The effect of simplifying assumptions in the bidomain model of cardiac tissue: application to ST segment shifts during partial ischaemia

Approaches for determining cardiac bidomain conductivity values: progress and challenges

Modelling the electrical activity of the heart is an important tool for understanding electrical function in various diseases and conduction disorders. Clearly, for model results to be useful, it is necessary to have accurate inputs for the models, in particular the commonly used bidomain model. However, there are only three sets of four experimentally determined conductivity values for cardiac ventricular tissue and these are inconsistent, were measured around 40 years ago, often produce different results in simulations and do not fully represent the three-dimensional anisotropic nature of cardiac tissue. Despite efforts in the intervening years, difficulties associated with making the measurements and also determining the conductivities from the experimental data have not yet been overcome. In this review, we summarise what is known about the conductivity values, as well as progress to date in meeting the challenges associated with both the mathematical modelling and the experimental techniques. Graphical abstract Epicardial potential distributions, arising from a subendocardial ischaemic region, modelled using conductivity data from the indicated studies.

Modelling far field pacing for terminating spiral waves pinned to ischaemic heterogeneities in cardiac tissue

In cardiac tissue, electrical spiral waves pinned to a heterogeneity can be unpinned (and eventually terminated) using electric far field pulses and recruiting the heterogeneity as a virtual electrode. While for isotropic media the process of unpinning is much better understood, the case of an anisotropic substrate with different conductivities in different directions still needs intensive investigation. To study the impact of anisotropy on the unpinning process, we present numerical simulations based on the bidomain formulation of the phase I of the Luo and Rudy action potential model modified due to the occurrence of acute myocardial ischaemia. Simulating a rotating spiral wave pinned to an ischaemic heterogeneity, we compare the success of sequences of far field pulses in the isotropic and the anisotropic case for spirals still in transient or in steady rotation states. Our results clearly indicate that the range of pacing parameters resulting in successful termination of pinned spiral waves is larger in anisotropic tissue than in an isotropic medium.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

Approximate solutions for certain bidomain problems in electrocardiography

The simulation of problems in electrocardiography using the bidomain model for cardiac tissue often creates issues with satisfaction of the boundary conditions required to obtain a solution. Recent studies have proposed approximate methods for solving such problems by satisfying the boundary conditions only approximately. This paper presents an analysis of their approximations using a similar method, but one which ensures that the boundary conditions are satisfied during the whole solution process. Also considered are additional functional forms, used in the approximate solutions, which are more appropriate to specific boundary conditions. The analysis shows that the approximations introduced by Patel and Roth [Phys. Rev. E 72, 051931 (2005)] generally give accurate results. However, there are certain situations where functional forms based on the geometry of the problem under consideration can give improved approximations. It is also demonstrated that the recent methods are equivalent to different approaches to solving the same problems introduced 20 years earlier.

A linear system of partial differential equations modeling the resting potential of a heart with regional ischemia

Ischemic ST-segment shift has been modeled using scalar stationary approximations of the bidomain model. Here, we propose an alternative simplification of the bidomain equations: a linear system modeling the resting potential, to be used in determining ischemic TP shift. Results of 2D and 3D simulations show that the linear system model is much more accurate than the scalar model. This improved accuracy is important if the model is to be used for solving the inverse problem of determining the size and location of an ischemic region. Furthermore, the model can provide insight into how the resting potential depends on the variations in the extracellular potassium concentration that characterize ischemic regions.

The effect of lesion size and tissue remodeling on ST deviation in partial-thickness ischemia

BACKGROUND

Myocardial ischemia causes ST segment elevation or depression in electrocardiograms and epicardial leads. ST depression in epicardium overlying the ischemic zone indicates that the ischemia is nontransmural. However, nontransmural ischemia does not always cause ST depression. Especially in animal models, ST depression is hard to reproduce.

OBJECTIVE

The purpose of this study was to determine the circumstances in which ST depression could be expected.

METHODS

We studied ischemia in a large-scale computer model of the human heart. A realistic representation of the ischemia-induced changes in resting membrane potential was used, which was based on diffusion of extracellular potassium. Ischemia diameter, transmural extent, and tissue conductivity were varied.

RESULTS

Our simulations confirm earlier work showing that partial-thickness ischemia, like full-thickness ischemia, typically causes ST elevation in an anisotropic model of the ventricles. However, we identified three situations in which ST depression can occur in overlying leads. The first is a reduced anisotropy ratio of the intracellular conductivity, which may result from hypertrophy and gap-junctional remodeling, circumstances that are likely to accompany ischemia. Second, an increase of the extracellular anisotropy has the same effect. Third, ST depression was found, independent of the anisotropy ratios, in very large and thin ischemic regions, resembling those that may occur in left-main or multivessel disease.

CONCLUSION

Both tissue remodeling and geometric factors can explain ST depression in overlying epicardial leads. We note at the same time that ST elevation is found in most circumstances, while depression occurs as a reciprocal effect, even in partial-thickness ischemia.

A new approach to the determination of cardiac potential distributions: Application to the analysis of electrode configurations

This paper presents a mathematical model and new solution technique for studying the electric potential in a slab of cardiac tissue. The model is based on the bidomain representation of cardiac tissue and also allows for the effects of fibre rotation between the epicardium and the endocardium. A detailed solution method, based on Fourier Series and a simple one-dimensional finite difference scheme, for the governing equations for electric potential in the tissue and the blood, is also presented. This method has the advantage that the potential can be calculated only at points where it is required, such as the measuring electrodes. The model is then used to study various electrode configurations which have been proposed to determine cardiac tissue conductivity parameters. Three electrode configurations are analysed in terms of electrode spacing, placement position and the effect of including fibre rotation: the usual surface four-electrode configuration; a single vertical analogue of this and a two probe configuration, which has the current electrodes on one probe and the measuring electrodes on the other, a fixed distance away. It is found that including fibre rotation has no effect on the potentials measured in the first two cases; however, in the two probe case, non-zero fibre rotation causes a significant drop in the voltage measured. This leads to the conclusion that it is necessary to include the effects of fibre rotation in any model which involves the use of multiple plunge electrodes.