The Restitution Portrait:. A New Method for Investigating Rate-Dependent Restitution

ForCEPSS-A framework for cardiac electrophysiology simulations standardization

BACKGROUND AND OBJECTIVE

Simulation of cardiac electrophysiology (CEP) is an important research tool that is increasingly being adopted in industrial and clinical applications. Typical workflows for CEP simulation consist of a sequence of processing stages starting with building an anatomical model and then calibrating its electrophysiological properties to match observable data. While the calibration stages are common and generalizable, most CEP studies re-implement these steps in complex and highly variable workflows. This lack of standardization renders the execution of computational CEP studies in an efficient, robust, and reproducible manner a significant challenge. Here, we propose ForCEPSS as an efficient and robust, yet flexible, software framework for standardizing CEP simulation studies.

METHODS AND RESULTS

Key processing stages of CEP simulation studies are identified and implemented in a standardized workflow that builds on openCARP1 Plank et al. (2021) and the Python-based carputils2 framework. Stages include (i) the definition and initialization of action potential phenotypes, (ii) the tissue scale calibration of conduction properties, (iii) the functional initialization to approximate a limit cycle corresponding to the dynamic reference state according to an experimental protocol, and, (iv) the execution of the CEP study where the electrophysiological response to a perturbation of the limit cycle is probed. As an exemplar application, we employ ForCEPSS to prepare a CEP study according to the Virtual Arrhythmia Risk Prediction protocol used for investigating the arrhythmogenic risk of developing infarct-related ventricular tachycardia (VT) in ischemic cardiomyopathy patients. We demonstrate that ForCEPSS enables a fully automated execution of all stages of this complex protocol.

CONCLUSION

ForCEPSS offers a novel comprehensive, standardized, and automated CEP simulation workflow. The high degree of automation accelerates the execution of CEP simulation studies, reduces errors, improves robustness, and makes CEP studies reproducible. Verification of simulation studies within the CEP modeling community is thus possible. As such, ForCEPSS makes an important contribution towards increasing transparency, standardization, and reproducibility of in silico CEP experiments.

Intracellular ion accumulation in the genesis of complex action potential dynamics under cardiac diseases

Intracellular ions, including sodium (Na^{+}), calcium (Ca^{2+}), and potassium (K^{+}), etc., accumulate slowly after a change of the state of the heart, such as a change of the heart rate. The goal of this study is to understand the roles of slow ion accumulation in the genesis of cardiac memory and complex action-potential duration (APD) dynamics that can lead to lethal cardiac arrhythmias. We carry out numerical simulations of a detailed action potential model of ventricular myocytes under normal and diseased conditions, which exhibit memory effects and complex APD dynamics. We develop a low-dimensional iterated map (IM) model to describe the dynamics of Na^{+}, Ca^{2+}, and APD and use it to uncover the underlying dynamical mechanisms. The development of the IM model is informed by simulation results under the normal condition. We then use the IM model to perform linear stability analyses and computer simulations to investigate the bifurcations and complex APD dynamics, which depend on the feedback loops between APD and intracellular Ca^{2+} and Na^{+} concentrations and the steepness of the APD response to the ion concentrations. When the feedback between APD and Ca^{2+} concentration is positive, a Hopf bifurcation leading to periodic oscillatory behavior occurs as the steepness of the APD response to the ion concentrations increases. The negative feedback loop between APD and Na^{+} concentration is required for the Hopf bifurcation. When the feedback between APD and Ca^{2+} concentration is negative, period-doubling bifurcations leading to high periodicity and chaos occurs. In this case, Na^{+} accumulation plays little role in the dynamics. Finally, we carry out simulations of the detailed action potential model under two diseased conditions, which exhibit steep APD responses to ion concentrations. Under both conditions, Hopf bifurcations leading to slow oscillations or period-doubling bifurcations leading to high periodicity and chaotic APD dynamics occur, depending on the strength of the ion pump-Na^{+}-Ca^{2+} exchanger. Using functions reconstructed from the simulation data, the IM model accurately captures the bifurcations and dynamics under the two diseased conditions. In conclusion, besides using computer simulations of a detailed high-dimensional action-potential model to investigate the effects of slow ion accumulation and short-term memory on bifurcations and genesis of complex APD dynamics in cardiac myocytes under diseased conditions, this study also provides a low-dimensional mathematical tool, i.e., the IM model, to allow stability analyses for uncovering the underlying mechanisms.

The electrical restitution of the non-propagated cardiac ventricular action potential

Sudden changes in pacing cycle length are frequently associated with repolarization abnormalities initiating cardiac arrhythmias, and physiologists have long been interested in measuring the likelihood of these events before their manifestation. A marker of repolarization stability has been found in the electrical restitution (ER), the response of the ventricular action potential duration to a pre- or post-mature stimulation, graphically represented by the so-called ER curve. According to the restitution hypothesis (ERH), the slope of this curve provides a quantitative discrimination between stable repolarization and proneness to arrhythmias. ER has been studied at the body surface, whole organ, and tissue level, and ERH has soon become a key reference point in theoretical, clinical, and pharmacological studies concerning arrhythmia development, and, despite criticisms, it is still widely adopted. The ionic mechanism of ER and cellular applications of ERH are covered in the present review. The main criticism on ERH concerns its dependence from the way ER is measured. Over the years, in fact, several different experimental protocols have been established to measure ER, which are also described in this article. In reviewing the state-of-the art on cardiac cellular ER, I have introduced a notation specifying protocols and graphical representations, with the aim of unifying a sometime confusing nomenclature, and providing a physiological tool, better defined in its scope and limitations, to meet the growing expectations of clinical and pharmacological research.

Activation of small conductance Ca2+ -activated K+ channels suppresses Ca2+ transient and action potential alternans in ventricular myocytes

At the cellular level, cardiac alternans is observed as beat-to-beat alternations in contraction strength, action potential (AP) morphology and Ca2+ transient (CaT) amplitude, and is a risk factor for cardiac arrhythmia. The (patho)physiological roles of small conductance Ca2+ -activated K+ (SK) channels in ventricles are poorly understood. We tested the hypothesis that in single rabbit ventricular myocytes pharmacological modulation of SK channels plays a causative role for the development of pacing-induced CaT and AP duration (APD) alternans. SK channel blockers (apamin, UCL1684) had only a minor effect on AP repolarization. However, SK channel activation by NS309 resulted in significant APD shortening, demonstrating that functional SK channels are well expressed in ventricular myocytes. The effects of NS309 were prevented or reversed by apamin and UCL1684, indicating that NS309 acted on SK channels. SK channel activation abolished or reduced the degree of pacing-induced CaT and APD alternans. Inhibition of KV 7.1 (with HMR1556) and KV 11.1 (with E4031) channels was used to mimic conditions of long QT syndromes type-1 and type-2, respectively. Both HMR1556 and E4031 enhanced CaT alternans that was prevented by SK channel activation. In AP voltage-clamped cells the SK channel activator had no effect on CaT alternans, confirming that suppression of CaT alternans was caused by APD shortening. APD shortening contributed to protection from alternans by lowering sarcoplasmic reticulum Ca2+ content and curtailing Ca2+ release. The data suggest that SK activation could be a potential intervention to avert development of alternans with important ramifications for arrhythmia prevention and therapy for patients with long QT syndrome. KEY POINTS: At the cellular level, cardiac alternans is observed as beat-to-beat alternations in contraction strength, action potential (AP) morphology and intracellular Ca2+ release amplitude, and is a risk factor for cardiac arrhythmia. The (patho)physiological roles of small conductance Ca2+ -activated K+ (SK) channels in ventricles are poorly understood. We investigated whether pharmacological modulation of SK channels affects the development of cardiac alternans in normal ventricular cells and in cells with drug-induced long QT syndrome (LQTS). While SK channel blockers have only a minor effect on AP morphology, their activation leads to AP shortening and abolishes or reduces the degree of pacing-induced Ca2+ and AP alternans. AP shortening contributed to protection against alternans by lowering sarcoplasmic reticulum Ca2+ content and curtailing Ca2+ release. The data suggest SK activation as a potential intervention to avert the development of alternans with important ramifications for arrhythmia prevention for patients with LQTS.

