Thèse De Doctorat De

THÈSE DE DOCTORAT DE

L’UNIVERSITÉ DE RENNES 1

ÉCOLE DOCTORALE NO 601 Mathématiques et Sciences et Technologies de l’Information et de la Communication Spécialité : Signal, Image,Vision

Par Gustavo GUERRERO Analyse à base de modèles des interactions cardiorespiratoires chez l’adulte et chez le nouveau-né

Unité de recherche : Laboratoire de Traitement du Signal et de l’Image - INSERM 1099

Thèse soutenue à Rennes le 23 Septembre 2020 devant le jury composé de :

Rapporteurs : Catherine MARQUE, Professeur, UTC Compiègne. Carolina VARÓN, Maître de Conférences, Université Technologique de Delft. Examinateurs : Erwan,L’HER, Professeur universitaire à l’Université de Bretagne Occidentale et Praticien Hospitalier au CHU de Brest. Dir. de thèse : Alfredo HERNÁNDEZ, Directeur de Recherche INSERM, Université de Rennes 1. Co-dir. de thèse : Virginie LE ROLLE, Maître de Conférences, Université de Rennes 1. Invité : Alain BEUCHÉE, Professeur, Université de Rennes 1.

ACKNOWLEDGEMENT

"Alfredo et Virginie sont trop forts. Spécialement Virginie..."

First, I would like to express my deepest appreciation to my thesis advisors, Alfredo Hernández and Virginie Le Rolle. Alfredo has been a mentor for me during all my professional career and he has always treated me like family. I am eternally grateful for the opportunities he has given me and I will always be in debt to him. Virginie is the perfect equilibrium between kindness and toughness. She kept me on the right path when I was not at my best, and she has always been lovely and generous. I would also like to extend my deepest gratitude to Lotﬁ Senhadji for being an example of leadership at the LTSI, earning the respect and esteem of all the under his guidance. Also, I would like to extend my sincere thanks to all the colleagues from the Lab, especially to Fabrice Tudoret for being easygoing and helpful since day one. To Guy, Oscar, Johanne, Mireille, Miguel, Gabriel, and many others for always having a smile in the halls of the LTSI and for letting me be part of their projects in many opportunities. I am also grateful to Patricia, Soizic, and Muriel for being so helpful every time I needed something and for the funniest conversations I had at the Lab. I also had the pleasure of working with Samuel Crand and Sophie Allain, during my ATER contract. Their passion for teaching is inspiring and it has made me consider it as a career path in the future. I cannot begin to express my thanks to my closest friends in Rennes. To Fatima, for being my roommate and my family in Rennes, and for being there for me every day no matter what. To Andrés, for adding a lot of fun, craziness, and spirituality to my existence. To Carole, for the immense support and aﬀection when I needed it the most. And to Luiz, Fausta, and Moises for all the happiness that you brought into my life. I am also grateful to all the other Ph.D. students and Postdocs that had the pleasure to spend time with. To Pablo, Karim, Nadine, Jenny, Matthieu, and Carlos for the laughs and the good times inside and outside the Lab. To the RU gang, Remo, Cristhyne, and Kimi for sharing a meal every day at 11h45, surrounded by good vibes and colorful conversations. Special thanks to all the people that I meet during my research stay at the University of Sherbrooke in Canada. Jean-Paul Praud and Nathalie Samson for their insights in

3 newborn physiology and their helpful advise to survive Canadian winter. To Kim, Émilie, and Cass, because they made me fall in love with Quebec and its people. And ﬁnally and the most important to my family, Yazmín, Humberto and, Nimzay. They are always alongside of me even if the distance and the circumstances keep us far away. I would not be here if it were not for them.

4 RÉSUMÉENFRANÇAIS

Le syndrome d’apnée du sommeil (SAS) est une maladie multifactorielle carac- térisée par des épisodes répétés d’arrêt de la respiration (apnées) ou d’importantes réduc- tions de l’amplitude de la respiration (hypopnées) pendant le sommeil du patient. Chez l’adulte, l’apnée est définie comme un arrêt de la respiration qui dure plus de 10 secondes. Ces événements provoquent des réponses cardio-respiratoires aiguës (désaturation en oxygène, hypoxie intermittente, réponses autonomes) et de fortes modifications de la structure du sommeil. À long terme, ces effets aigus augmentent la possibilité de souffrir de diverses pathologies chroniques, telles que l’hypertension artérielle, les accidents vasculaires cérébraux, les maladies cardiaques et certains troubles métaboliques. Il est estimé qu’entre 6 et 17% de la population adulte souffre de SAS, ce qui représente un problème majeur de santé publique. Les apnées sont également observées chez les nouveau-nés sous forme d’épisodes d’ap- née-bradycardie, en particulier chez les prématurés. Chez les nouveau-nés prématurés, le système de contrôle cardio-respiratoire est moins développé que chez les nouveau-nés à terme, ce qui entraîne une instabilité cardio-respiratoire. Par définition, l’apnée du prématuré est considérée cliniquement comme un arrêt de la respiration d’au moins 20 secondes ou de plus de 10 secondes si elle est associée à une bradycardie et/ou une désaturation en oxygène. Le syndrome d’apnée affecte plus de 50% des prématurés et tous les enfants dont le poids à la naissance est inférieur à 1000 g. En France, environ 7% des nouveau-nés sont des naissances prématurées. Le Chapitre 1 présente une introduction à la physiopathologie cardiorespiratoire, décrivant, en particulier, l’état de l’art sur les effets impliqués dans le SAS et dans l’apnée de la prématurité. En fait, les mécanismes physiologiques du SAS et des apnées du prématuré ne sont toujours pas complètement élucidés. L’interprétation des réponses cardio-respiratoires ai- guës aux apnées et hypopnées peut être difficile en raison de la variété des processus impliqués (régulation autonomique, cardiovasculaire, respiration, chémoréflexe, etc.), qui doivent être considérés conjointement pour une analyse appropriée. Une approche à base de modèles semble particulièrement adaptée à cette tâche, car elle permet d’intégrer ex- plicitement les connaissances physiologiques dans le traitement des données et d’analyser

5 les mécanismes sous-jacents qui sont difficiles à observer. Un état de l’art des modèles existants chez l’adulte et le nouveau-né est présenté au Chapitre 2. À notre connaissance, aucun modèle complet des interactions cardio-respiratoires n’a été utilisé pour l’étude des apnées de l’adulte de manière spécifique à chaque patient. Par ailleurs, aucun modèle adapté au nouveau-né n’a été proposé dans la littérature. L’objectif principal de cette thèse de doctorat est de proposer une approche, basée sur des modèles de la physiologie, afin d’améliorer la compréhension des réponses cardio-respiratoires aiguës aux événements d’apnée et d’hypopnée chez les adultes et les nouveau- nés. Afin de donner un cadre méthodologique à ce travail, le Chapitre 3 présente les outils de simulation, d’analyse de sensibilité et d’identification des modèles proposés dans cette thèse. Ces outils, développés depuis plusieurs années dans l’équipe SEPIA du LTSI, ont pu être exploités et améliorés dans le cadre de la thèse. L’environnement de modélisation et de simulation, précédemment développé dans notre équipe, appelé "Multiformalism Modeling and Simulation Library" (M2SL) a été utilisé pour l’implémentation de nos modèles. Cette librairie permet le couplage de modèles hétérogènes, définis par diffé- rents formalismes et échelles temporelles. Pour l’analyse paramétrique des modèles, des analyses de sensibilité en utilisant la méthode de "Screening" de Morris ont été utilisées. Le principe de cette méthode est d’effectuer une estimation de la sensibilité, un paramètre à la fois, en observant les effets élémentaires de chaque paramètre sur les sorties du modèle et de répéter cette évaluation dans plusieurs points de l’espace paramétrique. La méthode de Morris est spécialement utile dans des modèles avec un nombre important de paramètres et des temps de simulation élevés. L’identification des paramètres a été réalisée à l’aide d’algorithmes évolutionnaires, en particulier l’algorithme à évolu- tion différentielle (Differential evolution). Pour implémenter cette méthode, l’environnement M2SL a dû être adapté pour coupler le modèle avec une librairie d’optimisation parallèle massive. Cette librairie permet d’effectuer plusieurs identifications parallèles réparties dans plusieurs clusters de calcul. Le Chapitre 4 est dédié à la description d’un modèle original intégrant les interactions entre les fonctions cardiaques, respiratoires et autonomiques chez l’adulte. Ces travaux s’inscrivent dans la continuité des travaux antérieurs de notre équipe en modélisation physiologique intégrant de nouveaux composants et modèles existants dans la littérature nécessaire à l’étude des apnées. Le modèle proposé est composé de quatre composants interconnectés : i) le système respiratoire, ii) le système cardiovascu-

6 laire, iii) l’échange gazeux (dans les poumons et le métabolisme) et iv) le contrôle neural. Ce dernier est composé par le baroréflexe, le chémoréflexe et les récepteurs d’étirement pulmonaire. Le modèle complet intègre les principaux mécanismes physiologiques impli- qués dans la réponse cardio-respiratoire à l’apnée et constitue un exemple typique de modélisation intégrative en physiologie. Les simulations du modèle ont pu être confron- tées aux valeurs de référence de la littérature et reproduisent les signaux physiologiques ainsi que la dynamique dans le contexte des apnées obstructives. Le Chapitre 5 décrit une analyse quantitative, à base des modèles, des interactions cardio-respiratoires pendant l’apnée du sommeil. Une analyse pa- ramétrique complète du modèle adulte dans le contexte des apnées du sommeil a été réalisée. Les paramètres influents sur la réponse à une apnée obstructive ont pu être mis en évidence : fraction d’oxygène dans l’air inspiré, les taux métaboliques, le chémoré- flexe et la mécanique respiratoire. Ces résultats sont cohérents avec la littérature et avec les connaissances actuelles sur les approches thérapeutiques optimales. Cependant, l’approche proposée présente une hiérarchisation quantitative de la sensibilité de chacun de ces paramètres sur les réponses physiologiques. A partir des résultats de l’analyse de sen- sibilité, un sous-ensemble des paramètres du modèle a été sélectionné pour réaliser une identification spécifique-patient sur des données expérimentales, afin de reproduire la dynamique de la saturation d’oxygène (SaO2) pendant une apnée obstructive. La base de données cliniques étudiée est composée de 107 apnées obstructives réparties sur 10 patients (étude HYPNOS – projet ANR PASITHEA). A partir des paramètres identifiés et en utilisant des techniques de clustering, un phénotypage de patients apnéiques a été obtenu. Trois clusters ou phénotypes ont pu être mis en évidence, décrivant trois dynamiques différentes liées aux apnées du sommeil et à la respiration périodique. Deux clusters sont liés à une instabilité de la ventilation due à un gain de la boucle de contrôle respiratoire élevé et à une respiration périodique / Cheynes-Stokes. Le troisième cluster représente la réponse typique à un événement OSA. La pré- dominance de l’un de ces groupes chez un patient peut aider à la prise de décision entre différents traitements comme la CPAP ou l’oxygénothérapie. Ce travail constitue, à notre connaissance, la première modélisation spécifique-patient de la réponse cardio-respiratoire à une apnée. Dans le Chapitre 6, deux versions d’un modèle intégré de la physiologie cardio- respiratoire du nouveau-né à terme et prématuré ont été proposées. Les travaux antérieurs dans le domaine se sont concentrés sur des aspects isolés de la physiologie du

7 Résumé en français nouveau-né ou sur les interactions cardiorespiratoires chez l’adulte. Le principal défi était de combiner ces sous-modèles et les travaux antérieurs de notre équipe chez l’adulte afin de proposer un modèle intégré du nouveau-né à différents âges gestationnels. Une com- paraison qualitative entre les simulations d’une apnée centrale, avec les deux modèles, et les données expérimentales a été réalisée. Une analyse paramétrique complète de ces modèles dans le contexte de l’apnée-bradycardie du nouveau-né a été également proposée. Les résultats de l’analyse de sensibilité mettent en évidence les facteurs physiologiques les plus influents lors des épisodes d’apnée-bradycardie, comme la fraction d’oxygène inspiré, les taux métaboliques et le volume pulmonaire. Les équations et les paramètres des mo- dèles nouveau-né à terme et prematuré et du modèle adulte sont spécifiés dans les annexes du manuscrit. Les résultats de la thèse ouvrent de nouvelles perspectives pour la prise en charge, l’optimisation et la personnalisation de certaines thérapies (CPAP, PEEP, oxygénothéra- pie,...) en unités de soins intensifs néonatals et adultes. Les patients présentant un risque plus élevé d’instabilité de la ventilation et de respiration périodique pourraient notamment être identifiés afin d’améliorer et de personnaliser le traitement. L’approche adoptée dans ce travail, combinant la modélisation physiologique multi-résolution, analyse de sensibilité et identification des paramètres, offre un cadre méthodologique original et particulière- ment favorable à la proposition de nouveaux outils thérapeutiques et à l’évaluation in silico de dispositifs médicaux. De manière générale, les modèles proposés constituent de nouveaux outils pour la compréhension des mécanismes physiologiques sous-jacents et l’interprétation des données cliniques dans le cadre des syndromes d’apnée de l’adulte et de l’apnée du prématuré.

8 TABLEOF CONTENTS

List of acronyms 14

Introduction 19

1 Physiology of cardiovascular and respiratory control 25 1.1 Cardiovascular control ...... 25 1.1.1 The cardiovascular system ...... 25 1.1.2 The heart ...... 26 1.1.3 Cardiac electrical conduction ...... 28 1.1.4 The nervous system ...... 30 1.2 Respiratory control ...... 34 1.2.1 Respiratory mechanics ...... 34 1.2.2 Neural control of breathing ...... 35 1.2.3 Observation of the respiratory activity ...... 42 1.3 Cardio-respiratory interactions ...... 43 1.4 Pathologies involving deregulation of the cardio-respiratory control . . . . . 46 1.4.1 Sleep apnea syndrome (SAS) ...... 46 1.4.2 Cheyne-Stokes respiration (CSR) ...... 50 1.4.3 Apnea-bradycardia episodes on preterm infants ...... 52 1.5 Conclusion ...... 54

2 State of the art of cardio-respiratory modeling 69 2.1 Models of the cardiovascular system ...... 69 2.1.1 Model of the cardiac electrical activity ...... 69 2.1.2 Cardiac mechanics and circulatory model ...... 70 2.2 Respiratory models, gas exchange and gas transport ...... 71 2.3 Neural cardiorespiratory control ...... 72 2.3.1 Baroreﬂex modeling ...... 72 2.3.2 Chemoreﬂex modeling ...... 73 2.4 Metabolism ...... 74

9 TABLE OF CONTENTS

2.5 Cardio-respiratory modeling ...... 74 2.6 Model adaptation to newborns ...... 76 2.6.1 Cardiovascular model ...... 76 2.6.2 Respiration model ...... 76 2.6.3 Baroreﬂex ...... 76 2.6.4 Chemoreﬂex ...... 77 2.6.5 Gas exchange and gas transport ...... 77 2.6.6 Metabolism ...... 77 2.7 Conclusion ...... 77

3 Methods and tools for simulation and model analysis 87 3.1 Modeling and simulation ...... 87 3.1.1 Multi-formalism Modeling and Simulation Library (M2SL) . . . . . 92 3.2 Sensitivity Analysis ...... 97 3.2.1 Local sensitivity analysis ...... 101 3.2.2 Global sensitivity analysis ...... 101 3.2.3 Screening methods ...... 102 3.2.4 Proposed approach ...... 104 3.3 Parameter identiﬁcation ...... 105 3.3.1 Deterministic approaches ...... 106 3.3.2 Stochastic approaches ...... 106 3.3.3 Proposed approach ...... 112 3.4 Conclusion ...... 114

4 Adult cardio-respiratory model 121 4.1 Method ...... 121 4.1.1 Model structure ...... 121 4.1.2 Cardiovascular system model ...... 121 4.1.3 Respiratory model ...... 127 4.1.4 Gas exchange model ...... 129 4.1.5 Neural control ...... 133 4.2 Results and discussion ...... 135 4.2.1 Comparison between model simulation and physiology on baseline values ...... 135 4.2.2 Simulation of an obstructive apnea event ...... 141

10 TABLE OF CONTENTS

4.3 Model limitations ...... 144 4.4 Conclusion ...... 144

5 Model based analysis of cardio-respiratory interactions during sleep apnea 149 5.1 Experimental data ...... 149 5.1.1 Signal processing analysis ...... 152 5.2 Sensitivity analysis ...... 153 5.2.1 Methodology ...... 154 5.2.2 Results ...... 155 5.2.3 Discussion ...... 161 5.2.4 Conclusion ...... 164 5.3 Parameter identiﬁcation...... 165 5.3.1 Methodology ...... 165 5.3.2 Results ...... 168 5.3.3 Discussion ...... 174 5.3.4 Conclusion ...... 177

6 Newborn cardio-respiratory model 183 6.1 Methodology ...... 184 6.1.1 Integrated cardio-respiratory model adapted to the newborn physiology ...... 184 6.1.2 Sensitivity analysis ...... 189 6.1.3 Experimental protocol and data ...... 189 6.2 Results ...... 190 6.2.1 Comparison between model simulations and physiology baseline values191 6.2.2 Simulation of an apnea in newborns and qualitative comparison with experimental data ...... 196 6.2.3 Sensitivity analysis ...... 196 6.3 Discussion ...... 200 6.4 Conclusion ...... 202

Appendices 213

11 TABLE OF CONTENTS

A Equations 214 A.1 Cardiovascular system...... 214 A.1.1 Time-varying elastances driving functions ...... 214 A.1.2 Pressures, Volumes and Flows ...... 215 A.2 Respiratory system ...... 219 A.3 Baroreflex and pulmonary stretch receptors...... 220 A.3.1 Baroreflex dynamics ...... 220 A.3.2 Feedback regulation of the heart rate ...... 220 A.3.3 Sympathetic regulation mechanisms ...... 221 A.4 Chemoreflex ...... 223 A.4.1 Central chemoreflex ...... 223 A.4.2 Peripheral chemoreflex ...... 223 A.4.3 Outputs ...... 225 A.5 Lung gas exchange ...... 225 A.5.1 Lung gas exchange (dead volume, alveoli, pulmonary capillaries) . . 225 A.6 Gas exchange and gas transport ...... 226 A.6.1 Alveolar pressures and concentrations...... 226 A.6.2 Arterial pressures and concentrations...... 227 A.6.3 Arterial oxygen saturation...... 227 A.7 Tissue gas exchange...... 227 A.8 Gas transport ...... 227 A.9 Other cardio-respiratory interactions ...... 228

B List of parameters 229 B.1 Cardiovascular system ...... 229 B.2 Respiration system ...... 232 B.3 Baroreﬂex and pulmonary stretch receptors ...... 233 B.4 Chemoreﬂex ...... 236 B.5 Lung gas exchange ...... 239 B.6 Tissue gas exchange ...... 240 B.7 Gas transport ...... 241 B.8 Other cardio-respiratory interactions ...... 241

List of Figures 241

12 TABLE OF CONTENTS

List of Tables 245

List of Publications 246

13 LISTOFACRONYMS

Ach Acetylcholine AHI Apnea-hypopnea index ANS Autonomic nervous system AP Action Potential ARDS Acute respiratory distress syndrome ARP Absolute refractory period ATP Adenosine triphosphate AV Atrio ventricular AVN Atrioventricular node BMI Body mass index BR Breathing rate C Gas concentration CA Central Apnea CES Cardiac electrical conduction system CHF Cogestive heart failure CNS Central Nervous System

CO2 Carbon dioxide CPAP Continuous positive airway pressure CRC Cardio-respiratory coupling CSR Cheyne-Stokes Respiration CVS Cardiovascular system D Delay DAE Diﬀerential algebraic equations DE Diﬀerential evolution DRG Dorsal respiratory group DSS Decision support system E Elastance EA Evolutionary algorithms

14 TABLE OF CONTENTS

ECG Electrocardiogram ED Experimental design EE Elementary eﬀect EEG Electroencephalogram EMG Electromyogram EOG Electrooculogram FI Fraction of inspired gas FRC Functional residual capacity G Gain HR Heart rate HRR Baseline resting heart rate K Gain L Inertia LA Left Atrium LBB Left bundle branch LBH Lower bundle of His LG Loop Gain LV Left Ventricle M Metabolic rate M2SL Multi-formalism Modeling and Simulation Library NAVA Neurally adjusted ventilatory assist NE Norepinephrine NN Newborn NP Nasal pressure NTS Nucleus Tractus Solitarius

O2 Oxygen OA Obstructive apnea ODE Ordinary diﬀerential equation OH Hypopnea OSA Obstructive sleep apnea P Pressure Pa Partial pressure of a gas

15 TABLE OF CONTENTS

PB Periodic breathing PCA Principal component analysis PEEP Positive end-expiratory pressure pFRG Parafacial respiratory group PNS Peripheral nervous system PRG Pontine respiratory group PSG Polysomnography PSR Pulmonary stretch receptors PVR Pulmonary vascular resistance Q Flow R Resistance RA Right Atirum RBB Right bundle branch REM Rapid eye movement RMSE Root-mean-square error rRMSE Relative root-mean-square error RSA Respiratory sinus arrythmia RTN Retrotrapezoid nucleus RV Right Ventricle S Sympathetic SAN Sinoatrial node

SaO2 Oxygen saturation SAS Sleep apnea syndrom SCM Sternocleidomastoid muscles SDD Slow diastolic depolarization SNS Somatic nervous system SVR Systemic vascular resistance τ Time constant UBH Upper bundle of His UDP upstroke depolarization period V Volume Vag Vagal

16 TABLE OF CONTENTS

VRC Ventral respiratory column

INTRODUCTION

Apnea is defined as the cessation of respiratory airflow during a certain period of time and it could be of different origins (central, obstructive, or mixed). Hypopnea is defined as a significant reduction in respiratory amplitude. These respiratory events evoke acute cardio-respiratory responses, modifying heart rate, oxygen saturation and sleep structure (in the case of sleep apnea), as well as several other physiological variables. The repetition of breathing pauses episodes may have severe long-term consequences and are associated with an increased risk of hypertension, heart disease, stroke, and certain metabolic disorders [SS15]. In adults, the occurrence of apnea episodes is usually associated with sleep apnea ; whereas in neonates, apnea-bradycardia episodes are frequently observed in the preterm period. Sleep apnea syndrome (SAS) is a multifactorial condition characterized by repeated episodes of apnea and hypopnea during the patient’s sleep. Between 6% to 17% of the adult population suffers from SAS, but the syndrome is highly under-diagnosed [Sen+17; Ant+16]. The gold-standard test for the diagnosis of SAS is the clinical polysomnography (PSG), which consists of a complete multi-channel recording and monitoring of cardio-respiratory, neurological and sleep characterization signals during a whole night. Due to the long waiting-lists in sleep laboratories, SAS remains highly under-diagnosed. Therefore, there is a need for new methods and tools to better understand the pathophysiology of this condition and new reliable automatic diagnostic strategies for improving the detection and management of patients suffering from SAS [Eps+09]. Apnea-bradycardia of prematurity is a common problem in neonatal intensive care units (NICU) [Cha+18]. In France around 7% of newborns are preterm births. Prematu- rity, defined as birth before 37 weeks of gestation, is one of the leading causes of infant mortality [SL12]. It accounts for more than 70% of newborn deaths and more than half of long-term neurological disorders [RSG12]. The early postnatal period is critical because of the immaturity of the cardiovascular, respiratory and autonomic nervous systems [LSE11]. This predisposes newborns, especially preterm infants, to develop several pathologies such as apnea of prematurity, apparent life-threatening events and sudden infant death syndrome. A better understanding of the diseases affecting preterm newborns may help reduce

19 Introduction morbidity, mortality and health-care costs. In both SAS and newborn apnea-bradycardia, the underlying mechanisms that generate the occurrence of apneas are still not completely elucidated. This is partly due to a lack of observability of major components of the physiological system, but also to the difficulty of the interpretation of multisource data acquired during apnea episodes, because of the multidimensionality of the problem and the variety of processes involved (autonomic regulation, respiration, sleep mechanisms,...). All these mechanisms should be jointly considered for an appropriate interpretation. A model-based method could be particularly interesting in this case because: i) it allows the integration of physiological knowledge on data processing tasks, ii) it permits the analysis of underlying mechanisms that are difficult or impossible to observe, whilst avoiding invasive measurements (vagal and sympathetic activities, arterial pressure,...), iii) it could help to improve the therapy planning by evaluating different hypotheses or configuration scenarios of the system (mechanical ventilation strategy,...). Most of the existing models in the literature represent isolated aspects of the physiology (cardiovascular models [Ten+06; Sá +06; Le +08; LOH11], respiratory models [GBB67; BT00; Le +13; Fix+18], chemoreflex [UM02; ACU11; Mol+14; Bar+17] models, etc.). Only few integrated models of cardio-respiratory interactions have been proposed [Che+10; Alb+15]. However, these integrated models are either still limited by the low variety of physiological mechanisms that are integrated [UM00; MU01; Lu+01; Ell+13] or remain too complex to perform patient-specific analysis [Alb+15]. Moreover, to our knowledge, none of the existing integrated models have been used for a patient-specific analysis of the cardio-respiratory response to apneas or for a patient specific identification. This lack of model-based reasoning methods is even more pronounced in the case of newborn physiology, since, to our knowledge, no integrated cardio-respiratory models has yet been proposed in the literature. This work is in direct continuity of the previous contributions of our team in this field. The main objective of this thesis is to improve the understanding of the acute cardio-respiratory responses to apnea and hypopnea events by proposing a model-based analysis of cardiorespiratory interactions in adults and newborns. One of the main challenges here is to provide patient-specific and event-specific inter- pretations of apnea episodes, taking into account the morphology and the dynamics (duration, depth, etc) of the observed respiratory and oxygen saturation signals. In fact, repeated desaturation events are one of the main triggers for the development of complica-

20 Introduction tions associated with apnea episodes and the dynamics of these desaturation events have not been sufficiently studied in the literature. We hypothesize that a model-based analysis of these dynamics may provide new information to characterize the cardio-respiratory response to apnea, both in adults and newborns. This thesis is organized as follows: Chapter 1 presents a description of the main physiological functions that are studied in this work. In particular, a brief state of the art on cardio-respiratory physiology involved in sleep apnea syndrome, as well as in apneas of prematurity are described. Chapter 2 presents a state of the art on current mathematical modeling efforts towards the integration of cardio-respiratory functions in adults and newborns. Chapter 3 describes the methods and tools for modeling, simulation and analysis that are proposed and applied in this thesis, including a description of the modeling and simulation framework (M2SL), the sensitivity analysis methods (Morris screening method) and the parameter identification approach (Evolutionary algorithms). Chapter 4 describes an original integrated model of cardio-respiratory interactions in adults. The proposed model is composed of four interconnected components : i) the respiratory system, ii) the cardiovascular system, iii) the gas exchange (in the lungs and the tissue) and iv) the neural control. Chapter 5 presents a complete parameter analysis of our adult model, including sensitivity analysis and identification methods in the context of sleep apnea. The identification approach is applied to a clinical database (HYPNOS study), acquired during complete polysomnography, within the framework of French ANR project PASITHEA, which was dedicated to the analysis of sleep apnea syndrome. The dynamics of oxygen desaturation during OSA, during 107 apneas events distributed among 10 patients was analyzed on the model parameter space, and phenotypically different groups of OSA events were identified. Chapter 6 introduces two versions of a novel integrated model of the cardio-respiratory physiology of newborns at term and preterm infants. The results and tools developed in the context of newborns and prematurity were performed within the framework of EU Digi-NewB project (Horizon 2020 research and innovation program under grant agreement No 689260) and during a 3-month stay (from September to November 2018) in the Center of Excellence for Research on Maternal and Child Health of the University of Sherbrooke under the supervision of the professor of pediatric pulmonology Jean Paul Praud and his team.

21 Introduction

References

[ACU11] A. Albanese, N. W. Chbat, and M. Ursino, « Transient respiratory response to hypercapnia: Analysis via a cardiopulmonary simulation model », in: 2011 Annual International Conference of the IEEE Engineering in Medicine and Biology Society, IEEE, Aug. 2011, pp. 2395–2398. [Alb+15] Antonio Albanese et al., « An integrated mathematical model of the human cardiopulmonary system: Model development », in: American Journal of Physiology - Heart and Circulatory Physiology (2015), ajpheart.00230.2014. [Ant+16] Ulla Anttalainen et al., « Prolonged partial upper airway obstruction during sleep – an underdiagnosed phenotype of sleep-disordered breathing », in: European Clinical Respiratory Journal 3.1 (Jan. 2016), p. 31806. [Bar+17] William H. Barnett et al., « Chemoreception and neuroplasticity in respiratory circuits », in: Experimental Neurology 287 (Jan. 2017), pp. 153–164. [BT00] J.J. Batzel and H.T. Tran, « Modeling instability in the control system for human respiration: applications to infant non-REM sleep », in: Applied Mathematics and Computation 110.1 (Apr. 2000), pp. 1–51. [Cha+18] P. Chandrasekharan et al., « Apnea, bradycardia and desaturation spells in premature infants: Impact of a protocol for the duration of ’spell-free’ observation on interprovider variability and readmission rates », in: Journal of Perinatology (2018). [Che+10] Limei Cheng et al., « An integrative model of respiratory and cardiovascular control in sleep-disordered breathing », in: Respiratory Physiology and Neurobiology 174.1-2 (2010), pp. 4–28. [Ell+13] L. M. Ellwein et al., « Modeling cardiovascular and respiratory dynamics in congestive heart failure », in: Mathematical Biosciences 241 (2013), pp. 56– 74. [Eps+09] Lawrence J. Epstein et al., Clinical guideline for the evaluation, management and long-term care of obstructive sleep apnea in adults, June 2009. [Fix+18] Laura Ellwein Fix et al., « Theoretical open-loop model of respiratory mechanics in the extremely preterm infant », in: (2018), pp. 1–21.

22 Introduction

[GBB67] F S Grodins, J Buell, and A J Bart, « Mathematical analysis and digital simulation of the respiratory control system. », in: Journal of Applied Physiology 22.2 (Feb. 1967), pp. 260–276. [Le +08] Virginie Le Rolle et al., « Model-based analysis of myocardial strain data acquired by tissue Doppler imaging », in: Artiﬁcial Intelligence in Medicine 44.3 (Nov. 2008), pp. 201–219. [Le +13] Virginie Le Rolle et al., « Mathematical modeling of respiratory system mechanics in the newborn lamb . To cite this version : HAL Id : hal-00880028 », in: (2013). [LOH11] Virginie Le Rolle, David Ojeda, and Alfredo I. Hernández, « Embedding a Cardiac Pulsatile Model Into an Integrated Model of the Cardiovascular Reg- ulation for Heart Failure Followup », in: IEEE Transactions on Biomedical Engineering 58.10 (Oct. 2011), pp. 2982–2986. [LSE11] S. A. Lorch, L. Srinivasan, and G. J. Escobar, « Epidemiology of Apnea and Bradycardia Resolution in Premature Infants », in: PEDIATRICS 128.2 (Aug. 2011), e366–e373. [Lu+01] K Lu et al., « A human cardiopulmonary system model applied to the analysis of the Valsalva maneuver. », in: American journal of physiology. Heart and circulatory physiology 281.6 (2001), H2661–H2679. [Mol+14] Yaroslav I. Molkov et al., « A closed-loop model of the respiratory system: Focus on hypercapnia and active expiration », in: PLoS ONE 9.10 (Oct. 2014), ed. by Michael Koval, e109894. [MU01] Elisa Magosso and Mauro Ursino, « A mathematical model of CO 2 eﬀect on cardiovascular regulation », in: American Journal of Physiology-Heart and Circulatory Physiology 281.5 (2001), H2036–H2052. [RSG12] Francesco M. Risso, Andrea Sannia, and Diego Gazzolo, « Preterm and term newborn: Primary investigations », in: Journal of Maternal-Fetal and Neona- tal Medicine, 2012. [Sá +06] Carla D. Sá Couto et al., « A Model for Educational Simulation of Neonatal Cardiovascular Pathophysiology », in: Simulation in Healthcare: The Journal of the Society for Simulation in Healthcare 1.Inaugural (2006), pp. 4–9.

23 Introduction

[Sen+17] Chamara V. Senaratna et al., Prevalence of obstructive sleep apnea in the general population: A systematic review, Aug. 2017. [SL12] Carrie K. Shapiro-Mendoza and Eve M. Lackritz, Epidemiology of late and moderate preterm birth, 2012. [SS15] Robert C Stansbury and Patrick J Strollo, « Clinical manifestations of sleep apnea », in: Journal of Thoracic Disease 7.9 (Sept. 2015), E298–E310. [Ten+06] KHWJ Ten Tusscher et al., Progress in biophysics and molecular biology. Vol. 90, 1-3, Pergamon Press, 2006, pp. 326–345. [UM00] M Ursino and E Magosso, « Acute cardiovascular response to isocapnic hypoxia. I. A mathematical model. », in: American journal of physiology. Heart and circulatory physiology 279 (2000), H149–H165. [UM02] Mauro Ursino and Elisa Magosso, « A theoretical analysis of the carotid body chemoreceptor response to O2 and CO2 pressure changes », in: Respiratory Physiology & Neurobiology 130.1 (2002), pp. 99–110.

24 Chapter 1 PHYSIOLOGYOFCARDIOVASCULARAND RESPIRATORY CONTROL

The cardiovascular and respiratory systems are closely related to each other, to ensure the alveolar gas exchange, the supply of oxygen (O2) to all cells of the body, and to assure the transport and release of carbon dioxide (CO2), which is the waste of cell metabolism. The respiratory system includes all the anatomical structures and mechanisms allowing for pulmonary ventilation and gas exchange at the alveolar level. The cardiovascular system, meanwhile, ensures the transport of blood gases between the lungs and all the cells of the body and irrigates all the organs of the organism. The cardiac and respiratory functions are thus subject to fine regulation by a set of intertwined physiological control systems, which are functionally and anatomically linked. In this chapter, the physiology of the cardiovascular and respiratory systems and their relationship with the autonomic nervous system will be briefly presented. In addition, the physiology of cardio-respiratory interrelations will be detailed by exploring different cardio-respiratory pathologies in adults and newborns as sleep apneas, apneas of the preterm infant and bradycardias. These cardio-respiratory pathologies are the main application subject of this work, hence the importance of understanding their underlying physiology.

1.1 Cardiovascular control

1.1.1 The cardiovascular system

The cardiovascular system (CVS) is composed mainly of the heart and blood vessels (arteries, veins, blood capillaries). Its main function is to ensure the irrigation of all organs to ensure the supply of oxygen and nutrients and the elimination of wastes produced by the metabolism.

25 Chapter 1 – Physiology of cardiovascular and respiratory control

The CVS is divided into two circulatory paths: pulmonary and systemic circulations [GH10]. Oxygen-deprived blood is pumped from the heart to the lungs into the pulmonary circulation, via the pulmonary artery, and returns, oxygenated, back into the heart through the pulmonary vein. The systemic circulation transports oxygenated blood from the heart to the rest of the body, via the aorta, and returns oxygen-poor blood back into the heart through the venae cavae. Arteries, which are vessels with muscular and elastic thick walls, branch into smaller structures called arterioles and then into capillaries, which are tiny extremely thin-walled vessels connecting arterial and venous systems. After their passage through body tissues to exchange gases and nutrients, capillaries merge again into venules and then into veins, which are thin-walled vessels resisting lower pressures than arteries. Figure 1.1 illustrates the cardiovascular system, where blue represents oxygen-poor and CO2-rich blood circulation, whereas red refers to oxygen-rich and CO2-poor blood.

1.1.2 The heart

The heart is a muscular organ located in the anterior mediastinum, which ensures the circulation of blood in the vessels. The heart is divided into two parts, the left heart and the right heart including each an atrium and a ventricle (Figure 1.2). The right atrium (RA) receives deoxygenated blood from the systemic circulation, through the superior and inferior venae cavae; while oxygenated blood arrives to the left atrium (LA) back from the lungs, through the pulmonary veins [Whi14]. Each atrium communicates with the corresponding ventricle via atrioventricular orifices provided with a valve system, ensuring unidirectional blood flows. The tricuspid valve separates the RA from the right ventricle (RV) and the mitral valve separate the LA from the left ventricle (LV). Two additional semilunar valves are located at the exit of each ventricle to prevent blood backflow: the pulmonary valve between the RV and the pulmonary artery, and the aortic valve between the left ventricle (LV) and the aorta. Both atria and ventricles are separated by the interatrial and interventricular septums, respectively. Figure 1.2 shows the main anatomy of the heart. At the beginning of each cardiac cycle, both atria and ventricles are relaxed. Since blood circulates from areas of higher to lower pressures, atria begin to fill until pressure rises and blood moves into the ventricles. Likewise, this causes ventricular pressure to rise, which leads to the contraction of ventricles in order to pump blood from the RV into the pulmonary artery, as well as from the LV into the aorta. The mechanical cardiac activity

26 1.1. Cardiovascular control

Figure 1.1 – Cardiovascular system, composed of the heart and the pulmonary and systemic circulations. This ﬁgure is derived from Servier Medical Art by Servier and it is licensed under a Creative Commons Attribution 3.0 Unported License.

27 Chapter 1 – Physiology of cardiovascular and respiratory control

Figure 1.2 – Anatomical structure of the heart. This ﬁgure is derived from Servier Medical Art by Servier and it is licensed under a Creative Commons Attribution 3.0 Unported License. described above is initiated and controlled by an electrical cardiac activity, generated at the cellular level.

1.1.3 Cardiac electrical conduction

In normal conditions, the heart rhythm is established by the sinoatrial (SA) node, the heart’s pacemaker, where an electrical impulse causing heart muscles to contract is generated and propagated through the heart [And+09]. This electrical impulse is known as action potential (AP). The AP excites the RA and travels through Bachmann’s bundle to simultaneously excite the LA. Then, it propagates through internodal pathways in the RA to the atrioventricular (AV) node. From this node, the impulse travels through the bundle of His and down the bundle branches on either side of the interventricular septum: the right bundle branch (RBB) and the left bundle branch (LBB). Both branches lead into the Purkinje ﬁbers that ﬁnally transmit the electrical impulse to the muscle tissue (myocardium) of the ventricles. Figure 1.3 illustrates the cardiac electrical conduction pathway. In the cellular level, the cardiac AP is composed of the following 5 phases [NK05], represented in Figure 1.4 for both nodal and myocardial cell types: • Phase 0. When cardiac cells are in resting state, their membranes are negatively

28 1.1. Cardiovascular control

Figure 1.3 – Cardiac electrical conduction system. This ﬁgure is derived from Servier Medical Art by Servier and it is licensed under a Creative Commons Attribution 3.0 Unported License.

charged. However, a rapid inﬂux of sodium (Na+) ions causes the membrane potential to become positive. This phenomenon is called depolarization and, in pacemaker cells, occurs spontaneously. In ventricular cells, though, this fast depolarization or fast ﬂow of Na+ ions into the cell is caused by the electrical excitation of nearby cells.

• Phase 1. Only in non-pacemaker cells, the rapid inactivation of Na+ channels followed by an outgoing ﬂow of potassium (K+) ions causes a brief repolarization where the membrane potential slightly becomes more negative. This is referred to as a notch on the AP waveform.

• Phase 2. Known as the plateau phase due to a nearly balanced charge caused by the inﬂux of calcium (Ca2+) ions and the outgoing ﬂow of K+ ions, the membrane potential of non-pacemaker cells remains relatively constant.

• Phase 3. When Ca2+ channels close, the positive outward current of K+ ions that remains causes a negative change in the membrane potential and, thus, a rapid repolarization. Once the cell is suﬃciently negatively charged, K+ channels close and membrane potential is restored to a resting value.

• Phase 4. It refers to the resting state, when the cell is more easily excitable.

29 Chapter 1 – Physiology of cardiovascular and respiratory control

Figure 1.4 – Phases of the action potential from a SA node (left) and a ventricular (right) cell. Figure adapted from [Cal18].

Depolarization causes Na+ channels to close and inactivate. The refractory period of a cardiac cell is the time from phase 0 until the next possible depolarization. Indeed, diﬀer- ent degrees of refractoriness are encountered during an AP, depending on the amount of Na+ channels no longer inactivated and capable of reopening. During the absolute refractory period (ARP), which starts with phase 0 until halfway of phase 3, channels cannot be opened regardless of the strength of the stimulus received. The ARP is immediately followed, until the end of phase 3, by the relative refractory period (RRP), during which a stronger-than-usual stimulus can initiate another AP. This is due to the membrane potential’s hyperpolarization caused by the leaking of K+ ions at the end of phase 3, which opens the inactivation gates of Na+ channels, but still leaves those channels closed.

1.1.4 The nervous system

The nervous system is a complex network of billions of nerve cells called neurons that, together with the brain and spinal cord, monitor and control body functions by sending, receiving and interpreting information from the entire organism [GH10]. It is subdivided in the following subsystems: • The Central Nervous System (CNS),composed of the brain and spinal cord, and responsible for the supervision and coordination of the entire nervous system. • The Peripheral Nervous System (PNS), which consists of glands and nerves, in order to connect and transport information between the CNS and any part of the body. Nerves transmitting signals from the brain are called motor or eﬀerent nerves, while those transmitting information from the body to the CNS are sensory or aﬀerent nerves. The PNS is likewise divided into:

30 1.1. Cardiovascular control

— The somatic nervous system (SNS), which is composed of the nerves that mediate voluntary movement, and thus that innervate skeletal muscles and external sensory organs such as the skin. — The autonomic nervous system (ANS), which innervates internal organs and glands, in order to regulate involuntary body functions. It is further divided in the enteric, sympathetic and parasympathetic nervous systems. The former regulates bowel motility in digestion; the sympathetic nervous system is generally activated in cases of stress such as exercise to mobilize energy; and the parasympathetic nervous system is activated when organisms are in a relaxed state, in order to save energy.

Autonomic nervous system

The ANS regulates internal body functions such as the heart rate or respiration. It is mainly divided in two complementary branches: the sympathetic and the parasympathetic systems. In general, each eﬀector organ is connected with its respective sympathetic and/or parasympathetic system through preganglionic and postganglionic neurons, synapsed together in the autonomic ganglia (Figure 1.5). All preganglionic neurons are cholinergic, since they release a neurotransmitter called acetylcholine (Ach). In the parasympathetic system and in some cases of the sympathetic system, postganglionic neurons are also cholinergic. However, most sympathetic postganglionic neurons are adren- ergic, since they release norepinephrine (NE). Both sympathetic and parasympathetic systems can cause excitatory or inhibitory responses, depending on the eﬀector organ’s receptors. Moreover, for those organs having both sympathetic and parasympathetic in- nervations, their respective functions are typically opposed and it is the balance of the relative sympathetic to parasympathetic tone which dictates the organ action. For example, sympathetic stimulation of the SA node increases heart rate (HR), while parasympathetic stimulation does the contrary. Thus, the actual heart rate depends on the relative balance between these two systems.

Autonomic regulation of the heart function

The heart is permanently under the influence of the two branches of the SNA (sympathetic and parasympathetic [Swe13]. In particular, the autonomic nervous system has five effects on the functioning of the heart:

31 Chapter 1 – Physiology of cardiovascular and respiratory control

Figure 1.5 – Neural connection between the central nervous system (CNS) and eﬀector organs by the sympathetic and parasympathetic branches, through preganglionic and postganglionic neurons, synapsed together in the autonomic ganglia. This ﬁgure is derived from Servier Medical Art by Servier and it is licensed under a Creative Commons Attribution 3.0 Unported License.

• Chronotropic effect. If influences the HR at the level of the sinus node. The increase of sympathetic activity increases heart rate while increasing activity parasympathetic decreases it. • Dromotropic effect. It is manifested by a change in the conduction velocity between the sinus node and the atrioventricular node. Sympathetic activity increases the speed of conduction which will be diminished by the parasympathetic activity. • Inotropic effect. It is the effect on the contractility of the heart. The sympathetic influences the atria and the ventricles that contract more vigorously with the increase of sympathetic activity. The parasympathetic has more effect on the atria. • Bathmotropic effect. It is the effect on the excitability of the heart. Sympathetic activity increases excitability of the heart while the parasympathetic activity decreases it. • Lusitropic effect. It causes a faster relaxation after contraction. Regarding blood vessels, sympathetic neurons also cause vasoconstriction and, thus, a blood pressure rise through the increase of peripheral resistance. In the closed-loop system formed by both autonomic and cardiovascular control systems, the short-term regulation of blood pressure is known as the baroreflex arc (Figure 1.6). The nervous signals coming from arterial baroreceptors are transmitted to the ANS centers, which modulate heart rate, cardiac contractility and blood vessels resistance through the sympathetic and parasympathetic systems in order to keep blood pressure within physiological levels.

32 1.1. Cardiovascular control

Figure 1.6 – Autonomic and cardiovascular closed-loop system regulating blood vessels resistance, heart rate and cardiac contractility based on the pressure detected at arterial baroreceptors. This ﬁgure is derived from Servier Medical Art by Servier and it is licensed under a Creative Commons Attribution 3.0 Unported License.

33 Chapter 1 – Physiology of cardiovascular and respiratory control

Arterial baroreceptors are stretch receptors mainly located in the carotid sinus and in the aortic arch, ﬁring at a rate proportional to their stretch state and related to blood pressure in a sigmoid manner. When blood pressure is close to the point of maximum sensitivity and increases, stretch and, thus, baroreceptors ﬁring rate rises, making cardiac centers to decrease sympathetic stimulation and increase parasympathetic tone.

1.2 Respiratory control

The respiratory system consists of the upper and lower airways, the left and right lungs, the chest and abdominal walls and the respiratory muscles. During pulmonary ventilation, the inspired air passes through the trachea, enters the bronchi, passes through the bronchioles and reaches the alveoli, where gas exchange takes place. Pulmonary ventilation provides the necessary oxygen (O2) for the oxidation of energy substrates and the production of energy in the form of adenosine triphosphate (ATP). It also eliminates oxidation products in the form of carbon dioxide (CO2). It is located under the control of the respiratory center in the brainstem and it constantly adapts the breathing to the metabolic needs. Pulmonary ventilation is provided by the activity of skeletal muscles whose activity is controlled and modulated by diﬀerent central and peripheral aﬀerences integrated in the bulbopontic respiratory centers. In this section, the mechanical functioning of human ventilation and the nervous control of ventilation will be presented.

1.2.1 Respiratory mechanics

Breathing is generated by the rhythmic contraction of the respiratory muscles, the inspiratory and expiratory muscles [AP14]. During inspiration, an active phenomenon, the inspiratory muscles contract to increase the volume of the thoracic cavity which causes the lungs to dilate and creates a pressure gradient with the outside, allowing for the entry of oxygen-enriched air. During exhalation, the inspiratory muscles relax, causing a decrease in the volume of the chest cavity and the passive retraction of the lungs, which eliminates the air rich in carbon dioxide, produced by the metabolic reactions of organism.

Inspiratory muscles

The inspiratory muscles are divided into a main muscle (the diaphragm) and accessory muscles (external intercostal muscles, sternocleidomastoid muscles and scalene muscles)

34 1.2. Respiratory control

[Lan11]. Their goal is to ensure that adequate alveolar pressure is provided to guarantee an air entry into the lungs that meets the metabolic needs of the body. The diaphragm provides the majority of the inspiratory work. On inspiration, it contracts and moves downwards which leads to an increase in abdominal pressure. This is transmitted to the lower part of the chest wall that increases its diameter. In situations where the metabolic demand is greater (stress, exercise), accessory inspiratory muscles come into play and contract with the diaphragm. The contraction of external intercostal muscles elevates the ribs and increases the transverse diameter of the thorax. Sternocleidomastoid muscles (SCM) and scalenes are vertical inspiratory muscles of lesser importance. The SCM are inserted between the sternum, the clavicle and the mastoid. They raise the clavicle and move the sternum forward. The contraction of the scalenes, inserted on the transverse processes of the vertebrae C2 to C7 and on the ﬁrst and second ribs, raises this last two ribs.

Expiratory muscles

The expiratory muscles are the internal intercostal muscles and the abdominal muscles. At rest, these muscles do not contract; it is the elasticity of the lung parenchyma (elastic recoil) and the relaxation of the inspiratory muscles which make it possible to increase the intra-pulmonary pressure and to eliminate the air rich in CO2. During a forced expiration, the expiratory muscles participate in the expulsion of air. The internal intercostal muscles contract to lower the ribs which decreases the transverse diameter of the thorax. The contraction of the abdominal muscles increases the intra- abdominal pressure and pushes the diaphragm upwards.

