The dynamics of sustained reentry in a ring model of cardiac tissue

Double-wave reentry in excitable media

A monolayer of chick embryo cardiac cells grown in an annular geometry supports two simultaneous reentrant excitation waves that circulate as a doublet. We propose a mechanism that can lead to such behavior. The velocity restitution gives the instantaneous velocity of a wave as a function of the time since the passage of the previous wave at a given point in space. Nonmonotonic restitution relationships will lead to situations in which various spacings between circulating waves are possible. In cardiology, the situation in which two waves travel in an anatomically defined circuit is referred to as double-wave reentry. Since double-wave reentry may arise as a consequence of pacing during cardiac arrhythmias, understanding the dynamic features of double-wave reentry may be helpful in understanding the physiological properties of cardiac tissue and in the design of therapy.

Trigger vs. Substrate: Multi-Dimensional Modulation of QT-Prolongation Associated Arrhythmic Dynamics by a hERG Channel Activator

Background: Prolongation of the QT interval of the electrocardiogram (ECG), underlain by prolongation of the action potential duration (APD) at the cellular level, is linked to increased vulnerability to cardiac arrhythmia. Pharmacological management of arrhythmia associated with QT prolongation is typically achieved through attempting to restore APD to control ranges, reversing the enhanced vulnerability to Ca²⁺-dependent afterdepolarisations (arrhythmia triggers) and increased transmural dispersion of repolarisation (arrhythmia substrate) associated with APD prolongation. However, such pharmacological modulation has been demonstrated to have limited effectiveness. Understanding the integrative functional impact of pharmacological modulation requires simultaneous investigation of both the trigger and substrate. Methods: We implemented a multi-scale (cell and tissue) in silico approach using a model of the human ventricular action potential, integrated with a model of stochastic 3D spatiotemporal Ca²⁺ dynamics, and parameter modification to mimic prolonged QT conditions. We used these models to examine the efficacy of the hERG activator MC-II-157c in restoring APD to control ranges, examined its effects on arrhythmia triggers and substrates, and the interaction of these arrhythmia triggers and substrates. Results: QT prolongation conditions promoted the development of spontaneous release events underlying afterdepolarisations during rapid pacing. MC-II-157c applied to prolonged QT conditions shortened the APD, inhibited the development of afterdepolarisations and reduced the probability of afterdepolarisations manifesting as triggered activity in single cells. In tissue, QT prolongation resulted in an increased transmural dispersion of repolarisation, which manifested as an increased vulnerable window for uni-directional conduction block. In some cases, MC-II-157c further increased the vulnerable window through its effects on I_Na. The combination of stochastic release event modulation and transmural dispersion of repolarisation modulation by MC-II-157c resulted in an integrative behavior wherein the arrhythmia trigger is reduced but the arrhythmia substrate is increased, leading to variable and non-linear overall vulnerability to arrhythmia. Conclusion: The relative balance of reduced trigger and increased substrate underlies a multi-dimensional role of MC-II-157c in modulation of cardiac arrhythmia vulnerability associated with prolonged QT interval.

Geometrical measurement of cardiac wavelength in reaction-diffusion models

The dynamics of reentrant arrhythmias often consists in multiple wavelets propagating throughout an excitable medium. An arrhythmia can be sustained only if these reentrant waves have a sufficiently short wavelength defined as the distance traveled by the excitation wave during its refractory period. In a uniform medium, wavelength may be estimated as the product of propagation velocity and refractory period (electrophysiological wavelength). In order to accurately measure wavelength in more general substrates relevant to atrial arrhythmias (heterogeneous and anisotropic), we developed a mathematical framework to define geometrical wavelength at each time instant based on the length of streamlines following the propagation velocity field within refractory regions. Two computational methods were implemented: a Lagrangian approach in which a set of streamlines were integrated, and an Eulerian approach in which wavelength was the solution of a partial differential equation. These methods were compared in 1D/2D tissues and in a model of the left atrium. An advantage of geometrical definition of wavelength is that the wavelength of a wavelet can be tracked over time with high temporal resolution and smaller temporal variability in an anisotropic and heterogeneous medium. The results showed that the average electrophysiological wavelength was consistent with geometrical measurements of wavelength. Wavelets were however often shorter than the electrophysiological wavelength due to interactions with boundaries and other wavelets. These tools may help to assess more accurately the relation between substrate properties and wavelet dynamics in computer models.

Spatiotemporally controlled cardiac conduction block using high-frequency electrical stimulation

Background

Methods for the electrical inhibition of cardiac excitation have long been sought to control excitability and conduction, but to date remain largely impractical. High-amplitude alternating current (AC) stimulation has been known to extend cardiac action potentials (APs), and has been recently exploited to terminate reentrant arrhythmias by producing reversible conduction blocks. Yet, low-amplitude currents at similar frequencies have been shown to entrain cardiac tissues by generation of repetitive APs, leading in some cases to ventricular fibrillation and hemodynamic collapse in vivo. Therefore, an inhibition method that does not lead to entrainment – irrespective of the stimulation amplitude (bound to fluctuate in an in vivo setting) – is highly desirable.

Methodology/Principal Findings

We investigated the effects of broader amplitude and frequency ranges on the inhibitory effects of extracellular AC stimulation on HL-1 cardiomyocytes cultured on microelectrode arrays, using both sinusoidal and square waveforms. Our results indicate that, at sufficiently high frequencies, cardiac tissue exhibits a binary response to stimulus amplitude with either prolonged APs or no effect, thereby effectively avoiding the risks of entrainment by repetitive firing observed at lower frequencies. We further demonstrate the ability to precisely define reversible local conduction blocks in beating cultures without influencing the propagation activity in non-blocked areas. The conduction blocks were spatiotemporally controlled by electrode geometry and stimuli duration, respectively, and sustainable for long durations (300 s).

