process through simulation and (iii) evaluate the evolution of performance with the redesign case. This research is extended in (Aguirre, Parra, and Alvarado 2013), who added an extra layer of analysis by using data mining and root-cause analysis to improve the process. (Arnolds and Gartner 2018) and (Halawa, Chalil Madathil, and Khasawneh 2021) both used a combination of PM, simulation and optimization to determine the layout of a healthcare center facility. In both cases, PM was used to discover the pathway of patients between different locations and a simulation was used to evaluate the solutions generated by the optimization algorithms.

Frameworks using separate PM and simulation models have been developed recently. (Abohamad, Ramy, and Arisha 2017) combined an ED pathway analysis using the fuzzy miner algorithm with an Any-Logic® DES model. However, the results of the PM analysis were not automatically integrated into the simulation. (Camargo, Dumas, and González-Rojas 2020) proposed a complete framework to discover a process as a business process model network. Nonconforming traces were filtered from a log, and important parameters for the simulation (interarrival times or activity processing times) are simultaneously estimated.

Objective

In this paper, we propose an automatic framework that generates a population to feed the generic simulation model of a healthcare center and allows modelers to use their own simulated data, which has characteristics similar to those of actual patients. This framework uses historical clinical pathway data stored in a medicoadministrative database, filters and maps the clinical pathway of patients in a healthcare center, and generates and formats pathways for the simulation model. For this study, we use the data of multimorbid patients hospitalized in Centre Hospitalier Universitaire de Saint-Étienne (CHUSE) in 2017. The output population is prepared for an AnyLogic® DES model that is adapted from previous work led by our research group.

The remainder of this paper is organized as follows: Section 2 presents the current problems and describes the available data. The developed methodology is detailed in section 3. We describe the population used to test the framework and the numerical results obtained in section 4. Section 5 summarizes the main findings of this research, exposes the limitations of the study and lists opportunities to explore for future research.

CURRENT PROBLEMS

Our primary objective is to automatically populate a simulation model with filtered data. To achieve this goal, the proposed methodology needs to focus on two research problems. First, extracting a process map using process mining methods from the history of patient pathways. Numerous PM algorithms exist to generate process maps from event logs. We integrated these methods into our framework to accurately represent the hospitalization process. Secondly, we generate pathways using this process map to respect the stay characteristics of the original population. In particular, the most prevalent pathways and their proportion in the dataset need to be respected, and pathways that are not present in the original dataset need to be avoided. A series of filters must be tested to exclude most unusual pathways and include the most representative ones.

Generating a population from aggregated data avoids the use of personal data. Using this framework in hospitals would facilitate access to essential data (i.e., the most represented pathways and Length of stay (LoS) distribution in medical units) without disclosing personal data. This opportunity, particularly with the implementation of the new gdpr-related provisions, could facilitate the implementation of patient flow studies in hospitals.

The contribution of this article is twofold: (1) to establish a methodology generating a synthetic pool of patients from personal medico-administrative data using process mining and (2) to feed these data into a generic DES for health care process evaluation, allowing the automation of process studies.

METHODS

PM is a data mining technique allowing the better understanding and management of processes. The different applications identified are the discovery, analysis of conformance and identification of optimization potentials [START_REF] Aalst | Process Mining: Discovery, Conformance and Enhancement of Business Processes[END_REF]. Therefore, data mining is a bridge that links data science techniques and process sciences.

This section details the proposed framework and its constitutive elements. The complete workflow of this process is presented in Figure 1 and contains three distinct steps. First, process discovery is applied to patient data. Next, the output population is generated using a pathway generation module. Finally, the output population is passed on to a DES simulation.

Input Data

Medico-administrative databases contain essential information for analyzing the efficiency of patient pathways. Such data include administrative information (such as sex and age) and medical information (Ta-Figure 1: Schematics of the presented process from discovery to simulation.

ble 1). This type of data is considered sensible information. The list of variables accessed and their use in the present study is presented in Table 1. We only kept information related to the patient pathway. The occupation rate of each medical unit was calculated using the number of occupied beds every day and the theoretical number of beds per service in 2018 (additional data provided by the hospital). Each patient is modeled as an agent associated with the following parameters:

• the anonymous identifier of the patient's hospitalization;

• the pathway of the patient, given as a sequence of all medical units visited by the patient;

• the admission and discharge dates of the patient in each medical unit that compose the sequence; and • the occupation rate of each medical unit on the admission day in said unit.

An illustrative example is provided in Table 2. It displays two patients whose pathways are composed of 3 and 4 steps, respectively. Start and End steps were added for each stay, and the date of admission is converted to allow for these steps to be virtually instantaneous.

Process Discovery

All the described data were imported and processed using the PM4PY library (Berti, van Zelst, and Aalst 2019) in Python. Two discovery algorithms were tested to establish the process map, resulting in two different representations: Direct follow graphs (DFGs) and Petri net (PN). They were generated using the fuzzy We tested the following values for the local filter(percentage of paths we keep): [1, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1], combined with global filters of 100%, 95%, 90% or 85%.

Generating Pathways

Pathway generation is performed in two steps. First, we generated the sequence of visited medical units, and second, we generate the LoS for each step. Both the PN and DFGs models were tested to generate a new population using PM4PY, although the method is different for the two models. The DFGs graph generation is static, i.e. a given model will only generate one set of pathways. We generatec output logs considering the 10, 20, 50, 100 and 150 most frequent pathways. This method can also account for loops, allowing patients to go back and forth between two units. We allowed the generation of pathways with the same activity occurring 2, 5, 7 and 9 times. The PN model allows for dynamically generating logs from a model. We generated logs of 100, 500, 1000 and 5000 pathways, with traces of length 4, 6, 8, 10, 12 or 14 (the maximum number of activities observed). Loops can occur in the pathway but cannot be controlled.

In addition to the traditional metrics used in PM, we develop the 4 following indicators:

• Variant proportion: ratio between the number of variants generated and the number of variants observed in the input data; • Overlap: number of variants common to both input and output logs; • Under generation: proportion of variants of the input log not generated by the model; and • Overgeneration: proportion of variants of the output log not present in the input data.

We want to maximize the overlap, which shows how well we recreate the original data, and obtain a variant proportion close to one. In the similar approach, we want to minimize the over-and under-generation figures.

We used the available data to infer LoS distributions and draw a LoS value for each step of each pathway. Two possibilities were tested: (i) LoS distributions were calculated using the LoS history of each unit. (ii) calculate LoS distributions for each pair of units (A, B). This second option was studied to add more precision to the model in case of unit crowdings. Twelve different distributions were tested, and the best fitting distribution was kept for each unit or pair of units. The longest generated stays were limites 1.2 times the maximum duration observed in the data set to remain consistent with the initial observations.

In the simulation, we tracked the final LoS as the sum of a waiting period and a care period to check if it was consistant with the initial hospitalization LoS. We used the same methodology as previously and aggregated the LoS of the patients for each unit. We tested the two following hypotheses: (i) the waiting time in the current unit depends on whether the next unit is crowded, and (ii) the waiting time in the current unit depends on what the next service is and on whether it is crowded. We created both subsets of data and calculated the distribution for both cases. Generated patients, their pathways, and the calculated LoS distributions are automatically stored and exported in a normalized excel file for the simulation.

Simulation

A simple macroscopic simulation model of a generic hospital was developed using AnyLogic®software. Patients created in this model are randomly provided a pathway and a corresponding LoS sequence from the log generated previously. The patients' pathway are passed in the simulation as parameters. The only resource we considered is the number of beds per unit (staff utilization is not modeled). They were modeled using ResourcePool objects and dynamically chosen in seize blocks depending on the unit requested in the pathway. The model was thoroughly validated using different datasets to ensure its proper function.

RESULTS

The data used for the whole project relates to multimorbid patients hospitalized at the hospital of Saint Étienne in 2017. To identify patients in the database, multimorbidity was defined as having one diagnosis in at least three different chapters of the International Statistical Classification of Diseases and Related Health Problems 10th Revision (IDC-10) classification. A lot of diagnoses per patients are listed in medical units, and a vast majority of patients have diagnoses in 2 or more chapters. This first filter was made to exclude non multimorbid patients and was performed by the hospital administration. Data were provided by the CHUSE under Commission Nationale de l'Informatique et des Libertés (CNIL) authorization number 919300.

Process Discovery

Figure 3 shows the cumulative percentage of patients per variant. The first variant accounts for more than 23% of patients, and 90% of patients leave with only 21 variants. The DFG and PN graphs built with the fuzzy miner and inductive miner algorithms and an 85% filter (this filter value was chosen for readability reasons) on log variants are displayed in Figure 4 and Figure 5, respectively. These representations reveal three clusters of medical units. The first cluster groups units with patients who come for "simple" stays, with only one unit visited, such as the endocrinology, gastroenterology, or gerontology units. The second cluster is organized around post-emergency services in more complex pathways. The last cluster is composed of medical units with high resource utilization, such as glsICUs and surgery.

The fitness, precision, generalization and simplicity when using the global filters are displayed in Figure 6 for the two studied models. Fitness indicates how the model allows us to replay paths of the event log, simplicity allows us to select simpler representations, precision measures the ability to discriminate traces that are not in the original log, and generalization allows us to eliminate overly specific models. The two values of global filters that seem to meet the evaluation criteria are the 95% and 90% filters. These values allow the model to achieve a good balance between generalization and precision. For the local filter, keeping the 90% most frequent paths and activities on the log filtered to keep the 95% most frequent pathways gives satisfying results. The output fitness, accuracy and generalization values are close to 1, indicating good overall precision while keeping the simplicity metric at a satisfying level.

Log Generation

The results for the DFG and Petri net models, with the global filter set to 95% are shown in Table 8. Metrics are grouped into 4 cells as follows: variant proportion in top-left cells, overlap in top-right cells, undergeneration in bottom-left cells and overgeneration in bottom-right cells. The initial time values and the calculated distribution for the cardiology unit are displayed in Figure 9.

The distributions generated for pairs of units were rarely significant because of the insufficient number of observations for many of the paths considered. For statistically significant distributions, we obtained a shorter average LoS when the next unit was crowded. These counterintuitive results and the absence of results for some pairs of units prompted us to use distributions of units considered independently.

CONCLUSIONS AND PERSPECTIVES

The present paper develops an automated framework to use generates populations of patients in simulation.

The proposed framework generates process mining models from patients electronic records and generates a population mimicking patients pathways and LoS of the initial population. The resulting log is described using a set of metrics that can be used to fine tune the parameters that define the process model and population generation. The output population is automatically formatted to be ready to use in a generic DES. This simulation model accurately replays clinical pathways with realistic LoS (average obtained of 8.5 days, sim- ilar to the initial population) and allows modelers to account for phenomena that cannot be detected when using only process mining, such as the crowding of medical units.

This framework has several limitations. Although accurate when considering units, the LoS generation module cannot be used to model more precise transitions between units or waiting time, mainly because of the low number of patients that visited. We believe that considering only multimorbid patients may bias this dataset by over-or under-representing some pathways. Further investigations should be performed to remedy this problem and increase the possibilities of the framework. The generation module only allows for the generation of pathways and LoS. Other important features, such as age or diagnosis, are not considered, making it difficult to state the credibility of the pathway.

Improvements are also made to the simulation model. The admissions rates in the hospital were estimated using multimorbid patients' records and were corrected to account for non-multimorbid patients. In addition, the macroscopic layout of the simulation model and the fact that it only accounts for bed resources make it hard to use for microcosting analysis that is usually carried out with DES. The consumption of medical resources or patient admission constraints related to staff obligations (housekeeping or patient accompani- However, the model's ability to accurately replay pathways and to queue patients makes it a reliable tool for strategic decision making, for instance, to estimate the effects of a bed shortage in a unit or the change of care pathway for a particular type of patient.

In this chapter, we presented the conception of a Process-mining based framework to use representative pathway and length of stay information while avoiding to use personal data. Petri Net and DFG graphs were generated using PM4PY library and served to generate a population to feed a simulation implemented using AnyLogic®. This framework was tested using historical data of patients with multimorbidity hospitalized in the University Hospital of Saint-Étienne. The pathways and duration generated were thoroughly compared to the input data.

The implementation, and especially the fine tuning of pathway filters, still require technical expertise and direct access to the data. Further investigation is required to automate of this part of the process. Length of stay generation related to pathway information (direct successor) was attempted but the lack of transfer instances between some pairs of units made the generated distribution statistically non significant.

Motivation

To use the processed patients data, we need a simulation model accurately mimicking the organization of health care centers. The development of a macroscopic model of the hospital able to replay the physical pathway of patient is needed to assess the subsidiary effects of an organizational change. Such a model can serve as a base to study the hospital's patient flow under certain circumstances, for instance the temporary closure of a unit.

The massive arrival of COVID-19 patients to health care centers forced management to organize a parallel branch of the hospital dedicated for their care. This implementation of a "secondary" separated pathway is very similar to the creation of a multimorbid unit. Indeed both solutions require a redeployment of existing resources.

Summary

This chapter presents the development of a generic simulation model, able to replay the populations generated using process mining. The theoretical modeling of the healthcare center is presented. We implemented an early version of the process mining module presented in Chapter 2 to populate the part of the hospital not dedicated to COVID-19 patients. During the simulation, COVID epidemics is answered by the affectation of ICU beds from the usual care to the COVID-dedicated unit and the canceling of elective patients. This model is intended as a decision support tool for the evaluation of COVID-19 response scenarios.

COVID-19 Intensive Care Unit Optimization Using a Two-

Step Process Mining and Discrete-Event Simulation Approach Abstract-The sudden admission of many patients with similar needs caused by the COVID-19 (SARS-CoV-2) pandemic forced health care centers to temporarily transform units to respond to the crisis. This process greatly impacted the daily activities of the hospitals. In this paper, we propose a two-step approach based on process mining and discrete-event simulation for sizing a recovery unit dedicated to COVID-19 patients inside a hospital. A decision aid application is proposed to help hospital managers make crucial decisions, such as hospitalization cancellation and resource sizing, taking into account all units of the hospital. Three sources of patients are considered: (i) planned admissions, (ii) emergent admissions representing day-to-day activities, and (iii) COVID-19 admissions. Hospitalization pathways have been modeled using process mining based on synthetic medico-administrative data, and a generic model of bed transfers between units is proposed as a basis to evaluate the impact of those moves using discrete-event simulation. A practical case study in collaboration with a local hospital is presented to assess the robustness of the approach.

Note to Practitioners: Abstract-In this paper we develop and test a new decision-aid tool dedicated to bed management, taking into account exceptional hospitalization pathways such as COVID-19 patients. The tool enables the creation of a dedicated COVID-19 intensive care unit with specific management rules that are optimized by considering the characteristics of the pandemic. Health practitioners can automatically use medicoadministrative data extracted from the information system of the hospital to feed the model. Two execution modes are proposed: (i) optimization of the staffed beds assignments through a design of experiment and (ii) simulation of user-defined scenarios. A practical case study in collaboration with a local hospital is presented. The results show that for the majority of cases, our model found that the optimal strategy to minimize the number of transfers and the number of cancellations while maximizing the number of COVID-19 patients taken into care was to transfer beds to the COVID-19 ICU in batches of 12 and to cancel appointed patients using ICU when the department hit a 90% occupation rate. B ED management is a crucial matter in health care centers, and the occupancy ratio of medical units is an important indicator of performance for hospital managers. Keeping a small number of unoccupied beds is costly but necessary to address emergency admissions and patient surges. Resource sharing, through the creation of new medical units for example, makes sense to take into care patients with special requirements that do not fit in acute care and very specialized units, such as patients with multimorbidity.

Index Terms-Discrete

During spring and autumn 2020, SARS-CoV-2 (coronavirus disease 2019, abbreviated as COVID-19) caused a massive increase in recovery unit requirement, compelling hospital managers to open new beds and convert others. To avoid the contamination of patients and staff in health care centers, dedicated units were created and isolated from the rest of the facility. Simultaneously, to provide additional staff and beds for this new COVID-19 pathway, other medical units had to cancel their scheduled admissions. Operational research and industrial engineering techniques, such as discrete-event simulation (DES), are useful to optimize such new organizations and test response strategies and their impact on the whole facility via a systemic approach.

To apply a systemic approach for medical unit sizing optimization, we need to accurately model both COVID-19 and non-COVID-19 patients' hospitalization pathways. Process mining (PM) techniques can be used to analyze existing pathways based on available medico-administrative data in a health care center and to summarize the results into a graph. Such representation can be used to generate patient pathways for the simulation model.

The objective of this work consists of providing a new decision-aid tool for hospital managers and practitioners for the following decisions, taking into account the sudden surge of COVID-19 patients:

• How many and which planned patient hospitalizations should be canceled?

• How many resources from which medical units should be transferred to a dedicated COVID-19 unit? The scientific contribution of this article is threefold: 1) A new methodology using PM to learn from hospitalization pathway data and generate new patient matching with the population of interest is proposed. The proposed approach automatically generates patient replicas from cohorts using medico-administrative data available in the hospital. 2) A generic hospitalization meta-model allowing the simulation of sequences of care within a hospital is designed. 3) A medical unit sizing approach combining patients' hospitalization pathways generated using our PM approach (1) and our hospitalization meta-model ( 2) is developed. DES is used to run test scenarios to optimally size the COVID-19 unit. The remainder of this article is organized as follows. Section II presents insightful related works and provides the positioning of this paper against the literature. Section III describes the model of the hospital we built and the optimization problem at hand. Section IV presents the PM approach used to generate patients that will be used within DES. Section V describes the application of our method to the University Hospital of Saint-Étienne (France). Section VI concludes this paper and presents further research opportunities.

II. RELATED WORK

A. Healthcare Simulation and DES

As explained in [1], DES is a stochastic method of simulation in which individual entities pass through a network of queues and activities, and it is traditionally used at an operational level [2]. DES has been extensively covered in the literature, as [3] identifies DES as the second most commonly used simulation technique in health care systems and in a wide variety of scenarios. Indeed, in a systematic review on DES articles published between 1997 and 2017, [4] identifies 4 main areas of use for DES: health and care systems operation, disease progression modeling, screening modeling and health behavior modeling. However, 65% of the 211 studies focus on system operations, which enforces the idea that DES is mainly used for tactical-level decision making. This category groups papers covering problems such as staff or patient scheduling, capacity management and evaluation of operational changes. A good example of operation modeling can be found in [5], where 3 case studies illustrate the use of DES as a financial evaluation tool or a decision tool for capacity planning. Similarly, [6] studies the impact of integrating HIV patients and general consultation in one department on waiting times and sets into relief the causes of the observed increase in patient waiting times.

After emergency departments (ED), [4] notes that a high interest is taken in intensive care units (ICUs). Recently, [7] models an ICU department subject to planned and unplanned admissions. The DES model captures the patient flow in the ICU and assesses the capacities needed to handle patients in the different specialties of the medical unit. Departments like ICUs are deeply connected to other units, and the solution for improving the patient flow can be found in the patient's pathway management. [8] test the effect of different control policies on the performance of an ICU department. The assessed policies are applied to the ED, medical unit and intermediate care units. For instance, postponing planned surgeries is one of the policies evaluated and considered as a viable option for the short-term handling of the pandemic.

Global models are rarely designed but are becoming more common [4]. "Whole hospital" DES models tend to study the main areas of the hospital, namely, emergency, medicine and surgery, with different levels of detail. Based on previous experience and an extensive literature review, [9] establish a 4-level classification of models, from generic to specific: 1) Broad 'generic principle' model, e.g., a generalized theoretical queuing model; 2) Generic framework that can be developed into a toolkit; 3) Setting-specific generic model, where specificity is ensured by a change in the input data; 4) Setting-specific model, i.e., not necessarily transportable to another provider of the same service.

