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Blood and electrical cardiac cycles
The contractions of atria and ventricles are jointed to the electrical activity of the heart. The electrical activity is autorhythmic, i.e. it is independent from the blood supply and continues even if the heart is removed from the body, for instance for a transplantation. The source of the cardiac electrical activity is a network of muscle fibers, the so-called conduction system, that generates action potentials that trigger heart contractions. The conduction system defines the electrical pathway which the cardiac action potential propagate through and that allows the chambers to contract successively.
The cardiac cycle is summarized into five phases, schematized in Figures 1.4 and 1.5:
1. The cardiac excitation starts from the sinoatrial (SA) node, in the right atria (see Figure 1.3 top-right). The cells placed in the SA node act as a pacemaker: their resting potential is not stable, so they spontaneously depolarize to threshold. When the threshold is reached, the action potential is triggered and propagates throughout atria. Since atria are depolarized, they contract. Then, tricuspid and mitral valves open and let blood through into ventricles.
2. After propagating in the atria, the action potential reaches the atrioventricular (AV) node, located in the cardiac septum, between atria and ventricles (see Figure 1.3 bottom-left). The action potential propagates through the bundle of His, which is the unique electrical connection between atria and ventricles since a fibrous skeleton preventing conduction is present elsewhere. Through the bundle of His, the action potential enters the ventricles, propagates through the septum and reaches the apex of the heart. After the injection of the blood in the ventricles, tricuspid and mitral valves close. 3. The Purkinjie fibers conduct the action potential from the apex of the heart to the rest of the myocardium. Consequently, the ventricles contract, aortic and pulmonary valves open and the blood is injected into aorta and pulmonary arteries. Also, during this phase, atria are re-polarized and lead to atria relaxation.
4. At the end of ventricles contraction, their repolarization takes place. At the same time, atrial relaxation let the blood inject through venæ cava and pulmonary veins.
5. Finally, the whole heart is at resting potential. Ventricular muscle relaxes too, and let the blood flow in through tricuspid and mitral valves.
Electrical cycle at microscopic scale
At the microscopic scale, each cell follows an electrical cycle (see Figure 1.6). In order to describe the cell state, the so-called membrane potential is measured, which is the difference between the extra- and the intra-cellular potentials.
• First, the cell has a stable resting membrane potential. When the action potential reaches the cell, its sodium ion channels, also called voltage-gated fast Na+ channels, open. The opening of these channels let the Na+ flow into the cell and produces a rapid depolarization and contraction.
• Second, a plateau phase is observed, the cell remains depolarized and contracted. The partial opening of voltage-gated slow Ca2+ channels let calcium get into the cell while several voltage-gated K+ channels also open and let potassium ions leave the cell. Therefore, the plateau is given by the equilibrium between inflowing Ca2+ and outflowing K+ ions.
• Then, the action of Na/K pump induces the decreasing of the K+ ions concen-tration into the cell. The repolarization starts, the cell potential decreases and it relaxes again.
• Finally, potassium channels close and Na+/Ca2+ exchangers let sodium flow back into the cell and calcium flow out, bringing back the cell to its resting potential.
Cardiac electrophysiology
We introduce the reader to cardiac electrophysiology by presenting some historical notes on the discovery of electrical activity of the heart at each heart beat. Then, a detailed description of the standard 12 lead Electrocardiogram introduced by Einthoven is given. Finally, an overview on the numerical approximation advances is given.
From Descartes “animal spirits”. . .
Since the 16th century, physicists, medical doctors and philosophers have been interested in the causes of living movements. The first work stating that the blood and the nervous system are the causes of the body movements is due to René Descartes, French Philosopher (1596–1650) best known for the philosophical statement “Cogito ergo sum”. The post-mortem treatise Passions of the Soul (Les passions de l’âme) is published in 1662. Descartes defines the “animal spirits” as produced in the blood and responsible for the body movements. These spirits function in a capacity similar to modern medicine’s nervous system. The treatise was published after his death since he abandoned the work because of persecution of other radical thinkers such as Galileo Galilei. [Wik14b].
After Descartes work, many experiments have been conducted on the subject. Between 1664 and 1668 several studies on living muscles induced to give up the idea of an “animal spirit” and suggest the modern definition of nervous system. The mechanist theory of the french philosopher was first disproved by a Dutchman, Jan Swammerdam, who conduced studies on a frog. He observed that after removing the heart from a living frog, the frog was still capable of swimming, while it could not anymore after the ablation of the brain. It was during its dissection that the muscled twitched after the stimulation of several nerves. Few years later his first experiment on frogs, Swammerdam refines his theory on nerve conduction observing the movement of a muscle suspended in a glass tube caused by the irritation of the nerves with a silver wire. The movement of the muscle was probably due to the induction of a small electrical charge. [GJ96].
