Spiral Wave Breakup Research Articles

Effect of noise on defect chaos in a reaction–diffusion model

The influence of noise on defect chaos due to breakup of spiral waves through Doppler and Eckhaus instabilities is investigated numerically with a modified Fitzhugh-Nagumo model. By numerical simulations we show that the noise can drastically enhance the creation and annihilation rates of topological defects. The noise-free probability distribution function for defects in this model is found not to fit with the previously reported squared-Poisson distribution. Under the influence of noise, the distributions are flattened, and can fit with the squared-Poisson or the modified-Poisson distribution. The defect lifetime and diffusive property of defects under the influence of noise are also checked in this model.

Defibrillation Depends on Conductivity Fluctuations and the Degree of Disorganization in Reentry Patterns

Defibrillation depends on conductivity and disorganization. Cardiac fibrillation is the deterioration of the heart's normally well-organized activity into one or more meandering spiral waves, which subsequently break up into many meandering wave fronts. Delivery of an electric shock (defibrillation) is the only effective way of restoring the normal rhythm. This study focuses on examining whether higher degrees of disorganization requires higher shock strengths to defibrillate and whether microscopic conductivity fluctuations favor shock success. We developed a three-dimensional computer bidomain model of a block of cardiac tissue with straight fibers immersed in a conductive bath. The membrane behavior was described by the Courtemanche human atrial action potential model incorporating electroporation and an acetylcholine- (ACh) dependent potassium current. Intracellular conductivities were varied stochastically around nominal values with variations of up to 50%. A single rotor reentry was initiated and, by adjusting the spatial ACh variation, the level of organization could be controlled. The single rotor could be stabilized or spiral wave breakup could be provoked leading to fibrillatory-like activity. For each level of organization, multiple shock timings and strengths were applied to compute the probability of shock success as a function of shock strength. Our results suggest that the level of the small-scale conductivity fluctuations is a very important factor in defibrillation. A higher variation significantly lowers the required shock strength. Further, we demonstrated that success also heavily depends on the level of organization of the fibrillatory episode. In general, higher levels of disorganization require higher shock strengths to defibrillate.

Preventing alternans-induced spiral wave breakup in cardiac tissue: An ion-channel-based approach

The detailed processes involved in spiral wave breakup, believed to be one major mechanism by which tachycardia evolves into fibrillation, are still poorly understood. This has rendered difficult the proper design of an efficient and practical control stimulus protocol to eliminate such events. In order to gain new insights into the underlying electrophysiological and dynamical mechanisms of breakup, we applied linear perturbation theory to a steadily rotating spiral wave in two spatial dimensions. The tissue was composed of cells modeled using the Fenton-Karma equations whose parameters were chosen to emphasize alternans as a primary mechanism for breakup. Along with one meandering mode, not just one but several unstable alternans modes were found with differing growth rates, frequencies, and spatial structures. As the conductance of the fast inward current was increased, the instability of the modes increased, consistent with increased meandering and propensity for spiral breakup in simulations. We also explored a promising new approach, based on the theory, for the design of an energy efficient electrical stimulus protocol to control spiral wave breakup. The novelty lies in addressing the problem directly at the ion channel level and taking advantage of the inherent two dimensional nature of the rotating wave. With the help of the eigenmode method, we were able to calculate the exact timing and amplitude of the stimulus, and locate it optimally to maximize efficiency. The analysis led to a special-case example that demonstrated that a single, properly timed stimulus can have a global effect, suppressing all growing alternans modes over the entire tissue, thus inhibiting spiral wave breakup.

EFFECT OF HETEROGENEITY ON SPIRAL WAVE DYNAMICS IN SIMULATED CARDIAC TISSUE

There is considerable spatial heterogeneity in the electrical properties of the heart muscle and there are indications that anisotropic conduction may play an important role in the pathogenesis of clinical cardiac arrhythmias. Spiral waves of electrical activity are related to reentrant cardiac arrhythmias as ventricular tachycardia and ventricular fibrillation, and the generation of a wave breakup is hypothesized to underlie the transition from ventricular tachycardia to ventricular fibrillation — the leading cause of sudden cardiac death. Here we investigate the effect of heterogeneity on spiral wave reentry in a two-dimensional modified FitzHugh–Nagumo membrane model. Spiral wave breakup induced by the heterogeneity is found. The spiral wave dynamics is invariant under translational and rotational transformations in homogeneous tissue, but for heterogeneous tissue, this symmetry is broken due to the heterogeneity. The reentry dynamics depends on the degree of heterogeneity and the point where the reentry is initiated within the simulated tissue. This study may open potentially exciting new diagnostic and therapeutic possibilities in a clinical context.