[10] aims to provide a "macro level overview of the hospital system", where the main variable studied is the occupancy of the emergency, medicine and surgery medical units. Two additional units are modeled: the extended emergency care unit (designed for monitoring patients from the ED before being discharged). The model is validated for daily operations and is intended to act as a decision tool for implementing organizational changes. However, this model does not take into account sudden increases in arrivals that can create crowding in the hospital. [11] design a model to evaluate such situations, called surges. The model focuses on the emergency, medicine and surgery units and is designed to test the impact of different policies on a set of variables describing the hospital's state. This last model has been designed in collaboration with health care centers but is sufficiently generic to be applied in other facilities, provided access to the required data. Since most hospitals share the same concerns of optimizing the use of available resources and managing patient pathways while running near capacity, generic models allow developers to save time by tailoring these pre-existing models to their needs. The ongoing COVID-19 pandemic has caused a massive influx of patients in hospitals, most requiring prolonged hospitalizations in ICUs, thereby creating new challenges for hospital managers. In this context, [12] identify several areas where simulation modeling could support decision makers during the pandemic. Eleven decisions that could be assisted by simulation tools have been identified and mapped in terms of their area of impact and appropriate techniques. For operational management in the ICU, [13] propose an adaptation of their model to handle COVID and non-COVID patients in the ICU. In particular, they study the impact of capacity and arrival rates on the throughput of service to improve patient flow management.

The present article focuses mainly on challenges 6 and 8 identified in [12], i.e., hospital capacity and resource management, and aims to optimize the organization of a regional hospital, using DES as a decision tool.

B. Process Mining for Trace Generation

PM is a relatively recent technique in business process management that has become popular in the past 15 years [14]. PM combines data science and business process techniques to analyze operational processes from event data. The literature often divides PM into 3 main areas: process discovery, conformance checking and process business enhancement [15]. Processes in health care are complex, and decisions taken during a patient's stay depend strongly on the current state of the health care system and the patient's characteristics. Thus, PM can be seen as an opportunity to clearly identify processes using a comprehensible technique and to understand the global behavior of patients and underlying patterns. Overall, process discovery is dominant in the health care literature [16], [17]. As noted in [16], there is an opportunity to use PM techniques with multiple medical units or facilities, as it is most frequently applied for discovering health care processes in a single unit. This approach may be especially useful to identify macroscopic pathways and recurrent relations between services. Simulation is the most commonly used technique for process enhancement according to [16]. In particular, DES has been successfully used with PM on several occasions. In [18], event logs are transformed into DES models. From the study of the clinical pathway, [19] implement the resulting net of the PM approach in a hybrid agent-based and DES simulation model. The net is translated into an agent state-chart, and a simple DES model is then used for medical decisions in each state.

In this paper, we use PM in an original manner: we intend to learn from historical hospitalization pathways data using a discovered graph to generate new patients matching the population of interest, taking into account macroscopic pathways and relations among departments.

III. PROBLEM SETTING

A health care center is a complex system where patients, health practitioners, and administrative personnel interact frequently and use a great variety of supplies, such as medications, equipment, and numeric devices to access patients' electronic files, and other resources. Our objective is to provide practitioners a decision-aid tool to assess the effect of organizational changes on bed requirements in the hospital. Thus, we focus on a macroscopic model of the care processes of the hospital.

A. Hospital Description

Since we focus on macroscopic care processes, we disregard the treatment of patients themselves, such as the administration of drugs, the nurse care episodes, and appointments with doctors. Thus, the hospital is a set of entities (i.e., medical units) that deliver care to patients.

A medical unit µ is a medical entity dedicated to the care of patients with certain conditions. Each unit µ is characterized by its identifier in the database id, its capacity c measured as the number of beds available, its type t regarding the present problem (see section III-D2), and the length of stay distribution D in this unit. We formally define this agent as follows: µ = (id, c, t, D). Finally, H is the set of all medical units of the hospital, H = {µ i |i ∈ 1..N }, where N ∈ N is the number of units in the hospital.

B. Patient Description

A patient is an entity receiving care in the studied hospital, defined by the ordered sequence of stays in individual units.

Let π = {µ ij |i ∈ [[1 ; N ]], j ∈ [[1 ;
J]]} be the pathway of a patient with J ∈ N stays, where i is the unit index and j is the relative position of the stay in unit µ i in pathway π. The population of all patients is denoted as Π and is further divided into two subsets, Π em and Π p , corresponding to (i) patients admitted to the hospital after an ED visit and (ii) patients directly admitted into medical units. This distinction is made to take into account the difference in management of care between an emergency admission, which cannot be anticipated by the hospital, and planned stays, which are known before the admission of the patient.

C. Hospitalization Pathway

Figure 1 describes our generic hospitalization pathway using a Petri net. Planned admissions are modeled using source transition t p , while emergent admissions are modeled using source transition t e . Transition t ED models the stay of the patient in the ED, while planned patients transit directly to p 2 , which is connected to all possible medical units µ. At this point, depending on the trace of the patient, the entity loops through a sequence of medical units (transitions t 1 , . . . , t N ℓ ) until the discharge (transition t d ). In the present study, we aim to assess the effect of the admission of COVID-19 patients in the ICU on the care routine in the hospital. In particular, we are interested in showing how the creation of a hospitalization pathway dedicated to patients with COVID-19 has forced the hospital to reduce its usual activity.

The COVID-19 patient pathway is composed of a stay in the ICU (named COVID-19 ICU) and a follow-up stay in a medical unit bed. We focus on the ICU resource management and suppose that the resources of such unit come from the medical unit, emptied by the cancellation of nonsevere hospitalization. The length of stays for the ICU and followup stays are roughly evaluated by health professionals in the hospital and modeled using normal distributions.

In this paper, we vary the rate at which patients arrive and the timespan during which they are admitted to the hospital to test how these characteristics affect the system and our response strategy.

2) Optimization Problem: When confronted with the influx of COVID-19 patients, the hospital was forced to (i) reduce its day-to-day activities to treat those new patients and (ii) increase the capacity of ICU by opening beds and transferring resources from other units. With these considerations in mind, we can assimilate the present situation to a multiobjective optimization problem with the following objectives: O 1 Maximize the number of COVID-19 patient admissions. O 2 Minimize the impact on day-to-day activities by closing as few beds as possible in medical units. O 3 Minimize the number of time windows dedicated to bed transfer that the hospital has to set up. Objectives O 1 and O 2 ensure that the COVID situation is handled while reducing its impact on the day-to-day activities of the hospital. Objective O 3 reflects the fact that transfers are costly and arduous to implement, as they involve the administrative and medical staff, as well as the supply chain of the hospital to provide the additional furniture.

We identified the following levers of action for practitioners to reach this goal:

1) Cancel planned patients at their arrival at the hospital.

2) Transfer beds from other units and convert them into ICU beds. For this purpose, we divide medical units into three distinct types:

T 1 : units that cannot be affected by COVID-19 episodes. T 2 : units in which planned patients can be canceled to make staff available. T 3 : units in which beds can be converted to COVID-19 care. We defined two possibilities to cancel elective patients and ease the demand on ICU beds, as detailed in Section III-E2.

E. COVID-19 ICU management rules

1) Resource reassignment: COVID-19-assigned ICU beds are taken from 4 distinct sources: the ICU and the 3 different surgery units.

The number of beds transferred from each medical unit is determined during the simulation using dynamic assignment, depending on the number of available beds in each medical unit. The transfer mechanism is further described in Table I. We consider the transfer of beds and of the associated staff to be immediate. If the COVID service is full, the patients wait in a queue and are discharged if not admitted in the first 48 hours.

2) Planned hospitalization cancellation: The direct consequence of dedicating beds to the treatment of COVID-19 patients is a decrease in the number of beds available for the care of other patients. Indeed, we have seen during this pandemic that most hospitals had to delay planned hospitalizations to empty beds in short-stay medical units and make more staff available for COVID-19 ICUs.

We have modeled this situation by making it possible to cancel scheduled stays before their arrival to the medical unit. Two cancellation policies were considered (i) "By ICU": the patient is scheduled for a stay in the ICU and the ICU occupation rate exceeds a certain percentage, or (ii) "By Block": the patient is scheduled in a T 3 unit and the ICU occupation rate exceeds a certain percentage. These 2 solutions are tested in our experiment plan described in Section V-D.

F. Summary

In this section, we defined a generic hospitalization pathway using a formal Petri net model, which can be altered with one or more exceptional hospitalization pathway, such as the COVID-19 patient pathway.

IV. PATIENT HOSPITALIZATION PATHWAY GENERATION USING PROCESS MINING

The methods and tools proposed in this project are intended to be used with commonly available medico-administrative data in a hospital. This section describes the method we propose to automatically turn a medico-administrative database into data that can be used to populate our model.

To use process mining techniques, we first need to format the data information of all patients as event logs. The following section details this formatting and the further transformation into a graph. We also propose the trace generation process used in the simulation.

A. Process Mining

Definition 1 (Event): An event e = (l, θ) is a couple of a label l, representing an activity, and the time of the event θ, called a timestamp. Let N ℓ ∈ N be the number of possible labels.

In this study, an event models a stay in a medical unit, the label giving the name of the unit and the timestamp corresponding to the date of admission to the unit.

Definition 2 (Trace): A trace t =< e 1 , e 2 , . . . , e nt >, n t ∈ N is a chronological sequence of events linked by a unique identifier.

A trace represents the inpatient's sequence of stays in individual units that occurred between his or her admission and discharge.

Definition 3: An event log L = {t 1 , . . . , t |L| } is the set of all traces corresponding to the input data of the study, i.e., all included hospital stays.

We use the fuzzy miner algorithm from [20] to build a graphical representation of the process from the data stored in L. This graphical representation can be described as an oriented graph G = (V, C), with V being the set of nodes in the graph and C being the set of oriented arcs. In addition, we impose that (i) G should have only one source node and only one sink node, (ii) G should be acyclic, and (iii) V contains as many nodes as event labels from L.

Definition 4 (Frequency): The frequency f (n 1 , n 2 ) associated with an arc is the number of observations in L of transitions between medical unit µ 1 and µ 2 represented by nodes n 1 and n 2 .

Definition 5 (Hospitalization process map): A hospitalization process map H = (V, C, P ) is an oriented acyclic graph with

• V the set of nodes, |V | = N ℓ + 2,
• C the set of oriented arcs, • P : V × V → Z the function returning the probability to transit from one node to another,

P (n 1 , n 2 ) = f (n 1 , n 2 )/ k∈|V |\{n1} f (n 1 , k).
A node of set V corresponds to a medical unit of the studied hospital, the label of our event. In addition, there are two particular nodes: the source node n source , which is the fictitious origin point of all traces, and the sink node n sink , which is common to all discharged patients. Thus, we have

|V | = N ℓ + 2.
An oriented arc (n 1 , n 2 ) ∈ V × V represents possible transfers of patients between medical units n 1 and n 2 . From the data stored in log L, we are able to calculate the frequency f (n 1 , n 2 ) of each transfer and the probability for a patient in medical unit n 1 ∈ V \ {n sink } to be transferred to medical unit n 2 ∈ V \ {n source }.

Example 1 (Event log and associated hospitalization process map): Figure 2 is an example of the conversion of a simple event log into a hospitalization process map. This example uses an event log describing the cases of 2 patients in a process with 3 activities.

B. Trace Generation

Our hospitalization process map from Definition 5 obtained using our modified PM algorithm is used during the simulation to generate patients similar to the modeled cohort (e.g., scheduled, planned, or COVID-19 patient hospitalization pathways). We use an automatic procedure detailed in Algorithm 1 to determine the patient's pathway σ = (n 1 , n 2 , ..., n sink ) using a hospitalization process map as input.

Algorithm 1 Trace generation σ ← {n 1 , n 2 , ..., n i+1 } {Assign to σ} 10: end while {Repeat until discharge of patient} 11: Assign σ to the patient's trace π

In Algorithm 1, we build an empty trace σ for the generated patient. For the first step of the stay, i.e., the admission of the patient to a medical unit, we build an ensemble T 0 that contains all medical units directly following n source in H and their associated probabilities (see line 2). To complete the trace, we recursively build the set of candidate stays T i using the last element (stay in a medical unit) of the trace σ and select a new stay following the transition probabilities. The algorithm stops when the selected candidate element corresponds to the sink, i.e., when the patient is discharged.

C. Summary

In this section, we provided an automatic procedure to generate virtual patients similar to a cohort defined through an event log. For a real case application, the following procedure could be applied by practitioners for all cohorts of patients of interest:

1) Extract medico-administrative data following inclusion criteria defined by health practitioners (e.g., all adults admitted to the hospital through the ED). Necessary variables are listed in Appendix A. 2) Execute the modified process mining algorithm to obtain a hospitalization process map. 3) Execute Algorithm 1 to generate a virtual patient similar to the set of patients of the cohort of interest. In the following section, we define the generic hospitalization model used for discrete-event simulation.

V. CASE STUDY: OPTIMIZATION OF THE COVID-19 ICU OF THE SAINT-ÉTIENNE HOSPITAL A. Background

The Saint-Étienne hospital (abbreviated as CHUSE in the following) is a university hospital of 60 medical units at the head of a large health care network named the Loire Hospital Group (GHT, Groupement Hospitalier de Territoire in French). CHUSE has 1,802 beds in total. In 2020, CHUSE admitted 93,905 patients for hospitalization and 78,400 patients were seen in its Emergency Department.

Regarding COVID-19 activity, CHUSE hosted 2,161 COVID-19-related stays, including 285 stays in ICUs. At the peak of the crisis in November 2020, 365 beds (resp., 46 beds) were dedicated to COVID-19 patient in medical units (resp., in ICUs).

B. Patient Data

The patient data we used for the present study were generated specifically for the purpose of this study by mimicking adult patient pathways. The data comprise a case mix of rather simple pathways for each medical unit (direct admissions with no transfers or emergency admissions with a transfer to the unit) and complex stays (initial stay in a unit, transfer to surgery and ICU, or random transfers between units). The data use the format of the medico-administrative hospitalization database (PMSI, Programme de Médicalisation des Systèmes d'Information in French), to be easily applicable for other studies. We generated the hospitalization process map from this synthetic database and used it to populate our model with "regular patients".

Medico-administrative databases hold rich and useful information about health care pathways at an individual level: visited units, received medical exams and treatments, length of stay in each unit etc. In addition, historical data are readily available for longitudinal analysis of care trajectories.

Accessing this data for all patient and linking it to the pathway is technically complex. To address this situation, we hypothesized that the pathway of each patient was representative of his or her personal situation.

Thus, the sequence of individual stays in the units and their stay information is all the information required to model the patients admitted. We generated the following elements for each synthetic inpatient:

• anonymous patient identifier, • medical unit sequence to which the patient has been admitted,

• date of admission in each medical unit,

• length of stay (LoS) at each medical unit of the sequence,

• admission modalities and origin of the patient. Only adult patients were modeled: outpatients (e.g., patients coming to the hospital for chemotherapy or dialysis) and ambulatory visits that do not consume staffed beds were not considered. However, these patients were taken into account if they were transferred to another medical unit.

1) Medical Units Data: Our partners at the hospital provided us with a detailed list of the medical units. Ideally, we would use the exact staff and setting of all those units; however, those figures are constantly evolving. The capacities c were defined using the last reliable number of beds opened in each unit at the beginning of the project. This was during the last consolidation of numbers in December 2018. The length of stay distribution D and types t of each medical unit were defined by our medical experts.

C. Model implementation

The process mining approach was implemented using Disco® software, while the Petri net hospitalization models were implemented using AnyLogic® simulation software.

Patients and medical units are implemented as agents. Each step of the pathway is implemented using a delay block for which the delay time is determined by a triangular distribution. Parameters are calculated from the length of stay distribution for each unit. We keep track of the occupancy of each unit with the corresponding variable when patients enter the process block. If a medical unit is full, patients wait in a queue. In the case of a transfer from one medical unit to another, the patient remains in his or her current unit until a bed is available in the destination medical unit. [T 2 ] are noted on the simulation model using boolean parameters, while [T 3 ] are directly identified and used in the bed transfer functions.

Urgent patients are all admitted through the ED and are represented by a delay block with a delay time determined by a triangular distribution between 0 and 6 hours with a mean of 3 hours. These patients are redirected to the main part of the hospital to complete their stay.

Elective patients are directly routed toward the hospital's medical unit after the trace's assignment.

For both categories of patients, we estimated the mean number of arrivals for each day of the week with hospital professionals. Those figures are uploaded to the model's database in the form of two schedule objects that are used to set the arrival rates for the two sources of patients.

D. Experimental design

The total duration of the simulation is 365 days. The hospital is initially empty, and the first 90 days are used as a warm-up period, at the end of which we start to measure hospital status indicators, such as the mean occupancy rate in the medical units and indicators about the pandemic: length of stay in the ICU, number of non-served patients, etc.

We considered two sets of variables: variables describing the COVID-19 situation and variables modeling the response of the health care center. The first set is composed of the COVID-19 arrival rate and time frame, while the second groups the number of beds transferred from the ICU and the number of beds transferred from surgery medical units.

The beds transferred from ICU and surgery units represent the response of the hospital to the situation defined by COVID arrivals.

The transfer of beds from traditional units to the COVID sections is done dynamically. Once a day, a function checks the incoming and outgoing flux of COVID patients and determines whether the COVID service needs more beds or if some beds can be returned to their original medical units. The total number of beds that can be transferred from each medical unit is defined upstream, and the number of beds effectively transferred is re-evaluated to respect these figures. Beds are seized for the COVID ICU in batches, the quantity of which is also user-defined. Available beds are dispatched immediately; the rest are transferred when the patients occupying them are discharged. When the epidemic declines and the COVID-19 ICU empties, available beds are freed for their original units. Sufficient beds are kept in the COVID-19 ICU so that the occupation rate is maintained at 90 %. The epidemic state is calculated by comparing the number of arrivals and the number of discharges. The different conditions for deciding on the addition or removal of ICU beds are detailed in Table I.

̸ = 0 ⩾ 95 % Add beds -1 ⩽ . ⩽ 1 ̸ = 0 = 100% Add beds ̸ = 0 ⩽ 75 % Remove beds ⩽ -2 ̸ = 0 =100% Add beds ̸ = 0 ⩽ 80%
Remove beds a Daily evolution is calculated as the difference between the number of arrivals and the number of discharges over 24 hours.

The patient arrival rate is constant over a defined period, as explained in III-D, and is 2, 4 or 6 patients per day. These arrivals occur for the entirety of the simulation: 15, 30 or 45 days starting from day 210.

For performance evaluation, we measured the mean global occupancy in the hospital during the period of interest, as well as the mean occupation in the medical unit. The efficiency of the COVID-19 response scenario is measured through the following figures that corresponds to objectives described in III-D: the number of COVID-19 patients that could not be admitted, the number of elective hospitalizations that were canceled and the mean occupation ratio in the hospital. The global performance of the scenario was assessed by normalizing and summing these 3 figures and dividing by 3 to obtain a score between 0 and one (the aim is to minimize this value).

Our input variables are (1) the arrival rate and (2) time frame of the COVID-19 situation. The response variables are (i) the batch size of beds transferred to COVID-19 ICU units, (ii) the cancellation method for elective patients and (iii) the cut-off point of the ICU occupation rate beyond which we start canceling scheduled patients. The combination of these variables results in 480 experiments, and we replicated each experiment 100 times.

