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Karim El Houari
To cite this version:
Karim El Houari. Modeling and imaging of electrocardiographic activity. Signal and Image processing.
Université Rennes 1, 2018. English. �NNT : 2018REN1S063�. �tel-02066820�
THÈSE DE DOCTORAT DE
L’UNIVERSITE DE RENNES 1
COMUEUNIVERSITEBRETAGNELOIRE
Ecole Doctorale N°601
Mathèmatique et Sciences et Technologies de l’Information et de la Communication Spécialité :Signal, Image, Vision
Par
Karim EL HOUARI
Modélisation et imagerie électrocardiographiques
Thèse soutenue à RENNES, le 14 Décembre 2018
Unité de recherche : Laboratoire du Traitement du Signal et de l’Image Thèse N° :
Rapporteurs avant soutenance :
Christian Jutten (PU), GIPSA-LAB Freddy Odille (CR-INSERM), IADI
Composition du jury :
Examinateurs
Sofiane Boudaoud (MCF-HDR), BMBI Christian Jutten (PU), GIPSA-LAB Freddy Odille (CR-INSERM), IADI Directeur de thèse
Laurent Albera (MCF-HDR), LTSI Co-directeurs de thèse
Alfredo Hernández (DR-INSERM), LTSI
A CKNOWLEDGMENT
First and foremost, I would like to thank Professor Christian Jutten and Associate Professor Freddy Odille for giving their time and expertise to review this manuscript. I am also thankful to them and Associate Professor Sofiane Boudaoud for the interest they demonstrated for this work, and I very much appreciated the constructive discussion that followed my defense.
I would like to express my deep and sincere gratitude to my research supervisors in the LTSI laboratory: Associate Professor Laurent Albera, Research Director Alfredo Hernàndez and Research Engineer Amar Kachenoura. Not only their valuable critiques, advice and close guidance were prominent in the progress of my research, but their support, availability and kindness made this whole life experience unforgettable.
This project wouldn’t have been possible without the help and financial support of the company Ansys. In particular, I would like to thank Michel Rochette, head of Ansys research team in Lyon and Christelle Boichon-Grivot, Research Engineer within Ansys, first for their management and investment in this project, but also for offering such convenient working conditions. I would also like to extend my thanks to my collegues in Ansys: Valéry Morgenthaler, François Chapuis and Frédéric Thevenon for their valuable help in using Ansys Fluent and Ansys Mechanical software.
Many thanks go to Associate Professor Eric Darrigrand from the IRMAR laboratory for receiving me several times in his office and for removing some mathematical modeling ambiguities.
I also wish to acknowledge the help provided by my colleagues in the LTSI: Ahmad Karfoul, Siouar Bensaid and Paul Berraute, which saved me a lot of time in understanding and implementing various elements of this PhD such as Finite Element Method operators, data whitening and Morozov discrepancy principle. Special thanks go to Doctor Raphaël Martins and his colleagues of the university hospital of Rennes for their warm welcome in the electrophysiology room, the clinical data they provided, and the valuable explanations of the complex electrical mechanisms behind cardiac arrhythmia. More broadly, I truly consider myself lucky to experience such a pleasant professional environment, this is why I would like to thank all my colleagues in the LTSI for their warm welcome, their help, and the joyful time I have had in and outside the lab.
I must also thank my friends and my dear housemates for their moral support and for making these years I spent in Rennes such a thrill.
R ÉSUMÉ SUBSTANCIEL
Cette thèse est le résultat de la collaboration entre le Laboratoire Traitement du Signal et de l’Image (LTSI) et l’entreprise Ansys. Plus précisément, elle regroupe les compétences de deux équipes du LTSI: celles de l’équipe SEPIA en modéllisation de l’électrophysiologie cardiaque, et celles de l’équipe SESAME en termes de méthodes pour la localisation de sources épileptiques de manière non-invasive.
ANSYS est une entreprise qui est leader dans le domaine de la simulation numérique de systèmes multiphysiques. Ce travail est destiné à initier une activité de recherche dans le domaine de la modélisation électrique cœur-torse au sein de ces entités.
Motivation
Les maladies cardiovasculaires sont la cause majeure de mortalité dans le monde et représente un fardeau économique et sanitaire conséquent [106]. La moitié des décès causés par ces maladies est dûe à une mort cardiaque subite, qui est un arrêt soudain de la circulation du sang résultant de l’incapacité du cœur à pomper dans le corps [105]. Cette insuffisance cardiaque est la plus part du temps causée par une propagation anormale de l’activité électrique dans le cœur, phénomène connu sous le nom d’arythmie. Parmi les arythmies cardiaques, la Tachycardie Ventriculaire (TV) est la plus mortelle et représente 80% des morts subites. Malgré le développement de plus de 100 anti-arythmogènes, les efforts mis dans la prévention des arythmies ventriculaires sérieuses et fatales restent décevants. L’utilisation des Défibrillateurs Cardiaques Implantables (DCIs) est préconisé à la fois pour la prévention des morts subites chez les patients atteints de maladies structurelles et ceux en cas d’insuffisance de la fraction d’éjection du ventricule. Cependant, l’implantation des DCIs est très coûteuse et n’est pas curative. La perpétuation de la TV dépend de régions pathologiques dont la conduction est relativement lente, donnant la possibilité aux segments adjacents dans le myocarde de regagner de l’excitabilité lorsque l’impulsion se propage lentement dans la région pathologique. Le mélange des conductions normales (tissu sain) et anormales (tissu ischémique) peut causer la formation de circuits de ré-entrée et constitue le substrat arythmogène de la TV.