1.2.2 Neural control of breathing

Control of breathing is based on the control and coordination of respiratory muscle activity, to ensure optimal alveolar ventilation, regulate arterial blood gases and meet metabolic needs [MH91]. The respiratory control system is organized around control centers located in the brainstem. These are responsible for the automatic generation of the rhythm and the respiratory pattern. Their activity is modulated by sensory information from various receptors throughout the respiratory system and the brain, such as chemoreceptors (central and peripheral) and mechanoreceptors [Mol+14]. These centers are also under cortical (voluntary control) and subcortical (hypothalamus, cerebellum) inﬂuence

35 Chapter 1 – Physiology of cardiovascular and respiratory control

[Guz97].

Receptors and sensory aﬀerences

The afferent nerve fibers carry the sensory messages of the mechanoreceptors and chemoreceptors to the respiratory centers to inform them of the state of distension of the lungs and arterial blood gas pressures and pH. Two control loops are involved. The first is derived from the mechanoreceptors located in the upper airways and thoraco- pulmonary system, and whose afferent pathways are mediated by the vagus nerve. The second originates from peripheral and central chemoreceptors.

• Mechanoreceptors: Mechanoreceptors can be divided into three categories according to their anatomical location: bronchopulmonary receptors, upper airway receptors and thoracic receptors. — Bronchopulmonary receptors: These receptors are divided into three types, the slow-adapting receptors, fast-adapting receptors and bronchopulmonary C-fiber terminations[CU03]. The slow-adapting receptors are associated with smooth muscle fibers of the trachea and bronchi. These mechanoreceptors are sensitive to variations in lung volume and they are involved in the Hering- Breuer reflex, which prevents pulmonary over-distension and participates in the end of inspiration [BF06]. Fast-adapting receptors or receptors for irritation have free endings located in the tracheobronchial epithelium. These receptors are sensitive to rapid changes in lung volume and are involved in the mechanism of sighing. They are also stimulated by the irritants present in the bronchial lumen and are responsible for defense reflexes (cough, bronchospasm, secretion of mucus). Finally, the bronchopulmonary fibers C ends, the most abundant in the vagus nerve, are divided into two groups, the bronchials (at the level of the bronchial epithelium) and the pulmonary (juxta-alveolar, between the alveolar epithelium and pulmonary capillaries). Bronchial fibers are stimulated by irritants and are also involved, such as receptors for irritation, in defense reflexes. C-fibers of the pulmonary contingent are stimulated by pulmonary vessel con- gestion and pulmonary interstitial edema, during which they are involved in tachypnea [BF06]. — Airway receptors: Despite of their presence in all the upper airways, most of these receptors are located in the larynx [Wid01]. They are sensitive to cold,

36 1.2. Respiratory control

pressure and movement (contraction of the laryngeal muscles). They provide a fine regulation of the opening of the larynx. In addition, receptors for irritation and endings of C-fibers have been described in the mucus of the upper airways and more particularly in the larynx. Activated by various chemical and mechanical irritant stimuli, they are involved in several reflexes such as cough [Wid98] and laryngeal chemoreflex [RSP07].

— The thoracic receptors: They are located in the respiratory muscles and the joints of the thorax and make it possible to adapt the contraction of the inspiratory muscles to the load [Dan16].

• Chemoreceptors: They are located in the respiratory muscles and the joints of the thorax and make it possible to adapt the contraction of the inspiratory muscles to the load [Dan16].

— Peripheral chemoreceptors: Located at the arterial level, they are struc-

tures sensitive to changes in blood pressure in O2 (PaO2), CO2 (PaCO2) and arterial pH (pHa). They are involved in the response to hypercapnia (increase

of PaCO2) and hypoxemia (decrease of PaO2) to stimulate the ventilation. Peripheral chemoreceptors are divided according to their location in carotid chemoreceptors, located in the carotid glomerus at the bifurcation of the aorta, and aortic located in the aortic body at aortic arch [CK13]. It is at the level of the Nucleus Tractus Solitarius (NTS) that glossopharyngeal dopaminergic aﬀerents originating from peripheral chemoreceptors are projected [LMS77]. This information is then transmitted to the bulbar respiratory centers (specifically the ventral respiratory column) to adjust the respiratory motor drive [Kub+06].

— Central chemoreceptors: Located mainly in the brainstem, they play a

key role in the detection of PaCO2/pH variations. They are identiﬁed at the level of the ventrolateral surface of the bulb [GAS13], in the Nucleus Tac- tus Solitarius [Con+09], bulbar raphe (serotonergic neurons) [HR10] and the retrotrapezoid nucleus / parafacial respiratory group (pFRG / RTN) [GM10; Nat11]. Chemosensitive structures have also been described at the pons (locus coeruleus) and the hypothalamus [Nat11].

37 Chapter 1 – Physiology of cardiovascular and respiratory control

Respiratory centers

The respiratory centers located in the brainstem (bulb and bridge) regulate the rhythm and respiratory pattern. They are responsible for respiratory automatism, the automatic genesis of rhythm and respiratory pattern. They also control ventilation by adapting it to the needs of the body, through the integration of central and peripheral sensory aﬀerences. The neurons of the respiratory centers are arranged in three symmetrical and bilateral respiratory groups: the dorsal respiratory group (DRG), the ventral respiratory column (VRC) and the pontine respiratory group (PRG) (Figure 1.7).

Figure 1.7 – General diagram of interactions of the brainstem respiratory neural network. DRG, dorsal respiratory group; NTS, nucleus tractus solitarii; BötC, Bötzinger complex; PreBötC, preBötzinger complex; rVRG, rostral ventral respiratory column; cVRG, caudal ventral respiratory group; pFRG, parafacial respiratory group; RTN, retrotrapezoid nucleus; vRC, ventral respiratory column; PRG, pontine respiratory group; CVLM, caudal ventrolateral spinal bulge; RVLM, ventrolateral rostral spinal bulge. Adapted from [Mol+14; Bar+17].

• The dorsal respiratory group (DRG) The DRG is located in the dorsomedial part of the bulb and it is composed essentially of the neurons of the NTS. Caudal NTS form the first synapse of afferent nerve fibers from peripheral chemoreceptors and bronchopulmonary receptors via the vagus and glossopharyngeal nerves

38 1.2. Respiratory control

[Pat+01]. Afferences from central chemoreceptors and more particularly from RTN / pFRG also transit to the NTS [GAS13]. After integration, the NTS transmits this information to the respiratory nuclei modulating the rhythmogenesis in the VRC and PRG via second-order neurons [CA10]. NTS also includes inspiratory bulbospinal neurons that projects towards the motoneuron of the inspiratory muscles (phrenic nerves and external intercostals muscles) [MH91]. • The ventral respiratory column (VRC): The VRC is located in the ventrolateral part of the bulb and it is composed of several groups of organized neurons, from the caudal to rostral region, in caudal ventral respiratory group (cVRG), rostral ventral respiratory group (rVRG), preBötzinger complex (preBötC), Bötzinger complex (BötC) and RTN / pFRG nucleus. These last three nuclei generate the respiratory rhythm while the ventral respiratory group is responsible for the establishment of the respiratory pattern [Ryb+07]. The nuclei generating the respiratory rhythm are essentially grouped in the rostral part of the VRC. The preBötC is considered as an essential region for the generation of the inspiratory activity. It consists of pacemaker neurons initiating the respiratory rhythm [Smi+91]. The bulbar interneurons of the preBötC activate the inspiratory neurons of the rVRG. The neurons of the BötC, composed mainly of expiratory neurons, has been identified as the major source of expiratory activity during basic breathing. It also plays an important role in the phase-inspiration-expiration transition [ETK03]. The BötC sends projections on expiratory neurons present in the cVRG [Smi+09]. The role of the RTN / pFRG complex in rhythmogenesis is not yet clear. The RTN/pFRG complex was first identified in vitro as a center of genesis of inspiratory activity[OIK09], and then in vivo studies in rats suggest that it may play a role in the genesis of expiratory activity [JF06; FDG13]. Other more recent rat studies also highlight the role of RTN / pFRG in controlling inspiration and expiration [Huc+15; Sil+16]. The motor pattern of breathing is elaborated in the VRG. The rVRG ensures the motor pattern of inspiration. It consists mainly of inspiratory bulbospinal neurons that control the activity of the inspiratory muscles (diaphragm, external intercostal muscles, accessory inspiratory muscles) by stimulating the intercostal, phrenic and accessory muscle nuclei. CVRG contains rather, expiratory bulbospinal neurons, which provide the expiratory motor pattern by activating the motor neurons of expiratory muscles [MH91].

39 Chapter 1 – Physiology of cardiovascular and respiratory control

• Pontic Respiratory Group (PRG) It is located in the dorsolateral region of the bridge and it modulates the activity of bulbar nuclei generating rhythm. The PRG is composed of the parabrachial complex (PB) and Kölliker-Fuse nucleus (KF) and it is involved in the integration of sensorimotor aﬀerents, in the transition between inspiration and expiration and the post-inspiratory phase of the expiration [MD09].

Motor eﬀerents

The respiratory centers control the activity of motor neurons responsible for the contraction of the muscles involved in ventilatory mechanics. The eﬀerent pathways of the respiratory control system contain spinal motoneurons innervating the respiratory muscles (diaphragm, intercostal and abdominal muscles, accessory inspiratory muscles) and motor neurons of the cranial nerves controlling the muscles of the upper airways [MH91].

Characterization of the respiratory control: The loop gain (LG) concept

The complexity of the control mechanisms described in the previous section has mo- tivated research on the quantitative characterization of ventilatory regulation. One of the main approaches to perform such characterization is the concept of loop gain (LG). LG is an engineering term that describes the gain of the negative feedback loop that regulates ventilation [Wel+08b] [Wel+11]. Loop gain theory dictates there is a controller and a plant component, with a delay between the two [Kho00]. In ventilatory control, chemoreceptor sensitivity to blood gases reflects controller gain, and the effectiveness of the lungs to alter blood gases reflects plant gain [DM18]. Loop gain could be calculated as the ratio of the size of a response to the size of a disturbance in the ventilation [Nau10]:

∆Response LG = (1.1) ∆Disturbance

One way to produce a disturbance in the ventilation is a drop in the CPAP pressure [Wel+11]. If the magnitude of the response is greater than or equal to the magnitude of the disturbance, then the LG is ≥ 1 and ventilation will increase its instability, ﬂuctuating between hyperpnea and hypopnea/apnea. If LG≤1, or near zero, then ventilation will remain stable in response to a disturbance [Wel+08a]. Figure 1.8 shows a representation of the response of the ventilation to a disturbance with two diﬀerent loop gains.

40 1.2. Respiratory control

LG=0.72 7 Response= 6.5 0.72 L/min

5.5

Ventilation (L/min) 4.5 Disturbance= 1 L/min 4 -20 0 20 40 60 80 100 120 140 160 180 Time (s) LG=1 7 Response= 6.5 1 L/min

5.5

Ventilation (L/min) 4.5 Disturbance= 1 L/min 4 -20 0 20 40 60 80 100 120 140 160 180 Time (s)

Figure 1.8 – Representation of the response of the ventilation to a disturbance with diﬀerent loop gains. Top to bottom: LG=0.72; LG=1.

41 Chapter 1 – Physiology of cardiovascular and respiratory control

Calculating the loop gain has many clinical applications. LG estimations has been used to assess ventilatory control and instability and it can be used as decision criteria to use CPAP or oxygen therapy. It has been reported that OSA patients with high loop gain treated with oxygen therapy show a signiﬁcant reduction of AHI and loop gain, in contrast to low loop gain patients [Wel+08a]. Moreover, patients with heart failure and periodic breathing are more responsive to CPAP if they have a moderate loop gain, rather than a high loop gain [San+11]. High loop gain is recognized as one of the pathophysiological traits that contribute to cause obstructive sleep apnea (OSA) [DM18].

1.2.3 Observation of the respiratory activity

There are different methods to measure the ventilation or the respiratory activity [HSM05]. The pneumotachograph [HSM05; AlK+11] and the thermocouples/thermistors [Kri12] make possible to follow the variations of the air flow, while the pressure sensors make it possible to follow the variations of pressure with the mouth and/or the nose. The capnographer [PS19] tracks changes in PCO2 during the respiratory cycle at the entrance of the upper airways. Induction Respiratory Plethysmography [MD07; MS09] is used to monitor thoracoabdominal respiratory movements. Only the nasal pressure sensors will be detailed here, since this is the method used in the studies presented in this manuscript. The nasal pressure transducer is a sensor used to indirectly measure airflow by detect- ing pressure changes in the nose with an excellent response to airflow profile [Lee05]. It can be measured via nasal cannulae, mouthpiece or facemask [AlK+11]. The pressure transducer is a device which converts an applied pressure into a mea- surable electrical signal which is linear and proportional to the applied pressure. The transducer consists of an elastic material which will deform (diaphragm) when exposed to a pressurized medium and an electrical device which detects the deformation and converts it into a usable electrical signal[Tee19]. Nasal pressure monitoring is not recommended for patients who are predominantly mouth breathers or have nasal obstruction, but mouth breathing is uncommon [Bal+98]. Nasal pressure transducer has been validated as device for identifying apneas/hypopneas during sleep [Hei+02]. This representation can be quantitative if the acquisition system is calibrated, otherwise it is qualitative (Figure 1.9). Moreover, particularly for long-term recordings, pressure transducers provide an increased signal to noise ratio, when compared with other measurement methods, such as inductive or thermistor-based pneumotachog- raphy.

42 1.3. Cardio-respiratory interactions

Uncalibrated nasal pressure of an adult apneic patient 2000

1000

-1000

-2000 0 20 40 60 80 100 120 140 160 Time

Figure 1.9 – Nasal pressure signal acquired using a nasal pressure transducer in an adult apneic patient.

1.3 Cardio-respiratory interactions

The cardiovascular and respiratory systems must act jointly to maintain optimal oxygenation of the body. The coordination between these two functions is a crucial neces- sity and ensured by the link between the cardiovascular and respiratory control systems, anatomically and functionally. Anatomically, many central and peripheral interrelations have been described. The baroreceptors and peripheral chemoreceptors are located in the same regions at the carotid bifurcation and the aortic arch. On the other hand, the afferent messages from baroreceptors and peripheral chemoreceptors use the same nerves, the sinus nerve (or Hering) and the glossopharyngeal nerve from the carotid receptors, the aortic nerve (or Ludwig Cyon), and the vagus nerve from the receivers of the aortic arch. The afferent cardiovascular and respiratory endings make their first synapse in a common nucleus, the NTS. The neurons involved in cardiovascular control are located in the middle and caudal portions of the NTS and those involved in the regulation of respiration are localized in the caudal NTS [AM13]. In addition, the cardiovascular and respiratory control centers are located in the same ventrolateral bulbopontic region [Mor+11] (Figure 1.7). Functionally, arterial baroreceptor afferents have been shown to activate BötC neurons in the ventral respiratory column, modulating respiration during the arterial baroreflex [Bae+10]. Several studies have revealed a relationship between efferent cardiovascular and respiratory neurons. For example, the vagal neurons acting on the heart control also

43 Chapter 1 – Physiology of cardiovascular and respiratory control the bronchial smooth muscles [LSL06]. In addition, interactions between the two control systems have also been demonstrated at the Pontic level [Mor+11]. Moreover, it has been shown that in case of hypercapnia, there is an activation of respiration and the sympathetic system by the group A5 noradrenergic neurons located in the ventrolateral bridge. This suggests that these neurons contribute to cardio-respiratory stimulation associated with several respiratory disorders such as obstructive sleep apnea [Mor+11]. Among the cardio-respiratory interrelations, two mechanisms appear to be particularly important, i) respiratory sinus arrhythmia (RSA) and ii) cardio-respiratory coupling (CRC) Respiratory sinus arrhythmia (RSA) is characterized by a fluctuation of heart rate in phase with inspiration and expiration, with heart rate increasing during inspiration and decreasing during exhalation [HY03]. RSA has been shown to increase the efficiency of gas exchange [Hay+96]. RSA comes from a complex interaction at the central and peripheral level. In mammals, two main mechanisms have been described at the origin of RSA [Shy+91; Hor+95; DS] The first is direct modulation of preganglionic vagal cardiac neurons by central respiratory control centers. Efferent vagal fibers are more excited in expiration by stimulation of arterial chemoreceptors and baroreceptors [Kat+70; DGM76]. Indeed, it has been shown that the membrane potential of preganglionic cardiac vagal neurons is hyperpolarized during each inhalation due to the arrival of the inhibitory postsynaptic potential mediated by acetylcholine, which makes the neurons less subject to excitation during inspiration [Gil+84]. The second mechanism is inhibition of preganglionic vagal cardiac neurons by pulmonary inflation. Afferent activity in the lung is an important mechanism in the generation of RSA [DS; GIM87]. This results in a drop in the pressure detected by the baroreceptors. In response, they inhibit vagal activity. Activation of this regulatory loop results in an increase in heart rate [BKS85]. During expiration, the reverse mechanism occurs leading to a decrease in heart rate. Cardio-respiratory coupling has been less studied than RSA. CRC represents the effect of the cardiovascular system on respiration [Dic+14]. Statistical evaluation of the time interval distribution between respiratory phase transitions and the previous or subsequent heartbeat identified that the onset of inspiration occurs after a specific delay following the previous peak of systolic BP [GL99b; LG99; Fri+12]. CRC is less clear during awakening and more visible during quiet sleep and anesthesia [GL97] [LG99]. The proposed mechanism for CRC is that the peak of systolic pressure that occurs late in the ex-

44 1.3. Cardio-respiratory interactions halation initiates inspiration after a speciﬁc delay; this is the hypothesis of "baroreceptor- trigger hypothesis" [GL99a]. CRC is dependent on the sensory activity of the carotid sinus rather than an interaction between the neural networks of the brainstem that generate the cardiovascular and respiratory patterns [Dic+14].

Cardiovascular and respiratory control systems and their coordination ensure the maintenance of homeostasis. In adults, hypoxemia produces an increase in ventilation with modest accompanying increases in sympathetic nerve traffic to blood vessels [Som+89; SMA91]. Hypoxemia acts primarily through the peripheral chemoreceptors in the carotid sinus [MWS15]. During hypercapnia, both the central and peripheral chemoreceptors respond increasing alveolar ventilation [Vas12]. Hypercapnia also elicits increases in sympathetic outflow to peripheral blood vessels and this response is buffered by hyperventilation [MWS15]. Chronic intermittent hypoxic exposure activates repeatedly the chemoreflex producing transient and sustained changes in cardiovascular and respiratory functions. It can also initiate a vascular response, and activates the baroreflex which initially works against the sympathetic effects of the chemoreflex. [FST13].

In human and ovine term neonates, the cardiovascular response to hypoxia is characterized by an increase in heart rate and a moderate and transient decrease in arterial pressure [BC66; Sid+83; Pla+08]. In case of hypercapnia, the response to CO2 leads to an increase in respiratory activity, accompanied by a sympathetic increase of arterial pressure [Som+89]. In neonates, especially preterm infants, all aspects of the respiratory control system, including respiratory rhythmogenesis and the central CO2 and hypoxemia responsive chemosensitive response, are immature [CA10]. Although functional at birth, the arterial baroreﬂex is also immature in preterm infants [GDR02]. This immaturity contributes to the development of neonatal pathologies that aﬀect cardio-respiratory control.

A number of works have been recently dedicated to characterize, in a more advanced fashion than the standard linear, autoregressive modeling, the dynamics of cardio- respiratory variables, especially under certain pathologies. For instance, the cardio-respiratory coupling in patients with heart failure [Rad+18] or the inﬂuence of heart rate and respiration during sleep and sleep apnea have been studied through nonlinear methods [VV17][Sch+13; Var+19].

Despite all the progress presented in this section, the fundamental mechanisms and eﬀects of cardio-respiratory interactions are not fully understood yet.

45 Chapter 1 – Physiology of cardiovascular and respiratory control

1.4 Pathologies involving deregulation of the cardio- respiratory control

Several pathologies involve a deregulation of the cardio-respiratory control, but only sleep apnea syndrome (SAS) and Cheyne-Stokes respiration in adults and the apnea- bradycardia in newborns will be detailed here, since these are the pathologies studied in this manuscript.

1.4.1 Sleep apnea syndrome (SAS)

Sleep apnea syndrome (SAS) is characterized by repeated episodes of absent (apnea) or reduced (hypopnea) breathing. In adults an apnea is defined as a cessation of breathing that lasts longer than 10 seconds [Kim16]. These episodes often produce significant arterial hypoxemia and hypercapnia that usually lead to sleep fragmentation, transient sleep arousals and acute over-compensatory responses of the autonomic nervous system. In the long term, these repeated acute responses may be deleterious, being associated with higher cardiovascular and metabolic morbidities [Pep+13; Som+08; SS15]. It is estimated that 6% to 17% of the adult population suffer from SAS [Sen+17; Ant+16]. SAS is commonly divided into three event categories:

• Central events: Central apnea (CA) or hypopnea (CH) are denoted by an absence or marked reduction of brain stem respiratory output, and thus, neuromuscular respiratory drive to respiratory pump muscles.

• Obstructive events: Obstructive apnea (OA) or hypopnea (OH) are characterized by an extra-thoracic upper airway obstruction along with respiratory eﬀorts.

• Mixed events: Mixed apnea or hypopnea result from a combination of central and obstructive events.

Population-based studies have shown that most SAS events are driven by anatomical and neurochemical anomalies in the control of the upper airway and chest wall respiratory musculature. The dominant risk factors are the body weight, followed by male gender and craniofacial structure and aging [Dem+02; Par+88; Pep00; You+03]. Three physiological systems are primarily involved in SAS: i) the respiratory system, ii) the cardiovascular system and iii) the autonomic nervous system (ANS).

46 1.4. Pathologies involving deregulation of the cardio-respiratory control

Pathogenesis of Sleep Apnea

In humans, one of the major structures of the respiratory system is the upper airways. This structure performs complex motor behaviors required to generate speech. These behaviors are possible because of the presence of the hyoid bone, which is the key anchoring site for pharyngeal dilator muscles. This muscle is not rigidly attached to skeletal structures, leaving the human pharynx with no rigid support unlike in other animals [MR07]. This attribute gives to the upper airway the chance to narrow or collapse when the compensatory neural activation of dilator muscles is lost at sleep onset. However, these compensatory processes are the product of multiple factors and vary markedly among individuals [Dem+10]. A diagram of the pathogenesis of SAS is shown in Figure 1.10. First, the diagram shows the principal structural and functional determinants of an anatomical predisposition for airway closure. Generally, a patient having any of them is more prone to suﬀer from SAS. Then, after sleep, two paths to develop either an obstructive or central apnea are presented. The diagram emphasizes the mechanisms underlying both types of respiratory events, and integrates anatomical deﬁcits with mechanisms for central neurochemical control of breathing stability and compensatory neuromuscular control of upper airway muscles in order to explain the cyclical nature of SAS.

An illustration of these different mechanisms can also be appreciated in Figure 1.11. Figure 1.11-A shows the nasal pressure (NP) signal with the presence of several respiratory events. These events are evoked, as previously described, by a decrease of the tonic activity of the upper airway dilator muscles causing the upper airway to narrow/collapse (obstructive apnea) or by an unstable central respiratory motor output leading to a stoppage of the respiration (central apnea). After these alterations in the respiratory system, the lack of uptake of oxygen produces a decrease of oxygen concentration in the blood (hypoxemia) which induces intermittent hemodynamic variations in the cardiovascular system, this behavior can be observed in Figure 1.11-B, where the oxygen saturation (SaO2) signal is presented. Moreover, acute autonomic responses can also be seen in the heart rate (HR) signal (figure 1.11-C), where regulating mechanisms in the cardiovascular system are induced by recurrent activations of the baroreflex and chemoreflex as a response to hypoxia. Additionally, these respiratory events may provoke sleep micro-arousals (Figure 1.11-D) which are the main factor of the sleep fragmentation related to SAS.

47 Chapter 1 – Physiology of cardiovascular and respiratory control

Figure 1.10 – Scheme of the sleep apnea syndrome cyclical pathogenesis. Adapted from [D P18].

48 1.4. Pathologies involving deregulation of the cardio-respiratory control

Figure 1.11 – Example of a typical recording of a patient suffering from sleep apnea syndrome from PASITHEA Project [Her+16; D P18]. The first panel shows the nasal pressure (NP) signal, where the different apnea and hypopnea events are highlighted. The second panel represents the oxygen saturation (SaO2) signal with the intermittent hypoxia events associated with SAS. The third panel, presents the instantaneous heart rate (HR) signal, showing recurrent acute autonomic responses that are due to the respiratory events. Finally, the fourth panel presents a hypnogram signal (MA = micro-arousal, A = Awake, REM = Rapid eye movement, LS = Light sleep and DS = Deep sleep) representing the sleep structure of the patient during the recording. The hypnogram shows the presence of sleep arousals and sleep fragmentation produced by respiratory events. Figure adapted from [D P18].

49 Chapter 1 – Physiology of cardiovascular and respiratory control

1.4.2 Cheyne-Stokes respiration (CSR)

General deﬁnition

Cheyne–Stokes respiration is an abnormal pattern of breathing characterized by recurrent central apneas/hypopneas during sleep, alternating with a crescendo-decrescendo pattern of tidal volume [Har+34; BP92] as shown in Figure 1.12. CSR is observed in approximately 50% of patients with congestive heart failure (CHF) [Jav+95; Nau+95], with CHF being one of the major causes of mortality, and morbidity in developed coun- tries [RB19; Gaz05]. This respiration pattern appears predominantly during nonrapid eye movement (NREM) sleep, particularly in stages 1 and 2, during which breathing is normally regulated by changes in PaCO2 [HZG93]. Patients with CSR usually present symptoms of orthopnea, paroxysmal nocturnal dyspnea, daytime sleepiness [Nau98], increased sympathetic nervous activity [BRB12] and it is associated with memory and concentration problems [SA03]. Moreover, patients with congestive heart failure and Cheyne-Stokes respiration have a signiﬁcantly greater mortality than those without Cheyne-Stokes respiration [Fin+85; HZ96]. Polysomnography recordings is currently the gold standard for diagnosing CSR [SA03]. The risk factors of CSR are the severity of the CHF, age, being a male, low left- ventricular ejection fraction (LVEF) [Old+07], elevated pulmonary capillary wedge pressure and hypocapnia [Sol+99]. Main treatments for CSR are medical therapy directed at congestive heart failure, followed by CPAP and/or oxygen therapy [Nau98].

Pathogenesis of Cheyne-Stokes respiration

The precise mechanisms of CSR are not yet clearly elucidated, but the hypotheses are based on two main mechanisms: i) chronic hyperventilation and ii) an alteration in hemodynamics marked by an extension of the blood circulation time [Nau98; SA03]. Hyperventilation is common in the pathophysiology of every form of periodic breathing, causing PaCO2 to fall below the apneic threshold triggering a central apnea. The central apnea event rises PaCO2 above the apneic threshold, stimulating the peripheral chemoreceptors that trigger another hyperventilation driving the PaCO2 levels below the apnea threshold once again [BP92]. Left-heart failure that causes increased pulmonary venous pressure is regarded as a source of CSR [BRB12]. Elevated pulmonary venous pressure leads to pulmonary conges- tion that stimulates the pulmonary stretch receptors, which increases the sensitivity of

50 1.4. Pathologies involving deregulation of the cardio-respiratory control

Figure 1.12 – Example of Cheyne-Stokes respiration pattern.

peripheral chemoreceptors to CO2 through their vagal aﬀerents [Lor+05]. The increase of CO2 sensitivity leads the patient to hyperventilation, driving PaCO2 below the apnea threshold. Patients with heart failure are normally tachypneic and maintain lower PaCO2 levels, both awake and during sleep [RB19; Nau+93; HZG93]. Similarly, hypoxemia may contribute to hyperventilation and CSR in CHF through peripheral chemoreceptor stimulation [Nau98].

The circulation time of oxygenated blood leaving the pulmonary artery to reach the peripheral chemoreceptor is increased in patients with CHF [Nau98]. This introduces a delay to the feedback response of eﬀerent nerve compared to the actual gas composition of the blood. The extension of circulatory time is directly related to the length of the apnea-hyperpnea cycle and contributes to the crescendo-decrescendo respiratory pattern [Hal+96]. However, extension of the circulation time in animal models has not triggered CSR, so it is not thought to be a signiﬁcant precipitant of CSR.

51 Chapter 1 – Physiology of cardiovascular and respiratory control

Figure 1.13 – Age terminology during perinatal period. Age terminology during perinatal period. Adapted from [Bla+04].

1.4.3 Apnea-bradycardia episodes on preterm infants

At birth, neonates, even at term, can present cardio-respiratory instability resulting in short apneas, periodic breathing, and short, shallow cardiac delays. This instability is due to the immaturity of the cardio-respiratory system [Gar+13]. In preterm infants, the cardio-respiratory control system is less developed than in term infants. This immaturity is manifested by an increase in i) the duration and frequency of apnea, ii) the number of apneas associated with bradycardia and/or oxygen desaturations, and iii) the number of isolated bradycardias and desaturations (without apnea). Apnea of prematurity affects more than 50% of preterm infants and it affects all children with a birth weight of less than 1000 g [Smi+15]. The severity and frequency of apnea-bradycardia is correlated with the degree of immaturity of preterm infants [Fin+06; PM17b]. In order to understand the difference of immaturity, it is important to know the age terminology during the perinatal period [Bla+04] presented in Figure 1.13: — Gestational age: it is the time elapsed between the first day of the last normal menstrual period and the day of delivery. — Chronological age: it is the time elapsed after birth. — Post-menstrual age: it is the sum of gestational age and chronological age. — Corrected age: it represents the age of the child from the expected date of delivery. During postnatal maturation, apnea and bradycardia disappear [Ram+01; LSE11].

52 1.4. Pathologies involving deregulation of the cardio-respiratory control

Even in preterm infants with severe apnea-bradycardias, these cardio-respiratory events usually stop by 43 weeks of corrected age [Ram+01].

Apneas of prematurity

By definition, apnea of prematurity is clinically considered to be a cessation of breathing of at least 20 seconds or more than 10 seconds if it is coupled with bradycardia and/or oxygen desaturation [Fin+06; PM17a]. As in adults, there are three types of apnea: central (airflow interruption with stopping of breathing movements), obstructive (airflow interruption with persistent respiratory movements) and mixed [Eic16]. Most apnea episodes on preterm infants are of central or mixed nature [Eic16; PM17b].

Pathogenesis of apneas of prematurity

The transition from fetal life to neonatal life requires a sudden change in respiratory activity from an intermittent state, not associated with gas exchange, to a largely continuous state where survival will depend on the efficiency of pulmonary gas exchange [PM17b]. Several factors related to respiratory centers and peripheral reflex circuits are involved in the pathogenesis of apneas [Mat11; Eic16]. The immaturity of cardio-respiratory control systems, especially pontobulbar control centers, is the cause of basic cardio-respiratory instability in preterm infants [AM08] [CA10]. In addition, the nerve circuits involved in cardio-respiratory reflexes are immature. In preterm babies, the central chemosensitivity to CO2 is reduced, which results in a reduction of the ventilatory response to hypercapnia [MA05; Eic16]. This sensitivity increases with postnatal age [Fra+76]. Similarly, prematurity causes a change in the sensitivity of carotid chemoreceptors, so that hypoxia induces ventilatory depression and a bradycardic response [CK13; Eic16]. All these characteristics related to immaturity of the cardio-respiratory control system, contribute to the development of apnea of prematurity. The immaturity of certain vital functions such as deglutition, thermoregulation, sleep and wakefulness reactions and oesophagogastric motricity may also contribute to the development of apnea in preterm infants [RSP07; Tou+08; Dar13]. Several other factors may increase the occurrence of apneas such as stress, bacterial and viral infections, the environment (hyperthermia, excessive visual, auditory, tactile and olfactory stimulation), cerebral hypoxia, metabolic disorders (hypoglycemia, acidosis, anemia), gastroesophageal reflux [ZGM11] and hyperbilirubinemia [ABW14].

53 Chapter 1 – Physiology of cardiovascular and respiratory control

Bradycardias

Conventionally, bradycardia is deﬁned as a decrease in heart rate of at least 33% compared to an average value lasting more than 5 seconds [Poe+93]. In other words, it is deﬁned as a drop in heart rate to less than 70 to 100 beats per minute depending on gestational and postnatal age [PM17b]. Bradycardias are common events in preterm infants [AM08]. Although they are generally associated with apnea and/or desaturation (80% of cases), they can also be isolated. Their incidence increases when the length of apnea increases [Poe+93].

Pathogenesis of bradycardias

The explanation of the association of bradycardia with apnea of prematurity is un- clear[PM17b]. The presence of a correlation between decreased oxygen saturation and heart rate suggests that apnea-associated bradycardias are caused by hypoxemic stimulation of arterial chemoreceptors in response to cessation of pulmonary ventilation [HBE86; PM17b]. In addition, the simultaneous presence of bradycardia and apnea during stimulation of laryngeal chemoreceptors suggests a central origin of bradycardia, independent of hypoxia [AM08] and leading to an increase in vagal tone [PM17b].

1.5 Conclusion

Sleep apnea syndrome (SAS) is a multifactorial disease characterized by recurrent breathing pauses that can provoke significant arterial hypoxemia and hypercapnia leading to over compensatory cardio-respiratory responses. Many physiological sub-systems are involved in the pathogenesis and further development of SAS. Similarly, apnea-bradycardia events are a multidimensional problem that affects preterm newborns and evoke deleterious acute cardio-respiratory effects, leading to compensatory responses which efficiency depends highly on the level of maturity of the neonate. The physiology mechanisms behind the dynamics of oxygen desaturation and bradycardia produced by these events has not yet been completely elucidated. Cardiovascular and respiratory control are key mechanisms to understand these pathologies and their effects. They regulate respiration, heart rate and blood pressure to ensure the supply of oxygen and nutrients and the elimination of waste products generated by the metabolism. The autonomic nervous system (ANS) modulates the activity of the cardio-

54 REFERENCES respiratory system through both the sympathetic and the parasympathetic branches. The respiratory centers regulate the rhythm and respiratory pattern. One of the primary sensory receptors in the physiological closed-loop control involving the autonomic and cardio-respiratory systems are the baroreceptors and chemoreceptors. In the context of SAS, there is increasing evidence that recurrent respiratory events evoke over-compensatory autonomic activations and significantly contribute to the development of serious pathological conditions in the long term. In apneas of prematurity, several factors related to respiratory centers and peripheral reflex circuits are involved in its pathogenesis. Short-term consequences for the preterm newborn are hypoxemia and a rise in stroke volume. There is a lack of quantitative, explicative methods and tools to characterize patients at risk, or patients with a tendency to more complex respiratory events as periodic breathing or apnea-bradycardia. Improving the knowledge of the physiological variables behind the cardio-respiratory response to apnea can help to develop these tools, allowing for the development of innovative therapies or the improvement of existing ones. The next chapter will present a state of the art of the efforts performed in the literature to produce an integrated mathematical model of these different physiological mechanisms in adults and newborns. Our objective being to base our entire quantitative analysis method on such an integrated mathematical model.

References

[ABW14] Sanjiv B Amin, Vinod K Bhutani, and Jon F Watchko, « Apnea in acute bilirubin encephalopathy. », in: Seminars in perinatology 38.7 (Nov. 2014), pp. 407–11. [AlK+11] F. Q. Al-Khalidi et al., « Respiration rate monitoring methods: A review », in: Pediatric Pulmonology 46.6 (June 2011), pp. 523–529. [AM08] Jalal M. Abu-Shaweesh and Richard J. Martin, « Neonatal Apnea: What’s New? », in: Pediatric Pulmonology 43.10 (Oct. 2008), pp. 937–944. [AM13] Daniela Accorsi-Mendonça and Benedito H. Machado, « Synaptic transmission of baro- and chemoreceptors aﬀerents in the NTS second order neurons », in: Autonomic Neuroscience 175.1-2 (Apr. 2013), pp. 3–8.

55 Chapter 1 – Physiology of cardiovascular and respiratory control

[And+09] Robert H Anderson et al., « The anatomy of the cardiac conduction system », in: Clinical Anatomy 22.1 (Jan. 2009), pp. 99–113. [Ant+16] Ulla Anttalainen et al., « Prolonged partial upper airway obstruction during sleep – an underdiagnosed phenotype of sleep-disordered breathing », in: European Clinical Respiratory Journal 3.1 (Jan. 2016), p. 31806. [AP14] Andrea Aliverti and Antonio Pedotti, eds., Mechanics of Breathing, Milano: Springer Milan, 2014. [Bae+10] David M. Baekey et al., « Effect of baroreceptor stimulation on the respiratory pattern: Insights into respiratory–sympathetic interactions », in: Respiratory Physiology & Neurobiology 174.1-2 (Nov. 2010), pp. 135–145. [Bal+98] E Ballester et al., « Nasal prongs in the detection of sleep-related disordered breathing in the sleep apnoea/hypopnoea syndrome. », in: The European respiratory journal 11.4 (Apr. 1998), pp. 880–3. [Bar+17] William H. Barnett et al., « Chemoreception and neuroplasticity in respiratory circuits », in: Experimental Neurology 287 (Jan. 2017), pp. 153–164. [BC66] June P. Brady and Eliana Ceruti, « Chemoreceptor reflexes in the new-born infant: effects of varying degrees of hypoxia on heart rate and ventilation in a warm environment* », in: The Journal of Physiology 184.3 (June 1966), pp. 631–645. [BF06] E. Fiona Bailey and Ralph F. Fregosi, « Modulation of upper airway muscle activities by bronchopulmonary afferents », in: Journal of Applied Physiology 101.2 (Aug. 2006), pp. 609–617. [BKS85] R. W. de Boer, J. M. Karemaker, and J. Strackee, « Relationships between short-term blood-pressure fluctuations and heart-rate variability in resting subjects I: a spectral analysis approach », in: Medical and Biological Engi- neering and Computing 23.4 (July 1985), pp. 352–358. [Bla+04] Lillian R. Blackmon et al., Age terminology during the perinatal period, Nov. 2004. [BP92] T. D. Bradley and E. A. Phillipson, Central sleep apnea, 1992. [BRB12] Thomas Brack, Winfried Randerath, and Konrad E. Bloch, « Cheyne-Stokes Respiration in Patients with Heart Failure: Prevalence, Causes, Consequences and Treatments », in: Respiration 83.2 (Jan. 2012), pp. 165–176.

56 REFERENCES

[CA10] John L. Carroll and Amit Agarwal, « Development of ventilatory control in infants », in: Paediatric Respiratory Reviews 11.4 (Dec. 2010), pp. 199–207. [Cal18] M. Calvo, « Analysis of the cardiovascular response to autonomic nervous system modulation in Brugada syndrome patients », PhD thesis, Université de Rennes 1, 2018. [CK13] John L Carroll and Insook Kim, « Carotid chemoreceptor “resetting” revis- ited », in: Respiratory physiology & neurobiology 185.1 (Jan. 2013), pp. 30– 43. [Con+09] Susan C. Conrad et al., « Development of chemosensitivity in neurons from the nucleus tractus solitarii (NTS) of neonatal rats », in: Respiratory Phys- iology & Neurobiology 166.1 (Mar. 2009), pp. 4–12. [CU03] Michael J Carr and Bradley J Undem, « Bronchopulmonary afferent nerves. », in: Respirology (Carlton, Vic.) 8.3 (Sept. 2003), pp. 291–301. [D P18] D. Perez, « Optimal Control of Non-Invasive Neuromodulation for the Treat- ment of Sleep Apnea Syndromes », in: (2018). [Dan16] Laurence Dangers, « Application du principe de contre-irritation à l’étude des mécanismes neurophysiologiques de la dyspnée : de la physiologie à la thérapeutique », in: http://www.theses.fr (June 2016). [Dar13] Robert A Darnall, « The carotid body and arousal in the fetus and neonate. », in: Respiratory physiology & neurobiology 185.1 (Jan. 2013), pp. 132–43. [Dem+02] Jerome A Dempsey et al., « Anatomic Determinants of Sleep-Disordered Breathing Across the Spectrum of Clinical and Nonclinical Male Subjects », in: Chest 122.3 (Sept. 2002), pp. 840–851. [Dem+10] Jerome A Dempsey et al., « Pathophysiology of Sleep Apnea », in: Physio- logical Reviews 90.1 (Jan. 2010), pp. 47–112. [DGM76] N S Davidson, S Goldner, and D I McCloskey, « Respiratory modulation of barareceptor and chemoreceptor reflexes affecting heart rate and cardiac vagal efferent nerve activity. », in: The Journal of Physiology 259.2 (July 1976), pp. 523–530. [Dic+14] Thomas E. Dick et al., « Cardiorespiratory Coupling », in: Progress in brain research, vol. 209, 2014, pp. 191–205.

57 Chapter 1 – Physiology of cardiovascular and respiratory control

[DM18] Naomi Deacon-Diaz and Atul Malhotra, « Inherent vs. Induced Loop Gain Abnormalities in Obstructive Sleep Apnea », in: Frontiers in Neurology 9.NOV (Nov. 2018), p. 896. [DS] M De Burgh Daly and Mary J ’ Scottt, The effects of the stimulation of the carotid body chemoreceptors on heart rate in the dog, tech. rep., pp. 148–66. [Eic16] Eric C. Eichenwald, « Apnea of Prematurity », in: Pediatrics 137.1 (Jan. 2016), e20153757. [ETK03] Kazuhisa Ezure, Ikuko Tanaka, and Masahiro Kondo, « Glycine is used as a transmitter by decrementing expiratory neurons of the ventrolateral medulla in the rat. », in: The Journal of neuroscience : the official journal of the Society for Neuroscience 23.26 (Oct. 2003), pp. 8941–8. [FDG13] Jack L. Feldman, Christopher A. Del Negro, and Paul A. Gray, « Under- standing the Rhythm of Breathing: So Near, Yet So Far », in: Annual Review of Physiology 75.1 (Feb. 2013), pp. 423–452. [Fin+06] Neil N. Finer et al., « Summary Proceedings From the Apnea-of-Prematurity Group », in: Pediatrics 117.Supplement 1 (Mar. 2006), S47–S51. [Fin+85] Larry J. Findley et al., « Gheyne-Stokes Breathing During Sleep in Patients with Left Ventricular Heart Failure », in: Southern Medical Journal 78.1 (Jan. 1985), pp. 11–15. [Fra+76] I. D. Frantz et al., « Maturational effects on respiratory responses to carbon dioxide in premature infants », in: Journal of Applied Physiology 41.1 (July 1976), pp. 41–45. [Fri+12] Lee Friedman et al., « Cardio-ventilatory coupling in young healthy resting subjects », in: Journal of Applied Physiology 112.8 (Apr. 2012), pp. 1248– 1257. [FST13] Christopher S. Freet, James F. Stoner, and Xiaorui Tang, « Baroreflex and chemoreflex controls of sympathetic activity following intermittent hypoxia », in: Autonomic Neuroscience 174.1-2 (Mar. 2013), pp. 8–14. [Gar+13] Alfredo J. Garcia et al., « Cardiorespiratory coupling in health and disease », in: Autonomic Neuroscience 175.1-2 (Apr. 2013), pp. 26–37.

58 REFERENCES

[GAS13] Patrice G. Guyenet, Stephen B.G. Abbott, and Ruth L. Stornetta, « The respiratory chemoreception conundrum: Light at the end of the tunnel? », in: Brain Research 1511 (May 2013), pp. 126–137. [Gaz05] Thomas A. Gaziano, Cardiovascular disease in the developing world and its cost-eﬀective management, Dec. 2005. [GDR02] V Gournay, E Drouin, and J Roze, « Development of baroreﬂex control of heart rate in preterm and full term infants », in: Archives of Disease in Childhood Fetal and Neonatal Edition 86.3 (May 2002), F151. [GH10] Arthur C. Guyton and John E. Hall, Textbook of medical physiology, 2010, p. 1091. [Gil+84] M P Gilbey et al., « Synaptic mechanisms involved in the inspiratory modulation of vagal cardio-inhibitory neurones in the cat. », in: The Journal of Physiology 356.1 (Nov. 1984), pp. 65–78. [GIM87] A Guz, J A Innes, and K Murphy, « Respiratory modulation of left ventricular stroke volume in man measured using pulsed Doppler ultrasound. », in: The Journal of physiology 393 (Dec. 1987), pp. 499–512. [GL97] D C Galletly and P D Larsen, Cardioventilatory coupling during anaesthesia, tech. rep., 1997, pp. 35–40. [GL99a] D. C. Galletly and P. D. Larsen, « The determination of cardioventilatory coupling from heart rate and ventilatory time series », in: Research in Ex- perimental Medicine 199.2 (Nov. 1999), pp. 95–99. [GL99b] D Galletly and P Larsen, Ventilatory frequency variability in spontaneously breathing anaesthetized subjects, tech. rep. 4, 1999, pp. 552–63. [GM10] Patrice G. Guyenet and Daniel K. Mulkey, « Retrotrapezoid nucleus and parafacial respiratory group », in: Respiratory Physiology & Neurobiology 173.3 (Oct. 2010), pp. 244–255. [Guz97] A Guz, « Brain, breathing and breathlessness. », in: Respiration physiology 109.3 (Sept. 1997), pp. 197–204. [Hal+96] Michael J. Hall et al., « Cycle length of periodic breathing in patients with and without heart failure », in: American Journal of Respiratory and Critical Care Medicine 154.2 (1996), pp. 376–381.

59 Chapter 1 – Physiology of cardiovascular and respiratory control

[Har+34] T. R. Harrison et al., « Congestive heart failure: XX. Cheyne-stokes respiration as the cause of paroxysmal dyspnea at the onset of sleep », in: Archives of Internal Medicine 53.6 (June 1934), pp. 891–910. [Hay+96] J Hayano et al., « Respiratory sinus arrhythmia. A phenomenon improving pulmonary gas exchange and circulatory eﬃciency. », in: Circulation 94.4 (Aug. 1996), pp. 842–7. [HBE86] D J Henderson-Smart, M C Butcher-Puech, and D A Edwards, « Incidence and mechanism of bradycardia during apnoea in preterm infants. », in: Archives of Disease in Childhood 61.3 (Mar. 1986), pp. 227–232. [Hei+02] Steven J. Heitman et al., « Validation of Nasal Pressure for the Identiﬁcation of Apneas/Hypopneas during Sleep », in: American Journal of Respiratory and Critical Care Medicine 166.3 (Aug. 2002), pp. 386–391. [Her+16] A. I. Hernández et al., « PASITHEA: An Integrated Monitoring and Ther- apeutic System for Sleep Apnea Syndromes Based on Adaptive Kinesthetic Stimulation », in: Irbm 37.2 (2016), pp. 81–89. [Hor+95] R. L. Horner et al., « Respiratory-related heart rate variability persists during central apnea in dogs: mechanisms and implications », in: Journal of Applied Physiology 78.6 (June 1995), pp. 2003–2013. [HR10] Matthew R. Hodges and George B. Richerson, « The role of medullary sero- tonin (5-HT) neurons in respiratory control: contributions to eupneic ventilation, CO2 chemoreception, and thermoregulation », in: Journal of Applied Physiology 108.5 (May 2010), pp. 1425–1432. [HSM05] Q Hamid, J Shannon, and J Martin, Physiologic Basis of Respiratory Dis- ease, Pmph USA Ltd Series, BC Decker, Incorporated, 2005. [Huc+15] R. T. R. Huckstepp et al., « Role of Parafacial Nuclei in Control of Breathing in Adult Rats », in: Journal of Neuroscience 35.3 (Jan. 2015), pp. 1052–1067. [HY03] Junichiro Hayano and Fumihiko Yasuma, « Hypothesis: respiratory sinus arrhythmia is an intrinsic resting function of cardiopulmonary system. », in: Cardiovascular research 58.1 (Apr. 2003), pp. 1–9.