E. Results of the Experiments 1) Generation of Patients Using Process Mining:

The simplified process map that was generated using the synthetic data for emergency admissions is shown in Figure 3a, and that for elective admissions is shown in Figure 3b.

The emergency patient process map was generated using only the 90% most frequent cases to filter the most unusual cases. We display the 30% medical units and the 50% most frequent paths between medical units for readability purposes. For the same reasons, the elective process map displays only the 30% most frequent medical units.

The elective patient process map is dominated by a direct path: patients are admitted into their scheduled unit and are discharged. Cardiac surgery is also a well represented unit. This result was expected as our synthetic population was composed of mainly one-unit stays and elective surgery pathways. The urgent patient population requires a wider variety of medical units. Our synthetic population was targeted to consume ICU and surgery resources in priority, while other pathways were similar to elective patient pathways with an additional step for post-emergency treatment.

Full maps were exported in XML format, and we applied algorithm 1 to generate 100 000 patients from each map.

2) Result of the Sizing Experiments: Via simulation, we ran all the scenarios induced by the variations of parameters described in Section V-D. The exhaustive results of these experiments are displayed in Supplementary tables. Notably, the number of COVID-19 patients rejected is higher than the number of elective patients that must be canceled. This can be explained by the relatively low occupancy rate of the ICU (from 60 to 70 %), which allows us to transfer beds without affecting the unit's ability to host elective patients.

Table II displays the response scenario that is best suited to each situation when considering the trade-off between the number of elective stays canceled and the number of rejected COVID-19 patients. When taking a closer look at the results, we can observe that the 'By Block' policy is almost always less effective than the 'By ICU' policy, mainly because it logically results in greater cancellation of elective stays. The only situation where the results are comparable between the two policies is when COVID-19 occurs in the narrowest time frame window: between days 210 and 225, where the duration of the epidemic is short enough to be handled easily.

To conclude, the simulation model is intended to be used by practitioners as a decision-aid tool. Depending on the characteristics of the pandemic (COVID-19 arrival rate, time frame, etc.) and of the hospital, the tool provides meaningful insights to help practitioners make the best decisions regarding cancellation policy and the quantity of resources to transfer to COVID-19 units.

VI. CONCLUSIONS AND PERSPECTIVES

The present research describes the development of a macroscopic simulation model of a health care center for the study of patient pathways. Automated generation of pathways from a graph proved to be a viable option for the simulation studies and presents some advantages. First, using PM to generate an agent's pathway in the simulation increases the variety of pathways included in the model, as simulation studies often use a small number of pathways, usually the most prevalent pathways, while our approach allows modelers to take into account less frequent interactions. In addition, using PM could simplify the development of such models. When implementing the simulation in a new facility, applying a standardized PM approach provide the fuel the model with a formatted and comparable input. The impact of the different parameters used for PM on the simulation must be investigated further to determine the best settings of the simulation.

The application of the model when considering organizational changes is also interesting. For the example of the management of the pandemic, this model could be used to help the hospital decide whether opening a new medical unit in the next weeks would be beneficial. For instance, if an admission rate of 4 patients/day is observed for a few days, a close study using the model provides the modeler insight into how the resource allocation to the COVID-19 ICU will help absorb the sudden admission of COVID-19 patients.

When using the present model, one should note that the results described in Section V-E are not to be seen as gold standards when confronted with this type of situation. Each set up should be carefully studied and implemented in the simulation before being applied. We recommend using this model by first implementing actual predictions about the epidemic before proceeding in a two-step process: (i) create an experiment with a possible response and (ii) adjust the implemented response according to the results and to the available resources.

The present article aims to demonstrate the potential of the model when representing organizational problems. The experiments were designed to do so, and some of the simplifying assumptions are quite strong and not fully representative of the situation in the hospital.

For instance, we did not take into account some patients and medical units based on use, and we decided for the graph generation to take into account patients such that the result would account for 90% of the pathway variability. This assumption, made to simplify the graph, may have hidden some marginal interactions between medical units that could affect the hospital. The two cancellation policies are absolute scenarios, and decision makers in the field will want to modify them.

In addition, we considered the beds in medical units to be equivalent, so that a from a surgery medical unit could be transformed into an ICU bed. However, for many reasons, such as staff ability and availability of specific resources, this assumption is not operationally realistic. For use in real conditions, facility-specific adjustments might be necessary. Specifically, we recommend using an ability matrix to define the staff that needs to be re-assigned to the new unit to calculate the additional beds that must be closed.

APPENDIX A NECESSARY VARIABLES FOR TRACE GENERATION USING PROCESS MINING

To produce the process tree required for our patient generation process, we generated a population giving the following elements for each individual point:

• anonymous patient identifier, • medical unit sequence to which the patient has been admitted,

• date of admission in each medical unit. Jules Le Lay

Conclusion of the chapter

In this chapter, we described the use of a DES model as a decision aid tool for the management of the SARS-COV-2 epidemic. The health care center and hospitalization pathway are theoretically formalized. The simulated epidemic is dynamically answered through the canceling of appointed patients consuming ICU or surgery resource, and by transferring beds toward the COVID-19 unit. The main limitation of this study lies in the fact that we don't model the harm done by delaying the care of elective patients. Differentiate patients whose rescheduling only affects the hospital planning and the ones who are at risk to see their condition worsen is of great importance for the hospital. The organizational problematic to make up for canceled appointments if also not dealt with.

Chapter 4

Prediction of Hospital Readmission of Multimorbid Patients Using Machine Learning Models

Motivation

Polyvalent Unit (PU) is intended to care for multimorbid patients at risk of complications during their stay. The patients with multimorbidity frequenting the CHUSE are not precisely known. In particular, it is important to identify patients with multimorbidity and understand their profile to know on which patients we might act. The use of a quantitative multimorbidity index to single out patients with multimorbidity is widespread in short stay unit. However, many criticism have been emitted toward multimorbidity indexes and we need to verify the predictive power of multimorbidity indexes on readmission to the hospital, that is associated with lower quality of care.

Summary

The article presents the study of multimorbid patients hospitalized in the CHUSE during the year 2017. Detailed pathways and the diagnosis made at each step were made avail-

Introduction

The management for care of multimorbid patients, in hospitals is a rising concern among the scientific community. Multimorbid patients tend to have more complex needs and require coordinated care from several providers [1]. Multimorbidity, defined as the "co-occurrence of multiple chronic or acute diseases and medical conditions within one person" [2], is highly prevalent in Europe. Based on the Survey on Health, Aging and Retirement in Europe (SHARE) Nielsen et al. [3] Currently, there are multiple research projects to improve the overall quality of care both inside and outside of healthcare centers, establishing dedicated care pathways for multimorbid patients [4][5][6].

Barnett et al. [7] reported an association between age, sex, deprivation and multimorbidity based on a list of 40 medical conditions. This list was built using policy recommendations and important chronic conditions identified in [8]. However counting conditions can be quite limiting, and are a controversial measure of multimorbidity for these studies. [9] highlighted the importance of using a standard measure of multimorbidity to analyze and compare the results of studies in which different scores have been built to describe multimorbidity.

The most common measure of multimorbidity is the Charlson comorbidity index score, originally introduced in 1987 [10] and first updated in 1994 [11] and numerous times since to be applied with administrative databases [12,13] or to predict other outcomes [14]. In a recent systematic review, [15] explored the different multimorbidity measures developed outside of the counts of conditions. The hospital frailty risk score (HFRS), which uses weighted counts [16] or the Calderon-Larrañaga score, which counts clusters of diagnoses groups [17] are other ways to build a more efficient index.

Healthcare services can be monitored through several performance indicators. In this study we are interested in predicting the patients' readmission and length of stay (LoS)

indicators. Rehospitalization (or readmission) can be defined as "an admission to a hospital within a certain time frame (which can be 7, 15, 30,60, 90 days or even as long as a year) following an original (index) admission and discharge" [18]. According to [19],

monitoring readmission and predicting the readmission of patients during their initial hospitalization is essential for two main reasons. First, authorities use this metric to evaluate and report the efficiency of healthcare centers, where a higher readmission count is being associated with lower efficiency. Second, providing a clinically relevant readmission risk early in a hospitalization stay allows hospital workers to trigger preventive action and avoid subsequent admission, improving the consumption of medical supplies and the cost-effectiveness of the patients' care. This metric, from a June 22, 2022 3/23 cost-effectiveness perspective, is even more crucial for patients with additional chronic conditions as comorbidities that are associated with higher costs of care [20]. The LoS in hospitals is a key quality of care metric for patients and care providers, it relates to the occupancy rate of the service and is used to improve the care given.

Machine learning is a powerful set of data analysis techniques that identify and use patterns in data to realize predictions without explicitly specifying the procedure. Its use in healthcare over the past years has been extensive for the prediction of outcomes, as shown in [21]. A scoping review recently covered the use of machine learning algorithms for the prediction of hospital readmission [22]. According to this review there is a relatively high interest in tree-based methods (decision trees, random forest and boosted tree methods), although other techniques as neural networks and regularized logistic regression are also used. [23] predicted the LoS of stroke patients using J48 and a Bayesian network.

Materials and methods

Data description

The data used in the present study were extracted from the anonymized patients' electronic records of the hospital of Saint-Étienne (CHUSE) under 'Commission Nationale Informatique et Libertés' authorization number 919300. CHUSE is a university hospital at the heart of the regional healthcare network: -Groupement Hospitalier de la Loire. In 2019, CHUSE had more than one hundred thousand stays in one of the thousand beds in the medicine, surgery and obstetrics areas.

We focused on adult patients above 60 years old hospitalized in the CHUSE and discharged in 2017 with diagnoses in 2 different chapters of the ICD-10 classification (excluding chapters about pediatric or pregnancy-related conditions). We excluded patients following a highly controlled care pathway, such as dialysis or outpatient surgery, except when this outpatient care led to an extended hospital stay.

For each stay we analyzed variables related to the general information of the patient as well as information concerning their care pathway. These variables are listed below:

• anonymous patient identifier June 22, 2022 4/23

• age and sex of the patient

• service sequence

• date of admission in each unit

• length of stay (LoS) at each service of the sequence

• total length of stay (sum of the length of stays in individual services)

• admission modality and origin of the patient

• discharge modality and destination of the patient

• the list of diagnoses made at each service From this variables we were able to predict the same-hospital readmission within 30 and 365 days by linking the different stays of a unique patient using his/her anonymous identifier.

As previously mentioned there are numerous methods in the literature to describe multimorbidity. To compare the different approaches, we decided in this study capture multimorbidity by using all diagnoses information and using two multimorbidity scores:

the hospital frailty risk score [16] and the Calderon-Larrañaga score.

The hospital frailty risk score [16] was developed to identify older patients presenting frailty diagnoses. A higher risk score is associated with a higher risk of adverse outcomes and a higher use of medical resources. Weights were calculated using logistic regression targetting of the identified frail population and validated for the prediction of adverse outcomes.

Calderon-Larrañaga [17] explored a different approach. To build this multimorbidity score, [17] gathered a panel of medical experts to group diagnoses into categories according to "clinical criteria and relevance (pathophysiological pathway, treatment, prognosis, and prevalence)" and defined the score of a patient as the number of categories where the patients had at least one diagnosis.

In order to capture the multimorbidity by using all diagnosis information, we built a database containing the exhaustive list of diagnoses made to the patient during their stay (3-digit ICD-10 codes) to compare the performance obtained by creating thematic groups of diagnoses.

June 22, 2022 5/23

The Charlson comorbidity index score was computed for comparative purposes using the ICD-10 translation of the index established by [12]; however, this core was not used for predictive procedures.

Statistical analysis methodology

The use of different measures to synthesize multimorbidity in the literature and the tendency to use exhaustive data in process mining raises the question of the relevance of using aggregated scores in advanced statistical procedures. For this reason we compare the ability of several machine learning algorithms to predict readmission within 30 and 365 days and length of stay in the elderly multimorbid patient population using all diagnosis information and two multimorbidity scores: the hospital frailty risk score (HFRS) [16] and the Calderon-Larrañaga score [17]. In our study, we also use different categories of the Calderon-Larrañaga's and HFRS, to perform our statistical analysis and assess the loss of predictive power when aggregating the scores. For the Calderon-Larrañaga score we use the categories, and for HFRS we use groups of diagnoses having an equal weight in the score's calculation. We will refer to these nonaggregated versions of the scores as Calderon-Larrañaga portfolio and HFRS portfolio in the remainder of the paper. In the portfolio versions of the measures the category information was coded as a binary value equal to 1 if the category/diagnosis was active for the patient. We chose HFRS because it was designed to predict frailty and was validated for the prediction of 30-day readmission [16]. As mentioned before we built a database containing an exhaustive list of diagnoses made during the patients' stay (3-digit ICD-10 codes).

It is important to note that for the readmission analysis we excluded patients who died during their initial hospitalization, as readmission is not applicable for deceased patients. This resulted in the exclusion of 780 patients from the readmission database.

However, we kept the data from these patients to predict length of stay. Palliative care or complex pathologies and care resulting in the death of the patient might be associated with longer hospital stays.

However, it is also important to note that for this study we only have access to the patients' data during their hospitalization, which means that we do not have access to June 22, 2022 6/23 their status after one year.

The machine learning methods used in this study for prediction of hospital readmission were tree classifier, a random forest classifier and a k-nearest neighbor classifier.

For the length of stay prediction we used a tree regressor and a random forest regressor. All experiments were performed using Python 3.7, pandas [24,25] and scikit-learn [26].

For all learning experiments, the data were split between training and testing samples, with the training sample representing 75% of the original dataset. We used a grid search with cross-validation to parametrize the learning algorithms. The parameters that were tested and optimized were the depth and number of leaves for the decision tree approach, the depth, number of leaves and number of estimators for the random forest approach and the algorithm, leaf size and number of neighbors for the k-nearest neighbors approach.

As previously mentioned, we used the patients' anonymous identifiers to identify the patients stays. In addition to diagnosis information, we used the age, sex, length of stay, ED admission information and number of steps in the pathway to predict the same-hospital readmission within 30 days and 365 after discharge of the initial stay from the inpatient database. Those variables and the residential zip code (except the length of stay), were used for the prediction of length of stay.

Of the 12 391 multimorbid stays registered in our database, we found that only 1 965 (15.86%) used hospital services during the 30 days following the discharge. To address with this imbalance between the two patient classes, we used resampling techniques on our dataset and used appropriate metrics to evaluate the results.

Resampling is a method that changes the composition of the dataset to allow training on a balanced dataset. There are techniques that delete samples from the majority class, others that generate samples of the minority class and some that combines the two. We used the imbalanced-learn Python package presented in [27]. The different methods were applied to train the learning algorithm on the balanced dataset, and we selected the best performing combination of resampling and learning algorithm.

For the prediction of hospital readmission within 30 days we used the same 3 classifier algorithms: decision tree, random forest and k-nearest neighbor.

June 22, 2022 7/23 Some metrics such as accuracy are not an appropriate target to assess the performance of algorithms when dealing with unbalanced datasets. Therefore, we decided to focus on the F1-score, a weighted average of precision and recall, and receiving operating characteristic area under the curve (ROC-AUC) which compares sensitivity and specificity. Thus, we will have a better understanding of the classifier's efficiency.

To evaluate regression algorithms, we used two well-known metrics: mean absolute error and mean squared error. The classifiers' performances were assessed using accuracy (percentage of correctly predicted instance) and F1-score for rehospitalization within 365 days and receiver operating characteristic area under curve and F1-score for hospitalization within 30 days, where patients were unevenly distributed between positive and negative classes.

Database

Table 1 presents the general variables used by the prediction algorithm. Additionally the algorithms use information on the diagnoses made to the patients. Five versions of this database were coded in order to compare the diagnoses information and aggregated multimorbidity scores:

1. using raw information on the diagnoses: one column per diagnoses with binary values;

2. using Calderon-Larrañaga score;

3. using Calderon-Larrañaga portfolio (as explained in section Statistical Analysis Methodology);

4. using the HFRS score; and 5. using the HFRS portfolio.

These features were used to train and test prediction models targeting (i) readmission within 365 days, (ii) 30 days after the end of the initial hospitalization and (iii) the total length of stay.

June 22, 2022 8/23

Results

Epidemiological description

As specified above, we include patients above 60 years old with diagnoses in 2 different categories of ICD-10 classification who were discharged in 2017 in our study. This database includes 12 391 hospital stays of 8 882 unique patients (4 541 male patients and 4 341 female patients). The distribution of ages for this population is shown in We observe that the mean age for both male and female patients is quite high (74.73

June 22, 2022 9/23 We calculated the age-adjusted Charlson comorbidity index score [11], the Calderon-Larrañaga score [17] and the HFRS [16] for each patient. Problems related to care-provider dependency 1 626 a E78: "Disorders of lipoprotein metabolism and other lipidemias", contains more specific subcategories, such as "E780: Pure hypercholesterolemia", "E781: Pure hyperglyceridemia".

Prediction of readmission within 365 days

The results obtained from the different combination of algorithm and multimorbidity evaluation are displayed in table 3. We display the mean observed value and the 95% confidence interval calculated using a bootstrap method. Although using all diagnostic information provides the best performance, the computational time is higher than the computational time of the aggregated multimorbidity scores.

A good time-performance trade-off are obtained by using Calderon-Larrañaga portfolio and random forest, as the decrease in accuracy and F1-score is small (-0.096 and -0.108), with a net improvement in the computation time. Using the HFRS portfolio and random forest is the second best option. These performances gains can be explained by the thematic groups of diagnoses created by the experts specifically for the scores, which must result in easier analysis for the algorithms. Both metrics indicate that the random-forest algorithm, used on the Calderon-Larrañaga and HFRS portfolios are viable alternatives for the prediction of hospital rehospitalization within 365 days.

Prediction of readmission within 30 days

As presented in section , we implemented resampling methods to train the algorithms on balanced training sets before testing it on unbalanced sets. We used oversampling The performance is higher when the algorithm considers all information related to the patients' diagnoses, as seen in table 4. For example the combination of random forest and random undersampling with all diagnoses information provides an ROC-AUC score of 0.625 with a 95% Confidence Interval (CI) of [0.602 -0.649] on the testing samples. Similar to the results for readmission within 365 days, using Calderon-Larrañaga portfolio is more efficient than using the aggregated score alone, with a ROC-AUC score of 0.594 [0.573 -0.620]. In addition, Calderon-Larrañaga portfolio gives significantly better results than the experiments using HFRS. The ROC curves and calibration curves are displayed in appendix 3. We note that the results are June 22, 2022 13/23 quite different for HFRS, and the ROC-AUC and F1-score are comparable for the HFRS score and HFRS portfolio.

(

Prediction of length of stay

Overall, the random forest algorithm performed better than the decision tree, with an improvement of 10 to 30 square days in mean squared error from the decision tree results. This improvement is the most significant when using all available diagnoses, (a difference of -0.802 for MAE and -30.462 MSE). These results can not be considered conclusive as the best mean absolute error is still above 5 days, which represents half of the mean length of stay in the database. The raw results are displayed in table 5.

Discussion

For readmission within 365 days, prediction using all diagnosis gives the best results.

The Calderon-Larrañaga and HFRS give comparable results. Calderon-Larrañaga portfolio gives significantly better results than the experiments using HFRS scores and portfolio for the prediction of readmission within 30 days, but is still outperformed by the random forest with a random undersampling on all diagnosis information. The different experiments show that the machine learning algorithms for the prediction of length of stay give at best a of 5.208 mean absolute error and 72.687 mean square error (random forest used with HFRS score).

