Mathematical concepts of cardiac arrhythmias
Seminar Room 1, Newton Institute
How can mathematics help us to understand cardiac arrhythmias? The best known approach is to take a mathematical model of cell electrophysiology, insert it into a more-or-less complex model of cardiac architecture, and then study the resulting waves of activation that propagate through the myocardium. The mathematical formalism is a reaction-diffusion partial differential equation (PDE), with the local cell model serving as the reaction and the (anisotropic) diffusion of current as the spatial component. Since these PDEs are highly nonlinear, the main strategic approach is to simulate them on supercomputers. This approach forms the basis of most work in the mathematics of cardiac arrhythmias. But there is another role for mathematics, which is to provide us with concepts with which to understand the qualitative phenomena that emerge from our equation models. Ventricular fibrillation, the leading cause of sudden cardiac death, offers some good examples of this role. Fibrillation, mathematically, is spatiotemporal chaos, a phenomenon known to arise in these PDEs. We will discuss two principal mechanisms of fibrillation: reentrant scroll waves and ectopic focal activity. Each has a mathematical formulation that gives us insight into the causes of the phenomenon, as well as suggesting potential therapeutic targets.