Aslak Tveito

On the Computational Complexity of the Bidomain and the Monodomain Models of Electrophysiology

The bidomain model, coupled with accurate models of cell membrane kinetics, is generally believed to provide a reasonable basis for numerical simulations of cardiac electrophysiology. Because of changes occurring in very short time intervals and over small spatial domains, discretized versions of these models must be solved on fine computational grids, and small time-steps must be applied. This leads to huge computational challenges that have been addressed by several authors. One popular way of reducing the CPU demands is to approximate the bidomain model by the monodomain model, and thus reducing a two by two set of partial differential equations to one scalar partial differential equation; both of which are coupled to a set of ordinary differential equations modeling the cell membrane kinetics. A reduction in CPU time of two orders of magnitude has been reported. It is the purpose of the present paper to provide arguments that such a reduction is not present when order-optimal numerical methods are applied. Theoretical considerations and numerical experiments indicate that the reduction factor of the CPU requirements from bidomain to monodomain computations, using order-optimal methods, typically is about 10 for simple cell models and less than two for more complex cell models.

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.

Front tracking applied to a nonstrictly hyperbolic system of conservation laws

The application of a front tracking method to a nonstrictly hyperbolic system of conservation laws is described in one space dimension. The front tracking method is based on approximate solutions of Riemann problems. The method is compared with the random choice scheme and the upwind scheme.

A Front Tracking Method for Conservation Laws in One Dimension

Abstract We present a front tracking technique for conservation laws in one dimension. The method is based on approximations to the solution of Riemann problems where the solution is represented by piecewise constant states separated by discontinuities. The discontinuities are tracked until they interact, at this point a new Riemann problem is solved and so on. No finite differences are used. This method is tested on the system of nonstationary gas dynamics defined by the Euler equations, and three test cases are presented.

Optimal monodomain approximations of the bidomain equations

Abstract The bidomain equations are widely accepted to model the spatial distribution of the electrical potential in the heart. Although order optimal methods have been devised for discrete versions of these equations, it is still very CPU demanding to solve the equations numerically on a sufficiently fine mesh in 3D. Furthermore, the equations are hard to analyze; from a mathematical point of view, very little is known about the qualitative behavior of the solutions generated by these equations. It is well known that upon appealing to a certain relation between the extracellular and intracellular conductivities, the bidomain model can be rewritten in terms of a scalar reaction diffusion equation referred to as the monodomain model. This model is of course much easier to solve, and also the qualitative properties of the solutions are well known; such equations have been studied intensively for decades. It is the purpose of the present paper to show how the bidomain equations can be approximated in an optimal manner by the solution of a monodomain model. The key feature here is that this optimal solution can be computed without solving the bidomain model itself. The solution is obtained by putting the problem into a framework of parameter identification problems. Our results are illuminated by a series of numerical experiments.

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.

An order optimal solver for the discretized bidomain equations

The electrical activity in the heart is governed by the bidomain equations. In this paper, we analyse an order optimal method for the algebraic equations arising from the discretization of this model. Our scheme is defined in terms of block Jacobi or block symmetric Gauss–Seidel preconditioners. Furthermore, each block in these methods is based on standard preconditioners for scalar elliptic or parabolic partial differential equations (PDEs). Such preconditioners can be realized in terms of multigrid or domain decomposition schemes, and are thus readily available by applying ‘off-the-shelves’ software. Finally, our theoretical findings are illuminated by a series of numerical experiments. Copyright

The solution of nonstrictly hyperbolic conservation laws may be hard to compute

Some simple examples of nonstrictly hyperbolic conservation laws are presented that admit certain instability properties with respect to

A Second-Order Algorithm for Solving Dynamic Cell Membrane Equations

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