60 REFERENCES

[HZ96] Patrick J. Hanly and Naheed S. Zuberi-Khokhar, « Increased mortality associated with Cheyne-Stokes respiration in patients with congestive heart failure », in: American Journal of Respiratory and Critical Care Medicine 153.1 (Jan. 1996), pp. 272–276. [HZG93] P. Hanly, N. Zuberi, and R. Gray, « Pathogenesis of Cheyne-Stokes respiration in patients with congestive heart failure: Relationship to arterial PCO2 », in: Chest 104.4 (Oct. 1993), pp. 1079–1084. [Jav+95] Shahrokh Javaheri et al., « Occult sleep-disordered breathing in stable congestive heart failure », in: Annals of Internal Medicine 122.7 (Apr. 1995), pp. 487–492. [JF06] Wiktor A Janczewski and Jack L Feldman, « Distinct rhythm generators for inspiration and expiration in the juvenile rat. », in: The Journal of physiology 570.Pt 2 (Jan. 2006), pp. 407–20. [Kat+70] PG Katona et al., « Cardiac vagal efferent activity and heart period in the carotid sinus reflex », in: American Journal of Physiology-Legacy Content 218.4 (Apr. 1970), pp. 1030–1037. [Kho00] Michael C.K. Khoo, « Determinants of ventilatory instability and variability », in: Respiration Physiology, vol. 122, 2-3, Elsevier, Sept. 2000, pp. 167– 182. [Kim16] John Kimoff, « Obstructive Sleep Apnea », in: Murray and Nadel’s Textbook of Respiratory Medicine (Jan. 2016), pp. 1552–1568. [Kri12] Jyoti Krishna, « Polysomnography and MSLT », in: Principles and Prac- tice of Pediatric Sleep Medicine: Second Edition, Elsevier Inc., Jan. 2012, pp. 399–409. [Kub+06] Leszek Kubin et al., « Central pathways of pulmonary and lower airway vagal afferents », in: Journal of Applied Physiology 101.2 (Aug. 2006), pp. 618– 627. [Lan11] Michael A. Lane, « Spinal respiratory motoneurons and interneurons », in: Respiratory Physiology & Neurobiology 179.1 (Oct. 2011), pp. 3–13. [Lee05] Teofilo Lee-Chiong, ed., Sleep: A Comprehensive Handbook, Hoboken, NJ, USA: John Wiley & Sons, Inc., Nov. 2005.

61 Chapter 1 – Physiology of cardiovascular and respiratory control

[LG99] P. D. Larsen and D. C. Galletly, « Cardioventilatory coupling in the anaes- thetised rabbit, rat and guinea-pig », in: Pﬂ∆gers Archiv European Journal of Physiology 437.6 (Apr. 1999), pp. 910–916. [LMS77] J. Lipski, R. M. McAllen, and K. M. Spyer, « The carotid chemoreceptor input to the respiratory neurones of the nucleus of tractus solitarius », in: The Journal of Physiology 269.3 (Aug. 1977), pp. 797–810. [Lor+05] Geraldo Lorenzi-Filho et al., Cheyne-Stokes respiration in patients with congestive heart failure: causes and consequences. 2005. [LSE11] S. A. Lorch, L. Srinivasan, and G. J. Escobar, « Epidemiology of Apnea and Bradycardia Resolution in Premature Infants », in: PEDIATRICS 128.2 (Aug. 2011), e366–e373. [LSL06] M.J. Lewis, A.L. Short, and K.E. Lewis, « Autonomic nervous system control of the cardiovascular and respiratory systems in asthma », in: Respiratory Medicine 100.10 (Oct. 2006), pp. 1688–1705. [MA05] Richard J. Martin and Jalal M. Abu-Shaweesh, « Control of Breathing and Neonatal Apnea », in: Neonatology 87.4 (2005), pp. 288–295. [Mat11] O P Mathew, « Apnea of prematurity: pathogenesis and management strategies », in: Journal of Perinatology 31.5 (May 2011), pp. 302–310. [MD07] David J. Marlin and Christopher M. Deaton, « Pulmonary function testing », in: Equine Respiratory Medicine and Surgery, Elsevier Ltd, Jan. 2007, pp. 211–233. [MD09] Michael Mörschel and Mathias Dutschmann, « Pontine respiratory activity involved in inspiratory/expiratory phase transition. », in: Philosophical transactions of the Royal Society of London. Series B, Biological sciences 364.1529 (Sept. 2009), pp. 2517–26. [MH91] R Monteau and G Hilaire, « Spinal respiratory motoneurons. », in: Progress in neurobiology 37.2 (1991), pp. 83–144. [Mol+14] Yaroslav I. Molkov et al., « A closed-loop model of the respiratory system: Focus on hypercapnia and active expiration », in: PLoS ONE 9.10 (Oct. 2014), ed. by Michael Koval, e109894.

62 REFERENCES

[Mor+11] T S Moreira et al., « Central chemoreceptors and neural mechanisms of cardiorespiratory control. », in: Brazilian journal of medical and biological research = Revista brasileira de pesquisas medicas e biologicas 44.9 (Sept. 2011), pp. 883–9. [MR07] T D Morgan and J E Remmers, “Phylogeny and animal models: {An} unin- hibited survey”, In: Obstructive Sleep Apnea, 2007. [MS09] Reena Mehra and Kingman P. Strohl, « Evaluation and Monitoring of Res- piratory Function », in: Sleep Disorders Medicine, Elsevier Inc., Jan. 2009, pp. 188–197. [MWS15] Meghna P Mansukhani, Shihan Wang, and Virend K Somers, « Chemore- ﬂex physiology and implications for sleep apnoea: insights from studies in humans. », in: Experimental physiology 100.2 (Feb. 2015), pp. 130–5. [Nat11] Eugene Nattie, « Julius H. Comroe, Jr., Distinguished Lecture: Central chemoreception: then . . . and now », in: Journal of Applied Physiology 110.1 (Jan. 2011), pp. 1–8. [Nau+93] M Naughton et al., « Role of hyperventilation in the pathogenesis of central sleep apneas in patients with congestive heart failure. », in: The American review of respiratory disease 148.2 (Aug. 1993), pp. 330–8. [Nau+95] Matthew T. Naughton et al., « Treatment of congestive heart failure and Cheyne-Stokes respiration during sleep by continuous positive airway pressure », in: American Journal of Respiratory and Critical Care Medicine 151.1 (Jan. 1995), pp. 92–97. [Nau10] Matthew T. Naughton, Loop gain in apnea gaining: Control or controlling the gain?, Jan. 2010. [Nau98] M. T. Naughton, Pathophysiology and treatment of Cheyne-Stokes respiration, June 1998. [NK05] Jeanne M Nerbonne and Robert S Kass, « Molecular Physiology of Cardiac Repolarization », in: Physiological Reviews 85.4 (Oct. 2005), pp. 1205–1253. [OIK09] Hiroshi Onimaru, Keiko Ikeda, and Kiyoshi Kawakami, « Phox2b, RTN/pFRG neurons and respiratory rhythmogenesis », in: Respiratory Physiology & Neu- robiology 168.1-2 (Aug. 2009), pp. 13–18.

63 Chapter 1 – Physiology of cardiovascular and respiratory control

[Old+07] Olaf Oldenburg et al., « Sleep-disordered breathing in patients with symp- tomatic heart failure. A contemporary study of prevalence in and characteristics of 700 patients », in: European Journal of Heart Failure 9.3 (Mar. 2007), pp. 251–257. [Par+88] Markku Partinen et al., « Obstructive Sleep Apnea and Cephalometric Roentgenograms », in: Chest 93.6 (June 1988), pp. 1199–1205. [Pat+01] J F Paton et al., « Properties of solitary tract neurones responding to peripheral arterial chemoreceptors. », in: Neuroscience 105.1 (2001), pp. 231– 48. [Pep+13] Paul E. Peppard et al., « Increased prevalence of sleep-disordered breathing in adults », in: American Journal of Epidemiology 177.9 (2013), pp. 1006– 1014. [Pep00] Paul E Peppard, « Longitudinal Study of Moderate Weight Change and Sleep-Disordered Breathing », in: JAMA 284.23 (Dec. 2000), p. 3015. [Pla+08] Patrick Pladys et al., « Inﬂuence of Prematurity on Postnatal Maturation of Heart Rate and Arterial Pressure Responses to Hypoxia in Lambs », in: Neonatology 93.3 (2008), pp. 197–205. [PM17a] M E Patrinos and R J Martin, Apnea in the term infant, Semin. Fetal. Neonatal Med, 2017. [PM17b] M E Patrinos and R J Martin, Apnea, Bradycardia, and Desaturation, in: Manual of Neonatal Respiratory Care, 2017. [Poe+93] Christian F Poets et al., « The Relationship between Bradycardia, Apnea, and Hypoxemia in Preterm Infants », in: Pediatric Research 34.2 (Aug. 1993), pp. 144–147. [PS19] Nirzari K. Pandya and Sandeep Sharma, Capnography And Pulse Oximetry, StatPearls Publishing, Jan. 2019. [Rad+18] Nikola N. Radovanović et al., « Bidirectional cardio-respiratory interactions in heart failure », in: Frontiers in Physiology 9.MAR (Mar. 2018), p. 165. [Ram+01] R Ramanathan et al., « Cardiorespiratory events recorded on home monitors: Comparison of healthy infants with those at increased risk for SIDS. », in: JAMA 285.17 (May 2001), pp. 2199–207.

64 REFERENCES

[RB19] Mohan Rudrappa and Pradeep C. Bollu, Cheyne Stokes Respirations, 2019. [RSP07] Philippe Reix, Marie St-Hilaire, and Jean-Paul Praud, « Laryngeal sensitivity in the neonatal period: From bench to bedside », in: Pediatric Pul- monology 42.8 (Aug. 2007), pp. 674–682. [Ryb+07] Ilya A. Rybak et al., « Spatial organization and state-dependent mechanisms for respiratory rhythm and pattern generation », in: Progress in brain research, vol. 165, 2007, pp. 201–220. [SA03] D. Sistek and J. D. Aubert, Respiration de Cheyne-Stokes: Mécanismes et signification pathologique, 2003. [San+11] Scott A. Sands et al., « Loop gain as a means to predict a positive airway pressure suppression of Cheyne-Stokes respiration in patients with heart failure », in: American Journal of Respiratory and Critical Care Medicine (2011). [Sch+13] Steffen Schulz et al., « Cardiovascular and cardiorespiratory coupling analyses: a review », in: Philosophical Transactions of the Royal Society A: Mathe- matical, Physical and Engineering Sciences 371.1997 (Aug. 2013), p. 20120191. [Sen+17] Chamara V. Senaratna et al., Prevalence of obstructive sleep apnea in the general population: A systematic review, Aug. 2017. [Shy+91] B E Shykoff et al., « Respiratory sinus arrhythmia in dogs. Effects of phasic afferents and chemostimulation. », in: Journal of Clinical Investigation 87.5 (May 1991), pp. 1621–1627. [Sid+83] D. Sidi et al., « Developmental changes in oxygenation and circulatory responses to hypoxemia in lambs », in: American Journal of Physiology-Heart and Circulatory Physiology 245.4 (Oct. 1983), H674–H682. [Sil+16] Josiane N. Silva et al., « Neuroanatomical and physiological evidence that the retrotrapezoid nucleus/parafacial region regulates expiration in adult rats », in: Respiratory Physiology & Neurobiology 227 (June 2016), pp. 9–22. [SMA91] V K Somers, A L Mark, and F M Abboud, « Interaction of baroreceptor and chemoreceptor reflex control of sympathetic nerve activity in normal humans. », in: Journal of Clinical Investigation 87.6 (June 1991), pp. 1953– 1957.

65 Chapter 1 – Physiology of cardiovascular and respiratory control

[Smi+09] Jeffrey C Smith et al., « Structural and functional architecture of respiratory networks in the mammalian brainstem. », in: Philosophical transactions of the Royal Society of London. Series B, Biological sciences 364.1529 (Sept. 2009), pp. 2577–87. [Smi+15] V. C. Smith et al., « Stochastic Resonance Effects on Apnea, Bradycardia, and Oxygenation: A Randomized Controlled Trial », in: PEDIATRICS 136.6 (Dec. 2015), e1561–e1568. [Smi+91] J C Smith et al., « Pre-Bötzinger complex: a brainstem region that may generate respiratory rhythm in mammals. », in: Science (New York, N.Y.) 254.5032 (Nov. 1991), pp. 726–9. [Sol+99] P Solin et al., « Influence of pulmonary capillary wedge pressure on central apnea in heart failure. », in: Circulation 99.12 (Mar. 1999), pp. 1574–9. [Som+08] Virend K Somers et al., « Sleep Apnea and Cardiovascular Disease », in: Circulation 118.10 (Sept. 2008), pp. 1080–1111. [Som+89] V. K. Somers et al., « Influence of ventilation and hypocapnia on sympathetic nerve responses to hypoxia in normal humans », in: Journal of Applied Physiology 67.5 (Nov. 1989), pp. 2095–2100. [SS15] Robert C Stansbury and Patrick J Strollo, « Clinical manifestations of sleep apnea », in: Journal of Thoracic Disease 7.9 (Sept. 2015), E298–E310. [Swe13] C A Swenne, « Baroreflex sensitivity: mechanisms and measurement. », in: Netherlands heart journal : monthly journal of the Netherlands Society of Cardiology and the Netherlands Heart Foundation 21.2 (Feb. 2013), pp. 58– 60. [Tee19] Teesing, A Field Guide to Understanding Pressure Transducers Section I Basic Theory of Pressure Transducers, tech. rep., 2019. [Tou+08] Pierre Tourneux et al., « Influence of thermal drive on central sleep apnea in the preterm neonate. », in: Sleep 31.4 (Apr. 2008), pp. 549–56. [Var+19] Carolina Varon et al., « Quantification of Linear and Nonlinear Cardiorespi- ratory Interactions under Autonomic Nervous System Blockade », in: Com- puting in Cardiology, vol. 2019-September, IEEE Computer Society, Sept. 2019.

66 REFERENCES

[Vas12] Theodoros Vassilakopoulos, « Control of Ventilation and Respiratory Mus- cles », in: Clinical Respiratory Medicine (Jan. 2012), pp. 50–62. [VV17] Carolina Varon and Sabine Van Huffel, « Complexity and nonlinearities in cardiorespiratory signals in sleep and sleep apnea », in: Complexity and Non- linearity in Cardiovascular Signals, Springer International Publishing, Aug. 2017, pp. 503–537. [Wel+08a] Andrew Wellman et al., « Effect of oxygen in obstructive sleep apnea: Role of loop gain », in: Respiratory Physiology and Neurobiology (2008). [Wel+08b] Andrew Wellman et al., « Effect of oxygen in obstructive sleep apnea: Role of loop gain », in: Respiratory Physiology and Neurobiology 162.2 (July 2008), pp. 144–151. [Wel+11] Andrew Wellman et al., « A method for measuring and modeling the physiological traits causing obstructive sleep apnea », in: Journal of Applied Phys- iology 110.6 (June 2011), pp. 1627–1637. [Whi14] Robert H Whitaker, « Anatomy of the heart », in: Medicine 42.8 (Aug. 2014), pp. 406–408. [Wid01] J Widdicombe, « Airway receptors. », in: Respiration physiology 125.1-2 (Mar. 2001), pp. 3–15. [Wid98] J G Widdicombe, « Afferent receptors in the airways and cough. », in: Res- piration physiology 114.1 (Oct. 1998), pp. 5–15. [You+03] Terry Young et al., « Menopausal Status and Sleep-disordered Breathing in the Wisconsin Sleep Cohort Study », in: American Journal of Respiratory and Critical Care Medicine 167.9 (May 2003), pp. 1181–1185. [ZGM11] Jing Zhao, Fernando Gonzalez, and Dezhi Mu, « Apnea of prematurity: from cause to treatment », in: European Journal of Pediatrics 170.9 (Sept. 2011).

Chapter 2 STATE OF THE ART OF CARDIO-RESPIRATORY MODELING

Adult sleep apnea and pediatric apnea are both characterized by complex and dynamic interactions between the respiratory, the cardiovascular and the autonomic nervous systems. One way to tackle the complexity of these interactions is to create a mathematical representation of the intertwined function of these systems, to verify and to validate this modeling as much as possible and then to apply analytical methods to the proposed models. This integrative model-based approach seems particularly promising because it allows the integration of physiological knowledge on data processing tasks and it permits the analysis of underlying mechanisms that are diﬃcult or impossible to observe, while avoiding invasive clinical trials (neural activity, gas exchange,. . . ). The main objective of this chapter is to present a state of the art of the cardio-respiratory models currently available in the literature, representing adult and newborn physiology.

2.1 Models of the cardiovascular system

Models of the cardiovascular system can be divided in two main components: i) a component representing the electrical activity of the heart and ii) a component representing cardiac mechanics and circulation.

2.1.1 Model of the cardiac electrical activity

Since the 1950’s, the interest and knowledge of cardiac cell electrophysiology has been integrated in a wide range of models. These modeling eﬀorts can be categorized in three groups:

1. Realistic biophysical models of ion currents, deﬁning the continuous dynamics of the ions through the cardiac cell membranes [HH52; Nob62; Ten+06]; these de-

69 Chapter 2 – State of the art of cardio-respiratory modeling

Figure 2.1 – Diagram of the most important blocks of the cardio-respiratory system and its interactions.

tailed biophysical models require a high number of state variables and parameters to describe the ionic currents and action potential of a single cell, hindering their potential application in a global cardiovascular model, 2. Simplified ionic channel descriptions that are computationally cheaper models while reproducing the depolarization and repolarization activities of cardiac cells. These models are often based on the Hodgkin-Huxley simplification introduced by the FitzHugh-Nagumo model [Fit55; NAY62; KK12], 3. Simplified discrete models usually based on cellular automata, representing the electrical states of the action potential and their transitions [Her+00; Fle08; Le +08b]

2.1.2 Cardiac mechanics and circulatory model

In order to represent the whole CVS, the cardiac electrical activity models presented in section 2.1.1, must be coupled with mechanical descriptions of the cardiac tissue and hydraulic models allowing for the estimation of pressure and volume variations in each cardiac chamber. This electrical-mechanical and mechanical-hydraulic coupling has been largely addressed in the literature: from microscopic scales [HMK98], to macroscopic approaches [Gua+98; PN02], including intermediary formulations [LOH11]. A variety of mathematical models of the ventricular function have been proposed in

70 2.2. Respiratory models, gas exchange and gas transport other studies. The simplest models are based on a time-varying elastance [Gua+98; PN02]. These representations are usually implemented using lumped-parameter models, obtaining realistic simulations of cardiac pressure and volume and require low computational resources. However, the representation of the whole ventricle as a single element does not allow for an analysis of the regional ventricular function. Other approaches have been proposed in order to represent explicitly, at many different levels of detail, the mechanical activity of the ventricles. The simulation of these more complex models is often based on finite element methods (FEMs) [Ker+03; MM98; Ser+05; VM00]. While this kind of for- mulation provides a rather detailed description of the myocardium dynamics, it requires significant computational resources. Circulatory models have been also developed since the 1950’s. Most of these are based on the Windkessel model [Fra99] that describes the systemic arterial system as an anal- ogy with a hydraulic or an electric circuit in terms of resistance and compliance. The first model of a closed circulatory system with a constant amount of blood was developed by [Guy55; GLK55]. Following Guyton’s work, Grodins developed an uncontrolled cardiovascular system with systemic arteries and veins, pulmonary arteries and veins, and the left and right heart [Gro59]. Then, Beneken [Ben65; BD67] proposed a pulsatile and dynamic compartmental model of cardiovascular physiology in adults. This representation included the systemic circulation, establishing a difference between intrathoracic and extrathoracic arteries and veins. Each compartment is characterized by a hydraulic resistance, a compliance, and an unstressed volume. Inertial effects of blood are included in the systemic arterial compartments. This model has been complemented by the short, medium and long-term regulation of blood pressure by [Guy+72]. With the increase of computational power, a variety of other models of CVS can be found with different degrees of specificity as [UM00a; Smi04; Oje+15; Rom+16].

2.2 Respiratory models, gas exchange and gas transport

The modeling of the respiratory system can be categorized in four groups:

1. Simpliﬁed representations of the lungs as a constant-volume compartment with a continuous unidirectional ﬂow of air [BT00; GBB67; TPR04; UMA01]. The main parameter of these models is the minute ventilation.

71 Chapter 2 – State of the art of cardio-respiratory modeling

2. Lung mechanics models, the so-called "breathing models". They represent the airways and the diﬀerent compartments related to the respiration, such as the alveolar space and the chest wall [Fuk72; Rid91; Ava+01]. The ventilation drive of these models can be represented with a sinusoidal function [Ava+01] or another analytic function describing the activity of the respiratory muscles [Le +13; Fix+18]. 3. The gas exchange and gas transport models. These models describe the variations

of O2 and CO2 compartmental fractions due to changes in alveolar volume and pulmonary capillary blood [Ben+05; Ben+02]. They can also include the represen-

tation of O2 bonding and transport by hemoglobin and the transport of CO2 as bicarbonate [Ben06]. 4. The dynamical activity patterns of respiratory networks have also been modeled [BRS99; Smi+00; Ryb+04]. These representations emulate a central pattern generator of the brainstem respiratory networks using neuronal biophysical properties that have been proposed to account for the rhythmic activity patterns of inter- acting neuronal populations within distributed networks. In [Ryb+04], a detailed description of the respiratory network is coupled to a simpliﬁed model of the lungs.

Integrated models including the lung mechanics, gas exchange, gas transport and the control mechanisms can be found in the literature [KY89; TPR04; Bat07]. More recent models couple the brainstem respiratory network controlling the pulmonary subsystem, the lung biomechanics, and the gas exchange and transport [BS08; Mol+14; Bar+17; Mol+17].

2.3 Neural cardiorespiratory control

Most of the models of neural cardiorespiratory control are focused on specific sub- mechanisms separately. In this section, the state of the art of baroreflex and chemoreflex modeling is presented.

2.3.1 Baroreﬂex modeling

Models of baroreﬂex can be classiﬁed in two categories: behavioral and representative models. Behavioral models, which are based on auto-regressive representations [Bas+88; Bas+94], are particularly useful to analyze the spectral characteristics of heart rate signal. On the other hand, representative models integrate an explicit description of vagal

72 2.3. Neural cardiorespiratory control and sympathetic drive of the nervous systems on target organs [WS85; UGL98; UM03; Van+04]. Most of these models are based on transfer functions [Kaw+12]. Since these models describe the regulation of heart rate, contractility and peripheral resistance, they could be coupled to models of the cardiovascular systems [Smi+07]. The hemodynamic and nervous inﬂuences have been studied during controlled physiological tests, such as Valsalva maneuvers [Lu+01; Le +05] and orthostatic tests [Hel+02; Le +08a]. The inﬂu- ence of neuromodulation has also been studied through this kind of models by [Oje+15].

2.3.2 Chemoreﬂex modeling

Models of the chemoreflex have been mainly used to represent the control of the respiratory activity in closed-loop respiration models. Depending on the nature of the respiratory model, the chemoreflex efferents can modify i) the minute ventilation [Rev+89], ii) the amplitude and frequency of the ventilation drive signal [UM00a; Che+10; ACU11] or iii) the activity of specific neurons in the brainstem respiratory networks [Mol+14].

Inputs to the chemoreﬂex models are mostly the partial pressure of oxygen (PaO2) and carbon dioxide (PaCO2) in the blood at both central and peripheral sites. The peripheral chemoreceptors are sensitive to both PaO2 and PaCO2 and central chemoreceptors are sensitive to PaCO2 [Gou05]. Models of chemoreﬂex can be divided into three groups:

1. Analytic expressions or basic control loop feedback mechanisms that include all the chemoreceptor afferent contributions and directly modify the ventilation [Mil+65; Rev+89; Lu+03]. 2. Separated representations for the central chemoreceptors and the peripheral chemoreceptors in order to differentiate the dynamics of these components. Each chemoreceptor is usually composed of first order filters, with specific gains, time constants and delays. Central and peripheral responses are summed in order to obtain the global contribution of the chemoreflex to the respiratory drive [ACU11; Alb+15] 3. Models of the integration of the chemoreceptors within the brainstem’s respiratory

networks. In [Mol+14], the central chemoreception is represented via the CO2–dependent change of tonic excitatory drive from the RTN to the respiratory network. In [Bar+17] the peripheral chemoreﬂex is modeled with a population of 2nd-order cells in the NTS receiving peripheral chemoreceptor inputs and their eﬀerent projection to the respiratory and sympathetic circuits.

73 Chapter 2 – State of the art of cardio-respiratory modeling

Concerning only the chemoreceptor response, in [UM02], the transduction mechanisms of the peripheral chemoreceptors are explicitly modeled, where PaO2 and PaCO2 are transformed into electrical activity of the peripheral chemoreceptor ﬁbers.

2.4 Metabolism

Metabolism was usually modeled with a metabolic rate consumption of O2 and metabolic rate production of CO2 in the systemic tissue [Rev+89; UM00a; UM00b; San+09; Ell+13;

Alb+15; Che+16]. These rates modify the partial pressures or the concentrations of O2 and CO2 coming from the lung gas exchange.

The main diﬀerence between metabolism models is the location of O2 consumption and

CO2 production. In [Che+10] and [Rev+89] the metabolic rates are applied to compartments representing the brain and the body tissues; in [UM00a] and [Alb+15] metabolic rates are modeled in the coronaries, the brain, the skeletal muscles and the splanchnic and extrasplanchnic compartments.

2.5 Cardio-respiratory modeling

Some authors have worked to represent the interactions between the cardiovascular and respiratory systems, but with some limitations. Grodins et al. [Gro+54; Gro59; GBB67] and Guyton et al. [Guy+72] presented the pioneering works on cardio-respiratory modeling, but the limitations on the knowledge of the physiology and computational power at the time did not allow for a more complete description of the cardio-respiratory interactions. Ursino and Magosso [UM00a; MU01] proposed an integrated model of the cardiovascular control mechanisms but, these models lack a detailed description of the respiratory and gas exchange components. Ellwein [Ell+13] modeled the respiratory system and the CVS with a detailed description of the gas exchange and metabolism, but the control mechanisms are missing. Lu et al. [Lu+01; Lu+04] focused on the mechanical interactions between the respiratory and cardiovascular systems, including the gas exchange, but their work does not include a detailed description of the respiratory neural control. Cheng et al. [Che+10] presented a model that includes CVS and control models but also integrates a description of respiratory mechanics and gas exchange. This study is more focused on the interactions between sleep mechanisms and the autonomic nervous system

74 2.5. Cardio-respiratory modeling rather than on cardio-respiratory interactions. The model was not intended for patient- speciﬁc analysis and even if obstructive apneas, and periodic respiration were simulated, the results were not compared with real data. Albanese and Cheng [Alb+15; Che+16] developed one of the most advanced cardio- respiratory models to date, including Ursino and Magosso’s [UM00a; MU01] cardiovascular system and control models, a respiratory system with its control model, gas exchange and gas transport. Their study was mostly focused on acute perturbations aﬀecting the

O2 and CO2 levels in blood, but it was not tested in pathological conditions. The number of parameters of the model is another limiting factor of this model for the simulation of individual patients and model ﬁtting to pathological cases. To our knowledge, no integrated cardio-respiratory model has been proposed for the analysis of the cardio-respiratory response to apneas, having also the possibility to be numerically analyzed to provide patient-speciﬁc simulations.

Table 2.1 – Cardio-respiratory models in the literature with its components and the context where they were used. Ursino and Ellwein Lu Cheng Albanese Magosso (2013) (2001,2004) (2010) and Cheng (2000,2001) (2015) CVS X X X X X Respiration X X X X Baroreflex X X X X Chemoreflex X X X Gas XXX X exchange / metabolism Cardio- XXX X X respiratory interactions Context Response to Dynamics Analysis of Interactions Acute per- isocapnic during the Valsalva of sleep turbations hypoxia hypercapnia maneuver mechanisms affecting the and the O2 and CO2 ANS levels in blood

75 Chapter 2 – State of the art of cardio-respiratory modeling

2.6 Model adaptation to newborns

The number of models and sub-models of the cardio-respiratory physiology of infants and neonates remains very low compared to adult representations. While one may think that most of the adult models could be adapted to neonates with a relevant adaptation of parameters values and/or structural changes, these adaptations are far from being easy to represent. In the next sections, a state of the art of parameter adaptation and structural modiﬁcations of adult models to develop newborn models will be presented.

2.6.1 Cardiovascular model

The structure of CVS model for newborns presented by [Sá +06] was adapted from the adult CVS model proposed by [Ben65] and the infant CVS model proposed by [Goo+04]. Most of the parameters were scaled proportionally from adult/infant values to newborn values according to the blood volume and weight of a newborn at term. The rest of the parameters were adapted from values of the literature. The ventricular elastance curve was adapted based on the activation curves for a full-term fetus presented by [PBF97]. Speciﬁc structural modiﬁcations were introduced to represent four congenital heart defects that may be observed in newborns: (i) patent ductus arteriosus, (ii) tetralogy of Fallot, (iii) coarctation of the aorta, and (iv) transposition of the great arteries with atrial and ventricular septal defects. In [Jen+11] and [Dat+10], a similar approach was performed to develop a CVS preterm newborn model.

2.6.2 Respiration model

Respiratory models have been adapted for newborn lambs [Le +13] and preterm infants [Fix+18] based on adult models [Ava+01; Lu+01; Liu+98]. In [Le +13], a structural mod- iﬁcation was introduced, including the viscoelastic behavior of lung tissue. In [Fix+18], a nonlinear lung and chest wall compliances for the preterm infants was added.

2.6.3 Baroreﬂex

In [Sá +06; Jen+11] and [Dat+10] baroreﬂex control was included in the CVS model. In these cases, the parameters of the baroreﬂex submodel were scaled an adapted from adult models[WS85; UM03; UM02] based on the cardiovascular parameters of the newborn CVS model and values from the literature.

76 2.7. Conclusion

2.6.4 Chemoreﬂex

Few models of chemoreflex in newborns were found in the literature. In [Rev+89] the gains and time constants of the chemoreflex were manually fixed to fit the responses of the physiology.

2.6.5 Gas exchange and gas transport

For the adaptation of the gas exchange model, two main structural adaptations are needed: i) The oxygen dissociation curves and ii) a pulmonary shunt. The presence of fetal hemoglobin and the fast changes after birth induce important diﬀerences between the oxygen dissociation curves between adults an newborns. In [Rev+89] and [San+09], curves were ﬁtted in data describing the oxygen equilibrium curve in infants presented by [DRO71] and integrated in their models. The pulmonary shunt was included to represent the percentage of the blood in the pulmonary capillaries that is not oxygenated during the lung gas exchange. It was represented with a compartment added in parallel to the pulmonary peripheral circulation that does not participate in gas exchange [Rev+89].

2.6.6 Metabolism

Metabolic rates, time constants and delays of the sub-models are usually deﬁned from the physiology [San+09].

No complete, integrated cardio-respiratory model adapted for newborns were found in the literature.

2.7 Conclusion

This chapter presented the state of the art of cardio-respiratory models in adults and newborns. Only few models in the literature propose an integrated representation of the main cardio-respiratory physiological functions. In fact, it seems particularly difficult to simulate a particular phenomenon while maintaining a balance between model complexity, computational tractability and physiological reliability. To our knowledge, no complete cardio-respiratory model in adults has been used for the analysis of the cardio-respiratory response to apneas or for a patient specific identification.

77 Chapter 2 – State of the art of cardio-respiratory modeling

The number of works concerning the modeling of the cardio-respiratory physiology of infants and neonates remains very low compared to adult representations. The few existing ones are based on stabilized methodologies to scale parameter values from adult to newborn physiology and to include relevant structural differences of newborns. The fact that there is no complete, integrated cardio-respiratory model adapted to newborn physiology, opens the possibility of innovation in this topic. Based on the physiological processes involved in sleep apneas in adults and apnea- bradycardias in newborns, as presented in section 1, and cardio-respiratory models existing in the literature, the next part of this work will describe our contributions in this field, with the proposal and analysis of three cardio-respiratory models adapted for (i) adults, (ii) a newborn with a gestational age of 40 weeks and a post-menstrual age of 41 week and (iii) a preterm newborn with a gestational age of 28 weeks and a post-menstrual age of 29 weeks. The proposed adult model will be used for patient specific identification in the context of sleep apneas.

References

78 REFERENCES

[Bas+94] G. Baselli et al., « Model for the assessment of heart period and arterial pressure variability interactions and of respiration influences », in: Medical & Biological Engineering & Computing 32.2 (Mar. 1994), pp. 143–152. [Bat07] Jerry J. Batzel, Cardiovascular and respiratory systems : modeling, analysis, and control, Society for Industrial and Applied Mathematics, 2007, p. 274. [BD67] J E Beneken and B DeWit, A physical approach to hemodynamic aspects of the human cardiovascular system, in: Physical Bases of Circulatory Trans- port: Regulation and Exchange, 1967, pp. 1–45. [Ben+02] Habib Benallal et al., « Modeling of end-tidal and arterial PCO2 gradient: comparison with experimental data. », in: Medicine and science in sports and exercise 34.4 (2002), pp. 622–9. [Ben+05] Habib Benallal et al., « Evaluation of cardiac output from a tidally ventilated homogeneous lung model », in: European Journal of Applied Physiology 95.2- 3 (2005), pp. 153–162. [Ben06] Alona Ben-Tal, « Simplified models for gas exchange in the human lungs », in: Journal of Theoretical Biology 238.2 (2006), pp. 474–495. [Ben65] J E Beneken, A mathematical approach to cardio-vascular function: the uncontrolled human system, Rijksuniversiteit te Utrecht, Utrecht, 1965. [BRS99] R J Butera, J Rinzel, and J C Smith, « Models of respiratory rhythm generation in the pre-Bötzinger complex. I. Bursting pacemaker neurons », in: Journal Of Neurophysiology 82.1 (1999), pp. 382–397. [BS08] Alona Ben-Tal and Jeffrey C. Smith, « A model for control of breathing in mammals: Coupling neural dynamics to peripheral gas exchange and transport », in: Journal of Theoretical Biology 251.3 (2008), pp. 480–497. [BT00] J.J. Batzel and H.T. Tran, « Modeling instability in the control system for human respiration: applications to infant non-REM sleep », in: Applied Mathematics and Computation 110.1 (Apr. 2000), pp. 1–51. [Che+10] Limei Cheng et al., « An integrative model of respiratory and cardiovascular control in sleep-disordered breathing », in: Respiratory Physiology and Neurobiology 174.1-2 (2010), pp. 4–28.

79 Chapter 2 – State of the art of cardio-respiratory modeling

[Che+16] Limei Cheng et al., « An Integrated Mathematical Model of the Human Car- diopulmonary System: Model Validation under Hypercapnia and Hypoxia », in: American Journal of Physiology - Heart and Circulatory Physiology 2 (2016), ajpheart.00923.2014. [Dat+10] Marco Dat et al., « Modeling cardiovascular autoregulation of the preterm infant », in: Master, Department of cardiovascular biomechanics, Eindhoven University of Technology, Eindhoven (2010). [DRO71] Maria Delivoria-Papadopoulos, Nevenka P Roncevic, and Frank A Oski, « Postnatal Changes in Oxygen Transport of Term, Premature, and Sick In- fants: The Role of Red Cell 2,3-Diphosphoglycerate and Adult Hemoglobin », in: Pediatric Research 5.6 (June 1971), pp. 235–245. [Ell+13] L. M. Ellwein et al., « Modeling cardiovascular and respiratory dynamics in congestive heart failure », in: Mathematical Biosciences 241 (2013), pp. 56– 74. [Fit55] Richard FitzHugh, « Mathematical models of threshold phenomena in the nerve membrane », in: The Bulletin of Mathematical Biophysics 17.4 (Dec. 1955), pp. 257–278. [Fix+18] Laura Ellwein Fix et al., « Theoretical open-loop model of respiratory mechanics in the extremely preterm infant », in: (2018), pp. 1–21. [Fle08] J Fleureau, « Intégration de données anatomiques issues d’images MSCT et de modéles électrophysiologique et mécanique du coeur. », PhD thesis, Université de Rennes 1, 2008. [Fra99] Otto Frank, Die Grundform des arteriellen pulses: Mathematische Analyse. Erste Abhandlung - Otto Frank - Google Books, 1899, p. 44. [Fuk72] Y Fukui, « A Study of the Human Cardiovascular-Respiratory System Using Hybrid Computer Modeling. », PhD thesis, 1972. [GBB67] F S Grodins, J Buell, and A J Bart, « Mathematical analysis and digital simulation of the respiratory control system. », in: Journal of Applied Physiology 22.2 (Feb. 1967), pp. 260–276.

80 REFERENCES

[GLK55] Arthur C. Guyton, Arthur W. Lindsey, and Berwind N. Kaufmann, « Ef- fect of Mean Circulatory Filling Pressure and Other Peripheral Circulatory Factors on Cardiac Output », in: American Journal of Physiology-Legacy Content 180.3 (Feb. 1955), pp. 463–468. [Goo+04] Jane A. Goodwin et al., « A model for educational simulation of infant cardiovascular physiology », in: Anesthesia and Analgesia 99.6 (2004), pp. 1655– 1664. [Gou05] Alexander V Gourine, « On the peripheral and central chemoreception and control of breathing: an emerging role of ATP. », in: The Journal of physiology 568.Pt 3 (Nov. 2005), pp. 715–24. [Gro+54] Fred S. Grodins et al., « Respiratory Responses to CO2 Inhalation. A The- oretical Study of a Nonlinear Biological Regulator », in: Journal of Applied Physiology 7.3 (Nov. 1954), pp. 283–308. [Gro59] F S Grodins, « Integrative cardiovascular physiology: a mathematical syn- thesis of cardiac and blood vessel hemodynamics. », in: The Quarterly review of biology 34.2 (June 1959), pp. 93–116. [Gua+98] M. Guarini et al., « Estimation of cardiac function from computer analysis of the arterial pressure waveform », in: IEEE Transactions on Biomedical Engineering 45.12 (Dec. 1998), pp. 1420–1428. [Guy+72] Arthur C Guyton et al., « Systems analysis of arterial pressure regulation and hypertension », in: Annals of Biomedical Engineering 1.2 (1972), pp. 254– 281. [Guy55] Arthur C. Guyton, « Determination of Cardiac Output By Equating Venous Return Curves With Cardiac Response Curves », in: Physiological Reviews 35.1 (Jan. 1955), pp. 123–129. [Hel+02] Thomas Heldt et al., « Computational modeling of cardiovascular response to orthostatic stress », in: Journal of Applied Physiology 92.3 (Mar. 2002), pp. 1239–1254. [Her+00] A I Hernández et al., « Overview of CARMEM: a new dynamic quantitative cardiac model for ECG monitoring and its adaptation to observed signals. », in: Acta biotheoretica 48.3-4 (Dec. 2000), pp. 303–22.

81 Chapter 2 – State of the art of cardio-respiratory modeling

[HH52] A L Hodgkin and A F Huxley, « A quantitative description of membrane current and its application to conduction and excitation in nerve. », in: The Journal of physiology 117.4 (Aug. 1952), pp. 500–44. [HMK98] P J Hunter, A D McCulloch, and H E ter Keurs, « Modelling the mechanical properties of cardiac muscle. », in: Progress in biophysics and molecular biology 69.2-3 (1998), pp. 289–331. [Jen+11] Ward Jennekens et al., « Validation of a preterm infant cardiovascular system model under baroreflex control with heart rate and blood pressure data », in: 2011 Annual International Conference of the IEEE Engineering in Medicine and Biology Society (2011), pp. 896–899. [Kaw+12] T. Kawada et al., « Consideration on parameter determination of a new model describing dynamic vagal heart rate control in rats », in: 2012 Annual International Conference of the IEEE Engineering in Medicine and Biology Society, vol. 2012, IEEE, Aug. 2012, pp. 3809–3812. [Ker+03] ROY C.P. Kerckhoffs et al., « Timing of Depolarization and Contraction in the Paced Canine Left Ventricle: » in: Journal of Cardiovascular Electro- physiology 14.s10 (Oct. 2003), S188–S195. [KK12] S Karamcheti and J Kravitz, « Analyzing Cardiac Action Potentials with the Fitzhugh-Nagumo Model », in: Tech. rep. California State Summer School for Mathematics & Science Cluster 9 9 (2012). [KY89] MCK Khoo and SM Yamashiro, Models of control of breathing, New York: Chang M. P. HA, editor. Respiratory Physiology: An Analytical Approach. Mercer Dekker, 1989, pp. 799–829. [Le +05] V. Le Rolle et al., « A Bond Graph Model of the Cardiovascular System », in: Acta Biotheoretica 53.4 (Dec. 2005), pp. 295–312. [Le +08a] Virginie Le Rolle et al., « An autonomic nervous system model applied to the analysis of orthostatic tests », in: Modelling and Simulation in Engineering 2008.i (2008). [Le +08b] Virginie Le Rolle et al., « Model-based analysis of myocardial strain data acquired by tissue Doppler imaging », in: Artificial Intelligence in Medicine 44.3 (Nov. 2008), pp. 201–219.

82 REFERENCES

[Le +13] Virginie Le Rolle et al., « Mathematical modeling of respiratory system mechanics in the newborn lamb . To cite this version : HAL Id : hal-00880028 », in: (2013). [Liu+98] C. H. Liu et al., « Airway mechanics, gas exchange, and blood ﬂow in a nonlinear model of the normal human lung », in: Journal of Applied Physiology 84.4 (Apr. 1998), pp. 1447–1469. [LOH11] Virginie Le Rolle, David Ojeda, and Alfredo I. Hernández, « Embedding a Cardiac Pulsatile Model Into an Integrated Model of the Cardiovascular Reg- ulation for Heart Failure Followup », in: IEEE Transactions on Biomedical Engineering 58.10 (Oct. 2011), pp. 2982–2986. [Lu+01] K Lu et al., « A human cardiopulmonary system model applied to the analysis of the Valsalva maneuver. », in: American journal of physiology. Heart and circulatory physiology 281.6 (2001), H2661–H2679. [Lu+03] K. Lu et al., « Whole-Body Gas Exchange in Human Predicted by a Car- diopulmonary Model », in: Cardiovascular Engineering: An International Journal 3.1 (2003), pp. 1–19. [Lu+04] K. Lu et al., « Cerebral autoregulation and gas exchange studied using a human cardiopulmonary model », in: American Journal of Physiology-Heart and Circulatory Physiology 286.2 (Feb. 2004), H584–H601. [Mil+65] H T Milhorn et al., « A Mathematicla Model of the Human Respiratory Control System », in: Biophysical journal 5.1 (Jan. 1965), pp. 27–46. [MM98] K May-Newman and A D McCulloch, « Homogenization modeling for the mechanics of perfused myocardium. », in: Progress in biophysics and molecular biology 69.2-3 (1998), pp. 463–81. [Mol+14] Yaroslav I. Molkov et al., « A closed-loop model of the respiratory system: Focus on hypercapnia and active expiration », in: PLoS ONE 9.10 (Oct. 2014), ed. by Michael Koval, e109894. [Mol+17] Yaroslav I. Molkov et al., « Computational models of the neural control of breathing », in: Wiley Interdisciplinary Reviews: Systems Biology and Medicine 9.2 (2017), pp. 1–22.

83 Chapter 2 – State of the art of cardio-respiratory modeling

[MU01] Elisa Magosso and Mauro Ursino, « A mathematical model of CO 2 effect on cardiovascular regulation », in: American Journal of Physiology-Heart and Circulatory Physiology 281.5 (2001), H2036–H2052. [NAY62] J. Nagumo, S. Arimoto, and S. Yoshizawa, « An Active Pulse Transmission Line Simulating Nerve Axon », in: Proceedings of the IRE 50.10 (Oct. 1962), pp. 2061–2070. [Nob62] D Noble, « A modification of the Hodgkin–Huxley equations applicable to Purkinje fibre action and pace-maker potentials. », in: The Journal of physiology 160.2 (Feb. 1962), pp. 317–52. [Oje+15] David Ojeda et al., « Analysis of a baroreflex model for the study of the chronotropic response to vagal nerve stimulation », in: 2015 7th International IEEE/EMBS Conference on Neural Engineering (NER), IEEE, Apr. 2015, pp. 541–544. [PBF97] G Pennati, M Bellotti, and R Fumero, « Mathematical modelling of the human foetal cardiovascular system based on Doppler ultrasound data. », in: Medical engineering & physics 19.4 (June 1997), pp. 327–35. [PN02] Joseph L Palladino and Abraham Noordergraaf, A Paradigm for Quantifying Ventricular Contraction, tech. rep., 2002. [Rev+89] Michael Revow et al., « A Model of the Maturation of Respiratory Control in the Newborn Infant », in: IEEE Transactions on Biomedical Engineering 36.4 (1989), pp. 414–423. [Rid91] V Rideout, Mathematical and Computer Modeling of Physiological Systems, Upper Saddle River, {NJ}: Prentice Hall, 1991. [Rom+16] Hector M. Romero Ugalde et al., « Model-Based Design and Experimental Validation of Control Modules for Neuromodulation Devices », in: IEEE Transactions on Biomedical Engineering 63.7 (2016), pp. 1551–1558. [Ryb+04] I.A. Rybak et al., « Modeling the ponto-medullary respiratory network », in: Respiratory Physiology & Neurobiology 143.2-3 (Nov. 2004), pp. 307–319. [Sá +06] Carla D. Sá Couto et al., « A Model for Educational Simulation of Neonatal Cardiovascular Pathophysiology », in: Simulation in Healthcare: The Journal of the Society for Simulation in Healthcare 1.Inaugural (2006), pp. 4–9.

84 REFERENCES

[San+09] Scott A. Sands et al., « A model analysis of arterial oxygen desaturation during apnea in preterm infants », in: PLoS Computational Biology 5.12 (2009). [Ser+05] M. Sermesant et al., « Cardiac Function Estimation from MRI Using a Heart Model and Data Assimilation: Advances and Diﬃculties », in: Springer, Berlin, Heidelberg, 2005, pp. 325–337. [Smi+00] Jeﬀrey C Smith et al., « Respiratory rhythm generation in neonatal and adult mammals : the hybrid pacemaker – network model », in: Elsevier 122 (2000), pp. 131–147. [Smi+07] Bram W. Smith et al., « Simulation of cardiovascular system diseases by including the autonomic nervous system into a minimal model », in: Computer Methods and Programs in Biomedicine 86.2 (2007), pp. 153–160. [Smi04] B. W Smith, « Minimal haemodynamic modelling of the heart & circulation for clinical application. », in: January (2004). [Ten+06] KHWJ Ten Tusscher et al., Progress in biophysics and molecular biology. Vol. 90, 1-3, Pergamon Press, 2006, pp. 326–345. [TPR04] Zbigniew L Topor, Mariusz Pawlicki, and John E Remmers, « A computational model of the human respiratory control system: responses to hypoxia and hypercapnia. », in: Annals of biomedical engineering 32.11 (Nov. 2004), pp. 1530–45. [UGL98] Mauro Ursino, Marco Giulioni, and Carlo Alberto Lodi, « Relationships among cerebral perfusion pressure, autoregulation, and transcranial Doppler waveform: a modeling study », in: Journal of Neurosurgery 89.2 (Aug. 1998), pp. 255–266. [UM00a] M Ursino and E Magosso, « Acute cardiovascular response to isocapnic hypoxia. I. A mathematical model. », in: American journal of physiology. Heart and circulatory physiology 279 (2000), H149–H165. [UM00b] M Ursino and E Magosso, « Acute cardiovascular response to isocapnic hypoxia. II. Model validation. », in: American journal of physiology. Heart and circulatory physiology 279.1 (2000), H166–H175.

85 Chapter 2 – State of the art of cardio-respiratory modeling

[UM02] Mauro Ursino and Elisa Magosso, « A theoretical analysis of the carotid body chemoreceptor response to O2 and CO2 pressure changes », in: Respiratory Physiology & Neurobiology 130.1 (2002), pp. 99–110. [UM03] Mauro Ursino and Elisa Magosso, « Role of short-term cardiovascular regulation in heart period variability: a modeling study », in: American Journal of Physiology - Heart and Circulatory Physiology 284.4 (2003), H1479–H1493. [UMA01] M Ursino, E Magosso, and G Avanzolini, « An integrated model of the human ventilatory control system: the response to hypercapnia. », in: Clinical physiology (Oxford, England) 21.4 (July 2001), pp. 447–64. [Van+04] Arie M. Van Roon et al., « Introducing a baroreﬂex model for studying cardiovascular eﬀects of mental workload », in: Psychophysiology 41.6 (2004), pp. 961–981. [VM00] Frederick J. Vetter and Andrew D. McCulloch, « Three-Dimensional Stress and Strain in Passive Rabbit Left Ventricle: A Model Study », in: Annals of Biomedical Engineering 28.7 (July 2000), pp. 781–792. [WS85] KH Wesseling and JJ Settels, « Baromodulation explains short-term blood pressure variability », in: Psychophysiology of cardiovascular control (1985), pp. 69–97.

