The effect of conductivity values on ST segment shift in subendocardial ischaemia

Uncertainty quantification of the effect of cardiac position variability in the inverse problem of electrocardiographic imaging

Objective.Electrocardiographic imaging (ECGI) is a functional imaging modality that consists of two related problems, the forward problem of reconstructing body surface electrical signals given cardiac bioelectric activity, and the inverse problem of reconstructing cardiac bioelectric activity given measured body surface signals. ECGI relies on a model for how the heart generates bioelectric signals which is subject to variability in inputs. The study of how uncertainty in model inputs affects the model output is known as uncertainty quantification (UQ). This study establishes develops, and characterizes the application of UQ to ECGI.Approach.We establish two formulations for applying UQ to ECGI: a polynomial chaos expansion (PCE) based parametric UQ formulation (PCE-UQ formulation), and a novel UQ-aware inverse formulation which leverages our previously established 'joint-inverse' formulation (UQ joint-inverse formulation). We apply these to evaluate the effect of uncertainty in the heart position on the ECGI solutions across a range of ECGI datasets.Main results.We demonstrated the ability of our UQ-ECGI formulations to characterize the effect of parameter uncertainty on the ECGI inverse problem. We found that while the PCE-UQ inverse solution provided more complex outputs such as sensitivities and standard deviation, the UQ joint-inverse solution provided a more interpretable output in the form of a single ECGI solution. We find that between these two methods we are able to assess a wide range of effects that heart position variability has on the ECGI solution.Significance.This study, for the first time, characterizes in detail the application of UQ to the ECGI inverse problem. We demonstrated how UQ can provide insight into the behavior of ECGI using variability in cardiac position as a test case. This study lays the groundwork for future development of UQ-ECGI studies, as well as future development of ECGI formulations which are robust to input parameter variability.

Bidomain modeling of electrical and mechanical properties of cardiac tissue

Throughout the history of cardiac research, there has been a clear need to establish mathematical models to complement experimental studies. In an effort to create a more complete picture of cardiac phenomena, the bidomain model was established in the late 1970s to better understand pacing and defibrillation in the heart. This mathematical model has seen ongoing use in cardiac research, offering mechanistic insight that could not be obtained from experimental pursuits. Introduced from a historical perspective, the origins of the bidomain model are reviewed to provide a foundation for researchers new to the field and those conducting interdisciplinary research. The interplay of theory and experiment with the bidomain model is explored, and the contributions of this model to cardiac biophysics are critically evaluated. Also discussed is the mechanical bidomain model, which is employed to describe mechanotransduction. Current challenges and outstanding questions in the use of the bidomain model are addressed to give a forward-facing perspective of the model in future studies.

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.

Differences between models of partial thickness and subendocardial ischaemia in terms of sensitivity analyses of ST-segment epicardial potential distributions

Mathematical modelling is a useful technique to help elucidate the connection between non-transmural ischaemia and ST elevation and depression of the ECG. Generally, models represent non-transmural ischaemia using an ischaemic zone that extends from the endocardium partway to the epicardium. However, recent experimental work has suggested that ischaemia typically arises within the heart wall. This work examines the effect of modelling cardiac ischaemia in the left ventricle using two different models: subendocardial ischaemia and partial thickness ischaemia, representing the first and second scenarios, respectively. We found that it is possible, only in the model of subendocardial ischaemia, to see a single minimum on the epicardial surface above the ischaemic region, and this only occurs for low ischaemic thicknesses. This may help to explain the rarity of ST depression that is located over the ischaemic region. It was also found that, in both models, the epicardial potential distribution is most sensitive to the proximity of the ischaemic region to the epicardium, rather than to the thickness of the ischaemic region. Since proximity does not indicate the thickness of the ischaemic region, this suggests a reason why it may be difficult to determine the degree of ischaemia using the ST segment of the ECG.