Conclusion/Significance

Inhibition of cardiac excitation induced by high-frequency AC stimulation exhibits a binary response to amplitude above a threshold frequency, enabling the generation of reversible conduction blocks without the risks of entrainment. This inhibition method could yield novel approaches for arrhythmia modeling in vitro, as well as safer and more efficacious tools for in vivo cardiac mapping and radio-frequency ablation guidance applications.

MEMORY AND BISTABILITY IN A ONE-DIMENSIONAL LOOP OF MODEL CARDIAC CELLS

Unidirectional propagation has been studied in a one-dimensional loop of model cardiac cells represented as a homogeneous and isotropic cable. Membrane ionic currents were represented by a modified Beeler-Reuter model. The time constants of the gate variables of the slow inward current acting during the plateau of the action potential were divided by a parameter K ≥1. In the space-clamped model, increasing K shortens the action potential duration, changes the shape of the restitution curve and adds a slow memory component to the dynamics. In a paced regime, it promotes bistability in which period-1 and period-2 patterns coexist over an interval of pacing frequencies. In the loop, bistability is created between periodic and aperiodic modes of sustained reentry for an interval of loop length. The bistability of the space-clamped and loop model are both related to the form of the restitution curve.

Inhibition of renin-angiotensin system (RAS) reduces ventricular tachycardia risk by altering connexin43

Renin-angiotensin system (RAS) activation is associated with arrhythmias. We investigated the effects of RAS inhibition in cardiac-specific angiotensin-converting enzyme (ACE) overexpression (ACE 8/8) mice, which exhibit proclivity to ventricular tachycardia (VT) and sudden death because of reduced connexin43 (Cx43). ACE 8/8 mice were treated with an ACE inhibitor (captopril) or an angiotensin receptor type-1 blocker (losartan). Subsequently, electrophysiological studies were performed, and the hearts were extracted for Cx43 quantification using immunoblotting, immunohistochemistry, fluorescent dye spread method, and sodium current quantification using whole cell patch clamping. VT was induced in 12.5% of captopril-treated ACE 8/8 and in 28.6% of losartan-treated mice compared to 87.5% of untreated mice (P < 0.01). Losartan and captopril treatment increased total Cx43 2.4-fold (P = 0.01) and the Cx43 phosphorylation ratio 2.3-fold (P = 0.005). Treatment was associated with a recovery of gap junctional conductance. Survival in treated mice improved to 0.78 at 10 weeks (95% confidence interval 0.64 to 0.92), compared to the expected survival of less than 0.50. In a model of RAS activation, arrhythmic risk was correlated with reduced Cx43 amount and phosphorylation. RAS inhibition resulted in increased total and phosphorylated Cx43, decreased VT inducibility, and improved survival.

Effects of procainamide and sotalol on restitution properties, dispersion of refractoriness, and ventricular fibrillation activation patterns in pigs

BACKGROUND

Interest in combining antiarrhythmic drugs has been prompted by the lack of efficacy of monotherapies and the toxicity resulting from high doses of individual agents.

OBJECTIVES

We tested the hypothesis that procainamide and sotalol combined have greater beneficial effects on restitution, on the dispersion of refractoriness, and on decreasing the complexity of ventricular fibrillation (VF) than either drug alone.

METHODS

Six open-chest pigs received intravenous procainamide (15 mg/kg load and 50 microg/kg/min maintenance) followed by sotalol (1.5 mg/kg). Another six pigs received sotalol first and procainamide second. Before drugs and after each drug, 20-second episodes of electrically induced VF were recorded from a 21 x 24 unipolar electrode plaque (2 mm spacing) sutured on the lateral posterior left ventricular epicardium. Restitution properties and dispersion of refractoriness were estimated from activation recovery intervals during pacing.

RESULTS

The combination of the two drugs reduced the maximum slope of the restitution curve and during VF reduced the number of wavefronts, the activation rate, the percentage of wavefront families exhibiting reentry, and the conduction velocity more than either drug alone. In addition, in the group that received sotalol first, both drugs together reduced the SD and the coefficient of variation of the spatial dispersion of refractoriness compared with baseline.

CONCLUSIONS

Procainamide and sotalol combined have greater beneficial effects on restitution properties, dispersion of refractoriness, and the complexity of VF than either drug alone compared with baseline.

Analysis of damped oscillations during reentry: a new approach to evaluate cardiac restitution

Reentry is a mechanism underlying numerous cardiac arrhythmias. During reentry, head-tail interactions of the action potential can cause cycle length (CL) oscillations and affect the stability of reentry. We developed a method based on a difference-delay equation to determine the slopes of the action potential duration and conduction velocity restitution functions, known to be major determinants of reentrant arrhythmogenesis, from the spatial period P and the decay length D of damped CL oscillations. Using this approach, we analyzed CL oscillations after the induction of reentry and the resetting of reentry with electrical stimuli in rings of cultured neonatal rat ventricular myocytes grown on microelectrode arrays and in corresponding simulations with the Luo-Rudy model. In the experiments, P was larger and D was smaller after resetting impulses compared to the induction of reentry, indicating that reentry became more stable. Both restitution slopes were smaller. Consistent with the experimental findings, resetting of simulated reentry caused oscillations with gradually increasing P, decreasing D, and decreasing restitution slopes. However, these parameters remained constant when ion concentrations were clamped, revealing that intracellular ion accumulation stabilizes reentry. Thus, the analysis of CL oscillations during reentry opens new perspectives to gain quantitative insight into action potential restitution.