Ventricular Repolarization and Calcium Transient Show Resonant Behavior under Oscillatory Pacing Rate

Cardiac EC coupling is triggered by rhythmic depolarizing current fronts originating from the sino-atrial node, and the way variability in rhythm is associated with variability in action potential duration (APD) and, in turn, in the variability of calcium transient amplitude (CTA) and contraction is a key determinant of beating stability. Sinusoidal-varying pacing rate is adopted here in order to establish whether APD and CTA oscillations, elicited in a human ventricular AP model (OR) under oscillatory pacing, are consistent with the dynamics of two coupled harmonic oscillators, e.g., a two-degree-of-freedom system of mass and springs (MS model). I show evidence that this is the case, and that the MS model, preliminarily fitted to OR behavior, retains key features of the physiological system, such as the dependence of APD and CTA oscillation amplitudes from average value and from beat-to-beat changes in pacing rate, and the phase relationship between them. The bi-directionality of coupling between APD and CTA makes it difficult to discriminate which one leads EC coupling dynamics under variable pacing. The MS model suggests that the calcium cycling, with its greater inertia chiefly determined by the SR calcium release, is the leading mechanism. I propose the present approach to also be relevant at the whole organ level, where the need of compact representations of electromechanical interaction, particularly in clinical practice, remains urgent.

The development of L-type Ca2+ current mediated alternans does not depend on the restitution slope in canine ventricular myocardium

Cardiac alternans have crucial importance in the onset of ventricular fibrillation. The early explanation for alternans development was the voltage-driven mechanism, where the action potential (AP) restitution steepness was considered as crucial determining factor. Recent results suggest that restitution slope is an inadequate predictor for alternans development, but several studies still claim the role of membrane potential as underlying mechanism of alternans. These controversial data indicate that the relationship of restitution and alternans development is not completely understood. APs were measured by conventional microelectrode technique from canine right ventricular papillary muscles. Ionic currents combined with fluorescent measurements were recorded by patch-clamp technique. APs combined with fluorescent measurements were monitored by sharp microelectrodes. Rapid pacing evoked restitution-independent AP duration (APD) alternans. When non-alternating AP voltage command was used, Ca²⁺_i-transient (CaT) alternans were not observed. When alternating rectangular voltage pulses were applied, CaT alternans were proportional to I_CaL amplitude alternans. Selective I_CaL inhibition did not influence the fast phase of APD restitution. In this study we found that I_CaL has minor contribution in shaping the fast phase of restitution curve suggesting that I_CaL—if it plays important role in the alternans mechanism—could be an additional factor that attenuates the reliability of APD restitution slope to predict alternans.

CVAR-Seg: An Automated Signal Segmentation Pipeline for Conduction Velocity and Amplitude Restitution

BACKGROUND

Rate-varying S1S2 stimulation protocols can be used for restitution studies to characterize atrial substrate, ionic remodeling, and atrial fibrillation risk. Clinical restitution studies with numerous patients create large amounts of these data. Thus, an automated pipeline to evaluate clinically acquired S1S2 stimulation protocol data necessitates consistent, robust, reproducible, and precise evaluation of local activation times, electrogram amplitude, and conduction velocity. Here, we present the CVAR-Seg pipeline, developed focusing on three challenges: (i) No previous knowledge of the stimulation parameters is available, thus, arbitrary protocols are supported. (ii) The pipeline remains robust under different noise conditions. (iii) The pipeline supports segmentation of atrial activities in close temporal proximity to the stimulation artifact, which is challenging due to larger amplitude and slope of the stimulus compared to the atrial activity.

METHODS AND RESULTS

The S1 basic cycle length was estimated by time interval detection. Stimulation time windows were segmented by detecting synchronous peaks in different channels surpassing an amplitude threshold and identifying time intervals between detected stimuli. Elimination of the stimulation artifact by a matched filter allowed detection of local activation times in temporal proximity. A non-linear signal energy operator was used to segment periods of atrial activity. Geodesic and Euclidean inter electrode distances allowed approximation of conduction velocity. The automatic segmentation performance of the CVAR-Seg pipeline was evaluated on 37 synthetic datasets with decreasing signal-to-noise ratios. Noise was modeled by reconstructing the frequency spectrum of clinical noise. The pipeline retained a median local activation time error below a single sample (1 ms) for signal-to-noise ratios as low as 0 dB representing a high clinical noise level. As a proof of concept, the pipeline was tested on a CARTO case of a paroxysmal atrial fibrillation patient and yielded plausible restitution curves for conduction speed and amplitude.

CONCLUSION

The proposed openly available CVAR-Seg pipeline promises fast, fully automated, robust, and accurate evaluations of atrial signals even with low signal-to-noise ratios. This is achieved by solving the proximity problem of stimulation and atrial activity to enable standardized evaluation without introducing human bias for large data sets.

Electrophysiological Mechanisms Underlying T-Wave Alternans and Their Role in Arrhythmogenesis

T-wave alternans (TWA) reflects every-other-beat alterations in the morphology of the electrocardiogram ST segment or T wave in the setting of a constant heart rate, hence, in the absence of heart rate variability. It is believed to be associated with the dispersion of repolarization and has been used as a non-invasive marker for predicting the risk of malignant cardiac arrhythmias and sudden cardiac death as numerous studies have shown. This review aims to provide up-to-date review on both experimental and simulation studies in elucidating possible mechanisms underlying the genesis of TWA at the cellular level, as well as the genesis of spatially concordant/discordant alternans at the tissue level, and their transition to cardiac arrhythmia. Recent progress and future perspectives in antiarrhythmic therapies associated with TWA are also discussed.

Electrical Restitution and Its Modifications by Antiarrhythmic Drugs in Undiseased Human Ventricular Muscle

Introduction

Re-entry is a basic mechanism of ventricular fibrillation, which can be elicited by extrasystolic activity, but the timing of an extrasystole can be critical. The action potential duration (APD) of an extrasystole depends on the proximity of the preceding beat, and the relation between its timing and its APD is called electrical restitution. The aim of the present work was to study and compare the effect of several antiarrhythmic drugs on restitution in preparations from undiseased human ventricular muscle, and other mammalian species.

Methods

Action potentials were recorded in preparations obtained from rat, guinea pig, rabbit, and dog hearts; and from undiseased human donor hearts using the conventional microelectrode technique. Preparations were stimulated with different basic cycle lengths (BCLs) ranging from 300 to 5,000 ms. To study restitution, single test pulses were applied at every 20th beat while the preparation was driven at 1,000 ms BCL.

Results

Marked differences were found between the animal and human preparations regarding restitution and steady-state frequency dependent curves. In human ventricular muscle, restitution kinetics were slower in preparations with large phase 1 repolarization with shorter APDs at 1000 ms BCL compared to preparations with small phase 1. Preparations having APD longer than 300 ms at 1000 ms BCL had slower restitution kinetics than those having APD shorter than 250 ms. The selective I_Kr inhibitors E-4031 and sotalol increased overall APD and slowed the restitution kinetics, while I_Ks inhibition did not influence APD and electrical restitution. Mexiletine and nisoldipine shortened APD, but only mexiletine slowed restitution kinetics.

Discussion

Frequency dependent APD changes, including electrical restitution, were partly determined by the APD at the BCL. Small phase 1 associated with slower restitution suggests a role of I_to in restitution. APD prolonging drugs slowed restitution, while mexiletine, a known inhibitor of I_Na, shortened basic APD but also slowed restitution. These results indicate that although basic APD has an important role in restitution, other transmembrane currents, such as I_Na or I_to, can also affect restitution kinetics. This raises the possibility that ion channel modifier drugs slowing restitution kinetics may have antiarrhythmic properties by altering restitution.

Action potential shortening rescues atrial calcium alternans

KEY POINTS

Cardiac alternans refers to a beat-to-beat alternation in contraction, action potential (AP) morphology and Ca²⁺ transient (CaT) amplitude, and represents a risk factor for cardiac arrhythmia, including atrial fibrillation. We developed strategies to pharmacologically manipulate the AP waveform with the goal to reduce or eliminate the occurrence of CaT and contraction alternans in atrial tissue. With combined patch-clamp and intracellular Ca²⁺ measurements we investigated the effect of specific ion channel inhibitors and activators on alternans. In single rabbit atrial myocytes, suppression of Ca²⁺ -activated Cl^- channels eliminated AP duration alternans, but prolonged the AP and failed to eliminate CaT alternans. In contrast, activation of K⁺ currents (I_Ks and I_Kr ) shortened the AP and eliminated both AP duration and CaT alternans. As demonstrated also at the whole heart level, activation of K⁺ conductances represents a promising strategy to suppress alternans, and thus reducing a risk factor for atrial fibrillation.

