Action potential propagation in a thick strand of cardiac muscle

Impact of electrode orientation, myocardial wall thickness, and myofiber direction on intracardiac electrograms: numerical modeling and analytical solutions

Intracardiac electrograms (iEGMs) are time traces of the electrical potential recorded close to the heart muscle. We calculate unipolar and bipolar iEGMs analytically for a myocardial slab with parallel myofibers and validate them against numerical bidomain simulations. The analytical solution obtained via the method of mirrors is an infinite series of arctangents. It goes beyond the solid angle theory and is in good agreement with the simulations, even though bath loading effects were not accounted for in the analytical calculation. At a large distance from the myocardium, iEGMs decay as 1/R (unipolar), 1/R ² (bipolar and parallel), and 1/R ³ (bipolar and perpendicular to the endocardium). At the endocardial surface, there is a mathematical branch cut. Here, we show how a thicker myocardium generates iEGMs with larger amplitudes and how anisotropy affects the iEGM width and amplitude. If only the leading-order term of our expansion is retained, it can be determined how the conductivities of the bath, torso, myocardium, and myofiber direction together determine the iEGM amplitude. Our results will be useful in the quantitative interpretation of iEGMs, the selection of thresholds to characterize viable tissues, and for future inferences of tissue parameters.

A simple and efficient adaptive time stepping technique for low-order operator splitting schemes applied to cardiac electrophysiology

We present a simple, yet efficient adaptive time stepping scheme for cardiac electrophysiology (EP) simulations based on standard operator splitting techniques. The general idea is to exploit the relation between the splitting error and the reaction's magnitude-found in a previous one-dimensional analytical study by Spiteri and Ziaratgahi-to construct the new time step controller for three-dimensional problems. Accordingly, we propose to control the time step length of the operator splitting scheme as a function of the reaction magnitude, in addition to the common approach of adapting the reaction time step. This conforms with observations in numerical experiments supporting the need for a significantly smaller time step length during depolarization than during repolarization. The proposed scheme is compared with classical proportional-integral-differential controllers using state-of-the-art error estimators, which are also presented in details as they have not been previously applied in the context of cardiac EP with operator splitting techniques. Benchmarks show that choosing the time step as a sigmoidal function of the reaction magnitude is highly efficient and full cardiac cycles can be computed with precision even in a realistic biventricular setup. The proposed scheme outperforms common adaptive time stepping techniques, while depending on fewer tuning parameters.

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.

Generation of Monophasic Action Potentials and Intermediate Forms

The monophasic action potential (MAP) is a near replica of the transmembrane potential recorded when an electrode is pushed firmly against cardiac tissue. Despite its many practical uses, the mechanism of MAP signal generation and the reason it is so different from unipolar recordings are not completely known and are a matter of controversy. In this work, we describe a method to simulate realistic MAP and intermediate forms, which are multiphasic electrograms different from an ideal MAP. The key ideas of our method are the formation of compressed zones and junctional spaces-regions of the extracellular and bath or blood pool directly in contact with electrodes that exhibit a pressure-induced reduction in electrical conductivity-and the presence of a complex network of passive components that acts as a high-pass filter to distort and attenuate the signal that reaches the recording amplifier. The network is formed by the interaction between the passive tissue properties and the double-layer capacitance of electrodes. The MAP and intermediate forms reside on a continuum of signals, which can be generated by the change of the model parameters. Our model helps to decipher the mechanisms of signal generation and can lead to a better design for electrodes, recording amplifiers, and experimental setups.

Muscle Thickness and Curvature Influence Atrial Conduction Velocities

Electroanatomical mapping is currently used to provide clinicians with information about the electrophysiological state of the heart and to guide interventions like ablation. These maps can be used to identify ectopic triggers of an arrhythmia such as atrial fibrillation (AF) or changes in the conduction velocity (CV) that have been associated with poor cell to cell coupling or fibrosis. Unfortunately, many factors are known to affect CV, including membrane excitability, pacing rate, wavefront curvature, and bath loading, making interpretation challenging. In this work, we show how endocardial conduction velocities are also affected by the geometrical factors of muscle thickness and wall curvature. Using an idealized three-dimensional strand, we show that transverse conductivities and boundary conditions can slow down or speed up signal propagation, depending on the curvature of the muscle tissue. In fact, a planar wavefront that is parallel to a straight line normal to the mid-surface does not remain normal to the mid-surface in a curved domain. We further demonstrate that the conclusions drawn from the idealized test case can be used to explain spatial changes in conduction velocities in a patient-specific reconstruction of the left atrial posterior wall. The simulations suggest that the widespread assumption of treating atrial muscle as a two-dimensional manifold for electrophysiological simulations will not accurately represent the endocardial conduction velocities in regions of the heart thicker than 0.5 mm with significant wall curvature.

Electrical Pacing of Cardiac Tissue Including Potassium Inward Rectification

In this study cardiac tissue is stimulated electrically through a small unipolar electrode. Numerical simulations predict that around an electrode are adjacent regions of depolarization and hyperpolarization. Experiments have shown that during pacing of resting cardiac tissue the hyperpolarization is often inhibited. Our goal is to determine if the inward rectifying potassium current (I_K1) causes the inhibition of hyperpolarization. Numerical simulations were carried out using the bidomain model with potassium dynamics specified to be inward rectifying. In the simulations, adjacent regions of depolarization and hyperpolarization were observed surrounding the electrode. For cathodal currents the virtual anode produces a hyperpolarization that decreases over time. For long duration pulses the current-voltage curve is non-linear, with very small hyperpolarization compared to depolarization. For short pulses, the hyperpolarization is more prominent. Without the inward potassium rectification, the current voltage curve is linear and the hyperpolarization is evident for both long and short pulses. In conclusion, the inward rectification of the potassium current explains the inhibition of hyperpolarization for long duration stimulus pulses, but not for short duration pulses.

A brief history of tissue models for cardiac electrophysiology

The last four decades have produced a number of significant advances in the developments of computer models to simulate and investigate the electrical activity of cardiac tissue. The tissue descriptions that underlie these simulations have been built from a combination of clever insight and careful comparison with measured data at multiple scales. Tissue models have not only led to greater insights into the mechanisms of life-threatening arrhythmias but have been used to engineer new therapies to treat the consequences of cardiac disease. This paper is a look back at the early years in the cardiac modeling and the challenges facing the field as models move toward the clinic.