Dynamics of sustained reentry in a loop model with discrete gap junction resistances

The dynamics of reentry is studied in a one-dimensional loop of model cardiac cells with discrete intercellular gap junction resistance (R). Each cell is represented by a continuous cable with ionic current given by a modified Beeler-Reuter formulation. For R below a limiting value, propagation is found to change from period-1 to quasiperiodic (QP) at a critical loop length (L(crit)) that decreases with R. Quasiperiodic reentry exists from L(crit) to a minimum length (L(min)), which also shortens with R. The decrease of L(crit) (R) is not a simple scaling, but the bifurcation can still be predicted from the slope of the restitution curve giving the duration of the action potential as a function of the diastolic interval. However, the shape of the restitution curve changes with R. An increase of R does not seem to increase the number of possible QP solutions since, as in the continuous cable, only two QP modes of propagation were found despite an extensive search through alternative initial conditions.

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.

Conditions for propagation and block of excitation in an asymptotic model of atrial tissue

Detailed ionic models of cardiac cells are difficult for numerical simulations because they consist of a large number of equations and contain small parameters. The presence of small parameters, however, may be used for asymptotic reduction of the models. Earlier results have shown that the asymptotics of cardiac equations are nonstandard. Here we apply such a novel asymptotic method to an ionic model of human atrial tissue to obtain a reduced but accurate model for the description of excitation fronts. Numerical simulations of spiral waves in atrial tissue show that wave fronts of propagating action potentials break up and self-terminate. Our model, in particular, yields a simple analytical criterion of propagation block, which is similar in purpose but completely different in nature to the "Maxwell rule" in the FitzHugh-Nagumo type models. Our new criterion agrees with direct numerical simulations of breakup of reentrant waves.

Our search for the porcine mother rotor

A single stationary mother rotor, located in the fastest activating region and giving rise to activation fronts that propagate throughout the remainder of the myocardium, has been hypothesized to be responsible for the maintenance of ventricular fibrillation (VF). Others have reported a mother rotor in guinea pigs and rabbits. We wanted to see if a mother rotor exists in a larger heart, that is, pigs. Epicardial mapping studies have demonstrated that VF wavefronts in pigs tend to propagate from the posterior basal LV to the anterior LV and on to the anterior RV, raising the possibility of a mother rotor in the posterior LV. However, no sustained reentry consistent with a mother rotor was found on the posterior LV epicardium, even though an intramural mapping study showed that the fastest activating transmural layer was near the epicardium. Many wavefronts in the posterior LV entered the mapped region from the posterior boundary of the mapping array, adjacent to the posterior descending coronary artery, raising the possibility that a mother rotor is located in the right ventricle or septum. Since a previous study has shown that the RV activates more slowly than the LV during VF, the more likely site for a mother rotor was the septum. However, we then performed a study in which we recorded from the right side of the septum and found that reentry was uncommon there also and that the activation rate was slower than the posterobasal LV. Many of the VF wavefronts in the septum passed from the posterior septum toward the anterior septum. This fact coupled with the fact that many wavefronts passed from the posterior LV free wall toward the anterior LV free wall point to the region where the posterior free wall intersects with the septum, the region where the posterior papillary muscle is located, as the possible site of a mother rotor. Indeed, a recent abstract by others reports that, after propranolol, a stable reentrant circuit is present on the endocardium at the insertion of the posterior papillary muscle into the LV free wall in pigs.

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.

Head-tail interactions in numerical simulations of reentry in a ring of cardiac tissue

BACKGROUND

The relationships between action potential duration (APD) and conduction velocity (CV) to the previous diastolic interval (DI) are known as the APD and CV restitution relationships. There is considerable debate regarding the importance of these relationships in the development and stability of reentry.

OBJECTIVES

The purpose of this study was to increase the understanding of the ionic basis for restitution during reentry.

METHODS

APD and CV were studied numerically during one-dimensional reentry as ring length (L) was shortened. A three-state variable model (u, v, w) was used to analyze the effect of gating variables of the fast (v) and slow (w) currents on the spatial and temporal dynamics of transmembrane potential (u). Three parameter sets were used corresponding to three APD and CV restitution curves.

RESULTS

Sustained spatial oscillations of APD and CV larger than the ring length were observed in two of the parameter sets (cytochalasin-D model [CYTO] and model 3 [M3]) before block occurred at L = 6 cm. The last model (diacetyl monoxime [DAM]) resulted in uniform APD and CV for all L until block occurred at L = 3 cm. Multivalued APD and CV restitution relationships due to "dephasing" of w and v with DI were observed in M3 and CYTO simulations. Overall, these dynamics could be explained by the wavelength-to-ring length ratio and the sensitivity of APD on the value of the gating variables w and v.

CONCLUSION

Propagation stability is mostly controlled by APD sensitivity to w, but the APD restitution slope does not always reflect this sensitivity. The interaction of the dynamic history (i.e., memory) of the fast and slow currents and electrotonic effects resulted in multivalued restitution curves.