L’ablation par radiofréquence des substrats ventriculaires arythmogènes par cathéterization a montré un grand succès dans le traitement de la TV. Afin de délimiter le circuit critique (isthme), une cartographie
électro anatomique est effectuée dans une salle d’électrophysiologie. Les potentiels éléctriques de la surface intramyocardique sont acquis à l’aide d’un cathéter placé à l’intérieur de la cavité cardiaque, créant une cartographie partielle de l’activité électrique du cœur. Cette procédure est invasive, longue et difficile à la fois pour le patient et pour le chirurgien. Les taux de récurrence sont significatifs et le manque de consensus clinique sur la stratégie optimale d’ablation rendent cette option encore plus complexe [5]. D’un autre côté, une étude récente montre que le succès d’une ablation multiplie les chances de survie à l’arythmie par dix [62]. Cependant, la procédure n’aboutit que dans 41 à 89% des cas [189]. C’est pourquoi des avancées sont nécessaires pour une identification plus précise et une meilleure compréhension du substrat arythmogène ventriculaire.
Dans ce contexte, un intérêt grandissant pour l’Imagerie ElectroCardioGraphique (IECG) a emergé dans la communauté scientifique. Cette technique consiste à fournir des images 3D de la distribution spatiale de l’information électrique cardiaque, de manière non-invasive. Trois éléments sont requis pour effectuer une procédure d’IECG :
• un modèle géométrique du corps obtenu à partir de données acquises par Imagerie par Résonance Magnétique (IRM) ou par tomographie;
• de multiples mesures de potentiels à la surface du torse qui procurent une version attenuée de l’activité électrique cardiaque;
• un modèle mathématique qui traduit le lien physique entre les sources électriques cardiaques et les potentiels de surface observés.
Bien que l’IECG est prometteuse en théorie, elle demeure une tâche ardure pour trois raisons principales. 1) Elle implique un modèle direct qui traduit de manière adéquate le lien physique entre les ECGs observés et les sources électriques cardiaques à reconstruire. Ce modèle est soumis à plusieurs incertitudes dont: les imprécisions géométriques liées à l’acquisition des images anatomiques et à la déformation du cœur pendant la contraction et la relaxation, les conducitivités électriques des tissus et leur anisotropies, la difficulté de représenter la directionnalité des fibres cardiaques et enfin le bruit d’instrumentation des ECGs. 2) Elle requiert la résolution d’un problème mathématique rendu difficile à cause de son caractère mal-posé. Cette nature provient du faible nombre d’observations comparé nombre de sources à reconstruire mais aussi de l’atténuation de l’activité électrique lorsqu’elle se propage entre le cœur et la surface torse. Ainsi, une solution du problème n’est pas unique, et de petites variations sur les ECGs de surface peuvent engendrer des solutions divergentes. Afin de contourner ce caractère mal-posé, une régularisation est souvent utilisée pour restreindre l’espace des solutions acceptables de manière appropriée et garantir une solution unique au problème régularisé.
Plusieurs méthodes et algorithmes ont été proposés dans ce sens, mais des efforts restent à fournir
Résumé substanciel
dans l’incorporation d’a priori physiologiques sur la solution. 3) La validation clinique chez l’homme reste très limitée puisque l’enregistrement simulanné de l’activité électrique en tout point du cœur est inaccessible à la fois pour des raisons pratiques et éthiques. Une manière d’évaluer des algorithmes inverses est par l’étude de leur performance sur des données simulées par des modèles de propagation électriques cœur-torse. La première étape consiste à modéliser l’excitation et la propagation électrique cardiaque. Un modèle direct est alors utilisé afin de calculer des ECGs de surface correspondant à des cartographies de potentiels simulées connues. Certains ECGs et le modèle direct sont ensuite utilisés comme entrée des méthodes inverses. Les cartographies estimées par ces méthodes sont finalement comparées aux données de référence. De nombreuses études du problème inverse ont été basées sur des donées simulées. Cependant, les modèles impliqués sont soit trop complexes soit ne produisent pas des ECGs cohérents.
Figure 1:Etapes de résolution du problème inverse en utilisant des données simulées
Objectifs
L’objectif principal de cette thèse est l’évaluation quantitative de méthodes de résolution du problème inverse en ECG en utilisant des données obtenues par des simulations numériques contrôlées. Pour cela, nous procédons d’abord à i) la conception d’un modèle cœur-torse qui trouve un compromis entre réalisme des données générées et complexité, puis à ii) l’exploration de nouvelles méthodes de résolution du problème inverse en ECG et leur comparaison avec les méthodes classiques, en utilisant des données générées par notre modèle. Ce travail s’inscrit dans la perspective de pouvoir évaluer qualitativement les performances de plusieurs algorithmes de résolution du problème inverse sur une multitude de scénarios en termes de géométrie, de conductivité électrique et de pathologies, en utilisant divers modèles de sources.
Contributions
La première partie de cette thèse est dédiée à la conception d’un modèle de propagation cœur- torse rapide et simplifié générant des cartographies électriques cardiaques réalistes et des ECGs morphologiques. Ce modèle est bâti sur une géométrie idélisée construite en utilisant le logiciel Ansys.
Il implémente le formalisme monodomaine augmenté couplé avec le modèle ionique phénoménologique FitzHugh-Nagumo et utilise la méthode des élements finis pour la discrétisation des équations à dérivées partielles. Les paramètres du modèle sont identifiés par le moyen d’une stratégie évolutionnaire et leur influence est étudiée à travers une méthode de sensibilité de criblage. Une fois les paramètres identifiés dans le cas sain, nous montrons que ce modèle est un outil efficace pour générer des scénarios variés de cœurs sains et pathologiques produisant des ECGs cohérents. De plus, les résultats de l’analyse de sensibilité suggèrent que seuls quelques paramètres de ce modèle sont suffisants à inférer un comportement électrique cardiaque macroscopique à partir d’un ECG donné. Ce travail constitue un cadre contrôlé afin de réaliser des simulations de problème inverse et a été en partie publié dans la conférence internationale IEEE CAMSAP :
• EL HOUARI, K., KACHENOURA, A., ALBERA, L., BENSAID, S., KARFOUL, A., BOICHON-GRIVOT, C., ROCHETTE, M.,ANDHERNÁNDEZ, A. A fast model for solving the ecg forward problem based on an evolutionary algorithm. In Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 2017 IEEE 7th International Workshop on(2017), IEEE, pp. 1–5
Deuxièmement, nous implémentons et comparons la performance des approches classiques de résolution du problème inverse en ECG en utilisant la formulation en potentiels épicardiques.