Cardiac bidomain bath-loading effects during arrhythmias: interaction with anatomical heterogeneity

Cardiac tissue is always surrounded by conducting fluid, both in vivo (blood) and in experimental preparations (Tyrode's solution), which acts to increase conduction velocity (CV) close to the tissue-fluid interface, inducing transmural wavefront curvature. Despite its potential importance, computer modeling studies focused on arrhythmia mechanisms have previously not accounted for these bath-loading effects. Here, we investigate the increase in CV and concomitant change in transmural wavefront profiles upon both propagation and arrhythmia dynamics within models of differing anatomical complexity. In simplified slab models, in absence of transmural fiber rotation, bath-loading induced transmural wavefront curvature dominates, significantly increasing arrhythmia complexity compared to no bath. In the presence of fiber rotation, bath-loading effects are less striking and depend upon propagation direction: the bath accentuates natural concave curvature caused by transmurally rotating fibers, but attenuates convex curvature, which negates overall impact upon arrhythmia complexity. Finally, we demonstrate that the high degree of anatomical complexity within whole ventricular models modulates bath-loading induced transmural wavefront curvature. However, key is the increased surface CV that dramatically reduces both arrhythmia inducibility and resulting complexity by increasing wavelength and reducing the available excitable gap. Our findings highlight the importance of including bath-loading effects during arrhythmia mechanism investigations, which could have implications for interpreting and comparing simulation results with experimental data where such effects are inherently present.

The maximal downstroke of epicardial potentials as an index of electrical activity in mouse hearts

The maximal upstroke of transmembrane voltage (dV(m)/dt(max)) has been used as an indirect measure of sodium current I(Na) upon activation in cardiac myocytes. However, sodium influx generates not only the upstroke of V(m), but also the downstroke of the extracellular potentials V(e) including epicardial surface potentials V(es). The purpose of this study was to evaluate the magnitude of the maximal downstroke of V(es) (|dV(es)/dt (min)|) as a global index of electrical activation, based on the relationship of dV(m)/dt(max) to I(Na). To fulfill this purpose, we examined |dV(es)/dt(min)| experimentally using isolated perfused mouse hearts and computationally using a 3-D cardiac tissue bidomain model. In experimental studies, a custom-made cylindrical "cage" array with 64 electrodes was slipped over mouse hearts to measure V(es) during hyperkalemia, ischemia, and hypoxia, which are conditions that decrease I(Na). Values of |dV(es)/dt(min)| from each electrode were normalized (|dV(es)/dt (min)|(n)) and averaged (|dV(es)/dt(min)|(na)). Results showed that |dV(es)/dt(min)|(na) decreased during hyperkalemia by 28, 59, and 79% at 8, 10, and 12 mM [K(+)](o), respectively. |dV(es)/dt(min)| also decreased by 54 and 84% 20 min after the onset of ischemia and hypoxia, respectively. In computational studies, |dV(es)/dt(min)| was compared to dV(m)/dt(max) at different levels of the maximum sodium conductance G(Na), extracellular potassium ion concentration [K(+)](o), and intracellular sodium ion concentration [Na(+)](i), which all influence levels of I(Na). Changes in |dV(es)/dt(min)|(n) were similar to dV(m)/dt (max) during alterations of G(Na), [K(+)](o), and [Na(+)](i). Our results demonstrate that |dV(es)/dt(min)|(na) is a robust global index of electrical activation for use in mouse hearts and, similar to dV(m)/dt(max), can be used to probe electrophysiological alterations reliably. The index can be readily measured and evaluated, which makes it attractive for characterization of, for instance, genetically modified mouse hearts and drug effects on cardiac tissue.

Measurements of transmembrane potential and magnetic field at the apex of the heart

We studied the transmembrane potential and magnetic fields from electrical activity at the apex of the isolated rabbit heart experimentally using optical mapping and superconducting quantum interference device microscopy, and theoretically using monodomain and bidomain models. The cardiac apex has a complex spiral fiber architecture that plays an important role in the development and propagation of action currents during stimulation at the apex. This spiral fiber orientation contains both radial electric currents that contribute to the electrocardiogram and electrically silent circular currents that cannot be detected by the electrocardiogram but are detectable by their magnetic field, B(z). In our experiments, the transmembrane potential, V(m), was first measured optically and then B(z) was measured with a superconducting quantum interference device microscope. Based on a simple model of the spiral structure of the apex, V(m) was expected to exhibit circular wave front patterns and B(z) to reflect the circular component of the action currents. Although the circular V(m) wave fronts were detected, the B(z) maps were not as simple as expected. However, we observed a pattern consistent with a tilted axis for the apex spiral fiber geometry. We were able to simulate similar patterns in both a monodomain model of a tilted stack of rings of dipole current and a bidomain model of a tilted stack of spiraled cardiac tissue that was stimulated at the apex. The fact that the spatial pattern of the magnetic data was more complex than the simple circles observed for V(m) suggests that the magnetic data contain information that cannot be found electrically.

Probing field-induced tissue polarization using transillumination fluorescent imaging

Despite major successes of biophysical theories in predicting the effects of electrical shocks within the heart, recent optical mapping studies have revealed two major discrepancies between theory and experiment: 1), the presence of negative bulk polarization recorded during strong shocks; and 2), the unexpectedly small surface polarization under shock electrodes. There is little consensus as to whether these differences result from deficiencies of experimental techniques, artifacts of tissue damage, or deficiencies of existing theories. Here, we take advantage of recently developed near-infrared voltage-sensitive dyes and transillumination optical imaging to perform, for the first time that we know of, noninvasive probing of field effects deep inside the intact ventricular wall. This technique removes some of the limitations encountered in previous experimental studies. We explicitly demonstrate that deep inside intact myocardial tissue preparations, strong electrical shocks do produce considerable negative bulk polarization previously inferred from surface recordings. We also demonstrate that near-threshold diastolic field stimulation produces activation of deep myocardial layers 2-6 mm away from the cathodal surface, contrary to theory. Using bidomain simulations we explore factors that may improve the agreement between theory and experiment. We show that the inclusion of negative asymmetric current can qualitatively explain negative bulk polarization in a discontinuous bidomain model.

