# Incorporating Inductances in Tissue-Scale Models of Cardiac Electrophysiology

## TL;DR

This paper introduces a hyperbolic bidomain and monodomain model for cardiac electrophysiology that incorporates inductances, addressing the unrealistic infinite propagation speed in standard models and exploring nonlinear effects on conduction velocity.

## Contribution

It develops a hyperbolic formulation of cardiac electrophysiology models that includes inductances, providing a more realistic representation of wave propagation and nonlinear dynamics.

## Key findings

- Inductances can increase conduction velocity in nonlinear models.
- Hyperbolic models require finer spatial discretization for accurate velocity prediction.
- The models influence spiral wave dynamics and atrial fibrillation simulations.

## Abstract

In standard models of cardiac electrophysiology, including the bidomain and monodomain models, local perturbations can propagate at infinite speed. We address this unrealistic property by developing a hyperbolic bidomain model that is based on a generalization of Ohm's law with a Cattaneo-type model for the fluxes. Further, we obtain a hyperbolic monodomain model in the case that the intracellular and extracellular conductivity tensors have the same anisotropy ratio. In one spatial dimension, the hyperbolic monodomain model is equivalent to a cable model that includes axial inductances, and the relaxation times of the Cattaneo fluxes are strictly related to these inductances. A purely linear analysis shows that the inductances are negligible, but models of cardiac electrophysiology are highly nonlinear, and linear predictions may not capture the fully nonlinear dynamics. In fact, contrary to the linear analysis, we show that for simple nonlinear ionic models, an increase in conduction velocity is obtained for small and moderate values of the relaxation time. A similar behavior is also demonstrated with biophysically detailed ionic models. Using the Fenton-Karma model along with a low-order finite element spatial discretization, we numerically analyze differences between the standard monodomain model and the hyperbolic monodomain model. In a simple benchmark test, we show that the propagation of the action potential is strongly influenced by the alignment of the fibers with respect to the mesh in both the parabolic and hyperbolic models when using relatively coarse spatial discretizations. Accurate predictions of the conduction velocity require computational mesh spacings on the order of a single cardiac cell. We also compare the two formulations in the case of spiral break up and atrial fibrillation in an anatomically detailed model of the left atrium, and [...].