Analytical solutions for the Extracellular-Membrane-Intracellular model

TL;DR

This work advances cardiac electrophysiology modeling by deriving a family of analytical solutions for the Extracellular-Membrane-Intracellular (EMI) framework, covering a single cell, two coupled cells, and a manufactured multi-cell configuration. It pairs these analytical results with a robust numerical verification using mortar finite element discretization and operator splitting, implemented in open-source tools, to validate accuracy and convergence. The manufactured solutions further provide a general benchmark for N-cell EMI systems, enabling rigorous convergence analyses for high-order time-integration schemes. Overall, the paper delivers practical benchmarks and a validated computational workflow that support the development and validation of EMI-based cardiac simulations, with data and code publicly accessible for reproducibility.

Abstract

The Extracellular-Membrane-Intracellular (EMI) model is a novel mathematical framework for cardiac electrophysiology simulations. The EMI model provides a more detailed description of the heart's electrical activity compared to traditional monodomain and bidomain models, potentially making it better-suited for understanding the electrical dynamics of the heart under pathological conditions. In this paper, we derive and verify several analytical solutions for the EMI model. Specifically, we obtain a family of solutions for a single two-dimensional cell in polar coordinates and for a pair of coupled three-dimensional cells in spherical coordinates. We also introduce a manufactured solution for N three-dimensional cells in Cartesian coordinates. To verify the analytical solutions, we conduct numerical experiments using the mortar finite element method combined with operator splitting. The results demonstrate that the analytical solutions are effective for verifying the accuracy of numerical simulations of the EMI model.

Figures (7)

• Figure 1: Configuration of a two-dimensional domain with four cells (purple) surrounded by extracellular space (blue) with gap junctions (orange) Tveito2017.
• Figure 2: Domain representation for a single circular cell (purple) surrounded by a circular layer as extracellular space (blue).
• Figure 3: Domain representation of two coupled semi-spherical cells (purple) connected by a gap junction (orange) and surrounded by a spherical shell-shaped extracellular space (blue).
• Figure 4: Single three-dimensional cardiac cell and its dimensions: cell membranes (purple) and gap junctions (orange).
• Figure 5: Numerical and exact solutions for $u_e$, $\ui[1]$, and $v^{()}$ at $r=5$, in Experiment 1.