Which bidomain conductivity is the most important for modelling heart and torso surface potentials during ischaemia?

Mathematical simulations using the bidomain model, which represents cardiac tissue as consisting of an intracellular and an extracellular space, are a key approach that can be used to improve understanding of heart conditions such as ischaemia. However, key inputs to these models, such as the bidomain conductivity values, are not known with any certainty. Since efforts are underway to measure these values, it would be useful to be able to quantify the effect on model outputs of uncertainty in these inputs, and also to determine, if possible, which are the most important values to focus on in experimental studies. Our previous work has systematically studied the sensitivity of heart surface potentials to the bidomain conductivity values, and this was performed using a half-ellipsoidal model of the left ventricle. This study uses a bi-ventricular heart in a torso model and this time looks at the sensitivity of the torso surface potentials, as well as the heart surface potentials, to various conductivity values (blood, torso and the six bidomain conductivities). We found that both epicardial and torso potentials are the most sensitive to the intracellular longitudinal (along the cardiac fibres) conductivity (gil) with more minor sensitivity to the torso conductivity, and that changes in gil have a significant effect on the surface potential distributions on both the torso and the heart.

A modified approach to determine the six cardiac bidomain conductivities.

Accurate values for the six cardiac bidomain conductivities are crucial for meaningful computational studies of conduction in cardiac tissue, and are yet to be determined by experimental means. Although previous studies have proposed an approach using a multi-electrode array to measure potentials, from which the conductivities can be determined, it has been found that the conductivities cannot be retrieved consistently when the noise in the potentials varies. This paper presents a protocol, which not only has been shown to retrieve the conductivities to a reasonable accuracy, but does so under the presence of a more appropriate additive Gaussian noise model, while using fewer computational resources. Through repetitions of the protocol, a comparison of two pre-fabricated 128 electrode arrays, one array with a square arrangement of electrodes and the other with a rectangular arrangement, was made against a 75-electrode array proposed in previous studies. Results indicated that the two pre-fabricated arrays were generally more capable of obtaining the cardiac conductivities to a higher degree of accuracy than the 75-electrode array. The 128-electrode rectangular array was orientated such that the length of the array first ran along the direction of the fibres, then was reorientated such that the length of the array ran perpendicular to the direction of the fibres. The 128-electrode rectangular array, when orientated in this manner, was more capable of retrieving the conductivities than the remainder of the arrays tested, and thus we suggest this arrangement be used during experimental trials.

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.

Sensitivity analysis of ST-segment epicardial potentials arising from changes in ischaemic region conductivities in early and late stage ischaemia.

Although computational studies are increasingly used to gain insight into diseases such as myocardial ischaemia, there is still considerable uncertainty about the values for many of the parameters in these studies. This is particularly true for the bidomain conductivity values that are used in normal tissue and, even more so, in ischaemic tissue, when modelling ischaemia. In this work, we extended a previous study that used a half-ellipsoidal model and a realistic model to study subendocardial ischaemia during the ST segment, so that we could simulate both early and late stage ischaemia. We found that, for both stages of ischaemia, there was still the same connection between the degree of ischaemia and the development of features such as minima and maxima in the epicardial potential distribution (EPD), although the magnitudes of the potentials were very often less, which may be significant in terms of detecting them experimentally. Using uncertainty quantification associated with the ischaemic region conductivities, we also determined that the EPD features were sensitive to the ischaemic region extracellular normal and longitudinal conductivities during early stage ischaemia, whereas, during late stage ischaemia, the intracellular longitudinal conductivity was the most significant. However, since we again found that these effects were minor compared with the effects of fibre rotation angle and ischaemic depth, this might suggest that it is not necessary to use different conductivity values inside and outside the ischaemic region when modelling ST segment subendocardial ischaemia, unless the magnitudes of the potentials are an important part of the study.

Determining the most significant input parameters in models of subendocardial ischaemia and their effect on ST segment epicardial potential distributions.

There is considerable interest in simulating ischaemia in the ventricle and its effect on the electrocardiogram, because a better understanding of the connection between the two may lead to improvements in diagnosis of myocardial ischaemia. In this work we studied subendocardial ischaemia, in a simplified half-ellipsoidal bidomain model of a ventricle, and its effect on ST segment epicardial potential distributions (EPDs). We found that the EPD changed as the ischaemic depth increased, from a single minimum (min1) over the ischaemic region to a maximum (max) there, with min1 over the border of the region. Lastly, a second minimum (min2) developed on the opposite side of the ischaemic region, in addition to min1 and max. We replicated these results in a realistic ventricular model and showed that the min1 only case could be found for ischaemic depths of up to around 35% of the ventricular wall. In addition, we systematically examined the sensitivity of EPD parameters, such as the potentials and positions of min1, max and min2, to various inputs to the half-ellipsoidal model, such as fibre rotation angle, ischaemic depth and conductivities. We found that the EPD parameters were not sensitive to the blood or transverse bidomain conductivities and were most sensitive to either ischaemic depth and/or fibre rotation angle. This allowed us to conclude that the asynchronous development of the two minima might provide a way of distinguishing between low and high thickness subendocardial ischaemia, and that this method may well be valid despite variability in the population.