Les méthodes étudiées sont la famille des méthodes Tikhonov, et des méthodes basées sur une régularisation de norme`1. Nous étudions d’abord l’influence du pré-traitement des données et celle de l’algorithme d’optimisation utilisé.
Enfin, une nouvelle régularisation spatiotemporelle est proposée. Elle est basée sur la variation totale entre deux échantillons temporels consécutifs. Cette méthode utilise l’algorithme des directions alternées combinée avec le principe de contradiction de Morozov. Nous montrons que cette approche surpasse les méthodes classiques de résolution du problème inverse en ECG dans plusieurs scénarios.
Une étude préliminaire faisant la comparaison entre méthodes régularisantes basées sur la norme`1et
`2a été publiée dans la conférence internationale IEEE SAM:
• ELHOUARI, K., KACHENOURA, A., BCRRAUTE, P., BENSAID, S., KARFOUL, A., BOICHON-GRIVOT, C., HERNÁNDEZ, A., ALBERA, L., AND ROCHETTE, M. Investigating transmembrane current source formulation for solving the ecg inverse problem. In 2018 IEEE 10th Sensor Array and Multichannel Signal Processing Workshop (SAM)(2018), IEEE, pp. 371–375
Résumé substanciel
Organisation du manuscrit
Dans le premier chapitre, nous introduisons les notions de base en anatomie et en électrophysiologie cardiaque. Nous décrivons comment l’activité électrique est générée ainsi que la manière dont elle se propage dans le cœur et le torse. Nous expliquons ensuite les principales caractéristiques des ECGs mesurés dans le système standard d’ECG 12-Dérivations. Finalement, les méchanismes et les traitements relatifs à la TV sont présentés ainsi que les les défis reliés à la chirurgie par cathéterization par radiofréquence.
Le deuxième chapitre présente le modèle que nous proposons. Nous présentons d’abord un état de l’art des modèles électriques cœur-torse ainsi que des méthodes numériques de discreétisation impliquées. Dans un second temps nous présentons la géométrie conçue et nous décrivons de manière détaillée la dérivation et la discretisation spatiotemporelle des équations du modèle monodomaine.
Ensuite, nous présentons l’algorithme évolutionnaire utilisé pour identifier les paramètres et des résultats en termes de cartographies électriques cardiaques et ECGs. Nous abordons par la suite le sujet de la sensibilité du modèle vis à vis de ses paramètres et nous montrons que les paramètres non physiologiques ont une influence faible sur les simulations de propagation cardiaque. Enfin, nous montrons que le modèle est capable de produire des allures d’ECGs réalistes dans les cas d’une TV monomorphe et polymorphe ainsi que dans le cas d’un infarctus du myocarde.
Le troisième chapitre porte sur le problème inverse en ECG. Après un état de l’art des méthodes principales utilisées, nous analysons l’influence du pré-traitement des données et nous montrons qu’un blanchiment des données améliore les solutions. Ensuite, nous étudions la performance de différents algorithmes et de plusieurs termes de régularisation. Enfin, nous décrivons notre nouvelle approche basée sur une variation totale spatiotemporelle combinée avec le principe de contradiction de Morozov et nous étudions ses performances dans différents cas sains et pathologiques.
Le dernier chapitre contient une discussion générale des travaux présentés et les perspectives reliées en termes de modélisation, des méthodes inverses envisagées et de validation clinique.
T ABLE OF C ONTENTS
Résumé substanciel 5
List of Tables 14
List of Figures 17
List of Abbreviations 18
Introduction 21
1 Cardiac electrophysiology 25
1.1 Anatomy and function of the heart . . . . 25
1.2 Cardiac electrical activity . . . . 28
1.2.1 Cardiomyocytes . . . . 29
1.2.2 Pacemaker cells . . . . 30
1.3 Cardiac conduction system and Electrocardiograms . . . . 30
1.4 Electrocardiography . . . . 32
1.4.1 History . . . . 32
1.4.2 Modern Electrocardiography and 12-Lead ECG . . . . 33
1.5 Cardiac arrhythmia . . . . 37
1.5.1 Mechanisms underlying ventricular tachycardia . . . . 38
1.5.2 Treatment . . . . 39
1.5.3 Catheter ablation procedure . . . . 40
1.6 Catheter ablation and ECGI challenges . . . . 41
2 Cardiac modeling 43 2.1 State of the art . . . . 43
2.1.1 Cardiac cell models . . . . 44
2.1.2 Macroscopic cardiac source models and extra-cardiac potentials . . . . 45
2.1.3 Simulating propagation in the heart . . . . 47
2.1.4 Numerical methods . . . . 48
2.2 Proposal of a simplified heart-torso model . . . . 49
2.2.1 Heart-torso geometry and meshing . . . . 50
2.2.2 Model equations . . . . 52
2.2.3 Spatio-temporal discretization . . . . 56
2.3 Parameter identification using an evolutionary strategy . . . . 61
2.3.1 Optimization problem formulation . . . . 64
2.3.2 Evolutuionary Algorithms . . . . 66
2.3.3 Numerical simulations . . . . 67
2.4 Sensitivity analysis and model reduction . . . . 78
2.4.1 Morris method . . . . 78
2.4.2 Numerical simulations . . . . 80
2.5 Pathology simulation . . . . 81
2.6 Conclusions . . . . 93
3 ECG inverse problem 95 3.1 State of the art . . . . 96
3.1.1 Source formulations . . . . 96
3.1.2 Classical methods . . . . 97
3.2 Methods . . . 100
3.2.1 Data pre-processing . . . 100
3.2.2 Inverse problem formulation . . . 102
3.2.3 Optimization schemes . . . 104
3.2.4 Determination of the regularization parameter . . . 106
3.2.5 A new spatiotemporal approach with two`1constraints . . . 109