Models of cardiac tissue electrophysiology: progress, challenges and open questions

Models of cardiac tissue electrophysiology are an important component of the Cardiac Physiome Project, which is an international effort to build biophysically based multi-scale mathematical models of the heart. Models of tissue electrophysiology can provide a bridge between electrophysiological cell models at smaller scales, and tissue mechanics, metabolism and blood flow at larger scales. This paper is a critical review of cardiac tissue electrophysiology models, focussing on the micro-structure of cardiac tissue, generic behaviours of action potential propagation, different models of cardiac tissue electrophysiology, the choice of parameter values and tissue geometry, emergent properties in tissue models, numerical techniques and computational issues. We propose a tentative list of information that could be included in published descriptions of tissue electrophysiology models, and used to support interpretation and evaluation of simulation results. We conclude with a discussion of challenges and open questions.

Increased interstitial loading reduces the effect of microstructural variations in cardiac tissue

Electrical propagation in diseased and aging hearts is strongly influenced by structural changes that occur in both the intracellular and interstitial spaces of cardiac tissue; however, very few studies have investigated how interactions between the two spaces affect propagation at the microscale. In this study, we used one-dimensional microstructural computer models of interconnected ventricular myocytes to systematically investigate how increasing the effective interstitial resistivity (rho(oeff)) influences action potential propagation in fibers with variations in intracellular properties such as cell coupling and cell length. Changes in rho(oeff) were incorporated into a monodomain model using a correction to the intracellular properties that was based on bidomain simulations. The results showed that increasing rho(oeff) in poorly coupled one-dimensional fibers alters the distribution of electrical load at the microscale and causes propagation to become more continuous. In the poorly coupled fiber, this continuous state is characterized by decreased gap junction delay, sustained conduction velocity, increased sodium current, reduced maximum upstroke velocity, and increased safety factor. Long, poorly coupled cells experience greater loading effects than short cells and show the greatest initial response to changes in rho(oeff). In inhomogeneous fibers with adjacent well-coupled and poorly coupled regions, increasing rho(oeff) in the poorly coupled region also reduces source-load mismatch, which delays the onset of conduction block and reduces the dispersion of repolarization at the transition between the two regions. Increasing the rho(oeff) minimizes the effect of cell-to-cell variations and may influence the pattern of activation in critical regimes characterized by low intercellular coupling, microstructural heterogeneity, and reduced or abnormal membrane excitability.

A model of electrical conduction in cardiac tissue including fibroblasts

Fibroblasts are abundant in cardiac tissue. Experimental studies suggested that fibroblasts are electrically coupled to myocytes and this coupling can impact cardiac electrophysiology. In this work, we present a novel approach for mathematical modeling of electrical conduction in cardiac tissue composed of myocytes, fibroblasts, and the extracellular space. The model is an extension of established cardiac bidomain models, which include a description of intra-myocyte and extracellular conductivities, currents and potentials in addition to transmembrane voltages of myocytes. Our extension added a description of fibroblasts, which are electrically coupled with each other and with myocytes. We applied the extended model in exemplary computational simulations of plane waves and conduction in a thin tissue slice assuming an isotropic conductivity of the intra-fibroblast domain. In simulations of plane waves, increased myocyte-fibroblast coupling and fibroblast-myocyte ratio reduced peak voltage and maximal upstroke velocity of myocytes as well as amplitudes and maximal downstroke velocity of extracellular potentials. Simulations with the thin tissue slice showed that inter-fibroblast coupling affected rather transversal than longitudinal conduction velocity. Our results suggest that fibroblast coupling becomes relevant for small intra-myocyte and/or large intra-fibroblast conductivity. In summary, the study demonstrated the feasibility of the extended bidomain model and supports the hypothesis that fibroblasts contribute to cardiac electrophysiology in various manners.

Effect of nonuniform interstitial space properties on impulse propagation: a discrete multidomain model

This work presents a discrete multidomain model that describes ionic diffusion pathways between connected cells and within the interstitium. Unlike classical models of impulse propagation, the intracellular and extracellular spaces are represented as spatially distinct volumes with dynamic/static boundary conditions that electrically couple neighboring spaces. The model is used to investigate the impact of nonuniform geometrical and electrical properties of the interstitial space surrounding a fiber on conduction velocity and action potential waveshape. Comparison of the multidomain and bidomain models shows that although the conduction velocity is relatively insensitive to cases that confine 50% of the membrane surface by narrow extracellular depths (> or =2 nm), the action potential morphology varies greatly around the fiber perimeter, resulting in changes in the magnitude of extracellular potential in the tight spaces. Results also show that when the conductivity of the tight spaces is sufficiently reduced, the membrane adjacent to the tight space is eliminated from participating in propagation, and the conduction velocity increases. Owing to its ability to describe the spatial discontinuity of cardiac microstructure, the discrete multidomain can be used to determine appropriate tissue properties for use in classical macroscopic models such as the bidomain during normal and pathophysiological conditions.

A comparison of monodomain and bidomain propagation models for the human heart

A bidomain reaction-diffusion model of the human heart was developed and potentials resulting from normal depolarization and repolarization were compared with results from a compatible monodomain model. Comparisons were made for an empty isolated heart and for a heart with fluid-filled ventricles. Both sinus rhythm and ectopic activation were simulated. The model took 2 days on 32 processors to simulate a complete cardiac cycle. Differences between monodomain and bidomain results were very small, even for the extracellular potentials which, for the monodomain model, were computed with a high-resolution forward model. Electrograms computed with monodomain and bidomain models were visually indistinguishable. We conclude that, in the absence of applied currents, propagating action potentials on the scale of a human heart can be studied with a monodomain model.

Experimental and computational studies of strain-conduction velocity relationships in cardiac tissue

Velocity of electrical conduction in cardiac tissue is a function of mechanical strain. Although strain-modulated velocity is a well established finding in experimental cardiology, its underlying mechanisms are not well understood. In this work, we summarized potential factors contributing to strain-velocity relationships and reviewed related experimental and computational studies. We presented results from our experimental studies on rabbit papillary muscle, which supported a biphasic relationship of strain and velocity under uni-axial straining conditions. In the low strain range, the strain-velocity relationship was positive. Conduction velocity peaked with 0.59 m/s at 100% strain corresponding to maximal force development. In the high strain range, the relationship was negative. Conduction was reversibly blocked at 118+/-1.8% strain. Reversible block occurred also in the presence of streptomycin. Furthermore, our studies revealed a moderate hysteresis of conduction velocity, which was reduced by streptomycin. We reconstructed several features of the strain-velocity relationship in a computational study with a myocyte strand. The modeling included strain-modulation of intracellular conductivity and stretch-activated cation non-selective ion channels. The computational study supported our hypotheses, that the positive strain-velocity relationship at low strain is caused by strain-modulation of intracellular conductivity and the negative relationship at high strain results from activity of stretch-activated channels. Conduction block was not reconstructed in our computational studies. We concluded this work by sketching a hypothesis for strain-modulation of conduction and conduction block in papillary muscle. We suggest that this hypothesis can also explain uni-axially measured strain-conduction velocity relationships in other types of cardiac tissue, but apparently necessitates adjustments to reconstruct pressure or volume related changes of velocity in atria and ventricles.