COMPLICATED ELECTRICAL ACTIVITIES IN CARDIAC TISSUE

It has become widely accepted that ventricular fibrillation, the most dangerous cardiac arrhythmias, is a major cause of death in the industrialized world. Alternans and conduction block have recently been related to the progression from ventricular tachycardia to ventricular fibrillation. From the point of view in cellular electrophysiology, ventricular tachycardia is the formation of reentrant wave in cardiac tissue. And ventricular fibrillation arises from subsequent breakdown of reentrant wave into multiple drifting and meandering spiral waves. In this paper, we numerically study pulse and vortex dynamics in cardiac tissue. Our numerical results include 1:1 normal sinus rhythm, 2:1 conduction block, complete conduction block, spiral wave, and spiral breakup. All of our numerical findings can be corresponding to clinical measurements in electrocardiogram. Various electrical activities in cardiac tissue will be discussed in detail in the present manuscript.

The Shock Energy Required for Successful Defibrillation Depends on the Degree of Disorganization of the Reentrant Activation Pattern

Regardless of how it is initiated, the final stage of sudden cardiac death is ventricular fibrillation (VF), a completely disorganized electrical activity of the ventricles, characterized by multiple reentrant circuits. There is only one known therapy for VF, a strong electrical shock, normally referred to as electrical defibrillation. In this study we investigate the hypothesis that the energy level required to stop fibrillation is dependent on the actual level of organization of the underlying reentrant activity. A three-dimensional bidomain model with unequal anisotropy ratios incorporating stochastic variations of the conductivity tensor was used. The active membrane behavior was described by the Courtemanche model of the human atrial action potential incorporating an acetylcholin (ACh) dependent K+ current and electroporation. A spatial variation of [ACh] was used to obtain repolarization gradients allowing to control the degree of disorganization. Depending on the ACh settings, either a single rotor developed or spiral wave breakup evolving into fibrillatory like activity was observed. Shocks of varying strength and timing were delivered to both activation patterns and the probability of shock success as a function of shock strength was determined. Results indicate that arrhythmias with a higher degree of disorganization need significantly higher shock strengths to defibrillate.

Periodic stimulus and the single cardiac cell—getting more out of 1D maps

The response of an isolated cardiac cell to a periodic stimulus has traditionally been studied in terms of the duration of the action potential (APD) immediately following each stimulus. The APD approach offers explanations of several experimental observations, including the stability of the so-called 1:1 response which is thought to be relevant to the problem of spiral wave breakup and the onset of fibrillation. A discussion of some theoretical problems with the APD approach is given in order to motivate the derivation of a new type of map. This new one-dimensional map, which gives successive values of the prestimulus transmembrane potential instead of successive values of APD, relies on the presence of a one-dimensional slow manifold in the underlying dynamics. This slow manifold map extends the understanding offered by the APD approach to include an explanation of Wenckebach rhythms. In addition, the bifurcation structure of the map provides a unified description of the parameter dependence that agrees fairly well with experimental observation.

Breakup of spiral waves caused by radial dynamics: Eckhaus and finite wavenumber instabilities