3.3 Numerical experiments . . . 110
3.3.1 Experiment 1: influence of whitening and regularizers . . . 111
3.3.2 Experiment 2: evaluation of the ADMM combined with the discrepancy principle . 120 3.3.3 Experiment 3: Performance study of the double`1constraint . . . 122
3.4 Conclusions . . . 123
General conclusion & perspectives 125
A The full heart-torso bidomain model 129
B Quasi-static approximation 133
TABLE OF CONTENTS
C Epicardial formulation of the forward problem 135
Bibliography 139
1.1 Normal ECG characteristics . . . . 36
2.1 Reference ECG features. . . . 68
2.2 Parameters variation domain . . . . 69
2.3 Reference featuresciref and best solution featuresci(p). . . . 72
3.1 Variation grid for parameterλ . . . 112
3.2 Variation grid for parameterρ . . . 112
L IST OF F IGURES
1 Etapes de résolution du problème inverse en utilisant des données simulées . . . . 7
2 Steps for solving the inverse problem using simulated data . . . . 23
1.1 Human heart and body [13] . . . . 25
1.2 Human heart anatomy [185]. . . . 26
1.3 Heart layers[112] . . . . 27
1.4 Systole and Diastole in the atria and the ventricles[112] . . . . 28
1.5 Typical myocite action potential . . . . 29
1.6 Typical pacemaker cell action potential . . . . 30
1.7 Action potential spread and resulting ElectroCardioGram (ECG)[102] . . . . 32
1.8 An early commercial ECG device manufactured by Cambridge Scientific Instrument Company of London, 1911 [193] . . . . 33
1.9 Electrodes placement and normal 12-Lead ECG graph[1] . . . . 35
1.1012-Lead ECG angular representation [118] . . . . 35
1.11Typical ECG signal and its components [6] . . . . 36
1.12Cardioinsight ECG recording vest [2] . . . . 37
1.13Ventricular Tachycardia illustration and corresponding ECG signal . . . . 39
2.1 Heart-torso geometry . . . . 51
2.2 Heart geometry and its seven regions . . . . 52
2.3 Heart-torso mesh . . . . 53
2.4 SinoAtrial Node location . . . . 59
2.5 Heart-torso geometry and 12-Lead ECG electrode configuration . . . . 61
2.6 Left: phase plane representation of the nullclines ∂v∂tm = 0(blue), ∂u∂t = 0(red) and the obtained trajectory (green). Right: transmembrane potential (top) and recovery variable (bottom) . . . . 63
2.7 Action potentials generated by the FHN model by region . . . . 64
2.8 Target ECG signal and its features . . . . 65
2.9 Diagram explaining the basic idea of crossovers and mutations . . . . 67
2.10 Repolarization and repolarization times in an action potential . . . . 68
2.11 Boxplot of cost values evolution through generations . . . . 70
2.12 Estimated regional transmembrane potentials . . . . 71
2.13 Normalized ECG estimated by the EA (red) and reference ECG (blue) . . . . 72
2.15 Transmembrane potential (up) and corresponding bipolar potential (down) at node 100 . . 73
2.14 Simulated 12-Lead ECGs . . . . 74
2.16 Transmembrane potentials spatial distributions at 9 key instants of the ECG . . . . 76
2.17 Local activation times (ms) . . . . 77
2.18 Example of Morris method experiment plan in the simple case whereP = 2, L= 9and R= 4 . . . . 79
2.19 Sensitivity graphs of parameters by region . . . . 83
2.20 Sensitivity graphs when the most influential parameters are omitted . . . . 84
2.21 Sensitivity graphs obtained for five different model outputs, each color corresponds to a different ECG reference . . . . 85
2.22 Sensitivity graphs obtained for five different model outputs when the most influential parameters are omitted . . . . 86
2.23 Separation between the least and the most influential parameters by region . . . . 87
2.24 Reference (black) and simulated ECGs corresponding to the bestp(blue) and the bestp0 (red) . . . . 88
2.25 Monomorphic VT simulation . . . . 89
2.26 Polymorphic VT simulation . . . . 90
2.27 Comaparison of healthy 12-Lead ECGs (blue) and those in the case of myocardial infarction (red) . . . . 91
2.28 LAT mapping corresponding to the simulated ischemic condition . . . . 92
2.29 LAT mapping for the first ectopic beat of the simulated monomorphic VT . . . . 92
3.1 Selected times in Lead II for experiments 1 and 2 . . . 111
3.2 Graphical representation of `2 methods performance for raw and whitened data with respect toGand toX . . . 114
3.3 Graphical representation of `1 methods performance for raw and whitened data with respect toGand toX . . . 115
3.4 RE values when using`2 regularization at different times and optimal values ofλin the search grid. Best case scenarios are highlighted in shaded gray . . . 116
3.5 RE values when using`1regularization at different times and optimal values ofλandρin the search grids. Best case scenarios are highlighted in shaded gray. . . 117
LIST OF FIGURES
3.6 Estimated cardiac mappings at instant 5 by the different methods when data are whitened with respect toG . . . 118 3.7 Comparison of`2−V and`1−V along time when data are whitened with respect toG . 118 3.8 Influence of the TSVD index on the performance of`2−V and`1−V . . . 119 3.9 Comparison of `1 and`2 regularization when the gradient operator is used, for raw and
whitened data with respect toGand toX . . . 121 3.10Comparison of our proposed approach with the`1−V method when a pre-whitening of
Gis applied . . . 123
ADMM Alternating Direction Method of Multipliers.
ALM Augmented Lagrangian of Multipliers.
AVN Atrio Ventricular Node.
BEM Boundary Element Method.
CRESO Composite REsidual and Smoothing Operator.
CT Computed Tomography.