86 Chapter 3

METHODSANDTOOLSFORSIMULATION AND MODEL ANALYSIS

Implementation and analysis of integrated mathematical models are complex tasks that require a set of tools and methods to be appropriately implemented and applied. These tasks become particularly challenging in the case of complex, hybrid models, that may be composed of a set of submodels, developed under different mathematical formalisms. This chapter presents, in a first section, a state of the art of modeling and simulation tools, emphasizing the modeling toolkit developed by our group and used throughout this thesis: multi-formalism modeling and simulation environment (M2SL). A second section will focus on the methods for sensitivity parameter analysis used to evaluate and rank the parameters of a model under a specific situation, in our case sleep apnea or apneas of prematurity. The third section is focused on parameter identification methods, that will be applied in this work in order to fit the model to experimental data.

3.1 Modeling and simulation

The extended application of models in diﬀerent research disciplines has led to a vast choice of modeling and simulation tools. Primarily, industrial processes have driven the development of most simulation tools, but recent international initiatives in systems biology and physiological modeling, such as the IUPS Physiome [Mil10] or the Virtual Physiologi- cal Human [KN09], have encouraged the advancement of new modeling tools designed for applications in the life sciences. Unfortunately, describing all recent developments would be prohibitively long. This section summarizes the most important simulation tools, which are readily applicable to modeling and simulation in general, and those speciﬁcally useful for applications in physiology.

87 Chapter 3 – Methods and tools for simulation and model analysis

Generic integrated environments: A set of popular modeling and simulation tools are based on generic, graphical computing environments. In this category, commercial applications, led by MATLAB®/Simulink 1, are widespread for providing complete and extensive packages for numerous scientific domains (engineering, electronics, biology, mechanics, etc.). Currently, other commercial competitors offer good alternatives, including Wolfram Mathematica®/System Modeler 2, Dymola 3, MapleSim 4, Stella - Berkeley Madonna 5 and, in a lesser extent, an open-source free alternative, Scilab/Xcos 6. Simi- larly, several open source python-based libraries as SciPy 7 and SimPy 8 are available with multiple modules for optimization, linear algebra, integration, etc. These environments usually provide an embedded or independent graphical toolkit (such as Simulink, System Modeler, or Xcos) specifically designed for the creation of models using modular interconnected blocks. Only a few of these environments (Simulink, for instance) provide means for coupling different model formalisms (discrete models, ordinary differential equation ODE models), using specific libraries (such as StateFlow, for discrete models in Simulink) and the employment of advanced parameter analysis methods. Within this group of generic environments there is a set of tools specifically designed for building multi-physics and multi-scale models. ANSYS 9 solvers, COMSOL Multi- physics® 10 and the ADINA 11 systems stand out as the most popular commercial products. Although mainly applied to the automotive, aerospace, and fluid mechanics fields, these systems have also been successfully used in a number of biomedical applications. Although powerful and refined, most of these tools do not present an explicit approach to multi-formalism simulations. Simulations are performed using a centralized simulator approach that is specifically optimized for a given formalism, with globally-fixed simulation parameters. In this sense, these generic integrated environments are not optimal for handling formally heterogeneous systems.

1. http://www.mathworks.com 2. http://www.wolfram.com/system-modeler 3. http://www.dymola.com 4. http://www.maplesoft.com/products/maplesim 5. http://www.berkeleymadonna.com 6. http://www.scilab.org 7. https://scipy.org/scipylib/ 8. https://simpy.readthedocs.io/ 9. http://www.ansys.com 10. http://www.comsol.com 11. http://www.adina.com

88 3.1. Modeling and simulation

Generic modeling languages: In the pursuit of a machine and human readable description of a model, numerous modeling languages have been proposed, along the vast choice of simulation tools. Yet, a particular language emerged from the international coop- eration of key authors in the modeling field, the Modelica Association 12 and the Modelica language: a non-proprietary, equation-based modeling language for large hierarchical systems. The Modelica language allows for the definition of continuous (differential algebraic equations—DAE and ODE models) or discrete-time models. It has been applied to mul- tidomain models (robotics, mechanics, aerospace), but rarely to physiological modeling. Several commercial tools implement and profit from Modelica’s versatility to define their models, such as Dymola, MapleSim, Wolfram System Modeler, among others. An attrac- tive tool for educational purposes is the open-source implementation OpenModelica 13, providing a large set of tutorials, documentation and parameter analysis tools. Moreover, OpenModelica enjoys from large and very active developer and user communities.

Specific tools for physiological applications: Several simulation tools have been specifically designed for physiological modeling applications. Currently, most major efforts in providing specific physiological modeling tools are listed on the Physiome Project 14. Notable examples include the Continuum Mechanics, Image analysis, Signal processing and Signal identification (CMISS 15) toolkit, and its open source counterpart OpenCMISS, an environment specialized on finite element analysis on bioengineering problems. Conti- nuity 6 16 also offers a multi-scale modeling environment that has been used for cardiac mechanics and electrophysiology modeling. For fine scale applications on cancer, cardiac and soft-tissue, CHASTE [Mir+13] offers a simulation tool that permits the integration from cell to tissue models. However, these tools are not adapted to the integration of system-level physiological models that may be used to refine the boundary conditions of FEM models. A Java-based simulation system, JSim 17, deserves a special mention, not only due to the fact that it integrates ODE, PDE with discrete event models, but because it has grouped a set of more than 70000 models in a public online database. Moreover, PySb 18 offers a python-based framework for building mathematical models of biochemical

12. http://www.modelica.org 13. http://www.openmodelica.org 14. http://physiomeproject.org 15. http://www.cmiss.org 16. http://www.continuity.ucsd.edu 17. http://www.physiome.org/jsim 18. http://pysb.org/

89 Chapter 3 – Methods and tools for simulation and model analysis systems. In the particular context of biological and physiological modeling, a set of markup languages have been developed by researchers in order to ease the sharing, curation and testing phases. The most significant efforts are SBML [Huc+03], CellML [Gar+08] and FieldML [Bri+13]. The Systems Biology Markup Language (SBML) is a XML-based language designed from systems biology concepts that defines models as a description of chemical substances, reactions, parameters and mathematical expressions. CellML is also an XML-based format with a modular structure, which allows for model reuse, defining models as several interconnected components. Each component is defined by a set of variables and mathematical rules expressed in another markup language, MathML 19. Not constrained to cellular models, as incorrectly implied from its name, a large database of (partially) curated models are part of the CellML tools 20. Finally, FieldML [Chr+09] is an XML representation still under development that seeks to define multivariate field models, which is currently impossible with other markup languages.

Multi-formalism environments: Whereas most modeling environments are specialized in a particular formalism, a group of modeling and simulation tools handle multi- formalism systems explicitly. A special group can be deﬁned for these tools. The majority of multi-formalism frameworks adopt the formalism transformation approach, following the morphisms introduced by [ZPK00]. Zeigler focused most of his research on the transformation towards the DEVS formalism, but he also introduced the co-simulation of DEVS and DESS formalisms using interface objects, managed by coordinator objects. His work was continued with the DEVS-suite simulator 21, although this framework is so specialized in DEVS that its multi-formalism features are question- able. Meanwhile, Vangheluwe [Van01] worked on the formalism transformation concepts further, introducing the formalism transformation graph (FTG, Figure 3.1), illustrat- ing the possible model formalism morphisms, emphasizing the transformation towards a meta-formalism that incorporates DEVS and DESS. Vangheluwe’s initial eﬀorts were concentrated on a declarative modeling language named MSL, which was later renamed WEST++ and commercialized for water treatment plants. Later, his work led to the creation of AToM3 [DV02] and to the establishment of Modelica. Other notable simulation environments that use formalism-transformation are OsMoSys [Vit+04] and the

19. http://www.w3.org/Math 20. http://models.cellml.org 21. http://acims.asu.edu/software/devs-suite

90 3.1. Modeling and simulation

Virtual Laboratory Environment (VLE) [QDR09], a framework specialized in DEVS, parallel DEVS, Quantified State Systems (QSS), cellular automata and differential equations. Among the co-simulation implementations, a prominent effort is the High Level Architec-

Figure 3.1 – Formalism Transformation Graph (FTG), introduced by [Van01]: solid lines represent an existing morphism that transforms one formalism to another. Gray dashed lines indicate the availability of a simulator for a formalism. Adapted from [Van01]. ture (HLA), a general purpose architecture designed for distributed computer simulation systems. HLA is not a modeling environment, but a standard (in fact, it has become the IEEE 1516 standard) that deﬁnes how computer simulations communicate data and syn- chronize their actions by introducing coordination time points. Implementations of HLA are called Run-Time Infrastructures (RTI); a number of commercial and non-commercial RTI implementations exist today, although current eﬀorts are mostly oriented to the aspects and computational advantages of large distributed simulations. Our research team has developed, since more than 10 years, a multi-formalism modeling framework denominated M2SL. This tool can simulate and analyze multi-formalism and multi-resolution systems, based on the co-simulation method. Due to its capabilities and its previous use in several models of the group [Le +15; Rom+16; Cal18], M2SL was chosen as the modeling framework for this work.

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3.1.1 Multi-formalism Modeling and Simulation Library (M2SL)

The modeling and simulation toolkit used for all models in this thesis is the Multi- formalism Modeling and Simulation Library (M2SL). Works in this toolkit were initiated during a PhD of our team [Def06], with applications on the cardiac electrical system. Con- tinuously in evolution since that time, M2SL has been stabilized and registered [Her+09] and adapted to solve the problems encountered by multi-formalism and multi-scale modeling [Her+11; Oje13]. M2SL has been progressively improved, including a variety of coupled formalism-specific simulators, including discrete-time and continuous ordinary differential equations (ODE) [Def06], Bond-Graphs [Le 06], or low-resolution finite element methods [Fle08]. In order to solve the dynamics of the targeted heterogeneous models, M2SL uses the co-simulation principle. M2SL uses definitions of transformation objects and establish different synchronization strategies [Her+09; Her+11].

Model representation

A model in M2SL is a set of interconnected components; a combination of two types of model objects: atomic models (M a) and coupled models (M coup). Atomic models are the description of a specific component of a system using one particular formalism. Coupled models are the composition of two or more models that may be defined under different formalisms and the connections between them. A graphical representation of atomic and coupled models, including their organization, is presented as the model hierarchy in the left part of Figure 3.2. When a computational model is defined in M2SL, a global simulator S, called the root coordinator, is first created. This object analyzes the model hierarchy and creates a a a simulator Si for each atomic model Mi . The choice of the appropriate simulator type is automatically handled by the library. In this way, a model with formalism Fi is associated coup with a simulator designed for the same formalism Fi. For each coupled model Mi , a coup coordinator Si is created. Coordinators are a special kind of simulator that handle the connection of the internal components of a complex model and computes model outputs at the coupled level. Using an object-oriented methodology, models in M2SL are represented with different abstract classes, which define the structural elements of a model and its behaviors. The development of a model in M2SL consists in choosing a base abstract class, defining its data structures and then the programming of its behavior. The available data structures

92 3.1. Modeling and simulation

Figure 3.2 – Hierarchical structure of models and their corresponding simulators. Schema based on [ZPK00], adapted from [Oje13]. and behaviors of a model depends on the formalism of the model; however, it always complies with the following statement: a model is represented as a tuple M (F, I,O,E,P). The relation between each element of this tuple and the structures of M2SL is explained below:

Formalism F : The formalism of a model is deﬁned by the abstract class chosen as base class for its implementation. In other words, for each formalism, M2SL provides an abstract class. As of version 1.8.4, the available formalisms are summarized in Table 3.1. Following an object-oriented paradigm, a model in M2SL must inherit from one of these classes. Moreover, each formalism requires the implementation of particular behaviors, represented by the methods of each class. These behaviors will be presented later.

Table 3.1 – Formalisms supported in M2SL and their corresponding class. Adapted from [Oje13]. Formalism F Class Algebraic equations GenericModel Ordinary diﬀerential equations OdeModel Algebraic equations with discrete time DiscreteTimeModel

93 Chapter 3 – Methods and tools for simulation and model analysis

Table 3.2 – Metadata related to variables and parameters. Adapted from [Oje13]. Property Type Relevant to Description label String Any Unique identiﬁer of the variable units String Any Units of the variable (optional) printable Boolean Any Whether or not to include its trajec- tory in the output ﬁle errorScaleFactor Number State variable Weight used for the error associated with this variable

Variables I,O,E,P: The variables of a model are organized in four different groups according to their semantic definition: inputs, outputs, states, and parameters. Each single variable or parameter can be represented by any data structure provided by the C++ language 22. Variables and parameters are encapsulated in a class named GenericVariable, an object that aggregates metadata regarding the user configuration of each variable (Ta- ble 3.2).

Components: As explained before, in M2SL, models can be either atomic or complex. To permit the creation of complex models, the submodels container is also included in the deﬁnition of a model, which accommodates a list of references to other models.

Behaviors: The behavioral deﬁnition of a model comprises four diﬀerent procedures:

— Initialization: the calculation or simple assignment of initial values to all variables of the model.

— Variable synchronization: the update or modiﬁcation of the internal state of the model due to a change in the input variables.

— Output calculation: the computation of the output variables from the current internal state and the input variables.

— Termination: the ﬁnal procedure executed when the simulation ends.

This list serves as the base behavior set for all formalisms in M2SL; more behaviors may complement them when a particular formalism requires it.

22. The most natural choice among all C++ data structures would be int for any integer value or double for real values, but any other data type can be used.

94 3.1. Modeling and simulation

Figure 3.3 – Object oriented representation of models in M2SL. Adapted from [Oje13].

95 Chapter 3 – Methods and tools for simulation and model analysis

The simulation loop

All the objects, procedures and relations defined by M2SL are brought together in the simulation loop. A simulation in M2SL is conducted by a root coordinator, represented by the RootCoordinator class. This crucial element defines and updates the global time of the simulation, while coordinating the underlying simulators and their local simulation time. It consists of three procedures executed in a sequential fashion: initialization, simulation loop and finalization (Figure 3.4, left side).

Figure 3.4 – General execution ﬂow of a simulation in M2SL (left) and its detailed simulation loop (right). Adapted from [Oje13].

First, the initialization step prepares all models and simulators for the simulation. Then, the simulation loop repeats the following steps, illustrated in Figure 3.4 (right

96 3.2. Sensitivity Analysis side): i) synchronization of models, ii) simulation of models, iii) calculation of outputs, iv) advance global time, v) check stopping conditions. Lastly, when the simulation loop meets the stopping condition, the ﬁnalization step releases all resources acquired during the simulation.

Adaptive simulation and synchronization

Simulations in M2SL can follow diﬀerent temporal strategies for the step-by-step advancement of models’ dynamics, which aﬀect the internal execution of the simulation loop. There are three possible simulation strategies:

Fixed step simulation (FIXED): the user deﬁnes a global simulation step DT (δt). All simulators advance the state of a model using a single step of the same size. At the end of each simulation step, the synchronization of input and output variables is performed. Adaptive step with smallest synchronization step (ADAPT_SMALLEST): initially, each

simulator Si has an independent simulation step δti and the user speciﬁes an initial

synchronization step couplingDT (δtc). Each simulator advances with an adaptive

step until they all reach δtc, where the input and output synchronization occur. At this point, the error of each model is calculated and each simulator determines the minimum simulation step needed to meet the acceptable error ranges. The minimum

step throughout all simulators is selected as the next δtc. Adaptive step with ﬁxed synchronization step (ADAPT_FIXED): each simulator ad-

vances with an adaptive δti and the synchronization step δtc is ﬁxed by the user.

With the model and simulation toolkit deﬁned, a model can be implemented. In order to analyze and understand the dynamics of this new model and to identify the most relevant parameters under a speciﬁc simulation, a sensitivity analysis can be performed as showed in Section 3.2.

3.2 Sensitivity Analysis

Model parameters represent an element of the real system, or rather, a simpliﬁcation of such element. These parameters can sometimes be measured directly from the system, estimated from the observable data, or even guessed from prior knowledge. In any case,

97 Chapter 3 – Methods and tools for simulation and model analysis it is highly likely that the parameter value contains an intrinsic error or a level of uncertainty. Then, some questions arise regarding these parameters: What is the effect of this incertitude on the model outputs? Is it possible to measure quantitatively or qualitatively the effect of changes in parameter values on the outputs? The field of sensitivity analysis, along with the highly related area of uncertainty analysis, provides a set of tools that can answer these questions. There are many definitions to sensitivity analysis, mainly because it is a technique that has been used by different technical communities and because there are various known approaches. Saltelli et al. is probably one of the most influential authors in this field, whose extensive works present, formalize and classify most major sensitivity analysis topics and methods [Sal+08; SCS+00]. In [Sal+08] sensitivity analysis is defined as: The study of how the uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input. However, in the context the modeling applications presented in this manuscript, it is more appropriate to define sensitivity analysis as the measurement of the effect of changes in input values and model parameters on the outputs of a system. Sensitivity analysis can provide important information for modeling and simulation applications. The objectives include: — Factor prioritization: it can determine which inputs or parameters are more important, which can help guide the parameter estimation or motivate further attention in the observation of certain inputs. — Model simplification: it can identify which elements of the model have little effect and can be replaced with a simpler definition. — Parameter regions identification: it can pinpoint critical or interesting ranges in the parameter or input spaces. — Parameter interaction: not only it can measure the effect of changes of one parameter, it can also measure the effect of the interaction of parameters, i.e. the outcome of changes in two or more parameters. This information can be particularly important in the analysis of complex models like cardio-respiratory models, because it can reduce the number of parameters to consider for the analysis of a specific phenomenon. The factor prioritization can be significantly important for patient-specific identification and individual subject fitting, because it reduces the parameters taken into account, reducing the computing time.

98 3.2. Sensitivity Analysis

Sensitivity analysis are deﬁned for the study of a function y = f(X), where y denotes a single output, and X = [X1,X2,...,Xd] is a vector of d inputs or parameters. This notation can be easily translated to modeling notation, as it will be explained in Section 3.2.4. The approach followed by most sensitivity analysis methods is summarized in Figure 3.5. It consists in:

1. The deﬁnition of the distribution for each source of uncertainty each input or pa-

rameter Xi, or the deﬁnition of the relevant parameter space P. For simplicity it

will be assumed that there are d parameters denoted [X1,X2,...,Xd]. 2. The creation of an experimental design, which will be denoted ED, consisting of n sets of input values 23:

 (1) (1) (1)  x1 x2 . . . xd    (2) (2) (2)   x1 x2 . . . xd  ED =   , (3.1)  . . . .   . . .. .   (n) (n) (n)  x1 x2 . . . xd

where a row represents the values for each parameter. 3. The evaluation of each row of the experimental design ED, which yields a vector of outputs h iT Y = y(1), y(2), . . . , y(n) . (3.2)

4. The analysis of the outputs Y , identifying and associating the source of the variations in the outputs, with respect to the variations in the parameters.

The variety of the existing methods in sensitivity analysis lies on the diverse schemes to produce an experimental design and to analyze the variability of the evaluated outputs. However, the choice of the sensitivity analysis depends on several factors, such as the assumptions on the parameters of f(X) (linearity, independence or interaction) and the available computational resources for the evaluation of this function. Existing methods can be divided into three groups: local sensitivity methods, global sensitivity methods and screening methods. This categorization is not strict, considering that some methods can be considered as part of more than one of these groups.

23. Normally, this matrix is denoted M. Here, we changed this change of notation in order to avoid confusion with the notation of models M.

99 Chapter 3 – Methods and tools for simulation and model analysis

Figure 3.5 – Simpliﬁed diagram of the process of uncertainty and sensitivity analysis, based on [EPA09].

100 3.2. Sensitivity Analysis

3.2.1 Local sensitivity analysis

Local methods represent the simplest form of sensitivity analysis. This analysis studied a small region of the parameter space P. A natural approach consists in the selection of (0) (0) (0) (0) a working point X = [x0 , x1 , . . . , xd ], followed by the evaluation of the function f(X(0)) and at other points close to X(0). When the variations are introduced only in one parameter Xi at a time, the approach is termed a one-at-time (OAT) analysis. For example, a typical OAT experimental design for Xi would be:

 (0) (0) (0)  x0 . . . xi . . . xd    (0) (0) (0)   x0 . . . xi + δ . . . xd     (0) (0) (0)  EDi =  x0 . . . xi + 2δ . . . xd  , (3.3)    . . .   . . .   (0) (0) (0)  x0 . . . xi + (n − 1)δ . . . xd where δ is a predeﬁned perturbation of parameter Xi. In this example, only the variations 0 0 in [xi , xi +(n−1)δ] are explored, but this range can be deﬁned as evenly spaced variations from the minimum and maximum values of parameter Xi, or as an arbitrary variation of δ.

Once Y is obtained from the evaluation of matrix EDi (Equation 3.3), the results ∂Y can be analyzed in several ways. On one hand, the partial derivatives /∂Xi can be estimated or averaged, which can be normalized and compared to the partial derivatives ∂Y of other parameters /∂Xj . On the other hand, the results of the evaluation can be plotted with respect to the different values of the varying parameter. In that case, the effect of the parameter variation can be identified visually, or directly quantified using a linear regression and its coefficient of determination R2. Local sensitivity analyzes are practical for their simplicity and reduced number of evaluations. However, as their name imply, the parameter space is not fully explored, since it does not consider simultaneous variations of parameters. Consequently, local OAT approaches cannot detect interactions between parameters.

3.2.2 Global sensitivity analysis

In contrast with local methods, global sensitivity analysis focuses on the study of the eﬀect of the parameters but it does not constrain their values to the small region around a working point. Instead, it permits the parameters to take any value in a large region of

101 Chapter 3 – Methods and tools for simulation and model analysis interest. The most popular family of global sensitivity analysis methods is the variance-based approach. This kind of approach tries to identify what part of the variability of Y can be attributed to the variability of each parameter Xi (or groups of parameters). Its starting point is the following question: Does Y vary more or less when one ﬁxes one (or many) of its parameters? A detailed explanation of how this question is mathematically addressed is presented in [Sob93; M01; Sal+08]. In most of the global sensitivity analysis methods, the amount of evaluations, or model simulations, necessary to calculate the sensitivity indices is very high, which limits the application of global sensitivity analysis to models where the number of parameters is reduced and when one counts with a signiﬁcant computational budget. This is the main reason that drives another type of global sensitivity analysis that permits to cheaply identify and exclude unimportant parameters: screening methods.

3.2.3 Screening methods

Screening methods permit to identify qualitatively which parameters of a function are relatively influent on the function’s output and which parameters can be ignored. This information can help reduce the dimensionality of future analysis or estimation phases. The most common screening method is the Morris elementary effects method [Mor91b]. Morris’ method explores a subspace of the parameter space Ω ∈ P: a d-dimensional unit cube regularly divided as a grid of p levels 24. In this space, it calculates an elementary effect, defined as:

f(x1, . . . , xi, . . . , xd) − f(x1, . . . , xi + ∆, . . . , xd) EEi = , (3.4) ∆

1 where ∆ is a predefined multiple of /d−1, and (x1, x2, . . . , xd) is a randomly selected point, 1 2 such that each xi takes a value in {0, /(d−1), /(d−1),..., 1 − ∆}. The method starts with the calculation of r different elementary effects for all parameters, calculated with a clever experimental plan that uses r(d − 1) simulations [Mor91a]. For each parameter Xi, the mean and standard deviation (µi ± σi) of the elementary effects are computed and these two values are then studied in the µ vs. σ plane. In order to avoid the mutual cancellation of symmetrical elementary effects, more recent works [Cam+04] enhanced the method of

24. The original deﬁnition is constrained to a unit cube for simplicity, but it can be easily transformed to any uniform range.

102 3.2. Sensitivity Analysis

∗ Morris by using the mean over the absolute value of Equation 3.4, denoted µi . The analysis of the elementary eﬀects results in the µ∗ vs. σ plane, illustrated in Figure 3.6, derives the following information:

∗ — Parameters with low µi and σi can be considered as negligible parameters; a perturbation of this parameter does not cause a signiﬁcant eﬀect on the output.

∗ — Parameters with large µi but low σi reveal a linear eﬀect of parameter Xi over the output; a perturbation of this parameter causes a constant, non-negligible eﬀect over the output.

∗ — Parameter with large µi and large σi can be caused by a nonlinear eﬀect of parameter

Xi or by an important interaction with other parameters; a perturbation of this

parameter causes a non-negligible effect, but this effect varies for different Xi.

Figure 3.6 – Example of the results of the Morris elementary effects method. The elementary effects of all parameters are analyzed in the µ∗ −σ plane, identifying negligible effects, parameters with a linear effects and parameters that have a non-linear or interaction-related effect. Adapted from [Oje13].

The Morris elementary method is an advantageous tool to examine and identify important parameters of a function or model. Due to its relatively low computational requirements, it can be used prior to any heavy sensitivity analysis or extensive parameter exploration such as during a parameter estimation method. This method can quickly point out linear relations between parameters and outputs. On the other hand, the elementary effect method presents two specific disadvantages: it does not quantify the effect of a parameter, and it cannot discern between nonlinear relations and parameter interactions.

103 Chapter 3 – Methods and tools for simulation and model analysis

For this kind of analysis, one must turn to global sensitivity analysis using Sobol indices, as presented previously in Section 3.2.2.

3.2.4 Proposed approach

For the modeling framework and the clinical applications presented in this manuscript, sensitivity analyzes played an important role for the understanding of the underlying mechanisms of the modeled systems. The utilization of any sensitivity analysis for our modeling applications is straightforward: the function y = f(X) used in the formalization of all sensitivity analyzes can be easily adapted, according to our previous definition of h h h a simulator output, y = S (M ,PS,F ), where S is the simulation process applied to a model M h (F, I, O, E, P), using model parameter values P and inputs I, and y is an output of the model. The output functions y will be defined in every chapter to analyze a specific dynamic. Our problem becomes, thus, to analyze how the variations on y can be apportioned to changes in P and I, which are particularly difficult to define and measure on real physiological applications. The sensitivity analysis helps to identify the most important variables that need to be considered for a successful multi-formalism and multi-scale integration. For this analysis, the screening method of Morris was favored for its useful compromise between parameter space exploration and computational requirements. Moreover, in order to complement the qualitative identification of the nature of the parameter effects provided by Morris’ method, the following sensitivity index was used:

q ∗ 2 2 DMi = (µi ) + (σi) , (3.5) applied to all parameters Xi. Then, parameters are sorted according to their DMi, as illustrated in Figure 3.7. This index, which has been used in other modeling applications [SB11; Dua+03; Cal18] provides a rank of the parameter effects; parameters with a high sensitivity or strong interactions will have a high DMi, while unimportant ones are associated with a low DMi. A local sensitivity analysis will be applied to the most important parameters obtained with the Morris method, to observe the effects of the changes of these parameters on relevant simulated signals on the context of this work. Based on the results of the sensitivity analysis, a subgroup of parameters can be selected to try to fit model simulations to experimental data using the methodology presented in Section 3.3.

104 3.3. Parameter identiﬁcation

Figure 3.7 – Identiﬁcation of important and negligible parameters from the elementary eﬀects. The right plot shows the same results of a Morris analysis, but ranked according to the DMi index ( Equation 3.5). Adapted from [Oje13].

3.3 Parameter identiﬁcation

The parameter estimation of a model can be considered as an optimization problem, where the objective is to ﬁnd the parameter values Popt that minimizes an error function g between the experimental and the simulated data:

Popt = arg min g(Osim(P),Oobs) P∈P (3.6) subject to h(Osim(P)) ,

h h h where Osim(P) = S (M ,PS,F ), with M (F, I, O, E, P). The field of mathematical optimization offers a vast choice of methods and algorithms that solve this kind of problems: analytic approaches, iterative methods, gradient-based methods, deterministic and stochastic approaches, among others [NKT89]. However, not all of these methods are appropriate for the problem of parameter identification because i) for the clinical applications of this manuscript, the dimensionality of the problem is high enough to forbid the employment of methods whose computational complexity is exponential with respect to the number of parameters, ii) the nature of the underlying equations are either non-linear or not well understood, and iii) the objective functions and constraints are the result of complex model equations which complicate the calculation of their derivatives or partial derivatives. These limitations quickly discard classical optimization methods, such as Newton’s method, or Lagrange multipliers; linear pro-

105 Chapter 3 – Methods and tools for simulation and model analysis gramming approaches, such as the simplex algorithm [Dan98]; and exhaustive exploration approaches, such as branch-and-bound methods [LD60]. The remaining methods include approaches that approximate numerically the derivatives of the objective function, methods that use a heuristic to select interesting points in the parameter space, and methods based on a stochastic process.

3.3.1 Deterministic approaches

In this categorization of optimization techniques, deterministic approaches are deﬁned to provide a contrast to stochastic approaches: these methods ﬁnd the optimal or a sub- optimal solution to Equation 3.6 with a process that does not rely on a random behavior. Algorithms that calculate or approximate derivatives and gradients fall into this category (ex: gradient descent method). Other popular deterministic methods are hill-climbing algorithm [Min61] and the Nelder-Mead algorithm [NM65]. In general, deterministic methods are interesting because they eventually converge to a solution and do not need much information regarding the objective functions. However, the main disadvantages of these methods are i) the gradient estimations and the heuristics used require several evaluations of the objective function, which becomes problematic when the dimensionality of the parameter space is considerable, and ii) the convergence of these methods is not guaranteed.

3.3.2 Stochastic approaches

Stochastic search approaches are interesting when the parameter space and objective function are not well understood, or when the parameter exploration requires random perturbations in order to avoid local minima. A notable and popular stochastic approach is the particle swarm optimization [EK95]: an iterative procedure where a list of solutions is maintained and each candidate solution wanders the parameter space with a behavior that mixes exploration and attraction to good solutions. The converge of approaches that constantly evolve a list of candidate solutions is not guaranteed either; it mostly depends on a good choice of the algorithm parameters, principally the size of the candidate solution list and the number of iterations. However, stochastic approaches are praised for their ability to constantly explore the parameter space and avoid local minima.

106 3.3. Parameter identiﬁcation

Evolutionary algorithms

Within the stochastic approaches, evolutionary algorithms stand out for their original foundations. Evolutionary algorithms (EA) follow the approach of maintaining a set of candidate solutions, termed population, and repeatedly evolving this population with processes inspired by biological evolution: selection of the fittest, reproduction, recombination and mutation. Among the wide range of algorithms classified as EA, the most popular group used in optimization is the genetic algorithms (GA), initially conceived in [Hol75] and thoroughly formalized in [Gol89]. In this kind of algorithms, the following notation is used: a candidate solution is called an individual. An individual represents a solution by encoding it in the form of genetic information, or alleles. For example, in the case of parameter estimation, each allele can be a binary representation of the value of a parameter. The population evolves as a result of the following procedure, illustrated in Figure 3.8: 1. An initial population with N individuals is initialized, where each individual contains a random value for each one of its alleles. This generates a first generation of possible solutions. 2. Each individual of the population is assigned with a value that measures its fitness, a quantification of how good the individual is. The fitness value of an individual directly affects its chances to survive and reproduce. The calculation of the fitness

requires the evaluation of the target function g, but it can also be aﬀected by other variables. 3. An internal variable that counts the number of generations is incremented. This variable can be useful for the stopping criteria. 4. According to their ﬁtness and a stochastic process, a selection of individuals is performed. This phase designates pairs of individuals that will reproduce. 5. For each pair of selected individuals a reproduction operation generates two new individuals whose alleles are a combination of the two progenitors. This reproductive

process occurs with a predeﬁned probability pc for each pair of individuals. Newly generated individuals may go through a mutation process, with another predeﬁned

probability pm, which slightly modiﬁes one or more of its alleles. The probabili-

ties pc and pm directly control the exploration of new solutions. At the end of this stage, 2N individuals exist: the parent population of size N and a new oﬀspring population of the same size.

107 Chapter 3 – Methods and tools for simulation and model analysis

6. All new generated individuals are evaluated; their fitness is determined as well. 7. At this point, different strategies are possible: either the new population completely replaces the old population, or a replacement procedure that accounts for each individual fitness selects and discards all individuals to produce the next generation, a population of size N. 8. Finally, if a stopping criterion is met, the algorithm stops or, in the contrary, the algorithm restarts from step 3. Possible stopping criteria include a maximum number of generations (i.e. iterations) or when the individuals of the population have reached a certain fitness value.

As other stochastic approaches, EAs cannot assure convergence toward the optimal solution and their performance depend on a good choice of the EA parameters, Neverthe- less, they present an interesting compromise of space exploration, number of evaluations and quality of the solutions found, and they have been successfully used for parameter identification in other applications [Fle08]. Among this family of methods, there are several algorithms like Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) [Han06], Self-adaptive Differential Evolution [Bre+06; ESE11] and Differential Evolution [SP97]. The latter will be detailed, since it was used for a patient specific identification on this work.

Diﬀerential evolution algorithm (DE)

This algorithm (Figure 3.9) is a population based, heuristic optimizer developed by [SP97].

It uses a population of size N where each individual Pj (also denominated vector) of the population has d parameters [pj,1, pj,2, . . . , pj,d]. The ﬁrst population is generated randomly, within the bounds of the parameter space P. For each parameter pl with l ∈

[0, d], a range [pmin,l, pmax,l] is deﬁned to designate its minimum and maximum value.

Every individual Pj = [pj,1, . . . , pj,2, . . . , pj,d] is initialized with random alleles: pj,l ∼

U(pmin,l, pmax,l). DE starts with a mutation where new parameter vectors are generated by adding the weighted diﬀerence between two population vectors to a third vector denominated base vector. This new mutated vector is crossover with the parameters of another pre- determined vector, the target vector, to produce the so-called trial vector Uj. Then, the selection takes place, where if the trial vector has a lower error function g than the target

108 3.3. Parameter identiﬁcation

Figure 3.8 – General scheme of genetic algorithms. Adapted from [Oje13].

109 Chapter 3 – Methods and tools for simulation and model analysis

Figure 3.9 – Diagram of the Diﬀerential evolution algorithm.

110 3.3. Parameter identiﬁcation

vector Pj, the trial vector replaces the target vector in the following generation. Each individual of the population has to serve once as the target vector in a generation. There are several variants of the DE algorithm. In this work, the best/1/exp variant is used, where best imply that the base vector chosen for the mutation corresponds to the individual with the lower error g, 1 stand for the number of diﬀerence vectors used in the mutation, and exp denotes that an exponential crossover will be used. The mutation, crossover and selection for this variant will be explained.

Mutation: For each target vector from the current generation G (Pj,G with j = 1, 2...N) a mutant vector Vj is generated according to equation 3.7.

Vj = Pbv,G + F ∗ (Pr1,G − Pr2,G)with F ∈ [0; 2] (3.7) where r1, r2 are random indexes diﬀerent from index j; G is the current generation;

Pbv,G is the base vector that in this variant (best) corresponds to the individual with the lower error g in the population; F is a real and constant factor that corresponds to the weight coeﬃcient of the diﬀerential variation Pr1 − Pr2. Due to the number of variables in equation 3.7, the minimum population N for this algorithm is 4.

Crossover: In order to increase the diversity of the perturbed parameter vectors, crossover is introduced. DE crossover operator implements a discrete recombination of each mutated vector, Vj ,with its corresponding parent vector, Pj,G , to produce the trial vector Uj as oﬀspring. Vj, Pj,G and Uj are composed of sets of d parameters [vj,0, vj,1, . . . , vj,(d−1)],

[pj,G,0, pj,G,1, . . . , pj,G,(d−1)], and [uj,0, uj,1, . . . , uj,(d−1)] respectively. The variant best/1/exp uses an exponential crossover as implemented in Algorithm 1.

Algorithm 1 Exponential crossover 1: Uj = Pj,G; A parameter index m is randomly selected from [0, d − 1], L = 0; 2: repeat 3: uj,m = vj,m , m = (m + 1) modulo d, L = L + 1 4: until rand[0, 1] > CR or L = (k − 1)

Selection: In this stage, the algorithm has to take a decision whether or not the trial vector Uj should replace Pj,G and become a member of generation G + 1 (Pj,G+1). If the

111 Chapter 3 – Methods and tools for simulation and model analysis

error function g of Uj is smaller than the error function of Pj,G, then Pj,G+1 is set to Uj; otherwise, the old individual Pj,G is retained.

3.3.3 Proposed approach

During the parameter identifications of the modeling applications of this thesis, evolutionary algorithms were used. Based on the results of the sensitivity analysis, a reduced group of parameters are selected for patient-specific model identification, reducing computational cost and calculation time. Among the available evolutionary algorithms in the literature, the Differential Evolu- tionary (DE) algorithm was selected (Section 3.3.2), since it delivered a better performance in initial identifications. This algorithm follows the guidelines illustrated in Figure 3.9.

The objective function g(Osim(P),Oobs) used for this algorithm will be defined in each chapter where a patient-specific identification is performed, because its definition depends of the implemented model and the simulation to fit. To implement this methodology in an efficient way, parallel optimization library PAGMO was used.

Parallel optimization library PAGMO

Among the existing parallel optimization libraries, PAGMO (C++) was chosen. This is a research library for massively parallel optimization [BI19]. This tool provides eﬃ- cient implementations of bio-inspired and evolutionary algorithms and state-of the art optimization algorithms and it allow to implement new ones. Pagmo 25 provides automatic parallelization of the optimization process via a coarse- grained approach based on the island model [MS89]. An island I has a population of

N individuals, where each individual Pj has d parameters [pj,1, pj,2, . . . , pj,d]. Each island is an instance of the optimization problem an each of them are launched in a separate thread of execution, taking advantage of modern multiprocessor machines. A group of islands objects represent an archipelago class A that can asynchronously exchange information between islands to improve the overall convergence properties of the optimization. The connections between islands are denominated migrations and they were implemented by our group using a ring topology (Figure 3.3.3), where each node corresponds to an island and the edges represent routes through which the best individual Pbest can be communicated from one island to the contiguous one replacing the worst individual.

25. https://esa.github.io/pagmo2/

112 3.3. Parameter identiﬁcation

Figure 3.10 – Migration ring topology with h islands (I) in an archipelago A. A migration occurs each X generations, where the best individual Pbest of an island replace the worst individual on the contiguous island. Green circles represent islands/nodes and the arrows represent the edges of communication between islands. Inside each island there is a population on N individuals.

For this work, M2SL was integrated into the optimization problem class of PAGMO, in order to include the model simulation in the identification algorithm. An archipelago of 2 islands of 25 individuals each was defined. The population was continuously evolved for 100 generations, using differential evolutionary and doing a migration every 3 generations.

Quantiﬁcation of error between simulated and the experimental data

A metric should be defined to quantify the error between simulated and the experimental data. In this thesis, the main data to be compared are time series. Several methods have been proposed in the literature to measure a distance between two different time series, and they can be classified in four categories [EA12]: 1. Shape-based distances: they compare the overall shape of the series by measuring the closeness of the raw-values of two series[EA12]. Within this category, there are lock-step measures that compare the i-th point of a series to the i-th point of another

113 Chapter 3 – Methods and tools for simulation and model analysis

(e.g.: Euclidean distance) and elastic measures that allow flexible (one-to-many / one-to-none) comparison (e.g.: Dynamic Time Warping, Frechet distance)[Wan+13]. 2. Edit-based distances: they compare two time series on the basis of the minimum number of operations needed to transform one series into another one (e.g.: longest common subsequence, edit distance on real sequence). 3. Feature-based distances: they compare certain features extracted from the time series like coefficients from the discrete Fourier transform [SZ96], wavelet coefficients, or autocorrelation values. 4. Structure-based distances: They are mostly used on longer series and they aim to find higher-level structures in the series to compare them on a more global scale (e.g.: Piccolo distance [Pic90], Maharaj distance [Mah96]).

In our case, the simulated and experimental time series are synchronized, homogeneous (same physical units) and they have the same sample frequency. Therefore, a Euclidean distance can be used, in our case the root-mean-square error (RMSE). The RMSE is the square root of the average of squared errors between two signals and it can be deﬁned as:

v u N u 1 X RMSE = t (xsim(t) − xexp(t))2 (3.8) N t=0 where t corresponds to the current sample of the time series, xsim and xexp are, respectively, the simulated and experimental time series, N is the number of samples of the time series. In order to understand this metric regardless of the amplitude of the experimental data, the distance between the two signals can be scaled by normalizing the RMSE, obtaining a relative root-mean-square error (rRMSE):

v u N sim exp !2 u 1 X x (t) − x (k) t rRMSE = exp (3.9) N t=0 x (t)

3.4 Conclusion

This chapter presented a state-of-the-art on modeling and simulation methods and tools. In particular, the multi-formalism modeling and simulation library (M2SL) developed by our group, was introduced. M2SL has powerful capabilities like the implementation of temporal synchronization strategies and variable coupling implementation. Moreover, M2SL allows for an easy coupling of the produced model with model analysis

114 REFERENCES methods, for sensitivity analysis or parameter identification. M2SL was chosen as the tool for model development and analysis in this work. Sensitivity analysis and parameter identification are critical tasks in this work. A brief state of the art was proposed and a formal problem statement, for the application to this work was proposed in each case. Regarding sensitivity analysis and due to the dimensionality of the models usually produced in our field, the choice has been made to apply a two-step approach: firstly by using screening methods and then applying detailed, local sensibility tests. Concerning parameter identification, this problem was formulated as a mathematical optimization problem. The complex nature of the parameter identifications performed in this thesis and the previous experience of our team in this field led us to the selection of evolutionary algorithms (EA). Therefore, amid the existing EA algorithms, the Differential Evolution (DE) algorithm was presented along the general schema and concepts of EA. The set of methods and tools presented in this chapter constitute the methodological background for the development of this work.

References

[BI19] Francesco Biscani and Dario Izzo, esa/pagmo2: pagmo 2.10, Jan. 2019. [Bre+06] Janez Brest et al., « Self-adapting control parameters in diﬀerential evolution: A comparative study on numerical benchmark problems », in: IEEE Transactions on Evolutionary Computation (2006). [Bri+13] Randall D Britten et al., « FieldML, a proposed open standard for the physiome project for mathematical model representation », in: Medical & biological engineering & computing (2013), pp. 1–17. [Cal18] M. Calvo, « Analysis of the cardiovascular response to autonomic nervous system modulation in Brugada syndrome patients », PhD thesis, Université de Rennes 1, 2018. [Cam+04] F Campolongo et al., « Enhancing the Morris method », in: Proceedings of SAMO 2004, 4th Int. Conf. on Sensitivity Analysis of Model Output, 2004. [Chr+09] G Richard Christie et al., « FieldML: concepts and implementation », in: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 367.1895 (2009), pp. 1869–1884.

115 Chapter 3 – Methods and tools for simulation and model analysis

[Dan98] George B Dantzig, Linear programming and extensions, Princeton university press, 1998. [Def06] Antoine Defontaine, « Modélisation multirésolution et multiformalisme de l’activité électrique cardiaque », PhD thesis, Université de Rennes˜1, 2006. [Dua+03] J M Duarte et al., « A probabilistic sensitivity analysis of water-leaving radiance to water constituents in coastal shallow waters », in: Proceedings of SPIE, vol. 5155, 2003, p. 162. [DV02] Juan De Lara and Hans Vangheluwe, « AToM3: A Tool for Multi-formalism and Meta-modelling », in: Fundamental approaches to software engineering, Springer, 2002, pp. 174–188. [EA12] Philippe Esling and Carlos Agon, « Time-series data mining A Time series data mining », in: 45.1 (2012), p. 12. [EK95] Russell Eberhart and James Kennedy, « A new optimizer using particle swarm theory », in: Micro Machine and Human Science, 1995. MHS’95., Proceedings of the Sixth International Symposium on, IEEE, 1995, pp. 39– 43. [EPA09] EPA, Guidance on the Development, Evaluation, and Application of Envi- ronmental Models, tech. rep., U.S. Environmental Protection Agency, 2009. [ESE11] Saber M. Elsayed, Ruhul A. Sarker, and Daryl L. Essam, « Diﬀerential evolution with multiple strategies for solving CEC2011 real-world numerical optimization problems », in: 2011 IEEE Congress of Evolutionary Compu- tation, CEC 2011, 2011. [Fle08] J Fleureau, « Intégration de données anatomiques issues d’images MSCT et de modéles électrophysiologique et mécanique du coeur. », PhD thesis, Université de Rennes 1, 2008. [Gar+08] Alan Garny et al., « CellML and associated tools and techniques », in: Philo- sophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366.1878 (2008), pp. 3017–3043. [Gol89] David E Goldberg, Genetic algorithms in search, optimization, and machine learning, 1st, Addison-Wesley Longman Publishing Co., Inc., 1989.

116 REFERENCES

[Han06] Nikolaus Hansen, « The CMA Evolution Strategy: A Comparing Review », in: Towards a New Evolutionary Computation, Berlin, Heidelberg: Springer Berlin Heidelberg, 2006, pp. 75–102. [Her+09] A. I. Hernandez et al., « A multiformalism and multiresolution modelling environment: application to the cardiovascular system and its regulation », in: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 367.1908 (Dec. 2009), pp. 4923–4940. [Her+11] Alfredo I. Hernández et al., « Integration of detailed modules in a core model of body fluid homeostasis and blood pressure regulation », in: Progress in Biophysics and Molecular Biology (2011). [Hol75] John Holland, Adaptation in Natural and Artificial Systems, Michigan Press, 1975. [Huc+03] M Hucka et al., « The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models », in: Bioin- formatics 19.4 (2003), pp. 524–531. [KN09] Peter Kohl and Denis Noble, « Systems biology and the virtual physiological human », in: Molecular Systems Biology 5.1 (2009). [LD60] Ailsa H Land and Alison G Doig, « An automatic method of solving discrete programming problems », in: Econometrica: Journal of the Econometric So- ciety (1960), pp. 497–520. [Le +15] Virginie Le Rolle et al., « Recursive identification of an arterial baroreflex model for the evaluation of cardiovascular autonomic modulation », in: Com- puters in Biology and Medicine 66 (2015), pp. 287–294. [Le 06] Virginie Le Rolle, « Modélisation Multiformalisme du Système Cardiovascu- laire associant Bond Graph, Equations Différentielles et Modèles Discrets », PhD thesis, Rennes: Université de Rennes 1, Dec. 2006. [M01] Sobol Ilya M, « Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates », in: Mathematics and computers in simulation 55.1-3 (2001), pp. 271–280. [Mah96] Elizabeth Ann Maharaj, « A significance test for classifying ARMA models », in: Journal of Statistical Computation and Simulation 54.4 (1996), pp. 305– 331.

117 Chapter 3 – Methods and tools for simulation and model analysis

[Mil10] Katharine Miller, « Physiome THE », in: A mission imperative. Biomedical Computation Review (2010), pp. 8–15. [Min61] Marvin Minsky, « Steps toward artiﬁcial intelligence », in: Proceedings of the IRE 49.1 (1961), pp. 8–30. [Mir+13] Gary R Mirams et al., « Chaste: An Open Source C++ Library for Computa- tional Physiology and Biology », in: PLoS computational biology 9.3 (2013), e1002970. [Mor91a] M D Morris, « Factorial plans for preliminary computational experiments », in: Technometrics 33.2 (1991), pp. 161–174. [Mor91b] M D Morris, « Factorial sampling plans for preliminary computational experiments », in: Technometrics 33.2 (May 1991), pp. 161–174. [MS89] Bernanrd Manderick and Piet Spiessens, « Fine-Grained Parallel Genetic Algorithms », in: Proceedings of the 3rd International Conference on Genetic Algorithms, M. Kaufmann Publishers, 1989, pp. 428–433. [NKT89] George L Nemhauser, A H G Rinnooy Kan, and M J Todd, Optimization, volume 1 of Handbooks in operations research and management science, North- Holland, Amsterdam, 1989. [NM65] J A Nelder and R Mead, « A simplex method for function minimization », in: The computer journal 7.4 (1965), p. 308. [Oje13] David Ojeda, « Multi-resolution physiological modeling for the analysis of cardiovascular pathologies », PhD thesis, Dec. 2013. [Pic90] Domenico Piccolo, « A Distance Measure for Classifying Arima Models », in: Journal of Time Series Analysis 11.2 (Mar. 1990), pp. 153–164. [QDR09] Gauthier Quesnel, Raphaël Duboz, and Éric Ramat, « The Virtual Labora- tory Environment – An operational framework for multi-modelling, simulation and analysis of complex dynamical systems », in: Simulation Modelling Practice and Theory 17 (Apr. 2009), pp. 641–653. [Rom+16] Hector M. Romero Ugalde et al., « Model-Based Design and Experimental Validation of Control Modules for Neuromodulation Devices », in: IEEE Transactions on Biomedical Engineering 63.7 (2016), pp. 1551–1558. [Sal+08] Andrea Saltelli et al., Global Sensitivity Analysis. The Primer, 2008.