Quantifying the effect of uncertainty in input parameters in a simplified bidomain model of partial thickness ischaemia

Reduced blood flow in the coronary arteries can lead to damaged heart tissue (myocardial ischaemia). Although one method for detecting myocardial ischaemia involves changes in the ST segment of the electrocardiogram, the relationship between these changes and subendocardial ischaemia is not fully understood. In this study, we modelled ST-segment epicardial potentials in a slab model of cardiac ventricular tissue, with a central ischaemic region, using the bidomain model, which considers conduction longitudinal, transverse and normal to the cardiac fibres. We systematically quantified the effect of uncertainty on the input parameters, fibre rotation angle, ischaemic depth, blood conductivity and six bidomain conductivities, on outputs that characterise the epicardial potential distribution. We found that three typical types of epicardial potential distributions (one minimum over the central ischaemic region, a tripole of minima, and two minima flanking a central maximum) could all occur for a wide range of ischaemic depths. In addition, the positions of the minima were affected by both the fibre rotation angle and the ischaemic depth, but not by changes in the conductivity values. We also showed that the magnitude of ST depression is affected only by changes in the longitudinal and normal conductivities, but not by the transverse conductivities.

Computing ischemic regions in the heart with the bidomain model--first steps towards validation

We investigate whether it is possible to use the bidomain model and body surface potential maps (BSPMs) to compute the size and position of ischemic regions in the human heart. This leads to a severely ill posed inverse problem for a potential equation. We do not use the classical inverse problems of electrocardiography, in which the unknown sources are the epicardial potential distribution or the activation sequence. Instead we employ the bidomain theory to obtain a model that also enables identification of ischemic regions transmurally. This approach makes it possible to distinguish between subendocardial and transmural cases, only using the BSPM data. The main focus is on testing a previously published algorithm on clinical data, and the results are compared with images taken with perfusion scintigraphy. For the four patients involved in this study, the two modalities produce results that are rather similar: The relative differences between the center of mass and the size of the ischemic regions, suggested by the two modalities, are 10.8% ± 4.4% and 7.1% ± 4.6%, respectively. We also present some simulations which indicate that the methodology is robust with respect to uncertainties in important model parameters. However, in contrast to what has been observed in investigations only involving synthetic data, inequality constraints are needed to obtain sound results.

Mathematical modeling and simulation of ventricular activation sequences: implications for cardiac resynchronization therapy

Next to clinical and experimental research, mathematical modeling plays a crucial role in medicine. Biomedical research takes place on many different levels, from molecules to the whole organism. Due to the complexity of biological systems, the interactions between components are often difficult or impossible to understand without the help of mathematical models. Mathematical models of cardiac electrophysiology have made a tremendous progress since the first numerical ECG simulations in the 1960s. This paper briefly reviews the development of this field and discusses some example cases where models have helped us forward, emphasizing applications that are relevant for the study of heart failure and cardiac resynchronization therapy.

A nondimensional formulation of the passive bidomain equation

Simulation studies of ST depression arising from subendocardial ischemia show a marked difference in the resulting epicardial potential distributions depending on which of the 3 common experimentally determined bidomain conductivity data sets is chosen. Here, the governing equation is rendered nondimensional by dividing by the difference in normal and ischemic transmembrane potentials during the ST segment and by the sum of the intracellular and extracellular conductivities in the transverse direction, yielding the ratio of the sum of the intracellular and extracellular longitudinal conductivities divided by the sum of the intracellular and extracellular transverse conductivities as a dimensionless group. Averaging this ratio over the 3 sets of experimentally determined data gives the value of 3.21 ± 0.08. The effect of this narrow range means that the left-hand side of the governing equation can be considered, as a good approximation, to be the same for all these sets of conductivity data. Hence, the right hand of the nondimensional differential equation contains all the necessary information to compare the effect different conductivity data sets have on the epicardial potential distribution. As an example, an explanation is given as to why values from one data set give rise to epicardial distributions that are markedly different from those obtained from the other 2 data sets.

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.

The role of extracellular potassium transport in computer models of the ischemic zone

Ischemic heart disease is associated with large mortality and morbidity. Understanding of the relations between coronary artery occlusion, geometry of the ischemic region, physiology of ischemia, and the resulting changes in electrocardiogram (ECG) leads and catheter signals is important to support diagnosis and treatment. Computer models play an important role in understanding ischemia, by linking experimental to clinical results. In this paper we argue that the observed transport of extracellular potassium should be represented in such models. We used a diffusion equation to describe the transport mechanism. This model reproduced the measured spatial distribution of potassium, and its temporal development. We discuss the role of potassium transport next to other aspects of ischemia: the mechanism of changes in action potential and ECG, cellular coupling, anisotropic bidomain tissue conductivity, and the geometry of the ischemic zone.

ST elevation or depression in subendocardial ischemia?