The mean age of the studied patients is 76 years old. In addition, female multimorbid patients tend to be older than male patients. The mean age-adjusted June 22, 2022 14/23

Charlson morbidity score was 5.65, and the mean Calderon-Larrañaga score was 4.94.

The most prevalent diseases diagnosed in the patients were hypertension and type 2 diabetes mellitus.

In this study we included patients based on the number of diagnoses in different ICD-10 categories, before applying different scores (the Charlson comorbidity score, Calderon-Larrañaga score and HFRS).

Calderon-Larrañaga portfolio is a standardized and thorough tool, that accounts for most ICD-10 codes and chronic conditions, and we believe that it gives a quite accurate view on multimorbidity. Thus, it can be a good alternative to using the raw information on diagnoses for predicting readmission within 365 and 30 days.

The main limitations of this study are related to the lack of information on the vital status of patients after their hospitalization. This could represent a bias, as it is possible that patients died between the initial discharge and the end of the period of interest.

Thus, the absence of a hospital stay within 30 or 365 days after discharge may be caused by the death of the patient., In addition, we could not include socio-economical information in our study (availability of caregivers, socioeconomic status, and so on).

The prediction of medium-term readmission was efficient, with the best score achieved with random forest and all diagnoses information (accuracy of 0.826 [0.811 -0.840] and a F1-score of 0.812 [0.794 -0.829]). Using Calderon-Larrañaga portfolio resulted in a slight decrease in performance in both indicators (-0.096 and -0.108 respectively). We believe that the clusters of diseases used in Calderon-Larrañaga portfolio can be used as an efficient substitute for diagnoses information for predicting readmission within 365 days after initial discharge.

For the within 30 days all-cause readmission, we obtain at best an ROC-AUC score of 0.625, which is acceptable, although it is slightly lower than the recent results in the literature, as [22] reported a median AUC of 0.68 on the studies they identified. The accuracy on the unbalanced testing set is of 0.603. When using Calderon-Larrañaga portfolio, random forest and random undersampling, we obtained a mean ROC-AUC score of 0.594 and a mean accuracy of 0.556. The use of Calderon-Larrañaga portfolio can be a viable alternative for the prediction of the within 30 day readmission on a medico-administrative database.

Overall, the predictive power of our algorithms is quite low for the length of stay June 22, 2022 15/23 prediction.

A key component of multimorbidity according to Barnett et al. [7] that we could not grasp in this study is the socioeconomic aspect. Similar to [7], we can use geographical information as a proxy, but we do not have access in the present database to an evaluation of the socioeconomic status per area. In addition, we have access only to the residential zip-code of the patient, which can cover quite a large area and hide many disparities between patients. A favorable familial situation, with an available caregiver, is a key component of patients' care, and this information is not available to us. We believe that these two information of care for multimorbid patients would be a valuable addition to the methodology presented here.

Conclusion

In this study we described the general characteristics of the multimorbid population hospitalized in the CHUSE in 2017 and applied various machine learning techniques to predict key components of the hospitalization pathway: the length of stay and the rehospitalization within 30 and 365 days after initial discharge.

We built 5 versions of our database for each outcome, each taking into account various information on the diagnosis made. The raw diagnosis information was compared to 2 aggregation scores. First we used the HFRS, a score built to measure the frailty of patients based on diagnoses information. Second, we used the Calderon-Larrañaga score, a multimorbidity score based on groups of diagnoses.

Random forest classifiers provided better performance in predicting the within 30 and 365 day rehospitalization than the other models in this study. The second best performing combination (Calderon-Larrañaga's portfolio with random forrest) does not significantly outperforms HFRS on accuracy and F1-score. The F1-score (0.704 [0.685 -0.725]) obtained when using Calderon-Larrañaga portfolio shows that it can be used to predict one year all-cause readmission on this multimorbid population, and can be an effective substitute for the all-diagnosis approach with a small loss (10%).

We tested multiple resampling solutions to account for the imbalance in rehospitalization within the 30 day dataset. The most effective combination we found used the random forest classifier with random undersampling and all diagnoses, and it June 22, 2022 16/23 gave acceptable results, with an ROC-AUC score on the testing dataset of 0.625 [0.602 -0.649] slightly lower than the recent results in the literature. The results are mitigated for Calderon-Larrañaga score, HFRS score and HFRS portfolio (ROC-AUC between 0.561 and 0.571). Using Calderon-Larrañaga portfolio gives slightly better results (0.594 [0.573 -0.620]), which makes this method an easy-to-implement alternative from an exhaustive approach.

The use of a random forest regressor gave the best predictive results on LoS for the two metrics we used, MAE and MSE. Both the HFRS and the Calderon-Larrañaga score were outperformed by the use of all information on diagnosis. However, the HFRS performed better than the Calderon-Larrañaga score. The HFRS seems to be an acceptable alternative to the use of exhaustive information on diagnosis with a random forest algorithm for predicting length of stay.

For future research, we intend to apply prediction techniques on patient data to single out complicated pathways and combine this with the results obtained here in a discrete event simulation. Our goal will be to redirect those patients toward a modified integrated care pathway for multimorbid patients. It would also be of interest to combine the approach of this paper with prospective data at admission, which would provide additional valuable information, such as socioeconomic data and familial context, respecting the anonymity of patients. This could allow us to evaluate if the patient would be fit for the new multimorbid pathway and to track the decision process of the care team when deciding in which unit the patient must be routed.

Appendices

Conclusion of the chapter

This article set into relief the high age of patients with multimorbidity in CHUSE contrary to the observations found in the literature about presence of multimorbidity in younger population [25], [26], which might indicate that young persons with multimorbidity are less susceptible to be hospitalized. However, these observations stand on different definitions of multimorbidity and have been done on general population data. Further investigation is needed to conclude on this aspect. Most prevalent diagnosis indicated chronic conditions (hypertension, type 2 diabetes...). Overall decision-tree algorithms were dominant on other algorithms tested for predicting of all 3 outcomes. Using all available diagnosis gave optimal results in every case. However, deconstructing the 2 indexes to use the cluster of diagnosis they are using proved to be relevant. For all-cause readmission within 365 days, the algorithm using the clusters of the multimorbidity index performed almost as well as the algorithm using all diagnosis information.

Chapter 5

Sizing of a newly created medical unit for multimorbid care, a simulation-optimization approach

Motivation

The creation of new units, such as the PU, calls for extensive studies on the volume of patients expected to use it and the appropriate size. In reality, physicians in charge of the unit estimate the volume of patients and follow pre-established guidelines giving the unit's size. Economic studies are led afterward to validate these choices. Using statistical or ML methods on historical data we can estimate the volume of patients targeted for hospitalization in PU. The DES simulation could then help us determine the appropriate size for the unit. Moreover, it would allow to estimate the expected drop in attendance in other units. Using this additional information, optimization algorithms might be able to determine if this decrease could be sufficient to redirect beds toward the PU.

Chapter 5. Sizing of a newly created medical unit for multimorbid care, a simulation-optimization approach

Summary

This article presents the simulation-optimization experiment of the sizing of a PU for multimorbid patients. The simulation model developed in 4 have been improved and an optimization model have been developed for automatically size and compose the novel unit. Multimorbid patients set to receive care in the PU have been identified using a multimorbidity index (as used in reality) or using an advanced statistical method.

5.3 Sizing of a newly created medical unit for multimorbid care, a simulation-optimization approach

• Jules Le Lay, Vincent Augusto, Edgar Alfonso-Lizarazo, Malek Masmoudi, Baptiste Gramont, Xiaolan Xie, Bienvenu Bongue, and Thomas Célarier Prediction of Hospital Readmission of Multimorbid Patients Using Machine Learning Models, (submitted to the joint special issue of the Journal of Simulation and Health Systems on hybrid modelling and simulation).

Introduction

Multimorbidity is defined as the "co-occurrence of multiple chronic or acute diseases and medical conditions within one person" van den Akker, Buntinx, and Knottnerus (1996). This definition is derived from the term "comorbidity", a 50 year-old concept introduced by Feinstein (1970) as "any distinct additional entity that has existed or may occur during the course of a patient who has the index disease under study".

There is no single method to measure multimorbidity, and most studies on multimorbidity are based on counts of chronic conditions Huntley, Johnson, Purdy, Valderas, and Salisbury (2012).For example, [START_REF] Barnett | Epidemiology of multimorbidity and implications for health care, research, and medical education: a cross-sectional study[END_REF] counted diagnosed morbidities from a list of 40 conditions. This research has shifted towards more complex calculation modes, as highlighted in Stirland et al. (2020). These calculations include weighted counts in relation to a specific outcome, as recommended by Diederichs, Berger, and Bartels (2011), groups of diagnoses Calderón-Larrañaga et al. ( 2016) or the addition of biological test information Newman, Boudreau, Naydeck, Fried, and Harris (2008).

The care of multimorbid patients in hospitals is a growing concern for practitioners and needs to be addressed in a complex environment, as multiple factors can complicate the management of multidisciplinary pathways. From a clinical perspective, there has been a constant increase in hospital admissions through the Emergency Department (ED). Les établissements de santé -édition 2020 -Ministère des Solidarités et de la Santé (2021) reported a mean increase in emergency visits of 3.6% per year between 1996 and 2018, which resulted in the annual number of visits increasing from 10.1 million in 1996 to 21.1 million in 2018. This increase puts pressure on health care center staff, as unplanned stays are less manageable than elective stays from the perspectives of patients needs and the health care providers' constraints.

Another important parameter to take into account is the ageing of the population in the geographical region of Auvergne-Rhone-Alpes, where an increase in the elderly population is expected, especially in the Loire department. Between 2017 and 2050 Desgouttes and Gilbert (2017), based on demographic previsions by Institut National de la Statistique et des Études Économiques (INSEE) it is predicted that there will be an increase in population for every age group and a significant increase for the population over 65 years of age. According to [START_REF] Charpin | Perspectives démographique et financières de la dépendance[END_REF], also based on INSEE previsions, the number of dependent elderly persons in France is expected to double by 2070, causing an increase in associated healthcare expenditures and resource consumption.

The care of multimorbid patients is multidisciplinary by essence, and the careful management of care is required to simultaneously account for undesired interactions between treatments and coordinate multiple care pathways. Practitioners investigating new ways of managing multimorbidity in health care centres are urged to take action Rijken et al. (2017). A systematic review performed in 2018 identified 15 care management intervention studies targeting patients with 2 or more chronic conditions and high healthcare utilization Baker, Grant, and Gopalan (2018), and Rijken et al. (2018) found 112 integrated care practices in the Innovating Care for People with Multiple Chronic Conditions European (ICARE4EU) project. This study highlighted the growing number of nondisease-specific care practices. These practices put more emphasis on the patient's involvement, in their management of care and on the coordination between different disciplines.

Literature Review

Discrete Event Simulation (DES) is a modelling technique that belongs to the field of Operations Research (OR). It is particularly suited to model stochastic processes, using networks of queues and activities in distinct steps of the process. This technique has been widely used in healthcare for the past 20 years, with Salleh, Thokala, Brennan, Hughes, and Booth (2017) identifying no less than 37 reviews about DESs applied in healthcare. A simple search on the combination of terms "discrete event simulation" and "health care" results in more than four thousand hits, with more than 3300 published works in the last ten years. A thorough literature search performed in Vázquez-Serrano, Peimbert-García, and Cárdenas-Barrón (2021) confirmed this trend. Approximately one-third of the 231 journal papers they retrieved were published either in 2019, 2020 or between January and August 2021. This section presents a selection of articles from the literature on the use of DESs for healthcare modelling, and the possible combinations with optimization techniques.

DES in healthcare modelling

Developing digital twins of processes, i.e., "a mirror image of a physical process that is articulated alongside the process in question" Batty (2018), is of primary importance for organizations. It allows testing multiple changes in the existing process on a simplified model and measuring the effects they can have. As previously stated, defining the optimal use of existing resources in a healthcare system is extremely important, as process changes have effects on the health and lives of patients. Using a DES model as a digital twin is a viable solution to test new processes and modifications at a lower cost.

For instance, Ben-Tovim et al. ( 2016) developed a simulation model of an healthcare centre that models Flinders Medical Centre, a teaching hospital located in South Australia. A particular emphasis was placed on the ED, with a modelling of the screening process, emergency short stay and care units, and possible transfers to less detailed medicinal and surgical areas until patients eventually leave the hospital. This model is intended to serve as a decision-support tool for the hospital, and it tests alternative interventions for reducing congestion in the ED department. Similar considerations were found in Chavis, Cochran, Kocher, Washington, and Zayas-Cabán (2016), who studied the links between ED, post-emergency services and the decision to admit the patient into a medical service or to discharge the patient. Busby and Carter (2017) focused on the modelling of response policies in the case of ED surges. Their model included the ED, care and surgery unit in a generic simulation model, with great attention paid to the patients' flow between units and was tested on four different hospitals. The models described above intended to build data-driven generic simulation models, with a generic layout instantiated with information from the studied healthcare centre, or setting-specific generic models according to the classification established in Fletcher and Worthington (2009).

A vast part of the literature is dedicated to the ED. Its importance for the hospital as an entry door to the whole organization, its clearly defined layout and sequential process make it appropriate for DES modelling. Salmon, Rachuba, Briscoe, and Pitt (2018) identified 254 research publications (theses, journals and conference papers) that built ED simulation model. DES was the most represented simulation technique with 209 publications (more than one method could have been used for each considered paper). A majority of research focused on process modelling and on improving the performance of the ED from an organizational standpoint. Resource capacity and workforce planning are the next most investigated areas, which shows that there is a particular interest in how the allocation of material and human resources can improve the capability of department.

Uses of optimization techniques with simulation

The ability of stochastic simulation models to account for variability and randomness in complex processes allows the assessment of statistical significance of obtained results. However, complex scenarios often results in a high number of value combinations for the decision variables. Combining DES with optimization techniques appear to be a good solution to reduce the number of solutions the model needs to evaluate. Optimization is an OR technique that seek to maximize or minimize the performance of a system (here, a DES model) while respecting a set of constraints [START_REF] Lal | Simulation based optimization: Applications in healthcare[END_REF].

Most simulation-optimization studies in health care focus on Medical Unit (MU) bed requirements or on the staffing of units. Combining a hospital DES with an optimization model allowed Holm, Luras, and Dahl (2013) to reduce the number of overcrowded nights in the hospital from 6.5% to 4.2%. The entire hospital was modelled and the need for a reallocation of beds was formalized with the results.

As one of the most investigated units using DES, ED is logically the subject of many simulation-optimization studies. These studies often focus on finding an optimal staffing of ED. Mixed-Integer Linear Programming (MILP) searches for optimal combinations of integer value variables, and has been extensively used to plan staff shifts in the ED [START_REF] Lal | Simulation based optimization: Applications in healthcare[END_REF]. Cabrera, Taboada, Iglesias, Epelde, and Luque (2012) used MILP to find the staff composition that minimizes ED patients' length of stay while maintaining salary expenses below the allocated amount. Farahi and Salimifard (2021) and Zeinali, Mahootchi, and Sepehri (2015) both investigated the staff and bed requirements of an ED using a unique optimization model. Meta-heuristics such as genetic-algorithms have also been used for the same purposes by Sulis, Terna, Di Leva, Boella, and Boccuzzi (2020). A genetic algorithm is a stochastic technique that generates solutions and combines the best performing solutions to explore the search space.

Context of the study

Recently, the Centre Hospitalier Universitaire de Saint-Étienne (CHUSE) opened a Polyvalent medical Unit (PU) targeting multimorbid patients. Our goal is to investigate the creation of such a unit on the patients' pathway in the CHUSE and, more specifically, to build a generic DES model to assess the impact of the newly created unit targettng a specific part of the patient base. We then develop on an optimization model dedicated to composing this unit using beds that can be redirected from existing units involved in the care of multimorbid patients or created specifically.

The paper is organized as follows. Section 2 presents the dataset, the mathematical formulation of the optimization model and the simulation model. Section 3 presents our experimental plan and the main results obtained. Section 4 comments on the study and results and provides research leads that could be further investigated. Section 5 summarizes the main results and contributions of this study and concludes the paper.

Materials and methods

Generic Model of health care centre and patients

Simulation and modelling an entire health care centre is a complex task, mainly because of its high number of interconnected entities. The pathway of a certain type of patient can be modified due to an evolution of the pathway or due to unexpected events. Because of the level of complexity of health care centers, the effects of such changes on patient flow can be hard to predict. Our goal is to build a generic DES model representing a hospital at a macroscopic level that can be used as a decisionsupport tool for complex patient pathway changes.

Hospital Settings

The simulation model built in this study is largely based on the DES described in ?. Thus, we model hospital H as the set of all MU µ i , which characterized by a unique identifier i and a capacity c i corresponding to the number of beds available in this unit. For a total of N ∈ N distinct MU, we note H = {µ i |i ∈ 1..N }.

Patient pathway

As in ?, we modelled patients by the ordered sequence of stays individual units that compose their pathway. We make distinguish elective patients admitted directly in the necessary medical unit and urgent patients admitted through the ED to account for the differences in the management of care between the two types of patients.

We modelled multimorbid patients separately from the rest of the population with additional information about their multimorbidity status. Other patients were generated to set the hospital at its usual occupation level. Both patient types underwent the same generic hospitalization pathway as described in ?. We describe it using the petri net in Figure 1, with emergent admissions and planned admissions modelled by transitions t e and t p , respectively, and t ED is the admission to MU after the ED visit. Patients loop on place p 2 to model their stays in MU using transitions t 1 ...t N until discharge (transition t d ).

Non-multimorbid patients π o are formally modelled by their pathway in the hospital; we note that π o = {µ ij |i ∈ [[1 ; N ]], j ∈ [[1 ; J]]}, where i is the medical unit identifier and j is the relative position of unit µ i j in the pathway π o .

Multimorbid patients' modelling

Multimorbid patients are modelled separately from the usual patient base of the hospital to take into account the extra information needed for their care.

Multimorbid patients π m were modelled similarly to other patients but with extra information regarding their multimorbidity status, and detailed information on their length of stay in the hospital: π m = {M, s, Λ} where M is the pathway of the patient as described in 2.1.2, s is the multimorbidity score of the patient calculated using the method in Calderón-Larrañaga et al.

Optimization Problem

Multimorbid patients are taken into care in the same MU as other patients. Setting up an integrated and centralized care pathway for multimorbid patients who do not require highly specialized treatments, such as surgical intervention or intensive care, is a way to improve their support in hospitals. In this section, we describe the addition of a polyvalent medical unit to the hospital before presenting the optimization model we built to decide on its composition. Data were provided by the CHUSE under Commission Nationale de l'Informatique et des Libertés (CNIL) authorisation number 919300.

Creation of a unit centralizing multimorbid care

The polyvalent medical unit is a short stay unit dedicated to the general care of multimorbid patients, that are added to the existing hospital. It is intended as a platform for common pathologies that do not require complex care. We note that H is the healthcare centre with the addition of PU H = {µ i |i ∈ [[1 ; N + 1]]}, where i is the index of the medical unit and µ N +1 is the novel PU.

The exact capacity x of PU is not defined beforehand, and we use the optimization model described further in 2.2.3 to calculate the best allocation of resources to PU.

Modifying the existing MM Pathway

We suppose that a multi-MU stay of a multimorbid patient can be replaced by a stay in µ N +1 . We hypothesize that the length of stay in µ N +1 is equal to the sum of length of stays in individual units it replaces. Although the unit's polyvalent skills might ideally reduce this length of stay, we consider that this represents the upper bound of LoS in PU, where the care of patients could not be improved in any way.

