An Asymptotic Two-Layer Monodomain Model of Cardiac Electrophysiology in the Atria: Derivation and Convergence

Abstract

Abstract. We investigate a dimensional reduction problem of a reaction-diffusion system related to cardiac electrophysiology modeling in the atria. The atrial tissues are very thin. The physical problem is then routinely stated on a two-dimensional manifold. However, some electrophysiological heterogeneities are located through the thickness of the tissue. Despite their biomedical significance, the usual dimensional reduction techniques tend to average and erase their influence on the twodimensional propagation. We introduce a two-dimensional model with two coupled superimposed layers that allows us to take into account three-dimensional phenomena, but retains a reasonable computational cost. We present its mathematical derivation, show its convergence toward the threedimensional model, and check numerically its convergence speed.

Muscle Thickness and Curvature Influence Atrial Conduction Velocities

“…For example, the surface-based models cannot represent the dissociation of endocardial and epicardial electrical activities during fibrillation. Single layer surface models have been improved by introducing a second layer to account for a more three-dimensional character of the fibrillatory conduction (Gharaviri et al, 2012 ; Labarthe et al, 2014 ; Coudière et al, 2017 ). Comparisons of these bilayer models with three-dimensional simulations are very limited and do not consider the possible influence of geometrical curvature on the electrical propagation.…”

“…However, this surface representation of the atria cannot be used to describe the endo-epicardial electrical dissociation taking place during AFib (Gharaviri et al, 2012 ). To overcome this limitation, bilayer models have been proposed (Gharaviri et al, 2012 ; Labarthe et al, 2014 ; Coudière et al, 2017 ). Although these models have a reduced computational cost with respect to fully three-dimensional simulations, they fail to capture the loading of the muscle thickness and adjoining blood layer.…”

Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation

“…u k | Ω c k ≡ 0, then (3.19) also holds over R d \ Γ k . Taking interior traces yields 20) which is equivalent to…”

“…We believe the present work will open the path for such implementations though we will not tackle this here. Moreover, our approach could be extended to describe cardiac electrophysiology when adequately adapting non-linear models constructed via asymptotic expansions such as the ones given in [14,20].…”