ST-segment depression in epicardial electrograms can be a "reciprocal" effect of remote myocardial ischemia (MI), and can also be due to local partial-thickness or "subendocardial" MI. Experimental studies have shown either ST elevation or depression in leads overlying a subendocardial ischemic region. Those reporting elevation have shown depression over the lateral borders of the ischemia. Simulation studies with anisotropic models have explained the ST-elevation results. Presently, while experimentalists may have difficulty understanding the ST elevation, most model studies fail to explain ST depression in overlying leads during partial-thickness ischemia. We have simulated partial-thickness ischemia in a 3-dimensional model of the human heart. Our results show that the conductivity of the intracavitary blood, geometry of the ischemic region, and bidomain anisotropy ratios can all have a decisive influence on the sign of the ST deviation. We hypothesize that ST depression in leads overlying an ischemic zone is due to subendocardial ischemia in tissue where a redistribution of gap junctions has taken place.

Using models of the passive cardiac conductivity and full heart anisotropic bidomain to study the epicardial potentials in ischemia

In this paper we present a multi-scale approach for cardiac modeling. Based on the histology of cardiac tissue we created a geometrical model at a cellular scale to compute the effective conductivity of a piece of cardiac tissue. In turn, the conductivity values obtained from this cellular scale model were used in a whole heart model in which we simulated regional, subendocardial ischemia. Histological changes at a cellular level led to changes in the effective conductivity tensor of the tissue, which in turn resulted in changes in the epicardial potential patterns during the ST-interval. Two effects were studied using this multi-scale approach: (1) the influence of a dynamically growing ischemic region on the epicardial potentials, and (2) the influence of a dynamically changing conductivity in the ischemic zone due to changes in the underlying pathology. One specific finding was the presence of epicardial patterns consisting of a central elevation and two opposite depressions at the edges of the ischemic zone which rotated as the ischemia became more transmural. In addition, the epicardial potentials decreased in magnitude with the duration of the ischemia due to changes in the effective conductivity of the ischemic tissue predicted by the cellular level model.

Modeling cardiac ischemia

Myocardial ischemia is one of the main causes of sudden cardiac death, with 80% of victims suffering from coronary heart disease. In acute myocardial ischemia, the obstruction of coronary flow leads to the interruption of oxygen flow, glucose, and washout in the affected tissue. Cellular metabolism is impaired and severe electrophysiological changes in ionic currents and concentrations ensue, which favor the development of lethal cardiac arrhythmias such as ventricular fibrillation. Due to the burden imposed by ischemia in our societies, a large body of research has attempted to unravel the mechanisms of initiation, sustenance, and termination of cardiac arrhythmias in acute ischemia, but the rapidity and complexity of ischemia-induced changes as well as the limitations in current experimental techniques have hampered evaluation of ischemia-induced alterations in cardiac electrical activity and understanding of the underlying mechanisms. Over the last decade, computer simulations have demonstrated the ability to provide insight, with high spatiotemporal resolution, into ischemic abnormalities in cardiac electrophysiological behavior from the ionic channel to the whole organ. This article aims to review and summarize the results of these studies and to emphasize the role of computer simulations in improving the understanding of ischemia-related arrhythmias and how to efficiently terminate them.

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.

Analysis of Electrode Configurations for Measuring Cardiac Tissue Conductivities and Fibre Rotation

: This paper describes a multi-electrode grid, which could be used to determine cardiac tissue parameters by direct measurement. A two pass process is used, where potential measurements are made, during the plateau phase of the action potential, on a subset of these electrodes and these measurements are used to determine the bidomain conductivities. In the first pass, the potential measurements are made on a set of 'closely-spaced' electrodes and the parameters are fitted to the potential measurements in an iterative process using a bidomain model and a solver based on a modified Shor's r-algorithm. This first pass yields the extracellular conductivities. The second pass is similar except that a 'widely-spaced' electrode set is used and this time the intracellular conductivities are recovered. In addition, it is possible to determine the fibre rotation throughout the tissue, since the bidomain model used here is able to include the effects of fibre rotation. In the simulation studies presented here, the model is solved with known conductivities, on each of the two subsets of electrodes, to generate two sets of 'measured potentials.' Conductivities are then recovered by solving an inverse problem based on the measured potentials, to which various levels of noise are added. For example, simulations in the first pass are performed using an electrode spacing of 500 mum, for a situation where the longitudinal and transverse space constants are 769 and 308 mum, respectively. These give very accurate average percentage relative errors for the longitudinal and transverse extracellular conductivities, over five simulations with 1% noise added, of 0.3 and 0.2%. Twenty-five second pass simulations, on a 1 mm grid, yield average percentage relative errors of 3.8, 2.6 and 1.4% for the corresponding intracellular values and the fibre rotation angle, respectively.