ABSTRACT

At the cellular level alternans is observed as beat-to-beat alternations in contraction, action potential (AP) morphology and magnitude of the Ca²⁺ transient (CaT). Alternans is a well-established risk factor for cardiac arrhythmia, including atrial fibrillation. This study investigates whether pharmacological manipulation of AP morphology is a viable strategy to reduce the risk of arrhythmogenic CaT alternans. Pacing-induced AP and CaT alternans were studied in rabbit atrial myocytes using combined Ca²⁺ imaging and electrophysiological measurements. Increased AP duration (APD) and beat-to-beat alternations in AP morphology lowered the pacing frequency threshold and increased the degree of CaT alternans. Inhibition of Ca²⁺ -activated Cl^- channels reduced beat-to-beat AP alternations, but prolonged APD and failed to suppress CaT alternans. In contrast, AP shortening induced by activators of two K⁺ channels (ML277 for Kv7.1 and NS1643 for Kv11.1) abolished both APD and CaT alternans in field-stimulated and current-clamped myocytes. K⁺ channel activators had no effect on the degree of Ca²⁺ alternans in AP voltage-clamped cells, confirming that suppression of Ca²⁺ alternans was caused by the changes in AP morphology. Finally, activation of Kv11.1 channel significantly attenuated or even abolished atrial T-wave alternans in isolated Langendorff perfused hearts. In summary, AP shortening suppressed or completely eliminated both CaT and APD alternans in single atrial myocytes and atrial T-wave alternans at the whole heart level. Therefore, we suggest that AP shortening is a potential intervention to avert development of alternans with important ramifications for arrhythmia prevention and therapy.

The Visible Heart® project and methodologies: novel use for studying cardiac monophasic action potentials and evaluating their underlying mechanisms

INTRODUCTION

This review describes the utilization of Visible Heart® methodologies for electrophysiologic studies, specifically in the investigation of monophasic action potential (MAP) recordings, with the aim to facilitate new catheter/device design and development that may lead to earlier diagnosis, treatment, and ultimately a higher quality of life for patients with atrial fibrillation.

AREAS COVERED

We describe the historically proposed mechanisms behind which electrode is responsible for the MAP recording, new catheters for recording these signals, and how Visible Heart methodologies can be utilized to develop and test new technologies for electrophysiologic investigations.

EXPERT OPINION

When compared to traditional electrogram recordings, MAP waveforms provide clinical information vital to the understanding, diagnosis, and treatment of cardiac arrhythmias. New catheters and ablation technologies are routinely being assessed on reanimated large mammalian hearts (swine and human) in our laboratory. These abilities, combined with continued enhancements in imaging modalities and computational systems for electrical mapping, are being applied to the MAP catheter design process. Through this testing we are hopeful that the time from concept to product can be reduced, and that an array of MAP catheters can be placed in the hands of physicians, where they will improve patient outcomes.

Memory-induced nonlinear dynamics of excitation in cardiac diseases

Excitable cells, such as cardiac myocytes, exhibit short-term memory, i.e., the state of the cell depends on its history of excitation. Memory can originate from slow recovery of membrane ion channels or from accumulation of intracellular ion concentrations, such as calcium ion or sodium ion concentration accumulation. Here we examine the effects of memory on excitation dynamics in cardiac myocytes under two diseased conditions, early repolarization and reduced repolarization reserve, each with memory from two different sources: slow recovery of a potassium ion channel and slow accumulation of the intracellular calcium ion concentration. We first carry out computer simulations of action potential models described by differential equations to demonstrate complex excitation dynamics, such as chaos. We then develop iterated map models that incorporate memory, which accurately capture the complex excitation dynamics and bifurcations of the action potential models. Finally, we carry out theoretical analyses of the iterated map models to reveal the underlying mechanisms of memory-induced nonlinear dynamics. Our study demonstrates that the memory effect can be unmasked or greatly exacerbated under certain diseased conditions, which promotes complex excitation dynamics, such as chaos. The iterated map models reveal that memory converts a monotonic iterated map function into a nonmonotonic one to promote the bifurcations leading to high periodicity and chaos.

Short-term action potential memory and electrical restitution: A cellular computational study on the stability of cardiac repolarization under dynamic pacing

Electrical restitution (ER) is a major determinant of repolarization stability and, under fast pacing rate, it reveals memory properties of the cardiac action potential (AP), whose dynamics have never been fully elucidated, nor their ionic mechanisms. Previous studies have looked at ER mainly in terms of changes in AP duration (APD) when the preceding diastolic interval (DI) changes and described dynamic conditions where this relationship shows hysteresis which, in turn, has been proposed as a marker of short-term AP memory and repolarization stability. By means of numerical simulations of a non-propagated human ventricular AP, we show here that measuring ER as APD versus the preceding cycle length (CL) provides additional information on repolarization dynamics which is not contained in the companion formulation. We focus particularly on fast pacing rate conditions with a beat-to-beat variable CL, where memory properties emerge from APD vs CL and not from APD vs DI and should thus be stored in APD and not in DI. We provide an ion-currents characterization of such conditions under periodic and random CL variability, and show that the memory stored in APD plays a stabilizing role on AP repolarization under pacing rate perturbations. The gating kinetics of L-type calcium current seems to be the main determinant of this safety mechanism. We also show that, at fast pacing rate and under otherwise identical pacing conditions, a periodically beat-to-beat changing CL is more effective than a random one in stabilizing repolarization. In summary, we propose a novel view of short-term AP memory, differentially stored between systole and diastole, which opens a number of methodological and theoretical implications for the understanding of arrhythmia development.

Restitution slope is principally determined by steady-state action potential duration

Aims

The steepness of the action potential duration (APD) restitution curve and local tissue refractoriness are both thought to play important roles in arrhythmogenesis. Despite this, there has been little recognition of the apparent association between steady-state APD and the slope of the restitution curve. The objective of this study was to test the hypothesis that restitution slope is determined by APD and to examine the relationship between restitution slope, refractoriness and susceptibility to VF.

Methods and results

Experiments were conducted in isolated hearts and ventricular myocytes from adult guinea pigs and rabbits. Restitution curves were measured under control conditions and following intervention to prolong (clofilium, veratridine, bretylium, low [Ca]e, chronic transverse aortic constriction) or shorten (catecholamines, rapid pacing) ventricular APD. Despite markedly differing mechanisms of action, all interventions that prolonged the action potential led to a steepening of the restitution curve (and vice versa). Normalizing the restitution curve as a % of steady-state APD abolished the difference in restitution curves with all interventions. Effects on restitution were preserved when APD was modulated by current injection in myocytes pre-treated with the calcium chelator BAPTA-AM – to abolish the intracellular calcium transient. The non-linear relation between APD and the rate of repolarization of the action potential is shown to underpin the common influence of APD on the slope of the restitution curve. Susceptibility to VF was found to parallel changes in APD/refractoriness, rather than restitution slope.

Conclusion(s)

Steady-state APD is the principal determinant of the slope of the ventricular electrical restitution curve. In the absence of post-repolarization refractoriness, factors that prolong the action potential would be expected to steepen the restitution curve. However, concomitant changes in tissue refractoriness act to reduce susceptibility to sustained VF. Dependence on steady-state APD may contribute to the failure of restitution slope to predict sudden cardiac death.