In this paper, we link numerical observations of spiral breakup to a stability analysis of simple rotating spirals. We review the phenomenology of spiral breakup, important applications in pattern formation and the state of the art in numerical stability analysis of spirals. A strategy for the latter procedure is suggested. Phenomenologically, spiral breakup can occur near the centre of rotation (‘core breakup’) or far away from it (‘far-field breakup’). It may be accompanied by instabilities of the spiral core in particular spiral meandering that affect also the stability of waves in the far-field, because an unstable core acts as a moving source and introduces a (nonlinear) Doppler effect. In general, breakup of non-meandering spirals is related to an absolute instability of the planar wavetrain with the same wave number. To simplify the stability problem, we consider a one-dimensional cut (‘1D spiral’) with a fixed core position in simulations and compare the results with a stability analysis of planar wavetrains. These 1D spirals approximate the radial dynamics of non-meandering 2D spirals. To fully account for instabilities of 1D spirals, it is not sufficient to compute the direction of propagation of the unstable modes of the wavetrains, one also needs to compute the so-called absolute spectrum of these wavetrains. This allows us to decide whether the instability is of convective or absolute nature. Only the latter case implies an instability of spirals in finite domains. We carry out this programme for the case of core breakup in an excitable reaction–diffusion system, the modified Barkley model. Our analysis yields that core breakup can result from the absolute variant of a novel finite wavenumber instability of the radial dynamics where the critical perturbations are transported towards the core. From these results, we can confirm that a simple spiral breaks up if the wavetrain in the far-field is absolutely unstable. The central new result is the discovery of respective convective and absolute instabilities of wavetrains connected with modes of finite wavenumber that propagate in the direction opposite to the wavetrain. Hence, in spirals, perturbations are moving inward and core breakup becomes possible even in the absence of meandering.

Spiral wave break-up and planar front formation in two-dimensional reactive–diffusive media with straining

The propagation of spiral waves in two-dimensional reactive–diffusive media subject to a velocity field with straining is studied numerically. It is shown that, depending on the magnitude of the strain rate, the effects of straining on wave propagation can be classified in three regimes: the no break-up, the transitional and the break-up regimes. In the no break-up regime, the spiral wave preserves its integrity, whereas, in the transitional one, the spiral wave is deformed and transformed into a curved front which propagates almost along one of the principal directions of the strain rate tensor. In the break-up regime, spiral waves are broken up into many spiral waves, and planar fronts aligned with the principal direction of the strain rate tensor corresponding to elongation/stretching are formed. These fronts may be split into two new ones or are bands, the thickness of which decreases as the strain rate is increased.

Mechanism for spiral wave breakup in excitable and oscillatory media.

We study spiral wave breakup using a Fitzhugh-Nagumo-type system. We find that spiral wave breakup can occur near the core or far from it in both excitable and oscillatory regimes. There is a faraway breakup scenario in both excitable and oscillatory media that depends on long wavelength modulation modes. We observed three distinct scenarios, including one that involves breakup that does not develop into turbulence. However, we find that the mechanisms behind these three scenarios are the same: they are caused by the interaction between the dispersion relation and the asymptotic behavior of the modulation mode. The difference in phenomenology is due to the asymptotic behavior of the modulation mode.

Spiral wave stability in cardiac tissue with biphasic restitution

Human ventricular tissue as well as several animal ventricular preparations show a biphasic shape of the action potential duration restitution curve, with a local maximum at low diastolic intervals. We study numerically how the location and properties of this nonmonotonicity affect the stability of spiral waves. We find that, depending on the slopes of the ascending and of the descending parts of the restitution curve, we can have either stable rotation of the spiral wave or spiral breakup. We identify two types of spiral breakup: one due to a steep positive slope and another due to a steep negative slope in the restitution curve. We discuss the differences in their manifestation and possible implications. We also find that increasing the slope of the descending part of the restitution curve increases the meandering of the spiral wave, due to the repeated occurrence of conduction blocks near the spiral wave tip.

Transition from ventricular fibrillation to ventricular tachycardia: a simulation study on the role of Ca2+-channel blockers in human ventricular tissue

We study the effect of blocking the L-type Ca2+-channel on fibrillation in simulations in two-dimensional (2D) isotropic sheets of ventricular tissue and in a three-dimensional anisotropic anatomical model of human ventricles, using a previously developed model of human ventricular cells. Ventricular fibrillation (VF) was obtained as a result of spiral wave breakup and consisted of a varying number of chaotically wandering wavelets activating tissue at a frequency of about 6.0 Hz. We show that blocking the Ca2+-current by 75% can convert ventricular fibrillation into a periodic regime with a small number of stable spiral waves, ranging from six in 2D sheets of 25 × 25 cm to a single spiral in the anatomical model of human ventricles. The dominant frequency during this process changed to about 10.0 Hz in the 2D simulations, but to only 5.0 Hz in the whole heart simulations where a single spiral wave anchored around an anatomical obstacle. We show that the observed effects were due to a flattening of the electrical restitution curve, which prevented the generation of wave breaks and stabilized the activation patterns.