Head-tail interactions in numerical simulations of reentry in a ring of cardiac tissue

BACKGROUND

The relationships between action potential duration (APD) and conduction velocity (CV) to the previous diastolic interval (DI) are known as the APD and CV restitution relationships. There is considerable debate regarding the importance of these relationships in the development and stability of reentry.

OBJECTIVES

The purpose of this study was to increase the understanding of the ionic basis for restitution during reentry.

METHODS

APD and CV were studied numerically during one-dimensional reentry as ring length (L) was shortened. A three-state variable model (u, v, w) was used to analyze the effect of gating variables of the fast (v) and slow (w) currents on the spatial and temporal dynamics of transmembrane potential (u). Three parameter sets were used corresponding to three APD and CV restitution curves.

RESULTS

Sustained spatial oscillations of APD and CV larger than the ring length were observed in two of the parameter sets (cytochalasin-D model [CYTO] and model 3 [M3]) before block occurred at L = 6 cm. The last model (diacetyl monoxime [DAM]) resulted in uniform APD and CV for all L until block occurred at L = 3 cm. Multivalued APD and CV restitution relationships due to "dephasing" of w and v with DI were observed in M3 and CYTO simulations. Overall, these dynamics could be explained by the wavelength-to-ring length ratio and the sensitivity of APD on the value of the gating variables w and v.

CONCLUSION

Propagation stability is mostly controlled by APD sensitivity to w, but the APD restitution slope does not always reflect this sensitivity. The interaction of the dynamic history (i.e., memory) of the fast and slow currents and electrotonic effects resulted in multivalued restitution curves.

Complex-periodic spiral waves in confluent cardiac cell cultures induced by localized inhomogeneities

Spatiotemporal wave activities in excitable heart tissues have long been the subject of numerous studies because they underlie different forms of cardiac arrhythmias. In particular, understanding the dynamics and the instabilities of spiral waves have become very important because they can cause reentrant tachycardia and their subsequent transitions to fibrillation. Although many aspects of cardiac spiral waves have been investigated through experiments and model simulations, their complex properties are far from well understood. Here, we show that intriguing complex-periodic (such as period-2, period-3, period-4, or aperiodic) spiral wave states can arise in monolayer tissues of cardiac cell culture in vitro, and demonstrate that these different dynamic states can coexist with abrupt and spontaneous transitions among them without any change in system parameters; in other words, the medium supports multistability. Based on extensive image data analysis, we have confirmed that these spiral waves are driven by their tips tracing complex orbits whose unusual, meandering shapes are formed by delicate interplay between localized conduction blocks and nonlinear properties of the culture medium.

Restitution Properties During Ventricular Fibrillation in the In Situ Swine Heart

Background— Although restitution has been hypothesized to determine action potential duration (APD) during ventricular fibrillation (VF), cardiac memory may also be important.

Methods and Results— Transmembrane recordings were made with a floating microelectrode from the anterior right ventricular wall in 6 pigs during up to 60 seconds of VF. The recordings were divided into 5-second intervals, and APD 60 and the diastolic interval (DI) were calculated for each activation cycle throughout each interval. Stepwise linear regression was used to determine how well each APD 60 [APD 60 (n)] was predicted by the 4 previous DIs (n−1, n−2, n−3, n−4) and the 3 previous APD 60 s (n−1, n−2, n−3). A mean±SD of 3±1.5 of the variables entered the regression equation. DI(n−1) (70% of intervals) and APD 60 (n−1) (71% of intervals) appeared most frequently in the regression equations and were the first or second variables entered during the stepwise regression in 87% and 76% of the intervals in which they were present, respectively. The coefficients of DI(n−1) and APD 60 (n−1) were positive 89% and 98% of the time, respectively. R 2 of the regression for all entered variables during all intervals was 0.39±0.05.

Conclusions— The high incidence and positive coefficient of DI(n−1) indicate that restitution is important in determining APD during VF, whereas the similarly high incidence and positive coefficient of APD(n−1) indicate that cardiac memory is equally important. The finding that the regression equation accounts for only 39% of the variability of APD indicates that factors other than restitution and memory are also important in determining APD during VF.

Stimulation of unidirectional pulses in excitable systems

Using a judicious spatial shape of input current pulses (and electrodes), responses of an excitable system (FitzHugh-Nagumo) appear as unidirectional pulses (UDP's) instead of bidirectional ones (in one dimension) or circular ones (in two dimensions). The importance of the UDP's for a possible mechanism for pinpointing the reentry cycle position and for a possible use in tachycardia suppression is discussed.

Basic mechanisms of cardiac impulse propagation and associated arrhythmias

Propagation of excitation in the heart involves action potential (AP) generation by cardiac cells and its propagation in the multicellular tissue. AP conduction is the outcome of complex interactions between cellular electrical activity, electrical cell-to-cell communication, and the cardiac tissue structure. As shown in this review, strong interactions occur among these determinants of electrical impulse propagation. A special form of conduction that underlies many cardiac arrhythmias involves circulating excitation. In this situation, the curvature of the propagating excitation wavefront and the interaction of the wavefront with the repolarization tail of the preceding wave are additional important determinants of impulse propagation. This review attempts to synthesize results from computer simulations and experimental preparations to define mechanisms and biophysical principles that govern normal and abnormal conduction in the heart.

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.

Interactions Between Extracellular Stimuli and Excitation Waves in an Atrial Reentrant Loop

Extracellular Stimuli in an Atrial Reentrant Loop.