Mechanisms of stochastic onset and termination of atrial fibrillation studied with a cellular automaton model

Mathematical models of cardiac electrical excitation are increasingly complex, with multiscale models seeking to represent and bridge physiological behaviours across temporal and spatial scales. The increasing complexity of these models makes it computationally expensive to both evaluate long term (more than 60 s) behaviour and determine sensitivity of model outputs to inputs. This is particularly relevant in models of atrial fibrillation (AF), where individual episodes last from seconds to days, and interepisode waiting times can be minutes to months. Potential mechanisms of transition between sinus rhythm and AF have been identified but are not well understood, and it is difficult to simulate AF for long periods of time using state-of-the-art models. In this study, we implemented a Moe-type cellular automaton on a novel, topologically equivalent surface geometry of the left atrium. We used the model to simulate stochastic initiation and spontaneous termination of AF, arising from bursts of spontaneous activation near pulmonary veins. The simplified representation of atrial electrical activity reduced computational cost, and so permitted us to investigate AF mechanisms in a probabilistic setting. We computed large numbers (approx. 10⁵) of sample paths of the model, to infer stochastic initiation and termination rates of AF episodes using different model parameters. By generating statistical distributions of model outputs, we demonstrated how to propagate uncertainties of inputs within our microscopic level model up to a macroscopic level. Lastly, we investigated spontaneous termination in the model and found a complex dependence on its past AF trajectory, the mechanism of which merits future investigation.

Theory of the development of alternans in the heart during controlled diastolic interval pacing

The beat-to-beat alternation in action potential durations (APDs) in the heart, called APD alternans, has been linked to the development of serious cardiac rhythm disorders, including ventricular tachycardia and fibrillation. The length of the period between action potentials, called the diastolic interval (DI), is a key dynamical variable in the standard theory of alternans development. Thus, methods that control the DI may be useful in preventing dangerous cardiac rhythms. In this study, we examine the dynamics of alternans during controlled-DI pacing using a series of single-cell and one-dimensional (1D) fiber models of alternans dynamics. We find that a model that combines a so-called memory model with a calcium cycling model can reasonably explain two key experimental results: the possibility of alternans during constant-DI pacing and the phase lag of APDs behind DIs during sinusoidal-DI pacing. We also find that these results can be replicated by incorporating the memory model into an amplitude equation description of a 1D fiber. The 1D fiber result is potentially concerning because it seems to suggest that constant-DI control of alternans can only be effective over only a limited region in space.

Alternans in atria: Mechanisms and clinical relevance

Atrial fibrillation is the most common sustained arrhythmia and its prevalence is rapidly rising with the aging of the population. Cardiac alternans, defined as cyclic beat-to-beat alternations in contraction force, action potential (AP) duration and intracellular Ca²⁺ release at constant stimulation rate, has been associated with the development of ventricular arrhythmias. Recent clinical data also provide strong evidence that alternans plays a central role in arrhythmogenesis in atria. The aim of this article is to review the mechanisms that are responsible for repolarization alternans and contribute to the transition from spatially concordant alternans to the more arrhythmogenic spatially discordant alternans in atria.

Categorization and theoretical comparison of quantitative methods for assessing QT/RR hysteresis

BACKGROUND

In the human electrocardiogram, there is a lag of adaptation of the QT interval to heart rate changes, usually termed QT/RR hysteresis (QT-hys). Subject-specific quantifiers of QT-hys have been proposed as potential biomarkers, but there is no consensus on the choice of the quantifier.

METHODS

A comprehensive literature search was conducted to identify original articles reporting quantifiers of repolarization hysteresis from the surface ECG in humans.

RESULTS

Sixty articles fulfilled our inclusion criteria. Reported biomarkers were grouped under four categories. A simple mathematical model of QT/RR loop was used to illustrate differences between the methods. Category I quantifiers use direct measurement of QT time course of adaptation. They are limited to conditions where RR intervals are under strict control. Category IIa and IIb quantifiers compare QT responses during consecutive heart rate acceleration and deceleration. They are relevant when a QT/RR loop is observed, typically during exercise and recovery, but are not robust to protocol variations. Category III quantifiers evaluate the optimum RR memory in dynamic QT/RR relationship modeling. They estimate an intrinsic memory parameter independent from the nature of RR changes, but their reliability remains to be confirmed when multiple memory parameters are estimated. Promising approaches include the differentiation of short-term and long-term memory and adaptive estimation of memory parameters.

CONCLUSION

Model-based approaches to QT-hys assessment appear to be the most versatile, as they allow separate quantification of QT/RR dependency and QT-hys, and can be applied to a wide range of experimental settings.

Monophasic action potential recordings: which is the recording electrode?

The aim of this article is to provide an overview of current debate on the monophasic action potential (MAP) recording technique, specifically whether the depolarizing or the reference electrode is responsible for recording the MAP waveform. A literature search was made using key words including monophasic action potential, MAP, electrophysiological basis, recording electrode, depolarizing electrode, contact electrode, indifferent electrode, and reference electrode. References from articles were screened for additional relevant papers. Articles published by the different experimental groups claim that depolarizing electrode, but not reference electrode, records MAPs from the myocardium. This can be more accurately described when considering biophysical theory, which states that MAP is a bipolar signal with contributions from not only the depolarizing electrode but also remote activation at the reference electrode. It is not meaningful to claim that one is the recording electrode because potential differences must be measured between two points in space. Nevertheless, the MAP technique is useful for assessing the local electrical activity of the myocardium in contact with the depolarizing electrode. It is important to have the recording electrode in close proximity with the reference electrode to minimize contamination from far-field signals.

Effects of Na+ channel blockers on the restitution of refractory period, conduction time, and excitation wavelength in perfused guinea-pig heart

Na⁺ channel blockers flecainide and quinidine can increase propensity to ventricular tachyarrhythmia, whereas lidocaine and mexiletine are recognized as safe antiarrhythmics. Clinically, ventricular fibrillation is often precipitated by transient tachycardia that reduces action potential duration, suggesting that a critical shortening of the excitation wavelength (EW) may contribute to the arrhythmic substrate. This study examined whether different I_Na blockers can produce contrasting effects on the rate adaptation of the EW, which would explain the difference in their safety profile. In perfused guinea-pig hearts, effective refractory periods (ERP), conduction times, and EW values were determined over a wide range of cardiac pacing intervals. All I_Na blockers tested were found to flatten the slope of ERP restitution, indicating antiarrhythmic tendency. However, with flecainide and quinidine, the beneficial changes in ERP were reversed owing to the use-dependent conduction slowing, thereby leading to significantly steepened restitution of the EW. In contrast, lidocaine and mexiletine had no effect on ventricular conduction, and therefore reduced the slope of the EW restitution, as expected from their effect on ERP. These findings suggest that the slope of the EW restitution is an important electrophysiological determinant which can discriminate I_Na blockers with proarrhythmic and antiarrhythmic profile.

Cardiac dynamics: Alternans and arrhythmogenesis

Pre-existing heterogeneities present in cardiac tissue are essential for maintaining the normal electrical and mechanical functions of the heart. Exacerbation of such heterogeneities or the emergence of dynamic factors can produce repolarization alternans, which are beat-to-beat alternations in the action potential time course. Traditionally, this was explained by restitution, but additional factors, such as cardiac memory, calcium handling dynamics, refractory period restitution, and mechano-electric feedback, are increasingly recognized as the underlying causes. The aim of this article is to review the mechanisms that generate cardiac repolarization alternans and convert spatially concordant alternans to the more arrhythmogenic spatially discordant alternans. This is followed by a discussion on how alternans generate arrhythmias in a number of clinical scenarios, and concluded by an outline of future therapeutic targets for anti-arrhythmic therapy.

Nonlinear physics of electrical wave propagation in the heart: a review

The beating of the heart is a synchronized contraction of muscle cells (myocytes) that is triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media with applications to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact for cardiac arrhythmias.

Real-time feedback based control of cardiac restitution using optical mapping

Cardiac restitution is the shortening of the action potential duration with an increase in the heart rate. A shorter action potential duration enables a longer diastolic interval which ensures that the heart gets adequate time to refill with blood. At higher rates however, restitution becomes steep and thus, can lead to unstable electrical activity (alternans) in the heart, leading to fatal cardiac rhythms. It has been proposed that maintaining a shallow slope of cardiac restitution could have potentially anti-arrhythmic effects. Previous studies involved the control of action potential duration (APD) or diastolic interval (DI) in isolated tissue samples based on the feedback from single microelectrode recordings. This limited the spatial resolution of the feedback system. Here, we aimed to develop a real time feedback control system that enabled the detection of APDs from various single pixels based on optical mapping recordings. Stimuli were applied after a predefined fixed DI after detection of an APD. We validated our algorithm using optical mapping movies from an ex-vivo rabbit heart. Thus, we provide an optical mapping based approach for the control of cardiac restitution and a potential means to validate its anti-arrhythmic effects.

