Coupled Eikonal problems to model cardiac reentries in Purkinje network and myocardium

TL;DR

The paper tackles the challenge of modeling bidirectional electrical propagation between the His-Purkinje network and the myocardium to capture reentry phenomena. It introduces a partitioned Eikonal–diffusion framework with a novel pseudo-time method for the myocardial problem and a coupling algorithm that handles antidromic and orthodromic fronts, including reentries. The approach is demonstrated in a realistic biventricular geometry across healthy and pathological scenarios (WPW, LBBB, CRT), showing accurate activation patterns, PMJ classifications, and computational efficiency. This work lays the groundwork for fast, patient-relevant simulations and has potential to inform CRT optimization and cardiac digital twin development.

Abstract

We propose a novel partitioned scheme based on Eikonal equations to model the coupled propagation of the electrical signal in the His-Purkinje system and in the myocardium for cardiac electrophysiology. This scheme allows, for the first time in Eikonal-based modeling, to capture all possible signal reentries between the Purkinje network and the cardiac muscle that may occur under pathological conditions. As part of the proposed scheme, we introduce a new pseudo-time method for the Eikonal-diffusion problem in the myocardium, to correctly enforce electrical stimuli coming from the Purkinje network. We test our approach by performing numerical simulations of cardiac electrophysiology in a real biventricular geometry, under both pathological and therapeutic conditions, to demonstrate its flexibility, robustness, and accuracy.

Figures (9)

• Figure 1: Left: representation of the components of the CCS (picture adapted from https://commons.wikimedia.org/wiki/File:2018_Conduction_System_of_Heart.jpg). Focusing the attention on the ventricular electrical activity, the signal coming from the atrioventricular node (AV node) descends into the bundle of His, which divides into the right and the left bundle branches, finally branching into the Purkinje network. Right: a visualization of orthodromic (in blue) and antidromic propagation (in red) in the Purkinje network (represented in yellow) due to an intramuscular source (in black). The signal undergoes an orthodromic delay $d_\mathrm{o}$ before being transmitted from the Purkinje to the muscle, and an antidromic delay $d_\mathrm{a}~<~d_\mathrm{o}$ in the opposite direction.
• Figure 2: Ventricular myocardial domain $\Omega_\mathrm{mus}$ (left) with three sources at $S_0$ and corresponding Purkinje domain $\Omega_\mathrm{p}$ (right) with a source at $\Gamma_0$.
• Figure 3: Numerical solution returned by the novel pseudo-time method in a one-dimensional example, compared to the classic pseudo-time method. The spatial domain is $x \in [0,L]$. Three stimuli are imposed: two are classified as active stimuli by the novel algorithm (in blue) and one as inactive (in red). At each pseudo-time step the solution is updated and stimulation points are identified as either active or inactive according to \ref{['eq:active_condition']}. Notably, the steady-state solution obtained with the classic pseudo-time method is physically meaningless, because the latest stimulus should be disregarded, as its activation time is later than the one induced by the other two.
• Figure 4: PMJs classification in presence of opposite wavefronts propagating between the Purkinje network and the myocardium. Right: graphical representation of the function classify_pmjs(pmjs,$u_{\mathrm{p}}$,$u_{\mathrm{m}}$), defined in Algorithm \ref{['alg:pmj_classification']}. In the notation, AV node stands for atrioventricular node. Two types of orthodromic PMJs are introduced: $\mathcal{P}_{\text{OO}}$ is the collection of orthodromic PMJs activated by an electrical signal coming from the AV node, while $\mathcal{P}_{\text{OA}}$ is the collection of orthodromic PMJs activated by a signal coming from an antidromic PMJ.
• Figure 5: (a) Real biventricular geometry reconstructed from four-chamber heart of a patient strocchi, with tetrahedral mesh discretization. (b) A section of the ventricles with the generated Purkinje network. (c) Muscular fibers visualization. The fiber field is visualized as streamlines, colored according to the transmural distance, going from the endocardium (Endo, in blue) to the epicardium (Epi, in red).