Models of the spreading of excitation in myocardial tissue

We consider a macroscopic model of the excitation process in the anisotropic myocardium involving the transmembrane, extracellular, and extracardiac potentials v, ue, and u0. The model is described by a reaction-diffusion (R-D) system, and the component v exhibits a front-like behavior reflecting the features of the excitation process. In numerical simulations, the presence of a moving excitation layer imposes severe constraints on the time and space steps to achieve stability and accuracy; consequently, application of the model is very costly in terms of computer time. An approximate model has been derived from the R-D system by means of a singular perturbation technique, and it is described by an eikonal equation, nonlinear and elliptic, in the activation time psi (x). Larger space steps are possible with this equation. From psi (x), we can derive, for a given instant t, the transmembrane potential v and subsequently, by solving an elliptic problem, we can compute the corresponding extracellular and extracardiac potentials ue and u0. The results of the R-D and the eikonal models applied to a portion of the ventricular wall are in excellent agreement; moreover, the eikonal model requires only a small fraction of the computer time needed by the R-D system. Therefore, for large-scale simulations of the excitation process, only the eikonal model has been used, and we investigate its ability to cope with complex situations such as front-front collisions and related potential patterns.