On the passive cardiac conductivity

In order to relate the structure of cardiac tissue to its passive electrical conductivity, we created a geometrical model of cardiac tissue on a cellular scale that encompassed myocytes, capillaries, and the interstitial space that surrounds them. A special mesh generator was developed for this model to create realistically shaped myocytes and interstitial space with a controlled degree of variation included in each model. In order to derive the effective conductivities, we used a finite element model to compute the currents flowing through the intracellular and extracellular space due to an externally applied electrical field. The product of these computations were the effective conductivity tensors for the intracellular and extracellular spaces. The simulations of bi-domain conductivities for healthy tissue resulted in an effective intracellular conductivity of 0.16S/m (longitudinal) and 0.005 S/m (transverse) and an effective extracellular conductivity of 0.21S/m (longitudinal) and 0.06 S/m (transverse). The latter values are within the range of measured values reported in literature. Furthermore, we anticipate that this method can be used to simulate pathological conditions for which measured data is far more sparse.

Approximate solution to the bidomain equations for electrocardiogram problems

Simulating the electrocardiogram requires specifying the transmembrane potential distribution within the heart and calculating the potential on the surface of the body. Often, such calculations are based on the bidomain model of cardiac tissue. A subtle but fundamental problem arises when considering the boundary between the cardiac tissue and the surrounding volume conductor. In general, one finds that two potentials--the extracellular potential in the tissue and the potential in the surrounding bath--obey three boundary conditions, implying that the potentials are overdetermined. In this paper, we derive a general method for handling bidomain boundary conditions that eliminates this problem. The gist of the method is that we add an additional term to the transmembrane potential that falls exponentially with depth into the tissue. The purpose of this term is to satisfy the third boundary condition. Then, we take the limit as the length constant associated with this extra term goes to zero. Our result is two boundary conditions that approximately account for the full set of three boundary conditions at the tissue surface.

Mechanisms of ischemia-induced ST-segment changes

Many aspects of ischemia-induced changes in the electrocardiogram lack solid biophysical underpinnings although variations in ST segments form the predominant basis for diagnostic and monitoring of patients. This incomplete knowledge certainly plays a role in the poor performance of some forms of electrocardiogram-based detection and characterization of ischemia, especially when it is limited to the subendocardium. The focus of our recent studies has been to develop a comprehensive mechanistic model of the electrocardiographic effects of ischemia. The computational component of this model is based on highly realistic heart geometry with anisotropic fiber structure and allows us to assign ischemic action potentials to contiguous regions that can span a prescribed thickness of the ventricles. A separate, high-resolution model of myocardial tissue provides us with a means of setting electrical characteristics of the heart, including the status of gap junctional coupling between cells. The experimental counterpart of this model consists of dog hearts, either in situ or isolated and perfused with blood, in which we control coronary blood flow by means of a cannula and blood pump. By reducing blood flow through the cannula for various durations, we can replicate any phase of ischemia from hyper acute to early infarction. Based on the results of these models, there is emerging a mechanism of the electrocardiographic response to ischemia that depends strongly on the anisotropic conductivity of the myocardium. Ischemic injury currents flow across the boundary between healthy and ischemic tissue, but it is their interaction with local fiber orientation and the associated conductivity that generates secondary currents that determine epicardial ST-segment potentials. Results from experiments support qualitatively the findings of the simulations and underscore the role of myocardial anisotropy in electrocardiography.

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

In this study various electrical conductivity approximations used in bidomain models of cardiac tissue are considered. Comparisons are based on epicardial surface potential distributions arising from regions of subendocardial ischaemia situated within the cardiac tissue. Approximations studied are a single conductivity bidomain model, an isotropic bidomain model and equal and reciprocal anisotropy ratios both with and without fibre rotation. It is demonstrated both analytically and numerically that the approximations involving a single conductivity bidomain, an isotropic bidomain or equal anisotropy ratios (ignoring fibre rotation) results in identical epicardial potential distributions for all degrees of subendocardial ischaemia. This result is contrary to experimental observations. It is further shown that by assuming reciprocal anisotropy ratios, epicardial potential distributions vary with the degree of subendocardial ischaemia. However, it is concluded that unequal anisotropy ratios must be used to obtain the true character of experimental observations.