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.

Six Conductivity Values to Use in the Bidomain Model of Cardiac Tissue.

GOAL: The aim of this work is to produce a consistent set of six conductivity values for use in the bidomain model of cardiac tissue. METHODS: Studies in 2007 by Hooks et al. and in 2009 by Caldwell et al. have found that, in the directions longitudinal:transverse:normal (l:t:n) to the cardiac fibers, ratios of bulk conductivities and conduction velocities are each approximately in the ratio 4:2:1. These results are used here as the basis for a method that can find sets of six normalized bidomain conductivity values. RESULTS: It is found that the ratios involving transverse and normal conductivities are quite consistent, allowing new light to be shed on conductivity in the normal direction. For example, it is found that the ratio of transverse to normal conductivity is much greater in the intracellular (i) than the extracellular (e) domain. Using parameter values from experimental studies leads to the proposal of a new nominal six conductivity dataset: gil=2.4, gel=2.4, git=0.35, get=2.0, gin=0.08, and gen=1.1 (all in mS/cm). CONCLUSION: When it is used to model partial thickness ischaemia, this dataset produces epicardial potential distributions in accord with experimental studies in an animal model. It is, therefore, suggested that the dataset is suitable for use in numerical simulations. SIGNIFICANCE: Since the bidomain approach is the most commonly used method for modeling cardiac electrophysiological phenomena, new information about conductivity in the normal direction, as well as a consistent set of six conductivity values, is valuable for researchers who perform simulation studies.

Determining six cardiac conductivities from realistically large datasets.

Simulation studies of cardiac electrophysiological behaviour that use the bidomain model require accurate values for the bidomain extracellular and intracellular conductivities to produce useful results. This work considers an inversion algorithm, which has previously been shown, using simulated data, to be capable of retrieving six bidomain conductivities and the fibre rotation angle from measurements of electric potential made in the heart. The aim here is to see whether it is possible to improve the accuracy of the retrieved parameters. The scenario of retrieving only conductivities and not fibre rotation is examined but this does not lead to a worthwhile improvement in retrieval accuracy. It is also found that it is possible to retrieve the bidomain conductivities using not two but just one pass of the algorithm, made on a 'widely-spaced' electrode set. This appears to work because the algorithm is still very sensitive to the extracellular conductivities. However, the single-pass method is not recommended because the intracellular conductivities that are retrieved are not as accurate as those that are retrieved in the usual two-pass method, particularly for higher values of added noise. The second part of this work considers retrieving the six conductivities and fibre rotation from realistically large sets of potential measurements and identifies the best data analysis method. It is found that, even with added noise of up to 40%, the extracellular conductivities can still be retrieved extremely accurately (relative errors of around 2% on average) and so can the intracellular longitudinal conductivities and fibre rotation (errors less than 8% on average). The remaining intracellular conductivities have errors that are generally less than twice the added noise, particularly for the higher noise values.

A multi-electrode array and inversion technique for retrieving six conductivities from heart potential measurements.

A method for accurately finding cardiac bidomain conductivity parameters is a crucial part of efforts to study and understand the electrical functioning of the heart. The bidomain model considers current flowing along (longitudinal) and across (transverse) sheets of cardiac fibres, as well as between these sheets (normal), in both the extracellular and intracellular domains, which leads to six conductivity values. To match experimental studies, such a method must be able to determine these six conductivity values, not just the four where it is assumed that the transverse and normal conductivities are equal. This study presents a mathematical model, solution technique, multi-electrode array and two-pass inversion method, which can be used to retrieve all six conductivities from measurements of electrical potential made on the array. Simulated measurements of potential, to which noise is added, are used to demonstrate the ability of the method to retrieve the conductivity values. It is found that not only is it possible to accurately retrieve all six conductivity values, as well as a value for fibre rotation angle, but that the accuracy of such retrievals is comparable to the accuracy found in a previous study when only four conductivities (and fibre rotation) were retrieved.

Using a sensitivity study to facilitate the design of a multi-electrode array to measure six cardiac conductivity values.

When using the bidomain model to model the electrical activity of the heart, there are potentially six cardiac conductivity values involved: conductivity values in directions along and normal to the cardiac fibres with a sheet, as well as a conductivity value in the normal direction between the sheets, and these occur for both the extracellular and intracellular domains in the model. To date it has been common to assume that the two normal direction conductivity values are the same. However, recent work has demonstrated that six cardiac conductivity values, rather than four, are necessary for accurate modelling, which can then facilitate understanding of cardiovascular disease. To design a method to determine these conductivities, it is also necessary to design a suitable multi-electrode array, which can be used, in conjunction with an inversion technique, to retrieve conductivity values from measurements of potential made on the array. This work uses the results of a study, into the sensitivity of the measuring potentials to variability in the input conductivities, to facilitate the design of an array that could be used to retrieve six cardiac conductivity values, as well as fibre rotation angle. It is found that if an electrode in the array has a much lower value of potential than the other electrodes, then it tends to be much more sensitive to the input conductivities than the other electrodes. It also appears that inclusion of this type of electrode in the set of measuring electrodes is essential for accurately retrieving conductivity values. This technique is used to identify electrodes to be included in the array and using the final design it is demonstrated, using synthetic values of potential, that the six cardiac conductivity values, and the fibre rotation angle, can be retrieved very accurately.