To summarize, every sequence of stays identified as "nonspecialized" is replaced by a unique stay in PU of equal length. Table 1 displays examples of pathways where medical units are substituted with PU. MU A and MU B are conventional units that can be replaced with PU, while MU C, surgery and ICU provide specialized care that cannot be performed in PU. In this example, Patient 1 has a stay in MU A followed by a stay in MU B that can be replaced by a unique stay in PU. Patient 2 is similar to Patient 1, except that he goes through a surgery and ICU afterwards; in that case, only the two first steps are replaced by PU. Patient 3 and Patient 4 illustrate the fact that any stay in a "conventional" unit can be replaced by a stay in PU.

Multimorbid pathways are modified in the simulation before their admission to the hospital. The replacement is not performed on all patients, so the PU does not become overcrowded. We set up the two following rules to route patients towards the PU.

Rule 1: Using the multimorbidity score Descriptive scores are abundantly used in healthcare environment so we decided to implement a similar rule in the simulation. For each patient, we calculate a score s based on Calderón-Larrañaga et al. ( 2016). This score depends on the number of IDC-10 codes the patient has and ranges from 0 to 60.

The score conditions are summarized in Figure 2. Admitting PU patients with scores above 2 or 3 correspond to classical definition of multimorbidity. Other conditions aim at admitting patients who have multiple body systems affected, without redirecting all patients towards PU (in the original study, 55.8% of patients had a score of 4 or more). Approximately 25% of multimorbid patients fall within the range 5-15 score range.

Rule 2: Using a statistical analysis The objective of the PU is to care for multimorbid patients having complex pathways in the hospital, with transfers towards other units forced by the longer length of stay and additional needs of multimorbid patients.

We studied the pathway of all 809 patients having 4 or more stays in individual units. General information (such as age, sex, admission through ED and in-hospital mortality), sequence of stay, length of stay in each unit and available diagnoses were considered. Based on expert opinions, two physicians, one gerontologist and one internist, classified the patients into two categories: "justified" pathways (each stay in each MU was justified by a medical reason) and "complex" pathways (no obvious reasons for the patient to follow this pathway). Such analysis was double-blinded, each practitioner performed the analysis independently from each other, and a third doctor from the CHUSE ruled on the pathways when disagreements occurred.

A discriminant analysis was performed using STATA® software, categorical variables were recoded to dummy variables Babin and Watson ("2011"). The resulting function was implemented in the simulation. See Section 3 for the analysis of the DA function.

In the simulation, different cutoffs points were experimented between the two class centroids. Those values are presented in Figure 2 2

.2.3. Optimization problem formulation

The addition of a medical unit to an healthcare facility poses many organizational, logistical and financial problems. Here, we focus on the composition of beds of the newly created polyvalent unit. Bed capacity is usually defined according to a preexisting scale and the hospital defines the human resources assigned to each unit based on its level. The PU is set to open using this scale. The present optimization model intends to size the PU using two sources of beds: (i) the creation of new beds and (ii) the redirection of existing beds.

The PU of capacity c N +1 is composed of a mix of beds redirected from other MU and newly created beds. Routing multimorbid patients towards the PU will likely affect the other units of the hospital. Our optimization algorithm is based on the two following hypotheses: H1 Rerouting multimorbid patients towards the PU lowers the occupation level of other units, H2 The redirection of beds from a unit to the PU does not affect other resources dedicated to this MU.

The first hypothesis H1 is necessary as our study is based on past hospitalization and the generated population. We have no access to the real demand for each medical unit and do not know whether the observed occupation is the result of a strict management of the unit, with many patients rejected or guided towards other centres, or if it reflects the actual needs of its patient base. In the first case, the decrease in attendance of the unit induced by the management of multimorbid patients would not affect the MU's activity, an effect we cannot predict given the available data.

The second hypothesis H2 follows from the fact that we consider the transfer of a small number of beds from each unit, which will not require a thorough reorganization of the unit. We note each solution x as the set of created and redirected beds as follows: x = (x 0 , x 1 . . . x n ), where x i is the number of beds redirected from MU i to PU for i ∈ [[1 ; N ]] and x 0 is the number of beds created. The capacity of the PU for the solution is then given by the following formula: c N +1 = N i=0 x i Model 1: Naive Approach In this model, it is possible to transfer beds from every MU, with the exception of three small units having less than 5 beds each. This number of beds x imax is set to 0 for these units. We note that J is the set of all MU, and I is the subset MU from which we cannot take beds. For all units of H, we defined a maximum number of beds x imax that can be redirected towards PU to stay in line with hypothesis H2.

To evaluate the performance of the model, we approximated (i) the mean cost of the creation of beds c c from the economic study that preceded the creation of the PU, (ii) the cost of redirecting beds towards the PU c r , and (iii) the mean daily revenues generated per bed per day r pu . We note D as the total number of days for which the PU beds are occupied.

We decided to set the cost of a bed redirection to 10% of the bed creation cost to take into account the inconvenience related to its redirection. The objective f is to minimize the expenses of the hospital for creating this PU.

This optimization model 1 can thus be formulated as follows:

min x f (x) = c c * x 0 + N i=1 c r * x i -r pu * D s.t. x i ≥ 0 ∀i ∈ [[0 ; N ]] (ct1) x i ≤ x imax ∀i ∈ J (ct2) x i = 0 ∀i ∈ I (ct3) (1)
Model 2: Limiting the PU size We established a list of units from which it is not possible to redirect beds, either (i) because the MU is too small (as in model 1) or (ii) because the unit provides specialized care, which is incompatible with the care provided in the PU. The size of PU is also predefined. Additionally, we limit the PU size to meet the real-life constraints from CHUSE. The objective function is not affected by these changes. These modifications result in the Model 2:

min x f (x) = c c * x 0 + N i=1 c r * x i -r pu * D s.t. x i ≥ 0 ∀i ∈ [[0 ; N ]] (ct1) x i ≤ x imax ∀i ∈ J (ct2) x i = 0 ∀i ∈ I (ct3) N i=0 x i ≥ 17 (ct4) N i=0 x i ≤ 25 (ct5) (2)

Available data and the prediction of complexity

The CHUSE is a university hospital composed of 60 medical services and 1, 802 beds that is spread over 4 geographical installations. There were 93, 905 hospital stays in 2020 in medicinal, surgical and obstetrics unit. Additionally, there were 45, 093 visits to the adult emergency department that were recorded. The CHUSE is at the heart of a health care network of the Loire, which is composed of 16 public hospitals, the Groupement Hospitalier de Territoire (GHT) Loire.

The data that were provided to us by the CHUSE contained information about 19,112 stays in 2017 from a total of 10,566 unique adult anonymized patients. Each patient was diagnosed with codes in at least 3 different categories of the IDC-10. Available variables are listed and described in Table 3. In addition, the sequence of stays in independent units was made available to us, along with the length of stay at each step and a list of all diagnoses made to the patient at each step of their pathway.

We note that one third of the multimorbid patients are admitted through the emergency department, and that the in-hospital mortality remains quite low (approximately 5%). The mean age in this population was high, slightly above 68 years, but also included young patients. The mean length of stay was high (8.5 days), and the highest length of stay observed was 250 days, which skews the distribution of the LoS distribution.

These elements suggest that the vast majority of multimorbid patients follow controlled pathways, and only a small proportion of them can be considered difficult to manage.

Summary

This section presents the theoretical framework of the simulation. In particular, hospitals are presented as a collection of units, each having unique characteristics. Patients are defined as entities passing through units. The implementation of each element in the simulation model is detailed in this section. We also present the modifications of the multimorbid care pathway caused by the addition of the new PU and describe the optimization model that is used to size and compose the PU. A case study is introduced in the next section, and we present the statistical procedure that is implemented to help predict if a pathway was complex or not.

Results

Prediction of Pathway Complexity

The analysis of the 809 patients with the longest stays (4 or more steps in their clinical pathways) was performed by two physicians, one gerontologist and one internist. Disagreements on the classification of 101 cases were observed and settled with the help of a third doctor. As a result of this process, 115 stays were classified as complex based on the sequence of units composing their pathways and their lengths of stays at each step. Diagnosis was rarely consulted to state the complexity of the pathway.

We used a discriminant analysis to investigate the relationship between pathway information and a "complex' pathway". Table 4 displays the key information outputted by STATA for the discriminant function that was generated.

Wilk's lambda Λ calculation was statistically significant for the discriminant function (Λ = 0.6433, p < 0.0001), indicating a significant difference between the sub-groups. The 'group means on canonical variables' show that the centroid for the 'under control' pathway is relatively low at -1.827, while the centroid for complex pathways is 0.303. This indicates that the discriminant function efficiently discriminates those two subgroups. Subgroup 0 (the complex patients) tend to have higher scores in variables having a negative coefficients (largest coefficients are found for stays in internal medicine units, nephrology units and post-emergency surveillance unit or "unité d'hospitalisation de courte durée" in french), and subgroup 1 (non-complex patients) tend to have higher scores in variables having positive (the largest positive coefficients are found in psychology unit, outpatient visit, cardiology and intensive care units). We note that the length of stay and indicator of emergency department visits have positive coefficients and thus are used to predict noncomplex pathways.

The classification analysis presented in the classification matrix passes the true label for each patient as rows and the predicted subgroup as columns. We can see that the pathways are well classified overall (85.66% are correctly classified) and that each subgroup is quite well represented (83.48% for complex patient i.e., subgroup 0, and 86.02% for normal pathways) despite the relatively low number of complex pathways (15% of the 809 observations).

Optimization Experiment

A tabu search was coded in Java and included in the AnyLogic® simulation model to solve the optimization models presented in 2.2.3. The initial solution was arbitrarily set by redirecting 1 bed from each unit of J, both for mMdel 1 and Model 2. At each iteration, the algorithm generated a neighbourhood composed of solutions differing from the current solution by a distance of 1, meaning either that (i) an additional bed was transferred to the PU or created, or that (ii) one less bed was added to the PU (i.e., if the mean objective function value was significantly worse than the best solution found). Each solution was simulated several times to estimate the objective function and associated standard deviation. The maximum number of replications was set to 10, and replications were stopped once we ensured that the solution was worse than the one considered. The solution was added to the tabu list and served as the initial solution for the next iteration of the heuristic.

The maximum number of overall iterations was set to 100 in both models. Both for Model 1 and Model 2, the tabu search was stopped if the simulation did not improve the best solution found after 25 iterations. Those parameters were initially set to 1,000 and 200, respectively, and lowered after examining the duration of the tabu search. Indeed, simulation times exceeded 60 hours for about 300 iterations, and optimal solutions were found almost always in the first 50 iterations, both when using the score or the discriminant function to reroute multimorbid patients.

Evolution of the objective function value for Models 1 and 2 are displayed in figures 2 and 3, respectively. The x axis represents the iteration number. The y axis represents the maximum gain in the objective function normalized by the highest value of the objective function that we obtained for the two experiments.

For Model 1 the best identified solutions were promptly found by the algorithm, with between 50 and 75 iterations needed. Model 1 resulted in an effective gain from the initial solution of approximately 25% and Model 2 had a 27% effective gain (limiting the size of the PU). The discriminant function gives slightly better results than multimorbidity score rerouting. The tabu search with Model 2 gives slightly better performances. Limiting the search space at each iteration might improve the exploration of the solution space.

The best solution for Model 1 using rule 1 is obtained for rerouting of patients having scores 5 and 10. It consists of a PU of size 16 with 16 transferred beds and no created beds. For rule 2, the best results are obtained with a cutoff of for discriminant analysis, of a PU of size 12, with 12 transferred beds. This last solution is also the best overall result for Model 1.

Model 2's optimal solution is a PU of size 18 with 18 transferred beds and no created bed with rule 1 (multimorbidity scores between 3 and 15). With rule 2, the best solution gives a PU of size 17 without any created bed.

Discussion

The cost of redirecting a bed from a MU to the PU was set to 10% of the cost of a bed creation. Although it is likely that such a decision is less impactful than the net creation of bed, we did not have access to this information. We believe that one can estimate the cost of this redirection based on the mean occupation rate, mean length of stay and the mean revenue per stay in the MU as well as the forecasts for those figures. The best solutions we found did not include net creations of beds. The cost of redirection could be raised to incite the algorithm to create bed, and limit the changes on existing units.

As previously mentioned, the optimization running times are very long (the longest test was of approximately 71 hours for 302 iterations for the tabu search applied to Model 2). We measured the areas of the code that were the longest to execute, and without surprise, the AnyLogic® simulation is the main bottleneck, with 99.99% of the running time being taken by simulation runs. The average simulation time for the ten replications was approximately 32.93s when solving Model 2). Several factors increasing the simulation should be considered: (i) the length of the simulation (365 days), (ii) the high number of agents generated, and (iii) the input/output parameter management.

The mean duration of a simulation run also suggests that a very high number of solutions were explored to perform the entire optimization. Although we did not measure how many solutions were tested, we can estimate that on average, 26 solutions were evaluated at each iteration, and each was replicated 10 times.

To improve the efficiency of the optimization algorithm, a list of each explored solution with their score should be maintained. Then, the simulation could be launched only for nonrecorded solutions. We already implemented a solution to stop replications if the score was clearly not improving the global solution, without greatly impacting the results.

Conclusion

This article presented the development and validation of a macroscopic digital twin of a large French university hospital, CHUSE. A module generating synthetic patients using process mining was implemented to replicate the hospital's usual utilization level. An alternative clinical pathway for patients with multimorbidity was developed, focusing on a polyvalent unit that could avoid transferring unnecessarily patients. We used the historic data of multimorbid patients in the hospital, and solutions of rerouting were tested using medical (with multimorbidity scores) and statistical approaches (using a discriminant analysis). An optimization model was developed to size this novel unit, and a tabu search algorithm was implemented. The results show that our optimization module is able to find better performing solutions at the cost of high computation times.

Limitations of this study are mainly linked to the available knowledge on the population pool of medical units. We decided to model the creation of a novel unit using beds from other departments in the hospital, based on the assumption that this novel unit would absorb the flow of multimorbid patients frequenting this unit and thus lower the occupation. However we have no way to track the number of patients who needed care from this unit, and its occupation rate might not be lowered by the rerouting of multimorbid patients.

Further improvements to this model can be made. First, the simulation could be improved to include more details about the patient pathway, allowing the application of microcosting techniques to improve the evaluation of the solutions. A prospective analysis could be performed to evaluate these effects. A general framework for optimizing clinical decisions on clinical pathways could be derived from this model, with other optimization techniques implemented and ready-to-use. Chapter 5. Sizing of a newly created medical unit for multimorbid care, a simulation-optimization approach

Figures Captions Alone

Conclusion of the chapter

In this chapter, we explored the addition of the PU to the CHUSE. The discriminant analysis used to chose patients for the PU is thoroughly described. The sizing and composition of the unit was formulated as an optimization problem that minimizes the waiting time and number or beds (transferred or created), converted into costs for comparability reasons. Results show that the optimization module find significantly better solutions than obvious solutions, at the cost of long running times. Additional experiments should be performed to test different configurations of care, for instance the creation of a mobile polyvalent unit in the hospital to maintain the patients in their original units and lower the number of transfers. The estimation of bed redirection costs should also be improve to better model the different units activities.

Conclusion Summary

This thesis described a study on the improvement of the care for multimorbid patients in the hospital of Saint-Étienne. It resulted in the different contributions, both technical and scientific, detailed in this thesis and through the 4 publications constituting this manuscript.

In chapter 1, we presented the issue of multimorbidity. If the basic definition -the simultaneous occurrence of several diseases in one individual -is widely agreed upon, its operationalization in actual healthcare systems results in a high prevalence. Indeed, electronic healthcare records and coding systems like the ICD-10 allow the staff to systematically record diagnosis information. Several modifications of the definition have been proposed, and many indexes have been developed to provide physicians with tools to identify and quantify multimorbidity. A systematic search on online databases helped us identify new indexes. We chose to implement [14] index for its exhaustive coverage of ICD-10 codes. We also analyzed the global health care context in France to identify problematic behind the care for patients with multimorbidity.

Chapter 2 presented the pathway generation built during this thesis. PM was used to map the clinical pathway of a population of multimorbid patients and the resulting output was used to generate realistic pathways. We explored the generation of length of stay distribution at each step and formatted the result for an easy use with a generic simulation model of healthcare processes. The entire framework allows the pre-processing of patients data to generate a fictitious patient population, with respect to patients' privacy.

The generic simulation model and hospitalization pathway representation is further investigated in chapter-3. The model was developed using AnyLogic® to accurately replay the physical hospitalization sequence of the patient between medical units. We used the case of the COVID-19 pandemic to illustrate this. The evolution of the pandemic is evaluated in real time in the simulation and the hospital's response is dynamically adapted. An optimization by simulation experiment was conducted to chose the best combination of response variable values.

Multimorbid patients in the CHUSE is described in chapter 4. We used descriptive statistics to better understand the patients' profiles. Multimorbidity indexes ability to predict readmissions and length of stay have been assessed, with mitigated results.

The creation of a PU for the care of multimorbid patients have been covered in chapter 5. We formulated this problem as an optimization problem to target patients at risk of having unnecessary complex pathways. A sub-sample of multimorbid patients was Conclusion reviewed and classified by health practitioners, and a discriminant analysis was inferred from this data. A Tabu Search was then coded in Java and integrated to AnyLogic® as a custom experiment module. The output gives the size of the unit and its composition between net addition of beds to the hospital and beds redeployed from other units.

Those contributions altogether form a complete study of multimorbid patients' clinical care pathway. However, this work leaves several areas unexplored and some further developments can be considered.

Future work

This section compiles the research leads that were identified along this thesis and that still need to be investigated.

First and foremost, the process mining framework for patients generation could be completed. Indeed, this framework is sufficient to generate a population were only clinical pathway, and maybe length of stay when the quantity of data points is sufficient. However, we did not use this method to generate patients with multimorbidity in chapter 5. The additional medical information needed (diagnoses, multimorbidity indexes or even medical acts) can not be generated for the time being. We intend to complete the existing framework to associate the multimorbity index value from the pathway information, although two patients with the same pathway might have totally different health conditions.

In addition, other techniques of process mining could be applied to discover process maps. For instance the method developed in [START_REF] Oliveira | Optimal process mining of timed event logs[END_REF] could help to account for loops in historical patients pathways and avoid the generation of long and unrealistic pathways.

For the care of multimorbid pathways, it would be interesting to test other configurations for the PU. For instance, by targetting specific conditions susceptible to generate readmission or high consumption of resources. The tabu search should be able to identify the affected units and would give interesting insights as to how the units are connected and for which category of patients. Another interesting idea would be, in pure industrial engineering fashion, to target specifically the units that have high occupation rates and act as bottlenecks. However, the main limitation of our model is that it relies on historical data of the hospital and we do not have access to the patients that could not be admitted to this unit. The decrease of occupation caused by the modification of multimorbid patients pathway could be entirely compensated by those patients and modify their pathways. This limitation could be raised in two ways: (1) through a prospective study aiming at tracking the patients profiles that were rejected due to the lack of available beds, and (2) with a retrospective study, using the PU data generated in the next year and to compare to the data available for this study (2017) using clustering on patients information to pin out the pathways that would have been modified.

Finally, we strongly believe that the problem of multimorbidity need to be tackled at the regional level. The links between healthcare centers in the GHT are strong, and intercenter pathways should be carefully analyzed and included into the model. The PU was originally intended to care for patients of the GHT. Comparative studies of multimorbid population in each center should be led to better understand the impact of this novel unit on the global healthcare environment.

