Finite element solution of an eikonal equation for excitation wavefront propagation in ventricular mycodium

Abstract

An efficient finite element method is developed to model the spreading of excitation in ventricular myocardium by treating the thin region of rapidly depolarizing tissue as a propagating wavefront. The model is used to investigate the excitation sequence in the full canine ventricular myocardium. The solution to an eikonal–curvature equation for excitation time is shown to satisfy a reaction–diffusion equation for the bidomain myocardial model at the wavefront, while the solution to an eikonal–diffusion equation approximately satisfies the reaction–diffusion equation in the vicinity of the wavefront. The features of these two eikonal equations are discussed. A Petrov–Galerkin finite element method with cubic Hermite elements is developed to solve the eikonal–diffusion equation. The oscillatory errors seen when using the Galerkin weighted residual method with high mesh Péclet numbers are avoided by supplementing the Galerkin weights with C⁰ continuous functions based on derivatives of the interpolation functions. The ratio of the Galerkin and supplementary weights is a function of the Péclet number such that, for one-dimensional propagation, the error in the solution is within a small constant factor of the optimal error achievable in the trial space. An additional noinflow boundary term is developed to prevent spurious excitation initiating on the boundary. The need for discretization in time is avoided by using a continuation method to gradually introduce the non-linear term of the governing equation. A small amount of artificial diffusion is sometimes necessary. Simulations of excitation are performed using a model of the anisotropic canine ventricular myocardium with 23.55 degrees of freedom for the dependent variable, and results are compared with reported experimental observations. When it was assumed that Purkinje fibres influence propagation only on the endocardial surface, excitation of the entire myocardium was completed in 56 ms. Altering material parameters to represent penetration of the Purkinje fibres beneath the left endocardial surface reduced the completion time to 48 ms. Modelling the effects of the laminar structure of myocardium by reducing the propagation speed by 40% in the direction normal to the layers delayed completion of excitation by only 4%.