EA Evolutionary Algorithm.
ECG ElectroCardioGram.
ECGI ElectroCardioGraphy Imaging.
EDL Equivalent Double Layer.
FDM Finite Difference Method.
FEM Finite Element Method.
FHN FitzHugh-Nagumo.
FVM Finite Volume Method.
GCV Generalized Cross Validation.
IRN Iterative Reweighted Norm.
LAT Local Activation Time.
MRI Magnetic Resonance Imaging.
List of Abbreviations
PALM Proximal Alternating Linearized Minimization.
PDE Partial Differential Equation.
SAN Sino-Atrial Node.
SVD Singular Value Decomposition.
TSVD Truncated Singular Value Decomposition.
TV Total Variation.
UDL Uniform Double Layer.
VT Ventricular Tachycardia.
I NTRODUCTION
This thesis is the result of the collaboration between the “Laboratoire Traitement du Signal et de l’Image”
(LTSI) and ANSYS company, within the framework of a CIFRE industrial research agreement. More precisely, this work brings together two teams of the LTSI: the SEPIA team, with expertise in cardiac electrophysiology modeling, and the SESAME team, with expertise on algorithms for non-invasive cerebral source localization. ANSYS is a major company in the field of numerical simulation of 3D multiphysics systems. This work is intended to initiate research in the field of whole heart and torso modeling and electrocardiography imaging within these entities.
Motivation
Cardiovascular diseases are the leading cause of mortality worldwide and represent a huge health and economic burden [106]. About half of these deaths are due to sudden cardiac death, which is a sudden loss of blood flow resulting from the failure of the heart to effectively pump blood into the body [105].
This failure is in most cases caused by an abnormal propagation of electrical activity within the heart, a phenomenon referred to as cardiac arrhythmia. Among these conditions, Ventricular Tachycardia (VT) is the most life-threatening and is responsible for over 80% of sudden cardiac deaths [105]. VT is associated with a fast, abnormal heart rate that originates in the heart’s lower chambers.
Despite the development of over 100 anti-arrhythmic medications, efforts at prevention of serious and fatal ventricular arrhythmia have proved disappointing. Device-based therapies, involving the use of Implantable Cardioverter-Defibrillators (ICDs), are indicated for prevention of sudden cardiac death in patients with a structural disease. However, the implantation of such devices is expensive and non-curative. Perpetuation of VT strongly depends upon pathological regions of relatively slow conduction, enabling adjacent segments in the myocardium to regain excitability whilst the impulse travels slowly through the pathological area. This mixture of normal conduction (healthy tissue) and abnormal conduction (scar tissue) can lead to the formation of a re-entrant circuit and provide a substrate for VT.
Catheter radiofrequency ablation of the ventricular arrhythmogenic substrate has been proven to be a successful treatment for VT. In order to delineate the critical circuit (isthmus), conventional
electro anatomical mapping is performed in an interventional electrophysiology room. Voltages from the inner myocardial surface are acquired by a catheter placed inside the cardiac cavity, leading to a partial mapping of the electrical activity of the heart. This procedure is invasive, time consuming and challenging. There are significant recurrence rates and a lack of clinical consensus on the optimal ablation strategy [5]. A recent systematic review shows that a successful ablation increases the chance of future arrhythmia-free survival ten-fold [62]. However, only 41-89% of patients achieve the procedural endpoint [189]. Therefore, improvements are necessary to better identify and understand the ventricular arrhythmogenic substrate.
In this connection, ElectroCardioGraphy Imaging (ECGI) emerged as a novel tool for providing insight on cardiac electrical activity in a non-invasive fashion. The ultimate goal is to reproduce personalized cardiac spatiotemporal electrical patterns at a high resolution from ECG recordings and the acquisition of the patient’s anatomy. However, although this technique represents a considerable potential for enhancing VT catheter-based therapy, reducing exposure to angiographic X-ray radiation, and reducing the cost and time of the procedures, it remains an arduous task for three main reasons. 1) It requires an adequate model of the transfer between cardiac electrical activity and the ECG signals measured on the torso surface. 2) It requires solving a mathematical inverse problem for which no unique solution exists.
3) Clinical validation in humans is very limited since simultaneous whole heart electrical distribution recordings are inaccessible for both practical and ethical reasons.
One way to evaluate ECGI algorithms is by studying their performance on data simulated by heart-torso propagation models (Figure 2). The first step is modeling cardiac electrical excitation propagation. A forward model is then used for the synthesis of torso ECG potentials corresponding to known, simulated cardiac potential mappings. The surface ECGs and the forward model are used as an input to inverse methods. The estimated cardiac mappings by these methods are finally compared to the reference. Numerous ECG inverse problem studies have been based on simulated data. However, the involved models either suffer from their high complexity due to their high level of detail or do not produce realistic ECGs.
Objectives
The main objective of this thesis is to quantitatively compare existing and new methods for solving the inverse problem in the context of ECGI using a set of data obtained from controlled numerical whole-heart and torso simulations. In order to achieve this goal we proceed to i) the design of a heart-torso electrical model that finds a trade-off between the realism of the generated data and complexity and ii) a quantitative evaluation of reference-based performance criteria of classical and novel regularization-based ECGI algorithms.
Figure 2:Steps for solving the inverse problem using simulated data
Contributions
The first part of this thesis is dedicated to the design of a simplified and fast heart-torso propagation model that generates meaningful cardiac electrical patterns and is able to synthesize ECGs with realistic morphologies. This model is built on an idealized heart-torso geometry using ANSYS software. It implements the augmented monodomain formalism, coupled with the phenomenological FitzHugh-Nagumo (FHN) model and uses the Finite Element Method (FEM) for partial differential equations discretization. In order to produce realistic ECG signals, model parameters are identified by means of an evolutionary strategy and their influence is studied through a screening sensitivity analysis. Once parameters are identified for the healthy case, we show that this model is a reliable tool for generating various scenarios of the healthy and the diseased heart, producing realistic ECG signals. Moreover, sensitivity analysis results suggest that only few parameters of this model are sufficient to infer a desired macroscopic cardiac behavior from a given ECG observed from a patient.
