An S1 gradient of refractoriness is not essential for reentry induction by an S2 stimulus

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.

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.

How hyperpolarization and the recovery of excitability affect propagation through a virtual anode in the heart

Researchers have suggested that the fate of a shock-induced wave front at the edge of a "virtual anode" (a region hyperpolarized by the shock) is a key factor determining success or failure during defibrillation of the heart. In this paper, we use a simple one-dimensional computer model to examine propagation speed through a hyperpolarized region. Our goal is to test the hypothesis that rapid propagation through a virtual anode can cause failure of propagation at the edge of the virtual anode. The calculations support this hypothesis and suggest that the time constant of the sodium inactivation gate is an important parameter. These results may be significant in understanding the mechanism of the upper limit of vulnerability.

A mechanism for the upper limit of vulnerability

BACKGROUND

The strongest shock that induces reentry in the heart is the upper limit of vulnerability (ULV). In order to understand defibrillation, one must know what causes the ULV.

OBJECTIVE

The goal of this study was to examine the mechanism of the upper limit of vulnerability.

METHODS

Numerical simulations of cardiac tissue were performed using the bidomain model. An S2 shock was applied during the refractory period of the S1 action potential, and results using a smooth curving fiber geometry were compared with results using a smooth plus random fiber geometry.

RESULTS

When using a smooth fiber geometry only, no ULV was observed. However, when a random fiber geometry was included, the ULV was present. The difference arises from the fate of the shock-induced break wave front when it reaches the edge of the tissue hyperpolarized by the shock (the virtual anode).

CONCLUSION

Our numerical simulations suggest that local heterogeneities throughout the tissue may be crucial for determining the fate of the shock-induced wave front at the edge of the virtual anode, and therefore play an important role in the mechanism underlying the ULV.

The pinwheel experiment revisited: effects of cellular electrophysiological properties on vulnerability to cardiac reentry

In normal heart, ventricular fibrillation can be induced by a single properly timed strong electrical or mechanical stimulus. A mechanism first proposed by Winfree and coined the “pinwheel experiment” emphasizes the timing and strength of the stimulus in inducing figure-of-eight reentry. However, the effects of cellular electrophysiological properties on vulnerability to reentry in the pinwheel scenario have not been investigated. In this study, we extend Winfree's pinwheel experiment to show how the vulnerability to reentry is affected by the graded action potential responses induced by a strong premature stimulus, action potential duration (APD), and APD restitution in simulated monodomain homogeneous two-dimensional tissue. We find that a larger graded response, longer APD, or steeper APD restitution slope reduces the vulnerable window of reentry. Strong graded responses and long APD promote tip-tip interactions at long coupling intervals, causing the two initiated spiral wave tips to annihilate. Steep APD restitution promotes wave front-wave back interaction, causing conduction block in the central common pathway of figure-of-eight reentry. We derive an analytical treatment that shows good agreement with numerical simulation results.

Modeling cardiac ischemia

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

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.

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.

Refractory Gradient is Responsible for the Increase in Ventricular Vulnerability Under Sodium Channel Blockade

BACKGROUND

Previous studies have shown that sodium channel (I(Na)) blockade increases ventricular vulnerability; however, there were differences in the degree of the increase. Because the vulnerable window (VW) is altered by the type of preshock refractory gradient (RG), the hypothesis was that the differences in the arrhythmogenesis of I(Na) blockade result from the different types of preshock RG employed.

METHODS AND RESULTS

Simulations of regio(Na)l electric shock following constant pacing stimuli in 2-dimensional bidomain myocardial sheets under I(Na) blockade were conducted using 3 types of preshock RG: longitudinally tilted (LRG), transversely tilted (TRG), and non-tilted RG (NRG). The increase in the degree of I(Na) blockade almost linearly decreased the conduction velocity. The action potential duration in the LRG and TRG cases was non-linearly shortened with the increase in INa blockade because of electrotonic influences, whereas in the case of NRG it was slightly prolonged. In both LRG and TRG cases, the VW for reentry induction by electric shock was considerably widened by the INa blockade; however, this was not the case for NRG in which the VW was rather narrowed by the INa blockade.

CONCLUSION

The type of preshock RG alters the degree of the increase in ventricular vulnerability under INa blockade.

