A Finite Element Method for an Eikonal Equation Model of Myocardial Excitation Wavefront Propagation

• Published in: SIAM journal on applied mathematics 2002-08, Vol.63 (1), p.324-350
• Main Authors: Tomlinson, Karl A., Hunter, Peter J., Pullan, Andrew J.
• Format: Article
• Language: English
• Subjects: Approximation; Boundary conditions; Coefficients; Eigenvalues; Eikonal equation; Error rates; Finite element analysis; Finite element method; Interpolation; Myocardium; Peclet number; Propagation; Wave excitation; Weighting functions
• ISSN: 0036-1399; 1095-712X
• DOI: 10.1137/S0036139901389513

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 Péclet numbers are avoided by supplementing the Galerkin weights with C° 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 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.