Evidence for heterogeneous subsarcolemmal Na+ levels in rat ventricular myocytes.

The intracellular Na+ concentration ([Na+]) regulates cardiac contractility. Previous studies have suggested that subsarcolemmal [Na+] is higher than cytosolic [Na+] in cardiac myocytes, but this concept remains controversial. Here, we used electrophysiological experiments and mathematical modeling to test whether there are subsarcolemmal pools with different [Na+] and dynamics compared with the bulk cytosol in rat ventricular myocytes. A Na+ dependency curve for Na+-K+-ATPase (NKA) current was recorded with symmetrical Na+ solutions, i.e., the same [Na+] in the superfusate and internal solution. This curve was used to estimate [Na+] sensed by NKA in other experiments. Three experimental observations suggested that [Na+] is higher near NKA than in the bulk cytosol: 1) when extracellular [Na+] was high, [Na+] sensed by NKA was ~6 mM higher than the internal solution in quiescent cells; 2) long trains of Na+ channel activation almost doubled this gradient; compared with an even intracellular distribution of Na+, the increase of [Na+] sensed by NKA was 10 times higher than expected, suggesting a local Na+ domain; and 3) accumulation of Na+ near NKA after trains of Na+ channel activation dissipated very slowly. Finally, mathematical models assuming heterogeneity of [Na+] between NKA and the Na+ channel better reproduced experimental data than the homogeneous model. In conclusion, our data suggest that NKA-sensed [Na+] is higher than [Na+] in the bulk cytosol and that there are differential Na+ pools in the subsarcolemmal space, which could be important for cardiac contractility and arrhythmogenesis. NEW & NOTEWORTHY Our data suggest that the Na+-K+-ATPase-sensed Na+ concentration is higher than the Na+ concentration in the bulk cytosol and that there are differential Na+ pools in the subsarcolemmal space, which could be important for cardiac contractility and arrhythmogenesis. Listen to this article's corresponding podcast at https://ajpheart.podbean.com/e/heterogeneous-sodium-in-ventricular-myocytes/ .

Efficient solution of ordinary differential equations modeling electrical activity in cardiac cells.

The contraction of the heart is preceded and caused by a cellular electro-chemical reaction, causing an electrical field to be generated. Performing realistic computer simulations of this process involves solving a set of partial differential equations, as well as a large number of ordinary differential equations (ODEs) characterizing the reactive behavior of the cardiac tissue. Experiments have shown that the solution of the ODEs contribute significantly to the total work of a simulation, and there is thus a strong need to utilize efficient solution methods for this part of the problem. This paper presents how an efficient implicit Runge-Kutta method may be adapted to solve a complicated cardiac cell model consisting of 31 ODEs, and how this solver may be coupled to a set of PDE solvers to provide complete simulations of the electrical activity.

Multigrid block preconditioning for a coupled system of partial differential equations modeling the electrical activity in the heart.

The electrical activity of the heart may be modeled with a system of partial differential equations (PDEs) known as the bidomain model. Computer simulations based on these equations may become a helpful tool to understand the relationship between changes in the electrical field and various heart diseases. Because of the rapid variations in the electrical field, sufficiently accurate simulations require a fine-scale discretization of the equations. For realistic geometries this leads to a large number of grid points and consequently large linear systems to be solved for each time step. In this paper, we present a fully coupled discretization of the bidomain model, leading to a block structured linear system. We take advantage of the block structure to construct an efficient preconditioner for the linear system, by combining multigrid with an operator splitting technique.

Numerical solution of the bidomain equations.

Knowledge of cardiac electrophysiology is efficiently formulated in terms of mathematical models. However, most of these models are very complex and thus defeat direct mathematical reasoning founded on classical and analytical considerations. This is particularly so for the celebrated bidomain model that was developed almost 40 years ago for the concurrent analysis of extra- and intracellular electrical activity. Numerical simulations based on this model represent an indispensable tool for studying electrophysiology. However, complex mathematical models, steep gradients in the solutions and complicated geometries lead to extremely challenging computational problems. The greatest achievement in scientific computing over the past 50 years has been to enable the solving of linear systems of algebraic equations that arise from discretizations of partial differential equations in an optimal manner, i.e. such that the central processing unit (CPU) effort increases linearly with the number of computational nodes. Over the past decade, such optimal methods have been introduced in the simulation of electrophysiology. This development, together with the development of affordable parallel computers, has enabled the solution of the bidomain model combined with accurate cellular models, on geometries resembling a human heart. However, in spite of recent progress, the full potential of modern computational methods has yet to be exploited for the solution of the bidomain model. This paper reviews the development of numerical methods for solving the bidomain model. However, the field is huge and we thus restrict our focus to developments that have been made since the year 2000.