INTRODUCTION

The interactions between extracellular stimuli and excitation waves propagating in a reentrant loop are a complex function of stimulus parameters, structural properties, membrane state, and timing. Here the goal was a comprehensive understanding of the mechanisms and frequencies of the major interactions between the advancing excitation wave and a single extracellular stimulus, separated from issues of anatomic or geometric complexity.

METHODS AND RESULTS

A modernized computer model of a thin ring of uniform tissue that included a pair of extracellular stimulus electrodes (anode/cathode) was used to model one-dimensional cardiac reentry. Questions and results included the following: (1) What are the major interactions between a stimulus and the reentrant propagation wave, and are they induced near the cathode or near the anode; and, for each interaction, what are the initiating amplitude range and timing interval? At the cathode, the well-known mechanism of retrograde excitation terminated reentry; changes in timing or amplitude produced double-wave reentry or phase reset. At the anode, termination occurred at different cells depending on stimulus amplitude. (2) Relatively how often did termination occur at the anode? For most stimulus amplitudes, termination occurred more often at the anode than at the cathode, although not always at the same cell. (3) With random timing, what is the probability of terminating reentry? Stimulation for 5 msec terminated reentry with a probability from 0% to approximately 10%, as a function of increasing stimulus amplitude.

CONCLUSION

A single extracellular stimulus can initiate major changes in reentrant excitation via multiple mechanisms, even in a simple geometry. Termination of reentry, phase shifts, or double-wave reentry each occurs over well-defined ranges of stimulus amplitude and timing.

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.

Termination of reentry in an inhomogeneous ring of model cardiac cells

Reentrant waves propagating in a ring or annulus of excitable media are a model of the basic mechanism underlying a major class of irregular cardiac rhythms known as anatomical reentry. Such reentrant waves are terminated by rapid electrical stimulation (pacing) from an implantable device. Because the mechanisms of such termination are poorly understood, we study pacing of anatomical reentry in a one-dimensional ring of model cardiac cells. For realistic off-circuit pacing, our model-independent results suggest that circuit inhomogeneities, and the electrophysiological dynamical changes they introduce, may be essential for terminating reentry in some cases.

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

Resetting and annihilation of reentrant activity by a single stimulus pulse (S1) or a pair (S1-S2) of coupled pulses are studied in a model of one-dimensional loop of cardiac tissue using a Beeler-Reuter-type ionic model. Different modes of reentry termination are described. The classical mode of termination by unidirectional block, in which a stimulus produces only a retrograde front that collides with the activation front of the reentry, can be obtained for both S1 and S1-S2 applied over a small vulnerable window. We demonstrate that another scenario of termination-that we term collision block-can also be induced by the S1-S2 protocol. This scenario is obtained over a much wider range of S1-S2 coupling intervals than the one leading to a unidirectional block. In the collision block, S1 produces a retrograde front, colliding with the activation front of the pre-existing reentry, and an antegrade front propagating in the same direction as the initial reentry. Then, S2 also produces an antegrade and a retrograde front. However, the propagation of these fronts in the spatial profile of repolarization left by S1 leads to a termination of the reentrant activity. More complex behaviors also occur in which the antegrade fronts produced by S1 and S2 both persist for several turns, displaying a growing alternation in action potential duration ("alternans amplification") that may lead to the termination of the reentrant activity. The hypothesis that both collision block and alternans amplification depend on the interaction between the action potential duration restitution curve and the recovery curve of conduction velocity is supported by the fact that the dynamical behaviors were reproduced using an integro-delay equation based on these two properties. We thus describe two new mechanisms (collision block and alternans amplification) whereby electrical stimulation can terminate reentrant activity. (c) 2002 American Institute of Physics.

New approaches to antiarrhythmic therapy, Part I: emerging therapeutic applications of the cell biology of cardiac arrhythmias

Cardiac arrhythmias complicate many diseases affecting the heart and circulation, and they incorporate a multiplicity of underlying mechanisms. The evolution of scientific knowledge has made the complex changes produced by cardiovascular disease sufficiently understood at the organ, cellular, and molecular levels such that there is a diversity of therapeutic targets for pharmacological therapy and/or prevention. Moreover, the approach of rational drug design in mechanism-specific and disease-specific fashions facilitates the targeting of therapy using the methods of molecular, structural, and translational biology. Additional approaches, using similar drug design strategies but based on gene therapy and transcriptional and translational modification, are on the horizon. Hence, there is reason to be optimistic regarding the design, testing, and clinical availability of novel antiarrhythmic therapies.

Dynamic relationship of cycle length to reentrant circuit geometry and to the slow conduction zone during ventricular tachycardia

BACKGROUND

Knowledge of cycle-to-cycle changes in isthmus geometry is of potential importance for radiofrequency catheter ablation to stop reentrant ventricular tachycardia. It was hypothesized that isthmus geometry often undergoes continuous evolution throughout reentry and that cycle-length variability measurements could be used to segment reentry into distinct phases and to predict changes in isthmus geometry.

METHODS AND RESULTS

A canine infarct model of reentrant ventricular tachycardia in the epicardial border zone with a figure 8 pattern of conduction was used for analysis (25 monomorphic reentry episodes, 20 experiments). Tachycardias were segmented, on the basis of cycle-length variations, into 2 to 3 distinct phases corresponding to onset, maintenance, and spontaneous termination, when it occurred (6/25 episodes). Trends of linear cycle-length change occurred throughout the maintenance phase in all tachycardias. For each trend, quantitative geometric parameters of the isthmus were measured, and the following linear relationships were established. During a trend, the slow conduction zone activation interval and tachycardia cycle length increased, while isthmus length decreased. When isthmus length decreased, isthmus width decreased at its narrowed portion. Larger decreases in isthmus length corresponded to higher rates of linear cycle-length prolongation. Also, greater cycle-length variability tended to prolong tachycardia.