Monophasic action potentials generated by bidomain modeling as a tool for detecting cardiac repolarization times

Unipolar electrograms (EGs) and hybrid (or unorthodox or unipolar) monophasic action potentials (HMAPs) are currently the only proposed extracellular electrical recording techniques for obtaining cardiac recovery maps with high spatial resolution in exposed and isolated hearts. Estimates of the repolarization times from the HMAP downstroke phase have been the subject of recent controversies. The goal of this paper is to computationally address the controversies concerning the HMAP information content, in particular the reliability of estimating the repolarization time from the HMAP downstroke phase. Three-dimensional numerical simulations were performed by using the anisotropic bidomain model with a region of short action potential durations. EGs, transmembrane action potentials (TAPs), and HMAPs elicited by an epicardial stimulation close or away from a permanently depolarized site were computed. The repolarization time was computed as the moment of EG fastest upstroke (RTeg) during the T wave, of HMAP fastest downstroke (RTHMAP), and of TAP fastest downstroke (RTtap). The latter was taken as the gold standard for repolarization time. We also compared the times (RT90HMAP, RT90tap) when the HMAP and TAP first reach 90% of their resting value during the downstroke. For all explored sites, the HMAP downstroke closely followed the TAP downstroke, which is the expression of local repolarization activity. Results show that HMAP and TAP markers are highly correlated, and both markers RTHMAP and RTeg (RT90HMAP) are reliable estimates of the TAP reference marker RTtap (RT90tap). Therefore, the downstroke phase of the HMAP contains valuable information for assessing repolarization times.

Forward Euler Stability of the Bidomain Model of Cardiac Tissue

The bidomain model describes the anisotropic electrical properties of cardiac tissue. One common numerical technique for solving the bidomain equations is the explicit forward Euler method. In this communication we derive a relationship between the time and space steps that ensures the stability of this numerical method.

Small random fiber angle variations as a mechanism for far-field stimulation of cardiac tissue

The mechanism by which an electric field stimulates cardiac tissue far from the stimulating electrode is not known. Current theories rely on the inherent heterogeneity of cardiac tissue as contributing to this phenomenon. This paper explores a possible mechanism for far-field stimulation in relation to fiber curvature. We incorporate a randomized fiber angle geometry into a two-dimensional active cardiac tissue model with unequal anisotropy ratios already exhibiting smooth, curving fibers. We simulate cross-field stimulation to initiate reentry in the tissue model, and compare the electric field thresholds at different S1-S2 intervals for tissue with randomized fiber angles and tissue with a smooth fiber geometry. The tissue with random fiber angles has a significantly lower threshold for reentry at certain intervals on the strength-interval curve. We conclude that a random fiber geometry may have an effect on the mechanism for far-field stimulation.

A comparison of monodomain and bidomain reaction-diffusion models for action potential propagation in the human heart

A bidomain reaction-diffusion model of the human heart was developed, and potentials resulting from normal depolarization and repolarization were compared with results from a compatible monodomain model. Comparisons were made for an empty isolated heart and for a heart with fluid-filled ventricles. Both sinus rhythm and ectopic activation were simulated. The bidomain model took 2 days on 32 processors to simulate a complete cardiac cycle. Differences between monodomain and bidomain results were extremely small, even for the extracellular potentials, which in case of the monodomain model were computed with a high-resolution forward model. Propagation of activation was 2% faster in the bidomain model than in the monodomain model. Electrograms computed with monodomain and bidomain models were visually indistinguishable. We conclude that, in the absence of applied currents, propagating action potentials on the scale of a human heart can be studied with a monodomain model.

A convenient scheme for coupling a finite element curvilinear mesh to a finite element voxel mesh: application to the heart

Background

In some cases, it may be necessary to combine distinct finite element meshes into a single system. The present work describes a scheme for coupling a finite element mesh, which may have curvilinear elements, to a voxel based finite element mesh.

Methods

The method is described with reference to a sample problem that involves combining a heart, which is defined by a curvilinear mesh, with a voxel based torso mesh. The method involves the creation of a temporary (scaffolding) mesh that couples the outer surface of the heart mesh to a voxel based torso mesh. The inner surface of the scaffolding mesh is the outer heart surface, and the outer surface of the scaffolding mesh is defined by the nodes in the torso mesh that are nearest (but outside of) the heart. The finite element stiffness matrix for the scaffolding mesh is then computed. This stiffness matrix includes extraneous nodes that are then removed, leaving a coupling matrix that couples the original outer heart surface nodes to adjacent nodes in the torso voxel mesh. Finally, a complete system matrix is assembled from the pre-existing heart stiffness matrix, the heart/torso coupling matrix, and the torso stiffness matrix.

Results

Realistic body surface electrocardiograms were generated. In a test involving a dipole embedded in a spherical shell, relative error of the scheme rapidly converged to slightly over 4%, although convergence thereafter was relatively slow.

Conclusion

The described method produces reasonably accurate results and may be best suited for problems where computational speed and convenience have a higher priority than very high levels of accuracy.

The effect of the fiber curvature gradient on break excitation in cardiac tissue

BACKGROUND

Break excitation has been hypothesized as a mechanism for the initiation of reentry in cardiac tissue. One way break excitation can occur is by virtual electrodes formed due to a curving fiber geometry. In this article, we are concerned with the relationship between the peak gradient of fiber curvature and the threshold for break stimulation and the initiation of reentry.

METHODS

We calculate the maximum gradient of fiber curvature for different scales of fiber geometry in a constant tissue size (20x20 mm), and also examine the mechanisms by which reentry initiation fails.

RESULTS

For small peak gradients, reentry fails because break excitation does not occur. For larger peak gradients, reentry fails because break excitation fails to develop into full-scale reentry. For strong stimuli above the upper limit of vulnerability, reentry fails because the break excitation propagates through the hyperpolarized region and then encounters refractory tissue, causing the wave front to die.