This work constitutes a framework for building a controlled environment for ECGI simulations and was partly published in the IEEE CAMSAP international workshop:
• ELHOUARI, K., KACHENOURA, A., ALBERA, L., BENSAID, S., KARFOUL, A., BOICHON-GRIVOT, C., ROCHETTE, M.,ANDHERNÁNDEZ, A. A fast model for solving the ecg forward problem based on an evolutionary algorithm. InComputational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 2017 IEEE 7th International Workshop on(2017), IEEE, pp. 1–5
In the second part, we implement and compare the performance of classical approaches for solving the ECG inverse problem using the epicardial potential formulation. The studied methods are the family of Tikhonov methods, and`1 regularization based methods. We first study the influence of data pre-processing and that of the optimization algorithm used. We also propose a novel spatiotemporal
regularization based on the Total Variation (TV) norm of consecutive time samples. This method uses the Alternating Direction Method of Multipliers (ADMM) combined with Morozov’s discrepancy principle.
We show that this approach outperforms the classical methods for solving the inverse problem of ECG in several case-scenarios. A preliminary study that compares between `1 et `2 based methods was published in the IEEE SAM international workshop:
• ELHOUARI, K., KACHENOURA, A., BCRRAUTE, P., BENSAID, S., KARFOUL, A., BOICHON-GRIVOT, C., HERNÁNDEZ, A., ALBERA, L., AND ROCHETTE, M. Investigating transmembrane current source formulation for solving the ecg inverse problem. In 2018 IEEE 10th Sensor Array and Multichannel Signal Processing Workshop (SAM)(2018), IEEE, pp. 371–375
Manuscript organisation
In the first chapter, we introduce basic concepts of anatomy and cardiac electrophysiology. We describe how cardiac electrical activity is generated and propagated through the heart and the torso. We then explain the main characteristics of ECGs recorded in the standard 12-Lead ECG system. Finally, VT mechanisms and its treatment options are presented, as well as the underlying challenges of catheter ablation procedures.
The second chapter presents our proposed model. We first review heart-torso electrical models and the involved discretization methods. Secondly, we present the designed geometry and provide a detailed description of monodomain equations derivation and spatio-temporal discretization. Next, we describe the evolutionary algorithm used to identify model parameters and present results in terms of cardiac mappings and ECGs. Thereafter, we address the topic of model sensitivity to its parameters and show that non-physiological parameters have a low influence in cardiac simulations. Finally, we demonstrate that the model is capable of producing clinical patterns of ECGs related to monomorphic and polymorphic VTs as well as myocardial infarction.
The third chapter deals with the inverse problem in ECG. After a review of the main methods, we analyze the influence of data pre-processing and demonstrate that data spatial whitening enhances the solutions. Next, we study the performance of different algorithms with different regularization terms. Finally, we introduce our new approach based on spatio-temporal TV norm combined with the discrepancy principle and we study its performance through different case-scenarios.
The final chapter provides an overall discussion of this work and its perspectives in terms of modeling, the intended inverse methods to be investigated and clinical validation.
CHAPTER1
C ARDIAC ELECTROPHYSIOLOGY
This chapter describes the clinical context in which this thesis is framed. First, a brief review on the anatomy and function of the heart is provided. In particular, the generation of the cardiac electrical activity and the means for recording it are presented. Mechanisms underlying VT are described as well as the challenges raised by ablation-based procedures.
1.1 Anatomy and function of the heart
The heart is a hollow organ about the size of a closed fist located in the center of the chest, behind the sternum (figure 1.1). It is the most important muscle of the body for being the major actor for a proper blood circulation. Indeed, the heart is a muscle which regularly contracts and relaxes due to an electrical impulse that spreads through the heart tissue, and thus pumps blood through the vessels to the organs, muscles and nerves. The cardiac rhythm has to be sufficiently frequent in order to accomplish this task. In fact, most of human organs cannot survive more than few minutes if they are not supplied with oxygenated blood. An adult heart has a shape of a tilted cone which is about 12cm long, 9cm wide at its
Figure 1.1:Human heart and body [13]
furthest points, 6cm thick, and weighs 300g in average. It is composed of four major chambers, the right atrium, the left atrium, the right ventricle and the left ventricle. The right side of the heart receives blood that has a low concentration in oxygen from veins all over the body, it then pumps the blood through the pulmonary trunk into the lungs where it will become re-oxygenated. The left side of the heart receives oxygenated blood from the lungs, then it is pumped through the aorta back out to the rest of the body through a complex network of arteries, arterioles and capillaries. The veins return the deoxygenated blood to the right atrium and the cycle begins again (figure 1.2). The heart has four valves, each valve is
Figure 1.2: Human heart anatomy [185]
like a one-way door that keeps the blood flowing in the same direction. The valves are made up of two or three small but strong flaps of tissue called leaflets. Leaflets open to allow blood to flow through the valve and close to prevent blood from flowing backward. The openings and closings of the valves are controlled by blood pressure changes within each heart chamber. The tricuspid valve is positioned on the right side between the right atrium and the right ventricle, the pulmonary valve separates the right ventricle from the pulmonary artery, the mitral valve is positioned between the left atrium and the left ventricle, and the aortic valve separates the left ventricle from the aorta. The tricuspid and the mitral valves are called atrioventricular valves, the pulmonic and the aortic valves are called semilunar valves.
The heart is composed of three different layers, as shown in figure 1.3:
• the pericardium is a double walled bag that wraps the heart and the blood vessels filled with fluid that acts as a lubricant for the heart movements. The inner wall of the pericardium is a very dense
1.1. Anatomy and function of the heart
connective tissue that spreads the electrical stimulus of the heart;
• the myocardium is the muscle tissue of the heart, it forms a thick layer between the pericardium and the endocardium and is responsible for the heart contraction and relaxation;
• the endocardium is the inner wall of the heart, it protects the heart’s four chambers and is also made with connective tissue.
