Electrical waves in mathematical models of human-ventricular tissue: effects of different modeling formalisms, electrophysiological factors, and heterogeneities on ventricular arrhythmias and the termination of these arrhythmias via a deep-learning approach

Abstract

Sudden cardiac death (SCD) remains one of the significant causes of mortality in industrialized and developing countries. SCD is often caused by life-threatening cardiac arrhythmias like ventricular tachycardia and ventric- ular fibrillation, which are associated with the formation of spiral, broken- spiral (two dimensions), and scroll waves (in three dimensions) of electrical activation in cardiac tissue. To understand, and eventually control, such arrhythmias, it is important to carry out in vivo, ex vivo, in vitro, and in silico studies; the last of these has become increasingly important over the past three decades. In this thesis, I have carried out several in silico stud- ies of state-of-the-art mathematical models for cardiac tissue which focus mainly on four themes: (I) The effects of subcellular ion-channel modeling on electrical-wave dynamics in cardiac tissue; in particular, I have carried out a detailed comparison of wave dynamics in Hodgkin-Huxley and Markov- state formalisms for the Sodium (Na) channel in some mathematical models for human cardiac tissue. (II) The frequency and tip-trajectory of spiral waves and its dependence on electrophysiological parameters in a realistic mathematical model for human-ventricular tissue with and without fibrob- lasts. (III) The arrhythmogenicity of cardiac fibrosis and its dependence on the lacunarity parameter and Betti numbers of patterns of fibrotic tis- sue. (IV) The efficient elimination of pathological spiral and broken-spiral waves via a deep-learning-assisted detection and termination of spiral- and broken-spiral waves in mathematical models for cardiac tissue.

In Chapter 2 we investigate the effects of different subcellular model- ing formalsims for an ion-channel and on its properties, the action-potential of a single cell, and spiral and scroll waves in two- and three-dimensional human ventricular tissue. In particular, we compare and contrast the ex- citation properties of cardiac myocytes and cardiac tissue modelled by (a) a Hodgkin-Huxley-model (HHM) and (b) Markov-chain-model (MM) for- malisms for the sodium (Na) ion channel. Specifically, we bring out the differences between HHM and MM formalisms, for both wild-type (WT) and mutant (MUT ) models, for ion-channel kinetics, single-myocyte action potentials, and the spatiotemporal evolutions of spiral and scroll waves in different mathematical models of cardiac tissue. We show that the kinetic properties of Na ion channels are not the same for HHM and MM models; in particular, the range of values of the trans-membrane potential V m , in which there is a significant window current, depends significantly on these models, so there are marked differences in the opening times of the Na ion chan- nels, the maximal amplitude of the Na current, and the presence or absence of a late Na current. Furthermore, these changes lead to different excita- tion behaviours in cardiac tissue; specifically, two of the WT models showstable spiral waves, but the other one shows meandering and transiently breaking spiral waves. Our results are based on extensive direct numerical simulations of waves of electrical activation in these models, in two- and three-dimensional (2D and 3D) homogeneous simulation domains and also in domains with localised heterogeneities, either obstacles with randomly distributed inexcitable regions or mutant cells in a wild-type background. Our study brings out the sensitive dependence of spiral- and scroll-wave dy- namics on these five models and the parameters that define them. We list desiderata for a good model for the Na wild-type ion-channel; we use these desired properties to select one of the MM models that we study.

In Chapter 3 we study the effects of different electrophysiological pa- rameters in determining the frequency of the spiral waves in cardiac tissue by. Spiral waves of excitation in cardiac tissue are associated with life- threatening cardiac arrhythmias. It is, therefore, important to study the electrophysiological factors that affect the dynamics of these spiral waves. By using an electrophysiologically detailed mathematical model of a myocyte (cardiac cell), we study the effects of cellular parameters, such as membrane- ion-channel conductances, on the properties of the action-potential (AP) of a myocyte. We then investigate how changes in these properties, specifically the upstroke velocity and the AP duration (APD), affect the frequency ω of a spiral wave in the mathematical model that we use for human-ventricular tissue. We find that an increase (decrease) in this upstroke-velocity or a decrease (increase) in the AP duration increases (decreases) the spiral- wave frequency. We also study how other intercellular factors, such as the fibroblast-myocyte coupling, diffusive coupling strength, and the effective number of neighboring myocytes, modulate the ω. Finally, we demonstrate how a spiral wave can drift to a region with a high density of inexcitable cells called fibroblasts. Our results provide a natural explanation for the anchoring of spiral waves in highly fibrotic regions in fibrotic hearts.

In Chapter 4 we study diffuse fibrosis (DF), interstitial fibrosis (IF), patchy fibrosis (PF), and compact fibrosis (CF) and their arrhythmogenicity in cardiac tissue. We use mathematcal models for DF, IF, PF, and CF to study patterns of fibrotic cardiac tissue that have been generated by using Perlin noise and also by a simple model. We show that the fractal dimension D, the lacunarity L, and the Betti numbers β0 and β1 of such patterns, which we term as fibrotic-tissue markers, can be used to characterise the arrhythmogenicity of different types of cardiac fibrosis. We hypothesize, and then demonstrate by extensive in silico studies of detailed mathematical models for cardiac tissue, that the arrhytmogenicity of fibrotic tissue is high when β0 is large and the lacunarity parameter b is small.

In Chapter 5 we devise a new defibrillation scheme that uses a convo- lutional neural network (CNN), which we first train by using images from our simulations of waves of electrical activation in mathematical models for two-dimensional cardiac tissue. Unbroken- and broken-spiral waves, inpartial-differential-equation (PDE) models for cardiac tissue, are the mathe- matical analogs of life-threatening cardiac arrhythmias, namely, ventricular tachycardia (VT) and ventricular-fibrillation (VF). We develop (a) a deep- learning method for the detection of unbroken- and broken-spiral waves and (b) the elimination of such waves, e.g., by the application of low-amplitude control currents in the cardiac-tissue context. Our method is based on a convolutional neural network (CNN) that we train to distinguish between patterns with spiral-waves S and without spiral-waves N S . We obtain these patterns by carrying out extensive direct numerical simulations (DNSs) of PDE models for cardiac tissue in which the transmembrane potential V , when portrayed via pseudocolor plots, displays patterns of electrical acti- vation of types S and N S . We then utilize our trained CNN to obtain, for a given pseudocolor image of V , a heatmap that has high intensity in the regions where this image shows the cores of spiral waves and the associ- ated wave fronts. Given this heatmap, we show how to apply low-amplitude currents of 2D-Gaussian profile to eliminate spiral-waves efficiently. Our in silico results are of direct relevance to the detection and elimination of these arrhythmias because our elimination of unbroken or broken-spiral waves is the mathematical analog of low-amplitude defibrillation.