Resetting and annihilation of reentrant activity in a model of a one-dimensional loop of ventricular tissue

Varieties of reentrant dynamics

Experiments were carried out in monolayer tissue cultures of embryonic chick heart cells imaged using a calcium sensitive fluorescent dye. The cells were grown in annular geometries and in annular geometries with an isthmus connecting antipodal region of the annulus. We observed a large number of spatially different patterns of propagation consisting of one or more circulating waves. As well, we also observed rhythms in which rotors embedded in the annuli generated propagating pulses. These results demonstrate that many different patterns of excitation can be present in cardiac tissue with simple geometries.

Advantage of four-electrode over two-electrode defibrillators

Defibrillation is the standard clinical treatment used to stop ventricular fibrillation. An electrical device delivers a controlled amount of electrical energy via a pair of electrodes in order to reestablish a normal heart rate. We propose a technique that is a combination of biphasic shocks applied with a four-electrode system rather than the standard two-electrode system. We use a numerical model of a one-dimensional ring of cardiac tissue in order to test and evaluate the benefit of this technique. We compare three different shock protocols, namely a monophasic and two types of biphasic shocks. The results obtained by using a four-electrode system are compared quantitatively with those obtained with the standard two-electrode system. We find that a huge reduction in defibrillation threshold is achieved with the four-electrode system. For the most efficient protocol (asymmetric biphasic), we obtain a reduction in excess of 80% in the energy required for a defibrillation success rate of 90%. The mechanisms of successful defibrillation are also analyzed. This reveals that the advantage of asymmetric biphasic shocks with four electrodes lies in the duration of the cathodal and anodal phase of the shock.

Shock-induced termination of reentrant cardiac arrhythmias: comparing monophasic and biphasic shock protocols

In this article, we compare quantitatively the efficiency of three different protocols commonly used in commercial defibrillators. These are based on monophasic and both symmetric and asymmetric biphasic shocks. A numerical one-dimensional model of cardiac tissue using the bidomain formulation is used in order to test the different protocols. In particular, we performed a total of 4.8 × 10(6) simulations by varying shock waveform, shock energy, initial conditions, and heterogeneity in internal electrical conductivity. Whenever the shock successfully removed the reentrant dynamics in the tissue, we classified the mechanism. The analysis of the numerical data shows that biphasic shocks are significantly more efficient (by about 25%) than the corresponding monophasic ones. We determine that the increase in efficiency of the biphasic shocks can be explained by the higher proportion of newly excited tissue through the mechanism of direct activation.

Effects of pacing magnitudes and forms on bistability width in a modeled ventricular tissue

Bistability in periodically paced cardiac tissue is relevant to cardiac arrhythmias and its control. In the present paper, one-dimensional tissue of the phase I Luo-Rudy model is numerically investigated. The effects of various parameters of pacing signals on bistability width are studied. The following conclusions are obtained: (i) Pacing can be classified into two types: pulsatile and sinusoidal types. Pulsatile pacing reduces bistability width as its magnitude is increased. Sinusoidal pacing increases the width as its amplitude is increased. (ii) In a pacing period the hyperpolarizing part plays a more important role than the depolarizing part. Variations of the hyperpolarizing ratio in a period evidently change the width of bistability and its variation tendency. (iii) A dynamical mechanism is proposed to qualitatively explain the phenomena, which reveals the reason for the different effects of pulsatile and sinusoidal pacing on bistability. The methods for changing bistability width by external pacing may help control arrhythmias in cardiology.

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

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

Termination of reentrant cardiac action potential propagation using far-field electrical pacing

Several different types of rapid cardiac rhythm disorders, including atrial and ventricular fibrillation, are likely caused by multiple, rapidly rotating, action potential (AP) waves. Thus, an electrical pacing therapy, whose effectiveness is based on being delivered with a particular timing relative to one of these waves, is unlikely to be useful in terminating the remaining waves. Here, we develop pacing protocols that are designed to terminate rotating waves independently of when the sequences of stimuli are imposed or where each wave is in its rotation at the time the sequences are initiated. These protocols are delivered as far-field stimuli, and therefore are capable of simultaneously influencing all the waves present. The pacing intervals for these protocols are, in general, of unequal duration and are determined through examination of the dynamics of AP propagation in a 1-D ring model. Series of two or three stimuli with interstimulus intervals chosen in this way are shown to be effective in terminating these waves over a wide range of ring circumferences and AP dynamical parameters. Stimulus sequences of this type may form the basis for developing new defibrillation protocols to test in experiments or more realistic models of the electrical heart.

