Towards the modeling of the Purkinje/myocardium coupled problem: A well-posedness analysis

Abstract

The Purkinje network is the specialized conduction system in the heart. It ensures the physiological spread of the electrical wave in the ventricles. In this work, in an insulated heart framework, we model the free running Purkinje system, using the monodomain equation. The intra-myocardium part of the Purkinje fiber is coupled to the ventricular tissue using the bidomain equation. The coupling is performed through the extracellular potential. We discretize the problem in time using a semi-implicit scheme. Then, we write a variational formulation of the semi discrete problem in a non standard weighted Sobolev functional spaces. We prove the existence and uniqueness of the solution of the Purkinje/myocardium semi-discretized problem. We discretize in space by the finite element P 1 − Lagrange and conduct some numerical tests showing the anterograde and retrograde propagation of the electrical wave between the tissue and the Purkinje fibers.

Morphometric analysis of the His bundle (atrioventricular fascicle) in humans and other animal species, associated with electrophysiological variables. Histological and immunohistochemical study

“…Electrical impulses progress from the branches of HB towards PF to nally allow ventricular contraction, but these bers also have a determining role in generating ventricular arrhythmias, which can be observed at the electrocardiographic level (Li et al 2015;Aouadi et al 2019). For this reason, we reiterate the importance of the histological study of HB, as with a working knowledge of the cellular and tissue structure associated with cardiac physiology, it is possible to delineate the sites of arrhythmia production in different species.…”

Morphometric analysis of the His bundle (atrioventricular fascicle) in humans and other animal species. Histological and immunohistochemical study

“…Electrical impulses progress from the branches of HB towards PF to nally allow ventricular contraction, but these bers also have a determining role in generating ventricular arrhythmias, which can be observed at the electrocardiographic level (Li et al 2015;Aouadi et al 2019). For this reason, we reiterate the importance of the histological study of HB, as with a working knowledge of the cellular and tissue structure associated with cardiac physiology, it is possible to delineate the sites of arrhythmia production in different species.…”

Mathematical analysis of a 1D-3D coupled problem in cardiac electrophysiology

“…Mathematical model for the coupling Purkinje-myocardium. Our starting point is the model introduced in [1] and studied in [15,16]. The depolarization wave in the myocardium is governed by a reaction-diffusion equation coupled to an ODE system, modelling the gating variables W " pw 1 , .…”

“…In the paper by Bordas et al [18], a model for cardiac activity is analyzed, and an existence result is proven using the fixed-point approach from [2]. This approach was also adapted for the myocardium-Purkinje system in [16]. The authors utilized cross-sectional averaging of Purkinje fibers to formulate a one-dimensional model of the coupling conditions and prove the existence of a solution for the time semidiscretized problem.…”