Model of intracellular calcium cycling in ventricular myocytes

Mechanisms of ventricular arrhythmias: from molecular fluctuations to electrical turbulence

Ventricular arrhythmias have complex causes and mechanisms. Despite extensive investigation involving many clinical, experimental, and computational studies, effective biological therapeutics are still very limited. In this article, we review our current understanding of the mechanisms of ventricular arrhythmias by summarizing the state of knowledge spanning from the molecular scale to electrical wave behavior at the tissue and organ scales and how the complex nonlinear interactions integrate into the dynamics of arrhythmias in the heart. We discuss the challenges that we face in synthesizing these dynamics to develop safe and effective novel therapeutic approaches.

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.

Are SR Ca content fluctuations or SR refractoriness the key to atrial cardiac alternans?: insights from a human atrial model

Despite the important role of electromechanical alternans in cardiac arrhythmogenesis, its molecular origin is not well understood. The appearance of calcium alternans has often been associated to fluctuations in the sarcoplasmic reticulum (SR) Ca loading. However, cytosolic calcium alternans observed without concurrent oscillations in the SR Ca content suggests an alternative mechanism related to a dysfunction in the dynamics of the ryanodine receptor (RyR2). We have investigated the effect of SR release refractoriness in the appearance of alternans, using a mathematical model of a single human atrial cell, based on the model by Nygren et al. ( 30 ), where we modified the dynamics of the RyR2 and of SR Ca release. The genesis of calcium alternans was studied stimulating the cell for different periods and values of the RyR2 recovery time from inactivation. At fast rates cytosolic calcium alternans were obtained without concurrent SR Ca content fluctuations. A transition from regular response to alternans was also observed, changing the recovery time from inactivation of the RyR2. This transition was found to be hysteretic, so for a given set of parameters different responses were observed. We then studied the relevance of RyR2 refractoriness for the generation of alternans, reproducing the same protocols as in recent experiments. In particular, restitution of Ca release during alternans was studied with a S1S2 protocol, obtaining a different response if the S2 stimulation was given after a long or a short release. We show that the experimental results can be explained by RyR2 refractoriness, arising from a slow RyR2 recovery from inactivation, stressing the role of the RyR2 in the genesis of alternans.

Translating intracellular calcium signaling into models

The rich experimental data on intracellular calcium has put theoreticians in an ideal position to derive models of intracellular calcium signaling. Over the last 25 years, a large number of modeling frameworks have been suggested. Here, I will review some of the milestones of intracellular calcium modeling with a special emphasis on calcium-induced calcium release (CICR) through inositol-1,4,5-trisphosphate and ryanodine receptors. I will highlight key features of CICR and how they are represented in models as well as the challenges that theoreticians face when translating our current understanding of calcium signals into equations. The selected examples demonstrate that a successful model provides mechanistic insights into the molecular machinery of the Ca²⁺ signaling toolbox and determines the contribution of local Ca²⁺ release to global Ca²⁺ patterns, which at the moment cannot be resolved experimentally.