INTRODUCTION

Context

The management of key resources in healthcare centres has become a crucial matter for decision makers of the hospital. In particular, the bed occupancy ratio of a service is a useful performance indicator for the hospital. It is essential for caregivers to keep a small number of beds available to address emergencies while having too much unoccupied bed shows that the service is in overcapacity. A relevant management of beds will also have advantages for the patient, as it will improve the quality of care at the hospital and reduce the inpatients mean waiting time. The number of unoccupied beds to keep is the element in which management can rely to be responsive when unexpected events occur, such as the failure of a key device, or the departure of a surgeon. Other factors can impact the hospital attendance, like seasonality (epidemics such as flu in winter or heat waves in summer) or more persistent factors like the ageing of the population.

Before defining those figures it is essential to have a mid-term vision on the patients attendance to the hospital. The administration staff of the hospital needs to know the number of beds required to state on annual closing of beds during holidays or on the exceptional opening of extra beds for emergent patients. Industrial engineering and operational research approaches have been widely used to predict bed occupancy in hospital, and Discrete-Event Simulation (DES) have proven useful to manage resources in a process analysis. In this paper, we describe the development of a DES tool designed to predict bed occupancy ratios for a healthcare center taking into account the history of patients' attendance over the past years and predictions of future arrivals.

Related Work

The analysis of patients' stays in hospital is extensively covered in the literature. Predicting the attendance in a healthcare delivery structure allows managers to become proactive regarding resource management decisions. Forecasting models like time-series have been developed by [START_REF] Farmer | Models for Forecasting Hospital Bed Requirements in the Acute Sector[END_REF] to give health service planners this opportunity. Numerous analytical models were designed as decision support systems. For instance short-term bed occupancy can be predicted using autoregressive integrated moving average according to [START_REF] Abraham | Short-Term Forecasting of Emergency Inpatient Flow[END_REF]). Regression techniques also give good results [START_REF] Kumar | Predicting Bed Requirement for a Hospital Using Regression Models[END_REF]) when predicting short-term bed demands for several classes of beds. Similarly [START_REF] Tan | Data-Driven Decision-Support for Process Improvement through Predictions of Bed Occupancy Rates[END_REF] implemented Principal Component Analysis and Multiple Linear Regression methods to alert an Emergency Department (ED) of bed crunching risk on the short-term.

Beyond analytical prediction, simulation models have proven to be useful for modelling the patients' flow and a decision's impact on a healthcare system. For instance, queuing networks have been used by (El-Darzi et al. 1998) to understand the behavior of bed blockers in a geriatric department. Complete hospitals models are quite rare and researchers often focus on a department in particular. For instance, [START_REF] Chavis | A simulation model of patient flow through the emergency department to determine the impact of a short stay unit on hospital congestion[END_REF]) studies the impact of a post-emergency unit on ED patient's length of stay. Indeed, EDs are probably the most studied department using simulation. A review done by (Salmon et al. 2018) counts 254 journal or conferences publications and thesis using simulation to model ED between 2000 and 2016, with DES being the most used technique. Other techniques, like system dynamics (SD) or agent-based modelling (ABM) are more used when considering the interactions between ED and exterior factors. [START_REF] Cabrera | Optimization of Healthcare Emergency Departments by Agent-Based Simulation[END_REF]) modelled a Spanish ED with ABM to better characterise the operations of a complex system and (Chong et al. 2015) uses SD for its ability to account for the complex behaviour of such a service. However, when looking at the whole simulation area, [START_REF] Brailsford | Hybrid Simulation Modelling in Operational Research: A State-of-the-Art Review[END_REF]) notes that models combining different techniques are becoming more and more popular.

Building a model of the hospital as a whole is a challenge. The model proposed by (Ben-Tovim et al. 2016) simplifies the hospital's organisation to do so. The authors grouped together the ward into one unit, and did the same with surgical theatres. In (Busby and Carter 2017), DES is used to detail the links between ED, inpatients wards and surgical theatres. Such models are conceived to be generic and applicable to many hospitals.

Le Lay, Alfonso-Lizarazo, Augusto, Bongue, Célarier, Gonthier, Masmoudi, and Xie

Objective and Scientific Contribution

The objective of our work consists in proposing a decision-aid tool for hospital managers, allowing them to decide on the bed requirements for all medical units of a given hospital on a medium term horizon, using past hospitalization data such as medico-administrative data. The main contribution consists in the development of a data-driven DES model, allowing the user to test different resource scenarios and observe their consequences on bed occupancy prediction. The prediction of arrivals and length of stays will be done analytically and then passed to the simulation model. This model can be set up in different ways, to visualize the departure of a key resource at a precise time like the retirement of a surgeon, or a sudden influx of patients due to an unexpected epidemic. The scientific challenges are twofold:

• The model should consider all medical units: such tool is relevant if patient transfers are taken into account.

• The model should be fed with patient traces extracted from hospital data history to take into account patient's journey complexity. To do so we use a data mining approach to extract past patient traces that will be replayed in the model.

This paper is organised as follows. In Section 2 we present the modelling approach to create the forecast log and the simulation model. In Section 3 we present the results of the different experiments we ran. The analysis regarding those results are detailed in Sections 4. Conclusions and perspectives are given in Section 5.

MODELLING APPROACH

Modelling Hypothesis

The main objective of this paper consists in providing a simulation-based decision aid tool for hospital manager. This tool will help decide on bed allocation in a medium term horizon (next few months). The following hypothesis are taken into account:

• H1: A macroscopic model of the hospital is used. The lowest level of detail is the stay in a medical unit by a patient. To that extent, the patient journey in the hospital is modelled as a trace of stays in different medical unit between his/her admission and his/her discharge. Peripheral activities, such as transfers, tests and surgeries, are not explicitly modelled. • H2: We consider staffed-beds for resources. Each medical unit of the hospital has a predefined number of staffed-beds available. Human resources requirements are extrapolated from bed requirements. • H3: We consider that patients spending less than a day in a medical ward are not occupying a bed.

Indeed our main data source gives the length of stay of patients in days, causing wards with a low mean length of stay to be very easily overcrowded.

This model considers two main sources of patients:

1. Elective admissions patients. These are the patients already scheduled in the next few months. 2. Non elective and emergent patients. Such patients are admitted through the Emergency Department (ED) of the hospital, or directly in a medical unit under recommendation of the general practitioner or emergency services (e.g., Covid patients who are directly admitted in dedicated units).

For elective admissions patients, the model is fed using available data in the information system of the hospital. For non elective patients, we do not have any information apart from the previous admission. To model such patients, we build a case-mix of previous emergent patients and pick up randomly patients in this case-mix.

Le Lay, Alfonso-Lizarazo, Augusto, Bongue, Célarier, Gonthier, Masmoudi, and Xie

Formal Model

Patients

We decided to divide the admissions into 2 groups: (i) patients admitted directly into a ward, and (ii) patients admitted through the Emergency Department. We considered patients of the first group as being "planned" by the hospital: although their origins are varied, direct admission show that these pathways are under control. On the other hand, patients admitted through the ED are managed on the run and the medical staff is reacting to their pathologies. The information available through the patient record are the following:

• Anonymous patient identifier, • Wards in which the patient have been admitted, • Date of admission in each ward, • Length of Stay (LoS) in each visited ward, • Admission modalities and provenance.

From this we computed each patient's trace in the hospital, defined as the chronological succession of wards visited by a patient during his stay at the hospital.

We divided the available data in two subsets, patients admitted to the hospital through the ED (urgent patients), that we will be referring to as Data-set 1, and elective patients, similarly Data-set 2. We used those data-sets to compute several inputs for our simulation.

Medical Units

Medical Units (MU) are modelled as agents and their characteristics as parameters. Two kinds of parameters are used: constants giving information about the existing MU such as the capacity of the ward, and variable parameters used to keep track of the bed and staff demand. When a patient is admitted to a new ward, the ward's occupancy is incremented. The needed staff -doctors, nurses, and caregivers -is calculated by multiplying the occupancy by a theoretical ratio. Those ratios are given by a classification of wards established with health professionals.

Hospitalization Process Model

In our process, we use different sources for each patient type (elective and urgent). Following our hypothesis H1, the model is macroscopic, thus all medical units are modelled by a single delay block on AnyLogic. The only exception to this rule is the Emergency Department, which is placed after the urgent patient source. The patient is routed again to the unit's block if his stay is not finished, creating a loop on the medical unit block. When the stay is over, and all the medical units constituting the trace have been visited, the patient is discharged.

To account for the limited number of beds available, we designed a queuing process. A list of similar services was conceived for each unit. If the medical unit needed by the patient is full, we try to reroute the patient toward these services. The patient waits until a bed is available in a suitable unit. If no bed have been found after a week in the targeted services, the patients is routed to an unoccupied bed in a randomly chosen ward. This abusive choice is made in order to avoid the simulation from being blocked by an endless accumulation of patients. In similar situations, doctors would change the patient destination to ensure the care.

A poll of traces followed by emergency patients admitted during the first semester was created using the Data-set 1. This poll is used as a basis for the admissions of generated urgent patients, each generated agent will be randomly assigned a trace from this poll. We calculated admission rates for each day of the week from the daily number of admission in 2017/2018. Le Lay, Alfonso-Lizarazo, Augusto, Bongue, Célarier, Gonthier, Masmoudi, and Xie

In the final version of our model, planned admissions will be entered in the simulation from the hospital's information system. However in the experiment presented here, we used the 2018 patients in Data-set 2 to create the elective patients' admission log.

EXPERIMENTS AND RESULTS

Our objective is to assess the efficiency of bed-allocation policies in response to predefined situations. After assessing the consistency of the model with the hospital normal activities, we will consider two different scenarios. An increase of arrivals through the Emergency Department and the sudden arrival of patients positive to Covid-19. The Covid+ patient pathway is described here-under.

Case Study

This project is a collaboration with Saint-Étienne's University Hospital (CHUSE). This hospital is composed of 1,795 hospitalisation beds, in 60 departments, located on 3 different geographical sites across the city. It is also the heart of a large territory healthcare network (Groupement Hospitalier de Territoire Loire in French). This large group of healthcare centres is designed to pool the hospitals' resources and facilitate the coordination of care inside a territory.

In this study we excluded the paediatric sector as it is managed independently from other services. The main preoccupation of the hospital is the prediction of the bed demand on the medium term, which could help to predict the impact of unexpected events or policies on bed occupancy levels.

Our objective with this simulation is to provide the hospital with a decision-aid tool. Nowadays, decisions of bed allocations or seasonal bed reductions are based on experience, demands from medical wards or financial perspectives. This visualisation tool will allow the hospital to centralise admission previsions and see how these policies impact the patient's flow. In the actual context of pandemic, we decided to focus on recovery and intensive care units that are facing an unpredicted surge of patients.

Parameters of the Studied Hospital

The layout of the model is displayed in Figure 1. We used four independent sources to generate patients. ED and MU are modelled by delay blocks, the occupancy of each ward is calculated by incrementing the corresponding parameter.

As explained in 2.2.3, two sources generate urgent and elective patients. ED LoS is modelled by by triangular distribution with a mean of 6 hours. Elective and urgent patients are served by the same MU block. LoS for each patient is calculated with probability distributions specific to the ward. Using Arena Input Analyzer software we fitted probability distributions and the corresponding parameters for each MU.

Another source is generating warm-up patients. All patients admitted to the CHUSE in 2018 and still present on the 1st of January 2019 are generated using this source to initialise the simulation. Pathways and lengths of stay in each MU are directly used in the simulation. These patients go through a MU block with fixed LoS passed as patient's parameters.

The last source generates Covid-19 patients. Their pathway is fixed, they are admitted to the general recovery unit after a short ED stay. The LoS in this MU is determined by a normal law with a 12 days mean and a standard deviation of 3 days. This is a rough approximation based on discussions with professionals and preliminary data. 10% of the patients are discharged to account for in-hospital mortality. Others are redirected to an intensive care units for a similar time. We consider that after this ICU stay, the patients are discharged from the hospital.

Le Lay, Alfonso-Lizarazo, Augusto, Bongue, Célarier, Gonthier, Masmoudi, and Xie

Data

The data was extracted by the Department of Medical Information of the CHUSE. It includes all patients admitted to an adult care unit and discharged during the years 2017 and 2018. We also have the daily bed openings in 2018 for each ward to complete the model. We used the data (Santé Publique France 2020) regarding patients admitted to a recovery unit in the Loire department for coronavirus with respiratory complications from 03/18/2020 to 04/20/2020. We decided to set the upper bound for arrival rate to 10 patients per day. To simplify and simulate a short epidemic outbreak, Covid patients arrival is programmed from 03/18/2020 for 2 weeks before being set back to 0.

Design of Experiments

We ran different experiments to test the simulation model summarized in Table 1. First we tested the model with real data by using the available log on 2017/2018. This validation scenario will be run with infinite capacities, Scenario 0, and finite capacities, Scenario 0'.

A second set of experiments (Scenarios 10-43) is proposed to study a daily admission rate increase. We used our generated data predicting the attendance on 2019 with a varying arrival rate of urgent patients. According to [START_REF] Bergonzoni | Les Etablissements de Santé -Edition[END_REF], ED attendance in France has annually increased by 3.5% on average since 1996. Thus we decided to consider 4 ED admission rates (observed average, 1%, 3%, 5% increases).

Experiments will be run with infinite and finite capacities to see the impact of the rerouting possibilities on occupation indicator. Then we will redefine the MR capacity to 90% and 110% of the mean occupation of the finite capacity scenario to experiment a sizing of the ward.

For the last set of experiments (Scenarios 50-54), we extended the simulation to the first semester of 2020 and added the Covid patients outburst. We make set the admission rate of Covid patients to 5 per day for two weeks. Again, we will test this at finite, Scenario 5, and infinite capacities. However, we will allow the entry of Covid patients, even when the ward is fully occupied. We will test two response policies: the rerouting of urgent and elective patients to another ward, and a 25% increase of capacity of the MR ward, independently and combined.

In scenario 0 we consider a 1-year warm-up period (2017). For all other scenarios, the warm-up period of the model is set to 6 months using the warm-up patients described here-above.

Le Lay, Alfonso-Lizarazo, Augusto, Bongue, Célarier, Gonthier, Masmoudi, and Xie ward's capacity, explaining the small difference between the capacity (15) and the maximum occupancy (16).

Normal Activity Simulation

In scenarios 10, 20, 30 and 40 we notice that mean occupancy ratio is a lot smaller than the one in validation scenario, while the maximum occupancy across the 10 repetitions is higher (see Table 2). We believe this difference may be partly caused by the hypothesis H3. Indeed, this hypothesis may conduce the model to accept patients too quickly into certain wards, thus reducing the global occupation rate in critical services. The increase of admissions through the ED has a small impact on the ward's occupancy, but is still noticeable.

Finite Capacities and Ward Resizing

The results are summed up in Table 3. First, we see that taking into account finite capacities leads to higher occupation rates for the MR ward. We can also notice that the increase in emergency patients arrival does not affect the mean OR enough to have an impact on the wards' target values for our sizing experiment (8 and 10 beds) on such a short period of time. When the capacity is fixed to 8 beds, we can see that the MR ward occupation rate is really close to 100%, leaving no beds available for unexpected patients.

COVID Simulation

The massive arrival of patient logically causes an increase of the Occupancy Rate of the MR ward. At baseline, with a Covid patient arrival rate at 0, the system behaves as expected, with indicators similar to the one of scenario 1. The staff needed to respond to this surge is high as well.

Le Lay, Alfonso-Lizarazo, Augusto, Bongue, Célarier, Gonthier, Masmoudi, and Xie

In Table 4 a daily arrival rate of 5 Covid patients overcrowds the ward, with an average maximum occupancy of 65 patients at the same time. The bed and staff requirements are high, at baseline our medical partner indicates that about 8 doctors and 40 nurses work in the MR ward with 15 beds. Without any action on the admission policy, the hospital would need the equivalent of 5 MR units fully staffed to treat all patients during the pandemics. We can see that the combination of a temporary increase of the capacity and the routing of non-covid patients to other units are significantly reducing the Mean OR.

DISCUSSION

The main contribution of this paper is the conception of a digital twin of a health care centre, able to adequately mimic the behaviour of our partner hospital. We believe that our model can be a valuable decision-helping tool to estimate the resources needed for day-to-day operations and to assess the need for a special policy in the case of unpredicted events as we did in scenarios 50-54.

Although the results of the simulation are promising, we believe this model can be improved, in particular the wards' finite capacities and waiting queues. Indeed the relations between the ward can be more accurately described. For instance, a patient could be rerouted to another ward based on his diagnoses and the wards' specialities for instance.

Moreover, additional work should be done to clean the database and correct the problems posed by hypothesis H3. Indeed we noticed that wards highly demanded for short stays (less than 3 days in average) are not well modelled by our simulation, causing an accumulation of patients waiting for a bed, even in Scenario 0. We believe that obtaining more precise data for those few wards should help improve the model.

Response policies have been designed and tested to test our model. We believe that studying the actual response of the CHUSE to this crisis, with precise data on Covid patients' pathways, would improve this model. In addition, implementing dynamic surge policies, like allowing an admission without considering capacities under certain conditions would greatly improve the model.

CONCLUSIONS

In this paper, we described the creation of a simulation model of the hospital from patients' records. The study of patients' stays allowed us to build a macroscopic and representative model of the studied hospital. We produced a prospective analysis of patients arrivals on the year 2019 and the first semester of 2020. This allowed us to analyse bed occupancy when the hospital is running its usual activities.

Using data from the current coronavirus pandemic, we ran an experiment to demonstrate how this simulation could be used to assess the model's response to an unexpected variation of arrivals. We believe the results demonstrates how this tool can help the decision process concerning bed management policies when facing unexpected events. The strength of this data-driven DES model is its ability to account for the major part of the hospital. In addition, given a thorough data analysis, it can be used to test different organisations of the hospital in what-if scenarios, or applied to other health care centres.

Le Lay, Alfonso-Lizarazo, Augusto, Bongue, Célarier, Gonthier, Masmoudi, and Xie

We identified and detailed several perspectives of improvement that will help in providing a robust insight on the hospital current state and reassert its use as a decision-helping tool. First, implementing more realistic patients' pathways will improve the accuracy of the model in describing hospital's day-to-day activities. Secondly, the definition of dynamic surge responses is a necessary step analyze the impact of exceptional crises like Covid-19 epidemic. Once completed and reviewed by hospital's officials, a standalone version of the model is to be delivered as a decision-aid tool.

Abstract

La prise en charge des patients atteints de multimorbidité (ayant plusieurs pathologies chroniques simultanément) pose de nombreuses interrogations chez les professionnels de santé. Des dizaines de protocoles ont étés développés en France et en Europe pour faciliter la coordination des soins entre les différentes spécialités, à l'hôpital ainsi que pour les soins quotidiens à domicile. A cela s'ajoute un contexte hospitalier difficile, où de nombreux facteurs peuvent complexifier la prise en charge, comme la constante augmentation du nombre de visites aux urgences en France depuis 20 ans, et le vieillissement de la population, les patients âgés étant plus susceptibles d'être atteints de multimorbidité. Ce projet de recherche vise à construire un modèle de simulation généralisable du Centre Hospitalier Universitaire de Saint-Étienne (CHUSE) pour l'évaluation des parcours de soins. Composé de plus de 60 unités médicales pour un total de 1802 lits, le CHUSE a enregistré plus de 90 000 séjours en hospitalisation et 78 400 visites aux urgences en 2020.