118 REFERENCES

[SB11] S Schreider and R Braddock, « Application of the Morris algorithm for sensitivity analysis if the REALM model for the Goulburn Irrigation System », in: Environmental Modeling and Assessment 11.4 (2011), pp. 297–313. [SCS+00] Andrea Saltelli, Karen Chan, E Marian Scott, et al., Sensitivity analysis, Wiley New York, 2000. [Sob93] Ilya M Sobol, « Sensitivity estimates for nonlinear mathematical models », in: Mathematical Modelling and Computational Experiments 1.4 (1993), pp. 407– 414. [SP97] Rainer Storn and Kenneth Price, « Diﬀerential Evolution – A Simple and Eﬃcient Heuristic for global Optimization over Continuous Spaces », in: Journal of Global Optimization 11.4 (1997), pp. 341–359. [SZ96] Hagit Shatkay and Stanley B. Zdonik, « Approximate queries and representations for large data sequences », in: Proceedings - International Conference on Data Engineering, IEEE, 1996, pp. 536–545. [Van01] Hans Vangheluwe, « Multi-formalism modelling and simulation », PhD thesis, Universiteit Gent, 2001. [Vit+04] V Vittorini et al., « The OsMoSys approach to multi-formalism modeling of systems », in: Software and Systems Modeling 3.1 (2004), pp. 68–81. [Wan+13] Xiaoyue Wang et al., « Experimental comparison of representation methods and distance measures for time series data », in: Data Mining and Knowledge Discovery 26.2 (Mar. 2013), pp. 275–309. [ZPK00] Bernard P Zeigler, Herbert Praehofer, and Tag Gon Kim, Theory of modeling and simulation: Integrating discrete event and continuous complex dynamic systems, 2nd ed., Academic Press, 2000.

119

Chapter 4 ADULT CARDIO-RESPIRATORY MODEL

As described in Chapter 2, a limited number of integrated cardio-respiratory models has been identified in the literature. However, these models do not explicitly represent the physiological mechanisms underlying SAS and their structure is not adapted to patient- specific analysis. Moreover, to our knowledge, no integrated cardio-respiratory model has been used for the analysis of the cardio-respiratory response to apneas. In this chapter, an integrated adult cardio-respiratory model is proposed, representing the most significant mechanisms and interactions needed for the analysis of apneas. The proposed model has been developed using the M2SL library (Chapter 3) and is composed of a set of coupled sub-models based on previous works from our group, or inspired from models in the literature. Every sub-block of the model will be explained in detail in Section 4.1 and simulations will be presented to show the capabilities of the model and its correspondence to the values in the physiology in Section 4.2. The proposed adult model is the base of all the other models presented in this thesis. Chapter 6 will describe the adaptations and structural changes needed to adapt the adult model for newborns.

4.1 Method

4.1.1 Model structure

The proposed model (Figure 4.1) is composed of four interconnected components : i) the respiratory system, ii) the cardiovascular system, iii) the gas exchange (in the lungs and in peripheral tissue) and iv) the neural control.

4.1.2 Cardiovascular system model

The cardiovascular system (CVS) presented in Figure 4.2 is an adaptation of the previous models of our group [Oje+13; Oje+15; Rom+16; Cal+18; Cal+19]. It is constituted

121 Chapter 4 – Adult cardio-respiratory model

Figure 4.1 – General diagram of the proposed cardio-respiratory adult model. Boxes represent diﬀerent submodels. Dotted line arrows symbolize interactions between submodels. Pthor, thoracic pressure; Rpul pulmonary capillaries resistance; P mus, respiratory muscles pressure; BR, breathing rate; VA, alveolar volume; PaO2 and PaCO2, partial pressure of O2 and CO2 in the systemic arteries respectively; HR, heart rate; Rsys, systemic peripheral resistance; V usv, unstressed volume of the systemic veins; E, ventricle elastances; AP , arterial pressure.

122 4.1. Method of three coupled components: (i) the cardiac electrical system, (ii) cardiac mechanical activity and (iii) systemic and pulmonary circulation.

Figure 4.2 – Diagram of the proposed cardiovascular system model. Each cardiac compartment is characterized by an elastance (E) and the pressure (P ) and the volume (V ) are computed. Green dotted line blocks correspond to the interface with the gas exchange submodels. Q, ﬂow; R, resistance; L, inertia; ra, right atrium; rv, right ventricle; la, left atrium; lv, left ventricle; tv, tricuspid valve; mv, mitral valve; pa, pulmonary arteries; pp, pulmonary peripheral; pv, pulmonary veins; sa, systemic arteries ;sves, systemic peripheral vessels; sv, systemic veins; i, intrathoracic; e, extrathoracic; M, metabolic rates; V˙ , ﬂow.

Cardiac electrical system

The cardiac electrical conduction system (CES model), represented in Figure 4.3, is based on a discrete model adapted from [Her+02]. It is deﬁned as a set of interconnected cellular automata, each one explaining the electrical activation of a group of cardiac cells: the sinoatrial node (SAN), the left atrium (LA), the atrioventricular node (AVN),

123 Chapter 4 – Adult cardio-respiratory model the upper bundle of His (UBH), the lower bundle of His (LBH), left and right bundle branches (LBB and RBB),and left and right ventricles (LV and RV).

Figure 4.3 – Diagram of the simpliﬁed CES submodel. Top: automata involved on the cardiac electrical system. Yellow boxes represent nodal cell automata and orange boxes, myocardial cells. Bottom: state diagram representing the cycle of a nodal (yellow, left) and a myocardial automaton (orange, right); along with their respective action potential waveforms. Modiﬁed from [Cal18].

Each automaton state periodically changes among the four main electrophysiological activation periods: slow diastolic depolarization (SDD) for nodal cells or Idle for myocardial cells, upstroke depolarization period (UDP), absolute refractory period (ARP) and relative refractory period (RRP). When a given cell is activated (at the end of UDP phase), it sends an output activation signal to neighboring cells. The slope of the SDD phase in SAN, and thus its SDD period, depends on the HR that results from the barore- ﬂex model; as well as the output electrical activations of LV and LA are connected to the

124 4.1. Method

CVS model by triggering the mechanical contractions of the ventricles and the atrium chambers respectively. Values for most cell parameters were based on [Her00].

Cardiac cavities

The left and right heart with their corresponding ventricles and atria and the cardiac valves are represented in the CVS model (Figure 4.2). The cardiac valves are represented like diodes in order to simulate the one-way direction of blood when passing through. For each cardiac chamber, volumes (V ) are computed from the integral of their respective net

ﬂow (Qin − Qout).

— Atria: The atrial pressure Pa is a linear function of its instantaneous volume Va,

whose slope Ea represents the elastic properties of the atrial wall [Oje13].

Pa(Va, t) = Ea(t) · (V − Vua) + Pthor (4.1)

Ea(t) = (Ea,MIN + ea(t) · (Ea,MAX − Ea,MIN )) (4.2)

where Vua is the unstressed volume in the chamber, Pthor is the thoracic pressure

of the respiratory system and ea(t) is a Gaussian driving function triggered by the CES submodel that cycles between atrial diastole and systole:

HR 2 HR 2!! ea(t) = exp −Ba · · t − Ca (4.3) HRR HRR

Ba and Ca control the rise and peak of the atrial systole. These parameters are adjusted to reduce or enlarge the systolic period when the heart rate is diﬀerent than the baseline resting heart rate (HRR). The generic index a ∈ {ra, la} stands for right atrium and left atrium. — Ventricles: Concerning the ventricles , the elastance is deﬁned as a Two-hill driving

function ev(t) triggered by the CES system [SMW96]:

n  t 1    α1T 1 ev(t) = cte ·  n1  ·  n2  (4.4) 1 + t 1 + t α1T α2T

where α1 and α2 deﬁne the time of each ascending and descending part of the

elastance curve respectively within the heart period T . n1 and n2 determine the

125 Chapter 4 – Adult cardio-respiratory model

steepness of the ascending and descending parts of the periodic function. cte is a

positive constant. Blood pressure (Pv) of ventricles is calculated from the pressure- volume relationships associated with systole and diastole: the End Systolic Pressure- Volume Relationship and the End Diastolic Pressure-Volume Relationship:

Pv(Vv, t) = ev(t) · Pes(Vv) + (1 − ev(t)) · Ped(Vv) + Pthor (4.5)

Pes(Vv) = Ev,MAX · (Vv − Vuv) (4.6)

λ·(Vv−V0) Ped(Vv) = P0 · (e − 1) (4.7)

In these equations, the end systolic elastance (Ev,MAX ) and the unstressed volume

(Vuv) represent the slope and intercept of the linear relationship between pressure and volume during systole. During diastole, this relationship is non-linear and de-

scribed by a gradient (P0), curvature (λ) and volume at zero pressure (V0). The generic index v ∈ {rv, lv} stands for right ventricle and left ventricle

Cardiovascular system circulation

The cardiovascular circulation is divided into the pulmonary and systemic circulations. It also includes the diﬀerence between intrathoracic and extrathoracic compartments. Pressure (P ) in each compartment is calculated as a linear relationship between its volume (V ) and vascular elastance (E), following equation (4.8).

P (V ) = E · (V − Vu) (4.8)

where Vu is the unstressed volume of the compartment. Intrathoracic compartments are aﬀected by the thoracic pressure (Pthor) of the respiratory sub-model, hence equation (4.8) becomes equation (4.9)

P (V ) = E · (V − Vu) + Pthor (4.9)

These pressures are then used to calculate ﬂows between chambers by using:

∆P Q = (4.10) R

126 4.1. Method where ∆P is the pressure gradient between two chambers and R is the corresponding vascular resistance connecting them. In order to integrate the metabolic gas exchange sub-model, a new compartment representing the systemic peripheral vessels has been included in the systemic circulation, following the structure proposed in [Alb+15]. The parameters of this new compartment (elastance, unstressed volume) were determined proportionally, comparing the parameter of our model with the parameter of the compartments of [Alb+15]. A pulmonary shunt compartment parallel to the pulmonary peripheral vessels was included in the pulmonary circulation. Resistance (Rps;Rpp) and compliance (Cps,Cpp) values of these two compartments have been assigned to distribute the desired percentage of blood ﬂow (fs) coming out of the pulmonary arteries to the pulmonary shunt, as presented in eq. 4.11 and 4.12. Furthermore, the resistances of both compartments have been adapted to have an equivalent resistance equal to the total pulmonary resistance reported in the literature. (1.0 − fs) Rps = Rpp · (4.11) fs

fs Cps = Cpp · (4.12) (1.0 − fs) Moreover, the eﬀect of lung air volume in lung perfusion was included in the cardiovascular system using the following equation [Lu+01]:

2 VA Rpa(VA) = Rpa,0 · (4.13) VA,max where Rpa is the pulmonary arteries resistance, Rpa,0 is a constant to set the normal resistance value of the pulmonary arteries, VA is the alveolar volume and VA,max is the maximum Equation 4.13 represents how lung inﬂation increases pulmonary capillary resistance reducing blood ﬂow, thus facilitating gas exchange.

4.1.3 Respiratory model

The respiratory model (Figure 4.4) was adapted from previous work of our team [Le +13] and previous models in the literature [Ava+01]. It includes the upper airways, the intermediate airways, the lower airways, the alveolar compartment, the pleural cavity, the chest wall and the respiratory muscles. The tracheobronchial airways are characterized by

127 Chapter 4 – Adult cardio-respiratory model

Figure 4.4 – Electric diagram of the respiratory system model. P , pressure; R resistance; V , volume; C, compliance; V˙ , airflow; ao, airway opening; u, upper airway; c, intermediate airways; l, lower airways; pl, pleural; cw, chest wall; A, alveolar; mus, respiratory muscles. V˙ao_c, airflow from the airway opening to the lower airways, also denominated lung respiratory flow.

nonlinear relationships and the behavior of the alveolar space and chest wall are assumed to be linear. The upper airways are mainly composed of the nasal passages, the pharynx, the larynx and part of the trachea. In this complex structure, the upper airway resistance

Ru is represented by a non-linear Rohrer resistance:

Ru = K1 + K2 · V˙ao_c (4.14)

where the K1 represents resistive properties for laminar ﬂow, K2 is the resistance component that describes turbulence and V˙ao_c is the lung respiratory ﬂow.

The intermediate airways resistance (Rc) varies inversely with the square of airway volume Vc: 2 Vcmax Rc = K3 · (4.15) Vc where K3 is the resistance value when intermediate airway volume Vc is equal to its maximum Vcmax.

Lower airway resistance (Rl) is dependent of lung volume Vao_c because Rl decreases

128 4.1. Method with increasing lung volume [OCS76; Nun87]:

K4 Rl = (4.16) Vao_c where K4 is a positive constant. The alveolar space, the intermediate airway compartment and the chest wall were assumed to have a constant, purely elastic behavior and were represented by the constant compliances CA, Cc and Ccw respectively. In order to represent a realistic respiratory pressure and flow profiles, a function representing the pressure generated by respiratory muscles Pmus, defined by [Le +13], was used: β α ! t 1− tr r TI Pmus = Pmax · Bo + (1 − Bo) · · e (4.17) TI where Pmax corresponds to maximum muscle activity, B0 is the basal level at end-expiration, tr is the time elapsed from the onset of the current respiratory cycle and TI is the inspiration duration. In addition, parameters α and β characterize the Pmus signal profile during inspiration and expiration, respectively. The model allows for the simulation of ventilation through the airway opening pressure Pao. For spontaneous breathing, Pao is equal to zero. Thoracic pressure (Pthor) is assumed to be equal to pleural pressure (Ppl).

4.1.4 Gas exchange model

The gas exchange submodel is composed of three components: i) lung gas exchange, ii) peripheral tissue gas exchange and iii) gas transport.

Lung gas exchange:

The lung gas exchange model describes the exchanges of CO2 and O2 between the dead space compartment, the alveoli compartment and the pulmonary capillaries as represented in Figure 4.5. This submodel was adapted from [Ell+13; Alb+15]. The model has several inputs from other sub-models:

1. The intermediate airway volume (Vc), the dead space volume, the alveolar volume

(VA) and ﬂow V˙A and the lung respiratory ﬂow V˙l are obtained from the respiratory model.

2. The pulmonary peripheral volume (Vpp), the blood ﬂows through the pulmonary

129 Chapter 4 – Adult cardio-respiratory model

Figure 4.5 – Lung gas exchange model. F , gas fraction; V , volume; Q, blood ﬂow; C, concentration; I, inspired; D, dead space; A, alveolar; a, arterial; v, venous; out, pa, pulmonary capillaries; fs, shunt fraction; l, lung.

130 4.1. Method

artery, peripheral circulation, and shunt (Qpa, Qpp, and Qps) are obtained from the CVS model.

3. The delayed venous gas concentrations in the blood (C˜v,CO2,C˜v,CO2) are obtained from the metabolic gas exchange and gas transport model.

The fractions of inspired air (FIO2,FICO2) are an external input to the model. Frac- tions and partial pressures of each gas are computed applying conservation of mass to each of the three compartments for each gas, assuming that every compartment is homogeneous and perfectly mixed. The relation between fractions and partial pressure of a gas is determined using the ideal gas law. The conversion from gas concentrations to partial pressures and vice versa are calculated using the oxygen dissociation functions of [SFM79]. These dissociation functions take into account both the Haldane and the Bohr eﬀects. Equilibrium between pulmonary capillaries and alveoli in terms of gas partial pressures is assumed to happen instantaneously. The outputs of the model are the concentrations of gases in the pulmonary capillaries (CA,gas). These concentrations are converted to arterial blood gas concentrations (Ca,gas) mixing the pulmonary capillary and shunted blood. The pulmonary shunt represents the blood that does not take part in the lung gas exchange, hence it keeps the venous gas concentrations(C˜v,gas).

Peripheral tissue gas exchange:

A model of CO2 production and O2 consumption by the tissues and organs was integrated in the systemic peripheral vessels compartment of the CVS (Figure 4.6) as deﬁned by [Alb+15].

Figure 4.6 – Tissue gas exchange model. C, concentrations; Q, blood ﬂow; V , volume; M, metabolic rate; Out, Output; In, Input; lv, left ventricle; e, extrathoracic; sa, systemic arteries; sves, systemic peripheral vessels; ev, extrathoracic veins; iv, intrathoracic veins; T , tissue.

131 Chapter 4 – Adult cardio-respiratory model

This blood-tissue compartment is modelled as a simple container with a total volume given by the sum of the constant tissue volume (VT,sves) and the blood volume of the systemic peripheral vessels Vsves. Metabolic O2 consumption and CO2 production are assumed to happen at constant rates (MO2sves and MCO2sves respectively). The inputs of this model are the delayed arterial gas concentrations after the lung gas exchange model (CãO2,CãCO2) and the blood flows (Qout,sa,e) and volumes (Vsves) of the systemic peripheral vessels of the CVS.

The outputs are the gas concentrations in the mixed venous blood (CsvesO2, CsvesCO2) computed using conservation of mass principles in the venous compartments:

dCsvesO2 (VT,sves + Vsves) · = Qout,sa,e · (C˜a,O2 − CsvesO2) − MO2sves (4.18) dt

dCsvesCO2 (VT,sves + Vsves) · = Qout,sa,e · (C˜a,CO2 − CsvesCO2) + MCO2sves (4.19) dt

Gas transport:

This sub-model represents the convey of O2 and CO2 throughout the cardiovascular system as is presented in Figure 4.7.

Figure 4.7 – Gas transport model. C, concentration; M, metabolic rate; V , volume; a, arterial; v, venous; τLT and τLT , pure delays.

The time that blood takes to transport the gases from the pulmonary capillaries to the

132 4.1. Method systemic peripheral vessels and from the extra-thoracic veins to the pulmonary capillaries are deﬁned by pure delays τLT and τVL respectively. The transport throughout the intrathoracic and extra-thoracic veins is explicitly modelled applying conservation of mass principles to each systemic vein compartment.

4.1.5 Neural control

Chemoreﬂex model:

Peripheral and central chemoreﬂex submodels (Figure 4.8) , adapted from [Alb+15], were integrated in the model in order to represent modulations of breathing rhythm (BR) and the respiratory muscle maximal amplitude (Pmax) in response to PaO2 and PaCO2 modiﬁcations.

Figure 4.8 – Chemoreflex model. P a, arterial partial pressure; D, delay; BR, breathing rhythm; Pmus, activity of respiratory muscles; Pmax, maximum amplitude of the activity of respiratory muscles; fapc, afferent electrical activity of the peripheral chemoreceptors; c, central chemoreflex; p, peripheral chemoreflex; A, amplitude; f, frequency.

The central chemoreceptors are assumed to be sensitive to PaCO2 and its mechanism is represented as a combination of a pure delay (Dc) and ﬁrst-order systems deﬁned by some gains (Gc,A and Gc,f ) and time constants (τc,f and τc,A).

The peripheral chemoreceptors are assumed to be sensitive to PaCO2 and PaO2. This chemoreﬂex is described as a two-stage transduction mechanism. First, PaO2 and PaCO2 are transduced into the aﬀerent electrical activity of the peripheral chemoreceptors (facp).

133 Chapter 4 – Adult cardio-respiratory model

Then, the second stage is represented as a pure delay (Dp) and a set of first-order systems, defined by gains (Gp,A and Gp,f ) and time constants (τp,f and τp,A). Contributions of each branch are summed to the baseline response of each regulated variable in order to define the maximum pressure of the respiratory muscles(Pmax) and the breathing rythym (BR).

Baroreﬂex and pulmonary stretch receptors

Baroreflex model (Figure 4.9) is based on previous work of our laboratory [Rom+16; Le +08]. It includes the baroreceptors and afferent pathways, the cardiovascular control center, the efferent pathways (including the sympathetic and parasympathetic branches) and the pulmonary stretch receptors. The input pressure for the baroreceptors is the intrathoracic systemic arteries pressure

(Psa,i) of the CVS sub-model. The baroreceptors dynamical properties are represented by a ﬁrst-order transfer function, whose gain and time constant are denoted Kb and τB.

Figure 4.9 – Diagram of the baroreﬂex system. From the arterial pressure sensed at the intrathoracic systemic circulation, the baroreﬂex system regulates the heart rate, the systemic resistance, the venous unstressed volume and the ventricular elastance. K, gain; D, delay; τ , time constant; G, gain; VA, lung volume.

Then, the eﬀerent pathways controls heart rate, systemic peripheral resistance, unstressed volumes of the systemic veins and the elastances of the ventricles and each output is added to the baseline response of each regulated variable. Chronotropic control is divided in the sympathetic and parasympathetic branches, whereas systemic resistance, venous volume and ventricular elastance are only inﬂuenced

134 4.2. Results and discussion by the sympathetic system. The same structure, based on a normalization function, a delay and a first-order filter are applied to each efferent pathway. The normalization function is represented by the following sigmoid input-output relationship:

bx Fx(t) = ax + (4.20) 1 + eλx·(PB (t)−Mx) where the generic index x ∈ {V ag, S, R, V usv, EMax} stands for vagal heart rate, sympathetic heart rate, systemic peripheral resistance, unstressed volumes of the systemic veins and the maximum elastances of the ventricles, respectively. Pb is the baroreceptor output, and the parameters ax, bx, λx and Mx are used to adjust the sigmoidal shape. The pulmonary stretch receptors modulate the vagal branch of the baroreﬂex in relation to the changes of lung volume VA [Le +15; UM03]. This is one of the main generators of the respiratory sinus arrhythmia [RHI10].

4.2 Results and discussion

The combined model has 42 state variables and 151 parameters (Appendix B). The model was implemented using the M2SL simulation library [Her+09] as explained in Section 3.1.1. Model simulations in the following examples have a simulated duration of 1000 seconds. A fourth-order Runge-Kutta method was applied for all simulations, using an integration step (∆t = 1/1024 Hz = 0.000977 s) that was optimized in order to assure the numerical stability of the fastest sub-models.

4.2.1 Comparison between model simulation and physiology on baseline values

In order to verify the ability of the proposed model to reproduce variables typically observed on a healthy adult, a comparison between the outputs and waveform of a simulation of the model in normal resting conditions and average values in human adults was performed.

Hemodynamics

Table 4.1 presents some baseline values of important clinical hemodynamic variables, reported from the literature, in a healthy adult in resting conditions and it compares them

135 Chapter 4 – Adult cardio-respiratory model

Table 4.1 – Baseline values and simulated hemodynamics vital signs for an adult. Description Value (Adults) Simulation Cardiac output (L/min) 5.42 (3.06–9.00) [Cat+17] 5.71 Systemic systolic pressure(mmHg) 90-140 [Hel+02] 120.99 Systemic diastolic pressure (mmHg) 60-90 [Hel+02] 86.92 Pulmonary systolic pressure (mmHg) 15-28 [Hel+02] 15.64 Pulmonary diastolic pressure (mmHg) 5-16 [Hel+02] 5.22 Heart rate 72 [GH10] 72

Table 4.2 – Baseline values and simulated respiratory variables for an adult. Description Value (Adults) Simulation Tidal Volume (ml) 500 [GH10] 511 Respiratory rate (bpm) 12 [GH10] 12 FRC (l) 1.7-3.5 [Duk11] 2.41 Dead volume (ml) 150 [GH10] 148.6 to the simulation outputs of the cardio-respiratory adult model. Figure 4.10 shows the simulation of pulsatile blood pressure and volumes within the cardiovascular system. The simulated pressure of systemic intrathoracic arteries results in a systolic pressure of 120.99 mmHg and a diastolic pressure of 86.92 mmHg. The simulated pressure of pulmonary arteries goes from a systolic pressure of 15.64 mmHg and a diastolic pressure of 5.22 mmHg. The model is able to reproduce a typical PV loop, that reﬂects each phase of the cardiac cycle. The amplitude and duration of diﬀerent pressures in the cardio-vascular system are close to values reported in the literature.

Respiratory mechanics

Table 4.2 presents some baseline values of the respiratory mechanics measured in a healthy adult in resting conditions and it compares them to the simulation outputs of the adult model. Figure 4.11 shows the pressures, volumes and ﬂow waveforms generated by the respiratory model within the adult model. This behavior follows the normal physiological

136 4.2. Results and discussion

Pressures in the CVS 140 LV 120 RV PA 100 PV AO 80 SV

Pressure(mmHg) 40

0 0 0.2 0.4 0.6 0.8 1 Time(s)

Pressure-volume curves 140 LV 120 RV

100

Pressure(mmHg) 40

0 0 20 40 60 80 100 120 140 Volume(ml)

Figure 4.10 – Simulated blood pressures and volumes using the proposed adult model. Left ventricle (LV), right ventricle (RV), pulmonary artery (PA), pulmonary vein (PV), aorta (AO), systemic vein (SV).

137 Chapter 4 – Adult cardio-respiratory model response of an adult under quiet breathing conditions. At the beginning of inspiration, alveolar pressure (PA) and the activity of the respiratory muscles (Pmus) are equal to the atmospheric pressure (Patm) and the lung volume is equal to the FRC. During inspiration,

Pmus decreases, reducing the thoracic pressure (Pthor) from its resting value of -3.7 mmHg to approximately -6.2 mmHg, decreasing PA below Patm allowing air to ﬂow into the respiratory system. At the end of inspiration, respiratory muscles start relaxing, allowing air to ﬂow outside of the lungs until Pmus goes back to Patm.

2.8 [l])

ao_c 2.6 (V Lung Volume 2.4 0 2 4 6 8 10 12 14 16 18 20

0.5

[l/s]) 0 ao_c Lung flow (Q -0.5 0 2 4 6 8 10 12 14 16 18 20

0.18

0.16 [l]) c

(V 0.14 Dead Volume 0.12 0 2 4 6 8 10 12 14 16 18 20

-2

-4

[mmHg]) -6 thor (P

Thoracic pressure -8 0 2 4 6 8 10 12 14 16 18 20

-2 [mmHg]) -4 mus (P -6 respiratory muscles

Pressure generated by 0 2 4 6 8 10 12 14 16 18 20 Time(s)

Figure 4.11 – Pressure, volume, and ﬂow waveforms generated by the respiratory model using the adult model.

138 4.2. Results and discussion

Table 4.3 – Mean baseline values and mean simulated gas exchange variables for an adult. Variable Description Value (Adults) Simulation

PaO2 Arterial PO2(mmHg) 95 [GH10] 95.54

PaCO2 Arterial PCO2(mmHg) 40 [GH10] 40.94

SaO2 Oxygen saturation (%) 95-100 [CK19] 96.95

PAO2 Alveolar PO2(mmHg) 104 [Com77] 101.22

PACO2 Alveolar PCO2(mmHg) 40 [Com77] 40.99

CvO2 Mixed venous CO2 (ml/dl) 15 [Lif09] 15.45

CvCO2 Mixed venous CCO2 (ml/dl) 53 [AS05] 52.64

Gas exchange and transport

Table 4.3 presents some mean baseline values of the gas exchange pressures and saturation measured in a healthy adult in resting conditions and it compares them to the simulation outputs of the adult model. Figure 4.12 shows time proﬁles of partial pressures and the oxygen saturation for quiet breathing conditions. PaO2 oscillates between 93.66 mmHg to 97.37 mmHg and PaCO2 oscillates between 38.93 mmHg to 42.63 mmHg, both in synchrony with the respiratory cycle. PaO2 increase during inspiration and decrease during expiration. PaCO2 has the opposite behavior, decreasing during inspiration and increasing during expiration. Moreover, some faster oscillations can be observed in the partial pressures of gas in the arteries. This behavior is caused by the coupling of the pulmonary capillaries blood ﬂow from the CVS submodel in the gas exchange model. This phenomenon has been reported by previous investigators [Fuk72; FC68].

Cardio-respiratory interactions

In Chapter 1, it was stated that the cardiovascular and respiratory systems must act jointly to maintain optimal oxygenation of the body. The cardio-respiratory interactions represented in the model are: 1. Pulmonary stretch receptors.

2. Effect of thoracic pressure (Pthor) on cardiovascular intrathoracic compartments. 3. The influence of lung inflation in capillary resistance.

139 Chapter 4 – Adult cardio-respiratory model

2.8 [l])

ao_c 2.6 (V Lung Volume 2.4 0 2 4 6 8 10 12 14 16 18 20

96 [mmHg] 2 94 PaO 92 0 2 4 6 8 10 12 14 16 18 20

42 [mmHg] 2 40 PaCO 38 0 2 4 6 8 10 12 14 16 18 20

97.4

97.2 [%]

2 97

SaO 96.8

96.6 0 2 4 6 8 10 12 14 16 18 20 Time (s)

Figure 4.12 – Arterial partial pressure of O2 and CO2 (PaO2, PaCO2) and the arterial oxygen saturation (SaO2) simulated by the cardio-respiratory model.

140 4.2. Results and discussion

4. Respiratory sinus arrhythmia (RSA): the ﬂuctuation of heart rate in phase with inspiration and expiration, with heart rate increasing during inspiration and decreasing during exhalation [HY03].

Figure 4.13 presents the eﬀects of respiration on cardiovascular outputs in the adult model. When the thoracic pressure Pthor (equal to the pleural pressure Ppl) decreases, there is a drop in stroke volume. This is produced by the mechanical coupling between the thoracic pressure and the intrathoracic compartments of the cardiovascular model. In the other hand, the RSA is observed in the variation in HR in synchrony with the respiratory cycle. HR increases during inhalation and decreases during exhalation. This interaction is due to the mechanical coupling between the respiration and the cardiovascular system, but mainly by the action of the pulmonary stretch receptors (PSR). PSR modulate the vagal branch, increasing the vagal activity in expiration and decreasing heart rate. Similarly, PSR decrease vagal activity in inspiration, increasing heart rate. The magnitude of the variations of the model simulated HR (4.96 bpm) are similar to that typically observed in healthy adults (5 bpm [PD09]).

4.2.2 Simulation of an obstructive apnea event

The pathology of adults under study in this work is the sleep apnea syndrome, hence it is important to simulate the behavior of an apnea event using the adult model. An obstructive apnea event was simulated by increasing the resistance of the upper airways −1 Ru of the respiratory model to 100000 cmH2O · s · l . Figure 4.14 shows the behavior of the simulated partial pressure of oxygen and CO2, the oxygen saturation (SaO2) in the pulmonary capillaries and the tissues and the heart rate time series during a 20 seconds obstructive apnea by the model. Over the course of the apnea, the simulation follows the behavior described in the literature for an obstructive event [Whi95; Lev08]: i) the PaO2 falls (from 93.77 mmHg to 69.51 mmHg; ∆=24.26 mmHg), ii) the SaO2 falls (from 96.7 mmHg to 92.6 mmHg;

∆=4.1%) iii) the PaCO2 rises (from 42.62 mmHg to 47.29 mmHg ∆=4.67 mmHg) and iv) ventilatory eﬀort increases. This simulation is qualitatively comparable to clinical data we have observed in our works (see for example Figure 1.12). One of the main hypothesis of this work is that we can reproduce patient-speciﬁc apnea responses, by optimally tuning the parameters of our model (see Section 5.3).

141 Chapter 4 – Adult cardio-respiratory model

2.8 [l])

ao_c 2.6 (V Lung volume 2.4 0 2 4 6 8 10 12 14 16 18 20

-3 -4 -5

[mmHg]) -6

thor -7 (P

Thoracic pressure -8 0 2 4 6 8 10 12 14 16 18 20

72 Heart Rate (HR [bpm]) 70

0 2 4 6 8 10 12 14 16 18 20

Stroke volume (ml) 77 0 2 4 6 8 10 12 14 16 18 20 Time (s)

Figure 4.13 – Eﬀects of respiration on cardiovascular outputs. Top to bottom: time proﬁles of lung volume, thoracic pressure (Pthor), heart rate (HR), stroke volume.

142 4.2. Results and discussion

Lung Volume 4

l 3

AStart AEnd 2 0 20 40 60 80 100 120 SaO 2 100

% 95 SaO in Pulmonary capillaries 2 SaO in systemic tissue 2 90 0 20 40 60 80 100 120 PaO 2 100

mmHg 80

0 20 40 60 80 100 120 PaCO 2 50

40 mmHg

30 0 20 40 60 80 100 120 Heart rate 80

70 bpm

60 0 20 40 60 80 100 120 Time (s)

Figure 4.14 – Simulation of obstructive apnea event using the adult cardio-respiratory model. Top to bottom: Lung volume (Blue), Apnea Start (AStart) and Apnea End (AEnd) (Vertical red lines); SaO2 in pulmonary capillaries (Blue), SaO2 in systemic tissue (Or- ange); PaO2 in pulmonary capillaries (Blue); PaCO2 in pulmonary capillaries (Blue); Heart rate (Blue). The ﬁrst 40 seconds, there is a normal ventilation and then for 20 seconds there is an upper airway obstruction, producing an apnea event. This produces a drop in SaO2, PaO2 and an increase in PaCO2 during the event in the pulmonary capillaries. Then, ventilation resumes with an higher amplitude, returning SaO2, PaO2, PaCO2 to their baseline values. The diﬀerence between the SaO2 at the pulmonary capillaries and SaO2 at the systemic tissue is a delay.

143 Chapter 4 – Adult cardio-respiratory model

4.3 Model limitations

The main limitations of the model are mainly related to certain physiological mechanisms that have intentionally not been included. The model does not include long-term control mechanisms or thermoregulation or an adaptive metabolism module. We hypothesize that the effect of these functions on the short-term responses that we focus in this work is negligible. Another aspect that has not been integrated is sleep stage modeling, which may have an important role in the context of sleep apnea. Indeed, sleep states have important effects in the metabolism [SK10], sympathetic tone [AM16], cardiovascular function [SD13] and some studies affirm that REM sleep is a contributing factor for OSA [Per+13]. Some of these mechanisms will be included in future works.

4.4 Conclusion

Numerous mathematical models of physiological systems (cardiac and respiratory activity, renal function, autonomic nervous system...) at varying levels of detail have already been proposed in the literature. This chapter proposes a novel, integrated model of cardio- respiratory interactions in adults, which was based on both structural and functional integration through a multi-resolution approach. Formally heterogeneous models were coupled in order to represent original representations of: i) mechanisms involved in RSA, ii) interactions between discrete cardiac electrical conduction system and lumped- parameter cardio-respiratory models, iii) integration of metabolism in circulatory model and iv) an analytic representation of the respiration pattern adapted to adults. The proposed cardio-respiratory model is able to reproduce baseline cardiac and respiratory functions, with numerically stable physiological variables ﬂuctuating within physiological ranges. The integrated cardio-respiratory and metabolic sub-models allow for the simulation of an obstructive apnea, which is followed by the main known cardiac and respiratory acute events. This model represents one of the main contributions of this work and constitutes the cornerstone for all our further developments in this thesis. In particular, the next chapter presents a formal analysis of this model, leading to the estimation of patient-speciﬁc responses.

144 REFERENCES

References

[Alb+15] Antonio Albanese et al., « An integrated mathematical model of the human cardiopulmonary system: Model development », in: American Journal of Physiology - Heart and Circulatory Physiology (2015), ajpheart.00230.2014. [AM16] Mohammed Alzoubaidi and Babak Mokhlesi, Obstructive sleep apnea during rapid eye movement sleep: Clinical relevance and therapeutic implications, Oct. 2016. [AS05] GJ Arthurs and M Sudhakar, « Carbon dioxide transport », in: Continuing Education in Anaesthesia, Critical Care & Pain 5 (2005). [Ava+01] Guido Avanzolini et al., « Role of the mechanical properties of tracheobronchial airways in determining the respiratory resistance time course », in: Annals of Biomedical Engineering 29.7 (2001), pp. 575–586. [Cal+18] Mireia Calvo et al., « Model-based analysis of the autonomic response to head-up tilt testing in Brugada syndrome », in: Computers in Biology and Medicine 103 (Dec. 2018), pp. 82–92. [Cal+19] Mireia Calvo et al., « Recursive model identiﬁcation for the analysis of the autonomic response to exercise testing in Brugada syndrome », in: Artiﬁcial Intelligence in Medicine 97 (June 2019), pp. 98–104. [Cal18] M. Calvo, « Analysis of the cardiovascular response to autonomic nervous system modulation in Brugada syndrome patients », PhD thesis, Université de Rennes 1, 2018. [Cat+17] Giles N Cattermole et al., « The normal ranges of cardiovascular parameters measured using the ultrasonic cardiac output monitor », in: Physiol Rep 5.6 (2017). [CK19] Danny Castro and Michael Keenaghan, Arterial Blood Gas, StatPearls Pub- lishing, Feb. 2019. [Com77] JH Comroe, « Alveolar ventilation », in: Physiology of Respiration, Chicago: Year Book Medical Publisher, 1977, chap. 2, pp. 8–21. [Duk11] James Duke, Anesthesia secrets, 2011, p. 503.

145 Chapter 4 – Adult cardio-respiratory model

[Ell+13] L. M. Ellwein et al., « Modeling cardiovascular and respiratory dynamics in congestive heart failure », in: Mathematical Biosciences 241 (2013), pp. 56– 74. [FC68] Raymond W. Flumerfelt and Edward D. Crandall, « An analysis of external respiration in man », in: Mathematical Biosciences 3 (Aug. 1968), pp. 205– 230. [Fuk72] Y Fukui, « A Study of the Human Cardiovascular-Respiratory System Using Hybrid Computer Modeling. », PhD thesis, 1972. [GH10] Arthur C. Guyton and John E. Hall, Textbook of medical physiology, 2010, p. 1091. [Hel+02] Thomas Heldt et al., « Computational modeling of cardiovascular response to orthostatic stress », in: Journal of Applied Physiology 92.3 (Mar. 2002), pp. 1239–1254. [Her+02] Alfredo I. Hernández et al., « Model-based interpretation of cardiac beats by evolutionary algorithms: Signal and model interaction », in: Artiﬁcial Intelligence in Medicine 26.3 (2002), pp. 211–235. [Her+09] A. I. Hernandez et al., « A multiformalism and multiresolution modelling environment: application to the cardiovascular system and its regulation », in: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 367.1908 (Dec. 2009), pp. 4923–4940. [Her00] A. I. Hernández Rodriguez, « Fusion de signaux et de modèles pour la carac- térisation d’arythmies cardiaques », PhD thesis, Université Rennes 1., 2000. [HY03] Junichiro Hayano and Fumihiko Yasuma, « Hypothesis: respiratory sinus arrhythmia is an intrinsic resting function of cardiopulmonary system. », in: Cardiovascular research 58.1 (Apr. 2003), pp. 1–9. [Le +08] Virginie Le Rolle et al., « An autonomic nervous system model applied to the analysis of orthostatic tests », in: Modelling and Simulation in Engineering 2008.i (2008). [Le +13] Virginie Le Rolle et al., « Mathematical modeling of respiratory system mechanics in the newborn lamb . To cite this version : HAL Id : hal-00880028 », in: (2013).

146 REFERENCES

[Le +15] Virginie Le Rolle et al., « Recursive identification of an arterial baroreflex model for the evaluation of cardiovascular autonomic modulation », in: Com- puters in Biology and Medicine 66 (2015), pp. 287–294. [Lev08] Michael G. Levitzky, « Using the pathophysiology of obstructive sleep apnea to teach cardiopulmonary integration », in: American Journal of Physiology - Advances in Physiology Education 32.3 (2008), pp. 196–202. [Lif09] Edwards Lifesciences, Normal Hemodynamic Parameters and Laboratory Val- ues (Online), 2009. [Lu+01] K Lu et al., « A human cardiopulmonary system model applied to the analysis of the Valsalva maneuver », in: Am J Physiol Heart Circ Physiol 281.6 (Dec. 2001), pp. 2661–2679. [Nun87] J. F. (John Francis) Nunn, Applied respiratory physiology, Butterworths, 1987, p. 582. [OCS76] M F Olender, J W Clark, and P M Stevens, « Analog computer simulation of maximum expiratory flow limitation. », in: IEEE transactions on bio-medical engineering 23.6 (Nov. 1976), pp. 445–52. [Oje+13] David Ojeda et al., « Towards an Atrio-Ventricular Delay optimization as- sessed by a computer model for Cardiac Resynchronization Therapy », in: Proceedings of SPIE - The International Society for Optical Engineering 8922 (2013), pp. 333–336. [Oje+15] David Ojeda et al., « Analysis of a baroreflex model for the study of the chronotropic response to vagal nerve stimulation », in: 2015 7th International IEEE/EMBS Conference on Neural Engineering (NER), IEEE, Apr. 2015, pp. 541–544. [Oje13] David Ojeda, « Multi-resolution physiological modeling for the analysis of cardiovascular pathologies », PhD thesis, Dec. 2013. [PD09] E.L. Phillips and P.D. Donofrio, « Autonomic Disorders », in: Encyclopedia of Neuroscience (Jan. 2009), pp. 799–808. [Per+13] I Peregrim et al., « Does obstructive sleep apnea worsen during REM sleep? », in: Physiological research 62.5 (2013), pp. 569–75. [RHI10] Loren A. Rolak, Yadollah Harati, and Shahram Izadyar, « Autonomic Ner- vous System », in: Neurology Secrets (Jan. 2010), pp. 204–226.

147 Chapter 4 – Adult cardio-respiratory model

[Rom+16] Hector M. Romero Ugalde et al., « Model-Based Design and Experimental Validation of Control Modules for Neuromodulation Devices », in: IEEE Transactions on Biomedical Engineering 63.7 (2016), pp. 1551–1558. [SD13] Alessandro Silvani and Roger A L Dampney, Central control of cardiovascular function during sleep, Dec. 2013. [SFM79] J. L. Spencer, E. Firouztale, and R. B. Mellins, « Computational expressions for blood oxygen and carbon dioxide concentrations », in: Annals of Biomedical Engineering 7.1 (1979), pp. 59–66. [SK10] Sunil Sharma and Mani Kavuru, Sleep and metabolism: An overview, 2010. [SMW96] N. Stergiopulos, J. J. Meister, and N. Westerhof, « Determinants of stroke volume and systolic and diastolic aortic pressure », in: American Journal of Physiology-Heart and Circulatory Physiology 270.6 (June 1996), H2050– H2059. [UM03] Mauro Ursino and Elisa Magosso, « Role of short-term cardiovascular regulation in heart period variability: a modeling study », in: American Journal of Physiology - Heart and Circulatory Physiology 284.4 (2003), H1479–H1493. [Whi95] David P. White, « Pathophysiology of obstructive sleep apnoea », in: Thorax 50.7 (1995), pp. 797–804.

148 Chapter 5 MODEL BASED ANALYSIS OF CARDIO-RESPIRATORYINTERACTIONS DURINGSLEEPAPNEA

This chapter presents a model-based analysis of the acute cardio-respiratory interactions involved during sleep apnea, using the cardio-respiratory adult model proposed in Chapter 4. Equations and parameters characterizing this model can be found in Annexes A and B, respectively. The objectives of this chapter are: i) to perform a complete parametric analysis of an integrated cardio-respiratory model in the context of sleep apnea, ii) to propose a patient-specific and event-specific modeling for the cardio-respiratory response to an apnea, and iii) to analyze the dynamics of SaO2 during an OSA in a patient-specific manner. The structure of this chapter is divided in four parts. First, the experimental database used for the patient-specific identification is introduced. Second, the sensitivity analysis of the model in the context of obstructive sleep apnea is presented and specified with its methodology, results, discussion and conclusions. Third, the patient-specific identification of the parameters of the cardio-respiratory model to reproduce the experimental dynamics of the SaO2 during SAS is explained, and its results and discussion are shown. Finally, a general conclusion for the chapter is proposed.

5.1 Experimental data

The clinical database used for the patient-speciﬁc identiﬁcation in adults using the adult cardio-respiratory model was obtained within the framework of the HYPNOS study of the ANR TecSan project entitled “Personalized and Adaptive kinesthetic StIm- ulation Therapy, based on cardio-respiratory Holter monitoring, for slEep Apnea syndromes” (PASITHEA), supported by the French “Agence Nationale pour la Recherche”.

149 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea

The HYPNOS study was a multicenter, proof-of-concept, open-label, prospective study, conducted at 5 hospital-based university centers in France (Grenoble, Montpellier, Angers, Tours and Rennes). The study was conducted in accordance with applicable good clinical practice requirements in Europe, French law, ICH E6 recommendations, and the ethical principles of the Helsinki Declaration (1996 and 2000). The study was approved by an independent Ethics Committee (Comité de Protection des Personnes, Grenoble, France, IRB 2014-A00339-38) and registered at ClincalTrials.gov with the identifier NCT03300037 (date of registration: 03/10/2017). Subjects over 18 years, with severe OSA, defined as an apnea-hypopnea index (AHI) > 30/h and less than 20% of central events, were eligible. All patients provided informed consent. Pre-screening was performed via in-home sleep testing (polygraphy) or review of in-laboratory polysomnographs in patients previously referred for sleep apnea suspicion. Patients unable to give written consent or presenting any of the following criteria were not included: history of severe respiratory or cardiac failure, atrial fibrillation, morbid obesity with Body mass index (BMI > 40 kg/m2), sleep duration <4 hours/night, Parkinson’s disease, dysautonomia, pregnancy or lactation. Thirty four patients were eligible for the study [Her+18]. Of these, 8 patients were excluded from the analysis set due to technical problems with the polysomnography or the PASITHEA system, and 2 patients exited due to dysautonomia (1 patient with fi- bromyalgia and 1 patient with periodic leg movement). Thus, data from 24 patients are reported here. The study population is typical of patients with severe OSA, predominantly middle-age, male, obese with frequent co-morbidities. All patients underwent a full standard polysomnography (PSG) study at an in-hospital sleep laboratory. The PSG study consists on the acquisition, during a whole night, of a set of physiological signals (EEG, EMG, ECG, SaO2, ventilation,...) that are analyzed after the acquisition in order to quantify the sleep quality of the patients, characterize respiratory events and analyze their consequences [Pép+16]. Figure 5.1 presents a subset of the typical physiological signals recorded during PSG, from a patient included in the HYPNOS study. In this example, a 10 minutes segment is shown. During this support, 7 apnea and 9 hypopnea episodes were recorded. The acute consequences of these episodes on the cardiovascular and respiratory systems, as well as on the sleep structure of the patient can be easily observed (see legend for details). PSG data acquired during the HYPNOS study were evaluated by a single scorer at the core-laboratory of the project (CHU Grenoble), with quality assured by an intra-

150 5.1. Experimental data

Figure 5.1 – Example of a typical recording of a patient suffering from sleep apnea syndrome from PASITHEA Project [Her+16; D P18]. The first panel shows the nasal pressure (NP) signal, where the different apnea and hypopnea events are highlighted. The second panel represents the oxygen saturation (SaO2) signal with the intermittent hypoxia events associated with SAS. The third panel, presents the instantaneous heart rate (HR) signal, showing recurrent acute autonomic responses that are due to the respiratory events. Finally, the fourth panel presents a hypnogram signal (MA = micro-arousal, A = Awake, REM = Rapid eye movement, LS = Light sleep and DS = Deep sleep) representing the sleep structure of the patient during the recording. The hypnogram shows the presence of sleep arousals and sleep fragmentation produced by respiratory events. Figure adapted from [D P18].

151 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea scorer quality control process [Her+18]. Annotations included the instant of occurrence of obstructive, central and mixed apneas, hypopneas, the sleep state and body position, among others. In this work, the analysis was focused on selected annotated obstructive apnea events, during the periods in which no intervention (mechanical stimulation) was performed on the patients.

Apnea selection for patient-speciﬁc identiﬁcation

A subset of the annotated apnea events was selected for the phase of patient-specific model parameter identification. The following criteria was used to select the events: 1. Only obstructive apneas were chosen, since this was the target of the study and most of the recorded events were OSA. 2. The minimum duration of the event is 20 seconds, in order to increase the chances of observing acute cardio-respiratory effects. 3. A minimum separation of 20 seconds between the end of the previous apnea and the beginning of the current apnea was required, so as to minimize the cardio-respiratory effects due to the previous event on the event of interest. The simulation of such mixed events increases the complexity of the identification phase. Table 5.1 presents the number of selected apneas per patient from the HYPNOS data base, with a total of 107 obstructive apneas distributed among 10 patients. The mean desaturation during the selected apnea events (∆SaO2) and the apnea-hypopnea index (AHI) of the selected patient are included in this table. The AHI is the number of apneas or hypopneas per hour of sleep. Even if all the patients of this database have a severe OSA, only a few apnea events were selected for patients 3, 6 and 8, because most of their events did not meet the selection criteria. In order to identify the selected apneas that were part of successive respiratory events or periodic breathing, consecutive events within a 60 second window after the end of the selected OSAs were analyzed. Within this range of 60 seconds, 66% of the selected apneas were followed by another apnea, 21% by a hypopnea and 13% had no consecutive respiratory event.