Spatiotemporal dynamics of calcium-driven cardiac alternans

We investigate the dynamics of spatially discordant alternans (SDA) driven by an instability of intracellular calcium cycling using both amplitude equations [P. S. Skardal, A. Karma, and J. G. Restrepo, Phys. Rev. Lett. 108, 108103 (2012)] and ionic model simulations. We focus on the common case where the bidirectional coupling of intracellular calcium concentration and membrane voltage dynamics produces calcium and voltage alternans that are temporally in phase. We find that, close to the alternans bifurcation, SDA is manifested as a smooth wavy modulation of the amplitudes of both repolarization and calcium transient (CaT) alternans, similarly to the well-studied case of voltage-driven alternans. In contrast, further away from the bifurcation, the amplitude of CaT alternans jumps discontinuously at the nodes separating out-of-phase regions, while the amplitude of repolarization alternans remains smooth. We identify universal dynamical features of SDA pattern formation and evolution in the presence of those jumps. We show that node motion of discontinuous SDA patterns is strongly hysteretic even in homogeneous tissue due to the novel phenomenon of "unidirectional pinning": node movement can only be induced towards, but not away from, the pacing site in response to a change of pacing rate or physiological parameter. In addition, we show that the wavelength of discontinuous SDA patterns scales linearly with the conduction velocity restitution length scale, in contrast to the wavelength of smooth patterns that scales sublinearly with this length scale. Those results are also shown to be robust against cell-to-cell fluctuations due to the property that unidirectional node motion collapses multiple jumps accumulating in nodal regions into a single jump. Amplitude equation predictions are in good overall agreement with ionic model simulations. Finally, we briefly discuss physiological implications of our findings. In particular, we suggest that due to the tendency of conduction blocks to form near nodes, the presence of unidirectional pinning makes calcium-driven alternans potentially more arrhythmogenic than voltage-driven alternans.

Optical mapping of sarcoplasmic reticulum Ca2+ in the intact heart: ryanodine receptor refractoriness during alternans and fibrillation

RATIONALE

Sarcoplasmic reticulum (SR) Ca(2+) cycling is key to normal excitation-contraction coupling but may also contribute to pathological cardiac alternans and arrhythmia.

OBJECTIVE

To measure intra-SR free [Ca(2+)] ([Ca(2+)]SR) changes in intact hearts during alternans and ventricular fibrillation (VF).

METHODS AND RESULTS

Simultaneous optical mapping of Vm (with RH237) and [Ca(2+)]SR (with Fluo-5N AM) was performed in Langendorff-perfused rabbit hearts. Alternans and VF were induced by rapid pacing. SR Ca(2+) and action potential duration (APD) alternans occurred in-phase, but SR Ca(2+) alternans emerged first as cycle length was progressively reduced (217±10 versus 190±13 ms; P<0.05). Ryanodine receptor (RyR) refractoriness played a key role in the onset of SR Ca(2+) alternans, with SR Ca(2+) release alternans routinely occurring without changes in diastolic [Ca(2+)]SR. Sensitizing RyR with caffeine (200 μmol/L) significantly reduced the pacing threshold for both SR Ca(2+) and APD alternans (188±15 and 173±12 ms; P<0.05 versus baseline). Caffeine also reduced the magnitude of spatially discordant SR Ca(2+) alternans, but not APD alternans, the pacing threshold for discordance, or threshold for VF. During VF, [Ca(2+)]SR was high, but RyR remained nearly continuously refractory, resulting in minimal SR Ca(2+) release throughout VF.

CONCLUSIONS

In intact hearts, RyR refractoriness initiates SR Ca(2+) release alternans that can be amplified by diastolic [Ca(2+)]SR alternans and lead to APD alternans. Sensitizing RyR suppresses spatially concordant but not discordant SR Ca(2+) and APD alternans. Despite increased [Ca(2+)]SR during VF, SR Ca(2+) release was nearly continuously refractory. This novel method provides insight into SR Ca(2+) handling during cardiac alternans and arrhythmia.

Formation of spatially discordant alternans due to fluctuations and diffusion of calcium

Spatially discordant alternans (SDA) of action potential duration (APD) is a phenomenon where different regions of cardiac tissue exhibit an alternating sequence of APD that are out-of-phase. SDA is arrhythmogenic since it can induce spatial heterogeneity of refractoriness, which can cause wavebreak and reentry. However, the underlying mechanisms for the formation of SDA are not completely understood. In this paper, we present a novel mechanism for the formation of SDA in the case where the cellular instability leading to alternans is caused by intracellular calcium (Ca) cycling, and where Ca transient and APD alternans are electromechanically concordant. In particular, we show that SDA is formed when rapidly paced cardiac tissue develops alternans over many beats due to Ca accumulation in the sarcoplasmic reticulum (SR). The mechanism presented here relies on the observation that Ca cycling fluctuations dictate Ca alternans phase since the amplitude of Ca alternans is small during the early stages of pacing. Thus, different regions of a cardiac myocyte will typically develop Ca alternans which are opposite in phase at the early stages of pacing. These subcellular patterns then gradually coarsen due to interactions with membrane voltage to form steady state SDA of voltage and Ca on the tissue scale. This mechanism for SDA is distinct from well-known mechanisms that rely on conduction velocity restitution, and a Turing-like mechanism known to apply only in the case where APD and Ca alternans are electromechanically discordant. Furthermore, we argue that this mechanism is robust, and is likely to underlie a wide range of experimentally observed patterns of SDA.