In the latest 1700s, two italians gave the main contribution to modern electro-physiology. In 1786, the italian anatomist Luigi Galvani showed that a direct contact with an electrical generator leads to a muscle contraction. Later, he proved that an electrical stimulation of a frog’s heart leads to cardiac muscular contraction. Only In 1774, the electricity was used for the first time for a medical purpose. The Humane Society, later called Royal Humane Society, was funded in London to promulgate the idea of attempting to resuscitate the deads.
Few months after the foundation of the Human Society, the first case of “resuscitation” using electrical shock was published. A 3-year-old child named Catherine Sophie Greenhill had been pronounced dead after she felt from the first storey window. A society member, an apothecary named Squires, occurred to the scene within twenty minutes and, after the “consent of the parents”, he proceeded to give the child several shocks with a portable electrostatic generator to various parts of the body without any apparent success. After several minutes, upon transmitting a few shocks through the thorax, he perceived a small pulsation. This treatment caused her to regain pulse and respiration, and she recovered fully, after a time in coma.
The resuscitation of little Catherine Greenhill was probably the first successful cardiac defibrillation of a human being. In 1788, the Human Society member Charles Kite awarded a silver medal for advocating the resuscitation of victims in cardiac arrest and developing his own electrostatic revivifying machine. [Har90] few years later, in 1792, Alessandro Volta, Italian Scientist and inventor, showed that the electrical current is generated by the combination of two dissimilar metals, disproving the theory of “animal electricity” from Galvani. Of course, both of them were right, saying that the electrical current comes on one side from the animal tissue and on the other one from the metals. [GJ96]
Two other italians, Leopoldo Nobili, Professor of Physics at Florence, and Carlo Matteucci, Professor of Physics at the University of Pisa and student of Nobili, pursued the research on the field. Nobili, who was working to support the theory of animal electricity, felt demonstrated it: in 1827, he detected the current flow in the body of a frog from muscles to spinal cord. The main improvements are due to Matteucci who, in 1838, showed that an electric current accompanies each heart beat. He tried to demonstrate the conduction in nerves but his instruments were not sensitive enough. [GJ96].
The first description of “action potential” accompanying each muscular contraction was given by Emil Du Bois-Reymond, a German physiologist, in 1834. He was capable of detecting the small voltage potential of resting muscles and the decreasing of this potential with muscle contractions. Du Bois-Reymond divided the signal that he detected into different parts which he called “disturbance curves”: “o” was the stable equilibrium point, and p, q, r and s were the other points of the deflection. [GJ96].
In the early 1870s, two British physiologists, John Burdon Sanderson and Frederick Page, recorded for the first time the heart’s electrical current. In 1878, they reported that each heart beat is accompanied by an electrical variation consisting of two phases, this was the first description of ventricular depolarization and repolarization. The first phase, later called QRS, was characterized by a disturbance of short duration “in which the apex becomes positive”, while the second one “in which the apex tends to negativity” was much longer. [Fye94].
Cardiac electrophysiology models
In this section, we introduce mathematical models largely used in cardiac electro-physiology. In particular, we present the so-called bidomain equations [Tun78], which are the most used model for the cardiac electrical potential. Bidomain equations are coupled with some ionic models, hereafter we present the two that are the most used in the next chapters: the Fitzugh-Nagumo model [Fit61, NAY62], and the Mitchell and Schaeffer one [MS03]. In order to obtain an Electrocardiogram, a body potential model (called in what follows torso model) is presented and coupled to the bidomain equations. Finally, some specific approximation and the numerical schemes used in this thesis are presented.
The bidomain and the monodomain equations
As explained in Section 1.2, the heart tissue is composed of two parts: the cardiac muscle cells, whose domain is called the intra-cellular domain, and the rest of the media, which is called the extra-cellular domain. We denote by ΩH the heart domain, and Ωi and Ωe respectively the intra- and the extra-cellular domains such that ΩH = Ωi ∪ Ωe and Ωi ∩ Ωe = ∅. The membrane junction between the intra- and the extra-cellular domains is denoted by Γm = Ωi ∩ Ωe.
