Studies Of Spiral Turbulence And Its Control In Models Of Cardiac Tissue

Abstract

There is a growing consensus that life-threatening cardiac arrhythmias like ventricular tachycardia (VT) or ventricular fibrillation (VF) arise because of the formation of spiral waves of electrical activation in cardiac tissue; unbroken spiral waves are associated with VT and broken ones with VF. Several experimental studies have shown that inhomogeneities in cardiac tissue can have dramatic effects on such spiral waves. In this thesis we try to understand these experimental results by carrying out detailed and systematic studies of the interaction of spiral waves with different types of inhomogeneities in mathematical models for cardiac tissue. In Chapter 1 we begin with a general introduction to cardiac arrhythmias, the cardiac conduction system, and the connection between electrical activation waves in cardiac tissue and cardiac arrhythmias. As we have noted above, VT and VF are believed to be associated with spiral waves of electrical activation on cardiac tissue; such spiral waves form because cardiac tissue is an excitable medium. Thus we give an overview of excitable media, in which sub-threshold perturbations decay but super-threshold perturbations lead to an action potential that consists of a rapid stage of depolarization of cardiac cells followed by a slow phase of repolarization. During this repolarization phase the cells are refractory. We then give an overview of earlier studies of the effects of inhomogeneities in cardiac tissue; and we end with a brief description of the principal problems we study here. Chapter 2 describes the models we use in our work. We start with a general introduction to the cable equation and then discuss the Hodgkin-Huxley-formalism for the transport of ions across a cell membrane through voltage-gated ion channels. We then describe in detail the three models that we use for cardiac tissue, which are, in order of increasing complexity, the Panfilov model, the Luo Rudy Phase I (LRI) model, and the reduced Priebe Beuckelmann (RPB)model. We then give the numerical schemes we use for solving these model equations and the initial conditions that lead to the formation of spiral waves. For all these models we give representative results from our simulations and compare the states with spiral turbulence. In Chapter 3 we investigate the effects of conduction inhomogeneities (obstacles) in the three models introduced in Chapter 2. We outline first the experimental results that have provided the motivation for our study. We then discuss how we introduce obstacles in our simulations of the Panffilov, LRI, and RPB models for cardiac tissue. Next we present the results of our numerical studies of the effects, on spiral-wave dynamics, of the sizes, shapes, and positions of the obstacles. Our Principal result is that spiral-wave dynamics in these models depends sensitively on the position of the obstacle. We find, in particular, that, merely by changing the position of a conduction inhomogeneity, we may convert spiral turbulence (the analogue in our models of VF) to a single rotating spiral (the analogue of VT) anchored to the obstacle or vice versa; even more exciting is the possibility that, at the boundary between these two types of behaviour, we find a quiescent state Q with no spiral waves. Thus our study obtains all the possible qualitative behaviours found in experiments, namely, (1) VF might persist even in the presence of an obstacle, (2) it might be suppressed partially and become VT, or (3) it might be eliminated completely. In Chapter 4 we extend our work on conduction inhomogeneities (Chapter 3) to ionic inhomogeneities. Unlike conduction inhomogeneities, ionic inhomogeneities allow the conduction of activation waves. We find, nevertheless, that they too can lead to the anchoring of spiral waves or even the complete elimination of spiral-wave turbulence. Since spiral waves can enter the region in which there is an ionic inhomogeneity, their behaviours in the presence of such an inhomogeneity are richer than those with conduction inhomogeneities. We find, in particular, that a single spiral wave anchored at an ionic inhomogeneity can show temporal evolution that may be periodic, quasiperiodic, or even chaotic. In the last case the spiral wave shows a chaotic pattern inside the ionic inhomogeneity and a regular one outside it. Defibrillation is the control of arrhythmias such as VF. Most often defibrillation is effected electrically by administering a shock, either externally or via an internally implanted defibrillator. The development of low-amplitude defibrillation schemes, which minimise the deleterious effects of the applied shock, is a major challenge in the treatment of cardiac arrhythmias. Numerical studies of models for cardiac tissue provide us with convenient means of studying the elimination of spiral-wave turbulence by the application of external electrical stimuli; this is the numerical analogue of defibrillation. Over the years some low-amplitude defibrillation schemes have been suggested on the basis of such numerical studies. In Chapter 5 we discuss two such schemes that have been shown to suppress spiral-wave turbulence in two-dimensional models for cardiac tissue and also scroll-wave turbulence in three-dimensional models. One of these schemes uses local electrical pacing, typically in the centre of the simulation domain; the other applies the external electrical stimuli over a mesh. We study the efficacy of these schemes in the presence of conduction inhomogeneities. We find, in particular, that the local-pacing scheme, though effective in a homogeneous simulation domain, fails to control spiral turbulence in the presence of an obstacle and, indeed, might even facilitate spiral-wave break up. By contrast, the second scheme, which uses a mesh, succeeds in eliminating spiral-wave turbulence even in the presence of an obstacle. We end with some concluding remarks about the possible experimental implications of our study in Chapter 6.