Biphasic restitution of action potential duration and complex dynamics in ventricular myocardium

Dynamical effects of diffusive cell coupling on cardiac excitation and propagation: a simulation study

Cell coupling is considered to be important for cardiac action potential propagation and arrhythmogenesis. We carried out computer simulations to investigate the effects of stimulation strength and cell-to-cell coupling on action potential duration (APD) restitution, APD alternans, and stability of reentry in models of isolated cell, one-dimensional cable, and two-dimensional tissue. Phase I formulation of the Luo and Rudy action potential model was used. We found that stronger stimulation resulted in a shallower APD restitution curve and onset of APD alternans at a faster pacing rate. Reducing diffusive coupling between cells prolonged APD. Weaker diffusive currents along the direction of propagation steepened APD restitution and caused APD alternans to occur at a slower pacing rate in tissue. Diffusive current due to curvature changed APD but had little effect on APD restitution slope and onset of instability. Heterogeneous cell coupling caused APD inhomogeneities in space. Reduction in coupling strength either uniformly or randomly had little effect on the rotation period and stability of a reentry, but random cell decoupling slowed the rotation period and, thus, stabilized the reentry, preventing it from breaking up into multiple waves. Therefore, in addition to its effects on action potential conduction velocity, diffusive cell coupling also affects APD in a rate-dependent manner, causes electrophysiological heterogeneities, and thus modulates the dynamics of cardiac excitation. These effects are brought about by the modulation of ionic current activation and inactivation.

Model of intracellular calcium cycling in ventricular myocytes

We present a mathematical model of calcium cycling that takes into account the spatially localized nature of release events that correspond to experimentally observed calcium sparks. This model naturally incorporates graded release by making the rate at which calcium sparks are recruited proportional to the whole cell L-type calcium current, with the total release of calcium from the sarcoplasmic reticulum (SR) being just the sum of local releases. The dynamics of calcium cycling is studied by pacing the model with a clamped action potential waveform. Experimentally observed calcium alternans are obtained at high pacing rates. The results show that the underlying mechanism for this phenomenon is a steep nonlinear dependence of the calcium released from the SR on the diastolic SR calcium concentration (SR load) and/or the diastolic calcium level in the cytosol, where the dependence on diastolic calcium is due to calcium-induced inactivation of the L-type calcium current. In addition, the results reveal that the calcium dynamics can become chaotic even though the voltage pacing is periodic. We reduce the equations of the model to a two-dimensional discrete map that relates the SR and cytosolic concentrations at one beat and the previous beat. From this map, we obtain a condition for the onset of calcium alternans in terms of the slopes of the release-versus-SR load and release-versus-diastolic-calcium curves. From an analysis of this map, we also obtain an understanding of the origin of chaotic dynamics.

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.

Condition for alternans and its control in a two-dimensional mapping model of paced cardiac dynamics

We investigate a two-dimensional mapping model of a paced, isolated cardiac cell that relates the duration of the action potential to the two preceding diastolic intervals as well as the preceding action potential duration. The model displays rate-dependent restitution and hence memory. We derive a criterion for the stability of the 1:1 response pattern displayed by the model. This criterion can be written in terms of experimentally measured quantities-the slopes of restitution curves obtained via different pacing protocols. In addition, we analyze the two-dimensional mapping model in the presence of closed-loop feedback control. The control is initiated by making small adjustments to the pacing interval in order to suppress alternans and stabilize the 1:1 pattern. We find that the domain of control does not depend on the functional form of the map, and, in the general case, is characterized by a combination of the slopes. We show that the gain gamma necessary to establish control may vary significantly depending on the value of the slope of the so-called standard restitution curve (herein denoted as S12), but that the product gammaS12 stays approximately in the same range.

Toward an understanding of the molecular mechanisms of ventricular fibrillation

A major goal of basic research in cardiac electrophysiology is to understand the mechanisms responsible for ventricular fibrillation (VF). Here we review recent experimental and numerical results, from the ion channel to the organ level, which might lead to a better understanding of the cellular and molecular mechanisms of VF. The discussion centers on data derived from a model of stable VF in the Langendorff-perfused guinea pig heart that demonstrate distinct patterns of organization in the left (LV) and right (RV) ventricles. Analysis of optical mapping data reveals that VF excitation frequencies are distributed throughout the ventricles in clearly demarcated domains. The highest frequency domains are usually found on the anterior wall of the LV, demonstrating that a high frequency reentrant source (a rotor) that remains stationary in the LV is the mechanism that sustains VF in this model. Computer simulations predict that the inward rectifying potassium current (IK1) is an essential determinant of rotor stability and rotation frequency, and patch-clamp results strongly suggest that the outward component of the background current (presumably IK1) of cells in the LV is significantly larger in the LV than in the RV. These data have opened a new and potentially exciting avenue of research on the possible role played by inward rectifier channels in the mechanism of VF and may lead us toward an understanding of its molecular basis and hopefully lead to new preventative approaches.