A solution method for the determination of cardiac potential distributions with an alternating current source.

A recently presented solution method for the bidomain model (Johnston et al. 2006), which involves the application of direct current for studying electrical potential in a slab of cardiac tissue, is extended here to allow the use of an applied alternating current. The advantage of using AC current, in a four-electrode method for determining cardiac conductivities, is that instead of using 'close' and 'wide' electrode spacings to make potential measurements, increasing the frequency of the AC current redirects a fraction of the current from the extracellular space into the intracellular space. The model is based on the work of Le Guyader et al. (2001), but is able to include the effects of the fibre rotation between the epicardium and the endocardium on the potentials. Also, rather than using a full numerical technique, the solution method uses Fourier series and a simple one dimensional finite difference scheme, which has the advantage of allowing the potentials to be calculated only at points, such as the measuring electrodes, where they are required. The new alternating current model, which includes intracellular capacitance, is used with a particular four-electrode configuration, to show that the potential measured is affected by changes in fibre rotation. This is significant because it indicates that it is necessary to include fibre rotation in models, which are to be used in conjunction with measuring arrays that are more complex than those involving simply surface probes or a single vertical probe.

The relative effects of arterial curvature and lumen diameter on wall shear stress distributions in human right coronary arteries.

This study looks at blood flow in four different human right coronary arteries (RCAs), which have been reconstructed from bi-plane angiograms. A generalized power-law model of blood viscosity is used to study the blood flow at a particular point in the cardiac cycle. Large differences are found in the wall shear stress magnitude (WSS) distributions in the four arteries, leading to the conclusion that it is not possible to make generalizations based on the study of a single artery. The pattern of WSS is found to be related to the geometry of a particular artery, that is, lumen diameter and arterial curvature as well as a combination of these two factors. There is a strong correlation between WSS and reciprocal radius and a weaker correlation between high curvature and extremes of WSS, with high WSS on the 'inside' of a bend and low WSS on the 'outside' of a bend. This is in contrast to the situation for a simple curved tube with constant radius where the inverse is observed. For each artery, a region proximal to the acute margin is identified where low WSS is found and where WSS is lower on the 'inner' wall of the RCA than on the 'outer' wall. This region is one where several studies have found that the human RCA preferentially exhibits atherogenesis.

Possible four-electrode configurations for measuring cardiac tissue fiber rotation.

A number of electrode configurations, based on the usual four-electrode probe, are analysed in relation to the effect that changes in cardiac fiber rotation have on the potentials measured. Simulations are carried out using a mathematical model and a new solution technique, based on Fourier series followed by a simple one-dimensional finite difference scheme. This electrode analysis leads to the proposal of an in-principle method for determining cardiac fiber rotation.

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.

Non-Newtonian blood flow in human right coronary arteries: transient simulations.

This study looks at pulsatile blood flow through four different right coronary arteries, which have been reconstructed from biplane angiograms. A non-Newtonian blood model (the Generalised Power Law), as well as the usual Newtonian model of blood viscosity, is used to study the wall shear stress in each of these arteries over the entire cardiac cycle. The difference between Newtonian and non-Newtonian blood models is also studied over the whole cardiac cycle using the recently generalised global non-Newtonian importance factor. In addition, the flow is studied by considering paths of massless particles introduced into the flow field. The study shows that, when studying the wall shear stress distribution for transient blood flow in arteries, the use of a Newtonian blood model is a reasonably good approximation. However, to study the flow within the artery in greater detail, a non-Newtonian model is more appropriate.

Non-Newtonian blood flow in human right coronary arteries: steady state simulations.

This study looks at blood flow through four different right coronary arteries, which have been reconstructed from bi-plane angiograms. Five non-Newtonian blood models, as well as the usual Newtonian model of blood viscosity, are used to study the wall shear stress in each of these arteries at a particular point in the cardiac cycle. It was found that in the case of steady flow in a given artery, the pattern of wall shear stress is consistent across all models. The magnitude of wall shear stress, however, is influenced by the model used and correlates with graphs of shear stress versus strain for each model. For mid-range velocities of around 0.2 m s(-1) the models are virtually indistinguishable. Local and global non-Newtonian importance factors are introduced, in an attempt to quantify the types of flows where non-Newtonian behaviour is significant. It is concluded that, while the Newtonian model of blood viscosity is a good approximation in regions of mid-range to high shear, it is advisable to use the Generalised Power Law model (which tends to the Newtonian model in those shear ranges in any case) in order to achieve better approximation of wall shear stress at low shear.