The bidomain equations, introduced by Tung [Tun78] are based on the Ohm’s law and the conservation of the charge. Let us call ji and je respectively the electric currents in the intra- and extra-cellular domains. Applying the Ohm’s law, their expression is given by ji = σi∇ui, (1.7) je = σe∇ue.
where σi,e and ui,e are respectively the conductivity tensors (their expressions are given below, see Secetion 1.4.4) and the electrical potential in the intra- (resp. extra-) cellular domain.
The electric currents ji, je can be separated into two components: the surface charge µi,e changes due to the membrane capacitor behavior, and the total ionic current Iiontot that measures the current exchanges from Ωi to Ωe. Then, we obtain ∂µi + Iiontot = ji · n, ∂t (1.8) ∂µe − Iiontot = −je · n.
Inverse problems in cardiac electrophysiology
One of the goal of this work is to make some attempts in cardiac electrophysiology inverse problems. The aim of the inverse problems is to reconstruct the electrical activity of the heart assuming that we have access to some measurements of the electrical potential uT on part of the torso skin. This problem has been investigated in the last four decades and many strategies have been proposed.
The first approach that has been proposed was to estimate equivalent electrical dipoles [MKI+77, GRS84]. Later, the problem of recovering the heart surface potential, or the epicardial potential, was addressed. We will call this problem the “classical” inverse problem [BRS77, BTR00, WKJ11], was introduced. Finally, a new approach consists in estimating the transmembrane potential by solving an inverse problem on the whole (heart and torso) domain.
In this section, we give an overview on the classical inverse problem, which is known to be ill-posed, and on some regularization techniques. Second, we briefly present the transmembrane potential estimation problem. Finally, we introduce the problem of identification of some parameters useful to reconstruct the heart electrical activity which will be largely used in the next chapters.
Table of contents :
Introduction
Thesis general context
Thesis outline
Published and pre-print articles
Introduction (Français)
Contexte général de la thèse
Plan de thèse
Articles publiés et pre-print
1 Cardiac electrophysiology: model, equations, inverse problems and approximations
1.1 Introduction
1.2 Heart physiology
1.3 Cardiac electrophysiology
1.4 Cardiac electrophysiology models
1.5 Inverse problems in cardiac electrophysiology
1.6 Reduced Order Methods: a brief overview
2 Numerical simulations of full electrocardiogram cycle
2.1 Introduction
2.2 Whole heart mesh
2.3 Modeling assumptions
2.4 Healthy and pathological numerical simulations
2.5 Electrodes vest
2.6 Chapter conclusions
2.A Mitchell and Schaeffer ionic model
2.B Minimal Ventricular ionic model
2.C Courtemanche, Ramirez and Nattel ionic model
3 Estimation of some FitzHugh-Nagumo model parameters
3.1 Introduction
3.2 State of the art and motivation
3.3 Regularity of the solution
3.4 Estimation of reaction parameter
3.5 Estimation of a parameter in the second equation
3.6 Chapter conclusions
4 Reduced-order modeling and parameters identification with POD
4.1 Introduction
4.2 Presentation of the model
4.3 Proper Orthogonal Decomposition method
4.4 Application of POD to forward problems
4.5 Application of POD to the parameters identification
4.6 Chapter conclusions
4.A Genetic algorithm
5 Long-time simulations and Restitution Curves with POD
5.1 Introduction
5.2 Presentation of the models
5.3 Restitution Curve definition
5.4 Parameters identification in 0D case
5.5 Parameters identification with an ECG-based RC
5.6 Chapter conclusions
6 Reduced Order Model with Approximated Lax Pairs
6.1 Introduction
6.2 The ALP method
6.3 ALP in cardiac electrophysiology
6.4 Numerical experiments
6.5 Chapter conclusions
7 Inverse problems with ALP reduced-order method
7.1 Introduction
7.2 An overview on data assimilation
7.3 Application to Micro-Electrode Arrays measures
7.4 Application to epicardium potential reconstruction
7.5 Chapter conclusions
8 ROM with ALP and Discrete Empirical Interpolation Method
8.1 Introduction
8.2 The ALP-DEIM method
8.3 ALP-DEIM in cardiac electrophysiology
8.4 Numerical experiments
8.5 Perspectives
8.6 Chapter conclusions
Conclusions
Conlcusions (Français)
A FELiScE
A.1 FELiScE general principles
A.2 Structure of the code
A.3 Electrophysiology equations implementation
A.4 Reduced-Order Models implementation
A.5 Author’s contributions
B High performance computing for the reduced basis method
Bibliography