The Electrical Restitution Curve Revisited:

The electrical restitution curve (ERC) traditionally describes the recovery of action potential duration (APD) as a function of the interbeat interval or, more correctly, the diastolic interval (DI). Often overlooked in modeling studies, the normal ventricular ERC is triphasic, starting with a steep initial recovery at the shortest DIs, a transient decline, and a final asymptotic rise to a plateau phase reached at long DIs. Recent studies have proposed that it would be advantageous to lower the slope of the ERC by drug intervention, as this might reduce the potential for electrical alternans and ventricular fibrillation. This review discusses the pros and cons of a flat versus steep slope of the ERC and draws attention to mechanisms thatjustify the (physiologically) steep slope, rather than a flat slope, as a better design against arrhythmias. Five potential mechanisms are discussed, which allows for a different interpretation of the effect of the slope on arrhythmogenicity. The most important appears to be the physiologic rate adaptive shortening of APD that, by reciprocal lengthening of the DI, allows the subsequent APD to move more quickly from the steep initial ERC phase onto the flat phase. A less steep initial ERC phase would protract the transition toward more fully recovered APD and, in fact, may perpetuate electrical alternans. The triphasic ERC time course in normal myocardium cannot be explained by or fitted to single exponentials or single ion channel recovery kinetics. A simple tri-ionic model is suggested that may help explain the shape of the ERC at various repolarization levels and place APD recovery into perspective with intracellular calcium recycling and recovery of contractile force.

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.

Condition for alternans and stability of the 1:1 response pattern in a "memory" model of paced cardiac dynamics

We analyze a mathematical model of paced cardiac muscle consisting of a map relating the duration of an action potential to the preceding diastolic interval as well as the preceding action potential duration, thereby containing some degree of "memory." The model displays rate-dependent restitution so that the dynamic and S1-S2 restitution curves are different, a manifestation of memory in the model. We derive a criterion for the stability of the 1:1 response pattern displayed by this model. It is found that the stability criterion depends on the slope of both the dynamic and S1-S2 restitution curves, and that the pattern can be stable even when the individual slopes are greater or less than one. We discuss the relation between the stability criterion and the slope of the constant-BCL restitution curve. The criterion can also be used to determine the bifurcation from the 1:1 response pattern to alternans. We demonstrate that the criterion can be evaluated readily in experiments using a simple pacing protocol, thus establishing a method for determining whether actual myocardium is accurately described by such a mapping model. We illustrate our results by considering a specific map recently derived from a three-current membrane model and find that the stability of the 1:1 pattern is accurately described by our criterion. In addition, a numerical experiment is performed using the three-current model to illustrate the application of the pacing protocol and the evaluation of the criterion.

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.

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.

Mathematical analysis of dynamics of cardiac memory and accommodation: theory and experiment

Decreasing the slope of the dynamic, but not conventional, restitution curves is antifibrillatory. Cardiac memory/accommodation underlies the difference. We measured diastolic interval (DI) and action potential duration (APD) in epicardial, endocardial, and Purkinje tissue from eight dogs. Consecutive 100-stimulus trains were given to study transitions between basic cycle lengths (BCL) ranging from 400 to 1,300 ms. (DI,APD) pairs aligned immediately on the line DI + APD = BCL (64/67) or oscillated (3/67). The shifting effect of up to 10 extrastimuli on restitution curves was also measured. These curves were fit with the equation APD = alpha + beta exp(-DI/tau), where alpha is asymptote, beta is drop, and tau is time constant. Linear regression of the parameters against the number of extrastimuli showed that premature and postmature stimuli decreased and increased alpha and beta and increased and decreased tau, respectively. Analysis of a mathematical model treating memory as an exponentially decreasing shift of restitution curves shows that oscillatory DI,APD is expected with large DeltaBCL, steep restitution slope, or increased cardiac accommodation. The model explains phase shifts and suggests a common mechanism for Purkinje and myocardial electrical alternans.

