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Boundary Integral Formulation of the Cell-by-Cell Model of Cardiac Electrophysiology

Giacomo Rosilho de Souza, Rolf Krause, Simone Pezzuto

TL;DR

The study tackles the cell-by-cell EMI equations governing intra-, extra-, and membrane potentials in cardiac tissue using a boundary-element formulation to remove degeneracy and reduce degrees of freedom. By recasting the EMI system as a parabolic equation on the membrane and performing a boundary discretization with discrete Poincaré–Steklov operators, the authors show an equivalence to an ODE system on the transmembrane boundary. A constrained minimization and linear-mapped variables yield the ODE in terms of the transmembrane potential $V_0$ and gating variables $z$, with matrices $F$, $G$ and augmented operators defining the coupling across domains. Time integration is performed with the fully explicit multirate stabilized Runge–Kutta method (mRKC), enabling stable explicit stepping for stiff membrane terms and ionic dynamics, while numerical experiments report exponential spatial convergence in the single-cell case and demonstrate biologically relevant propagation and gap-junction interactions. Overall, the approach reduces degrees of freedom, improves numerical stability, and enables efficient simulations of cell-by-cell cardiac electrophysiology using a boundary-integral framework.

Abstract

We propose a boundary element method for the accurate solution of the cell-by-cell bidomain model of electrophysiology. The cell-by-cell model, also called Extracellular-Membrane-Intracellular (EMI) model, is a system of reaction-diffusion equations describing the evolution of the electric potential within each domain: intra- and extra-cellular space and the cellular membrane. The system is parabolic but degenerate because the time derivative is only in the membrane domain. In this work, we adopt a boundary-integral formulation for removing the degeneracy in the system and recast it to a parabolic equation on the membrane. The formulation is also numerically advantageous since the number of degrees of freedom is sensibly reduced compared to the original model. Specifically, we prove that the boundary-element discretization of the EMI model is equivalent to a system of ordinary differential equations, and we consider a time discretization based on the multirate explicit stabilized Runge-Kutta method. We numerically show that our scheme convergences exponentially in space for the single-cell case. We finally provide several numerical experiments of biological interest.

Boundary Integral Formulation of the Cell-by-Cell Model of Cardiac Electrophysiology

TL;DR

The study tackles the cell-by-cell EMI equations governing intra-, extra-, and membrane potentials in cardiac tissue using a boundary-element formulation to remove degeneracy and reduce degrees of freedom. By recasting the EMI system as a parabolic equation on the membrane and performing a boundary discretization with discrete Poincaré–Steklov operators, the authors show an equivalence to an ODE system on the transmembrane boundary. A constrained minimization and linear-mapped variables yield the ODE in terms of the transmembrane potential and gating variables , with matrices , and augmented operators defining the coupling across domains. Time integration is performed with the fully explicit multirate stabilized Runge–Kutta method (mRKC), enabling stable explicit stepping for stiff membrane terms and ionic dynamics, while numerical experiments report exponential spatial convergence in the single-cell case and demonstrate biologically relevant propagation and gap-junction interactions. Overall, the approach reduces degrees of freedom, improves numerical stability, and enables efficient simulations of cell-by-cell cardiac electrophysiology using a boundary-integral framework.

Abstract

We propose a boundary element method for the accurate solution of the cell-by-cell bidomain model of electrophysiology. The cell-by-cell model, also called Extracellular-Membrane-Intracellular (EMI) model, is a system of reaction-diffusion equations describing the evolution of the electric potential within each domain: intra- and extra-cellular space and the cellular membrane. The system is parabolic but degenerate because the time derivative is only in the membrane domain. In this work, we adopt a boundary-integral formulation for removing the degeneracy in the system and recast it to a parabolic equation on the membrane. The formulation is also numerically advantageous since the number of degrees of freedom is sensibly reduced compared to the original model. Specifically, we prove that the boundary-element discretization of the EMI model is equivalent to a system of ordinary differential equations, and we consider a time discretization based on the multirate explicit stabilized Runge-Kutta method. We numerically show that our scheme convergences exponentially in space for the single-cell case. We finally provide several numerical experiments of biological interest.
Paper Structure (4 sections, 2 theorems, 28 equations, 1 figure)

This paper contains 4 sections, 2 theorems, 28 equations, 1 figure.

Table of Contents

  1. Discretization of the full cell-by-cell model
  2. Spatial discretization of the cell-by-cell model
  3. Reduction to an ordinary differential equation
  4. Time integration

Key Result

Theorem 1.1

The linear maps $\mathcal{\psi}_i$ from eq:defphii satisfy with $\bm{\lambda}\in\mathbb{R}^M$ and $\bm{\beta}\in\mathbb{R}^N$ solution to The matrices $F\in\mathbb{R}^{M\times M}$, $G\in\mathbb{R}^{M\times N}$ are defined by $\bm{e}_i\in\mathbb{R}^{M_i}$ is the vector of ones and with $\alpha_i>0$, $i=0,\ldots,N$. If needed, $\bm{u}_{i}$ for $i=0,\ldots,N$ is computed with where $\bm{\beta}=(

Theorems & Definitions (4)

  • Theorem 1.1
  • proof
  • Theorem 1.2
  • proof