CONCLUSIONS

Cycle-length alterations occur throughout reentry in this canine model and are predictive of isthmus geometry changes. Because similar reentry dynamics, which affect catheter ablation efficacy, have been observed clinically, estimation of changes in geometry during electrophysiological study may help target ablation sites.

Dynamics of action potential head-tail interaction during reentry in cardiac tissue: ionic mechanisms

In a sufficiently short reentry pathway, the excitation wave front (head) propagates into tissue that is partially refractory (tail) from the previous action potential (AP). We incorporate a detailed mathematical model of the ventricular myocyte into a one-dimensional closed pathway to investigate the effects of head-tail interaction and ion accumulation on the dynamics of reentry. The results were the following: 1) a high degree of head-tail interaction produces oscillations in several AP properties; 2) Ca(2+)-transient oscillations are in phase with AP duration oscillations and are often of greater magnitude; 3) as the wave front propagates around the pathway, AP properties undergo periodic spatial oscillations that produce complicated temporal oscillations at a single site; 4) depending on the degree of head-tail interaction, intracellular [Na(+)] accumulation during reentry either stabilizes or destabilizes reentry; and 5) elevated extracellular [K(+)] destabilizes reentry by prolonging the tail of postrepolarization refractoriness.

Influence of propafenone on resetting and termination of canine atrial flutter

Previous studies on atrial flutter (AF) presumed that resetting was due to the prematurity effect (PE) in which the stimulated antegrade wavefront travels in the tail of the AF preexisting wavefront. We studied the collision effect (CE) between the AF and the stimulated retrograde wavefronts, its contribution to resetting, and its relationship to AF termination and how they are affected by the Class IC agent propafenone (PPF). A canine model of AF was created using a Y-shaped lesion in the right atrium in 14 dogs (33 +/- 3 kg). Five atrial bipolar electrodes were positioned around the tricuspid valve. In a subsequent set of 11 dogs, we used 16 bipolar electrodes for recording. AF was induced by burst pacing. Single and multiple stimuli were applied to measure conduction time and reset-response curves (RRCs). This was repeated after the administration of PPF (1 mg/kg loading dose for 10 minutes, followed by 1.8 mg/kg/per hour infusion). Three distinct mechanisms were found to contribute to the RRC: the PE, the CE, and heterogeneity. PPF stabilized the RRC, increased significantly the cycle length (CL), the duration of the effective refractory period, as well as the duration of the excitable gap. However, PPF did not alter the duration of the fully excitable portion. We studied 36 annihilations without and 48 with PPF. Transient fibrillation was found in 75% of the episodes without, compared to 22% with PPF. Other types of termination such as conduction block, CL oscillations, and reversal of activation were found for 25% of the episodes without and 78% with PPF. In many cases, conduction block and CL oscillations were associated with a failure of propagation of the stimulated antegrade wavefront in the region of collision. Termination by reversal of activation suggests that propagation was two dimensional and could not be represented by a one dimensional movement. The average coupling interval (in percent of CL), that induced fibrillation was not significantly different from that at which conduction block occurred. This suggests that transient fibrillation is associated with a weak CE rather than with rapid pacing. The CE is amplified by multiple stimuli and PPF. The incidence of transient fibrillation in AF annihilation diminishes with PPF as the CE becomes more important. This suggests that the evaluation of PE and CE in AF may be an indication of the risk of atrial fibrillation.

A mechanism of transition from ventricular fibrillation to tachycardia : effect of calcium channel blockade on the dynamics of rotating waves

Abbreviation of the action potential duration and/or effective refractory period (ERP) is thought to decrease the cycle length of reentrant arrhythmias. Verapamil, however, paradoxically converts ventricular fibrillation (VF) to ventricular tachycardia (VT), despite reducing the ERP. This mechanism remains unclear. We hypothesize that the size and the dynamics of the core of rotating waves, in addition to the ERP, influence the arrhythmia manifestation (ie, VF or VT). The objectives of this study were (1) to demonstrate functional reentry as a mechanism of VF and VT in the isolated Langendorff-perfused rabbit heart in the absence of an electromechanical uncoupler and (2) to elucidate the mechanism of verapamil-induced conversion of VF to VT. We used high-resolution video imaging with a fluorescent dye, ECG, frequency and 2-dimensional phase analysis, and computer simulations. Activation patterns in 10 hearts were studied during control, verapamil perfusion (2x10(-6) mol/L), and washout. The dominant frequency of VF decreased from 16.2+/-0.7 to 13.5+/-0.6 Hz at 20 minutes of verapamil perfusion (P<0.007). Concomitantly, phase analysis revealed that wavefront fragmentation was reduced, as demonstrated by a 3-fold reduction in the density of phase singularities (PSs) on the ventricular epicardial surface (PS density: control, 1.04+/-0.12 PSs/cm(2); verapamil, 0.32+/-0.06 PSs/cm(2) [P=0.0008]). On washout, the dominant frequency and the PS density increased, and the arrhythmia reverted to VF. The core area of transiently appearing rotors significantly increased during verapamil perfusion (control, 4.5+/-0.6 mm(2); verapamil, 9.2+/-0.5 mm(2) [P=0.0002]). In computer simulations, blockade of slow inward current also caused an increase in the core size. Rotating waves underlie VF and VT in the isolated rabbit heart. Verapamil-induced VF-to-VT conversion is most likely due to a reduction in the frequency of rotors and a decrease in wavefront fragmentation that lessens fibrillatory propagation away from the rotor.