Nous présentons une méthode de génération de population pour la simulation en utilisant les techniques d'exploration de processus (Process Mining). Un jumeau numérique de l'hôpital a été développé sous la forme d'une simulation en utilisant le logiciel AnyLogic®, testé sur la COVID-19. Nous avons évalué la performance de score de multimorbidité pour prédire les réadmissions à l'hôpital. L'ensemble de ces éléments ont permis d'étudier la création d'une unité de soins polyvalent dédié notamment aux seau sein du CHUSE, et un algorithme d'optimisation a été ajouté pour dimensionner cette unité.

1 . 1

 11 Multimorbidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Multimorbidity concept and definition . . . . . . . . . . . . . . . . . 1.1.2 Identification of multimorbidity in general population . . . . . . . . 1.2 Measurement of multimorbidity . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Measuring Multimorbidity in the Literature . . . . . . . . . . . . . . 1.2.2 Generic Formulation for Multimorbidity Indexes . . . . . . . . . . . 1.3 Choice of an index: Systematic Review . . . . . . . . . . . . . . . . . . . . . 1.3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Care of Multimorbid patients . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Integrated Care Pathways . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Aging population in AURA . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Context of care in health care centers . . . . . . . . . . . . . . . . . . 1.5 Positioning of the present research . . . . . . . . . . . . . . . . . . . . . . . 2 Automated Generation of Patient Population for Discrete Event Simulation using Process Mining 2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Automated Generation of Patient Population for Discrete Event Simulation using Process Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conclusion of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents 3 COVID-19 Intensive Care Unit Optimization Using a Two-Step Process Mining and Discrete-Event Simulation Approach 3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 COVID-19 Intensive Care Unit Optimization Using a Two-Step Process Mining and Discrete-Event Simulation Approach . . . . . . . . . . . . . . . 3.4 Conclusion of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Prediction of Hospital Readmission of Multimorbid Patients Using Machine Learning Models 4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Prediction of Hospital Readmission of Multimorbid Patients Using Machine Learning Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conclusion of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Sizing of a newly created medical unit for multimorbid care, a simulationoptimization approach 5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Sizing of a newly created medical unit for multimorbid care, a simulationoptimization approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Conclusion of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Conclusion A First appendix: Impact of COVID-19 Epidemics on Bed Requirements in a Healthcare Center Using Data-Driven Discrete-Event Simulation A.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 A.2 Impact of COVID-19 Epidemics on Bed Requirements in a Healthcare Center Using Data-Driven Discrete-Event Simulation . . . . . . . . . . . . . . . 120 A.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Figure 1 : 1 Contextualization 1 . 1

 1111 Figure 1: Articulation of the 4 articles of this manuscript and their associated contributions.

Figure 1 . 1 :

 11 Figure 1.1: Examples of patients for calculations of multimorbidity scores.

Figure 1 . 2 :

 12 Figure 1.2: Schematic overview of the different steps of the articles' selection process.

Figure 1 . 3 :

 13 Figure 1.3: Multimorbidity Index choice guide as found in Stirland et al.[START_REF] Stirland | Measuring multimorbidity beyond counting diseases: Systematic review of community and population studies and guide to index choice[END_REF]

Chapter 1 .

 1 Contextualization (a) Evolution of emergency visits between 1996 and 2018. (b) Evolution of the number of hospitalization beds between 2013 and 2020.

Figure 1 . 4 :

 14 Figure 1.4: Evolution of health care context in recent years.

Figure 2 :

 2 Figure 2: Example of local filters

Figure 3 :

 3 Figure 3: Percentage of patients by number of variants.

Figure 4 :

 4 Figure 4: Graph obtained with an 85% global filter on log variants.

Figure 5 :

 5 Figure 5: Petri net obtained with an 85% global filter on log variants.

Figure 6 :

 6 Figure 6: Global filter effect on metrics.

Figure 7 :

 7 Figure 7: Evaluation of the DFG model based on the global filter on activities and local filters on the paths and activities of the DFG model.

  (a) Metrics obtained for the DFG model. (b) Metrics obtained for the PN model (results for variant length 14 are not presented to improve the readability).

Figure 8 :

 8 Figure 8: Evaluation metrics results table for the DFG and PN model with a 95% global filter.

Figure 9 :

 9 Figure 9: Example of the fitted distribution. ment) cannot be properly modeled currently. Here, we used data aggregated at the unit level, and accessing analyses such as those performed in (Arias et al. 2020) would allow us to generate patient pathways with more details and perform cost analysis accounting for a large number of resources.

Chapter 3 .

 3 COVID-19 Intensive Care Unit Optimization Using a Two-Step Process Mining and Discrete-Event Simulation Approach

Fig. 1 :

 1 Fig. 1: Generic hospitalization pathway Petri net

Fig. 2 :

 2 Fig. 2: Simple Event Log and Associated PM Graph Generated with Disco Software Case ID Timestamp Activity Patient 1 01/01/2021 A1 Patient 1 05/01/2021 A3 Patient 2 03/01/2021 A1 Patient 2 05/01/2021 A2

Fig. 3 :

 3 Fig. 3: Simplified Process Map Representing our Synthetic Data, Generated Using Disco® . (a) Simplified Process Map of Emergency Patients.

4. 1

 1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3 Prediction of Hospital Readmission of Multimorbid Patients Using Machine Learning Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.4 Conclusion of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Fig 1 .

 1 Fig 1. On the left side of the graph are male patients, and on the right side of the graph are female patients.

Fig 1 .

 1 Fig 1. Age pyramid of the multimorbid patients of the hospital of Saint-Étienne (male patients on the left side of the graph and female patients on the right side of the graph).

:Fig 2 . 23 Fig 3 .

 2233 Fig 2. Results of the different resampling techniques and classifiers for the prediction of readmission within 30 days

5. 1 96 5. 4

 1964 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.3 Sizing of a newly created medical unit for multimorbid care, a simulationoptimization approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

  (2016) and Λ = {λ ij |i ∈ [[1 ; N ]], j ∈ [[1 ; J]]} is the set of lengths of stay in MU at each step, where i the unit index and j the relative position of the stay in the pathway. We use the multimorbidity score defined by Calderón-Larrañaga et al. (2016) because it was developed recently by a panel of medical experts and includes the large majority of International Classification of Diseases and Related Health Problems 10th Revision (IDC-10) codes of chronic conditions, which allow an easy implementation with medico-administrative data.

Figure 1 .

 1 Figure 1.: Generic hospitalization pathway petri net.

Figure 2 :

 2 Figure 2:

Figure 2 .

 2 Figure 2.: Results of the optimization experiments for Model 1.

Figure 3 :

 3 Figure 3:

Figure 3 .

 3 Figure 3.: Results of the optimization experiments for Model 2.

  Figure 1 caption: Generic hospitalization pathway petri net. Figure 2a caption: Model 1 using rule 2 (score).Figure 2b caption: Model 1 using rule 2 (discriminant analysis).Figure 2 caption: Results of the optimization experiments for Model 1. Figure 3a caption: Model 2 using rule 2 (score).Figure 3b caption: Model 2 using rule 2 (discriminant analysis).Figure 3 caption: Results of the optimization experiments for Model 2.

  Figure 1 caption: Generic hospitalization pathway petri net. Figure 2a caption: Model 1 using rule 2 (score).Figure 2b caption: Model 1 using rule 2 (discriminant analysis).Figure 2 caption: Results of the optimization experiments for Model 1. Figure 3a caption: Model 2 using rule 2 (score).Figure 3b caption: Model 2 using rule 2 (discriminant analysis).Figure 3 caption: Results of the optimization experiments for Model 2.

Figure 2 caption:

 2 Figure 1 caption: Generic hospitalization pathway petri net. Figure 2a caption: Model 1 using rule 2 (score).Figure 2b caption: Model 1 using rule 2 (discriminant analysis).Figure 2 caption: Results of the optimization experiments for Model 1. Figure 3a caption: Model 2 using rule 2 (score).Figure 3b caption: Model 2 using rule 2 (discriminant analysis).Figure 3 caption: Results of the optimization experiments for Model 2.

  Figure 1 caption: Generic hospitalization pathway petri net. Figure 2a caption: Model 1 using rule 2 (score).Figure 2b caption: Model 1 using rule 2 (discriminant analysis).Figure 2 caption: Results of the optimization experiments for Model 1. Figure 3a caption: Model 2 using rule 2 (score).Figure 3b caption: Model 2 using rule 2 (discriminant analysis).Figure 3 caption: Results of the optimization experiments for Model 2.

Figure 3 caption:

 3 Figure 1 caption: Generic hospitalization pathway petri net. Figure 2a caption: Model 1 using rule 2 (score).Figure 2b caption: Model 1 using rule 2 (discriminant analysis).Figure 2 caption: Results of the optimization experiments for Model 1. Figure 3a caption: Model 2 using rule 2 (score).Figure 3b caption: Model 2 using rule 2 (discriminant analysis).Figure 3 caption: Results of the optimization experiments for Model 2.

Figure 1 :

 1 Figure 1: Hospitalization process model representation.

  ÉCOLE DES MINES DE SAINT-ÉTIENNE 158, cours Fauriel -CS 62362 42023 Saint-Étienne Cedex 2 -France Tél. +33(0)4 77 42 01 23 www.mines-stetienne.fr École Nationale Supérieure des Mines de Saintdes Parcours de Soins de Patients Multimorbides à l'aide de Techniques de Process Mining Spécialité: Génie Industriel Mots-Clefs: Simulation Évènements Discrets, Exploration de Processus, Découverte de Processus, Analyse de Données

  CHUSE Centre Hospitalier Universitaire de Saint-Étienne. 19, 37, 69, 70, 94, 116, 117 DES Discrete Event Simulation. 17, 18, 19, 37, 39, 95 ED Emergency Department. 37 ESPS Enquête Santé et Protection Sociale 2010 et 2014. 23 GHT Groupement Hospitalier de Territoire. 37, 118 HSM Enquête Handicap-Santé Ménage. 23 I4S Ingénierie des Systèmes de Soins et des Services de Santé. 37 ICU Intensive Care Unit. 19 INSEE Institut National de la Statistique et des Études Économiques. 17

	Glossary
	ML Machine Learning. 17, 19, 95
	OR Operations Research. 19
	PM Process Mining. 17, 18, 117
	PMSI the national hospital discharge database or "Programme de Médicalisation des
	Systèmes d'Information". 28, 39
	PU Polyvalent Unit. 69, 95, 96, 116, 117, 118

  2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3 Automated Generation of Patient Population for Discrete Event Simulation using Process Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.4 Conclusion of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

  Traces are grouped into collections called event logs. Discovery algorithms translate event logs into comprehensible business models, such as Petri nets. Process mining (PM) is extensively used in health care studies and clinical pathway discovery. Of 172 studies, Erdogan et al. reported 156 studies that apply some sort of process discovery methodology, but only 5 involve multiple department pathways (Erdogan and Tarhan 2018). Another review by Rojas et al. reported that 60% of the 74 papers studied investigate healthcare activities from a control-flow perspective (Rojas et al. 2016). The mapping of clinical pathways for multimorbid patients (patients with several conditions at the same time) has been studied for several purposes. Zhang et al. used multimorbid patients medical records to identify the most used services (Zhang and Chen 2012), while Aali et al. was interested in the visualization of the evolution of multimorbidity through a patient's diagnoses over time (Aali, Mannhardt, and Toussaint 2022).

Table 1 :

 1 Variable description.

Variable Comment Administrative Anonymous identifier Linking stays information Age (no use in the present study) Sex (no use in the present study) Residence geographical zone (no use in the present study) Medical Clinical origin Analysis and modeling of care pathway information Visited Medical Units Analysis, modeling and simulation LoS in each unit Analysis, modeling and simulation Admission date Simulation (arrival law) LoS in the hospital Analysis, modeling and simulation

Table 2 :

 2 Example of event log after pre-processing.

	Id Case 1 Case 2	MU Start Internal Medicine Cardiology Neurology End Start Post-ED unit Clinical gerontology 2017.09.25 00:00:00 Admission date 2017.04.12 00:00:00 2017.04.12 00:00:01 2017.04.15 00:00:00 2017.04.26 00:00:00 2017.05.02 00:00:01 2017.09.03 00:00:00 2017.09.03 00:00:01 Cardiology 2017.09.26 00:00:00 Clinical gerontology 2017.10.02 00:00:00 End 2017.10.05 00:00:01	Current occupation NA 76 % 112 % 1 84 % NA NA 85 % 92 % 87 % 95 % NA

1

In some circumstances, the number of beds in one unit can be temporarily increased

  Event Simulation, Healthcare Applications, Process Mining.

	I. INTRODUCTION
	Jules Le Lay, Vincent Augusto and Xiaolan Xie are with Mines Saint-Etienne, Univ Clermont Auvergne, INP Clermont Auvergne, CNRS, UMR 6158 LIMOS, Centre CIS, F -42023 Saint-Etienne France Edgar Alfonso-Lizarazo is with Université de Lyon, Univ Jean Monnet Saint-Étienne, LASPI, EA3059, F-42023 Saint-Étienne, France Malek Masmoudi is with University of Sharjah, College of Engineering, United Arab Emirates Bienvenu Bongue is with Université Jean Monnet, Chaire de santé des ainés, Centre Technique d'Appui et de Formation des centers d'Examens de Santé (CETAF) Saint-Étienne, 42000 France Thomas Celarier and Baptiste Gramont are with University hospital, Uni-versity Jacques Lisfranc, Department of Clinical Gerontology, Saint-Étienne, 42000, France

  Select n 1 among elements e candidate in T 0 following probabilities P {Select the first element of σ} 4: σ ← {n 1 } {Assign to σ} 5: while n i+1 ̸ = n sink do Build T i = {e candidate = (e i , P(n i , e))|e ∈ V, P(n i , e) ̸ = 0} {Set of potential transfers}

	1: σ ← ∅ 2: T 0 ← {e candidate V, P(n source , e) ̸ = 0} 3: 6: i ← card(σ) 7:	{Initialize σ} = (e i , P(n source , e))|e ∈ {Build T 0 }

8:

Select n i+1 among elements e candidate in T i following probabilities P {Select the next stay in the trace} 9:

TABLE I :

 I Details of the necessary conditions to trigger the transfer of beds to or from the COVID-19 ICU.

	Daily evolution a	COVID ICU Capacity	Occupation Rate	Action
	⩾ 2	0	-	Add beds

TABLE II :

 II Estimated best response scenarios to answer each situation.

	COVID Arrival Rate	COVID time frame	Bed transfer batch size	Cancellation policy	Cancellation point cutoff	Score
	2 2 2 2 4 4 4 4 6 6 6 6 ⋆ several solutions have the same score; we display the solutions that 0 -365 12 By ICU 0.9 0.159 210 -225 12 By ICU 0.9 0.052 ⋆ 210 -240 12 By ICU 0.9 0.051 210 -255 12 By ICU 0.9 0.051 0 -365 12 By ICU 0.8 0.303 210 -225 12 By Block 0.9 0.084 210 -240 12 By ICU 0.85 0.086 ⋆ 210 -255 12 By ICU 0.9 0.086 0 -365 12 By ICU 0.9 0.439 210 -225 12 By Block 0.9 0.117 210 -240 12 By ICU 0.9 0.119 ⋆ 210 -255 12 By ICU 0.9 0.119 ⋆ cancel the smallest number of patients. See supplementary tables for the exhaustive results in Supplementary tables.

  Jules Le Lay is currently a Ph.D Student at the Center for Health Engineering at Mines Saint-Étienne, FRANCE. He received an MsC in Engineering from Mines Saint-Étienne, Auvergne-Rhône-Alpes in 2017. His research interests include modeling, simulation and optimization of health care systems.Thomas Célarier is an associate professor, geriatric doctor and head of the geriatric medical unit and clinical gerontology of the University Hospital Center of St Etienne. He is also a medical consultant for the AURA G´erontopˆole, and he coordinates many studies on interventional research and works on health care modeling.Xiaolan Xie is a professor and head of the department of Healthcare Engineering of the Center for Biomedical and Healthcare Engineering, Ecole Nationale Sup´erieure des Mines (EMSE), Saint Etienne, France. His research interests include data analytics, design, planning and scheduling, supply chain optimization, and performance evaluation of health care and manufacturing systems. He is author/coauthor of 300+ publications, including 110+ journal articles and six books. He has rich industrial application experience with European industries and is PI for various national and international projects. Pr. Xie is a fellow of IEEE, the founding chair of the IEEE RAS-Technical Committee on Automation in Health Care Management and an IEEE RAS 2017 Distinguished Lecturer. He has been an editor/associate editor for IEEE Transactions on Automation Science & Engineering, IEEE Transactions on Automatic Control, IEEE Transactions on Robotics & Automation and the International Journal of Production Research. He has been a Guest Editor of various special issues on health care engineering and manufacturing systems. He is general chair of the 2021 IEEE Conference on Automation Science and Engineering held in Lyon, France.

	Edgar Alfonso-Lizarazo holds a Ph.D. in industrial engineering from the École Nationale Superieure des Mines de Saint-Étienne, France. Currently, he is researcher member of the LASPI laboratory and Associate Professor at Université Jean Monnet, Saint-´Etienne, France. His research interests include modeling, simulation and optimization of health care systems and medical decision making.
	Vincent Augusto is a professor and a permanent faculty member of Mines Saint-Étienne, Center for Biomedical and Healthcare Engineering. His areas of research are performance evaluation, engineering of health care systems (industrial engineering, simulation), optimization and management of health care systems (operational research), health data and medical decision aids (process mining, machine learning). He defended his thesis in 2008 related to the modeling, analysis and control of flows in healthcare systems using UML and Petri nets (industrial engineering). He defended his "Habilitation 'a Diriger des Recherches" in 2016 on the topic of Health Territories Engineer-
	Malek Masmoudi is engineer and PhD in industrial engineering. He is currently associate professor at Sharjah University, UAE, and was the head

ing. He was a visiting professor at CIRRELT (Interuniversity Research Center on Business Networks, Logistics and Transport), Laval University, Québec, Canada in 2009 and 2015. He is co-responsible for the living lab MedTechLab, where approach design, experiments and innovation are combined with health care engineering. He is director of the Center for Biomedical and Healthcare Engineering of Mines Saint-Étienne since January 2020 (70 people). Bienvenu Bongue holds a Ph.D. in pharmacoepidemiology from Toulouse III University. He is also holder of a Habilitation to supervise research since 2013. Mr Bongue is a research member of the SNA-EPIS Laboratory; EA 4607 of Jean Monnet University (Saint-Étienne, France) and Associate Professor at Montréal University (Québec, Canada). He is also the Chair of "Seniors health -Prevention engineering" of Jean Monnet University of St-Etienne. He is currently supervising 4 PhDs students. Mr. Bienvenu Bongue is a member of numerous scientific councils and the author of numerous scientific papers and reports. of the industrial engineering Master at Jean-Monnet University, France, from 2013 to 2021. He received the Doctorate of Sciences (Habilitation) from Jean-Monnet University in 2017. His main research interests are the applications of data science and optimization techniques to various engineering and management fields. He published two springer books in data science and more than twenty-five international journal papers in operations research and industrial engineering.

Table 1 .

 1 General variables regarding the patient and descriptive information on their

	values
	Variable	Nature	Signification		Mean Range (min-	Std dev
						max)
	pat id	Integer	Anonymous	patient	-	-	-
			identifier
	sex	Binary	Sex of the patient (1 =
			M)
	age	Integer	Age of the patient (in	76.35 60 -104	9.28
			years)
	duration	Integer	Total length of stay of	10.66 0 -250	11.81
			the patient in hospital
			(in days)
	geo	Zip Code Residential zip code of	-	-	-
			the patient
	nb steps	Integer	Number of medical	1.68	1 -14	1.02
			units visited during
			pathway
	mortality a Binary	True if the patient died	0.05	-	-
			during the index stay
	ed adm	Binary	True if the patient was	0.36	-	-
			admitted through the
			ED (Emergency depart-
			ment)
	re hosp30	Binary	True if the patient was	0.16	-	-
			readmitted within 30
			days after index hospi-
			talization
	re hosp365 Binary	True if the patient was	0.47	-	-
			readmitted within 365
			days after index hospi-
			talization

a Predictive features for length of stay.

