A Cell-Based Framework for Numerical Modeling of Electrical Conduction in Cardiac Tissue

Abstract

In this paper, we study a mathematical model of cardiac tissue based on explicit representation of individual cells. In this EMI model, the extracellular (E) space, the cell membrane (M), and the intracellular (I) space are represented as separate geometrical domains. This representation introduces modeling flexibility needed for detailed representation of the properties of cardiac cells including their membrane. In particular, we will show that the model allows ion channels to be non-uniformly distributed along the membrane of the cell. Such features are difficult to include in classical homogenized models like the monodomain and bidomain models frequently used in computational analyses of cardiac electrophysiology. The EMI model is solved using a finite difference method (FDM) and two variants of the finite element method (FEM). We compare the three schemes numerically, reporting on CPU-efforts and convergence rates. Finally, we illustrate the distinctive capabilities of the EMI model compared to classical models by simulating monolayers of cardiac cells with heterogeneous distributions of ionic channels along the cell membrane. Because of the detailed representation of every cell, the computational problems that result from using the EMI model are much larger than for the classical homogenized models, and thus represent a computational challenge. However, our numerical simulations indicate that the FDM scheme is optimal in the sense that the computational complexity increases proportionally to the number of cardiac cells in the model. Moreover, we present simulations, based on systems of equations involving ∼117 million unknowns, representing up to ∼16,000 cells. We conclude that collections of cardiac cells can be simulated using the EMI model, and that the EMI model enable greater modeling flexibility than the classical monodomain and bidomain models.

Bifurcation Analysis of a Certain Hodgkin-Huxley Model Depending on Multiple Bifurcation Parameters

“…In addition, the general aim is to extend this research to models of the complete heart, cf. [18,19]. The main novelty of this paper is the consideration of a bifurcation problem depending on two bifurcation parameters to investigate the ion current interaction and the occurrence of EADs related to the calcium current.…”

Multigrid Methods for Discrete Fractional Sobolev Spaces

“…In this section we apply the multilevel algorithm (4.4) to construct mesh independent preconditioners for a coupled multidomain problem originating from a geometrically accurate model of electric signal propagation in cardiac tissue, the EMI model, [33].We remark that the EMI model is simple in a sense that it is a single-physics problem where two elliptic equations are coupled. However, the interface problems encountered here are identical to those found in multiphysics applications, e.g.…”

“…Therein the fractionality s is dictated by the mapping properties of the Schur complement operator. Some further examples of coupled systems with domains of different dimensionality include Babuška's problem for enforcing Dirichlet boundary conditions on an elliptic operator [5], flow stabilization by removal of tangential velocity at the boundary through Lagrange multipliers [8], the no-slip condition on the surface of a falling solid in the Navier-Stokes fluid [16], inextensibility constraint in the complex model of vesicle formation [1], and the potential jump on a membrane of a cardiac cell [33]. We note that in these applications the fractional Laplace problem has to be solved with both positive and negative exponent.…”

Inversion and computational maturation of drug response using human stem cell derived cardiomyocytes in microphysiological systems

“…It should also be noted that it is possible to measure the extracellular field potential in the microphysiological systems using a multi-electrode array (MEA) system, see e.g., [40,1]. Such data can be incorporated in our method by using the EMI model (see e.g., [41]) instead of the common AP models. In this case, the function H given by (4) would have to be extended to include the EFPs.…”