Computational Simulations of Vortex Waves Interacting with Heterogeneities

Vortex waves are striking spatial-temporal patterns and observed in diverse systems, such as active galaxies, catalytic reactions, and even biological systems. In the heart, vortex patterns of electrical waves are linked to ventricular arrhythmia. The fast rotating vortices, once created, repetitively excite the cardiac tissues and act as a pacemaker with a high frequency. The resulting heartbeat is much faster than the normal rhythm and termed ventricular tachycardia (VT). If the single vortex is unstable and fragmented, it leads to an often fatal ventricular fibrillation (VF). In this case, the heart does not pump but quivers asynchronously and causes sudden cardiac death, which accounts for 325,000 deaths in the United States each year. Most excitable media, particularly biological systems, are not homogeneous. For example, in the heart, the heterogeneity can refer to dead tissues, collagen fibers, or blood vessels. Indeed, the inhomogeneity of the media plays a critical role in the behavior of the vortex wave. Many experimental results have shown that the presence of the uncoupling substrates changes the activation pattern of excitation waves. In both chemical media and cardiac tissues, these changes result in abnormal conductance locally and a rise in the susceptibility to vortices. On the contrary, an unexcitable or impermeable heterogeneity can anchor the vortex wave. The anchored nonlinear waves are stabilized and even suppress the turbulence in their surrounding space. To date, the complex relationship between the uncoupling/insulating heterogeneity and the vortex wave has not been elucidated and clearly requires more in-depth studies. A major driving force is computer simulations that have greatly contributed to a deeper understanding of observations and continue to guide the design of new experiments. With even greater technological advances in the future, computer simulations will be indispensable tools to interpret dynamics of excitable media. In this thesis, I described computer simulations of both chemical and cardiac models in two- as well as three-dimensional spaces to study the dynamics of the vortex waves in presence of heterogeneities. For chemical media, my studies focus on the interaction between the vortex wave and a moving inert heterogeneity. For two-dimensional systems, I designed simulations to investigate the impact of a moving disk with varying velocities on a rigidly rotating wave. Many other factors that may contribute to the dynamics were also discussed, such as the phase of the spiral wave and the radius of the disk. Additionally, I integrated local fluid dynamics into the model to accord with the realistic chemical solution. The results suggested that the Stokes flow generated by the moving disk can be a very weak perturbation to the wave pattern. In three-dimensional simulations, I further explored the dynamics of a scroll wave that was partially pinned to the cylindrical heterogeneity. The scroll wave was thus stretched and deformed. Moreover, for systems with step-like heterogeneities, the simulations predicted that over short distances scroll waves are attracted towards the step and then rapidly commence a steady drift along the step line. For my studies of cardiac systems, I primarily modeled a static heterogeneity in a slab of cardiac tissue with a single reentrant wave or turbulence. Important properties of the heterogeneity as well as biological features of the cardiac fibers were considered in my research. For example, I locally changed the conductions within the cylindrical heterogeneity, which represents the tissues damaged by gap junctional uncoupling in the real heart. In addition, the role of an insulating heterogeneity in systems with scroll wave turbulence was also investigated. By anchoring to a thin cylinder, the stabilized wave rotates sufficiently fast to repel the free segments of the turbulent filament tangle, which then annihilate at the system boundaries. Furthermore, the results show that even thicker cylinders can suppress analogous forms of tachycardia by forming pinned multi-armed vortices. In this process, the observed number of wave arms depends on the cylinder radius.