Figure 1.3: Heart layers[112]
Blood pressure and its circulation rhythm is defined by the precise moments at which ventricles and atria contract and relax. This mechanical activity can be decomposed into two key periods for each chamber: the systole and the diastole, described in figure 1.4.
Ventricular contraction causes the atrioventricular valves to close, which signals the beginning of the ventricular systole. The semilunar valves were closed during this period. Continued ventricular contraction increases the pressure above the pressure in the aorta and the pulmonary trunk, causing the semilunar valves to open, which is the late ventricular systole. When the ventricles relax and their pressure drop, blood flowing back towards the relaxed ventricles causes the semilunar valves to close, which is the beginning of ventricular diastole. The atrioventricular valves remain closed during this period. When the pressure in the ventricles becomes lower than the pressure in the atria, the atrioventricular valves open and blood flows into the relaxed ventricles, the atria then contract and complete the ventricular filling, which is the late diastole. As seen in figure 1.4, these mechanical
Figure 1.4: Systole and Diastole in the atria and the ventricles[112]
phases can be observed in the electrical signal recorded by an electrode on the chest. Indeed, the cardiac mechanical activity is triggered by a complex electrical activity that follows a specific pattern.
1.2 Cardiac electrical activity
Cardiac cells are filled and surrounded by a ionic solution, mostly sodium N a+, potassium K+, and calcium Ca2+. These charged atoms move between the inside and the outside of the cell through proteins called ion channels. Cells are connected through gap junctions which form channels that allow ions to flow from one cell to another. Two types of cardiac muscle cells can be discriminated: myocytes and pacemaker cells. The heart is mostly constituted of long, tubular spirally oriented myocytes referred to as muscle fibers.
1.2. Cardiac electrical activity
1.2.1 Cardiomyocytes
When exposed to electrical or chemical stimuli, small voltage variations occur across myocytes membrane due to ionic exchanges. Theses variations define anaction potential that can be described in five steps [122] (figure 1.5):
• phase 4: during its resting period, the cell is polarized. The inside of the cellular membrane is charged negatively if we take the exterior as a reference. The potential difference between the intracellular and the extracellular spaces, called the transmembrane potential, is at this stage about -90mV. This resting state is due to regulated outward leak ofK+and inward current from rectifier channels;
• phase 0: when a cell is stimulated by a current coming from an adjacent cell and its voltage exceeds a certain threshold, sodium fast channels open, which causes a quick depolarization.
The potential difference reaches about 30mV. The rise of Ca2+ concentration in the cytoplasm induces changes in the molecules constituting the cell and causes its contraction;
• phase 1: this phase occurs with the inactivation of the fast N a+ channels. A small downward deflection of the action potential is due to the movement ofK+ions;
• phase 2: a plateau phase is then observed, in which the cell remains depolarized and contracted, caused by calcium ions flowing into the cell and potassium ions outside of it;
• phase 3: characterized by a slow repolarization, caused by the closing of calcium channels, while potassium channels remain open.
Figure 1.5: Typical myocite action potential
When a cell is depolarized, gap junctions act as low resistive pathways interconnecting cardiomyocytes, causing neighboring cells, in turn, to depolarize and generate an action potential.
1.2.2 Pacemaker cells
Pacemaker cells on the other hand are self excitable and fire action potentials on their own. This group of cells constitute the natural pacemaker of the heart and determine heart rate in a healthy subject.
Pacemaker cells also pass through the same five phases as myocardial cells (figure 1.6). However, they have a different shape and do not require external stimulation to fire action potentials.
Figure 1.6: Typical pacemaker cell action potential
In both types of cells, the period from phase 0 until the end of phase 3 is called the refractory period.
It is characterized by the fact that cells cannot produce another action potential during this period. This is due to the closing of sodium channels as transmembrane voltage increases. They remain locked no matter how strong the excitatory stimulus is.
1.3 Cardiac conduction system and Electrocardiograms
Action potentials shape and duration depend on local properties of the cardiac tissue in terms of ionic channels in the cell membrane, electrical conductivity and wavefront curvature. Given cardiac cell properties and cardiac action potentials, a depolarization wave travels through the heart followed by repolarization, at each heartbeat. At a given time instant, the depolarization wave, also called the activation wavefront, separates depolarized cells from those still at rest and is located at the narrow transition region. If we take the extracellular space as a reference, the resting region is uniformly positively charged, whereas the depolarized region is uniformly negatively charged. The depolarization wavefront thus acts as a double layer of current sources or dipoles pointing towards resting cells and originating from depolarized ones. Unlike the depolarization process, repolarization is not a propagating phenomenon. It occurs because action potentials always have a finite duration and is not caused by neighbouring stimulation. However, if action pulses had the same durations, the repolarization can be also be approximated with a waveform-like phenomenon. This is the case in atria but not in ventricles.
1.3. Cardiac conduction system and Electrocardiograms
Indeed, the action potentials of the epicardial cells are shorter than those of endocardial cells, thus acting as a waveform from epicardium to endocardium, contrarily to depolarization. Following the same reasoning as that of the depolarization wave but with opposite charges, the cardiac repolarization process can also be considered as a propagation of a double layer of dipoles, but with opposite polarity than that of the activation wave.
Figure 1.7 describes the initiation and propagation of cardiac electrical activity and the corresponding action potentials in each cardiac region. In a healthy heart, each heartbeat begins in the SinoAtrial Node (SAN) or sinus node, located in the right atrium, which is mostly constituted of pacemaker cells.
The generated action potential causes neighbouring cells to depolarize. It then travels following the orientation of the fiber structure of the heart at a velocity that depends on local electrical conductivity.
Fibers are organized in a counterclockwise rotation from epi- to endocardium, they have a laminar organization and can be seen as a set of muscle sheets running radially from epi- to endocardium.