5.1.1 Signal processing analysis

The obstructive apnea events selected from the HYPNOS database had to be processed in order to be used in this work. The RR interval time series of each patient was extracted

152 5.2. Sensitivity analysis

Table 5.1 – List of patients and selected apneas for patient-specific identification. Patients Selected apneas ∆SaO2(%) AHI 2 32 5.26 54.99 3 1 2.40 62.46 5 10 7.33 20.20 6 2 5.65 35.55 8 2 3.51 18.60 11 29 6.68 47.90 12 7 9.42 17.19 15 14 8.56 71.23 16 6 5.26 46.10 17 4 2.82 26.42 Total 10 patients 107 apneas from the ECG using a QRS complex detector developed by our team [Doy+19]. The raw ECG time series had a sampling frequency of 1024 Hz and it was resampled to a sampling frequency of 1000 Hz to be used as an input of the QRS detector. These QRS detection instants were used to observe the evolution of the instantaneous heart rate during an apnea event. The raw SaO2 signal acquired from the PSG system presents a "sample-and-hold" shape. Hence, a moving average filter was applied using a

5 second window to smooth the signal. This filtered SaO2 is the one used in the cost function of the patient-specific identification phase (Section 5.3)

5.2 Sensitivity analysis

A possible concern of a complex model like the proposed adult cardio-respiratory model is the high number of parameters used (151 parameters). This high number of degrees of freedom has a direct consequence on the identifiability of the model to a given individual. However, not all parameters have to be fine-tuned to reproduce a given, only the ones related to the specific physiological aspects under investigation are critical. A sensitivity analysis was performed in order to determine which inputs or parameters are more important in the context of the sleep apneas and to select a subgroup of parameters for an eventual patient-specific identification.

153 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea

5.2.1 Methodology

In order to find the most influential model parameters affecting the SaO2, PaO2,

PaCO2 and heart rate during an obstructive apnea event in adults, a sensitivity analysis was performed following the approach proposed in Section 3.2.4. A Morris’s screening method [Mor91a] was applied on 151 parameters of the model (Section B). Parameter ranges were selected from the nominal values ±30%. The output function y of Morris’ method was speciﬁed for the analysis of an obstructive apnea using the adult model.

Output functions used for the Morris’ screening method

The conditions under study in this work is the SAS, hence for the sensitivity analysis of the adult model, an obstructive apnea event has to be simulated. A 20 second apnea was simulated increasing the upper airways resistance as shown in Section 4.2.2. The output function is deﬁned to observe the changes in the waveform shape of the most relevant Mean simulated signals during the apnea event. Two output functions ∆X and ∆X , with

X ∈ {SaO2, P aO2, P aCO2}, were deﬁned in order to calculate the elementary eﬀects for the sensitivity analysis (Figure 5.2.1):

1. between the start and the end of an apnea event:

XEndApnea − XStartApnea ∆X = (5.1) XStartApnea

2. in 15 second’s window before the apnea and a 15 second’s window after the apnea:

Mean XMeanAfter − XMeanBefore ∆X = (5.2) XMeanBefore

Mean The elementary effects are calculated with each output ∆X and ∆X using the definition given by [Mor91b] (Equation 3.4). The three most important parameters, obtained from the Morris analysis, were selected and a local sensitivity analysis was performed in order to analyze the influence of these parameters on other relevant cardio-respiratory signals.

154 5.2. Sensitivity analysis

Lung volume (l) 3.4 Apnea Start Apnea End 3.2 20 s

3 l 2.8

2.6

2.4 0 10 20 30 40 50 60 70 80

X=PaO 2 110

15 s Mean PaO 100 PaO at the start 2 2 after of the apnea 90

mmHg Mean PaO 80 2 before 15 s 70 PaO at the end 2 of the apnea 60 0 10 20 30 40 50 60 70 80 Time (s)

Figure 5.2 – Example of a variable used for the sensitivity analysis (X) and how the ∆X markers are obtained from a simulation of a 20 second obstructive apnea. In this case, X=PaO2, which is represented as simulated at the output of pulmonary capillaries. The green circles represent the values of X at the start and at the end of the apnea event. These values are used to calculate ∆X . The orange dashed-line rectangle represents the Mean 15 second window before and after the apnea event use to calculate ∆X .

5.2.2 Results

Sensitivity analysis using the Morris’s screening method

The results of the Morris’ screening method will be separated into two subsections, Mean one for the output function ∆X and the other one for the output function ∆X .

155 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea

Output functions ∆X

Figure 5.3 presents the results of the sensitivity analysis performed on the adult model during an obstructive apnea of 20 seconds, using the output functions ∆X with X ∈

{SaO2, P aO2, P aCO2,HR}. Only the 6 most sensitive model parameters are shown for each ∆X . These subsets of parameters are considered the most relevant for variations of each variable X during an obstructive apnea event.

0.3 0.3

0.2 0.2

0.1 0.1

0 0 2 2 A A fs uA uA C C V V FIO FIO T,sves O2sves O2sves V CO2sves CO2sves M M M M

0.2 1

0.1 0.5

0 0 b 2 1 uA cw K K Rin sa,e V Rout C K FIO E a T,sves Vusv,e O2sves V K CO2sves M M

Figure 5.3 – Most inﬂuential parameters of the proposed adult model obtained through the Morris screening method, using the output function ∆X , with X ∈ {SaO2, P aO2, P aCO2,HR}. Green bars: Morris distance Di. For each parameter, the ∗ absolute mean µi (purple bar) and the standard deviation σi (yellow bars) of the elementary eﬀects are also displayed. Descriptions of each parameter are given later in this section or in Section B.

There are several common, sensitive parameters in the obtained results for ∆SaO2,

∆P aO2 and ∆P aCO2 :

— Metabolic rates: oxygen consumption rate MO2sves and CO2 production rate MCO2sves.

156 5.2. Sensitivity analysis

— Respiratory mechanics parameters: unstressed volume of air in the alveoli VuA and

lung compliance CA.

— Fraction of oxygen in inspired air (FIO2)

For ∆SaO2 , the total volume in the systemic tissue (VT,sves) is also relevant. The most relevant parameter for SaO2 during the apnea is FIO2. For ∆P aO2 , the fraction of pulmonary shunt (fs) is signiﬁcant too. The most important parameters for PaO2 during the apnea are MO2sves and FIO2. For ∆P aCO2 , the gain of the baroreceptors (Kb) and

VT,sves are part of the sensitive subgroup of parameters. The most important parameter for PaCO2 during the apnea is MCO2sves.

For ∆HR, there is no parameter that stands out in the sensitivity analysis, because the

Morris distance Di in Figure 5.3 is almost the same for all the parameters. This might be due to the fact that the baseline of the heart rate does not change strongly in the model simulations during the apnea. The two more important parameters for the heart rate are the compliance of the chest wall (Ccw) and the elastance of the extrathoracic systemic arteries (Esa,e).

Mean Output functions ∆X

Figure 5.4 presents the result of the sensitivity analysis performed on the adult model Mean for an obstructive apnea of 20 seconds using the output functions ∆X with X =

{SaO2, P aO2, P aCO2,HR}. Only the 6 most important parameters were taken for each Mean ∆X . These subsets of parameters are the most relevant after the apnea. There are several common parameters in the results of the sensitivity analysis on Mean Mean Mean ∆SaO2 , ∆P aO2 and ∆P aCO2 :

— Metabolic rates: oxygen consumption rate MO2sves and CO2 production rate MCO2sves.

— Fraction of oxygen in inspired air (FIO2). — Chemoreﬂex parameters: chemoreﬂex gains for the amplitude of the respiratory

muscles (Gc,A and Gp,A) and the time constants (τc,A and τp,A) for the central and peripheral chemoreﬂex respectively.

The most important parameter for SaO2 and PaO2 after the apnea is FIO2. For Mean ∆P aCO2 , the unstressed volume of air in the alveoli (VuA) is part of the sensitive subgroup of parameters. The most important parameters for PaCO2 after the apnea are the central

157 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea

0.015 0.1

0.01 0.05 0.005

0 0 2 2 c,A c,A c,A c,A p,A p,A p,A G G G FIO FIO O2sves O2sves CO2sves M M M

0.06 0.3

0.04 0.2

0.02 0.1

0 0 c uA ula O2 c,A c,A D sa,e V V Rout FI G E O2sves EMAX,lv CO2sves M K M

Figure 5.4 – Most inﬂuential parameters of the Morris screening method using the output Mean function ∆X on SaO2, PaO2, PaCO2 and HR on the adult model. Green bars: Morris ∗ distance Di. For each parameter, the absolute mean µi (purple bar) and the standard deviation σi (yellow bars) of the elementary eﬀects are also displayed.

158 5.2. Sensitivity analysis

chemoreﬂex gain (Gc,A) and time constant (τc,A) for the amplitude of the respiratory muscles. Mean For ∆HR , the relevant parameters come from diﬀerent mechanisms. The most important is a sigmoid parameter of the sympathetic regulation mechanisms of the systemic resistance (λRout). The unstressed volume of the left atrium(Vula), a parameter characterizing Pmus during expiration (β), the gain of sympathetic regulation of the end-systolic elastance in the left ventricle (KEMAX,lv), delay from the systemic arteries to the central chemosensitive area (Dc) and the elastance of the extrathoracic systemic arteries (Esa,e).

Local sensitivity analysis

In order to understand and illustrate the results of the Morris sensitivity analysis (Section 5.2.2) a local sensitivity analysis was performed using three of the most influential parameters of the adult model during and after the apnea: the fraction of oxygen in the inspired air (FIO2), the metabolic rate of oxygen consumption (MO2sves) and the central chemoreflex gain for the amplitude of the respiratory muscles (Gc,A). This analysis showed the effects of these parameters in several simulated signals of the model (SaO2, PaO2,

PaCO2, HR).

Figure 5.5 shows the eﬀects of changes of FIO2 and MO2sves. These two local sensitivity analyses are presented together, because both has inversely proportional eﬀects in the analyzed simulated signals during an OSA event. A smaller FIO2 or a higher

MO2sves decreases directly the baseline of the PaO2 and SaO2. This induces an increase of the amplitude of the activity of respiratory muscles through the peripheral chemoreflex, decreasing PaCO2. Moreover, a smaller FIO2 results in a more profound desaturation, because the reserve of air in the lungs during the apnea has less oxygen, so it is less capable of slowing down the desaturation. Similarly, a higher MO2sves produces a more profound desaturation, because during the apnea there is no airflow to the lungs, so the metabolism is the only mechanisms changing the amounts of O2 and CO2. Heart rate does not variate importantly during apnea. After the apnea event the HR has higher oscillations if FIO2 is smaller or if MO2sves is higher, because of the higher respiratory efforts to recover from the hypoxia, increasing the activity of the pulmonary stretch receptors and the thoracic pressure, amplifying the RSA.

Figure 5.6 presents the effect of changes of Gc,A. There is not important changes of amplitude or baseline value in any of the observed signals before or during the apnea, just a phase shift. During an obstructive apnea, the chemoreflex has no relevant influence,

159 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea

FIO2 = 0.15 FIO2 = 0.175 MO2sves = 2 A FIO2 = 0.2 B MO2sves = 4 FIO2 = 0.225 MO2sves = 6 FIO2 = 0.25 MO2sves = 8 Alveoli Volume FIO2 = 0.275 Alveoli Volume MO2sves = 10 4 4

3.5 3.5

3 3 cmH2O cmH2O 2.5 2.5 AStart AEnd AStart AEnd 2 2 0 20 40 60 80 100 0 20 40 60 80 100 SaO2 SaO2 100 100

80 90

% 60 % 80 40

70 20 0 20 40 60 80 100 0 20 40 60 80 100 PaO2 PaO2 150 150

100 100

mmHg 50 mmHg 50

0 0 0 20 40 60 80 100 0 20 40 60 80 100

PaCO2 PaCO2 50 50

45 40 40

mmHg mmHg 30 35

30 20 0 20 40 60 80 100 0 20 40 60 80 100

Heart rate Heart rate 76 76

74 74 bpm bpm 72 72

70 70 0 20 40 60 80 100 0 20 40 60 80 100 Time (s) Time (s)

Figure 5.5 – Local sensitivity analysis of: A) the fraction of oxygen of inspire air (FIO2) and B) the metabolic rate of oxygen consumption (MO2sves) on the adult model during 160 an obstructive apnea. 5.2. Sensitivity analysis because even if there is an adaptation of the respiration, the closed upper airways block any ﬂow of air to the lungs. The important inﬂuence of Gc,A is after the apnea, when a Gc,A with higher amplitude produce a faster recovery of the partial pressure of gases to their baseline value, changing the SaO2. A higher Gc,A produce a fast increase in the alveolar volume after the apnea, increasing the activity of the pulmonary stretch receptors and the thoracic pressure (Pthor), amplifying the RSA. In cases when the amplitude of

Gc,A is too important, an overcompensation phenomenon can be observed after the apnea, with a ﬁrst increase of lung volume followed by a compensating reduction of lung volume that can generate a subsequent apnea. If the overcompensation continues after the OSA event, a periodic breathing could be induced.

5.2.3 Discussion

A possible concern highlighted in previous models with similar complexity [Alb+15] is that the number of parameters of the model will complicate the task of ﬁtting any individual subject, dynamic or pathology. To overcome this concern, and knowing that the pathology of study is the sleep apnea, a sensitivity analysis of the model was performed in the context of a 20 seconds apnea. This analysis evaluated the relative signiﬁcance of model parameters on SaO2, PaO2, PaCO2 and heart rate (HR) before, during and after an apnea event. Four groups of parameter appeared as variables of interest for SaO2,

PaO2, PaCO2: i) the fraction of oxygen in inspired air (FIO2), ii) metabolic rates: oxygen consumption rate MO2sves and CO2 production rate MCO2sves; iii) chemoreflex parameters: chemoreflex gains for the amplitude of the respiratory muscles (Gc,A and Gp,A) and the time constants (τc,A and τp,A) for the central and peripheral chemoreflex respectively and iv) respiratory mechanics parameters: unstressed volume of air in the alveoli VuA and lung compliance CA. These results are consistent with several factors highlighted in the literature on the dynamics of sleep apnea and existing treatments.

— Fraction of oxygen in inspired air (FIO2): As shown in the local sensitivity

analysis of Section 5.2.2 (Figure 5.5), a smaller FIO2 decreases the baseline value

of the PaO2 and SaO2. It also decreases the amount of oxygen in the reserve of air in the lungs during the apnea, reducing the ability of the system to slow down the desaturation. The relevance of this parameter agrees with the use of oxygen therapy as a treatment of sleep apnea [Meh+13] [Got+14].

— Metabolic oxygen consumption and CO2 production rate (MO2sves, MCO2sves):

161 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea

Alveoli Volume 4

3.5

cmH2O 2.5 AStart AEnd 2 0 10 20 30 40 50 60 70 80 90 100 SaO2 100

GcA = -1

% 95 GcA = -3 GcA = -5 GcA = -7 GcA = -9 90 0 10 20 30 40 50 60 70 80 90 100 PaO2 120

100

mmHg 80

60 0 10 20 30 40 50 60 70 80 90 100 PaCO2 50

mmHg 30

20 0 10 20 30 40 50 60 70 80 90 100 Heart rate 80

75 bpm

70 0 10 20 30 40 50 60 70 80 90 100 Time (s)

Figure 5.6 – Local sensitivity analysis of the central chemoreﬂex gain for the amplitude of the respiratory muscles (Gc,A) on the adult model during an obstructive apnea.

162 5.2. Sensitivity analysis

During an apnea, the lack of ventilation induces the stop of the ﬂow of O2 and CO2 and the chemoreﬂex cannot modify the mechanics of the respiration. Hence, the

factors that are going to aﬀect the oxygen desaturation and the changes of PaCO2

during an apnea are the O2 metabolic consumption rate (MO2sves) and the CO2

metabolic production rate (MCO2sves) respectively.

A higher MO2sves (as shown in Figure 5.5) will lead to a deeper desaturation during

an apnea. A higher MCO2sves will lead to a faster increase of the PaCO2. This conﬁrm the important role of the metabolism in the acceleration of the desaturation during apneas as stated by the literature [SW09; LJH12].

— Chemoreflex parameters (gains: Gc,A, Gp,A; time constants: τc,A and τp,A): During an obstructive apnea, the respiratory flow is completely stopped, so the chemoreflex cannot influence the respiration. But after the apnea, due to the hy-

poxia and the increase of PaCO2 generated by the event, the chemoreﬂex will try

to restore the O2/ CO2 changing the amplitude of the respiration. The behavior of this compensation depends of the chemoreﬂex gains for the amplitude of the

respiratory muscles (Gc,A and Gp,A) and the time constants (τc,A and τp,A) of the

central and peripheral chemoreflex. A higher amplitude of Gc,A or Gp,A produces a substantial recovery of the partial pressure of gases to their baseline value (as shown in Figure 5.6). In cases when the gain amplitude is too high, an overcompensation phenomenon can be observed after the apnea, with a first important increase of lung volume followed by a compensating reduction of lung volume that can generate a subsequent apnea. The time constants have a similar influence in the compensation after an apnea event, but instead of changing the amplitude of the respiration, they define how fast the chemoreflex will react after the apnea. A small time constant ac- celerates the reaction of the chemoreflex and it can generate overcompensations with subsequent apneas. This confirm the increased chemoreflex sensitivity in obstructive sleep apneas [Tro+13; Nar+99]. — Respiratory mechanics parameters (unstressed volume of air in the alveoli

VuA and lung compliance CA): changes of the unstressed volume of the lungs (VuA) leads to signiﬁcant changes of the functional residual capacity (FRC). The FRC is the reserve of air that is going to be used during an apnea and a higher reserve

means a smaller desaturation. Similarly, the lung compliance CA is a measure of the lung’s ability to stretch and expand. A high compliance increases the FRC [HS19]. These results corroborate the important role of therapies like CPAP and expiratory

163 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea

positive airway pressure as a treatment for apneas in adults because one of the eﬀect of these therapies is to increase functional residual capacity [ACK89] [Hei+08].

During apnea there is no parameter that stands out in the sensitivity analysis of the heart rate and this is due to the fact that the baseline of the heart rate does not change strongly in the model simulations during the apnea. Moreover, there is not relevant RSA during the apnea event because of the lack of changes in lung volume during the apnea. After the apnea event, the most important parameter for the HR is a sigmoid parameter controlling how strong the sympathetic regulation mechanisms of the systemic resistance

(λRout) is going to be. After the apnea, there are important changes of thoracic pressure that will affect the intrathoracic compartments in the CVS, affecting the arterial pressure and stimulating the baroreceptors. Depending of the value of (λRout), a stronger or a weaker sympathetic response is going to be observed in the HR. With the results of the sensitivity analysis and the physiology of obstructive sleep apneas, a reduce subgroup of parameters will be selected for a patient-specific identification. This reduction in the number of parameters can be realistically fine-tuned to a specific patient and a specific apnea event. This methodology can be used to simplify the problem of fitting any other pathology that involves the mechanisms represented in this integrated model. These parameters could be potentially useful for monitoring therapy.

5.2.4 Conclusion

The sensitivity analysis highlighted the most important parameters of the adult cardio- respiratory model during an OSA event. Four groups of parameters appeared as relevant

:i) the fraction of oxygen in inspired air (FIO2), ii) metabolic rates; iii) chemoreflex parameters: chemoreflex gains and time constants and iv) respiratory mechanics parameters. These results are consistent with several factors highlighted in the literature on the dynamics of sleep apnea and existing treatments. This method helps to reduce the number of parameters used for an eventual patient-specific identification (Section 5.3), reducing the eventual compute power and time to use. This kind of analysis could be performed with any other condition under study to understand the most important mechanisms involved and to reduce the amount parameters to fit the model to experimental data.

164 5.3. Parameter identiﬁcation.

5.3 Parameter identiﬁcation.

5.3.1 Methodology

Objective functions

The objective function g(Osim(P),Oobs) is deﬁned to minimize the error between the simulated and experimental SaO2 in a time support going from 10 seconds before the beginning of the apnea and 60 seconds after the end of the apnea. The function to be minimized is:

g(Osim(P),Oobs) = ω1 ∗ (d + r) + ω2 ∗ Min + ω3 ∗ MinLoc (5.3) with: Tmin 1 X sim exp d = SaO2 (te) − SaO2 (te) (5.4) T f min te=0

Tend 1 X sim exp r = SaO2 (te) − SaO2 (te) (5.5) T − T f end min te=Tmin

sim exp Min = min(SaO2 ) − min(SaO2f ) (5.6)

sim exp MinLoc = SaO2 (Tmin) − SaO2f (Tmin) (5.7) where te corresponds to the time elapsed since the onset of the identiﬁcation period, sim exp SaO2 and SaO2f are, respectively, the simulated and experimental SaO2, Tmin is the exp time associated with the minimum value of SaO2f and Tend is the end of the identiﬁ- cation period. d accounts for the desaturation segment of the experimental SaO2 and r represents the recovery of the SaO2 due to the resuming of breathing. Min accounts for the minimum value of the desaturation and MinLoc for the location of the minimum. ωi correspond to the weight given to each component of the objective function, with ω1=0.7;

ω2=0.4; ω1=0.4 giving more importance to d and r.

Parameters used for the identiﬁcation

The subgroup of parameters used for the identiﬁcations were selected from the results of the Morris analysis, the local sensitivity analysis and the physiology. The parameters

165 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea

chosen are directly related to changes of SaO2 and other directly related gas exchange variables like PaO2 and PaCO2 during and after an obstructive apnea. 9 parameters were selected:

— The metabolic rates (MO2Sves and MCO2Sves).

— The gains (Gc,A and Gp,A) and the time constants (τc,A and τp,A) of the central and peripheral chemoreﬂex for the control of the amplitude of the activity of respiratory muscles.

— The unstressed volume of the alveoli (VuA).

— The gas transport constant between the lung and the systemic tissues τLT was included to identify the time that it takes for the blood to travel from the lung to the pulse oximeter sensor site [Poe10]. — The parameter ApneaBegin was deﬁned to indicate at what stage of the respiration cycle, the obstructive apnea was triggered, determining the reserve of air in the lungs during the apnea event. ApneaBegin = 0 corresponds to an obstructive event that starts at the beginning of inspiration. ApneaBegin = 1 corresponds to an obstructive event that starts at the end of expiration.

Even if FIO2 appeared as one of the most relevant parameters in the sensitivity analysis, it was not included in the parameter identification, because the patients of the database were not under oxygenation, hence the FIO2 was assumed as the nature air oxygen percentage (21%). The lower and upper bounds used per parameter during the identification were fixed manually. ±80% for VuA, τLT , MO2Sves and MCO2Sves. Larger bounds were taken for the chemoreflex gains due to the lack of information about the magnitude of these parameters in the literature with +300%/-90% for Gc,A and Gp,A. The bounds of the time constant of the chemoreflex were narrower to ±30% for τc,A and τp,A. Larger bounds were taken for the chemoreflex gains due to the lack of information about the magnitude of these parameters in the literature with +300%/-90% for Gc,A and

Gp,A. The bounds of the time constant of the chemoreﬂex were narrower to ±30% for τc,A and τp,A.

Identiﬁcation using the adult cardio-respiratory model

For the identiﬁcation, the adult cardio-respiratory model described in Chapter 4 is used. The only information taken from the experimental data for the simulation is the

166 5.3. Parameter identiﬁcation. duration of the apnea event to ﬁt and baseline simulation reference of the heart-rate

(HR0) determined with the mean of the patient heart rate during the apnea event. The

OSA is produced by increasing the resistance of the upper airways Ru of the respiratory −1 model to 100000 cmH2O · s · l (as presented in Section 4.2.2) during the duration of the experimental apnea event.

Clustering

In order to identify clusters of phenotypically-similar responses to apnea, the 107 identified parameter sets were clustered using K-means method [Llo82]. The objective is to observe if there are similarities within the identified sets and to analyze if each cluster corresponds to a specific dynamic to OSA. The basic idea of K-means method is: given a d dataset of n observations (X1,X2,...,Xn) such that each observation Xi ∈ R , k-means clustering aims to partition the n observations into k sets S = {S1, S2, ..., Sk} such that the distance from the observations to the assigned cluster centers is minimized. In our context, observations Xi = Pi, representing the identified model parameter set i, with d being the number of parameters per set. Algorithm 2 shows the procedure of K-means clustering.

Algorithm 2 K-means clustering algorithm. Adapted from [Man+11] d Require: k: number of clusters; Data set of n points (X1,X2,...,Xn) ∈ R Ensure: k subset clusters S = {S1, S2, ..., Sk} 1: Initialization: Form the initial K clusters, with K initial data point centers 2: repeat 3: for each point Xi do 4: find the nearest center and assign xi to the corresponding cluster. 5: end for 6: update clusters by calculating new centers using mean of the members. 7: until stop-iteration criteria satisfied (number of iterations, difference on the value of the distortion function) 8: return clustering result

The number of cluster k is obtained using the Gap Statistic Method deﬁned by [TWH01]. In order to visualize the results of the clustering, a principal component analysis (PCA) was performed in the parameter space for unsupervised dimension reduction. The two principal components were used to visualize the ﬁrst factorial plane, with a color assigned per cluster.

167 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea

Table 5.2 – Median and Interquartile range of the RMSE and rRMSE between the iden- tiﬁed model simulation of SaO2 and the experimental SaO2 per patient and for the total database. Patient Apneas Median RMSE (%) Iqr RMSE (%) Median rRMSE Iqr rRMSE 2 32 1.216 0.398 0.013 0.004 3 1 1.422 0 0.015 0 5 10 1.350 0.529 0.015 0.006 6 2 2.133 0.910 0.023 0.011 8 2 2.078 1.233 0.023 0.014 11 29 1.736 0.917 0.018 0.010 12 7 1.299 0.357 0.015 0.004 15 14 2.010 0.511 0.021 0.005 16 6 1.349 0.543 0.015 0.007 17 4 0.819 0.238 0.009 0.003 Total 107 1.431 0.6597 0.0153 0.0073

5.3.2 Results

Parameter identiﬁcation of the database

107 obstructive apneas distributed among 10 patients (Section 5.1) were identified using the cardio-respiratory model. Table 5.2 presents the median and interquartile range of the root mean square error (RMSE) and the relative root mean square error (rRMSE) between the identified model simulations of SaO2 and the experimental SaO2 of OSA events per patient and for the total database. The identification methodology achieved a median RMSE of 1.431% for all the identifications for the PASITHEA database. The simulated SaO2 reproduces closely the experimental SaO2 in all the model identifications. This is the first successful parameter identification of a cardio-respiratory model in the context of OSA.

SaO2 dynamics reproduced by the identiﬁed model

The model was able to reproduce three diﬀerent dynamics observed in the experimental OSA events (Section 5.1). In the following list, a description of these three dynamics will be presented, accompanied by ﬁgures comparing the SaO2 and the respiration generated with the model and the experimental data for every case:

1. Example 1: OSA with normal recovery. Figure 5.7 shows a typical response to OSA,

with a signiﬁcant decrease of the SaO2 during apnea and an increase of ventilatory

168 5.3. Parameter identiﬁcation.

eﬀort after the event [Lev08]. The RMSE between the experimental SaO2 and the

simulated SaO2 for this example is 0.7302%. 2. Example 2: OSA followed by a hypopnea. In Figure 5.8, after the OSA event, there is an increase of the ventilatory eﬀort followed by a signiﬁcant reduction of the

ventilation resulting in a hypopnea. The RMSE between the experimental SaO2

and the simulated SaO2 for this example is 1.603%. 3. Example 3: OSA followed by another apnea. Figure 5.9 shows an apnea event followed by a fast increase of the ventilation and a subsequent apnea. The RMSE for this example is 1.26%.

Patient 17 3000 Uncalibrated nasal pressure 5 Simulated lung volume 2000 4

3 (l)

1000 L V

NP (AU) 2 0 1 Start End -1000 0 0 10 20 30 40 50 60 70 80 90

100

94 RMSE=0.7302% SaO2 % rRMSE=0.0076 Filtered SaO2 92 Raw SaO2 Start End Simulated SaO2 90 0 10 20 30 40 50 60 70 80 90 s

Figure 5.7 – Example 1. OSA with normal recovery. Comparison between apnea event of PASITHEA database and simulation with identiﬁed parameters using the adult model. Top to bottom: Experimental uncalibrated nasal pressure (Blue) and simulated lung volume (Red); Filtered experimental SaO2 (Blue), raw experimental SaO2 (Dark blue) and simulated SaO2 (Red).

In these three cases, it is remarkable that even if the experimental nasal pressure was not included in the objective function of the identiﬁcation algorithm, the simulated lung volume was able to reproduce correctly the observed respiratory activity, in particular

169 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea

Patient 2 4000 4 Uncalibrated nasal pressure Simulated lung volume 3000 3.5 (l)

2000 3 L V NP (AU)

1000 2.5

Start End 0 2 0 20 40 60 80 100 120

100

RMSE=1.603% SaO2 % 90 rRMSE=0.017 Filtered SaO2 Raw SaO2 Start End Simulated SaO2 85 0 20 40 60 80 100 120 s

Figure 5.8 – Example 2. OSA followed by a hypopnea. Comparison between apnea event of PASITHEA database and simulation with identiﬁed parameters using the model. Top to bottom: Experimental uncalibrated nasal pressure (Blue) and simulated lung volume (Red); Filtered experimental SaO2 (Blue), raw experimental SaO2 (Dark blue) and simulated SaO2 (Red). after the apnea event. However, in other cases the model desaturation is closed to the experimental SaO2 but the simulated respiration did not match the experimental nasal pressure.

Characterization of the cardio-respiratory response to OSA

Using the results of the parameter identiﬁcation and the insights given by the model, a characterization of the cardio-respiratory response to OSA was proposed. In order to do this, a K-mean clustering method was used on the 107 sets of parameters identiﬁed and three clusters were found (Figure 5.10). The set of parameter of the barycenters of the three clusters were extracted as shown by Figure 5.11. Barycenter 1 has high gains and short time constants for the chemore-

ﬂex, long lung-tissue gas transport time (τLT ), low metabolic rates and regular lung un-

170 5.3. Parameter identiﬁcation.

Patient 12 3.5 Uncalibrated nasal pressure 1500 Simulated lung volume

1000 (l) L V NP(AU) 2.5 500 Start End 2 0 20 40 60 80 100 120

100

85 RMSE=1.26% SaO2 % rRMSE=0.0139 Filtered SaO2 80 Raw SaO2 Start End Simulated SaO2 75 0 20 40 60 80 100 120 Time (s)

Figure 5.9 – Example 3. OSA followed by another apnea. Comparison between apnea event of PASITHEA database and simulation with identiﬁed parameters using the adult model. Top to bottom: Experimental uncalibrated nasal pressure (Blue) and simulated lung volume (Red); Filtered experimental SaO2 (Blue), raw experimental SaO2 (Dark blue) and simulated SaO2 (Red).

stressed volume. Barycenter 2 has similar parameters for the chemoreﬂex and τLT , but high metabolic rates and high lung unstressed volume. Parameters of Barycenter 3 are closed to the reference parameters used by the model for a normal adult. Taking advantage of the explicability provided by the model, 3 simulations were performed with each of the barycenters (Figure 5.12) to understand the physiology behind of each cluster response to OSA and thus be able to characterize them. The physiological explanation of each simulation is listed below:

1. Cluster 1: The simulated OSA generates a central apnea and ventilation instability similar to Cheynes-Stokes respiration. Previous to the OSA, the respiration tidal vol-

ume (VT) is almost half of the normal value (0.5L [GH10]) resulting in a low PaO2.

During the OSA event, the PaO2 and the SaO2 have a signiﬁcant drop and the

PaCO2 has a small increase. The deep hypoxia stimulates the peripheral chemore-

171 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea

Figure 5.10 – K-mean clustering on the 107 set of parameters identiﬁed for the adult cardio-respiratory model using the PASITHEA database. Three clusters were found: 1 (red), 2 (green), 3 (blue).

V M M G G Apnea uA O2sves CO2sves c,A c,A p,A p,A LT Begin 2 6 6 0 0 100 0.6 100 20 4 4 -2 -2 0.4 -1 -1 O/v 2 l

1 s s 50 s O/mmHg 50 10 2 ml.s 2 ml.s 2 -4 -4 0.2 cmH

0 0 0 cmH -6 0 -6 0 0 0 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 Clusters Clusters Clusters Clusters Clusters Clusters Clusters Clusters Clusters

Figure 5.11 – Comparison of the set of parameters of each one of the barycenters of the clusters obtained with the K-means.

ceptors and due to the important gain and the short time constants of both the central and peripheral chemoreceptors the ventilation starts to oscillate.

2. Cluster 2: The simulated OSA generates hyperventilation followed by ventilation instability similar to a Cheynes-Stokes respiration. In this case, previous the OSA

the functional residual capacity (FRC) and the VT have amplitudes slightly above

normal values, resulting in a normal PaO2. During the OSA, the PaO2 and the

SaO2 drop, and the PaCO2 has a signiﬁcant increase due to the high metabolic

CO2 production rate. This hypercapnia stimulates the central and the peripheral

chemoreceptors, this last ones being stimulated by the change of PaO2 too. Due to the big gains and the small time constants of the central and peripheral chemoreﬂex,

there is hyperventilation after the OSA, producing a signiﬁcant drop of the PaCO2,

172 5.3. Parameter identiﬁcation.

SaO PaO PaCO Lung volume 2 2 2 6 100

5 Cluster 1 100 50

4 95 L

% 80 40 3 mmHg mmHg

2 S E 60 30 S E 90 S E S E 1 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 SaO PaO PaCO Lung volume 2 2 2 6 100

5 Cluster 2 100 50

4 95 L

% 80 40 3 mmHg mmHg

2 S E 60 30 S E 90 S E S E 1 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 SaO PaO PaCO Lung volume 2 2 2 6 100

5 Cluster 3 100 50

4 95 L

% 80 40 3 mmHg mmHg

2 S E 60 30 S E 90 S E S E 1 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 Time (s) Time (s) Time (s) Time (s)

Figure 5.12 – Simulations of an OSA using the adult cardio-respiratory model with the parameters of the barycenters of the three cluster found with the K-means clustering method. Each row presents the simulation of the lung volume, the SaO2, the arterial partial pressure of oxygen (PaO2), the arterial partial pressure of CO2 and the heart rate per each of three clusters.

starting an oscillatory respiration. 3. Cluster 3: The simulated OSA has the typical response to obstructive apnea. Before

the event, the lung volume is normal. During the obstruction the SaO2 and PaO2

decrease and the PaCO2 increase. After the apnea there is an increase of ventilatory eﬀort and the respiration stabilizes.

OSA dynamics for patients during a night of sleep

Each one of the 107 set of parameters identified were assigned with their correspondent cluster. We wanted to observe the evolution of these clusters during the night of sleep of each patient to determine if there was a pattern or changes during different stages of the night. In Patients 3, 6, 8, 12, 16 and 17, it was difficult to perform this analysis because they showed 7 or less selected apneas during a night of sleep ( 8 hours of sleep) and they are scattered during the whole night.

173 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea

Table 5.3 – Apneas per Cluster for each Patient. Apneas per Cluster Patient Cluster 1 Cluster 2 Cluster 3 2 14 8 10 5 8 0 2 11 9 7 13 15 5 9 0

The evolution of the clusters during the night of sleep and the number of apneas per cluster for Patients 2, 5, 11 and 15 are illustrated in Figure 5.13 and Table 5.3 respectively. They can be analyzed as follows:

1. Patient 5: The set of parameters identified in most of the apneas belong to Cluster 1. 70% of these events are within 10 minutes window. Cluster 1 is related to ventilation instability, high loop gain and periodic respiration. In the experimental data of this patient, 90% of the selected apneas are followed by either an apnea or a hypopnea. 2. Patient 15: 64% of the set of parameters identified belong to Cluster 2 and they are scattered during the whole night. The other 36% are part of Cluster 1. Both of these clusters are related to ventilatory instability, high loop gain and periodic respiration. In the experimental data of this patient, 93% of the selected OSAs are followed by either an apnea or a hypopnea. 3. Patient 2 and Patient 11: These patients show the largest number of events. For both cases, there is not a clear, distinct pattern during the night. Even if the chemoreflex and the metabolism can change during the night, these changes cannot be as fast or abrupt as proposed by the model. In many cases there are contiguous OSAs that belong to different clusters, meaning that the identified parameters are related to different dynamics.

Hence, there is a pattern of high loop gain and ventilatory instability in patients 15 and 5. For patient 2 and 11 there is no clear distribution of the clusters during the night.

5.3.3 Discussion

Patient-speciﬁc parameter identiﬁcation on a PSG database.

To our knowledge, this work proposes the first successful patient-specific and event- specific parameter identification of the cardio-respiratory response to obstructive sleep

174 5.3. Parameter identiﬁcation.

Clusters vs. Time 3 2 1 Patient 2 0 2 2.5 3 3.5 4 4.5 3 Cluster 1 2 Cluster 2 1 Cluster 3 Patient 5 0 3 3.5 4 4.5 5 5.5 6 3

Patient 11 0 3.5 4 4.5 5 3 2 1

Patient 15 0 3.5 4 4.5 5 5.5 6 Time(Hours)

Figure 5.13 – Comparison of the set of parameters of each one of the barycenters of the clusters obtained with the K-means.

apneas on a polysomnography database using an integrated adult model. The identiﬁed model was capable to reproduce the desaturation and in several cases the nasal pressure of 107 obstructive apneas only using the duration of the apnea as an input parameter.

In several opportunities the model not only reproduced the selected OSA, but also the following respiratory events (apnea, hypopnea, normal recovery). This result is directly related with the objective function used in the identification. Indeed, since a 60 second window after the end of the obstructive apnea is included in the objective function, the identification method takes into consideration the eventual desaturation produced by consecutive events. This demonstrates the potential of the identification approach but also how the shape of the SaO2 during and after an apnea event can be useful to estimate the ventilatory dynamics of a patient.

The parameter identification of integrated cardio-respiratory models is usually considered as challenging task [Alb+15]. Results of the sensitivity analysis provide valuable information about the most influential parameters on SaO2, which could be selected for the identification (9 parameters out of 151). This methodology can be used to reproduce other conditions.

175 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea

Interpretation of patient-speciﬁc cardio-respiratory event

In Section 5.3.2, a characterization of the cardio-respiratory response to OSA was proposed using the set of parameters identified in the database. This resulted in three different clusters with different physiological responses to apnea. A phenotype can be assigned to each of these clusters thanks to the observability given by the cardio-respiratory model:

1. Cluster 1: The response to OSA corresponds to a subject with ventilatory instability. The disturbance produced by the OSA event produces a periodic breathing

(PB)/Cheyne-Stokes respiration (CSR) due to low PaO2. This behavior resembles to Cheyne-Stokes respiration due to hypoxemia in congestive heart failure through peripheral chemoreceptor stimulation [Nau98]. This cluster is also associated to long circulation delay. 2. Cluster 2: The response to OSA is related to ventilatory instability. The disturbance produced by the OSA event generated a hyperventilation that dropped the value of

PaCO2 signiﬁcantly producing a PB and CSB. This dynamic is the common patho-

physiology of all forms of periodic breathing, where hyperventilation drops PaCO2 levels below the apneic threshold triggering a central apnea. Once the chemorecep-

tors sense the rise in the PaCO2 level above the apnea threshold, hyperventilation

recurs driving the PaCO2 level below the apnea threshold once again [BP92; Sol+00; JC98]. Similarly, increased central hypercapnic ventilatory responsiveness has been reported to occur in Cheyne-Stokes respiration [Wel+11]. Moreover, this cluster presents long circulation delay, a characteristic that has been reported as relevant to produce periodic breathing [CL06] 3. Cluster 3: It describes a typical response to OSA, with moderate chemoreﬂex gains and metabolic rates [Lev08].

The dynamics observed in Cluster 1 and 2 are similar to the ones observed in patients with Cheyne-Stokes respiration and periodic breathing. The high gains and short time constants for the chemoreﬂex presented in this two clusters have a direct relation with high controller loop gain and chemoreﬂex sensitivity. These two aspects are highly relevant for periodic breathing and ventilation instability [Kho00; Fra+00]. Moreover, the metabolic rates characterize each of the clusters, modifying the stable value of PaO2 and PaCO2 and producing conditions related to the pathogenesis of periodic breathing like hypocapnia

[Nau+92; JC98]. These parameters can also produce more abrupt changes of PaO2 and

176 5.3. Parameter identiﬁcation.

PaCO2 during the OSA where the metabolism plays the main role. Metabolic rates are directly related to the plant gain of the ventilatory system, that is an important factor for sleep apnea [Nau10].

Identiﬁcation limitations

Most of the identification limitations are linked to the model limitations mentioned in Section 4.3. Events like sighs and arousals cannot be simulated by the model and they can have an impact in the dynamic of sleep apneas. Sighs are related to posterior apneas, hypoventilation, or slowing the respiratory frequency and they can produce up to 2% change of the SaO2 [PWK83]. The autonomic responses linked to sleep arousals are an important cause of the resume of breathing after the apnea [Spa+05; Tay+16]. Similarly, it was mentioned that the model is able to simulate central apneas after the forced OSA, but it is not capable to produce consecutive obstructive or mixed apneas. The mechanisms to spontaneously generate an increase in upper airway resistance until it collapses and produces an obstructive apnea have not been integrated into the model. Moreover, even if in the selection criteria for the experimental apneas used for the identification there was a minimum of 20 seconds separation between the end of the previous apnea and the beginning of the selected apnea, it was observed that the dynamics of previous events were still affecting the initial conditions at the beginning of the selected obstructive apnea. This affects the response of the chemoreflex, the lung volume and the shape of the desaturation. In the other hand, the objective function used for the identification was mainly based on the shape of SaO2. In several cases, the identified model was able to reproduce the experimental SaO2 and the uncalibrated nasal pressure. But in other cases, the simulated SaO2 resembles the experimental SaO2 but the respiration is different. In future works, the respiration will be taken into account in the objective function.

5.3.4 Conclusion

This section presented the ﬁrst successful parameter identiﬁcation of a cardio-respiratory model in the context of OSA. The cardio-respiratory model was able to reproduce the experimental SaO2 during and after an apnea event in a database of 107 obstructive apneas. In some cases, the model reproduced the experimental nasal pressure too. Obstructive apneas with a normal recovery or with consecutive apneas/hypopneas were simulated by

177 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea the model. From all the identifications, three clusters were found using K-means, describing three different dynamics related to sleep apnea and periodic breathing. The first two clusters were related to ventilation instability due to high loop gain and periodic/Cheynes- Stokes breathing. The third cluster represented the typical response to an OSA event. The predominance of any of these clusters in a patient can help in the decision making between different treatments like CPAP or oxygen therapy. The methodology used for the identification can be applied to study other conditions that involve the same physiology described in our cardio-respiratory model.

References

[ACK89] N C Abbey, K R Cooper, and J A Kwentus, « Beneﬁt of nasal CPAP in obstructive sleep apnea is due to positive pharyngeal pressure. », in: Sleep 12.5 (Oct. 1989), pp. 420–2. [Alb+15] Antonio Albanese et al., « An integrated mathematical model of the human cardiopulmonary system: Model development », in: American Journal of Physiology - Heart and Circulatory Physiology (2015), ajpheart.00230.2014. [BP92] T D Bradley and E A Phillipson, « Central sleep apnea. », in: Clinics in chest medicine 13.3 (Sept. 1992), pp. 493–505. [CL06] Neil S. Cherniack and Guy S. Longobardo, « Mathematical models of periodic breathing and their usefulness in understanding cardiovascular and respiratory disorders », in: Experimental Physiology 91.2 (Mar. 2006), pp. 295– 305. [D P18] D. Perez, « Optimal Control of Non-Invasive Neuromodulation for the Treat- ment of Sleep Apnea Syndromes », in: (2018). [Doy+19] Matthieu Doyen et al., « Robust, real-time generic detector based on a multi- feature probabilistic method », in: PLoS ONE 14.10 (2019). [Fra+00] Darrel P. Francis et al., « Quantitative general theory for periodic breathing in chronic heart failure and its clinical implications », in: Circulation 102.18 (Oct. 2000), pp. 2214–2221. [GH10] Arthur C. Guyton and John E. Hall, Textbook of medical physiology, 2010, p. 1091.

178 REFERENCES

[Got+14] Daniel J. Gottlieb et al., « CPAP versus Oxygen in Obstructive Sleep Ap- nea », in: New England Journal of Medicine 370.24 (2014), pp. 2276–2285. [Hei+08] Raphael Heinzer et al., « Effect of expiratory positive airway pressure on sleep disordered breathing », in: Sleep 31.3 (2008), pp. 429–432. [Her+16] A. I. Hernández et al., « PASITHEA: An Integrated Monitoring and Ther- apeutic System for Sleep Apnea Syndromes Based on Adaptive Kinesthetic Stimulation », in: Irbm 37.2 (2016), pp. 81–89. [Her+18] Alfredo I. Hernández et al., « Kinesthetic stimulation for obstructive sleep apnea syndrome: An "on-off" proof of concept trial », in: Scientific Reports 8.1 (2018), pp. 1–7. [HS19] Erin Hopkins and Sandeep Sharma, Physiology, Functional Residual Capac- ity, 2019. [JC98] S Javaheri and W S Corbett, « Association of low PaCO2 with central sleep apnea and ventricular arrhythmias in ambulatory patients with stable heart failure. », in: Annals of internal medicine 128.3 (Feb. 1998), pp. 204–7. [Kho00] Michael C.K. Khoo, « Determinants of ventilatory instability and variability », in: Respiration Physiology, vol. 122, 2-3, Elsevier, Sept. 2000, pp. 167– 182. [Lev08] Michael G. Levitzky, « Using the pathophysiology of obstructive sleep apnea to teach cardiopulmonary integration », in: American Journal of Physiology - Advances in Physiology Education 32.3 (2008), pp. 196–202. [LJH12] Ivan T. Ling, Alan L. James, and David R. Hillman, « Interrelationships between Body Mass, Oxygen Desaturation, and Apnea-Hypopnea Indices in a Sleep Clinic Population », in: Sleep 35.1 (Jan. 2012), pp. 89–96. [Llo82] Stuart P. Lloyd, « Least Squares Quantization in PCM », in: IEEE Trans- actions on Information Theory 28.2 (1982), pp. 129–137. [Man+11] Shie Mannor et al., « K-Means Clustering », in: Encyclopedia of Machine Learning, Springer US, 2011, pp. 563–564. [Meh+13] Vanita Mehta et al., Obstructive sleep apnea and oxygen therapy: A systematic review of the literature and meta-analysis, 2013.

179 Chapter 5 – Model based analysis of cardio-respiratory interactions during sleep apnea

[Mor91a] M D Morris, « Factorial plans for preliminary computational experiments », in: Technometrics 33.2 (1991), pp. 161–174. [Mor91b] M D Morris, « Factorial sampling plans for preliminary computational experiments », in: Technometrics 33.2 (May 1991), pp. 161–174. [Nar+99] Krzysztof Narkiewicz et al., « Human Obesity Is Characterized by a Selec- tive Potentiation of Central Chemoreflex Sensitivity », in: Hypertension 33.5 (May 1999), pp. 1153–1158. [Nau+92] Matthew Naughton et al., « Role of Hyperventilation in the Pathogenesis of Central Sleep Apneas in Patients with Congestive Heart Failure », in: http://dx.doi.org/10.1164/ajrccm/148.2.330 (Oct. 1992). [Nau10] Matthew T. Naughton, Loop gain in apnea gaining: Control or controlling the gain?, Jan. 2010. [Nau98] M. T. Naughton, Pathophysiology and treatment of Cheyne-Stokes respiration, June 1998. [Pép+16] J L Pépin et al., « Fixed-pressure CPAP versus auto-adjusting CPAP: comparison of efficacy on blood pressure in obstructive sleep apnoea, a randomised clinical trial », in: Thorax 71.8 (Aug. 2016), pp. 726–733. [Poe10] Christian F. Poets, Apnea of prematurity: What can observational studies tell us about pathophysiology?, Aug. 2010. [PWK83] R. Perez Padilla, P. West, and M. H. Kryger, « Sighs during sleep in adult humans », in: Sleep (1983). [Sol+00] Peter Solin et al., « Peripheral and central ventilatory responses in central sleep apnea with and without congestive heart failure », in: American Jour- nal of Respiratory and Critical Care Medicine 162.6 (Dec. 2000), pp. 2194– 2200. [Spa+05] Jonas Spaak et al., « Muscle sympathetic nerve activity during wakefulness in heart failure patients with and without sleep apnea », in: Hypertension 46.6 (Dec. 2005), pp. 1327–1332. [SW09] R. Sirian and Jonathan Wills, « Physiology of apnoea and the benefits of preoxygenation », in: Continuing Education in Anaesthesia, Critical Care and Pain 9.4 (2009), pp. 105–108.