Conduction block in one-dimensional heart fibers.

We present a nonlinear dynamical systems analysis of the transition to conduction block in one-dimensional cardiac fibers. We study a simple model of wave propagation in heart tissue that depends only on the recovery of action potential duration and conduction velocity. If the recovery function has slope >or=1 and the velocity recovery function is nonconstant, rapid activation causes dynamical heterogeneity and finally conduction block away from the activation site. This dynamical mechanism may play a role in the initiation and breakup of spiral waves in excitable media.

Termination of spiral wave breakup in a Fitzhugh–Nagumo model via short and long duration stimuli

Rotating spiral waves have been observed in a variety of nonlinear biological and physical systems. Spiral waves are found in excitable and oscillatory systems and can be stationary, meander, or even degenerate into multiple unstable rotating waves (a process called "spiral wave breakup"). In the heart, spiral wave breakup is thought to be the underlying mechanism of cardiac fibrillation. The spatiotemporal complexity of multiple unstable spiral waves is difficult to control or terminate. Here, the mechanisms of the termination of spiral wave breakup in response to global stimulation are investigated. A modified Fitzhugh-Nagumo model was used to represent cellular kinetics to study the role of the fast (activation) and slow (recovery) variables. This simplified model allows a theoretical analysis of the termination of spiral wave breakup via both short and long duration pulses. Simulations were carried out in both two-dimensional sheets and in a three-dimensional geometry of the heart ventricles. The short duration pulses affected only the fast variable and acted to reset wave propagation. Monophasic pulses excited tissue ahead of the wave front thus reducing the amount of excitable tissue. Biphasic shocks did the same, but they also acted to generate new wave fronts from the pre-existing wave tails by making some active regions excitable. Thus, if the short duration stimuli were strong enough, they acted to fill in excitable tissue via propagating wave fronts and terminated all activity. The long duration wave forms were selected such that they had a frequency spectrum similar to that of the pseudoelectrocardiograms recorded during fibrillation. These long duration wave forms affected both the recovery and activation variables, and the mechanism of unstable multiple spiral wave termination was different compared to the short duration wave forms. If the long duration stimuli were strong enough, they acted to alter the "state" (i.e., combination of fast and slow variables) of the tissue throughout 1.5 cycles, thus "conditioning" the tissue such that by the end of the stimuli almost no excitable tissue remained. The peak current, total energy, and average power of stimuli required to terminate spiral wave breakup were less for the long duration wave forms compared to the short duration wave forms. In addition, closed loop feedback via stimulation with a wave form that was the difference of the pseudoelectrocardiogram and a strongly periodic chaotic signal was successful at terminating spiral wave breakup. These results suggest that it may be possible to improve cardiac defibrillation efficacy by using long duration wave forms to affect recovery variables in the heart as opposed to the traditional brief duration wave forms that act only on the fast variables. (c) 2002 American Institute of Physics.

Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity.

It has become widely accepted that the most dangerous cardiac arrhythmias are due to reentrant waves, i.e., electrical wave(s) that recirculate repeatedly throughout the tissue at a higher frequency than the waves produced by the heart's natural pacemaker (sinoatrial node). However, the complicated structure of cardiac tissue, as well as the complex ionic currents in the cell, have made it extremely difficult to pinpoint the detailed dynamics of these life-threatening reentrant arrhythmias. A simplified ionic model of the cardiac action potential (AP), which can be fitted to a wide variety of experimentally and numerically obtained mesoscopic characteristics of cardiac tissue such as AP shape and restitution of AP duration and conduction velocity, is used to explain many different mechanisms of spiral wave breakup which in principle can occur in cardiac tissue. Some, but not all, of these mechanisms have been observed before using other models; therefore, the purpose of this paper is to demonstrate them using just one framework model and to explain the different parameter regimes or physiological properties necessary for each mechanism (such as high or low excitability, corresponding to normal or ischemic tissue, spiral tip trajectory types, and tissue structures such as rotational anisotropy and periodic boundary conditions). Each mechanism is compared with data from other ionic models or experiments to illustrate that they are not model-specific phenomena. Movies showing all the breakup mechanisms are available at http://arrhythmia.hofstra.edu/breakup and at ftp://ftp.aip.org/epaps/chaos/E-CHAOEH-12-039203/ INDEX.html. The fact that many different breakup mechanisms exist has important implications for antiarrhythmic drug design and for comparisons of fibrillation experiments using different species, electromechanical uncoupling drugs, and initiation protocols. (c) 2002 American Institute of Physics.