Pacing-induced spatiotemporal dynamics can be exploited to improve reentry termination efficacy

Some potentially fatal cardiac arrhythmias may be terminated by a series of premature stimuli. Monomorphic ventricular tachycardia, which may be modeled as an excitation wave traveling around in a ring, is one such arrhythmia. We investigated the mechanisms and requirements for termination of such reentry using an ionic cardiac ring model. Termination requires conduction block, which in turn is facilitated by spatial dispersion in repolarization and recovery time. When applying short series of two or three stimuli, we found that for conduction block to robustly occur, the magnitude of the spatial gradient in recovery time must exceed a critical value of 20 ms/cm. Importantly, the required spatial gradient can be induced in this homogeneous system by the dynamics of the stimulus-induced waves-we show analytically the necessary conditions. Finally, we introduce a type of pacing protocol, the "aggressive ramp," which increases the termination efficacy by exploiting such pacing-induced heterogeneities. This technique, which is straightforward to implement, may therefore have important clinical implications.

Liberation of a pinned spiral wave by a single stimulus in excitable media

The unpinning of a spiral wave from an anatomic obstacle by the application of a single stimulus near the core of the rotating wave was studied experimentally in a cell culture of cardiomyocyte monolayers as well as by computer simulations. It is shown that, with suitable positioning and timing, a single stimulus is sufficient for the successful unpinning of a pinned spiral wave. Successful unpinning is achieved when two conditions are fulfilled: (1) The stimulus is delivered in the vulnerable window of the rotating wave, and (2) the stimulus is delivered in a spatial zone in proximity to the obstacle, where the shape of the zone is defined by the phase of the anchored spiral wave. Two different scenarios for successful unpinning are discussed, which are distinguished by the distance to the stimuli applied to the obstacle.

Resetting and termination of reentry in a loop-and-tail cardiac model

Ventricular tachycardia is a type of cardiac arrhythmia that can be associated with a wave circulating around in a loop. Because this rhythm is potentially fatal, its termination is highly desirable. Theoretical approaches have suggested that discontinuity in the phase resetting response to electrical stimuli is indicative of the ability for such termination. We investigate the usefulness of such theoretical predictions when the stimulus site is located at some distance away from the reentrant loop, as would typically be the case during antitachycardia pacing in the heart. We show that there exists a critical tail length below which termination of reentry occurs over a range of stimulus timing values as predicted theoretically by a discontinuous window in the phase resetting curve. Above this critical length, however, a paradoxical situation exists: termination of reentry appears impossible, yet there is a point discontinuity in the phase resetting curve. These findings offer insight into termination using a single stimulus in a loop-and-tail model, an important step toward understanding the mechanism of antitachycardia pacing.

Alternans amplification following a two-stimulus protocol in a one-dimensional cardiac ionic model of reentry: from annihilation to double-wave quasiperiodic reentry

Electrical pacing is a common procedure in both experimental and clinical settings to study and/or annihilate anatomical reentry. A previous study [Comtois and Vinet, Chaos 12, 903 (2002)] has described new ways to terminate reentry in a one-dimensional loop model by a protocol consisting of only two stimulations. Annihilation of the reentrant activity was much more likely with these new scenarios than through a unidirectional block. This paper investigates the sensitivity of these scenarios of annihilation to the length of the pathway. It shows that double-pulse stimulation can stop the reentry if the circuit is shorter than a limiting length. Beyond this upper limit, stimulation rather yields sustained double-wave reentry. The same dynamical mechanism, labeled alternans amplification, is found to be responsible for these two types of post-stimulus dynamics.

Multistability of reentrant rhythms in an ionic model of a two-dimensional annulus of cardiac tissue

The dynamics of reentry in a model of a two-dimensional annulus of homogeneous cardiac tissue, with a Beeler-Reuter type formulation of the membrane ionic currents, is examined. The bifurcation structure of the sustained propagated solutions is described as a function of Rin and Rout, the inner and outer radii of the annulus. The transition from periodic to quasiperiodic reentry occurs at a critical Rin, which first diminishes and then saturates as Rout is increased. The reduction of the critical Rin is a consequence of the increase of the wave-front curvature. There is a range of Rin below the critical radius in which two distinct quasiperiodic solutions coexist. Each of these solutions disappears at its own specific value of Rin, and their annihilation is preceded by a new type of bifurcation leading to a regime of propagation with transient successive detachments of the wave front from the inner border of the annulus.