Cellular basis for dispersion of repolarization underlying reentrant arrhythmias

Substantial heterogeneity in ion channel density and expression exists in cells isolated from various regions of the heart. Cell-to-cell coupling in the intact heart, however, is expected to attenuate the functional expression of the ion channel heterogeneities. Due to limitations of conventional electrophysiological recording techniques, the extent to which cellular electrical heterogeneities are functionally present in intact myocardium remains unknown. High-resolution optical mapping with voltage-sensitive dyes was used to measure transepicardial and transmural repolarization gradients in the Langendorff perfused guinea pig ventricle and the canine wedge preperation, respectively. Diversity of repolarization kinetics in the transepicardial direction modulated dispersion of repolarization in a biphasic fashion as premature coupling interval was shortened. Moreover, modulation of repolarization paralleled arrhythmia vulnerability in a predictable fashion. Transmural optical mapping revealed significant gradients of repolarization across the ventricular wall that were markedly increased in a surrogate model of LQTS. Transmural gradients of repolarization in LQTS were associated with an enhanced susceptibility to TdP. Therefore, despite strong cell-to-cell coupling in the normal heart, heterogeneities in the ionic make-up of cells across the epicardial and transmural surfaces result in functional heterogeneities of repolarization leading to arrhythmias.

Effects of [K(+)](o) on electrical restitution and activation dynamics during ventricular fibrillation

To test whether hyperkalemia suppresses ventricular fibrillation (VF) by reducing the slope of the action potential duration (APD) restitution relation, we determined the effects of the extracellular K(+) concentration ([K(+)](o)) ([KCl] = 2.7-12 mM) on the restitution of APD and maximum upstroke velocity (V(max)) the magnitude of APD alternans and spatiotemporal organization during VF in isolated canine ventricle. As [KCl] was increased incrementally from 2.7 to 12 mM, V(max) was reduced progressively. Increasing [KCl] from 2.7 to 10 mM decreased the slope of the APD restitution relation at long, but not short, diastolic intervals (DI), decreased the range of DI over which the slope was >/=1, and reduced the maximum amplitude of APD alternans. At [KCl] = 12 mM, the range of DI over which the APD restitution slope was >/=1 increased, and the maximum amplitude of APD alternans increased. For [KCl] = 4-8 mM, the persistence of APD alternans at short DI was associated with maintenance of VF. For [KCl] = 10-12 mM, the spontaneous frequency during VF was reduced, and activation occurred predominantly at longer DI. The lack of APD alternans at longer DI was associated with conversion of VF to a periodic rhythm. These results provide additional evidence for the importance of APD restitution kinetics in the development of VF.

Simultaneous maps of optical action potentials and calcium transients in guinea-pig hearts: mechanisms underlying concordant alternans

1. The mechanisms underlying electro-mechanical alternans caused by faster heart rates were investigated in perfused guinea-pig hearts stained with RH237 and Rhod-2 AM to simultaneously map optical action potentials (APs) and intracellular free Ca2+ (Ca2+i). 2. Fluorescence images of the heart were focused on two 16 x 16 photodiode arrays to map Ca2+i (emission wavelength (lamdda;em) = 585 +/- 20 nm) and APs (lamdda;em > 715 nm) from 252 sites. Spatial resolution was 0.8 mm x 0.8 mm per diode and temporal resolution 4000 frames s-1. 3. The mean time-to-peak for APs and [Ca2+]i was spatially homogeneous (8.8 +/- 0.5 and 25.6 +/- 5.0 ms, respectively; n = 6). The durations of APs (APDs) and Ca2+i transients were shorter at the apex and progressively longer towards the base, indicating a gradient of ventricular relaxation. 4. Restitution kinetics revealed increasingly longer delays between AP and Ca2+i upstrokes (9.5 +/- 0.4 to 11.3 +/- 0.4 ms) with increasingly shorter S1-S2 intervals, particularly at the base, despite nearly normal peak [Ca2+]i. 5. Alternans of APs and Ca2+i transients were induced by a decrease++ in cycle length (CL), if the shorter CL captured at the pacing site and was shorter than refractory periods (RPs) near the base, creating heterogeneities of conduction velocity. 6. Rate-induced alternans in normoxic hearts were concordant (long APD with large [Ca2+]i) across the epicardium, with a magnitude (difference between odd-even signals) that varied with the local RP. Alternans were initiated by gradients of RP, producing alternans of conduction that terminated spontaneously without progressing to fibrillation.