Measuring surface potential components necessary for transmembrane current computation using microfabricated arrays

This study was designed to test the feasibility of using microfabricated electrodes to record surface potentials with sufficiently fine spatial resolution to measure the potential gradients necessary for improved computation of transmembrane current density. To assess that feasibility, we recorded unipolar electrograms from perfused rabbit right ventricular free wall epicardium ( n = 6) using electrode arrays that included 25-μm sensors fabricated onto a flexible substrate with 75-μm interelectrode spacing. Electrode spacing was therefore on the size scale of an individual myocyte. Signal conditioning adjacent to the sensors to control lead noise was achieved by routing traces from the electrodes to the back side of the substrate where buffer amplifiers were located. For comparison, recordings were also made using arrays built from chloridized silver wire electrodes of either 50-μm (fine wire) or 250-μm (coarse wire) diameters. Electrode separations were necessarily wider than with microfabricated arrays. Comparable signal-to-noise ratios (SNRs) of 21.2 ± 2.2, 32.5 ± 4.1, and 22.9 ± 0.7 for electrograms recorded using microfabricated sensors ( n = 78), fine wires ( n = 78), and coarse wires ( n = 78), respectively, were found. High SNRs were maintained in bipolar electrograms assembled using spatial combinations of the unipolar electrograms necessary for the potential gradient measurements and in second-difference electrograms assembled using spatial combinations of the bipolar electrograms necessary for surface Laplacian (SL) measurements. Simulations incorporating a bidomain representation of tissue structure and a two-dimensional network of guinea pig myocytes prescribed following the Luo and Rudy dynamic membrane equations were completed using 12.5-μm spatial resolution to assess contributions of electrode spacing to the potential gradient and SL measurements. In those simulations, increases in electrode separation from 12.5 to 75.0, 237.5, and 875.0 μm, which were separations comparable to the finest available with our microfabricated, fine wire, and coarse wire arrays, led to 10%, 42%, and 81% reductions in maximum potential gradients and 33%, 76%, and 96% reductions in peak-to-peak SLs. Maintenance of comparable SNRs for source electrograms was therefore important because microfabrication provides a highly attractive methods to achieve spatial resolutions necessary for improved computation of transmembrane current density.

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.

How the spatial frequency of polarization influences the induction of reentry in cardiac tissue

Influences of spatial frequency of polarization.

INTRODUCTION

The mechanism by which an electric field induces a rotor during cross-field stimulation of cardiac tissue is not entirely known. Different heterogeneous aspects of cardiac tissue have been offered as possible theories, a prominent one being fiber curvature. The polarization produced when an electric field is applied to a sheet of tissue is varied over many spatial frequencies, depending upon the fiber angle. This article compares the effect of high and low spatial frequencies of polarization on reentry induction.

METHODS AND RESULTS

We incorporate a randomized fiber angle geometry into a two-dimensional active cardiac tissue model with unequal anisotropy ratios already exhibiting smooth, curving fibers. We simulate cross-field stimulation to initiate reentry in the tissue model, and compare the electric field thresholds at different S1-S2 intervals for tissue with randomized fiber angles, tissue with a smooth fiber geometry, and tissue with randomized fiber angles plus smooth, curving fibers. The tissue with both small, random fiber angles and curving fibers has a significantly lower threshold for reentry at certain intervals on the strength-interval curve than for the two cases individually.

CONCLUSION

Cardiac tissue exhibiting a random fiber geometry in conjunction with a smooth fiber geometry includes high and low spatial frequencies of polarization that may have an effect on the mechanism for reentry at certain S1-S2 intervals. Low spatial frequency regions of hyperpolarization carve out excitable pathways, and high spatial frequency regions provide the large gradient of transmembrane potential required to initiate break excitation.

Effect of plunge electrodes in active cardiac tissue with curving fibers

OBJECTIVES

Our goal is to determine if plunge electrodes change how the heart responds to electrical stimulation.

BACKGROUND

Several experiments designed to study the induction of a rotor in cardiac tissue have used plunge electrodes to measure the transmural potential. It is our hypothesis that these electrodes may have affected the electrical response of the tissue to a shock.

METHODS

We previously have shown that a single plunge electrode in two-dimensional, passive cardiac tissue induces a significant transmembrane potential when stimulated by a large shock. In this study, we expand our simulation to include an array of nine electrodes in active tissue with curving fibers. We compare the thresholds for rotor induction in tissue with and without electrodes by initiating a planar S1 wavefront and then stimulating the tissue at different intervals with a uniform S2 electric field perpendicular to S1. In tissue without plunge electrodes, virtual electrode polarization due to the curving fibers is generally widespread over the entire tissue, whereas polarization tends to be localized around the electrodes in tissue including them.

RESULTS

Our results show that at some S1-S2 intervals, the presence of plunge electrodes can result in reentry when it otherwise would not be possible. For other S1-S2 intervals, such as during the vulnerable period when the reentry threshold is at a minimum, the induction of reentry is unaffected by the presence of plunge electrodes.

CONCLUSIONS

Plunge electrodes can play an important role during the stimulation of cardiac tissue, but this is highly dependent on the chosen S1-S2 interval.

Feasibility of cardiac microimpedance measurement using multisite interstitial stimulation

This study was designed to test the hypothesis that analyses of central interstitial potential differences recorded during multisite stimulation with a set of interstitial electrodes provide sufficient data for accurate measurement of cardiac microimpedances. On theoretical grounds, interstitial current injected and removed using electrodes in close proximity does not cross the membrane, whereas equilibration of intracellular and interstitial potentials occurs distant from electrodes widely separated. Multisite interstitial stimulation should therefore give rise to interstitial potential differences recorded centrally that depend on intracellular and interstitial microimpedances, allowing independent measurement. Simulations of multisite stimulation with fine (25 microm) and wide (400 microm) spacing in one-dimensional models that included Luo-Rudy dynamic membrane equations were performed. Constant interstitial and intracellular microimpedances were prescribed for initial analyses. Discrete myoplasmic and gap-junctional components were prescribed intracellularly in later simulations. With constant microimpedances, multisite stimulation using 29 total electrode combinations allowed interstitial and intracellular microimpedance measurements at errors of 0.30% and 0.34%, respectively, with errors of 0.05% and 0.40% achieved using 6 combinations and 10 total electrodes. With discrete myoplasmic and junctional components, comparable accuracy was maintained following adjustments to the junctions to reflect uncoupling. This allowed uncoupling to be quantified as relative increases in total junctional resistance. Our findings suggest development of microfabricated devices to implement the procedure would facilitate routine measurement as a component of cardiac electrophysiological study.