Comparison of Detailed and Simplified Models of Human Atrial Myocytes to Recapitulate Patient Specific Properties

Computer studies are often used to study mechanisms of cardiac arrhythmias, including atrial fibrillation (AF). A crucial component in these studies is the electrophysiological model that describes the membrane potential of myocytes. The models vary from detailed, describing numerous ion channels, to simplified, grouping ionic channels into a minimal set of variables. The parameters of these models, however, are determined across different experiments in varied species. Furthermore, a single set of parameters may not describe variations across patients, and models have rarely been shown to recapitulate critical features of AF in a given patient. In this study we develop physiologically accurate computational human atrial models by fitting parameters of a detailed and of a simplified model to clinical data for five patients undergoing ablation therapy. Parameters were simultaneously fitted to action potential (AP) morphology, action potential duration (APD) restitution and conduction velocity (CV) restitution curves in these patients. For both models, our fitting procedure generated parameter sets that accurately reproduced clinical data, but differed markedly from published sets and between patients, emphasizing the need for patient-specific adjustment. Both models produced two-dimensional spiral wave dynamics for that were similar for each patient. These results show that simplified, computationally efficient models are an attractive choice for simulations of human atrial electrophysiology in spatially extended domains. This study motivates the development and validation of patient-specific model-based mechanistic studies to target therapy.

Global alternans instability and its effect on non-linear wave propagation: dynamical Wenckebach block and self terminating spiral waves

The main mechanism of formation of reentrant cardiac arrhythmias is via formation of waveblocks at heterogeneities of cardiac tissue. We report that heterogeneity and the area of waveblock can extend itself in space and can result formation of new additional sources, or termination of existing sources of arrhythmias. This effect is based on a new form of instability, which we coin as global alternans instability (GAI). GAI is closely related to the so-called (discordant) alternans instability, however its onset is determined by the global properties of the APD-restitution curve and not by its slope. The APD-restitution curve relates the duration of the cardiac pulse (APD) to the time interval between the pulses, and can easily be measured in an experimental or even clinical setting. We formulate the conditions for the onset of GAI, study its manifestation in various 1D and 2D situations and discuss its importance for the onset of cardiac arrhythmias.

Restitution analysis of alternans using dynamic pacing and its comparison with S1S2 restitution in heptanol-treated, hypokalaemic Langendorff-perfused mouse hearts

Action potential duration (APD) and conduction velocity restitution explain the dependence of these parameters on the previous diastolic interval (DI). It is considered to be an adaptive mechanism for preserving diastole at fast heart rates. Hypokalaemia is known to induce ventricular arrhythmias that could be prevented by heptanol, the gap junction uncoupler, mediated through increases in ventricular refractory period (VERP) without alterations in APDs. The present study investigated alternans and restitution properties during normokalaemia, hypokalaemia alone or hypokalaemia with heptanol (0.1 mM) in Langendorff-perfused mouse hearts using a dynamic pacing protocol. APD90 alternans were elicited in the epicardium and endocardium during normokalaemia. Hypokalaemia increased the amplitudes of epicardial APD90 alternans when basic cycle lengths (BCLs) were ≤65 msec, which was associated with increases in maximum APD90 restitution gradients, critical DIs and APD90 heterogeneity. Heptanol (0.1 mM) did not exacerbate or reduce the APD90 alternans or alter these restitution parameters further. By contrast, endocardial APD90 alternans did not show increases in amplitudes during hypokalaemia at short BCLs studied, and restitution parameters were also unchanged. This was true whether in the presence or absence of 0.1 mM heptanol. The study demonstrates that anti-arrhythmic effects of heptanol exerted during hypokalaemia occurred despite exacerbation of APD90 alternans. This would suggest that even in the presence of arrhythmogenic APD90 alternans, arrhythmias could still be prevented by influencing VERP alone. Restitution data obtained here by dynamic pacing were compared to previous data from S1S2 pacing.

Stochastic pacing reveals the propensity to cardiac action potential alternans and uncovers its underlying dynamics

KEY POINTS

Beat-to-beat alternation (alternans) of the cardiac action potential duration is known to precipitate life-threatening arrhythmias and can be driven by the kinetics of voltage-gated membrane currents or by instabilities in intracellular calcium fluxes. To prevent alternans and associated arrhythmias, suitable markers must be developed to quantify the susceptibility to alternans; previous theoretical studies showed that the eigenvalue of the alternating eigenmode represents an ideal marker of alternans. Using rabbit ventricular myocytes, we show that this eigenvalue can be estimated in practice by pacing these cells at intervals varying stochastically. We also show that stochastic pacing permits the estimation of further markers distinguishing between voltage-driven and calcium-driven alternans. Our study opens the perspective to use stochastic pacing during clinical investigations and in patients with implanted pacing devices to determine the susceptibility to, and the type of alternans, which are both important to guide preventive or therapeutic measures.

ABSTRACT

Alternans of the cardiac action potential (AP) duration (APD) is a well-known arrhythmogenic mechanism. APD depends on several preceding diastolic intervals (DIs) and APDs, which complicates the prediction of alternans. Previous theoretical studies pinpointed a marker called λalt that directly quantifies how an alternating perturbation persists over successive APs. When the propensity to alternans increases, λalt decreases from 0 to -1. Our aim was to quantify λalt experimentally using stochastic pacing and to examine whether stochastic pacing allows discriminating between voltage-driven and Ca(2+) -driven alternans. APs were recorded in rabbit ventricular myocytes paced at cycle lengths (CLs) decreasing progressively and incorporating stochastic variations. Fitting APD with a function of two previous APDs and CLs permitted us to estimate λalt along with additional markers characterizing whether the dependence of APD on previous DIs or CLs is strong (typical for voltage-driven alternans) or weak (Ca(2+) -driven alternans). During the recordings, λalt gradually decreased from around 0 towards -1. Intermittent alternans appeared when λalt reached -0.8 and was followed by sustained alternans. The additional markers detected that alternans was Ca(2+) driven in control experiments and voltage driven in the presence of ryanodine. This distinction could be made even before alternans was manifest (specificity/sensitivity >80% for -0.4 > λalt  > -0.5). These observations were confirmed in a mathematical model of a rabbit ventricular myocyte. In conclusion, stochastic pacing allows the practical estimation of λalt to reveal the onset of alternans and distinguishes between voltage-driven and Ca(2+) -driven mechanisms, which is important since these two mechanisms may precipitate arrhythmias in different manners.

Spatial discordance and phase reversals during alternate pacing in discrete-time kinematic and cardiomyocyte ionic models

Alternans, a beat-to-beat alternation in the cardiac action potential duration (APD), is a dynamical instability linked with the initiation of arrhythmias and sudden cardiac death, and arises via a period-doubling bifurcation when myocytes are stimulated at fast rates. In this study, we analyze the stability of a propagating electrical wave in a one-dimensional cardiac myocyte model in response to an arrhythmogenic rhythm known as alternate pacing. Using a discrete-time kinematic model and complex frequency (Z) domain analysis, we derive analytical expressions to predict phase reversals and spatial discordance in the interbeat interval (IBI) and APD, which, importantly, cannot be predicted with a model that neglects the influence of cell coupling on repolarization. We identify key dimensionless parameters that determine the transition from spatial concordance to discordance. Finally, we show that the theoretical predictions agree closely with numerical simulations of an ionic myocyte model, over a wide range of parameters, including variable IBI, altered ionic current gating, and reduced cell coupling. We demonstrate a novel approach to predict instability in cardiac tissue during alternate pacing and further illustrate how this approach can be generalized to more detail models of myocyte dynamics.