Existence of bistability and correlation with arrhythmogenesis in paced sheep atria

INTRODUCTION

Studies of the electrical dynamics of cardiac tissue are important for understanding the mechanisms of arrhythmias. This study uses high-frequency pacing to investigate the dynamics of sheep atria.

METHODS AND RESULTS

A 504-electrode mapping plaque was affixed to the right atrium in six sheep. Cathodal pacing stimuli were delivered to the center of the plaque. Pacing period (Tp) was decreased from 275 +/- 25 msec to 75 +/- 25 msec and then increased to 230 +/- 70 msec in steps of either 5 or 10 msec. In all 21 trials in six sheep, the atrium responded 1:1 at longer Tps and 2:1 at shorter Tps. As Tp was decreased, the response switched to 2:1 at a particular Tp. Conversely, as Tp was increased, the response switched back to 1:1 at a particular Tp. Over 21 trials, the 1:1-to-2:1 and 2:1-to-1:1 transitions occurred at 119.5 +/- 18.8 msec and 130.0 +/- 19.1 msec, respectively. This hysteretic behavior yielded bistability windows, 10.5 +/- 7.2 msec wide, wherein 1:1 and 2:1 responses existed at the same Tp. In 15 trials and in all animals, idiopathic wavefronts emanating from outside the mapped region passed through the mapped region. In 13 of those trials, the idiopathic wavefronts occurred at Tps within the bistability window or within 35 msec of its upper or lower limit.

CONCLUSION

Bistability windows and idiopathic wavefronts were observed and found to be correlated with each other, suggesting a connection between bistability and arrhythmogenesis.

Cardiac electrical restitution properties and stability of reentrant spiral waves: a simulation study

Spiral wave breakup is a proposed mechanism underlying the transition from ventricular tachycardia to fibrillation. We examined the importance of the restitution of action potential duration (APD) and of conduction velocity (CV) to the stability of spiral wave reentry in a two-dimensional sheet of simulated cardiac tissue. The Luo-Rudy ventricular action potential model was modified to eliminate its restitution properties, which are caused by deactivation or recovery from inactivation of K+, Ca2+, and Na+ currents (IK, ICa, and INa, respectively). In this model, we find that 1) restitution of ICa and INa are the main determinants of the steepness of APD restitution; 2) for promoting spiral breakup, the range of diastolic intervals over which the APD restitution slope is steep is more important than the maximum steepness; 3) CV restitution promotes spiral wave breakup independently of APD restitution; and 4) "defibrillation" of multiple spiral wave reentry is most effectively achieved by combining an antifibrillatory intervention based on altering restitution with an antitachycardia intervention. These findings suggest a novel paradigm for developing effective antiarrhythmic drugs.

Dynamic restitution of action potential duration during electrical alternans and ventricular fibrillation

The restitution kinetics of action potential duration (APD) were investigated in paced canine Purkinje fibers (P; n = 9) and endocardial muscle (M; n = 9), in isolated, perfused canine left ventricles during ventricular fibrillation (VF; n = 4), and in endocardial muscle paced at VF cycle lengths (simulated VF; n = 4). Restitution was assessed with the use of two protocols: delivery of a single extrastimulus after a train of stimuli at cycle length = 300 ms (standard protocol), and fixed pacing at short cycle lengths (100-300 ms) that induced APD alternans (dynamic protocol). The dynamic protocol yielded a monotone increasing restitution function with a maximal slope of 1.13 +/- 0.13 in M and 1.14 +/- 0.17 in P. Iteration of this function reproduced the APD dynamics found experimentally, including persistent APD alternans. In contrast, the standard protocol yielded a restitution relation with a maximal slope of 0.57 +/- 0.18 in M and 0.84 +/- 0.20 in P, and iteration of this function did not reproduce the APD dynamics. During VF, the restitution kinetics at short diastolic interval were similar to those determined with the dynamic protocol (maximal slope: 1.72 +/- 0.47 in VF and 1.44 +/- 0.49 in simulated VF). Thus APD dynamics at short coupling intervals during fixed pacing and during VF were accounted for by the dynamic, but not the standard, restitution relation. These results provide further evidence for a strong relationship among the kinetics of electrical restitution, the occurrence of APD alternans, and complex APD dynamics during VF.