Wave block formation in homogeneous excitable media following premature excitations: dependence on restitution relations

Spiral wave formation and disorganized activity in excitable media require the existence of broken waves and are related to partial wave block. The determinants of wave block in excitable systems are incompletely understood, especially for cardiac excitable tissue. Previous work in one-dimensional cardiac models has suggested that wave break of a premature excitation (PE) requires critical timing and that the conditions for broken waves are improbable. We analyzed the mechanism of unidirectional wave block that occurs when two consecutive PEs interact with a normal plane wave in a generic one-dimensional spatial excitable medium. A nondimensional coupled-map model built from mesoscopic characteristics of the substrate (the velocity and action potential duration restitution functions) shows that block can occur over a large interval of timing between the two PEs and leads to wave break in two-dimensional media. This mechanism may be an important determinant of spiral wave formation by the response to premature excitations.

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.

Stability and bifurcation in an integral-delay model of cardiac reentry including spatial coupling in repolarization

We present the bifurcation analysis of a revised version of the integral-delay model [Courtemanche et al., Siam J. Appl. Math. 56, 119 (1996)] of reentry in a one-dimensional ring that includes a spatial coupling in the calculation of the action potential duration. This coupling is meant to reproduce the modulation of repolarization by the diffusive current flowing through the intercellular resistance. We show that coupling modifies the criterion for the stability of the period-1 solution, which is no longer uniquely related to the action potential restitution curve, but depends also on the degree of coupling between cells and on the dispersion relation of the velocity. Coupling also changes the scenario from an infinite-dimension Hopf bifurcation to a finite sequence of Hopf bifurcations that take place at different ring lengths.

Reentrant waves in a ring of embryonic chick ventricular cells imaged with a Ca2+ sensitive dye

According to the classic model initially formulated by Mines, reentrant cardiac arrhythmias may be associated with waves circulating in a ring geometry. This study was designed to study the dynamics of reentry in a ring geometry of cardiac tissue culture. Reentrant calcium waves in rings of cultured embryonic chick cardiac myocytes were imaged using a macroscope to monitor the fluorescence of intracellular Calcium Green-1 dye. The rings displayed a variety of stable rhythms including pacemaker activity and spontaneous reentry. Waves originating from a localized pacemaker could lead to reentry as a consequence of unidirectional block. In addition, more complex patterns were observed due to the interactions between reentrant and pacemaker rhythms. These rhythms included instances in which pacemakers accelerated the reentrant rhythm, and instances in which the excitation was blocked in the vicinity of pacemakers. During reentrant activity an appropriately timed electrical stimulus could induce resetting of activity or cause complete annihilation of the propagating waves. This experimental preparation reveals many spontaneously occuring complex rhythms. These complex rhythms are hypothesized to reflect interactions between spontaneous pacemakers, wave propagation, refractory period, and overdrive suppression. This preparation may serve as a useful model system to further investigate complex dynamics arising during reentrant rhythms in cardiac tissue.

Critical role of inhomogeneities in pacing termination of cardiac reentry

Reentry around nonconducting ventricular scar tissue, a cause of lethal arrhythmias, is typically treated by rapid electrical stimulation from an implantable cardioverter defibrillator. However, the dynamical mechanisms of termination (success and failure) are poorly understood. To elucidate such mechanisms, we study the dynamics of pacing in one- and two-dimensional models of anatomical reentry. In a crucial realistic difference from previous studies of such systems, we have placed the pacing site away from the reentry circuit. Our model-independent results suggest that with such off-circuit pacing, the existence of inhomogeneity in the reentry circuit is essential for successful termination of tachycardia under certain conditions. Considering the critical role of such inhomogeneities may lead to more effective pacing algorithms. (c) 2002 American Institute of Physics.

Introduction: Mapping and control of complex cardiac arrhythmias

This paper serves as an introduction to the Focus Issue on mapping and control of complex cardiac arrhythmias. We first introduce basic concepts of cardiac electrophysiology and describe the main clinical methods being used to treat arrhythmia. We then provide a brief summary of the main themes contained in the articles in this Focus Issue. In recent years there have been important advances in the ability to map the spread of excitation in intact hearts and in laboratory settings. This work has been combined with simulations that use increasingly realistic geometry and physiology. Waves of excitation and contraction in the heart do not always propagate with constant velocity but are often subject to instabilities that may lead to fluctuations in velocity and cycle time. Such instabilities are often treated best in the context of simple one- or two-dimensional geometries. An understanding of the mechanisms of propagation and wave stability is leading to the implementation of different stimulation protocols in an effort to modify or eliminate abnormal rhythms. (c) 2002 American Institute of Physics.