Cell-specific cardiac electrophysiology models

The traditional cardiac model-building paradigm involves constructing a composite model using data collected from many cells. Equations are derived for each relevant cellular component (e.g., ion channel, exchanger) independently. After the equations for all components are combined to form the composite model, a subset of parameters is tuned, often arbitrarily and by hand, until the model output matches a target objective, such as an action potential. Unfortunately, such models often fail to accurately simulate behavior that is dynamically dissimilar (e.g., arrhythmia) to the simple target objective to which the model was fit. In this study, we develop a new approach in which data are collected via a series of complex electrophysiology protocols from single cardiac myocytes and then used to tune model parameters via a parallel fitting method known as a genetic algorithm (GA). The dynamical complexity of the electrophysiological data, which can only be fit by an automated method such as a GA, leads to more accurately parameterized models that can simulate rich cardiac dynamics. The feasibility of the method is first validated computationally, after which it is used to develop models of isolated guinea pig ventricular myocytes that simulate the electrophysiological dynamics significantly better than does a standard guinea pig model. In addition to improving model fidelity generally, this approach can be used to generate a cell-specific model. By so doing, the approach may be useful in applications ranging from studying the implications of cell-to-cell variability to the prediction of intersubject differences in response to pharmacological treatment.

Mathematical models of cardiac cell electrophysiology are widely used as predictive and illuminatory tools, but have been developed for decades using a suboptimal process. The models are typically constructed by manual adjustment of parameters to fit simple data and therefore often underperform when used to predict complex behavior such as arrhythmias. We present a novel method of model parameterization using automated optimization and dynamically rich fitting data and then demonstrate that this approach is better at finding the “real” model of a cell. Application of the method to cardiac myocytes leads to cell-specific models, which may enable well-controlled studies of both cellular- and subject-level population heterogeneity in disease propensity and response to therapies.

Cardiac myocyte alternans in intact heart: Influence of cell-cell coupling and β-adrenergic stimulation

BACKGROUND

Cardiac alternans are proarrhythmic and mechanistically link cardiac mechanical dysfunction and sudden cardiac death. Beat-to-beat alternans occur when beats with large Ca(2+) transients and long action potential duration (APD) alternate with the converse. APD alternans are typically driven by Ca(2+) alternans and sarcoplasmic reticulum (SR) Ca(2+) release alternans. But the effect of intercellular communication via gap junctions (GJ) on alternans in the intact heart remains unknown.

OBJECTIVE

We assessed the effects of cell-to-cell coupling on local alternans in intact Langendorff-perfused mouse hearts, measuring single myocyte [Ca(2+)] alternans synchronization among neighboring cells, and effects of β-adrenergic receptor (β-AR) activation and reduced GJ coupling.

METHODS AND RESULTS

Mouse hearts (C57BL/6) were retrogradely perfused and loaded with Fluo8-AM to record cardiac myocyte [Ca(2+)] in situ with confocal microscopy. Single cell resolution allowed analysis of alternans within the intact organ during alternans induction. Carbenoxolone (25 μM), a GJ inhibitor, significantly increased the occurrence and amplitude of alternans in single cells within the intact heart. Alternans were concordant between neighboring cells throughout the field of view, except transiently during onset. β-AR stimulation only reduced Ca(2+) alternans in tissue that had reduced GJ coupling, matching effects seen in isolated myocytes.

CONCLUSIONS

Ca(2+) alternans among neighboring myocytes is predominantly concordant, likely because of electrical coupling between cells. Consistent with this, partial GJ uncoupling increased propensity and amplitude of Ca(2+) alternans, and made them more sensitive to reversal by β-AR activation, as in isolated myocytes. Electrical coupling between myocytes may thus limit the alternans initiation, but also allow alternans to be more stable once established.

Voltage and calcium dynamics both underlie cellular alternans in cardiac myocytes

Cardiac alternans, a putative trigger event for cardiac reentry, is a beat-to-beat alternation in membrane potential and calcium transient. Alternans was originally attributed to instabilities in transmembrane ion channel dynamics (i.e., the voltage mechanism). As of this writing, the predominant view is that instabilities in subcellular calcium handling are the main underlying mechanism. That being said, because the voltage and calcium systems are bidirectionally coupled, theoretical studies have suggested that both mechanisms can contribute. To date, to our knowledge, no experimental evidence of such a dual role within the same cell has been reported. Here, a combined electrophysiological and calcium imaging approach was developed and used to illuminate the contributions of voltage and calcium dynamics to alternans. An experimentally feasible protocol, quantification of subcellular calcium alternans and restitution slope during cycle-length ramping alternans control, was designed and validated. This approach allows simultaneous illumination of the contributions of voltage and calcium-driven instability to total cellular instability as a function of cycle-length. Application of this protocol in in vitro guinea-pig left-ventricular myocytes demonstrated that both voltage- and calcium-driven instabilities underlie alternans, and that the relative contributions of the two systems change as a function of pacing rate.

Nonlinear and Stochastic Dynamics in the Heart

In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.

Change of short-term memory effect in acute ischemic ventricular myocardium: a computational study

The ionic mechanism of change in short-term memory (STM) during acute myocardial ischemia has not been well understood. In this paper, an advanced guinea pig ventricular model developed by Luo and Rudy was used to investigate STM property of ischemic ventricular myocardium. STM response was calculated by testing the time to reach steady-state action potential duration (APD) after an abrupt shortening of basic cycling length (BCL) in the pacing protocol. Electrical restitution curves (RCs), which can simultaneously visualize multiple aspects of APD restitution and STM, were obtained from dynamic and local S1S2 restitution portrait (RP), which consist of a longer interval stimulus (S1) and a shorter interval stimulus (S2). The angle between dynamic RC and local S1S2 RC reflects the amount of STM. Our results indicated that compared with control (normal) condition, time constant of STM response in the ischemic condition decreased significantly. Meanwhile the angle which reflects STM amount is less in ischemic model than that in control model. By tracking the effect of ischemia on intracellular ion concentration and membrane currents, we declared that changes in membrane currents caused by ischemia exert subtle influences on STM; it is only the decline of intracellular calcium concentration that give rise to the most decrement of STM.

Spatiotemporal evolution and prediction of [Ca(2+) ]i and APD alternans in isolated rabbit hearts

INTRODUCTION

Action potential duration (APD) alternans can be accompanied by alternans in intracellular calcium transients ([Ca(2+) ]i ), leading to electromechanical alternans. Electromechanical alternans is considered a substrate for ventricular fibrillation. Although some techniques have been developed to predict APD alternans, the onset of [Ca(2+) ]i alternans has never been predicted.

METHODS AND RESULTS

Simultaneous mapping of voltage and calcium was performed in 8 Langendorff-perfused rabbit hearts. APD, [Ca(2+) ]i amplitude (CaA) and duration (CaD) alternans were induced using a perturbed downsweep protocol. Local onset of alternans (B(onset) ) was defined as the cycle length (BCL) at which at least 10% of the RV exhibited alternans. We observed that the local onset of CaA alternans always occurred first, followed by APD and then CaD alternans. We constructed APD, CaD, and CaA restitution portraits for 2 regions of the heart defined at B(onset) : the 1:1alt region, which developed alternans, and the 1:1 region, which did not. Our results also show that the slopes S12 Max and SDyn were higher in 1:1alt region (SDyn  = 0.99 ± 0.04 vs 0.73 ± 0.06; S12 Max  = 0.95 ± 0.13 vs 0.65 ± 0.1, P < 0.05) prior to onset of CaD alternans, while S12 and S12 Max were significantly higher in the 1:1alt region (S12  = 0.59 ± 0.19 vs 0.19 ± 0.02; S12 Max  = 1.09 ± 0.1 vs 0.61 ± 0.08, P < 0.05) prior to onset of CaA alternans.

CONCLUSION

We successfully applied the restitution portrait technique to the prediction of [Ca(2+) ]i (both CaA and CaD) alternans. The slopes of the APD/CaD/CaA restitution portrait are definitive indicators of APD, CaD, and CaA alternans.