The Effect of Conductivity on ST-Segment Epicardial Potentials Arising from Subendocardial Ischemia

We quantify and provide biophysical explanations for some aspects of the relationship between the bidomain conductivities and ST-segment epicardial potentials that result from subendocardial ischemia. We performed computer simulations of ischemia with a realistic whole heart model. The model included a patch of subendocardial ischemic tissue of variable transmural thickness with reduced action potential amplitude. We also varied both intracellular and extracellular conductivities of the heart and the conductivity of ventricular blood in the simulations. At medium or high thicknesses of transmural ischemia (i.e., at least 40% thickness through the heart wall), a consistent pattern of two minima of the epicardial potential over opposite sides of the boundary between healthy and ischemic tissue appeared on the epicardium over a wide range of conductivity values. The magnitude of the net epicardial potential difference, the epicardial maximum minus the epicardial minimum, was strongly correlated to the intracellular to extracellular conductivity ratios both along and across fibers. Anisotropy of the ischemic source region was critical in predicting epicardial potentials, whereas anisotropy of the heart away from the ischemic region had a less significant impact on epicardial potentials. Subendocardial ischemia that extends through at least 40% of the heart wall is manifest on the epicardium by at least one area of ST-segment depression located over a boundary between ischemic and healthy tissue. The magnitude of the depression is a function of the bidomain conductivity values.

Mechanism for ST depression associated with contiguous subendocardial ischemia

INTRODUCTION

A mechanism for ST depression arising on the epicardial surface over the border between normal and ischemic tissue is proposed. Depression is caused by current flowing in a transmural loop that begins and ends at the lateral boundary between healthy and ischemic tissue and passes through the transmural boundary between healthy and ischemic tissue. The result is ST depression at the epicardium over the lateral boundary. The size and direction of current flow are dictated by differences in the magnitude and orientation of anisotropic conductivity between those boundaries.

METHODS AND RESULTS

Computer simulations verified and quantified the relationship between ST depression and conductivity differences. We used computer simulations based on an anatomically accurate, anisotropic model of canine ventricles and a bidomain representation of the effects of ischemia to verify the biophysical basis of this mechanism.

CONCLUSION

ST depression at the epicardium appears over a lateral boundary between healthy and ischemic tissue.

Mechanisms of ST change in partial thickness ischemia

The origin of ST depression in ischemia remains poorly understood. The accepted source is of intracellular current flowing between the ischemic and non ischaemic muscle both in systole and diastole such that the AC recorded electrocardiogram shows ST elevation over the ischemic area. The difficulty comes with partial thickness ischemia where the body surface changes do not allow localisation of the ischemic region. In an animal model we have shown that the reason one cannot see the region on the body surface is that the epicardial distribution of ST segment is almost identical for partial thickness ischaemia in the left anterior descending coronary artery, (LAD) and circumflex coronary artery (Cx) territories. Dissection of the reasons for this finding has lead to 3 contributing factors. The first is the role of the right ventricular blood mass, the second the boundary between ischemia and normal and the third the presence of anisotropy and its contribution. In a block of myocardium with anisotropy included we have shown marked differences between the distributions depending on the anisotropy. We have also shown that the published values of conductivity for use in the bidomain model produce unacceptably disparate results.

A cylindrical model for studying subendocardial ischaemia in the left ventricle

In this paper a mathematical model of a left ventricle with a cylindrical geometry is presented with the aim of gaining a better understanding of the relationship between subendocardial ischaemia and ST depression. The model is formulated as an infinite cylinder and takes into account the full bidomain nature of cardiac tissue, as well as fibre rotation. A detailed solution method (based on Fourier series, Fourier transforms and a one dimensional finite difference scheme) for the governing equations for electric potential in the tissue and the blood is also presented. The model presented is used to study the effect increasing subendocardial ischaemia has on the epicardial potential distribution as well as the effects of changing the bidomain conductivity values. The epicardial potential distributions obtained with this cylindrical geometry are compared with results obtained using a previously published slab model. Results of the simulations presented show that the morphologies of the epicardial potential distributions are similar between the two geometries, with the main difference being that the cylindrical model predicts slightly higher potentials.