Curvature effects on activation speed and repolarization in an ionic model of cardiac myocytes

Reentry is a major mechanism underlying the initiation and perpetuation of many cardiac arrhythmias. Stimulated ventricular myocytes give action potential characterized by a fast upstroke, a long-lasting plateau, and a late repolarization phase. The plateau phase determines the action potential duration (APD) during which the system remains refractory, a property essential to the synchronization of the heart cycle. The APD varies much with prematurity and this change has been shown to be the main determinant of the dynamics in models of paced cells and cable, and during reentry in the one-dimensional loop. Curvature has also been shown to be an important factor for propagation in experimental and theoretical cardiac extended tissue. The objective of this paper is to combine both curvature and prematurity effects in a kinematical model of propagation in cardiac tissue. First, an approximation of the ionic model is used to obtain the effects of curvature and prematurity on the speed of propagation, the APD, and the absolute refractory period. Two versions of the ionic model are studied that differ in their rate of excitability recovery. The functions are used in a kinematical model describing the propagation of period-1 solutions around an annulus.

Spiral waves in two-dimensional models of ventricular muscle: formation of a stationary core

Previous experimental studies have clearly demonstrated the existence of drifting and stationary electrical spiral waves in cardiac muscle and their involvement in cardiac arrhythmias. Here we present results of a study of reentrant excitation in computer simulations based on a membrane model of the ventricular cell. We have explored in detail the parameter space of the model, using tools derived from previous numerical studies in excitation-dynamics models. We have found appropriate parametric conditions for sustained stable spiral wave dynamics (1 s of activity or approximately 10 rotations) in simulations of an anisotropic (ratio in velocity 4:1) cardiac sheet of 2 cm x 2 cm. Initially, we used a model that reproduced well the characteristics of planar electrical waves exhibited by thin sheets of sheep ventricular epicardial muscle during rapid pacing at a cycle length of 300 ms. Under these conditions, the refractory period was 147 ms; the action potential duration (APD) was 120 ms; the propagation velocity along fibers was 33 cm/s; and the wavelength along fibers was 4.85 cm. Using cross-field stimulation in this model, we obtained a stable self-sustaining spiral wave rotating around an unexcited core of 1.75 mm x 7 mm at a period of 115 ms, which reproduced well the experimental results. Thus the data demonstrate that stable spiral wave activity can occur in small cardiac sheets whose wavelength during planar wave excitation in the longitudinal direction is larger than the size of the sheet. Analysis of the mechanism of this observation demonstrates that, during rotating activity, the core exerts a strong electrotonic influence that effectively abbreviates APD (and thus wavelength) in its immediate surroundings and is responsible for the stabilization and perpetuation of the activity. We conclude that appropriate adjustments in the kinetics of the activation front (i.e., threshold for activation and upstroke velocity of the initiating beat) of currently available models of the cardiac cell allow accurate reproduction of experimentally observed self-sustaining spiral wave activity. As such, the results set the stage for an understanding of functional reentry in terms of ionic mechanisms.

Wave propagation simulation in normal and infarcted myocardium: computational and modelling issues

Simulation of propagating action potentials (PAP) in normal and abnormal myocardium is used for the understanding of mechanisms responsible for eliciting dangerous arrhythmias. One- and two-dimensional models dealing with PAP properties are reviewed in this paper viewed both from the computational and mathematical aspects. These models are used for linking theoretical and experimental results. The discontinuous nature of the PAP is demonstrated through the combination of experimental and theoretically derived results. In particular it can be shown that for increased intracellular coupling resistance the PAP upstroke phase properties (Vmax, dV/dtmax and tau foot) change considerably, and in some cases non-monotonically with increased coupling resistance. It is shown that tau foot) is a parameter that is very sensitive to the cell's distance to the stimulus site, the stimulus strength and the coupling resistance. In particular it can be shown that in a one-dimensional structure the tau foot value can increase dramatically for lower coupling resistance values near the stimulus site and subsequently can be reduced as we move to distances larger than five resting length constants from the stimulus site. The tau foot variability is reduced with increased coupling resistance, rendering the lower coupling resistance structures, under abnormal excitation sequences, more vulnerable to conduction block and arrhythmias. Using the theory of discontinuous propagation of the PAP in the myocardium it is demonstrated that for specific abnormal situations in the myocardium, such as infarcted tissue, one- and two-dimensional models can reliably simulate propagation characteristics and explain complex phenomena such as propagation at bifurcation sites and mechanisms of block and re-entry. In conclusion it is shown that applied mathematics and informatics can help in elucidating electrophysiologically complex mechanisms such as arrhythmias and conduction disturbances in the myocardium.