Effects of pacing site and stimulation history on alternans dynamics and the development of complex spatiotemporal patterns in cardiac tissue

Alternans of action potential duration has been associated with T wave alternans and the development of arrhythmias because it produces large gradients of repolarization. However, little is known about alternans dynamics in large mammalian hearts. Using optical mapping to record electrical activations simultaneously from the epicardium and endocardium of 9 canine right ventricles, we demonstrate novel arrhythmogenic complex spatiotemporal dynamics. (i) Alternans predominantly develops first on the endocardium. (ii) The postulated simple progression from normal rhythm to concordant to discordant alternans is not always observed; concordant alternans can develop from discordant alternans as the pacing period is decreased. (iii) In contrast to smaller tissue preparations, multiple stationary nodal lines may exist and need not be perpendicular to the pacing site or to each other. (iv) Alternans has fully three-dimensional dynamics and the epicardium and endocardium can show significantly different dynamics: multiple nodal surfaces can be transmural or intramural and can form concave/convex surfaces resulting in islands of discordant alternans. (v) The complex spatiotemporal patterns observed during alternans are very sensitive to both the site of stimulation and the stimulation history. Alternans in canine ventricles not only exhibit larger amplitudes and persist for longer cycle length regimes compared to those found in smaller mammalian hearts, but also show novel dynamics not previously described that enhance dispersion and show high sensitivity to initial conditions. This indicates some underlying predisposition to chaos and can help to guide the design of new drugs and devices controlling and preventing arrhythmic events.

Estimating eigenvalues of dynamical systems from time series with applications to predicting cardiac alternans

The stability of a dynamical system can be indicated by eigenvalues of its underlying mathematical model. However, eigenvalue analysis of a complicated system (e.g. the heart) may be extremely difficult because full models may be intractable or unavailable. We develop data-driven statistical techniques, which are independent of any underlying dynamical model, that use principal components and maximum-likelihood methods to estimate the dominant eigenvalues and their standard errors from the time series of one or a few measurable quantities, e.g. transmembrane voltages in cardiac experiments. The techniques are applied to predicting cardiac alternans that is characterized by an eigenvalue approaching −1. Cardiac alternans signals a vulnerability to ventricular fibrillation, the leading cause of death in the USA.

The role of fine-scale anatomical structure in the dynamics of reentry in computational models of the rabbit ventricles

Fine-scale anatomical structures in the heart may play an important role in sustaining cardiac arrhythmias. However, the extent of this role and how it may differ between species are not fully understood. In this study we used computational modelling to assess the impact of anatomy upon arrhythmia maintenance in the rabbit ventricles. Specifically, we quantified the dynamics of excitation wavefronts during episodes of simulated tachyarrhythmias and fibrillatory arrhythmias, defined as being respectively characterised by relatively low and high spatio-temporal disorganisation.Two computational models were used: a highly anatomically detailed MR-derived rabbit ventricular model (representing vasculature, endocardial structures) and a simplified equivalent model, constructed from the same MR-data but lacking such fine-scale anatomical features. During tachyarrhythmias, anatomically complex and simplified models showed very similar dynamics; however, during fibrillatory arrhythmias, as activation wavelength decreased, the presence of fine-scale anatomical details appeared to marginally increase disorganisation of wavefronts during arrhythmias in the complex model. Although a small amount of clustering of reentrant rotor centres (filaments) around endocardial structures was witnessed in follow-up analysis (which slightly increased during fibrillation as rotor size decreased), this was significantly less than previously reported in large animals. Importantly, no anchoring of reentrant rotors was visibly identifiable in arrhythmia movies. These differences between tachy- and fibrillatory arrhythmias suggest that the relative size of reentrant rotors with respect to anatomical obstacles governs the influence of fine-scale anatomy in the maintenance of ventricular arrhythmias in the rabbit. In conclusion, our simulations suggest that fine-scale anatomical features play little apparent role in the maintenance of tachyarrhythmias in the rabbit ventricles and, contrary to experimental reports in larger animals, appear to play only a minor role in the maintenance of fibrillatory arrhythmias. These findings also have important implications in optimising the level of detail required in anatomical computational meshes frequently used in arrhythmia investigations.

Mechanisms of ventricular arrhythmias: a dynamical systems-based perspective

Defining the cellular electrophysiological mechanisms for ventricular tachyarrhythmias is difficult, given the wide array of potential mechanisms, ranging from abnormal automaticity to various types of reentry and kk activity. The degree of difficulty is increased further by the fact that any particular mechanism may be influenced by the evolving ionic and anatomic environments associated with many forms of heart disease. Consequently, static measures of a single electrophysiological characteristic are unlikely to be useful in establishing mechanisms. Rather, the dynamics of the electrophysiological triggers and substrates that predispose to arrhythmia development need to be considered. Moreover, the dynamics need to be considered in the context of a system, one that displays certain predictable behaviors, but also one that may contain seemingly stochastic elements. It also is essential to recognize that even the predictable behaviors of this complex nonlinear system are subject to small changes in the state of the system at any given time. Here we briefly review some of the short-, medium-, and long-term alterations of the electrophysiological substrate that accompany myocardial disease and their potential impact on the initiation and maintenance of ventricular arrhythmias. We also provide examples of cases in which small changes in the electrophysiological substrate can result in rather large differences in arrhythmia outcome. These results suggest that an interrogation of cardiac electrical dynamics is required to provide a meaningful assessment of the immediate risk for arrhythmia development and for evaluating the effects of putative antiarrhythmic interventions.

Using noise to determine cardiac restitution with memory

Variation in cardiac pacing cycles, as seen, for example, in heart rate variability, has been observed for decades. Contemporarily, various mathematical models have been constructed to investigate the electrical activity of paced cardiac cells. Yet there has not been a study of these cardiac models when there is variation in the pacing cycles such as noise. We present a method that uses the stochasticity of pacing cycles to determine approximate models of the dynamics of cardiac cells, and use these models to detect bifurcations to alternans.

Nonlinear dynamics in cardiology

The dynamics of many cardiac arrhythmias, as well as the nature of transitions between different heart rhythms, have long been considered evidence of nonlinear phenomena playing a direct role in cardiac arrhythmogenesis. In most types of cardiac disease, the pathology develops slowly and gradually, often over many years. In contrast, arrhythmias often occur suddenly. In nonlinear systems, sudden changes in qualitative dynamics can, counterintuitively, result from a gradual change in a system parameter-this is known as a bifurcation. Here, we review how nonlinearities in cardiac electrophysiology influence normal and abnormal rhythms and how bifurcations change the dynamics. In particular, we focus on the many recent developments in computational modeling at the cellular level that are focused on intracellular calcium dynamics. We discuss two areas where recent experimental and modeling work has suggested the importance of nonlinearities in calcium dynamics: repolarization alternans and pacemaker cell automaticity.

Uncovering the dynamics of cardiac systems using stochastic pacing and frequency domain analyses

Alternans of cardiac action potential duration (APD) is a well-known arrhythmogenic mechanism which results from dynamical instabilities. The propensity to alternans is classically investigated by examining APD restitution and by deriving APD restitution slopes as predictive markers. However, experiments have shown that such markers are not always accurate for the prediction of alternans. Using a mathematical ventricular cell model known to exhibit unstable dynamics of both membrane potential and Ca²⁺ cycling, we demonstrate that an accurate marker can be obtained by pacing at cycle lengths (CLs) varying randomly around a basic CL (BCL) and by evaluating the transfer function between the time series of CLs and APDs using an autoregressive-moving-average (ARMA) model. The first pole of this transfer function corresponds to the eigenvalue (λ(alt)) of the dominant eigenmode of the cardiac system, which predicts that alternans occurs when λ(alt) ≤ -1. For different BCLs, control values of λ(alt) were obtained using eigenmode analysis and compared to the first pole of the transfer function estimated using ARMA model fitting in simulations of random pacing protocols. In all versions of the cell model, this pole provided an accurate estimation of λ(alt). Furthermore, during slow ramp decreases of BCL or simulated drug application, this approach predicted the onset of alternans by extrapolating the time course of the estimated λ(alt). In conclusion, stochastic pacing and ARMA model identification represents a novel approach to predict alternans without making any assumptions about its ionic mechanisms. It should therefore be applicable experimentally for any type of myocardial cell.