180 REFERENCES

[Tay+16] Keri S Taylor et al., « Arousal From Sleep and Sympathetic Excitation Dur- ing Wakefulness Obstructive Sleep Apnea », in: Hypertension 68 (2016), pp. 1467–1474. [Tro+13] Ivani C. Trombetta et al., « Obstructive Sleep Apnea is Associated with Increased Chemoreﬂex Sensitivity in Patients with Metabolic Syndrome », in: Sleep 36.1 (Jan. 2013), pp. 41–49. [TWH01] Robert Tibshirani, Guenther Walther, and Trevor Hastie, « Estimating the number of clusters in a data set via the gap statistic », in: Journal of the Royal Statistical Society. Series B: Statistical Methodology 63.2 (Jan. 2001), pp. 411–423. [Wel+11] Andrew Wellman et al., « A method for measuring and modeling the physiological traits causing obstructive sleep apnea », in: Journal of Applied Phys- iology 110.6 (June 2011), pp. 1627–1637.

181

Chapter 6

NEWBORNCARDIO-RESPIRATORY MODEL

In a similar fashion as disclosed in the previous chapter, we hypothesize that a model- based approach may be useful for the analysis and improved understanding of the acute responses of apnea-bradycardia events in newborns. However, the development of a cardio- respiratory model adapted to the newborn physiology is a particularly challenging task. Firstly, both the anatomy and the physiology of newborns are different from adults, leading to a need of significant modifications on both the structure and the parameters used on adult models. Secondly, the anatomy and the physiology of newborns changes substantially though time, especially for preterm newborns. This makes it is impossible to propose a single cardio-respiratory model for all stages of prematurity and newborn evolution. Third, some mechanisms are poorly known, in particular those related to the maturation process. Finally, there is a lack of high-quality, annotated data to verify and validate these models.

In this context, this chapter covers two main objectives. The first objective is to present the first integrated cardio-respiratory models adapted to term and preterm newborns. Two models are proposed: i) a model of a newborn at term (Mterm), with a gestational age of 40 weeks and a post-menstrual age of 41 weeks and ii) a model of a preterm newborn (M28) with a gestational age of 28 weeks and a post-menstrual age of 29 weeks. The second objective is to perform the first complete parametric sensitivity analysis of an integrated cardio- respiratory model of newborns in the context of apnea of prematurity/apnea-bradycardias. To our knowledge, none of these objectives have been addressed in the literature.

183 Chapter 6 – Newborn cardio-respiratory model

6.1 Methodology

6.1.1 Integrated cardio-respiratory model adapted to the newborn physiology

In this section, the adaptations performed in the adult model to obtain the newborn models (Mterm and M28) are explained: structural and parameter adaptation.

Structural adaptation

Three structural adaptations were performed on the adult cardio-respiratory model (Chapter 4) in order to develop a model adapted to the physiology of the newborn.

Models Mterm and M28 share the same structural modifications. 1. Because of the lack of references to adapt the transduction stage presented in the adult model, the structure of the peripheral chemoreflex was redefined (Figure 6.1). The new sub-model consists of two control loops, similar to the ones defined for the

central chemoreﬂex, one for the changes of PaO2 and another one for the changes

of PaCO2. The gains and time constants were adjusted manually.

2. An efferent pathway dependent of the peripheral chemoreceptors (upO2, see peripheral chemoreceptors in Figure 6.1) activity was added to the vagal pathway of the baroreflex (Figure 6.2). Previous research suggests that bradycardia occurs during apnea as a response to desaturation, through the activation of a peripheral chemoreceptor reflex [HBE86]. Moreover, [Lag+90] affirms that infants with recurrent prolonged apnea and bradycardia have an increased parasympathetic nervous activity. The adapted structure is similar to the pulmonary stretch receptors representation. 3. In the gas exchange, the method to convert partial pressures of gases to concentrations of gases was adjusted to follow the oxygen dissociation curves of newborns and preterms. The equations of [Rev+89] were used, fitting its parameters to the oxygen dissociation curves reported by [DRO71] for each newborns and preterms (Figure 6.3).

Parameter adaptation

Parameters from previous studies The parameters already deﬁned or measured in the literature for term and preterm infants can be found in the following list:

184 6.1. Methodology

Respiratory central pattern generator

5@AB? =>?

'(,* ∆5 5@AB 5678 ± !"#$.& @AB,( Σ 4 1 + - . . (,* Respiratory 5@D& Lung muscles mechanics 56784,9: '(,0 ∆=> => ( Σ Central 1 + -(,0. . Chemoreceptors ∆5@AB,234

E234 '2,*34 "#1.& 5684 ± ! 1 + -2,*34. .

5684,9; '2,034 ∆=>234 1 + -2,034. .

'2,*:34 "#1.& 56784 ± ! 1 + -2,*:34. . ∆5@AB,2:34

56784,9; '2,0:34 Peripheral 1 + - . . ∆=> Chemoreceptors 2,0:34 234

Figure 6.1 – Chemoreflex model for newborns. Pa, arterial partial pressure; D, delay; BR, breathing rhythm; Pmus, activity of respiratory muscles; Pmax, maximum amplitude of the activity of respiratory muscles; c, central chemoreflex; p, peripheral chemoreflex; A, amplitude; f, frequency.

Figure 6.2 – Diagram of the baroreﬂex system for newborns. From the arterial pressure registered at the intrathoracic systemic circulation, the baroreﬂex system regulates the heart rate, the systemic resistance, the venous unstressed volume and the ventricular elastance. K, gain; D, delay; τ , time constant; G, gain; VA, lung volume.

185 Chapter 6 – Newborn cardio-respiratory model

A B

Figure 6.3 – Oxygen dissociation curves of blood from (A) term infants and (B) preterm infants at diﬀerent postnatal ages; each curve represents the mean value of the infants studied in each age group. Curves extracted from [DRO71].

186 6.1. Methodology

— Parameters of the CVS of the Mterm model were adapted from [Sá +06].

— The delay of the vagal and sympathetic pathways of the baroreﬂex for the Mterm

and M28 model were extracted from [Jen+11; Dat+10; Urs98].

— The central and peripheral delays of the chemoreceptors for the Mterm and M28 model were obtained from [Jen+11; Dat+10; Urs98].

— The gas transport delays (τLT and τVL) for the Mterm and M28 model were extracted from [Rev+89; BT00].

— The lung compliance (CA) was determined using [TBG05] and [Chu+64] for the

Mterm and M28 model respectively.

— The Chest wall compliance (Ccw) was extracted from [TBG05] and [GB80] for the

Mterm and M28 model respectively.

The values of VuA and Vuc were deﬁned to produce an end-expiratory lung volume equal to normal function residual capacity [TBG05; Tho+04] following the method described by [Alb+15]. At the end of expiration, Vc is equal to the reference dead volume [Das+18;

Neu+15] and VuA and Vuc follows the equations below:

VuA = FRC − Vc + Ppl ∗ CA (6.1)

Vuc = Vc + Ppl ∗ Cc (6.2)

With the pleural pressure Ppl at end-expiration equal to -3.75 cmH2O for newborns and

-3 cmH2O for preterms. The value, units and description of every parameter of the each model are listed in Appendix B.

Scaling methods Scaling methods were applied to obtain a blood volume of 310 ml and 110 ml for a 3.5 kg term and 1 kg preterm infants, respectively [Sai+72].

1. Systemic peripheral vessels: As mentioned in Chapter 4, in order to integrate the metabolic gas exchange sub-model, a new compartment representing the systemic peripheral vessels is included in the systemic circulation following the structure of [Alb+15]. The proportion between the parameters of the systemic peripheral vessels and the systemic arteries of [Alb+15] model were used to determine the parameters

of this new compartment (elastance Esves, unstressed volume Vusves) in the Mterm model using the parameters of the systemic arteries of [Sá +06] model. This new

187 Chapter 6 – Newborn cardio-respiratory model

compartment added an unstressed volume to the CVS, increasing the total blood volume of the newborn at term from 310 ml to 354 ml.

2. Unstressed volumes and elastances: The unstressed volumes and elastances of every compartment of the CVS were adapted to obtain the reference total blood volume for newborns at term and preterms following the method presented by [Jen+11;

Dat+10]. A scaling factor Fvol was deﬁned:

VReference Fvol = (6.3) VUnscaled

with VReference as the reference total blood volume and VUnscaled as the total blood volume of the unscaled model. The unstressed volumes and the elastances of every

compartment were multiplied and divided respectively by Fvol. As mentioned in the previous point, the integration of the systemic peripheral vessels increased the total blood volume of the newborn at term model from 310 ml to 354 ml. To scale the 310ml model back to the reference total blood volume, a Fvol of 354ml was applied. To 110ml obtain the parameters of the M28 model, the Mterm was scaled using a Fvol = 310ml to obtain a total blood volume of 110 ml.

3. Vascular resistances: Following [Jen+11] and [Dat+10], the vascular resistances of

the Mterm model are multiplied by a scaling factor of 2 to obtain the vascular

resistances of the M28 model. This scaling factor is based on the proportions of the total systemic vascular resistance (SVR) and a total pulmonary vascular resistance (PVR) between newborns at term and preterms.

4. Blood volume in the tissue of the systemic peripheral vessels (VT,sves): VT,sves was

adapted from the adult model of [Alb+15] to newborns model. A scaling factor FBV was deﬁned: WReference FBV = (6.4) WUnscaled

where WReference is the reference weight of the patient of the target model and

WUnscaled is the weight of the patient of the unscaled model. The VT,sves is multiplied

by FBV . The adult model is deﬁned for a patient of 70 kg, the Mterm model for a

patient of 3.5 kg and the M28 model for a patient of 1 kg. Hence, the adult VT,sves 3.5kg 1kg is multiplied by a FBV = 70kg and a FBV = 70kg to obtain the parameters for the Mterm and M28 model respectively.

188 6.1. Methodology

Manual parameter adaptation

Gains of the baroreflex and the parameters of the efferent pathway dependent on the peripheral chemoreceptors (Uchem and Gchem) were adjusted manually to the baseline values of the afferents and the baseline value of the arterial intrathoracic pressure, but they have to be identified with experimental data. The gains and time constants of the chemoreflex were defined to ensure numerical stability of the simulations. They have to be confirmed with experimental data.

6.1.2 Sensitivity analysis

In order to find the most influential model parameters affecting the SaO2 and heart rate during an apnea event in newborns, a sensitivity analysis was performed following the approach proposed in Section 3.2.4. A Morris’s screening method [Mor91] was applied on 154 parameters of the model (Section B). Parameter ranges were selected from the nominal Mean values ±30%. The same output functions ∆X and ∆X defined for the sensitivity analysis in adults (Section 5.2.1) were used. In this case, a central apnea event was simulated instead of an obstructive apnea, because most of the apneas of newborn are central. This was achieved by reducing to 0 the activity of the respiratory muscles (Pmus=0) at the instant of the beggining of the simulated apnea event. The most important parameterw, obtained through the Morris analysis, were selected and local sensitivity analyses were performed in order to analyze the influence of these parameters on other relevant cardio-respiratory signals.

6.1.3 Experimental protocol and data

A database from the Center of Excellence for Research on Maternal and Child Health of the University of Sherbrooke was used to compare the model simulations to experimental data. The database was the result of a observational study entitled "Volume pulmonaire et apnées centrales du nouveau-né et du nourrisson" lead by the main researcher Jean Paul Praud with the collaboration of Marie St-Hilaire (MD PhD) et Nathalie Samson (PhD). The study was approved by an independent Ethics Committee (Comité d’éthique de la recherche du Centre Intégré Universitaire de Santé et des Services Sociaux de l’Estrie - Centre Hospitalier Universitaire de Sherbrooke, Projet 2012-379). All parents provided informed consent. For each newborn subject, the ECG, the electroencephalogram (EEG), the electrooculogram (EOG), the respiration (abdominal and thoracic) and the SaO2

189 Chapter 6 – Newborn cardio-respiratory model

Table 6.1 – Summarize of the characteristics of each patient of the database of Sherbrooke University at birth and at the recording and the amount of annotated apneas per patient. Gestational age Post-menstrual Weight at Weight at Patient Apneas (weeks) age (weeks) birth (g) recording (g) 1 55 34 34 (4/7) 2410 2215 2 102 30 (1/7) 37 (4/7) 1400 2275 3 7 33 (1/7) 33 (5/7) 1980 1820 4 20 27 (6/7) 38 (6/7) 990 2885 5 25 35 (2/7) 38 (3/7) 2460 2825 6 20 30 34 (4/7) 1260 1930 7 12 30 34 (4/7) 1210 1690 8 25 29 (2/7) 34 (2/7) 1480 2225 9 56 28 (3/7) 34 (5/7) 1100 1988 10 1 28 (3/7) 34 (5/7) 1090 1970 11 10 27 (5/7) 36 (1/7) 1230 2310 12 28 38 (3/7) 40 (5/7) 3015 3215 13 45 40 (1/7) 46 3700 4250 14 145 33 (6/7) 34 (6/7) 2250 2240 15 15 32 34 (5/7) 1150 1520 16 41 32 34 (5/7) 1590 1955 time series were acquired for 1 to 3 hours. The newborns had diﬀerent gestational ages and chronological ages at the time of the acquisition. The data are annotated, including apnea events (start, end and duration), sighs, desaturation and periodic breathing. Table 6.1 summarize the characteristics of each patient at birth, at the recording and the number of annotated apneas per patient.

6.2 Results

The Mterm and M28 models have 45 state variables and 154 parameters (Appendix B). Models were implemented using the M2SL simulation library [Her+09] as explained in Section 3.1.1. Model simulations were performed with a runtime of 1500 seconds and an

190 6.2. Results

Table 6.2 – Baseline values and simulated hemodynamics vital signs for newborn at term and preterm. Simulation Simulation Description Value (at term) Value (Preterm) (Mterm) (M28) Cardiac output (ml/min) 500±165 [PC55] 480 197±80[Sol+09] 160.97 Systemic systolic pressure (mmHg) 73±11 [Tan87] 77.25 56±7[Ken+09] 57.37 Systemic diastolic pressure (mmHg) 45±12 [Tan87] 43.97 33±5[Ken+09] 32.06 Pulmonary systolic pressure (mmHg) 38.5±5.8[Emm+64] 37.72 35±5 [Sep+94] 26.65 Pulmonary diastolic pressure (mmHg) 13±6[Emm+64] 10 N/A 10.3 Heart rate (bpm) 135±20 [Sá +06] 130 148±19[Sel+11] 141.57 integration step of ∆t = 0.001 seconds to assure that each submodel arrives at a stable state. Simulations of Mterm and M28 models were ﬁrst confronted to reference values from the literature and were compared qualitatively to clinical data of 2 patients.

6.2.1 Comparison between model simulations and physiology baseline values

Hemodynamics

Table 6.2 presents some baseline values of important clinical hemodynamic variables, reported from literature, in a newborn at term and preterm in resting conditions and it compares them to the simulation outputs of the Mterm and M28 models. Figure 6.4 shows simulations of pulsatile blood pressures and volumes within the cardiovascular system.

In the Mterm model, the simulated pressure of systemic intrathoracic arteries results in a systolic pressure of 77.25 mmHg and a diastolic pressure of 43.97 mmHg and the simulated pressure of pulmonary arteries goes from a systolic pressure of 37.72 mmHg and a diastolic pressure of 10 mmHg. In the M28, the simulated pressure of systemic intrathoracic arteries results in a systolic pressure of 57.37 mmHg and a diastolic pressure of 32.06 mmHg and the simulated pressure of pulmonary arteries goes from a systolic pressure of 26.65 mmHg and a diastolic pressure of 10.3 mmHg. The pulmonary systolic pressure is slightly above of the reference value found in the literature. No reference value of the pulmonary diastolic pressure in preterm newborns was found. The model is able to reproduce a typical PV loop, that reﬂects each phase of the cardiac cycle. The amplitude and duration of diﬀerent pressures in the cardio-vascular system are close to values reported in the literature.

191 Chapter 6 – Newborn cardio-respiratory model

M M term 28 80 80 LV LV RV RV 60 PA 60 PA PV PV AO AO SV SV 40 40

Pressure(mmHg) 20 20 Pressure(mmHg)

0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 Volume(ml) Time(s)

80 80 LV LV 70 70 RV RV 60 60

50 50

40 40

30 30 Pressure(mmHg) Pressure(mmHg) 20 20

10 10

0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 Volume(ml) Volume(ml)

Figure 6.4 – Simulated blood pressures and volumes using Mterm and M28 models. Left ventricle (LV), right ventricle (RV), pulmonary artery (PA), pulmonary vein (PV), aorta (AO), systemic vein (SV).

192 6.2. Results

Table 6.3 – Baseline values and simulated respiratory variables for newborns at term and preterm. Simulation Simulation Description Value (at term) Value (Preterm) (Mterm) (M28) Tidal Volume (ml/kg) 4.8[GM03] 4.3 6[Pan+00] 6 Respiratory rate (bpm) 50.9[GM03] 49 60[Don98] 60 FRC (ml/kg) 30[TBG05] 30.1 26.7[Tho+04] 26.7 Dead volume (l) 2.4[Das+18] 2.4 2.5[Neu+15] 2.5

Respiratory mechanics

Table 6.3 presents some baseline values of the respiratory mechanics measured in a newborns at term and preterm in resting conditions and it compares them to the simulation outputs of the Mterm and M28 models. Figure 6.5 shows the pressures, volumes and

ﬂow waveforms generated by the respiratory model within the Mterm and M28 models. These physiological signals are consistent with normal physiological behavior of a newborn at term and preterm in quiet breathing conditions. At the beginning of inspiration, alveolar pressure (PA) and the activity of the respiratory muscles (Pmus) are equals to the atmospheric pressure (Patm) and the lung volume is equal to the FRC. During inspiration,

Pmus decreases, reducing the thoracic pressure (Pthor) from its resting value of -2.7 mmHg to approximately -6.14 mmHg in the Mterm model and from -2.15 mmHg to -4.36 mmHg in the M28 model, decreasing PA below Patm allowing air to ﬂow into the respiratory system. At the end of inspiration, respiratory muscles start relaxing, allowing air to ﬂow outside of the lungs until Pmus go back to Patm.

Gas exchange and transport

Table 6.4 presents some mean baseline values of the gas exchange pressures and saturation measured in a newborn at term and a preterm in resting conditions and it compares them to the simulation outputs of the Mterm and M28 models. Figure 6.6 shows time proﬁles of partial pressures and the oxygen saturation for quiet breathing conditions. The PaO2 and PaCO2 dynamics are synchronous with the respiratory cycle. PaO2 oscillates between 73.8 mmHg to 74.74 mmHg for Mterm and between

78.87 mmHg and 79.04 in M28. PaCO2 goes from 36.23 mmHg to 38.41 mmHg for Mterm and from 32.81 mmHg to 36.8 mmHg for M28. PaO2 increases during inspiration and de-

193 Chapter 6 – Newborn cardio-respiratory model

M M term 28 0.14 0.035 [l]) 0.12 0.03 ao_c (V 0.1

Lung Volume 0.025 0 1 2 3 4 0 1 2 3 4

0.1 0.1

[l/s]) 0 0 ao_c Lung flow (Q -0.1 -0.1 0 1 2 3 4 0 1 2 3 4

-3 10-3 10 10 2.8

[l]) 2.6

c 9

(V 2.4

Dead Volume 8 2.2 0 1 2 3 4 0 1 2 3 4

-2 -2

-4 -4

-6 -6 [mmHg])

thor 0 1 2 3 4 0 1 2 3 4 (P Time(s) Time(s) Thoracic pressure

Figure 6.5 – Pressure, volume, and ﬂow waveforms generated by the respiratory model using the Mterm and M28 models.

Table 6.4 – Mean baseline values and mean simulated gas exchange variables for a newborn at term and a preterm. Simulation Simulation Variable Description Value (at term) Value (Preterm) (Mterm) (M28)

PaO2 Arterial PO2(mmHg) 77.5±7.5 [GM03] 74.3 77.5±2.5[GM03] 78.96

PaCO2 Arterial PCO2(mmHg) 40±5[BKR04] 37.48 34±2[GM03] 35.02

SaO2 Oxygen saturation (%) 96.5±3.5 [Poe98] 97.3 94.45±5.55 [Poe+91] 98.49

194 6.2. Results

creases during expiration. PaCO2 has the opposite behavior, decreasing during inspiration and increasing during expiration. Moreover, the partial pressures of gas in the arteries have other frequency components diﬀerent of the respiratory rhythm. This behavior is caused by the coupling of the pulmonary capillaries blood ﬂow from the CVS submodel in the gas exchange model.

M M term 28 0.13 0.035

0.12 [l]) 0.03 ao_c 0.11 (V

Lung volume 0.1 0.025 0 1 2 3 4 0 1 2 3 4

75 79.1

74.5 79

(mmHg) 78.9 2

73.5

PaO 78.8 0 1 2 3 4 0 1 2 3 4

39 38

(mmHg) 38 36 2

37 34 PaCO

36 32 0 1 2 3 4 0 1 2 3 4

97.4 98.52

98.5 (%) 2 97.3 98.48 SaO

97.2 98.46 0 1 2 3 4 0 1 2 3 4 Time(s) Time(s)

Figure 6.6 – Arterial partial pressure of O2 and CO2 (PaO2, PaCO2) and the arterial oxygen saturation (SaO2) simulated by the Mterm and M28 models.

195 Chapter 6 – Newborn cardio-respiratory model

6.2.2 Simulation of an apnea in newborns and qualitative comparison with experimental data

A central apnea event was simulated by reducing the activity of the respiratory muscles to zero (Pmus=0). In some cases, a sigh was observed before the apnea event. A sigh is deﬁned as an inspiration of at least 2 times greater than the usual tidal volume [WTR01]. Therefore, in the case of a sigh before the central apnea, an inspiration of ≥2 times the tidal volume was applied. Patients 3 and 4 from the database of Sherbrooke University were used to compare the model simulations to the experimental data. Patient 3 has a gestational age of 33 weeks and 1 day, a post-menstrual age of 33 weeks and 5 day and a weight of 1820 g at the time of recording. Patient 4 has a gestational age of 27 weeks and 6 days, a post-menstrual age of 38 weeks and 6 days and a weight of 2885 g at the time of recording. Patient 3 was compared to the M28 model and Patient 4 to the Mterm model. 4 parameters were adapted manually in the Mterm and the M28 model to obtain simulations similar to the experimental data: the metabolic oxygen consumption rate (MO2sves), the lung-tissue gas transport delay (τLT ), the baseline value of the heart rate (HR0) and the set-point value of PaO2 of the peripheral chemoreﬂex (P aO2,np).

Figure 6.7 shows the respiration, the SaO2 and the heart rate of Patient 3 and 4 during and apneic event and the qualitative comparison with M28 and Mterm model simulations respectively. In both cases the dynamics of the SaO2 time series are similar, with a gas transport delay of 13 seconds for Patient 3 and 11 seconds for Patient 4. The respiration rate of the experimental data and the simulations is similar. A sigh was simulated for Patient 4 before the central apnea. Simulated and experimental heart rates possess similar order of magnitude.

6.2.3 Sensitivity analysis

Figure 6.8 presents the result of the sensitivity analysis performed on Mterm and M28 models. During the apnea the parameters that aﬀects the most the desaturation in the

Mterm model are the metabolic oxygen consumption rate (MO2sves), the fraction of inspired oxygen (FIO2), the set-point value of PaO2 of the peripheral chemoreﬂex (P aO2,np) and the metabolic CO2 consumption rate (MCO2sves). In the M28 model, the most relevant parameters are FIO2, MO2sves, P aO2,np, and the unstressed volume of the lungs (VuA). In the 15-seconds window after the apnea the parameters that aﬀects the most the

196 6.2. Results

Patient 3 Patient 4 0.8 -1 0.3 Start End Start End Experimental data 0.045 L Experimental data L 0.6 0.25 Preterm newborn Newborn at tem model simulation 0.04 model simulation 0.2

0.4 (l) -1.5 (l) 0.035 0.15 0.2 Experimental Experimental Respiration (AU) Respiration (AU) 0.03 Lung volume. V 0.1 Lung volume. V 0 -2 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

100 100

95 (%) (%)

2 90 2 95

SaO 85 SaO

80 90 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

160 160

140 140

120 120

Heart rate (bpm) 100 Heart rate (bpm) 100

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 Time (s) Time (s)

Figure 6.7 – Comparison between experimental data and model output signals during an apnea. Left: Apnea of Patient 3 compared to the M28 model; Right: Apnea of Patient 4 compared to the Mterm model.

SaO2 in the Mterm and M28 model are the FIO2, P aO2,np, MO2sves, MCO2sves, VuA and at what point of the respiration cycle the apnea started (ApneaBegin).

The parameters that aﬀect the most the heart rate, during the 15-seconds window after the apnea, in the Mterm model are the P aO2,np, FIO2, MO2sves and VuA. In the

M28 model the more influential parameters are FIO2, MO2sves, P aO2,np and the delay of the peripheral chemoreflex (Dp). In order to understand and illustrate the results of the sensitivity analysis a local sensitivity analysis (Figure 6.9) was performed on the fraction of inspired oxygen (FIO2) which is the most influential parameters of the Mterm and the

M28 model during and after the apnea.

In both models, a smaller FIO2 results in a more profound desaturation. A decrease of FIO2 decreases the PaO2 and the SaO2. Also, the reserve of air in the lungs during the apnea has less oxygen, so it is less capable of slowing down the desaturation. In the preterm, the eﬀect in the desaturation during the apnea is stronger for a FIO2 smaller than normal (<21%) than in a newborn at term. This must be due to the oxygen dissociation curve in preterms, because the relationship between PaO2 and SaO2 falls more rapidly in the preterms than in the newborns at term at low levels of oxygen [DRO71].

197 Chapter 6 – Newborn cardio-respiratory model

0.4 0.4

0.2 0.2

2 0 0 2 A uA uA V C HR V FIO FIO O2sves aO2,nP O2sves aO2,nP CO2sves P M aCO2,nC P M M P

0.04 0.02

0.02 0.01

0 0 2 2 uA uA V V Begin FIO Begin FIO aO2,nP O2sves aO2,nP O2sves CO2sves P CO2sves P M M M M Apnea Apnea

0.4 0.3

0.2 0.2 0.1

0 0 v 2 2 p uA M D V chem FIO FIO aO2,nP O2sves O2sves aO2,nP G P CO2sves P usv,e,Min M M V M

Figure 6.8 – Most inﬂuential parameters of the Morris screening method using the out- Mean put function ∆X on SaO2 and ∆X on SaO2 and HR on the Mterm and M28 models. ∗ Green bars: Morris distance Di. For each parameter, the absolute mean µi (purple bar) and the standard deviation σi (yellow bars) of the elementary eﬀects are also displayed. Descriptions of each parameter are given late in this section or in Section B.

198 6.2. Results

Lung Volume. M Lung Volume. M term 28 0.12 0.035

0.11 0.03 l l 0.1 0.025

0.09 0.02 Start End Start End 0.08 0 20 40 60 80 100 0 20 40 60 80 100 SaO SaO 2 2 100 100

95 95

90 90

% 85 % 85

80 80

75 75

70 70 0 20 40 60 80 100 0 20 40 60 80 100

Heart rate Heart rate 200 160

140 150

120 100 bpm bpm 100

50 80

0 60 0 20 40 60 80 100 0 20 40 60 80 100 Time (s) Time (s) FIO2 = 0.15 FIO2 = 0.175 FIO2 = 0.2 FIO2 = 0.225 FIO2 = 0.25 FIO2 = 0.275

Figure 6.9 – Local sensitivity analysis of the fraction of oxygen of inspire air (FIO2) on the Mterm model and the M28 model. 199 Chapter 6 – Newborn cardio-respiratory model

6.3 Discussion

This chapter presents the first integration and analysis of a mathematical model representing the cardiovascular and respiratory systems and short-term nervous regulation loops for newborns. Two versions of the integrated model were proposed concerning: a newborn at term (Mterm) and a preterm newborn (M28). The methodology used to modify our adult model performing structural and parameter adaptation using references from the literature resulted in a model that is capable of simulating the behavior of newborn at terms and preterm. To our knowledge, this is the first work performing a formal sensitivity analysis to evaluate the relative significance of physiological parameters on saturation and heart rate in response to an apnea event. The quantified evaluation of effects involved in saturation and HR responses permits the extraction of several variables of interests: the fraction of inspired oxygen (FIO2), metabolic consumption and production rate (MO2sves,

MCO2sves), the unstressed volume of the lungs (VuA), the set-point value of PaO2 of the peripheral chemoreﬂex (P aO2,np) and delay of the peripheral chemoreﬂex (Dp).

— Fraction of oxygen of inspired oxygen (FIO2): For the Mterm model, the

changes of FIO2 do not modify the heart rate signiﬁcantly. But in the M28 model,

a smaller FIO2 produces a decrease in the baseline value of the heart rate before the apnea and a deeper bradycardia during the apnea event. This is due to the in- ﬂuence of the peripheral chemoreceptors’ activity in the vagal response in a hypoxic

condition. The relevance of the FIO2 in the sensitivity analysis for newborns and preterm agrees with the use of oxygen therapy in neonatal care environment for the treatment of apneas [Sim+02].

— Metabolic consumption and production rate (MO2sves, MCO2sves): During an

apnea, the lack of ventilation halts the ﬂow of O2 and CO2 and the chemoreﬂex cannot modify the mechanics of the respiration. Hence, the factors that are going

to aﬀect the oxygen desaturation and the changes of PaCO2 during an apnea are

the O2 metabolic consumption rate (MO2sves) and the CO2 metabolic production

rate (MCO2sves) respectively. A higher MO2sves will lead to a deeper desaturation

during an apnea. A higher MCO2sves will lead to a faster increase of the PaCO2. This conﬁrms the important role of the metabolism in the acceleration of the desaturation during apnea, as stated by the literature [UMS91], in previous models in adults [GLH18] and preterms [San+09] and in the application of hypothermia therapy

in this population. In the M28 model, a higher MO2sves will produce an increase

200 6.3. Discussion

in the activity of the peripheral chemoreceptors, increasing the deepness of the bradycardia.

— Unstressed volume of the lungs (VuA): Changes of the unstressed volume of the

lungs (VuA) leads to signiﬁcant changes of the functional residual capacity (FRC). The FRC is the reserve of air that will be used during an apnea. A higher reserve evokes a smaller desaturation. This is an important factor for the presence and deepness of apneas in preterm infants [UMS91]. This result corroborates the important role of the CPAP and PEEP as a treatment for apneas in newborns at term and in preterms [Pan+09; HS10], because they control the baseline value of the FRC.

— Set-point value of PaO2 of the peripheral chemoreﬂex (P aO2,np): The set-

point value of PaO2 of the peripheral chemoreﬂex (P aO2,np) deﬁnes the target

value of PaO2 for the respiratory control loop. Changing this parameter will lead

to changes in the baseline value of the PaO2 and the activity of the peripheral chemoreceptors, modifying the response during and after the apnea event. In the

M28 model, this will aﬀect the deepness of the bradycardia and the baseline value of the heart rate due to the eﬀerent pathway of the peripheral chemoreceptors. The chemosensitivity and the response of the peripheral chemoreceptors have been reported as relevant factors in the response to hypoxia in preterms [KL94] and in neonatal apnea [Gau+04].

— Delay of the peripheral chemoreﬂex (Dp): The delay of the peripheral chemore-

ﬂex (Dp) aﬀects how fast the respiration is will respond to the hypoxia produced by

an apnea event, as it has been reported in the literature [Kat04]. In the M28 model, this parameter affects the efferent pathway of the vagal branch, dependent on the peripheral chemoreceptors, shifting the occurrence of the bradycardia. Results concerning the relative significance of the model parameters provide valuable information for the application of such an integrated model for the analysis of the nervous response induced by apnea in term newborn and preterm. Simulated data show the ability of the model to simulate baseline values in preterm and newborn at term and the coherence with physiology in the response to apnea events. Sensitivity analysis results highlight the physiological factors that are highly influent during apnea-bradycardia episodes and it corroborates the relevance of most of traditional treatments in neonatal apnea (CPAP, PEEP, oxygen therapy). These parameters must be carefully analyzed in future treatments performed in the clinical database and they should be considered in potential applications for newborn monitoring.

201 Chapter 6 – Newborn cardio-respiratory model

Model limitations

The main limitations of the models are related to certain mechanisms that are not included or are poorly known in the literature. The models do not include thermoregulation, metabolic adaptation or laryngeal chemoreflexes. Maintaining a neutral thermal environment is one of the key physiologic challenges a newborn must face after delivery [CMN13]. Changes in the temperature of the newborn can change oxygen consumption, respiratory rate, glucose use, pulmonary vasoconstriction, etc, leading to the variation of highly sensitive parameters. In general, this is however a minor limitation on preterm newborns followed in NICUs, since they are kept on a temperature-controlled environment. Moreover, after birth, the neonate has to adapt the metabolism from the transplacental supply of glucose to an exogenous nutritional intake [PD05]. These adaptations are highly dependent on growth, feeding, prematurity, etc. Laryngeal chemoreflexes have been related to cardiorespiratory control in the neonatal period [PSD08] and these aspects have not been represented into the model. Finally, newborns often present comorbidities, such as tetralogy of Fallot or the coarctation of the aorta, leading to significant modifications of the cardiovascular system. Some of this mechanisms will be included in future works.

6.4 Conclusion

The inclusion of the maturation factor in personalized models of newborn was a major challenge that multi-scale models needed to resolve. To our knowledge, the model, presented in this chapter, is the first integrated representation of the cardio- respiratory interactions, adapted to the neonatal period. Previous works in the field have been focused on isolated aspects of the newborn physiology or on the cardiorespiratory interactions in adults. The main challenge was to combine these sub-models and previous works of our group in adults in order to propose an integrated model, which allows the creation of a Digital Newborn at different gestational levels. The main contributions of this work are: 1) the integration of physiological functions involved in complex interactions between the respiratory and cardiac systems, 2) the adaptation of the model to a newborn at term (Mterm) a preterm infant (M28) based on a literature review of neonatal physiology, 3) the comparisons of simulations with experimental data and 4) the sensitivity analysis of physiological parameters. Our results show the importance of several physiological factors in apnea of prematurity: the fraction of inspired oxygen, metabolic O2 consumption rate, metabolic CO2 consumption

202 REFERENCES rate and a lung volume parameter. These parameters should be considered in potential applications for newborn. The future applications of these two versions of the model are particularly promising concerning: i)parameter identification in different stages of the newborn period or prematurity ii) the proposal of an interactive educational tool for the understanding of newborn physiology. — Parameter identification: the evolution of identified parameters in different stages of maturity should provide a better comprehension of the changes and dynamics of newborns and could lead to potential applications for newborn monitoring. — Proposal of an interactive educational tool: the proposed virtual newborn patient could be integrated into a set of interactive educational tools, which could be used in medical schools or hospitals, for the teaching of newborn physiology. The

Mterm and M28 models can form the basis of an educational simulation software, that would aim to improve clinical education of cardio-respiratory interactions.

References

[Alb+15] Antonio Albanese et al., « An integrated mathematical model of the human cardiopulmonary system: Model development », in: American Journal of Physiology - Heart and Circulatory Physiology (2015), ajpheart.00230.2014. [BKR04] W Boemke, M O Krebs, and R Rossaint, « [Blood gas analysis]. », in: Der Anaesthesist 53.5 (2004), pp. 471–92. [BT00] J.J. Batzel and H.T. Tran, « Modeling instability in the control system for human respiration: applications to infant non-REM sleep », in: Applied Mathematics and Computation 110.1 (Apr. 2000), pp. 1–51. [Chu+64] Josephine S. Chu et al., « Lung compliance and lung volume measured con- currently in normal full-term and premature infants », in: Pediatrics 34.4 (1964). [CMN13] CMNRP, Newborn Thermoregulation Self-Learning Module, tech. rep., Inter- professional Education and research Committee of the Champlain Maternal Newborn Regional Program, 2013.

203 Chapter 6 – Newborn cardio-respiratory model

[Das+18] Theodore Dassios et al., « Physiological and anatomical dead space in me- chanically ventilated newborn infants », in: Pediatric Pulmonology 53.1 (Jan. 2018), pp. 57–63. [Dat+10] Marco Dat et al., « Modeling cardiovascular autoregulation of the preterm infant », in: Master, Department of cardiovascular biomechanics, Eindhoven University of Technology, Eindhoven (2010). [Don98] Steven M Donn, Neonatal and pediatric pulmonary graphics: principles and clinical applications, Futura Publishing Company, 1998. [DRO71] Maria Delivoria-Papadopoulos, Nevenka P Roncevic, and Frank A Oski, « Postnatal Changes in Oxygen Transport of Term, Premature, and Sick In- fants: The Role of Red Cell 2,3-Diphosphoglycerate and Adult Hemoglobin », in: Pediatric Research 5.6 (June 1971), pp. 235–245. [Emm+64] George C Emmanouilides et al., « Pulmonary arterial pressure changes in human newborn infants from birth to 3 days of age », in: The Journal of Pediatrics 65.3 (1964), pp. 327–333. [Gau+04] Estelle B Gauda et al., « Maturation of peripheral arterial chemoreceptors in relation to neonatal apnoea », in: Seminars in Neonatology 9.3 (June 2004), pp. 181–194. [GB80] T Gerhardt and E Bancalari, « Chestwall compliance in full-term and premature infants. », in: Acta paediatrica Scandinavica 69.3 (May 1980), pp. 359– 64. [GLH18] Gustavo Guerrero, Virginie Le Rolle, and Alfredo I. Hernández, « Sensitivity Analysis of a Cardiorespiratory Model for the Study of Sleep Apnea », in: CINC (2018). [GM03] Anne. Greenough and A. D. Milner, Neonatal respiratory disorders, Arnold, 2003, p. 550. [HBE86] D J Henderson-Smart, M C Butcher-Puech, and D A Edwards, « Incidence and mechanism of bradycardia during apnoea in preterm infants. », in: Archives of Disease in Childhood 61.3 (Mar. 1986), pp. 227–232.

204 REFERENCES

[Her+09] A. I. Hernandez et al., « A multiformalism and multiresolution modelling environment: application to the cardiovascular system and its regulation », in: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 367.1908 (Dec. 2009), pp. 4923–4940. [HS10] David J Henderson-Smart and Peter A Steer, « Caffeine versus theophylline for apnea in preterm infants », in: Cochrane Database of Systematic Reviews 1 (Jan. 2010), p. CD000273. [Jen+11] Ward Jennekens et al., « Validation of a preterm infant cardiovascular system model under baroreflex control with heart rate and blood pressure data », in: 2011 Annual International Conference of the IEEE Engineering in Medicine and Biology Society (2011), pp. 896–899. [Kat04] M Katz-Salamon, « Delayed chemoreceptor responses in infants with apnoea. », in: Archives of disease in childhood 89.3 (Mar. 2004), pp. 261–6. [Ken+09] Alison L. Kent et al., « Normative blood pressure data in non-ventilated premature neonates from 28–36 weeks gestation », in: Pediatric Nephrology 24.1 (Jan. 2009), pp. 141–146. [KL94] M Katz-Salamon and H Lagercrantz, « Hypoxic ventilatory defence in very preterm infants: attenuation after long term oxygen treatment. », in: Archives of disease in childhood. Fetal and neonatal edition 70.2 (Mar. 1994), pp. 90– 5. [Lag+90] H. Lagercrantz et al., « Autonomic Reflexes in Preterm Infants », in: Acta Paediatrica 79.8-9 (Aug. 1990), pp. 721–728. [Mor91] M D Morris, « Factorial plans for preliminary computational experiments », in: Technometrics 33.2 (1991), pp. 161–174. [Neu+15] Roland P. Neumann et al., « Influence of gestational age on dead space and alveolar ventilation in preterm infants ventilated with volume guarantee », in: Neonatology 107.1 (2015), pp. 43–49. [Pan+00] Paresh B. Pandit et al., « Lung resistance and elastance in spontaneously breathing preterm infants: effects of breathing pattern and demographics », in: Journal of Applied Physiology 88.3 (Mar. 2000), pp. 997–1005.

205 Chapter 6 – Newborn cardio-respiratory model

[Pan+09] T Pantalitschka et al., « Randomised crossover trial of four nasal respiratory support systems for apnoea of prematurity in very low birthweight infants », in: Archives of Disease in Childhood - Fetal and Neonatal Edition 94.4 (July 2009), F245–F248. [PC55] K J Prec and D E Cassels, « Dye dilution curves and cardiac output in newborn infants. », in: Circulation 11.5 (May 1955), pp. 789–98. [PD05] Martin Ward Platt and Sanjeev Deshpande, « Metabolic adaptation at birth », in: Seminars in Fetal and Neonatal Medicine 10.4 (2005), pp. 341–350. [Poe+91] C. F. Poets et al., « Oxygen saturation and breathing patterns in infancy. 2: Preterm infants at discharge from special care », in: Archives of Disease in Childhood 66.5 (May 1991), pp. 574–578. [Poe98] Christian F. Poets, « When do infants need additional inspired oxygen? A review of the current literature », in: Pediatric Pulmonology 26.6 (Dec. 1998), pp. 424–428. [PSD08] J. P. Praud, M. St-Hilaire, and D. Dorion, « Laryngeal chemoreﬂexes and cardiorespiratory control in the neonatal period », in: Archives de Pediatrie 15.5 (June 2008), pp. 620–622. [Rev+89] Michael Revow et al., « A Model of the Maturation of Respiratory Control in the Newborn Infant », in: IEEE Transactions on Biomedical Engineering 36.4 (1989), pp. 414–423. [Sá +06] Carla D. Sá Couto et al., « A Model for Educational Simulation of Neonatal Cardiovascular Pathophysiology », in: Simulation in Healthcare: The Journal of the Society for Simulation in Healthcare 1.Inaugural (2006), pp. 4–9. [Sai+72] Saroj Saigal et al., « Placental transfusion and hyperbilirubinemia in the premature », in: Pediatrics 49.3 (1972). [San+09] Scott A. Sands et al., « A model analysis of arterial oxygen desaturation during apnea in preterm infants », in: PLoS Computational Biology 5.12 (2009). [Sel+11] Fabio Augusto Selig et al., « Heart rate variability in preterm and term neonates. », in: Arquivos brasileiros de cardiologia 96.6 (June 2011), pp. 443– 9.

206 REFERENCES

[Sep+94] Marko P. Seppänen et al., « Doppler-Derived Systolic Pulmonary Artery Pressure in Acute Neonatal Respiratory Distress Syndrome », in: Pediatrics 93.5 (1994). [Sim+02] N. Simakajornboon et al., « Effect of Supplemental Oxygen on Sleep Archi- tecture and Cardiorespiratory Events in Preterm Infants », in: PEDIATRICS 110.5 (Nov. 2002), pp. 884–888. [Sol+09] V Soloveychik et al., « Acute hemodynamic effects of caffeine administra- tion in premature infants », in: Journal of Perinatology 29.3 (Mar. 2009), pp. 205–208. [Tan87] K.L. Tan, « Blood Pressure in Full-term Healthy Neonates », in: Clinical Pediatrics 26.1 (Jan. 1987), pp. 21–24. [TBG05] H. William. Taeusch, Roberta A. Ballard, and Christine A. Gleason, Avery’s diseases of the newborn. Elsevier Saunders, 2005, p. 1633. [Tho+04] Mark R. Thomas et al., « Pulmonary Function at Follow-up of Very Preterm Infants from the United Kingdom Oscillation Study », in: American Journal of Respiratory and Critical Care Medicine 169.7 (Apr. 2004), pp. 868–872. [UMS91] C J Upton, A D Milner, and G M Stokes, « Apnoea, bradycardia, and oxygen saturation in preterm infants. », in: Archives of disease in childhood 66.4 Spec No (Apr. 1991), pp. 381–5. [Urs98] M Ursino, « Interaction between carotid baroregulation and the pulsating heart: a mathematical model. », in: Am J Physiol 275.5 Pt 2 (Nov. 1998). [WTR01] Frank H. Wilhelm, Werner Trabert, and Walton T. Roth, « Characteristics of sighing in panic disorder », in: Biological Psychiatry 49.7 (2001), pp. 606– 614.

207

CONCLUSION

The integration of accurate mathematical models within the management of human health issues is currently of paramount importance for better prediction and treatment of diseases, in order to provide patients with an affordable, personalized and predictive care. European efforts towards the "Digital Patient" are focused on this direction. The first contribution of this work was thus the proposal of a set of integrated models of the cardio-respiratory interactions, for the analysis of the acute responses to apnea and hypopnea events, adapted to adults, term and preterm newborns. These models were validated using reference values from the literature and clinical data. To our knowledge, this is the first such model set in the literature. Indeed, previous works in the field have been mainly focused on isolated aspects of the physiology or on the cardiorespiratory interactions in adults. A common methodology was applied for the proposal of these integrated models, strongly based on methods and tools proposed in our team during the last decade. The main challenge was to combine sub-models and previous works of our group in cardiac and respiratory function, with novel sub-models developed in this thesis concerning gas transport, gas exchange and metabolism, in order to propose an integrated cardio-respiratory model. Another major challenge was the adaptation of the adult model to the physiology of term and preterm newborns. The second contribution of this work was the selection of the most sensitive parameters involved in the acute cardio-respiratory response to apnea, both in adults and term and preterm newborns. These results, obtained from the application of screening and local sensitivity analysis methods to the proposed integrated models, pointed out significant physiological variables that may be particularly interesting for the development of novel diagnosis our therapeutic devices. Moreover, these analyses were used as a method for dimensionality reduction, as a first step towards patient-specific parameter identification. Concerning the sensitivity analysis, the results highlighted physiological factors related to the metabolic rates, the chemoreflex, the respiratory mechanics and the fraction of oxygen in the inspired air (FIO2), with different degrees of importance for adults, newborns at term and preterm infants. These factors are highly influential in

209 sleep apneas and apnea-bradycardia episodes and they corroborate the relevance of most of traditional treatments in adult and neonatal apnea (CPAP, PEEP, oxygen therapy). The third contribution concerns the patient-specific and event-specific analysis of cardio-respiratory interactions of adult sleep apnea. Selected, particularly sensitive parameters of the proposed adult model were identified in a patient-specific and event-specific (during a whole night) fashion in order to characterize the acute cardiorespiratory response of these patients to apnea events in a quantitative, integrative and explainable manner. The identification method is based on a particular evolutionary algorithm class, named differential evolution, that was adapted, integrated with M2SL and firstly deployed in this work. This method was applied to a clinical database composed of 107 obstructive apneas distributed among 10 patients (HYPNOS clinical study) in order to study the dynamics of SaO2 during an obstructive sleep apnea. Once the parameters have been identified, a number of transformations can be applied on the parameter space to further characterize the patient’s response. K-means clustering was applied in this sense, which allowed for the identification of three different clusters, describing three different phenotypes related to sleep apnea and periodic breathing. These phenotypes can be used to determine patients with higher risk of ventilation instability and periodic respiration, in order to improve and personalize treatment. To our knowledge, this is the first patient- and event-specific model-based analysis of apnea events in the literature and we expect a number of applications of this approach. Overall, the obtained results provide new insights into the underlying mechanisms of SAS and apnea of prematurity improving physiopathology and prognosis interpretation, with a potential future impact on diagnostic and therapeutic strategies. The proposed approach may be used as an instrument for the identification of patients with higher risk of ventilation instability and periodic respiration in order to improve and personalize treatment. The three proposed models have a number of potential applications within a digital patient perspective and may be used as an educational tool to understand human physiology. A number of limitations still exist in this work, and most of them are related to physiological mechanisms that are not represented in our models, such as sleep mechanisms, thermoregulation, sighs or upper airway resistance control. Moreover, only the shape of experimental SaO2 and the apnea duration are exploited in the parameter identification phase, instead of using all the available experimental data. Further developments will be directed towards model refinement and the integration of complementary observations, in

210 order to obtain more accurate results. In future works, Digital Patient representations obtained as proposed in this work may be used to conduct virtual physiological experiments and to test new diagnostic or theraopeytic hypotheses, before intitiating pre-clinical or clinical investigations. In the context of SAS and apneas bradycardias, new treatments and clinical maneuvers could be tested to predict the possible outcomes, without endanger any real patient. New ventilation methods could be easily applied to our models, adding a controlled pressure in the input of the upper airways in the respiratory sub-model. Invasive methods like the NAVA (Neurally adjusted ventilatory assist), where the ventilation is controlled by the electrical activity of the diaphragm, could be substituted by a synchronized and well identified model that will pilot the ventilation using the simulated respiratory muscles activity. Similarly, oxygen therapy could be simulated controlling the fraction of oxygen in inspired air (FIO2) in the model. New methods using mixed gases for the therapy or adaptative oxygen percentages using simulated signals as a reference are hypothesis that could be tested. In the case of newborn modeling, patient-specific identification using experimental data of newborns in different stages of maturity should provide a better comprehension of the changes and dynamics of newborns and could lead to potential applications for newborn monitoring. Moreover, increasing the observability of the model using information from the patient’s electronic health record (EHR), along with further acquired physioological data is considered in a future perspective. Parameters like lung compliance, lung resistance or the functional residual capacity could be measured or obtained from the EHR, reducing the number of parameters to identify. In some cases, time series of experimental data (e.g. nasal pressure, lung volume) could be injected into the model, reducing the number of sub-models needed in the integrated cardio-respiratory model and the number of parameters to identify. National efforts on the "Health Data Hub" are currently directed towards this direaction and we think that our models are a way to optimally combine these heterogeneous data. Finally, in the current context, the adult model could be adapted and applied to study the cardio-respiratory mechanisms of COVID-19 and to study the evolution and severity in some patients. The elderly and patients with preexisting health conditions (cardiovascular disease, diabetes, hypertension, chronic respiratory disease) are at the greatest risk of dying of this virus. Our model is able to take into account several of the cardio-respiratory factors involved in this pathology and to simulate effects in late stages

211 of the virus, such as an acute respiratory distress syndrome (ARDS). Finally, the approach adopted in this work, combining multi-resolution physiological modeling, sensitivity analysis and parameter identiﬁcation, as well as their application to concrete medical problems, is particularly promising and can be generalized to other clinical applications. Model-based studies could help to minimize the risk of clinical research programs and optimize clinical development. These approaches may well gradually lead to signiﬁcant changes in the way of treatment, but also in the way of understanding the origins of disease.

