Mathematical
aspects and numerical methods for reaction-diffusion modeling in eletrocardiology
P. Colli Franzone*
Dipartimento di Matematica, Istituto di Analisi Nuerica-C.N.R., Università di Pavia, Pavia, Italy
We shall present a summary of a methodological approach to the macroscopic description of the cardiac excitation and recovery phases in the ventricular myocardium.
A mathematical model on a periodic cellular assembling of the tissue is introduce and the macroscopic model called ``Bidomain'' is derived by homogenization techniques. The Bidomain model consists in a degenerate reaction-diffusion system, coupled with ordinary differential equations, having solutions characterized by steep fronts propagating through the myocardium during the excitation phase.
The simulation of the Bidomain reaction-diffusion problem, in 3-dimensional geometry with twisted anisotropy, exhibits considerable numerical difficulties and complexity in order to cover the entire range of cardiac electrodynamics behavior. We briefly describe some relaxed approaches, related to anisotropic geometric evolution laws, also called eikonal-curvature models, and a portable parallel code for numerical simulations in computational electrocardiology in three dimensions.
Simulation studies are an important tool toward a better understanding of how anisotropy in myocardial architecture affects the action potential duration contributing to the spatial inhomogeneity of refractoriness, a major determinant of reentrant arrhythmias. We present some large scale simulations of a full heartbeat in Bidomain and Monodomain models on 3--D block of myocardial tissue using a parallel solver showing the effects of the intramural fiber rotation, anisotropic conductivity and of the transmural heterogeneity.
Recovery sequences play an important role in the study of the mechanisms underlying the genesis of cardiac arrhythmias, but there are still uncertainties about the best method for determining recovery times from extracellular recordings. An analysis of the reliability and accuracy provided by extracellular markers of the transmembrane repolarization times are presented based on Bidomain simulations.
This talk reports on joint works with L. F. Pavarino, G. Savarè, S. Sacchi and with B. Taccardi.