Isoprenaline increases the slopes of restitution trajectory in the conscious rabbit with ischemic heart failure

Roughly speaking, restitution is the dependence of recovery time of cardiac electrical activity on heart rate. Increased restitution slope is theorized to be predictive of sudden death after heart injury such as from coronary artery occlusion (ischemia). Adrenaline analogs are known to increase restitution slope in normal hearts, but their effects in failing hearts are unknown. Twenty-six rabbits underwent coronary ligation (n = 15) or sham surgery (n = 11) and implantation of a lead in the heart for recording electrocardiograms. Eight weeks later, unanesthetized rabbits were given 0.25-2.0 ml of 1 μmol/L isoprenaline intravenously, which increased heart rate. Heart rate was quantified by time between QRS peaks (RR) and heart activity duration by R to T peak time (QTp). Ligated rabbits (n = 6) had lower ejection fraction than sham rabbits (n = 7, p < 0.0001) indicative of heart failure, but similar baseline RR (269 ± 15 vs 292 ± 23 ms, p = 0.07), QTp (104 ± 17 vs 91 ± 9 ms, p = 0.1), and isoprenaline-induced minimum RR (204 ± 11 vs 208 ± 6 ms, p = 0.4). The trajectory of QTp vs TQ plots displayed hysteresis and regions of negative slope. The slope of the positive slope region was >1 in ligated rabbits (1.27 ± 0.66) and <1 in sham rabbits (0.35 ± 0.14, p = 0.004). The absolute value of the negative slope was greater in ligated rabbits (- 0.81 ± 0.52 vs - 0.35 ± 0.14, p = 0.04). Isoprenaline increased heart rate and slopes of restitution trajectory in failing hearts. The dynamics of restitution trajectory may hold clues for sudden death in heart failure patients.

Toward prediction of the local onset of alternans in the heart

A beat-to-beat variation in the cardiac action potential duration is a phenomenon known as alternans. Alternans has been linked to ventricular fibrillation, and thus the ability to predict the onset of alternans could be clinically beneficial. Theoretically, it has been proposed that the slope of a restitution curve, which relates the duration of the action potential to the preceding diastolic interval, can predict the onset of alternans. Experimentally, however, this hypothesis has not been consistently proven, mainly because of the intrinsic complexity of the dynamics of cardiac tissue. It was recently shown that the restitution portrait, which combines several restitution curves simultaneously, is associated with the onset of alternans in isolated myocytes. Our main purpose in this study was to determine whether the restitution portrait is correlated with the onset of alternans in the heart, where the dynamics include a spatial complexity. We performed optical mapping experiments in isolated Langendorff-perfused rabbit hearts in which alternans was induced by periodic pacing at different frequencies, and identified the local onset of alternans, B(onset). We identified two regions of the heart: the area that exhibited alternans at B(onset) (1:1(alt)) and the area that did not (1:1). We constructed two-dimensional restitution portraits for the epicardial surface of the heart and measured the spatial distribution of three different slopes (the dynamic restitution slope, S(dyn)(RP), and two local S1-S2 slopes, S(12) and S(12)(max)) separately for these two regions. We found that the S(12) and S(12)(max) slopes differed significantly between the 1:1(alt) and 1:1 regions just before the onset of alternans, and S(dyn)(RP) slopes were statistically similar. In addition, we found that the slopes of the dynamic restitution curve S(dyn) were also statistically similar between these two regions. On the other hand, the quantitative values of all slopes were significantly different from the theoretically predicted value of one. These results demonstrate that the slopes measured in the restitution portrait correlate with the onset of alternans in the heart.

Short-term memory and electrical restitution in the canine transmural ventricle

Cardiac short-term memory is an intrinsic property of paced myocardium that reflects the influence of pacing history. Using an optical mapping method to record membrane voltage and intracellular calcium (Ca(2 +)(i)), this study investigated the properties and mechanisms of short-term memory in isolated and perfused canine wedge preparations. In addition to the dynamic and S1S2 pacing protocols, a perturbed downsweep pacing protocol was used to get a complete overview of the restitution portrait. Abrupt changes in basic cycle length (BCL) were applied to investigate the accommodation process of action potential duration (APD). The results showed unobvious differences of memory among the epi-, mid- and endo-myocytes, implying an insignificant memory-induced transient heterogeneity in APD across the transmural canine hearts. With the decrease of pacing rate S1, memory gradually elevated and achieved a maximum around 400 ms, and then reduced as S1 decreased further, indicating a non-monotonic relationship between memory and the pacing rate. After suppressing the Ca(2 +)(i) transient with ryanodine (3 µmol l(-1)), the accommodation process of APD to a new BCL significantly abbreviated (τ = 37.41 ± 4.42 stimuli before ryanodine, τ = 15.84 ± 4.74 stimuli after ryanodine, p < 0.01). Therefore, Ca(2 +)(i) cycling was suggested to play an important role in memory during dynamic pacing.

T-wave alternans and arrhythmogenesis in cardiac diseases

T-wave alternans, a manifestation of repolarization alternans at the cellular level, is associated with lethal cardiac arrhythmias and sudden cardiac death. At the cellular level, several mechanisms can produce repolarization alternans, including: (1) electrical restitution resulting from collective ion channel recovery, which usually occurs at fast heart rates but can also occur at normal heart rates when action potential is prolonged resulting in a short diastolic interval; (2) the transient outward current, which tends to occur at normal or slow heart rates; (3) the dynamics of early after depolarizations, which tends to occur during bradycardia; and (4) intracellular calcium cycling alternans through its interaction with membrane voltage. In this review, we summarize the cellular mechanisms of alternans arising from these different mechanisms, and discuss their roles in arrhythmogenesis in the setting of cardiac disease.

Vulnerable windows define susceptibility to alternans and spatial discordance

Electrophysiological alternans is a beat-to-beat alternation of the action potential duration and/or Ca(2+) transient amplitude and is linked to ventricular arrhythmias. We investigated the significance of various rate parameters under different experimental conditions with respect to alternans incidence and the propensity for spiral wave formation. Voltage and Ca(2+) were optically mapped in monolayers of neonatal rat ventricular myocytes. Alternans did not occur at physiological temperature, but its incidence increased significantly at lowered temperatures. Pacing cycle length for spatially concordant alternans onset (PCL(C)), PCL for spatially discordant alternans onset (PCL(D)), and minimum cycle length for loss of 1:1 or 2:2 capture (MCL) also significantly increased with lower temperature but in a way such that the differences between PCL(C) and MCL and between PCL(D) and MCL widened. These results provided the rationale to identify the former difference as the alternans vulnerable window (AVW; in ms) and the latter difference as the discordant alternans vulnerable window (AVW(D); in ms). Computational simulations showed that interventions that widen AVW, including altered Ca(2+) cycling and enhanced K(+) currents, also promote alternans, regardless of whether PCL(C) or MCL increased or decreased. The simulation results were confirmed experimentally by addition of the ATP-sensitive K(+) channel agonist pinacidil. Mathematical analysis provided a theoretical basis linking the size of AVW to the incidence of alternans. Finally, experiments showed that the size of AVW(D) is related to the incidence of spatially discordant alternans and, additionally, to the incidence of spiral wave formation. In conclusion, vulnerable windows can be defined that are strongly correlated with alternans incidence, spatial discordance, and spiral wave formation.

A new approach to measure the contribution of restitution to repolarization alternans

Several studies suggest link between repolarization alternans and arrhythmia. A potential target for minimization of alternans amplitude is pharmacological flattening of restitution function, which links a diastolic interval (DI) and subsequent action potential duration (APD). While our recent studies have shown that DI dependent restitution is not a necessary mechanism for alternans, in circumstances of nearly invariant activation intervals, restitution contributes to alternans. Determination of the degree to which restitution contributes to alternans during stable alternans, which requires determination of the gain between DI and APD, is not possible because it always is unity. We propose that the rate of change of alternans along the length of the tissue may provide an estimate of the degree to which restitution contributes to alternans amplitude. We conducted experiments with swine to demonstrate the above approach. In a linear strand of tissue, we paced such that DIs for successive activations were invariant at one end, which eliminates the restitution dependent mechanism of alternans at this end. Due to conduction delays, at the distal end, both restitution dependent and independent mechanisms manifest. Action potentials recorded from right ventricular endocardial tissue from swine (n = 3) showed an average difference in amplitudes of alternans between the two ends to be 11.99, 25.49, and 39.37 msec. Rates of change of alternans amplitude as a function of distance, computed using linear interpolation, were 0.36, 1.69 and 0.97. We propose that this rate of change may provide an indirect measure of degree of contribution of restitution to alternans and thus may be useful in evaluating therapeutic approaches to minimize its amplitude.