212 Appendices

213 Appendix A EQUATIONS

A.1 Cardiovascular system.

A.1.1 Time-varying elastances driving functions

For adults: HR 2 HR 2!! ea(t) = exp −Ba · t − Ca (A.1) HRR HRR

n  t 1    α1T 1 ev(t) = cte  n1   n2  (A.2) 1 + t 1 + t α1T α2T

For newborns:

Tas = HP ∗ 0.3 (A.3)

Tvs = 0.16 + HP ∗ 0.3 (A.4)

∆T = HP ∗ 0.02 (A.5)

 2 E E − E · π∗(t−∆T ) , T < t < T T  Min,lv + ( Max,lv Min,lv) sin( Tvs ) if ∆ ∆ + vs Elv(t) = EMin,lv, otherwise (A.6)

 2 E E − E · π∗(t−∆T ) , T < t < T T  Min,rv + ( Max,rv Min,rv) sin( Tvs ) if ∆ ∆ + vs Erv(t) = EMin,rv, otherwise (A.7)

214  2 E E − E · π∗(t) , T < t < T T  Min,ra + ( Max,ra Min,ra) sin( Tvs ) if ∆ ∆ + vs Era(t) = (A.8) EMin,ra, otherwise

 2 E E − E · π∗(t) , T < t < T T  Min,la + ( Max,la Min,la) sin( Tvs ) if ∆ ∆ + vs Ela(t) = (A.9) EMin,la, otherwise

A.1.2 Pressures, Volumes and Flows

• Left chamber

Pla(Vla, t) = (EMIN,la + ea(t) · (EMAX,la − EMIN,la)) · (Vla − Vua) + Pthor (A.10)

Plv(Vlv, t) = ev(t) · Pes,lv(V ) + (1 − ev(t)) · Ped,lv(Vlv) + Pthor (A.11)

Pes,lv(Vlv) = EMax,lv ∗ (Vlv − Vulv) (A.12)

λ(Vlv−V0) Ped,lv(Vlv) = P0,l · (e − 1) (A.13)

  Pla−Plv ,P − P ≥  Rmv la lv 0 Qin,lv = (A.14) 0,Pla − Plv < 0  (Plv−Psa,i  ,Plv − Psa,i ≥ 0 Rsa,i Qout,lv = (A.15) 0,Plv − Psa,i < 0

dVla = Qout,pv − Qin,lv (A.16) dt

dVlv = Qin,lv − Qout,lv (A.17) dt • Systemic circulation

215 Psa,i = Esa,i · (Vsa,i − Vusa,i) + Pthor (A.18)

Psa,e = Esa,e · (Vsa,e − Vusa,e) (A.19)

Psves = Esves · (Vsves − Vusves) (A.20)

Psv,e = Esv,e · (Vsv,e − Vusv,e) (A.21)

Psv,i = Esv,i · (Vsv,i − Vusv,i) + Pthor (A.22)

dQout,sa,i Psa,i − (Rsa,e · Qout,sa,i) − Psa,e = (A.23) dt Lsa,e

 Psa,e−Psves  ,Psa,e − Psves ≥ 0 Rin,sves Qout,sa,e = (A.24) 0,Psa,e − Psves<0   Psves−Psv,e ,P − P ≥  Rout,sves sves sv,e 0 Qout,sves = (A.25) 0,Psves − Psv,e < 0)

  Psv,e−Psv,i ,P − P ≥  Rsv,e sv,e sv,i 0 Qout,sv,e = (A.26) 0,Psv,e − Psv,i < 0)

 Psv,i−Pra  ,Psv,i − Pra ≥ 0 Rsv,i Qout,sv,i = (A.27) 0,Psv,i − Pra < 0

dVsa,i = Qout,lv − Qout,sa,i (A.28) dt

dVsa,e = Qout,sa,i − Qout,sa,e (A.29) dt

dVsves = Qout,sa,e − Qout,sves (A.30) dt

216 dVsv,e = Qout,sves − Qout,sv,e (A.31) dt

dVsv,i = Qout,sv,e − Qout,sv,i (A.32) dt • Right chamber

Pra(Vra, t) = (EMIN,ra + ea(t) · (EMAX,ra − EMIN,ra)) · (Vra − Vura) + Pthor (A.33)

Prv(Vrv, t) = ev(t) · Pes,rv(Vrv) + (1 − ev(t)) · Ped,rv(Vrv) + Pthor (A.34)

Pes,rv(Vrv) = Ees,rv · (Vrv − Vuv) (A.35)

λ·(Vrv−V0) Ped,rv(Vrv) = P0,r · (e − 1) (A.36)

  Pra−Prv ,P − P ≥  Rtv ra rv 0 Qin,rv = (A.37) 0,Pra − Prv < 0)

  Prv−Ppa ,P − P ≥  Rpa rv pa 0 Qout,rv = (A.38) 0,Prv − Ppa < 0)

dVrv = Qin,rv − Qout,rv (A.39) dt

dVra = Qout,sv,i − Qin,rv (A.40) dt • Pulmonary circulation

Ppa = Epa · (Vpa − Vupa) + Pthor (A.41)

Ppv = Epv · (Vpv − Vupv) + Pthor (A.42)

217   Ppv−Pla ,P − P ≥  Rpv pv la 0 Qout,pv = (A.43) 0,Ppv − Pla < 0)

dVpa = Qout,rv − Qpa (A.44) dt

For adults: (1.0 − fs) Rps = Rpp · (A.45) fs

fs Cps = Cpp · (A.46) (1.0 − fs)

Ppp = Pps (A.47)

(Ppa − Ppp) Qpa = (A.48) Rpa

(Ppp − Ppv) Qpp = (A.49) Rpp

(Pps − Ppv) Qps = (A.50) Rps

Cpp · (Qpa − Qps − Qpp) QCpp = (A.51) (Cps + Cpp)

Cps · (Qpa − Qps − Qpp) QCps = (A.52) (Cps + Cpp)

dPpp (Qpa − Qps − Qpp) dPpl = + 0.73556) (A.53) dt (Cps + Cpp) dt

dVpv = Qpp + Qps − Qout,pv (A.54) dt

dVpp = QCpp (A.55) dt

218 dVps = QCps (A.56) dt

For newborns: Ppa − Ppv Qpa = (A.57) Rpa

dVpv = Qpa − Qout,pv (A.58) dt

A.2 Respiratory system

Ru = K1 + K2 · V˙ (A.59)

2 Vcmax Rc = K3 · (A.60) Vc

K4 Rl = (A.61) V

Vc − Vuc Pc = + Ppl (A.62) Cc

VA − VuA PA = + Ppl (A.63) CA

V Pcw = (A.64) Ccw

Ptm = Pc + Ppl (A.65)

β α ! t 1− tr r TI Pmus = Pmax Bo + (1 − Bo) · · e (A.66) TI

Pao − Pc Qao_c = (A.67) Ru − Ri

219 Pc − PA Qc_A = (A.68) Ru − Ri

Qc_pl = Qao_c − Qc_A (A.69)

dVao_c = Qao_c (A.70) dt

dVA = Qc_A (A.71) dt

dPpl Qao_c dPmus = + (A.72) dt Ccw dt

Pthor = Ppl · 0.735559 (A.73)

A.3 Baroreﬂex and pulmonary stretch receptors.

A.3.1 Baroreﬂex dynamics

dPb Kb · Psa,i − Pb = (A.74) dt τb

A.3.2 Feedback regulation of the heart rate

bv Nv = av + (A.75) eλv(Pb−Mv) + 1

For adults:

Nv,d = Nv(t − Dv) · (1 − Gpsr · (VA − Vpsr)) (A.76)

For newborns:

Nv,d = Nv(t − Dv) · (1 − Gpsr · (VA − Vpsr) − Gchem · (upO2 − Uchem) (A.77)

220 dV ag Kv · Nv,d − V ag = (A.78) dt τv

bs Ns = as + (A.79) eλs·(Pb−Ms) + 1

Ns,d = Ns(t − Ds) (A.80)

dS Ks · Ns,d − S = (A.81) dt τs

1 HR = = HR0 + S − V ag (A.82) HP

A.3.3 Sympathetic regulation mechanisms

bRin NRin = aRin + (A.83) eλRin·(Pb−MRin) + 1

NRin,d = NRin(t − DRin) (A.84)

dSRin KRin · NRin,d − SRin = (A.85) dt τRin

Rin,sves = Rin,sves,Min + SRin (A.86)

bRout NRout = aRout + (A.87) eλRout·(Pb−MRout) + 1

NRout,d = NRout(t − DRout) (A.88)

dSRout KRout · NRout,d − SRout = (A.89) dt τRout

Rout,sves = Rout,sves,Min + SRout (A.90)

221 bV usv,i NV usv,i = aV usv,i + (A.91) eλV usv,i·(Pb−MV usv,i) + 1

NV usv,i,d = NV usv,i(t − DV usv,i) (A.92)

dSV usv,i KV usv,i · NV usv,i,d − SV usv,i = (A.93) dt τV usv,i

Vusv,i = Vusv,i,Min + SV usv,i (A.94)

bV usv,e NV usv,e = aV usv,e + (A.95) eλV usv,e·(Pb−MV usv,e) + 1

NV usv,e,d = NV usv,e(t − DV usv,e) (A.96)

dSV usv,e KV usv,e · NV usv,e,d − SV usv,e = (A.97) dt τV usv,e

Vusv,e = Vusv,e,Min + SV usv,e (A.98)

bEMAX,lv NEMAX,lv = aEMAX,lv + (A.99) eλEMAX,lv·(Pb−MEMAX,lv) + 1

NEMAX,lv,d = NEMAX,lv(t − DEMAX,lv) (A.100)

dSEMAX,lv KEMAX,lv · NEMAX,lv,d − SEMAX,lv = (A.101) dt τEMAX,lv

EMAX,lv = EMAX,lv,Min + SEMAX,lv (A.102)

bEMAX,rv NEMAX,rv = aEMAX,rv + (A.103) eλEMAX,rv·(Pb−MEMAX,rv) + 1

NEMAX,rv,d = NEMAX,rv(t − DEMAX,rv) (A.104)

222 dSEMAX,rv KEMAX,rv · NEMAX,rv,d − SEMAX,rv = (A.105) dt τEMAX,rv

EMAX,rv = EMAX,rv,Min + SEMAX,rv (A.106)

A.4 Chemoreﬂex

A.4.1 Central chemoreﬂex

uc(t) = Pa,CO2(t − Dc) − Pa,CO2,nC (A.107)

d∆Pmax,c −∆Pmax,c + Gc,A · uc = (A.108) dx τc,A

d∆BRc −∆BRc + Gc,f · uc = (A.109) dx τc,f

A.4.2 Peripheral chemoreﬂex

For adults:

xO2 = Ach · (1 − SaO2) + Bch (A.110)

xO2 −1∗ K φO2 = K02 ∗ (1 − e 02 ) (A.111)

φCO2 = KCO2 ∗ (Ca,CO2 − Ct) (A.112)

φT = φO2 · φCO2 (A.113)

 φT −1∗ K Kstat ∗ (1 − e stat ) if C˜a,CO2 > Ct fcstat = (A.114) 0 otherwise

φ φT −1∗ CO2dyn −1∗ K K φ˜C = Kstat ∗ (1 − e stat ) + Kdyn ∗ (1 − e dyn ) (A.115)

223 dCa,CO2 dφCO2dyn τzh ∗ − φCO2dyn = dt (A.116) dt τph

dφC φ˜C − φC = (A.117) dt τpl

 φC if φC > 0 fapc = (A.118) 0

up(t) = fapc(t − Dp) − facp,n (A.119)

d∆Pmaxp −∆Pmaxp + Gp,A · up = (A.120) dx τp,A

d∆BRp −∆BRp + Gp,f · up = (A.121) dx τp,f

For newborns:

upO2(t) = Pa,O2(t − Dp) − Pa,O2,nP (A.122)

d∆Pmaxp,O2 −∆Pmaxp,02 + Gp,A,O2 · upO2 = (A.123) dx τp,A,02

d∆BRpO2 −∆BRpO2 + Gp,f,02 · up,02 = (A.124) dx τp,f,O2

upCO2(t) = Pa,CO2(t − Dp) − Pa,CO2,nP (A.125)

d∆Pmaxp,CO2 −∆Pmaxp,C02 + Gp,A,CO2 · upCO2 = (A.126) dx τp,A,C02

d∆BRpCO2 −∆BRpCO2 + Gp,f,CO2 · up,CO2 = (A.127) dx τp,f,CO2

224 A.4.3 Outputs

For adults:

Pmax = Pmax0 + ∆Pmaxc + ∆Pmaxp (A.128)

BR = BR0 + ∆BRc + ∆BRp (A.129)

For newborns:

Pmax = Pmax0 + ∆Pmaxc + ∆Pmaxp,O2 + ∆Pmaxp,CO2 (A.130)

BR = BR0 + ∆BRc + ∆BRpO2 + ∆BRpCO2 (A.131)

A.5 Lung gas exchange

A.5.1 Lung gas exchange (dead volume, alveoli, pulmonary capillaries)

  ˙ dFD,O2 V · (FI,O2 − FD,O2), Inspiration Vc · = (A.132) dt V˙A · (FD,O2 − FA,O2) Expiration

  ˙ dFD,CO2 V · (FI,CO2 − FD,CO2), Inspiration Vc · = (A.133) dt V˙A · (FD,CO2 − FA,CO2) Expiration

 V˙A · (FD,O2 − FA,O2) + K · ((Qpp + QCpp) · (C˜v,O2 − Ca,O2)    dCAO2 dFA,O2 −V pp · dt ), Inspiration VA · = dt K · ((Qpp + QCpp) · (C˜v,O2 − Ca,O2)    dCAO2 −V pp · dt ), Expiration (A.134)

225  V˙A · (FD,CO2 − FA,CO2) + K · ((Qpp + QCpp) · (C˜v,CO2 − Ca,CO2)    dCACO2 dFA,CO2 −V pp · dt ), Inspiration VA· = dt K · ((Qpp + QCpp) · (C˜v,CO2 − Ca,CO2)    dCACO2 −V pp · dt ), Expiration (A.135)

A.6 Gas exchange and gas transport

A.6.1 Alveolar pressures and concentrations.

PA,O2 = FA,O2 · (Patm − PWS) (A.136)

PA,CO2 = FA,CO2 · (Patm − PWS) (A.137)

For adults: 1 + β1 · PA,CO2 XA,O2 = PA,O2 · (A.138) K1,O2 · (1 + α1 · PA,CO2)

1 + β2 · PA,O2 XA,CO2 = PA,CO2 · (A.139) K2,CO2 · (1 + α2 · PA,O2)

1 h1 XA,O2 CA,O2 = Z · Csat,O2 · 1 (A.140) h1 1 + XA,O2

1 h2 XA,CO2 CA,CO2 = Z · Csat,CO2 · 1 (A.141) hS 1 + XA,CO2

For newborns at term and preterms:

c PA,O2 ! b CA,O2 = a · 1 − e (A.142)

226 0.415 CA,CO2 = 0.107 · (PA,CO2) (A.143)

A.6.2 Arterial pressures and concentrations.

For adults: (Qpp · CA,O2) + (Qps · C˜v,O2) Ca,O2 = (A.144) Qpp + Qps

(Qpp · CA,CO2) + (Qps · C˜v,CO2) Ca,CO2 = (A.145) Qpp + Qps

For newborns:

(Qpa · (1 − fs) · CA,O2) + (Qpa · (fs) · C˜v,O2) Ca,O2 = (A.146) Qpa

(Qpa · (1 − fs) · CA,CO2) + (Qpa · (fs) · C˜v,CO2) Ca,CO2 = (A.147) Qpa

A.6.3 Arterial oxygen saturation.

Ca,O2 · 100 − Pa,O2 · 0.003 SaO2 = (A.148) (Hgb · 1.39)

A.7 Tissue gas exchange.

dCsvesO2 (VT,sves + Vsves) · = QInsves · (C˜a,O2 − CsvesO2) − MO2sves (A.149) dt

dCsvesCO2 (VT,sves + Vsves) · = QInsves · (C˜a,CO2 − CsvesCO2) + MCO2sves (A.150) dt

A.8 Gas transport

dCsevO2 Vsev · = QOutsves · (CsvesO2 − CsevO2) (A.151) dt

227 dCsivO2 Vsiv · = QOutev · (CsevO2 − CsivO2) (A.152) dt

dCsevCO2 Vsev · = QOutsves · (CsvesCO2 − CsevCO2) (A.153) dt

dCsivCO2 Vsiv · = QOutev · (CsevCO2 − CsivCO2) (A.154) dt

C˜a,O2 = Ca,02(t − τLT ) (A.155)

C˜a,CO2 = Ca,C02(t − τLT ) (A.156)

C˜v,O2 = CsivO2(t − τVL) (A.157)

C˜v,CO2 = CsivCO2(t − τVL) (A.158)

A.9 Other cardio-respiratory interactions

2 VA Rpp = Rpp0 · (A.159) Vthres

228 Appendix B LIST OF PARAMETERS

B.1 Cardiovascular system

Symbol Description Value Value Value NN PM Adult

Vtot Total blood volume 310.682 110 4740 HP Heart period 0.46 0.42 0.833 HR Heart rate 130 142 72

Time-varying elastances

EMin,lv Diastolic elastance of the left ventricle 3.003 8.464 0.0995

EMax,lv Maximum systolic elastance of the 60.637 170.885 3.76 left ventricle

EMin,la Diastolic elastance of the left atrium 3.997 11.264 0.1326

EMax,la Maximum systolic elastance of the 4.248 11.972 0.31 left atrium

EMin,rv Diastolic elastance of the right 2.992 8.432 0.063 ventricle

EMax,rv Maximum systolic elastance of the 39.283 110.705 0.72 right ventricle

EMin,ra Diastolic elastance of the right atrium 2.581 7.273 0.055

EMax,ra Maximum systolic elastance of the 11.534 32.504 0.165 right atrium

Ba Constant controlling the rise and N/A N/A 84.375 peak of the atrial systole

Ca Constant controlling the rise and N/A N/A 0.32 peak of the atrial systole

229 α1 Constant deﬁning the time of the N/A N/A 0.3698 ascending part of the elastance curve

α2 Constant deﬁning the time of the N/A N/A 0.3644 descending part of the elastance curve n1 Constant determining the steepness N/A N/A 1.1993 of the elastance curve n2 Constant determining the steepness N/A N/A 21.87 of the elastance curve HRR Baseline resting heart rate N/A N/A 60

P0,r Gradient N/A N/A 0.8

P0,l Gradient N/A N/A 0.85 α Curvature N/A N/A 0.015

V0 Zero pressure N/A N/A 18.088

Elastances

Esa,i Elastance of the intrathoracic 15.53 43.76 1.58 systemic arteries

Esa,e Elastance of the extrathoracic 6.62 18.66 0.615 systemic arteries

Esves Elastance of the systemic peripheral 2.52 7.11 0.629 vessels

Esv,i Elastance of the extrathoracic 0.57 1.6 0.02 systemic veins

Esv,e Elastance of the extrathoracic 0.285 0.8 0.0187 systemic arteries

Epa Elastance of the pulmonary arteries 12.45 35.08 0.258

Epv Elastance of the pulmonary veins 0.548 1.54 0.05

Pressures, volumes and ﬂows

Left chamber

Vula Unstressed volume of left atrium 0.219 0.078 27.13

Vulv Unstressed volume of left ventricle 0.438 0.155 18.1

230 Rmv Mitral valve resistance 0.06 0.121 0.003

Systemic circulation

Rsa,i Resistance of the aortic valve and 0.018 0.0362 0.008 intrinsic artery

Vusa,i Unstressed volume of the 8.021 2.8463 126.61 intrathoracic arteries

Vusa,e Unstressed volume of the 21.192 7.5198 334.63 extrathoracic arteries

Vusves Unstressed volume of the systemic 25.825 9.1631 407.88 peripheral vessels

Vusv,e Unstressed volume of the 54.408 22.3542 904.40 extrathoracic veins

Vusv,i Unstressed volume of the 64.723 26.5927 1076.24 intrathoracic veins

Rsa,e Resistance of the extrathoracic 1.5 3.0152 0.06 arteries

Lsa,e Blood ﬂow inertia 0.0018 0.0018 0.0007

Rin,sves Resistance of peripheral vessels 2.468 4.96 0.5

Rout,sves Resistance of peripheral vessels 2.468 4.96 0.5

Rsv,e Resistance of the extrathoracic veins 0.21 0.422 0.09

Rsv,i Resistance to forward ﬂow of the 0.015 0.03 0.003 right atrium

Right chamber

Vura Unstressed volume of right atrium 0.219 0.078 27.13

Vurv Unstressed volume of right ventricle 0.289 0.103 18.088

Rtv Resistance of tricuspid valve 0.06 0.121 0.003

Pulmonary circulation

Rpa Resistance of the pulmonary arteries 0.018 0.036 0.046

Vupa Unstressed volume of the pulmonary 2.864 1.016 45.22 arteries

231 Vupv Unstressed volume of the pulmonary 20.054 7.116 306.31 veins

Rpp Resistance of the pulmonary vessels 0.991 1.457 0.0652

Rps Resistance of the pulmonary shunt N/A N/A 3.1947

Rpv Resistance to forward ﬂow in the left 0.015 0.03 0.003 atrium *Units : heart rate, BPM; heart period, s; volumes, ml;,resistances , mmHg.ml-1.s; elastances, mmHg.ml-1;inertia„mmHg.ml-1.s2

B.2 Respiration system

Symbol Description [Units] Value Value Value NN PM Adult

K1 Resistance component for laminar 2.323 2.323 0.5 ﬂow [cmH20.s.l-1]

K2 Resistance component describing 30 30 0.2 turbulance [cmH20.s2.l-2]

K3 Resistance value at Vcmax 15.330 15.330 0.2 [cmH20.s.l-1]

K4 Positive constant of the lower airway 1 1 9.5 resistance[cmH20.s]

Vcmax Maximum intermediate airway 0.0130 0.0026 0.1822 volume[l]

Vuc Unstressed intermediate airway 0.0069 0.00167 0.069 volume[l]

Cc Compliance of the intermediate 0.0004 0.0002 0.0131 airways[cmH20.l-1]

VuA Unstressed volume of the lungs[l] 0.0820 0.0173 1.265 -1 CA Lung compliance[cmH20.l ] 0.0039 0.0023 0.2 -1 Ccw Chest wall compliance[cmH20.l ] 0.020 0.0064 0.2445

Pmax Maximum muscle activity[cmH20] -5 -3.73 -5

B0 Basal level at end expiration 0 0 0

232 TI Inspiration time[s] 0.500 0.375 1.875

β Parameter characterizing Pmus during 2.3 2.3 2.3 expiration

α Parameter characterizing Pmus during 2.6 2.6 2.6 inspiration

B.3 Baroreﬂex and pulmonary stretch receptors

Symbol Description Value Value Value NN PM Adult

Baroreﬂex dynamics

Kb Gain of the baroreceptors 1 1 1

τb Time constant of the baroreceptors[s] 2 2 2

Feedback regulation of the heartrate

aV Sigmoid parameter 0 0 0

bV Sigmoid parameter 1 1 1

MV Sigmoid parameter[mmHg] 55 42 80 -1 λV Sigmoid parameter[mmHg ] -0.040 -0.04 -0.04

DV Delay of vagal response[s] 0.2 0.2 0.2 -1 Gpsr Pulmonary stretch receptors gain[l ] 27 18.9 0.66

Vpsr Pulmonary stretch receptors set-point 0.0966 0.0267 2.265 value[l] -1 Gchem Chemoreﬂex gain[mmHg ] 0 0.1 N/A

Uchem Chemoreﬂex set-point value[mmHg] 0 1.878 N/A

Kv Gain of the vagal response[Hz] 0.800 0.8 0.8

τv Time constant of the vagal response[s] 1.500 1.5 1.5

aS Sigmoid parameter 0.300 0.3 0.3

bS Sigmoid parameter 0.700 0.7 0.7

MS Sigmoid parameter[mmHg] 50 37 75 -1 λS Sigmoid parameter[mmHg ] 0.090 0.09 0.09

DS Delay of sympathetic response[s] 2 2 2

233 KS Gain of the sympathetic response[Hz] 1 1 1

τs Time constant of the sympathetic 2 2 2 response[s] HR0 Basal value of heart rate[Hz] 2.380 2.36 1.35

Sympathetic regulation mechanisms aRin Sigmoid parameter 0 0 0 bRin Sigmoid parameter 1 1 1

MRin Sigmoid parameter[mmHg] 55 42 80 -1 λRin Sigmoid parameter[mmHg ] 0.040 0.04 0.04

DRin Delay of sympathetic regulation of 3 3 3 the systemic peripheral resistances[s]

KRin Gain of sympathetic regulation of the 2.205 4.432 0.525 systemic peripheral resistances[mmHg.ml-1.s]

τRin Time constant of sympathetic 6 6 6 regulation of the systemic peripheral resistances[s]

Rin,sves,Min Minimum value of the systemic 1.365 2.744 0.325 peripheral resistances [mmHg.ml-1.s] aRout Sigmoid parameter 0 0 0 bRout Sigmoid parameter 1 1 1

MRout Sigmoid parameter[mmHg] 55 42 80 -1 λRout Sigmoid parameter[mmHg ] 0.040 0.04 0.04

DRout Delay of sympathetic regulation of 3 3 3 the systemic peripheral resistances[s]

KRout Gain of sympathetic regulation of the 2.205 4.432 0.525 systemic peripheral resistances[mmHg.ml-1.s]

τRout Time constant of sympathetic 6 6 6 regulation of the systemic peripheral resistances[s]

Rout,sves,Min Minimum value of the systemic 1.365 2.744 0.325 peripheral resistances[mmHg.ml-1.s]

234 aV usv,i Sigmoid parameter 0 0 0 bV usv,i Sigmoid parameter 1 1 1

MV usv,i Sigmoid parameter[mmHg] 55 42 80 -1 λV usv,i Sigmoid parameter[mmHg ] -0.040 -0.04 -0.04

DV usv,i Delay of sympathetic regulation of 10 10 6 the intrathoracic venous unstressed volume[s]

KV usv,i Gain of sympathetic regulation of 40.878 21.758 645.75 intrathoracic venous unstressed volume[ml]

τV usv,i Time constant of sympathetic 6 6 6 regulation of intrathoracic venous unstressed volume[s]

Vusv,i,Min Minimum value of the intrathoracic 44.284 15.714 699.56 venous unstressed volume[ml] aV usv,e Sigmoid parameter 0 0 0 bV usv,e Sigmoid parameter 1 1 1

MV usv,e Sigmoid parameter[mmHg] 55 42 80 -1 λV usv,e Sigmoid parameter[mmHg ] -0.040 -0.04 -0.04

DV usv,e Delay of sympathetic regulation of 10 10 6 the extrathoracic venous unstressed volume[s]

KV usv,e Gain of sympathetic regulation of 34.363 18.290 542.64 extrathoracic venous unstressed volume[ml]

τV usv,e Time constant of sympathetic 6 6 6 regulation of extrathoracic venous unstressed volume[s]

Vusv,e,Min Minimum value of the extrathoracic 37.226 13.209 587.86 venous unstressed volume[ml] aEMAX,lv Sigmoid parameter 0 0 0 bEMAX,lv Sigmoid parameter 1 1 1

MEMAX,lv Sigmoid parameter[mmHg] 55 42 80

235 -1 λEMAX,lv Sigmoid parameter[mmHg ] 0.040 0.04 0.04

DEMAX,lv Delay of sympathetic regulation of 3 3 3 the end-systolic elastance in the left ventricle[s]

KEMAX,lv Gain of sympathetic regulation of the 24.255 68.354 1.5 end-systolic elastance in the left ventricle [mmHg.ml-1]

τEMAX,lv Time constant of sympathetic 10 10 10 regulation of the end-systolic elastance in the left ventricle[s]

EMax,lv,Min Minimum value of the end-systolic 48.509 136.708 3 elastance in the left ventricle [mmHg.ml-1]

aEMAX,rv Sigmoid parameter 0 0 0

bEMAX,rv Sigmoid parameter 1 1 1

MEMAX,rv Sigmoid parameter[mmHg] 55 42 80 -1 λEMAX,rv Sigmoid parameter[mmHg ] 0.040 0.04 0.04

DEMAX,rv Delay of sympathetic regulation of 3 3 3 the end-systolic elastance in the right ventricle[s]

KEMAX,rv Gain of sympathetic regulation of the 15.713 44.282 0.289 end-systolic elastance in the right ventricle[mmHg.ml-1]

τEMAX,rv Time constant of sympathetic 10 10 10 regulation of the end-systolic elastance in the right ventricle[s]

EMax,rv,Min Minimum value of the end-systolic 31.426 88.564 0.577 elastance in the right ventricle[mmHg.ml-1]

B.4 Chemoreﬂex

236 Symbol Description Value Value Value NN PM Adult

Central chemoreﬂex

Dc Delay from the systemic arteries to 4 4 8 the central chemosensitive area[s]

Pa,CO2,nC Pa,CO2 set-point value[mmHg] 34.500 34 40

Gc,A Central chemoreﬂex gain for the -0.240 -0.24 -2 amplitude of the respiratory muscles

τc,A Time constant of the central 105 105 105.0 chemoreﬂex controlling the amplitude of the respiratory muscles[s]

Gc,f Central chemoreﬂex gain for the 0.070 0.07 0.015 respiratory frequency [Hz.mmHg-1]

τc,f Time constant of the central 400 400 400 chemoreﬂex controlling the respiratory frequency[s]

Peripheral chemoreﬂex

Dp Delay from the systemic arteries to 7 7 7 the peripheral chemosensitive area[s]

Pa,CO2,nP Pa,CO2 set-point value[mmHg] 34.500 34 N/A

Gp,A,CO2 Peripheral chemoreﬂex gain for the -0.060 -0.06 N/A amplitude of the respiratory muscles[cmH2O.mmHg-1]

τp,A,CO2 Time constant of the peripheral 83 83 N/A chemoreﬂex controlling the amplitude of the respiratory muscles[s]

Gp,f,CO2 Peripheral chemoreﬂex gain for the 0.030 0.03 N/A respiratory frequency[Hz.mmHg-1]

τp,f,CO2 Time constant of the peripheral 147.780 147.78 N/A chemoreﬂex controlling the respiratory frequency[s]

Pa,O2,nP Pa,O2 set-point value[mmHg] 72 77.5 N/A

237 Gp,A,O2 Peripheral chemoreﬂex gain for the 0.300 0.3 N/A amplitude of the respiratory muscles[cmH2O.mmHg-1]

τp,A,O2 Time constant of the peripheral 83 83 N/A chemoreﬂex controlling the amplitude of the respiratory muscles[s]

Gp,f,O2 Peripheral chemoreﬂex gain for the -0.100 -0.1 N/A respiratory frequency[Hz.mmHg-1]

τp,f,O2 Time constant of the peripheral 147.780 147.78 N/A chemoreﬂex controlling the respiratory frequency[s]

Pmax0 Basal value of the maximum muscle -5 -3.73 N/A activity[cmH2O]

BR0 Basal value of the breathing rate [Hz] 0.750 1 0.2

Ach Constant parameters tuned to ﬁt N/A N/A 600 experimental data []

Bch Constant parameters tuned to ﬁt N/A N/A 10.18 experimental data []

KO2 Constant parameters tuned to ﬁt N/A N/A 200 experimental data []

KCO2 Constant parameters tuned to ﬁt N/A N/A 1 experimental data [s-1]

Kstat Upper saturation for the static N/A N/A 20 response of the peripheral chemoreceptors [s-1]

τph Time constants of the pole in the N/A N/A 3.5 high-pass transfer function [s]

τzh Time constants of the zero in the N/A N/A 600 high-pass transfer function [s]

τpl Time constant of the real pole in the N/A N/A 3.5 low-pass ﬁlter [s]

238 Kdyn Upper saturation for the N/A N/A 45 rate-dependent response of the peripheral chemoreceptors components, [s-1]

Gp,A Peripheral chemoreﬂex gain for the N/A N/A -2 amplitude of the respiratory muscles

τp,A Time constant of the peripheral N/A N/A 105.0 chemoreﬂex controlling the amplitude of the respiratory muscles[s]

Gp,f Peripheral chemoreﬂex gain for the N/A N/A 0.015 respiratory frequency [Hz.mmHg-1]

τp,f Time constant of the peripheral N/A N/A 400 chemoreﬂex controlling the respiratory frequency[s]

Ct Time constant of the peripheral N/A N/A 0.46 chemoreﬂex controlling the respiratory frequency[s]

facp,n Aﬀerent peripheral chemoreceptor N/A N/A 3.7 activity set-point value [spikes/s]

B.5 Lung gas exchange

Symbol Description Value Value Value NN PM Adult

Lung gas exchange (Dead Volume, Alveoli, pulmonary capillaries)

FI,O2 Partial fraction of inspired O2 0.2103 0.2103 0.2103

FI,CO2 Partial fraction of inspired CO2 0.000421 0.000421 0.00042 K Constant including the shunt fraction 1.21 1.21 1.21 and the convertion to BTPS(Body temperature and pressure saturated

239 Patm Atmospheric pressure[mmHg] 760 760 760

PWS Saturated vapour pressure of water at 47 47 47 body temperature[mmHg] a Fitted parameter to the oxygen 0.226 0.186 N/A equilibrium curve b Fitted parameter to the oxygen 12.5 11.35 N/A equilibrium curve[mmHg-1] c Fitted parameter to the oxygen 3.1 3.266 N/A equilibrium curve fs Shunt fraction 0.022 0.01 0.02 Hgb Hemoglobin rate[g.dl-1] 16.200 13.4 15

Csat,O2 Constant parameter [mM.l] N/A N/A 9

Csat,CO2 Constant parameter [nm.l] N/A N/A 86.11

h1 Constant parameter [] N/A N/A 0.3836

K1,O2 Constant parameter [mmHg] N/A N/A 14.99 -1 α1 Constant parameter [mmHg ] N/A N/A 0.03198 -1 β1 Constant parameter [mmHg ] N/A N/A 0.008275

h2 Constant parameter [] N/A N/A 1.819

K2,CO2 Constant parameter [mmHg] N/A N/A 194.4 -1 α2 Constant parameter [ mmHg ] N/A N/A 0.05591 -1 β2 Constant parameter [mmHg ] N/A N/A 0.03255 Z Conversion factor [l.mM-1] N/A N/A 0.0227

B.6 Tissue gas exchange

Symbol Description Value Value Value NN PM Adult

VT,sves Tissue volume [ml] 1785.950 510.271 35719

MO2sves Metabolic O2 comsumption 0.467 0.15 4.1 rate[ml.s-1]

MCO2sves Metabolic CO2 production 0.292 0.15 3.45 rate[ml.s-1]

240 B.7 Gas transport

Symbol Description Value Value Value NN PM Adult

τLT Time to transport gases from the 8 8 18 lungs to the systemic tissues[s]

τVL Time to transport gases from the 9 9 10 systemic veins to the lungs[s]

B.8 Other cardio-respiratory interactions

Symbol Description Value Value Value NN PM Adult

Rpp0 Basal value of the peripheral vessels 38.250 0.099 0.276 [mmHg.ml-1.s]

Vthres Threshold parameter[l] 0.6 0.0063 5.8

241 LISTOF FIGURES

1.1 Cardiovascular system, composed of the heart and the pulmonary and systemic circulations...... 27 1.2 Anatomical structure of the heart...... 28 1.3 Cardiac electrical conduction system...... 29 1.4 Phases of the action potential from a SA node and a ventricular cell. . . . 30 1.5 Neural connection between the central nervous system (CNS) and effector organs by the sympathetic and parasympathetic branches...... 32 1.6 Autonomic and cardiovascular closed-loop system regulation...... 33 1.7 General diagram of interactions of the brainstem respiratory neural network. 38 1.8 Representation of the response of the ventilation to a disturbance with different loop gains...... 41 1.9 Nasal pressure signal acquired using a nasal pressure transducer in an adult apneic patient...... 43 1.10 Scheme of the sleep apnea syndrome cyclical pathogenesis...... 48 1.11 Example of a typical recording of a patient suffering from sleep apnea syndrome from PASITHEA Project...... 49 1.12 Example of Cheyne-Stokes respiration pattern...... 51 1.13 Age terminology during perinatal period...... 52

2.1 Diagram of the more important blocks of the cardio-respiratory system and its interactions...... 70

3.1 Formalism Transformation Graph (FTG)...... 91 3.2 Hierarchical structure of models and simulators...... 93 3.3 Object oriented representation of models in M2SL...... 95 3.4 General execution and simulation loop...... 96 3.5 Uncertainty and sensitivity analysis process...... 100 3.6 Example of sensitivity analysis with the Morris method...... 103 3.7 Identiﬁcation of important parameters with the Morris method...... 105

242 3.8 General scheme of genetic algorithms...... 109 3.9 Diagram of the Diﬀerential evolution algorithm...... 110 3.10 Migration ring topology...... 113

4.1 Diagram of the cardio-respiratory adult model...... 122 4.2 Diagram of the cardiovascular system model...... 123 4.3 Diagram of the simplified cardiac electrical system submodel...... 124 4.4 Electric diagram of the respiratory system model...... 128 4.5 Lung gas exchange model...... 130 4.6 Tissue gas exchange model...... 131 4.7 Gas transport model...... 132 4.8 Chemoreflex model...... 133 4.9 Diagram of the baroreflex system...... 134 4.10 Simulated blood pressures and volumes using the adult model...... 137 4.11 Pressure, volume, and flow waveforms generated by the respiratory model using the adult model ...... 138

4.12 Arterial partial pressure of O2 and CO2 and the arterial oxygen saturation simulated by the cardio-respiratory model...... 140 4.13 Eﬀects of respiration on cardiovascular outputs...... 142 4.14 Simulation of an obstructive apnea event using the adult cardio-respiratory model...... 143

5.1 Example of a typical recording of a patient suﬀering from sleep apnea syndrome from PASITHEA Project...... 151 5.2 Example of a variable used for the sensitivity analysis (X) and how the

∆X markers are obtained from a simulation of a 20 second obstructive apnea.155 5.3 Most inﬂuential parameters of the proposed adult model obtained through

the Morris screening method, using the output function ∆X , with X ∈

{SaO2, P aO2, P aCO2,HR}...... 156 5.4 Most inﬂuential parameters of the Morris screening method using the out- Mean put function DeltaX on SaO2, PaO2, Pa cO2 and HR on the adult model.158 5.5 Local sensitivity analysis of: A) the fraction of oxygen of inspire air (FIO2)

and B) the metabolic rate of oxygen consumption (MO2sves) on the adult model during an obstructive apnea...... 160

243 5.6 Local sensitivity analysis of the central chemoreﬂex gain for the amplitude

of the respiratory muscles (Gc,A) on the adult model during an obstructive apnea...... 162 5.7 Example of parameter identification. OSA with normal recovery...... 169 5.8 Example of parameter identification. OSA followed by a hypopnea. . . . . 170 5.9 Example of parameter identification. OSA followed by another apnea. . . . 171 5.10 K-mean clustering on the 107 set of parameters identified for the adult cardio-respiratory model using the PASITHEA database...... 172 5.11 Comparison of the set of parameters of each one of the barycenters of the clusters obtained with the K-means...... 172 5.12 Simulations of an OSA using the adult cardio-respiratory model with the parameters of the barycenters of the three cluster found with the K-means clustering method...... 173 5.13 Comparison of the set of parameters of each one of the barycenters of the clusters obtained with the K-means...... 175

6.1 Chemoreflex model for newborns...... 185 6.2 Diagram of the baroreflex system for newborns...... 185 6.3 Oxygen dissociation curves of blood from (A) term infants and (B) preterm infants at different postnatal ages...... 186

6.4 Simulated blood pressures and volumes using Mterm and M28 models. . . . 192 6.5 Pressure, volume, and ﬂow waveforms generated by the respiratory model

using the Mterm and M28 models...... 194

6.6 Arterial partial pressure of O2 and CO2 (PaO2, PaCO2) and the arterial

oxygen saturation (SaO2) simulated by the Mterm and M28 models. . . . . 195 6.7 Comparison between experimental data and model output signals during an apnea...... 197 6.8 Most inﬂuential parameters of the Morris screening method using the out- Mean put function ∆X on SaO2 and ∆X on SaO2 and HR on the Mterm and

M28 models...... 198

6.9 Local sensitivity analysis of the fraction of oxygen of inspire air (FIO2) on

the Mterm model and the M28 model ...... 199

244 LISTOF TABLES

2.1 Cardio-respiratory models in the literature with its components and the context where they were used...... 75

3.1 Formalisms supported in M2SL...... 93 3.2 Metadata related to variables and parameters...... 94

4.1 Baseline values and simulated hemodynamics vital signs for an adult. . . . 136 4.2 Baseline values and simulated respiratory variables for an adult...... 136 4.3 Mean baseline values and mean simulated gas exchange variables for an adult...... 139

5.1 List of patients and selected apneas for patient-speciﬁc identiﬁcation. . . . 153 5.2 Median and Interquartile range of the RMSE and rRMSE between the

identiﬁed model simulation of SaO2 and the experimental SaO2 per patient and for the total database...... 168 5.3 Apneas per Cluster for each Patient...... 174

6.1 Summarize of the characteristics of each patient of the database of Sher- brooke University at birth and at the recording and the amount of annotated apneas per patient...... 190 6.2 Baseline values and simulated hemodynamics vital signs for newborn at term and preterm...... 191 6.3 Baseline values and simulated respiratory variables for newborns at term and preterm...... 193 6.4 Mean baseline values and mean simulated gas exchange variables for a newborn at term and a preterm...... 194

245 LISTOF PUBLICATIONS

Journal articles 1. Hernández, A., Guerrero, G., Feuerstein, D., Graindorge, L., Perez, D., Amblard, A., Mabo, P., Pépin, J. L., Senhadji, L., (2016). "PASITHEA: An Integrated Mon- itoring and Therapeutic System for Sleep Apnea Syndromes Based on Adaptive Kinesthetic Stimulation." IRBM 37(2): 81-89. 2. Hernández, A., Pérez, D., Feuerstein, D., Loiodice, C., Graindorge, L., Guerrero, G., et al. ”Kinesthetic stimulation for obstructive sleep apnea syndrome: An “on-oﬀ” proof of concept trial.”. 2018. Scientiﬁc reports.

International conferences 1. Feuerstein D., Graindorge, L., Amblard, A., Tatar, A., Guerrero G., Christophle- Boulard, S., Loiodice, C., Hernandez, A., Pépin, J. L., (2015) "Real-time detection of sleep breathing disorders," 2015 Computing in Cardiology Conference (CinC), pp. 317-320. 2. Perez, D., Guerrero, G.,Feuerstein, D., Graindorge, L., Amblard, A., Senhadji, L., Pepin, J., Hernandez, A.,(2016). "Closed loop Kinesthetic Stimulation for the Treatment of Sleep Apnea Syndromes." 2016 Computing in Cardiology Conference (CinC), Vol 43. 3. Guerrero, G., Le Rolle, V., Hernández, A. "Sensitivity Analysis of a Cardiorespira- tory Model for the Study of Sleep Apnea". 2018 Computing in Cardiology Conference (CinC). 4. Guerrero, G., Le Rolle, V., Pépin, J.L., Hernández, A. Parameter identiﬁcation of a cardio-respiratory model for the study of obstructive sleep apnea. 2020 VPH2020 Conference.

246

Titre : Analyse à base de modèles des interactions cardiorespiratoires chez l’adulte et chez le nouveau-né

Mot clés : Apnée, prématuré, Identiﬁcation spéciﬁque-patient, Analyse de sensibilité

Résumé : Les mécanismes physiologiques à tats, un sous-ensemble de paramètres a été l’origine des épisodes d’apnée, chez l’adulte et sélectionné pour réaliser la première identifi- le prématuré, ne sont pas encore entièrement cation spécifique-patient d’un modèle adulte élucidés. L’objectif principal de cette thèse est pour étudier la dynamique de SaO2 pendant de proposer une approche, basée sur des mo- une apnée obstructive à partir d’une base de dèles computationnels, afin de mieux com- données clinique composée de 107 apnées prendre la réponse cardio-respiratoire aiguë à obstructives réparties sur 10 patients. À par- un épisode d’apnée. Un modèle original des tir des paramètres identifiés, un phénotypage interactions cardio-respiratoires a été proposé des patients a été obtenu, différenciant les et adapté selon 3 versions : adulte, nouveaux- patients présentant un risque accru d’instabi- né à terme et prématuré. Les analyses de sen- lité respiratoire et de respiration périodique. sibilité, réalisées sur ces modèles, ont mis en Les résultats de la thèse ouvrent de nouvelles en évidence l’importance de certaines gran- perspectives pour la prise en charge et l’opti- deurs physiologiques : fraction d’oxygène ins- misation de certaines thérapies (CPAP, PEEP, piré, les taux métaboliques, le chemoréflexe et oxygénothérapie,...) en unités de soins inten- le volume pulmonaire. À partir de ces résul- sifs néonatals et adultes.

Title: Model-based analysis of cardio-respiratory interactions in adults and newborns

Keywords: Apnea, Prematurity, Patient-speciﬁc identiﬁcation, Sensitivity analysis

Abstract: The physiological mechanisms be- results, a subset of parameters was selected hind apnea episodes in adults and prema- to perform the first patient-specific identifica- ture infants are not yet fully elucidated. The tion of an adult model to study the dynamics main objective of this thesis is to propose of SaO2 during an obstructive apnea from a an approach, based on computational mod- clinical database composed of 107 obstructive els, in order to better understand the acute apneas distributed over 10 patients. From the cardio-respiratory response to an episode of parameters identified, a phenotyping of the pa- apnea. An original model of cardio-respiratory tients was obtained, differentiating the patients interactions has been proposed and adapted with an increased risk of respiratory instabil- in 3 versions: adult, newborn at term and ity and periodic breathing. The results of the preterm infant. Sensitivity analyses, performed thesis open up new perspectives for the man- on these models, have highlighted the impor- agement and optimization of certain therapies tance of certain physiological variables: the (CPAP,PEEP,oxygen therapy, etc.) in neonatal fraction of inspired oxygen, metabolic rates, and adult intensive care units. the chemoreflex and lung volume. From these