The topology of defibrillation

We describe how Art Winfree's ideas about phase singularities can be used to understand the response of cardiac tissue with a random preexisting pattern of reentrant waves (fibrillation) to a large brief current stimulus. This discussion is organized around spatial dimension, beginning with a discussion of reentry on a periodic ring, followed by reentry in a two-dimensional planar domain (spiral waves), and ending with consideration of three-dimensional reentrant patterns (scroll waves). In all cases, we show how reentrant activity is changed by the application of a shock, describing conditions under which defibrillation is successful or not. Using topological arguments we draw the general conclusion that with a generic placement of stimulating electrodes, large-scale virtual electrodes do not give an adequate explanation for the mechanism of defibrillation.

Dynamics of virtual electrode-induced scroll-wave reentry in a 3D bidomain model

Functional reentry in the heart can be caused by a wave front of excitation rotating around its edge. Previous simulations on the basis of monodomain cable equations predicted the existence of self-sustained, vortex-like wave fronts (scroll waves) rotating around a filament in three dimensions. In our simulations, we used the more accurate bidomain model with modified Beeler-Reuter ionic kinetics to study the dynamics of scroll-wave filaments in a 16 x 8 x 1.5-mm slab of ventricular tissue with straight fibers. Wave fronts were identified as the areas with inward current. Their edges represented the filaments. Both transmural and intramural reentries with I- and U-shaped filaments, respectively, were obtained by the S1-S2 point stimulation protocol through the virtual electrode-induced phase singularity mechanism. The filaments meandered along elongated trajectories and tended to attach to the tissue boundaries exposed to air (no current flow) rather than to the bath (zero extracellular potential). They completely detached from electroporated (zero transmembrane potential) boundaries. In our simulations, the presence of the bath led to generation of only U-shaped filaments, which survived for the 1.5-mm-thick slab but not for the slabs of 0.5- or 3-mm thicknesses. Thus boundary conditions may be another determinant of the type and dynamics of reentry.

Art Winfree and the bidomain model of cardiac tissue

This paper reviews Art Winfree's contributions to the bidomain model of cardiac tissue. Specifically, he first predicted quatrefoil reentry, he showed that an S1 refractory gradient is not required for an S2 stimulus to induce reentry, and his work on spiral wave meandering led to studies on how the path of the tip of a spiral wave is influenced by tissue anisotropy.

Effect of acute global ischemia on the upper limit of vulnerability: a simulation study

The goal of this modeling research is to provide mechanistic insight into the effect of altered membrane kinetics associated with 5-12 min of acute global ischemia on the upper limit of cardiac vulnerability (ULV) to electric shocks. We simulate electrical activity in a finite-element bidomain model of a 4-mm-thick slice through the canine ventricles that incorporates realistic geometry and fiber architecture. Global acute ischemia is represented by changes in membrane dynamics due to hyperkalemia, acidosis, and hypoxia. Two stages of acute ischemia are simulated corresponding to 5-7 min (stage 1) and 10-12 min (stage 2) after the onset of ischemia. Monophasic shocks are delivered in normoxia and ischemia over a range of coupling intervals, and their outcomes are examined to determine the highest shock strength that resulted in induction of reentrant arrhythmia. Our results demonstrate that acute ischemia stage 1 results in ULV reduction to 0.8A from its normoxic value of 1.4A. In contrast, no arrhythmia is induced regardless of shock strength in acute ischemia stage 2. An investigation of mechanisms underlying this behavior revealed that decreased postshock refractoriness resulting mainly from 1) ischemic electrophysiological substrate and 2) decrease in the extent of areas positively-polarized by the shock is responsible for the change in ULV during stage 1. In contrast, conduction failure is the main cause for the lack of vulnerability in acute ischemia stage 2. The insight provided by this study furthers our understanding of mechanisms by which acute ischemia-induced changes at the ionic level modulate cardiac vulnerability to electric shocks.

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 pinwheel experiment re-revisited

Virtual electrode induced phase singularity hypothesis explains the origin of cardiac arrhythmias caused by artificial electrical induction of rotors, i.e. vortex-like self-sustained sources of activity. This mechanism is thought to underlie both stimulus-induced arrhythmias and shock defibrillation therapy. In this paper, we extend this hypothesis to three dimensions using the bidomain model of cardiac tissue. We predict that virtual electrode polarization can produce three topologically distinct types of rotors anchored to: (1) transmural I-shaped scroll wave filaments; (2) near-surface U-shaped scroll wave filaments; and (3) intramural O-shaped scroll wave filaments.