An eikonal-diffusion solver and its application to the interpolation and the simulation of reentrant cardiac activations

Electrical propagation of the cardiac impulse in the myocardium can be described by the eikonal-diffusion equation. This equation governs the field of activation times in a domain where conduction properties are specified. This approach has been applied to knowledge-based interpolation of sparse measurements of activation times and to the creation of initial conditions for detailed ionic models of cardiac propagation. This paper presents the mathematical basis, matrix formulation, and compact Matlab implementation of an iterative finite-element solver (triangular meshes) for the eikonal-diffusion equation extended to reentrant activations, which automatically identifies the period of reentry and computes the resulting isochrones. An iterative algorithm is designed to perform Laplacian interpolation of reentrant activation maps to be used as initial estimate for the eikonal-diffusion solver. The performance of the algorithm is analyzed in test-case geometries (ventricular slice and simplified atrial surface model).