Numerical Simulation in Electrocardiology Using an Explicit Generalized Finite Difference Method
Journal Title: Biomedical Journal of Scientific & Technical Research (BJSTR) - Year 2019, Vol 13, Issue 5
Abstract
In this paper we present a fast, accurate and conditionally stable algorithm to solve a monodomain model in 2D, which describes the electrical activity in the heart. The model consists of a parabolic anisotropic Partial Differential Equation (PDE), which is coupled to systems of Ordinary Differential Equations (ODEs) describing electrochemical reactions in the cardiac cells. The resulting system is challenging to solve numerically, because of its complexity. We propose a simple method based on operator splitting and an explicit meshless method for solving the PDE together with an adaptive method for solving the system of ODE’s for the membrane and ionic currents. The obvious difficulty of performing direct measurements in electro cardiology has motivated wide interest in the numerical simulation of cardiac models. In this paper we will present a numerical scheme for the equations of the monodomain model, which describes the electrical activity in the heart. The approach presented herein including the use of a meshless method differs to the best of our knowledge from other numerical approaches in the literature. An operator-splitting algorithm [1] is used to split the monodomain model into PDE and systems of ODEs. Because of the irregular geometry of the cardiac muscle we use an Generalized Finite Difference Method (GFDM) for spatial discretization of the PDE together with an explicit method for time discretization. The explicit method includes a stability limit formulated for the case of irregular clouds of nodes that can be easily calculated.The GFDM is evolved from classical Finite Difference Method (FDM). However, GFDM can be applied over general or irregular clouds of points. The basic idea is to use Moving Least Squares (MLS) approximation to obtain explicit difference formulae which can be included in the Partial Differential Equations. Benito, Ureña and Gavete have made interesting contributions to the development of this method [2,3,4]. In [5] the extension of the GFDM to the solution of anisotropic elliptic and parabolic equations is given. In [6] an FDM is presented to solve the bidomain equations. Althought this method is referred as a GFD scheme it is derived from a Taylor series expansion and a classic Least Squares (LS) method solved by a Moore-Penrose pseudo-inverse method together with a singular value decomposition method. This LS-GFDM uses elements to determine the nodal support templates. In our method a Taylor series expansion is used but the GFD scheme is obtained using Moving Least Squares (MLS) approximation. Then, the explicit difference formulae for irregular clouds of points can be easily obtained using a simple Cholesky method.
Authors and Affiliations
María Lucía Gavete, Francisco Vicente, Luis Gavete, Juan José Benito
Keywords
Related Articles
- EP ID EP585678
- DOI 10.26717/BJSTR.2019.13.002454
- Views 161
- Downloads 0
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