The electrical signal first spreads through the left atrium wall from the SA node to the AtrioVentricular node (AVN) where it is delayed until the end of atrial contraction. The AVN is also mostly made up of pacemaker cells and can take over electrical propagation when atrial rhythm is disrupted.
Action potentials then pass rapidly along atrioventricular bundle which extends from the AVN to the interventricular septum. The atrioventricular bundle, called His bundle, divides into right and left bundle branches with very high conductivity. The depolarization wave is thus rapidly shifted to the apex of each ventricle. Action potentials are then carried by Purkinje fibers from the bundles to the ventricular walls.
This propagation of the action potential (figure 1.7) in a wavefront-like manner is responsible for the synchronized mechanical contraction described in 1.1.
When electrodes are placed on the torso surface, the recorded signal with respect to a certain reference on the torso is referred to as the ElectroCardioGram (ECG). Roughly speaking, it measures the contributions of all potential variations in the whole heart scale. Figure 1.7 relates a normal ECG and its key events to the action potential spread at the heart level. One can notice that 3 waves arise from the trace of all the action potentials contributions:
• the P wave which represents the atria depolarization and contraction;
• the QRS complex corresponds to the ventricular depolarization and contraction;
• the T wave represents ventricular repolarization and release.
The durations and the amplitudes of these waves and the linking segments are very important for cardiac pathologies diagnosis, as detailed in section 1.4.
Figure 1.7: Action potential spread and resulting ElectroCardioGram (ECG)[102]
1.4 Electrocardiography
1.4.1 History
The first time that electrical current was recorded from skeletal muscles was in 1786 by the italian physician Luigi Galvani. Yet, it is only in 1842 thanks to the work of Dr. Carlo Matteucci that cardiac electrical potentials existence is known. He demonstrated that an electrical current accompanies every heartbeat in a frog. The first experiments were conducted in 1878 by John Burden Sanderson and Frederick Page who recorded a frog’s heart electrical current with a capillary electrometer. They were the first to detect what will be later called the QRS complex and the T wave. In 1887, Augustus D.
Waller published the first human ECG recorded by electrodes placed on the chest and the back. A breakthrough came in 1895 when Willem Einthoven highlights the five deflections P, Q, R, S and T using a correction formula and an improved electrometer. He then invented the string galvanometer in 1901, which was the first electrocardiograph, and published the first organised presentation of normal and abnormal ECGs in 1906. He was awarded the Nobel prize in 1924 for his work in electrocardiography.
The three lead electrocardiogram recorded from the two arms and one foot quickly became a widespread tool for detecting cardiac arrhythmias from the 1900s till the 1930s. At that time, it was commonly admitted that some myocardial infections could not be detected because some electrical information
1.4. Electrocardiography
were missing in the recordings. The reason was that, in theory, the recorded leads were defined with respect to a remote electrode placed in the infinite medium, which was clearly not the case in practice.
In 1934, Dr. Frank N. Wilson introduced the ’Central Terminal’ concept. It consists in considering a reference electrode defined by connecting a 5 kΩ resistor from each terminal of the limb leads to a common point called the ’Central Terminal’. In 1938, the American Heart Association and the Cardiac Society of Great Britain published their recommendation for recording exploring leads from six sites named V1 through V6 across the chest, these were used in addition to the existing three limb leads.
In 1942, Emanuel Goldberger adds three signals to the existing 9 ECGs used in clinical practice. This
Figure 1.8: An early commercial ECG device manufactured by Cambridge Scientific Instrument Company of London, 1911 [193]
marked the birth of the standard 12-Lead ECG.
Since then, many advances have been made in the instrumentation in terms of device compactness, numerical and electronic improvements. It is now an inevitable tool in any clinical setting and broadened considerably the scope of possibilities in cardiac pathologies diagnosis and monitoring.
1.4.2 Modern Electrocardiography and 12-Lead ECG
Electrocardiography consists in tracing the heart’s electrical activity over a period of time via electrical signals recorded on probes placed on the torso surface. It has revealed itself to be a fundamental tool for cardiac pathologies diagnosis and cardiac monitoring since it is non-invasive, inexpensive and easy
to use. An ECG can help detect:
• arrhythmia: alteration of cardiac electrical system;
• coronary heart disease: obstruction of blood supply in arteries due to fatty substances;
• heart attacks: sudden block of blood supply to the heart;
• cardiomyopathy: enlargement and thickening of cardiac muscle.
The ECG is a tracing of projections of cardiac electrical potentials, called leads, on specific axes, depending on probes placement. Those leads represent a view of the electrical activity of the heart from a particular angle across the body. The 12-Lead ECG became a standard in clinical practice since the American Heart Association published their recommendation in 1954. It consists in recording signals from 10 electrodes respecting the following placement (see figure 1.9):
• V1:4thintercostal space1to the right of the sternum;
• V2:4thintercostal space to the left of the sternum;
• V3: midway between V2 and V4;
• V4:5thintercostal space at the midclavicular line;
• V5: anterior axillary line at the same level as V4;
• V6: midaxillary line at the same level as V4 and V5;
• RL: anywhere above the ankle and below the torso;
• RA: anywhere between the shoulder and the elbow;
• LL: anywhere above the ankle and below the torso;
• LA: anywhere between the shoulder and the elbow.
This configuration enables one to visualize cardiac activity from 12 different angles (figure 1.10) at each moment of each recorded heartbeat. The observed leads can be decomposed into three main categories:
• 6 precordial leads defined by the 6 electrodes V1, V2, V3, V4 , V5 , V6
• 3 limb leads I, II and III. Lead I is the voltage between the left arm electrode (LA) and the Right Arm electrode (RA): I = LA - RA, lead II is the voltage between the Left Leg electrode (LL) and RA:
II = LL - RA and lead III is the voltage between LL and LA: III = LL - LA.
1space between ribs 4 and 5
