A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology

This work deals with the numerical solution of the monodomain and bidomain models of electrical activity of myocardial tissue. The bidomain model is a system consisting of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time-dependent ODE modeling the evolution of the so-called gating variable. In the simpler sub-case of the monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple models for the membrane and ionic currents are considered, the Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical examples demonstrates thatthese methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, an optimalthreshold for discarding non-significant information in the multiresolution representation of the solution is derived, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed-up, memory compression, and errors in different norms.Comment: 25 pages, 41 figure

The cardiac bidomain model and homogenization

We provide a rather simple proof of a homogenization result for the bidomain model of cardiac electrophysiology. Departing from a microscopic cellular model, we apply the theory of two-scale convergence to derive the bidomain model. To allow for some relevant nonlinear membrane models, we make essential use of the boundary unfolding operator. There are several complications preventing the application of standard homogenization results, including the degenerate temporal structure of the bidomain equations and a nonlinear dynamic boundary condition on an oscillating surface.Comment: To appear in Networks and Heterogeneous Media, Special Issue on Mathematical Methods for Systems Biolog

Convergence of discrete duality finite volume schemes for the cardiac bidomain model

We prove convergence of discrete duality finite volume (DDFV) schemes on distorted meshes for a class of simplified macroscopic bidomain models of the electrical activity in the heart. Both time-implicit and linearised time-implicit schemes are treated. A short description is given of the 3D DDFV meshes and of some of the associated discrete calculus tools. Several numerical tests are presented

An introduction to mathematical and numerical modeling in heart electrophysiology

The electrical activation of the heart is the biological process that regulates the contraction of the cardiac muscle, allowing it to pump blood to the whole body. In physiological conditions, the pacemaker cells of the sinoatrial node generate an action potential (a sudden variation of the cell transmembrane potential) which, following preferential conduction pathways, propagates throughout the heart walls and triggers the contraction of the heart chambers. The action potential propagation can be mathematically described by coupling a model for the ionic currents, flowing through the membrane of a single cell, with a macroscopical model that describes the propagation of the electrical signal in the cardiac tissue. The most accurate model available in the literature for the description of the macroscopic propagation in the muscle is the Bidomain model, a degenerate parabolic system composed of two non-linear partial differential equations for the intracellular and extracellular potential. In this paper, we present an introduction to the fundamental aspects of mathematical modeling and numerical simulation in cardiac electrophysiology

Reaction-Diffusion systems for the macroscopic Bidomain model of the cardiac electric field

The paper deals with a mathematical model for the electric activity of the heart at macroscopic level. The membrane model used to describe the ionic currents is a generalization of the phase-I Luo-Rudy, a model widely used in 2-D and 3-D simulations of the action potential propagation. From the mathematical viewpoint the model is made up of a degenerate parabolic reaction diffusion system coupled with an ODE system. We derive existence, uniqueness and some regularity results

Well-posedness for a modified bidomain model describing bioelectric activity in damaged heart tissues

We prove the existence and the uniqueness of a solution for a modified bidomain model, describing the electrical behaviour of the cardiac tissue in pathological situations. The main idea is to reduce the problem to an abstract parabolic setting, which requires to introduce several auxiliary differential systems and a non-standard bilinear form. The main difficulties are due to the degeneracy of the bidomain system and to its non-standard coupling with the diffusion equation

On uniqueness theorems for the inverse problem of Electrocardiography in the Sobolev spaces

We consider a mathematical model related to reconstruction of cardiac electrical activity from ECG measurements on the body surface. An application of recent developments in solving boundary value problems for elliptic and parabolic equations in Sobolev type spaces allows us to obtain uniqueness theorems for the model. The obtained results can be used as a sound basis for creating numerical methods for non-invasive mapping of the heart.Comment: arXiv admin note: substantial text overlap with arXiv:2106.0412