Spiral Wave Generation in Heterogeneous Excitable Media

As the coupling in a heterogeneous excitable medium is reduced, three different types of behavior are encountered: plane waves propagate without breaking up, plane waves break up into spiral waves, and plane waves block. We illustrate these phenomena in monolayers of chick embryonic heart cells using calcium sensitive fluorescent dyes. Following the addition of heptanol, an agent that reduces the electrical coupling between cells, we observe breakup of spiral waves. These results are modeled in a heterogeneous cellular automaton model in which the neighborhood of interaction is modified.

Afterdepolarizations promote the transition from ventricular tachycardia to fibrillation in a three-dimensional model of cardiac tissue.

Recent experimental results regarding the action potential duration restitution curve have explained the transition from ventricular tachycardia (VT) to fibrillation (VF) in terms of spiral wave (SW) meandering and breakup. However, it remains unclear whether VF always has a steep restitution curve. The present study was designed to test the hypothesis that afterdepolarizations occur at excitable gaps during VF and affect the SW dynamics, even if the restitution curve is gentle. Homogeneous and isotropic 3-dimensional tissue was simulated with a LRd model. Because of the gentle restitution curve, it was not expected that SW instabilities would occur in this condition. In the tissue, a stationary SW reentry was initially observed; however, afterdepolarizations erupted from the excitable gap near the SW tip, and the SW then meandered widely. Following that, afterdepolarizations erupted far from the SW tip, resulting in SW breakup. In this manner, the wave dynamics degenerated into a chaotic state within a few seconds. Furthermore, not only triggered activity but also subthreshold afterdepolarizations were found to cause SW instabilities. These results suggest that afterdepolarizations may play an important role in the transition to VF and that the mechanism is independent of restitution properties.

Dynamics of spiral waves in excitable media with local time-periodic modulation

The propagation of spiral waves in excitable media with locally inhomogeneous time-periodic modulations is studied numerically, and it is shown that the size of the local inhomogeneity and the amplitude and frequency of the periodic forcing play a major role in determining the spiral wave dynamics, suppression and breakup. It is also shown that, in the absence of breakup, spiral waves flatten and thicken as the size of the inhomogeneity is increased, and that spiral waves may penetrate into the modulation region if the forcing frequency is sufficiently small. At high frequencies, however, it is shown that the inhomogeneity behaves as an obstacle. It is also reported that, for sufficiently large modulation regions, the spiral wave may be annihilated and become an almost cylindrical wave with initially high concentration of the activator in the center of the inhomogeneity.

Absolute versus convective instability of spiral waves

Absolute and convective instabilities of spirals are investigated using the continuous and the so-called absolute spectrum. It is shown that the nature of transport, induced by an absolute instability, is determined by spectral data of the asymptotic wave trains. The results are applied to core and far-field breakup of spiral waves in excitable and oscillatory media.

Origins of spiral wave meander and breakup in a two-dimensional cardiac tissue model.

We studied the stability of spiral waves in homogeneous two-dimensional cardiac tissue using phase I of the Luo-Rudy ventricular action potential model. By changing the conductance and the relaxation time constants of the ion channels, various spiral wave phenotypes, including stable, quasiperiodically meandering, chaotically meandering, and breakup were observed. Stable and quasiperiodically meandering spiral waves occurred when the slope of action potential duration (APD) restitution was < 1 over all diastolic intervals visited during reentry; chaotic meander and spiral wave breakup occurred when the slope of APD restitution exceeded 1. Curvature of the wave changes both conduction velocity and APD, and their restitution properties, thereby modulating local stability in a spiral wave, resulting in distinct spiral wave phenotypes. In the LRI model, quasiperiodic meander is most sensitive to the Na+ current, whereas chaotic meander and breakup are more dependent on the Ca2+ and K+ currents.