Underdetection of ventricular tachycardia using a 40 ms stability criterion: effect of antiarrhythmic therapy

Inappropriate shocks can complicate cardioverter defibrillator therapy. Among solutions proposed to avoid oversensing are algorithms to reduce inappropriate detection of atrial fibrillation (AF) or sinus tachycardia. In patients not on antiarrythmic drugs, an interval stability criterion of 40 ms has been validated with the Medtronic PCD to discriminate ventricular tachycardia (VT) from AF. With this algorithm, VT is considered stable if no interval varies from one of the three preceding intervals by more than 40 ms. If an interval does not fulfill this criterion, the VT event counter is reset to zero. The aim of this study was to investigate the incidence of underdetection when this criterion is applied in patients treated with antiarrhythmic drugs. We studied 132 sustained monomorphic VTs induced in 42 patients during 101 electrophysiological studies (EPS). EPS were performed without treatment (group I, 24 patients, 44 VTs); on Class Ia drug (group II, 17 patients, 24 VTs); Class Ic drug (group III, 22 patients, 39 VTs); or sotalol (group IV, 17 patients, 25 VTs). The endocardial electrogram of all VT episodes was digitized and the stability algorithm was applied. The reset arrhythmias were distributed among no delay, small, moderate (< 10 s) and important (> 15 s) delay in VT detection. The relation between drug use and reset was analyzed. Reset was found in 86 (65%) of induced VTs. No difference in heart rate or induction mode was shown between reset and nonreset VTs. There was a significative association between drug use and reset probability (Chi2 significantly different, P < 0.05). In patients treated with Class Ic drugs, the probability of finding an important delay in VT detection was 12.5% versus 0% in nontreated patients or in patients treated with sotalol. We conclude that a stability criterion of 40 ms is probably safe in nontreated patients but should be used with caution in patients treated with antiarrhythmics, especially in the presence of Class Ic drugs.

Cycle length dynamics and spatial stability at the onset of postinfarction monomorphic ventricular tachycardias induced in patients and canine preparations

BACKGROUND

The aim of this study was to determine whether cycle length (CL) variations at the onset of monomorphic ventricular tachycardias follow distinctive patterns.

METHODS AND RESULTS

We retrospectively analyzed 59 monomorphic ventricular tachycardias induced in 40 patients in whom intraoperative mapping was performed with 63 epicardial and 64 endocardial electrograms recorded simultaneously. Activation times and CL were determined at each electrode site over several beats (36+/-10 beats, mean+/-SD) starting with the first after programmed stimulation. In the majority of the tachycardias, CL variations were accounted for by fitting to an exponential function: CL=CLs+Ae-b/tau, where CLs is the stable CL, b is beat number, tau is the time constant (in beat number), and A is the magnitude of CL relaxation. A decelerating trend (with reference to rate) (negative A) accounted for 21 tachycardias, an accelerating trend in rate (positive A) accounted for 12 tachycardias, and 4 others displayed a double dynamic behavior, with an initial acceleration followed by a decelerating trend in rate. Among the ventricular tachycardias that were not fitted to exponential models, 12 showed a constant trend and 10 others showed irregular CL fluctuations. The monomorphic character of the tachycardias was established by principal-component analysis, which also indicated that CL dynamics associated with the accelerating and decelerating trends may be related to shortening and prolongation of activation times, respectively, occurring in equal proportion at all recording sites. In canine preparations in which reentry circuits could be mapped with high resolution, CL showed an accelerating trend in rate when circus movement of excitation occurred around a transmural scar in muscle generating unipolar electrograms with relatively high -dV/dtmax, and a decelerating trend in rate occurred when functional reentry occurred in muscle generating unipolar electrograms with depressed -dV/dtmax.

CONCLUSIONS

Beat-to-beat CL variations may occur at the onset of sustained monomorphic ventricular tachycardia as a result of uniform acceleration or deceleration of activation times while the overall activation pattern remains constant. The associated initial trends in the rate of sustained monomorphic ventricular tachycardia follow typical patterns that might provide "signatures" corresponding to reentry substrates with distinctive functional properties.

Reentry: insights from theoretical simulations in a fixed pathway

This review article summarizes theoretical insights into the principles and mechanisms associated with reentrant activity in cardiac tissue. A mathematical ring model is used in computer simulations to investigate, at the cellular level, mechanistic aspects of initiation, perpetuation, and termination of reentry. Taking advantage of the ability to compute membrane processes in this model, we relate dynamic properties of the reentrant action potential (e.g., beat-to-beat alternans) to the underlying kinetics of membrane ionic channels. Effects on reentry of inhomogeneities in refractoriness, excitability, cellular coupling at gap junctions, and fiber cross-section are also studied.

Simulation of two-dimensional anisotropic cardiac reentry: effects of the wavelength on the reentry characteristics

A two-dimensional sheet model was used to study the dynamics of reentry around a zone of functional block. The sheet is a set of parallel, continuous, and uniform cables, transversely interconnected by a brick-wall arrangement of fixed resistors. In accord with experimental observations on cardiac tissue, longitudinal propagation is continuous, whereas transverse propagation exhibits discontinuous features. The width and length of the sheet are 1.5 and 5 cm, respectively, and the anisotropy ratio is fixed at approximately 4:1. The membrane model is a modified Beeler-Reuter formulation incorporating faster sodium current dynamics. We fixed the basic wavelength and action potential duration of the propagating impulse by dividing the time constants of the secondary inward current by an integer K. Reentry was initiated by a standard cross-shock protocol, and the rotating activity appeared as curling patterns around the point of junction (the q-point) of the activation (A) and recovery (R) fronts. The curling R front always precedes the A front and is separated from it by the excitable gap. In addition, the R front is occasionally shifted abruptly through a merging with a slow-moving triggered secondary recovery front that is dissociated from the A front and q-point. Sustained irregular reentry associated with substantial excitable gap variations was simulated with short wavelengths (K = 8 and K = 4). Unsustained reentry was obtained with a longer wavelength (K = 2), leading to a breakup of the q-point locus and the triggering of new activation fronts.