Optimized Schwarz Methods and Model Adaptivity in Electrocardiology Simulations

Abstract

The Bidomain model is nowadays one of the most accurate mathematical descriptions of the action potential propagation in the heart. However, its numerical approximation is in general fairly expensive as a consequence of the mathematical features of this system, and several works have been devoted to devise effective solvers and preconditioners, [10,11,15] among others. A simplification of this model, called Monodomain problem is often adopted in order to reduce computational costs of the numerical solution of the cardiac potential. The latter model is however less accurate. A possible trade-off between accuracy and cost is a model adaptive strategy. The computational domain is subdivided into regions, coupled through an Optimized Schwarz Method, in which either the Bidomain or the Monodomain problem is solved, according to an a-posteriori model error estimator following the spatio-temporal evolution of the action potential propagation. Here we present a possible implementation of this approach, following up previous works on the error estimation and the Optimized Schwarz coupling.

An Introduction to Mathematical and Numerical Modeling of Heart Electrophysiology

“…By letting λ m = min σ l e /σ l i , σ t e /σ t i and λ M = max σ l e /σ l i , σ t e /σ t i , an alternative PE formulation can be obtained by linear combinations of the equations in (24), with coefficients λ 1+λ , − 1 1+λ , λ m ≤ λ ≤ λ M , and (1, 1):…”

“…If the constant λ in (29) is properly chosen, the block (1,1) of (37) is actually the discretization of the Monodomain model. Following this consideration, an improved version of the model adaptive strategy has been introduced in [24], where only the block (1,1) of (37) is solved in the Monodomain regions. The coupling between regions was based on Optimized Schwarz Methods [26,25], a popular technique in the field of Domain Decomposition algorithms (see e.g.…”

“…The stiffness of the problem, due to the presence of a steep propagation front, led to the introduction of adaptive schemes, in both time [46], and time and space [7]. Another approach has been aiming at a simplification of the original problem, by using a somehow optimized Monodomain model [39], and by developing model adaptive techniques, where the costly Bidomain model is replaced by the Monodomain one (or an extended version of it) far from the depolarization front and the recovery tail of the action potential [26,25,24]. In the rest of the paper we provide a general survey of the mathematical and numerical aspects of the cardiac electrophysiology modeling.…”