Complex Structure and Dynamics of the Heart

Abstract

Excitable media are a class of spatially extended biological and chemical systems, which display a characteristic type of pattern formation by supporting the propagation of nonlinear excitation waves. In the heart, plane waves are associated with the normal heartbeat. They spread from specialized pacemaker cells and travel through the muscle tissue to trigger the coordinated contraction of the four chambers of the heart. Self-excited activation patterns, such as spiral waves and high-dimensional spatio-temporal chaos, underlie cardiac arrhythmias, such as tachycardia and life-threatening ventricular fibrillation. For lack of a better strategy, high-energy electrical shocks remain the only reliable way to terminate fibrillation, despite severe side effects including tissue damage and intolerable pain. To enable low-energy approaches, it is necessary to quantitatively characterize the dynamics of arrhythmias and to identify the mechanisms governing the interaction of electric fields with the cardiac muscle. This work addresses both aspects and is focused on the effect of structural substrate heterogeneity. This heterogeneity is inherent to the cardiac muscle and results from, e.g., spatially varying cell properties, different cell types, lesions and complex cardiac anatomy. The thesis consists of three parts. The aim of the first is to establish a method of nonlinear dynamics as a new tool for the characterization of activation patterns in excitable media. It adopts the generic view of cardiac tissue as a heterogeneous excitable medium and utilizes Lyapunov stability analysis to assess, in a numerical model, the stability and complexity of undesired activation patterns under the influence of spatially varying substrate properties. The results show that heterogeneity can both stabilize and destabilize spiral wave dynamics. Furthermore, it can lead to emergent effects on the complexity of spatio-temporal chaos, which may prove to be significant in the attempt to control fibrillation. Methodologically, Lyapunov stability analysis is shown to provide information that is not otherwise accessible: symmetry breaks in the system can be detected, spatial domains controlled by different spiral waves can be objectively defined, (de-)stabilizing effects of parameter changes and heterogeneity can be identified before they result in qualitatively altered dynamics. In the second part of the thesis, a theory of the fundamental mechanisms governing the interaction of weak electric fields with complex anatomical shape of cardiac tissue is developed. The devised mathematical framework indicates that the curvature and shape of tissue boundaries resulting from cardiac anatomy determine the locations that are most sensitive to electric-field stimulation. In particular, electric fields are shown to cause strong tissue depolarization near convex outer tissue boundaries, an effect that is confirmed in cell-culture experiments, though not expected from previous theories. Other effects of boundary curvature are also discussed, e.g. on the time scale of tissue depolarization. The identified mechanisms determine where in the tissue the local excitation threshold can be overcome by stimulation with a global electric field. The theory accordingly explains the emergence of virtual electrodes that induce localized wave sources in response to weak electric fields during low-energy control approaches. One promising control strategy is low-energy anti-fibrillation pacing (LEAP), which uses a series of weak electric-field pulses to terminate fibrillation and achieves an energy reduction of more than 80% compared to standard, high-energy defibrillation. The physical mechanisms underlying this control strategy are investigated in the third part of this work. It is shown that wave patterns in canine hearts, triggered by pulsed weak electric fields and observed on the cardiac surface in optical mapping experiments, result from a field strength tunable number of wave nucleation sites in the tissue. The heterogeneities that give rise to these virtual electrodes are hypothesized to stem from the branches of coronary vessels, which penetrate the tissue, create internal tissue boundaries and serve as wave nucleation sites at different field strengths depending on their size. To test this hypothesis, a model is constructed that predicts the characteristics of activation patterns from the size distribution of coronary vessels, which is found to follow a power law. Outstanding quantitative agreement is found between the model prediction and the experimentally observed activation patterns for different field strengths. The inherent heterogeneity of the muscle therefore enables multi-site control of chaos in the heart. In summary, the results presented in this work show that, while structural heterogeneity can stabilize malignant activity and may lead to unexpected emergent effects on the complexity of chaotic activity, it also provides the substrate for promising low-energy control methods.