Table 2 .

 2 The ten most frequent diagnoses appearing in the database

	ICD-10	diagnoses	Number
	Code		of occur-
			rences
	I10	Essential hypertension	6 391
	E11	Type 2 diabetes mellitus	3 189
	I48	Atrial fibrillation	2 681
	E78	Disorders of lipoprotein metabolism a	2 512
	N18	Chronic kidney disease	2 063
	I50	Heart failure	2 035
	J96	Respiratory failure	1 718
	E66	Overweight and obesity	1 658
	I25	Chronic ischemic heart disease	1 643
	Z74

Table 3 .

 3 Accuracy, F1-score and computation times obtained for the prediction of readmission within 365 days.

	Metric	Accuracy	F1-score	Computation time
	DT 1 -All Diags RF 2 -All Diags KNN 3 -All Diags DT -CL 4 score RF -CL 4 score KNN -CL 4 score DT -CL 4 portfolio RF -CL 4 portfolio KNN -CL 4 portfolio DT -HFRS score RF -HFRS score KNN -HFRS score DT -HFRS portfolio RF -HFRS portfolio KNN -HFRS portfolio 0.533 [0.515 -0.552] 0.548 [0.527 -0.568] 0.597 [0.580 -0.614] 0.540 [0.518 -0.563] 0.826 [0.811 -0.840] 0.812 [0.794 -0.829] 0.549 [0.532 -0.568] 0.551 [0.532 -0.572] 0.547 [0.529 -0.564] 0.527 [0.505 -0.551] 0.616 [0.598 -0.634] 0.632 [0.612 -0.651] 0.534 [0.517 -0.552] 0.542 [0.521 -0.563] 0.595 [0.578 -0.613] 0.570 [0.548 -0.592] 0.730 [0.716 -0.748] 0.704 [0.685 -0.725] 0.544 [0.527 -0.560] 0.551 [0.528 -0.560] 0.553 [0.536 -0.570] 0.523 [0.502 -0.545] 0.594 [0.577 -0.611] 0.607 [0.587 -0.627] 0.550 [0.531 -0.567] 0.568 [0.544 -0.588] 0.576 [0.559 -0.594] 0.583 [0.562 -0.602] 0.719 [0.702 -0.734] 0.696 [0.676 -0.715] 1 DT = Decision Tree, 2 RF = Random forest, 3 KNN = K-nearest neighbors, 557.7s 5 291.4 s 335.1 s 6.7s 1 472.2s 2.8s 29.5s 1 754.8s 98.9s 8.2s 1 581.6s 21.8s 18.3s 1 583.3s 96.3s
	4 CL = Calderon-Larrañaga.
	Random forest appears to be the best performing algorithm. The best performance
	measures, accuracy (0.826) and f1-score (0.812), are obtained by using all diagnosis
	June 22, 2022			11/23

Table 4 .

 4 Accuracy, F1-score and ROC-AUC results obtained for the prediction of readmission within 30 days TrS: Training Set,3 TeS: Testing Set.

	Best combi-	Set	Accuracy	F1-score	ROC AUC
	nation
	All diags: RF and RU 1	Balanced TrS 2 TrS	0.891 [0.879 -0.902] 0.703 [0.694 -0.712]	0.889 [0.877 -0.901] 0.500 [0.483 -0.515]	0.891 [0.879 -0.902] 0.772 [0.763 -0.781]
		TeS 3	0.603 [0.586 -0.620]	0.358 [0.328 -0.387]	0.625 [0.602 -0.649]
	CL score: RF and RU	Balanced TrS 0.636 [0.620 -0.652] TrS 0.544 [0.535 -0.555]	0.649 [0.632 -0.667] 0.334 [0.319 -0.349]	0.636 [0.619 -0.653] 0.596 [0.583 -0.608]
		TeS	0.526 [0.507 -0.545]	0.307 [0.275 -0.334]	0.565 [0.535 -0.589]
	CL portfolio: RF and RU	Balanced TrS 0.829 [0.816 -0.842] TrS 0.634 [0.624 -0.645]	0.834 [0.822 -0.848] 0.444 [0.427 -0.460]	0.829 [0.817 -0.842] 0.725 [0.713 -0.735]
		TeS	0.556 [0.537 -0.575]	0.331 [0.304 -0.358]	0.594 [0.573 -0.620]
	HFRS: RF and RU	Balanced TrS 0.633 [0.616 -0.649] TrS 0.535 [0.525 -0.546]	0.653 [0.634 -0.670] 0.335 [0.320 -0.349]	0.633 [0.615 -0.650] 0.597 [0.583 -0.610]
		TeS	0.510 [0.491 -0.532]	0.304 [0.275 -0.328]	0.561 [0.536 -0.584]
	HFRS' portfolio: RF	Balanced TrS 0.835 [0.821 -0.850]	0.830 [0.813 -0.843]	0.835 [0.822 -0.849]
	and RU	TrS	0.645 [0.644 -0.664]	0.440 [0.422 -0.455]	0.713 [0.702 -0.723]
		TeS	0.562 [0.544 -0.580]	0.309 [0.282 -0.336]	0.571 [0.547 -0.594]
	1 RU: Random Undersampling, 2

Table 5 .

 5 Length of stay prediction results.

	Algorithm	MAE	MSE
	Decision Tree -All diags	6.010 103.149
	Random forest -All diags	5.208 72.687
	Decision Tree -Calderon-Larrañaga's score	6.297 97.854
	Random forest -Calderon-Larrañaga's score	6.146 88.163
	Decision Tree -Calderon-Larrañaga's portfolio	5.911 82.767
	Random forest -Calderon-Larrañaga's portfolio 5.894 81.903
	Decision Tree -HFRS score	5.849 104.728
	Random forest -HFRS score	5.609 77.532
	Decision Tree -HFRS portfolio	6.137 100.737
	Random forest -HFRS portfolio	5.728 82.811

Table 1 .

 1 : Examples of the substitution of PU in patients pathways.

	Tables
			Patient ID unit 1 unit 2	unit 3 unit 4
			Patient 1 MU A MU B	-	-
			Patient 1'		PU	-	-
			Patient 2 MU A MU B surgery	ICU
			Patient 2'		PU	surgery	ICU
			Patient 3 MU A surgery	ICU	MU A
			Patient 3'	PU	surgery	ICU	PU
			Patient 4 MU A MU C	MU A MU C
			Patient 4'	PU	MU C	PU	MU C
	Rule	1	2	3		4	5	6	7
	Rule 1 s ≥ 2 Rule 2 ≥ -1 ≥ -0.7 s ≥ 3	2 ≥ s ≥ 5 5 ≥ s ≥ 15 5 ≥ s ≥ 10 3 ≥ s ≥ 10 3 ≥ s ≥ 15 ≥ -0.8 ≥ -0.9 ≥ -0.762 ≥ -1.1 ≥ -1.2

Table 2 .

 2 : Different cut-offs for PU patients rerouting using multimorbidity scores and discriminant analysis.

	Variable	Nature Signification	Mean	Range (min-max) Std dev
	pat id	Integer Anonymous patient identifier	-	-	-
	sex	binary Sex of the patient (0 = M, 1 = F)
	age	Integer Age of the patient (in years)	68.45 years	18 -104	15.31
	duration	Integer Total length of stay of the patient in	8.49	0 -250	11.44
		hospital (in days)
	mortality binary 1 if the patient died during the index	4.33 %	-	-
		stay
	ed adm	binary 1 if the patient was admitted through	13.36 %	-	-
		the ED (Emergency department)

Table 3 .

 3 : General variables regarding the patient and descriptive information on their values. Canon. corr. Eigen-value Variance Likelihood ratio F statistic Df1 a Df2 b Prob > F effect degrees of freedom of the function; b error degrees of freedom of the function

	0.5973	0.5546	1.0000	0.6433	5.9394	69	739	0.0000 e

a

Table 4 .

 4 : Information of the discriminant function calculated by STATA

	True labels	0	1	Total
	0	96 83.49 % 16.52 % 100 % 19 115
	1	97 13.98	597 86.02	694 100 %
	Total	193 23.86 % 76.14 % 100 % 616 809
	Priors	0.5000	0.5000

Table 5 .

 5 : Classification matrix resulting of the discriminant function

	Figures
	Figures with captions
					Discharge
						t d
	Planned					MU 1
	admissions
		t p			.	t 1
					.
					.
	Emergent					MU N l
	admissions	t e	p 1	Emergency t ED	p 2	t Nl
				department

Table 2 :

 2 Occupancy ratio and maximum occupancy in the MR ward at infinite capacities.

		Mean OR (%) (std dev) Average maximum Occupancy (std dev)
	Scenario 0	83.90 (-)	16 (-)
	Scenario 10	52.19 (1.53)	18.0 (1.63)
	Scenario 20	52.94 (1.65)	18.2 (1.23)
	Scenario 30	54.00 (2.49)	18.7 (1.50)
	Scenario 40	54.48 (2.52)	18.1 (1.52)

Table 3 :

 3 Occupancy ratio and maximum occupancy in the MR ward with finite capacities.

		MR Capacity Mean OR (%) (std dev) Maximum Occupancy (std dev)
	Scenario 0'	15	84.84 (-)	15 (-)
	Scenario 11	15	58.94 (2.00)	15 (-)
	Scenario 12	8	99.97 (0.04)	8 (-)
	Scenario 13	10	88.75 (3.54)	10 (-)
	Scenario 21	15	57.38 (1.51)	15 (-)
	Scenario 22	8	99.95 (0.09)	8 (-)
	Scenario 23	10	91.72 (4.69)	10 (-)
	Scenario 31	15	59.86 (2.25)	15 (-)
	Scenario 32	8	99.98 (0.03)	8 (-)
	Scenario 33	10	92.31 (3.87)	10 (-)
	Scenario 41	15	60.40 (1.83)	15 (-)
	Scenario 42	8	99.96 (0.07)	8 (-)
	Scenario 43	10	91.57 (4.22)	10 (-)

Table 4 :

 4 Occupancy ratio and maximum occupancy in the MR ward with Covid patients.

		Mean OR (%) (std dev) Average maximum Occupancy (std dev)
	Scenario 0	83.9 (-)	16.0 (-)
	Scenario 50	60.8 (1.52)	65.0 (9.37)
	Scenario 51	69.0 (2.12)	63.3 (10.12)
	Scenario 52	67.1 (2.43)	62.5 (9.96)
	Scenario 53	66.2 (1.99)	59.5 (7.06)
	Scenario 54	64.6 (1.96)	61.5 (7.15)

[START_REF] Aubert | Best Definitions of Multimorbidity to Identify Patients With High Health Care Resource Utilization[END_REF].4% on the 1-year time frame of the Enquête Santé et Protection Sociale

et 2014 (ESPS) database and 39.0% using the Enquête Handicap-Santé Ménage (HSM) database.

SF-36 PF = Short Form

questions physical functioning scale.

The present analysis was done before the publication of Stirland et al. review.

Authorized licensed use limited to: ECOLE DES MINES DE SAINT ETIENNE. Downloaded on August 30,2021 at 08:14:51 UTC from IEEE Xplore. Restrictions apply.

Remerciements

Acknowledgements

The authors would like to thank Dr. Laure Martinez for her patience and expertise in the analysis of complex pathways and for arbitrating the disagreements between the two authors labelling complex pathways.

Above 70 years old Department of Veterans Affairs Preventive Care Trial -DVA PCT HRQoL (SFAbove 60 years old Swedish National study of Aging and Care-Kungsholmen and Swedish National

Funding

This work was funded by the Agence Nationale de la Recherche (ANR), grant number ANR-18-CE19-0016.

Contents

AUTHOR BIOGRAPHIES

JULES LE LAY is currently a Ph.D. student at the Center for Health Engineering at Mines Saint-Étienne. He received an MSc in engineering from Mines Saint-Étienne in 2017. His research interests include modeling, simulation and optimization of healthcare systems. His email address is jules.le-lay@emse.fr.

JULIA NEVEU is an MSc student in engineering at ESME Sudria. She performed an internship at Mines Saint-Étienne on this project. Her research interests include computer sciences and data analysis.

BENJAMIN DALMAS is an assistant professor in healthcare data science in the Department of Biomedical and Healthcare Engineering. He holds a Ph.D. from Clermont-Auvergne University for his work on the modeling of healthcare pathways. His research interests are the modeling and prediction of patient pathways through the analysis of medical and administrative data using a wide range of techniques, from process and data mining to machine learning. His email address is benjamin.dalmas@emse.fr.

VINCENT AUGUSTO is a professor and a permanent faculty member of Mines Saint-Étienne, Center for Biomedical and Healthcare Engineering. His areas of research comprise performance evaluation, optimization and management of healthcare systems, operational research, health data and medical decision aid (process mining, machine learning). He holds a Ph.D. for his work on the modeling, analysis and control of flows in healthcare systems. He defended his "Habilitation à Diriger des Recherches" on Health Territories Engineering in 2016. He was a visiting professor at CIRRELT (Inter-university Research Center on Business Networks, Logistics and Transport), Laval University, Québec, Canada, in 2009 and 2015. He has been co-responsible for the living lab MedTechLab and director of the Center for Biomedical and Healthcare Engineering of Mines Saint-Étienne since January 2020. His email address is augusto@emse.fr.

Chapter 4. Prediction of Hospital Readmission of Multimorbid Patients Using Machine

Learning Models able by CHUSE. We use descriptive statistics to understand the profile of this population. Two different indicators have been calculated using diagnosis information available: the multimorbidity index developed by Calderon-Larrañaga et al. [14] and a frailty score developed by Gilbert et al. to predict readmission of older patients The ability of those indexes to predict short and long term readmissions and length of stay were assessed and compared to the use of all available diagnoses.

Prediction of Hospital Readmission of Multimorbid Patients

Using Machine Learning Models

ABSTRACT

The care of multimorbid patients in health care centres is a complex task. Interactions between diseases and treatments, iatrogenic dependence in elderly patients and the decompression of secondary chronic diseases during hospitalization may force the care management team to modify the patient's care pathway and transfer the patient to other units. To address this problem, the University Hospital of Saint-Étienne opened a polyvalent unit to manage the care of multimorbid patients inside the unit. Operations research techniques such as discrete event simulation have been proven to be useful for modelling health care processes. In this study, we present a simulation model to assess the effects of adding such a unit on the multimorbid patient's flow through the hospital.

KEYWORDS

Discrete Event Simulation, Optimization, Health care analytic, multiborbidity.

Word Count: 5300 words.

Declaration of interests

The authors report that there are no conflicts of interest to declare.

A.1 Motivation

Bed management in health care centers is a complex task, as decision makers as they need to find balance between two elements: make full use of the available resource of the department and prevent emergencies by keeping available a small number of beds. As part of our collaboration with the hospital, we developed a DES model intended as a digital twin of the hospital, able to mimic the global behavior of the health care center and realistically represent the changes in bed occupancy when a perturbation occurs, whether those perturbations are predictable and benign (staff on leave during holidays, modification of a pathway) or not, and represent an unusual situation (failure of an equipment, global pandemic). In particular, we study the impact of the increase of admissions through the emergency department. This model was to serve as a base for the devel-Appendix A. First appendix: Impact of COVID-19 Epidemics on Bed Requirements in a Healthcare Center Using Data-Driven Discrete-Event Simulation opment of new care pathways for multimorbid patients. This work coincided with the beginning of the COVID-19 pandemic, which represented an opportunity to test the ability of our model to evaluate the impact of a new COVID pathway on the occupation rate of other medical units.

The conference publication here under presents the hospitalization process model established and explored two scenarios: the increase of patients admissions through emergency department, motivated by the evidence of a constant increase of ED visits in France, and a first response to COVID-19 through the reorganization of the Multipurpose Recovery ward.

A.

Key Performance Indicators

The output variables of the simulation are the mean occupancy ratio of the "Multi-purpose Recovery" (MR) ward and the maximum occupancy over the studied period. The first figure will allow us to compare the global impact of the epidemic on the ward, while the second gives us the maximum amount of resources that are required to face the demand and thus quantify how the service is overwhelmed. Those indicators were calculated over different periods depending on the scenario. In scenario 0 we consider the year 2018, for scenarios 10-43 we consider the year 2019, and for scenarios 50-54 regarding Covid patients, the indicators were calculated between the 01-01-2020 and 06-30-2020.

Results

For each scenario we set 10 replications and calculated the mean and standard deviation of the two KPIs. The results are summed up in Tables 2, 3 and 4.

Validation Scenario

In this scenario we inject history log and compare the output of the simulation with real KPIs. The results in Table 2 show a Mean OR of 83.90%. This result is coherent with the real observed occupation rate. The maximum occupancy in the service is 16. This ward is required by a lot of patients and sudden surges can occur. However doctors and nurses will manage patients pathways to keep the number of patients under the AUTHOR BIOGRAPHIES JULES LE LAY is currently a Ph.D Student at the Center for Health Engineering at Mines Saint-Etienne, FRANCE. He received a MsC in Engineering from Mines Saint-Etienne, Auvergne-Rhône-Alpes in 2017. His research interests include modelling, simulation and optimisation of health-care systems. His e-mail address is jules.le-lay@emse.fr.

Appendix A. First appendix: Impact of COVID-19 Epidemics on Bed Requirements in a Healthcare Center Using Data-Driven Discrete-Event Simulation

A.3 Conclusion

This communication presented a macroscopic DES model of the hospital of Saint-Étienne, able to accurately replay patients pathways. This model can serve as a digital twin to monitor the hospital's patients flow. We studied the case of COVID-19, which greatly disrupted care in 2020, and tested several response scenarios to absorb the additional patients' flow.

The model discussed here is an early version of the one developed in chapter 2. The two identified improvement possibilities, use realistic patients pathway generated from the history, and dynamically respond to the epidemics evolution, were implemented and are described in this chapter.

List of publications

Journal papers

Abstract

The care for multimorbid patients (patients having multiple chronic diseases) in health care centers is a growing concern for health professional. Multiple research on care pathways in Europe aim to ease the simultaneous treatment of multiple diseases, both inside and outside the hospital. This has to be dealt with in a complex environment, where multiple factors interfere with the management of multidisciplinary pathways. The constant increase of hospital admissions through the emergency department delays care and support from the pertinent units of the hospital. The expected increase of the elderly population in the Loire department, in France, and the established link between age and multimorbidity raises concerns of an increase in multimorbid patient admissions. The hospital of Saint-Étienne CHUSE is a university hospital at the head of the Loire Hospital Group (GHT, Groupement Hospitalier de Territoire in French), a large network of health care centers. CHUSE has 1,802 beds dispatched in more than 60 medical units. In 2020, CHUSE admitted more than 90 000 patients for hospitalization and 78 400 patients were seen in its Emergency Department.

We developed a methodology to generate a population with realistic clinical pathways using Process Mining techniques. A digital twin of the hospital was built using AnyLogic®and tested using the casestudy of the COVID-19 pandemic management. We evaluated the ability of multimorbidity indexes to predict hospital readmission. Those elements allowed to investigate the creation of a Polyvalent Unit for the care of multimorbid patients in the CHUSE through simulation. An optimization algorithm was implemented to size this unit using the simulation model.