Genesis of the monophasic action potential: role of interstitial resistance and boundary gradients

The extracellular potential at the site of a mechanical deformation has been shown to resemble the underlying transmembrane action potential, providing a minimally invasive way to access membrane dynamics. The biophysical factors underlying the genesis of this signal, however, are still poorly understood. With the use of data from a recent experimental study in a murine heart, a three-dimensional anisotropic bidomain model of the mouse ventricular free wall was developed to study the currents and potentials resulting from the application of a point mechanical load on cardiac tissue. The applied pressure is assumed to open nonspecific pressure-sensitive channels depolarizing the membrane, leading to monophasic currents at the electrode edge that give rise to the monophasic action potential (MAP). The results show that the magnitude and the time course of the MAP are reproduced only for certain combinations of local or global intracellular and interstitial resistances that form a resting tissue length constant that, if applied over the entire domain, is smaller than that required to match the wave speed. The results suggest that the application of pressure not only causes local depolarization but also changes local tissue properties, both of which appear to play a critical role in the genesis of the MAP.

Electrode systems for measuring cardiac impedances using optical transmembrane potential sensors and interstitial electrodes--theoretical design

The cardiac electrical substrate is a challenge to direct measurement of its properties. Optical technology together with the capability to fabricate small electrodes at close spacings opens new possibilities. Here, those possibilities are explored from a theoretical viewpoint. It appears that with careful measurements from a well-designed set of electrodes one can obtain structural conductivities, separating intracellular from interstitial values, and longitudinal from transverse. Resting membrane resistance also can be obtained.

Approximate analytical solutions of the Bidomain equations for electrical stimulation of cardiac tissue with curving fibers

The mechanism by which an applied electric field stimulates cardiac tissue far from the stimulating electrodes is not wholly understood. One possible mechanism relates the curving cardiac fibers to the induced membrane currents and transmembrane potentials. However, we lack a qualitative understanding of where these areas of polarization will occur when an electric field is applied to a sheet of cardiac tissue with curving fibers. In our study, we derive an analytical model for the transmembrane potential, dependent on the gradient of the fiber angle theta, for a two-dimensional passive sheet of cardiac tissue exhibiting various fiber geometries. Unequal anisotropy ratios are crucial for our results. We compare the results from our analytical solution to a numerical calculation using the full bidomain model. The results of our comparison are qualitatively consistent, albeit numerically different. We believe that our analytical approximation provides a reliable prediction of the polarization associated with an electric field applied to cardiac tissue with any fiber geometry and a qualitative understanding of the mechanisms behind the virtual electrode polarization.

Vortex cordis as a mechanism of postshock activation: arrhythmia induction study using a bidomain model

INTRODUCTION

The ventricular apex has a helical arrangement of myocardial fibers called the "vortex cordis." Experimental studies have demonstrated that the first postshock activation originates from the ventricular apex, regardless of the electrical shock outcome; however, the related underlying mechanism is unclear. We hypothesized that the vortex cordis contributes to the initiation of postshock activation. To clarify this issue, we numerically studied the transmembrane potential distribution produced by various electrical shocks.

METHODS AND RESULTS

Using an active membrane model, we simulated a two-dimensional bidomain myocardial tissue incorporating a typical fiber orientation of the vortex cordis. Monophasic or biphasic shock was delivered via two line electrodes located at opposite tissue borders. Transmembrane potential distribution during the monophasic shock at the center of the vortex cordis showed a gradient high enough to initiate postshock activation. The postshock activation from the center of the vortex cordis was not suppressed, regardless of the initiation of spiral wave reentry. Spiral wave reentry was induced by the monophasic shock when the center area of the vortex cordis was partially excited by the nonuniform virtual electrode polarization. Postshock activation following the biphasic shock also originated from the center of the vortex cordis, but it tended to be suppressed due to the narrower excitable gap around the center of the vortex cordis. The electroporation effect, which was maximal at the center of the vortex cordis, is another possible mechanism of postshock activation.

CONCLUSION

Our simulations suggest that the vortex cordis may cause postshock activation.

The importance of anisotropy in modeling ST segment shift in subendocardial ischaemia

In this paper, a simple mathematical model of a slab of cardiac tissue is presented in an attempt to better understand the relationship between subendocardial ischaemia and the resulting epicardial potential distributions. The cardiac tissue is represented by the bidomain model where tissue anisotropy and fiber rotation have been incorporated with a view to predicting the epicardial surface potential distribution. The source of electric potential in this steady-state problem is the difference between plateau potentials in normal and ischaemic tissue, where it is assumed that ischaemic tissue has a lower plateau potential. Simulations with tissue anisotropy and no fiber rotation are also considered. Simulations are performed for various thicknesses of the transition region between normal and ischaemic tissue and for various sizes of the ischaemic region. The simulated epicardial potential distributions, based on an anisotropic model of the cardiac tissue, show that there are large potential gradients above the border of the ischaemic region and that there are dips in the potential distribution above the region of ischaemia. It could be concluded from the simulations that it would be possible to predict the region of subendocardial ischaemia from the epicardial potential distribution, a conclusion contrary to observed experimental data. Possible reasons for this discrepancy are discussed. In the interests of mathematical simplicity, isotropic models of the cardiac tissue are also considered, but results from these simulations predict epicardial potential distributions vastly different from experimental observations. A major conclusion from this work is that tissue anisotropy and fiber rotation must be included to obtain meaningful and realistic epicardial potential distributions.

Distributed computing for membrane-based modeling of action potential propagation

Action potential propagation simulations with physiologic membrane currents and macroscopic tissue dimensions are computationally expensive. We, therefore, analyzed distributed computing schemes to reduce execution time in workstation clusters by parallelizing solutions with message passing. Four schemes were considered in two-dimensional monodomain simulations with the Beeler-Reuter membrane equations. Parallel speedups measured with each scheme were compared to theoretical speedups, recognizing the relationship between speedup and code portions that executed serially. A data decomposition scheme based on total ionic current provided the best performance. Analysis of communication latencies in that scheme led to a load-balancing algorithm in which measured speedups at 89 +/- 2% and 75 +/- 8% of theoretical speedups were achieved in homogeneous and heterogeneous clusters of workstations. Speedups in this scheme with the Luo-Rudy dynamic membrane equations exceeded 3.0 with eight distributed workstations. Cluster speedups were comparable to those measured during parallel execution on a shared memory machine.

