A finite element method for an eikonal equation model of myocardial excitation wavefront propagation

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 excitation propagation in the full canine ventricular myocardium. An eikonal-curvature equation and an eikonal-diffusion equation for excitation time are compared. A Petrov Galerkin finite element method with cubic Hermite elements is developed to solve the eikonal-diffusion equation on a reasonably coarse mesh. The oscillatory errors seen when using the Galerkin weighted residual method with high mesh Peclet numbers are avoided by supplementing the Galerkin weights with C 0 functions based on derivatives of the interpolation functions. The ratio of the Galerkin and supplementary weights is a function of the Peclet 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 no-inflow boundary term is developed to prevent spurious excitation from initiating on the boundary. The need for discretization in time is avoided by using a continuation method to gradually introduce the nonlinear term of the governing equation. A simulation is performed in an anisotropic model of the complete canine ventricular myocardium, with 2355 degrees of freedom for the dependent variable.