A numerically efficient model for simulation of defibrillation in an active bidomain sheet of myocardium

Presented here is an efficient algorithm for solving the bidomain equations describing myocardial tissue with active membrane kinetics. An analysis of the accuracy shows advantages of this numerical technique over other simple and therefore popular approaches. The modular structure of the algorithm provides the critical flexibility needed in simulation studies: fiber orientation and membrane kinetics can be easily modified. The computational tool described here is designed specifically to simulate cardiac defibrillation, i. e., to allow modeling of strong electric shocks applied to the myocardium extracellularly. Accordingly, the algorithm presented also incorporates modifications of the membrane model to handle the high transmembrane voltages created in the immediate vicinity of the defibrillation electrodes.

Time dependence of anodal and cathodal refractory periods in cardiac tissue

Mehra et al. (PACE 1980; 3:526) observed that immediately after implantation of a pacing electrode in a dog heart, the anodal refractory period (RP) is shorter than the cathodal RP, but after several weeks the anodal RP becomes longer than the cathodal RP. We examine this experiment using numerical simulations based on the bidomain model of cardiac tissue and a Beeler-Reuter membrane. Our hypothesis is that accumulation of inexcitable tissue around the electrode following implantation causes the effective size of the electrode to increase and that this increase is the mechanism underlying the change in RP. We calculate that the anodal RP is shorter than the cathodal RP for both large and small electrodes. However, for large electrodes the threshold for anode "break" stimulation is greater than 8 mA. Mehra et al. defined RP experimentally as the interval at which the threshold stimulus strength becomes greater than 8 mA. If we restrict the stimulus current in our calculations to less than 8 mA, we exclude anode break stimulation from our calculation of the RP. In that case, our results are consistent with Mehra et al. and suggest that their observation resulted from their definition of RP.

Effects of spatial segmentation in the continuous model of excitation propagation in cardiac muscle

INTRODUCTION

Spatial segmentation is essential for the numerical simulation of excitation propagation in cardiac muscle.

METHODS AND RESULTS

This study evaluated the effects of spatial segmentation on action potential and on the velocity of propagation in a continuous one-dimensional model of cardiac muscle [intracellular and extracellular resistivities along (L) and transverse (T) to the muscle fibers: 402 omega(cm) (R(e), L), 3,620 omega(cm) (R(e), T), 48 omega(cm) (R(e), L), and 126 omega(cm) (R(e), T), J of Physiol 255:335-346, 1976) and either Luo-Rudy (L-R, Circ Res 68:1501-1526, 1991) or Beeler-Reuter (B-R, J Physiol 268:177-210, 1977) ionic currents. Related cable equations for active membrane are derived. Spatial segmentations of < 31.2 microm (L, L-R), < 11.5 microm (T, L-R), < 44.7 microm (L, B-R), and < 16.5 microm (T, B-R) were required for < 1% errors in the characteristic parameters of action potential. Similarly, spatial segmentations of < 54.5 microm (L, L-R), < 20.1 microm (T, L-R), < 84.3 microm (L, B-R), and < 31.2 microm (T, B-R) were required for < 1 % errors in the velocity of conduction.

CONCLUSION

In general, spatial segmentations of < 26.9% and < 50.8% of the space constant of a fully activated membrane gave < 1.0% errors in the characteristic parameters of action potential and in the velocity of propagation, respectively, for both membranes. The action potential duration was relatively insensitive to the spatial segmentation. Our analysis suggests that lambda(full is a better criterion for the selection of spatial segmentation in numerical simulation than the space constant of the resting membrane.

Optical transmembrane potential recordings during intracardiac defibrillation-strength shocks

BACKGROUND

The prolongation of the action potential after defibrillation-strength shocks is believed to be a critical component of defibrillation. The response of the transmembrane potential to the shock may affect this prolongation. We studied the effects of an intracardiac shock on the transmembrane potential and action potential duration at multiple sites on the epicardium using a voltage-sensitive dye and optical mapping system.

METHODS AND RESULTS

A laser scanner recorded optical action potentials with voltage-sensitive dye at 63 spots on both the left and right ventricles of six isolated, perfused rabbit hearts. Hearts were paced with epicardial point stimulation followed by the delivery of a 2 A and 20 ms rectangular waveform shock during the relative refractory period. The shock was given between right atrial and right ventricular electrodes. Of 621 total spots analyzed, 241 spots hyperpolarized and 76 spots depolarized with a right ventricular anode, whereas 159 spots hyperpolarized and 145 spots depolarized with a right ventricular cathode (P < 0.05). Both hyperpolarized and depolarized spots exhibited prolonged action potential duration, although prolongation was greater with depolarizing responses (16.7 +/- 9 ms vs. 13.3 +/- 13.4 ms, p<0.001). Hyperpolarized and depolarized spots were not randomly distributed, but clustered into regions. The size of the hyperpolarized regions was larger than the depolarized regions with RV anodal stimulation (27 +/- 20 spots/hyperpolarized region vs. 8.5 +/- 9 spots/depolarized region, p < 0.03) but not with RV cathodal stimulation. With reversal of electrode polarity, spots hyperpolarized near the shocking electrodes frequently did not reverse polarization but remained hyperpolarized.

CONCLUSIONS

Distinct regions of either polarization occur during intracardiac defibrillation-strength shocks. Although hyperpolarizing membrane responses were observed more often than depolarizing responses, depolarizing membrane polarization resulted in greater action potential prolongation. The absence of sign change in polarization in some regions with shocks of opposite polarities suggests that nonlinear intrinsic membrane properties are operative during strong electrical stimulation.

Validation of three-dimensional conduction models using experimental mapping: are we getting closer?

The anisotropic material properties, irregular geometry, and specialized conduction system of the heart all affect the three-dimensional (3D) spread of electrical activation. A limited number of research groups have tried accounting for these features in 3D conduction models to investigate more thoroughly their observations of cardiac electrical activity in 3D experimental preparations. The full potential of these large scale conduction models, however, has not been realized because of a lack of quantitative validation with experiment. Such validation is critical in order to use the models to predict the electrical response of the myocardium to drugs or electrical stimulation. In this paper, a quantitative, experimental validation of paced activation in a 3D conduction model of a 3 cm x 3 cm x 1 cm section of the ventricular wall is presented. Epicardial and intramural pacing stimuli were applied in the center of a 528 channel electrode plaque sutured to the left ventricle in dogs. Unipolar electrograms were recorded at 2 kHz during and after pacing. Fiber directions within the tissue below the electrodes were estimated histologically and from pace-mapping. Simulated epicardial electrograms were computed for surface paced beats using our 3D bidomain model of the mapped tissue volume incorporating the measured fiber directions. Extracellular potentials and isochronal maps resulting from paced activations in both model and experiment were directly compared. Preliminary results demonstrate that our 3D model reproduces qualitatively such key features of the experimental data as electrogram morphologies and epicardial conduction velocities. Though quantitative agreement between model and experiment was only moderate, the validation approach described herein is an essential first step in assessing the predictive capability of present day conduction models.

Bipolar stimulation of a three-dimensional bidomain incorporating rotational anisotropy

A bidomain model of cardiac tissue was used to examine the effect of transmural fiber rotation during bipolar stimulation in three-dimensional (3-D) myocardium. A 3-D tissue block with unequal anisotropy and two types of fiber rotation (none and moderate) was stimulated along and across fibers via bipolar electrodes on the epicardial surface, and the resulting steady-state interstitial (phi e) and transmembrane (Vm) potentials were computed. Results demonstrate that the presence of rotated fibers does not change the amount of tissue polarized by the point surface stimuli, but does cause changes in the orientation of phi e and Vm in the depth of the tissue, away from the epicardium. Further analysis revealed a relationship between the Laplacian of phi e, regions of virtual electrodes, and fiber orientation that was dependent upon adequacy of spatial sampling and the interstitial anisotropy. These findings help to understand the role of fiber architecture during extracellular stimulation of cardiac muscle.

A numerical method for the solution of the bidomain equations in cardiac tissue

A numerical scheme for efficient integration of the bidomain model of action potential propagation in cardiac tissue is presented. The scheme is a mixed implicit-explicit scheme with no stability time step restrictions and requires that only linear systems of equations be solved at each time step. The method is faster than a fully explicit scheme and there is no increase in algorithmic complexity to use this method instead of a fully explicit method. The speedup factor depends on the timestep size, which can be set solely on the basis of the demands for accuracy. (c) 1998 American Institute of Physics.

A new three-dimensional finite-difference bidomain formulation for inhomogeneous anisotropic cardiac tissues

Bidomain modeling of cardiac tissues provides important information about various complex cardiac activities. The cardiac tissue consists of interconnected cells which form fiber-like structures. The fibers are arranged in different orientations within discrete layers or sheets in the tissue, i.e., the fibers within the tissue are rotated. From a mathematical point of view, this rotation corresponds to a general anisotropy in the tissue's conductivity tensors. Since the rotation angle is different at each point, the anisotropic conductivities also vary spatially. Thus, the cardiac tissue should be viewed as an inhomogeneous anisotropic structure. In most of the previous bidomain studies, the fiber rotation has not been considered, i.e., the tissue has been modeled as a homogeneous orthotropic medium. In this paper, we describe a new finite-difference bidomain formulation which accounts for the fiber rotation in the cardiac tissue and hence allows a more realistic modeling of the cardiac tissue. The formulation has been implemented on the data-parallel CM-5 which provides the computational power and the memory required for solving large bidomain problems. Details of the numerical formulation are presented together with its validation by comparing numerical and analytical results. Some computational performance results are also shown. In addition, an application of this new formulation to provide activation patterns within a tissue slab with a realistic fiber rotation is demonstrated.

A generalized activating function for predicting virtual electrodes in cardiac tissue

To fully understand the mechanisms of defibrillation, it is critical to know how a given electrical stimulus causes membrane polarizations in cardiac tissue. We have extended the concept of the activating function, originally used to describe neuronal stimulation, to derive a new expression that identifies the sources that drive changes in transmembrane potential. Source terms, or virtual electrodes, consist of either second derivatives of extracellular potential weighted by intracellular conductivity or extracellular potential gradients weighted by derivatives of intracellular conductivity. The full response of passive tissue can be considered, in simple cases, to be a convolution of this "generalized activating function" with the impulse response of the tissue. Computer simulations of a two-dimensional sheet of passive myocardium under steady-state conditions demonstrate that this source term is useful for estimating the effects of applied electrical stimuli. The generalized activating function predicts oppositely polarized regions of tissue when unequally anisotropic tissue is point stimulated and a monopolar response when a point stimulus is applied to isotropic tissue. In the bulk of the myocardium, this new expression is helpful for understanding mechanisms by which virtual electrodes can be produced, such as the hypothetical "sawtooth" pattern of polarization, as well as polarization owing to regions of depressed conductivity, missing cells or clefts, changes in fiber diameter, or fiber curvature. In comparing solutions obtained with an assumed extracellular potential distribution to those with fully coupled intra- and extracellular domains, we find that the former provides a reliable estimate of the total solution. Thus the generalized activating function that we have derived provides a useful way of understanding virtual electrode effects in cardiac tissue.

Nonsustained reentry following successive stimulation of cardiac tissue through a unipolar electrode

INTRODUCTION

Using numerical simulations, we predict that nonsustained reentry occurs following a strong, premature stimulus through a unipolar electrode.

METHODS AND RESULTS

Our simulations were based on the bidomain model of cardiac tissue, and the active membrane properties were represented by the Beeler-Reuter model. An outwardly propagating wavefront was excited by an initial stimulus (S1). A second stimulus (S2) was then applied through the same electrode. Nonsustained reentry or reentrant-like behavior followed the S2 stimulus for both cathodal and anodal stimulation, and were associated with "break" stimulation but not with "make" stimulation. The direction of spiral-wave rotation was reversed when the polarity of the stimulus was reversed. These complex dynamics occur only for a narrow window of S1-S2 intervals. During anodal S2 stimulation, two different modes of reentry exist. Our simulations also explain the "no response" phenomenon.

CONCLUSION

Our mathematical model predicts that both anodal and cathodal unipolar S2 stimulation results in reentry. This behavior arises from an interaction of virtual anodes and cathodes surrounding the stimulating electrode.