Using dominant eigenvalue analysis to predict formation of alternans in the heart

Ventricular fibrillation at the whole heart level is often preceded by the alternation of action potential duration (APD), i.e., alternans, at the cellular level. As proven in many experiments, traditional approaches based on the slope of the restitution curve have not been successful in predicting alternans formation. Recently, a technique has been theoretically developed based on dominant eigenvalue analysis to predict alternans formation in isolated cardiac myocytes. Here, we aimed to demonstrate that this technique can be applied to predict alternans formation at the whole heart level. Optical mapping was performed in Langendorff-perfused hearts from New Zealand white rabbits (n = 4), which were paced at decreasing basic cycle lengths to introduce APD alternans. In each heart, the basic cycle length corresponding to the local onset of alternans, B(onset), was determined and two regions of the heart were identified at B(onset): one region which exhibited alternans (1:1(alt)) and one which did not (1:1). Corresponding two-dimensional eigenvalue (λ) maps were generated using principal component analysis by analyzing action potentials after short perturbations from the steady state, and mean eigenvalues (λ[over ¯]) were calculated separately for the 1:1 and 1:1(alt) regions. We demonstrated that λ[over ¯] calculated at B(onset) was significantly different (p<0.05) between the two regions. Our results suggest that this dominant eigenvalue technique can be used to successfully predict the local alternans formation in the heart.

Cardiac electrophysiological dynamics from the cellular level to the organ level

Cardiac alternans describes contraction of the ventricles in a strong-weak-strong-weak sequence at a constant pacing frequency. Clinically, alternans manifests as alternation of the T-wave on the ECG and predisposes individuals to arrhythmia and sudden cardiac death. In this review, we focus on the fundamental dynamical mechanisms of alternans and show how alternans at the cellular level underlies alternans in the tissue and on the ECG. A clear picture of dynamical mechanisms underlying alternans is important to allow development of effective anti-arrhythmic strategies.

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.

Calcium alternans in cardiac myocytes: order from disorder

Calcium alternans is associated with T-wave alternans and pulsus alternans, harbingers of increased mortality in the setting of heart disease. Recent experimental, computational, and theoretical studies have led to new insights into the mechanisms of Ca alternans, specifically how disordered behaviors dominated by stochastic processes at the subcellular level become organized into ordered periodic behaviors. In this article, we summarize the recent progress in this area, outlining a holistic theoretical framework in which the complex effects of Ca cycling proteins on Ca alternans are linked to three key properties of the cardiac Ca cycling network: randomness, refractoriness, and recruitment. We also illustrate how this '3R theory' can reconcile many seemingly contradictory experimental observations.

Bifurcation theory and cardiac arrhythmias

In this paper we review two types of dynamic behaviors defined by the bifurcation theory that are found to be particularly useful in describing two forms of cardiac electrical instabilities that are of considerable importance in cardiac arrhythmogenesis. The first is action potential duration (APD) alternans with an underlying dynamics consistent with the period doubling bifurcation theory. This form of electrical instability could lead to spatially discordant APD alternans leading to wavebreak and reentrant form of tachyarrhythmias. Factors that modulate the APD alternans are discussed. The second form of bifurcation of importance to cardiac arrhythmogenesis is the Hopf-homoclinic bifurcation that adequately describes the dynamics of the onset of early afterdepolarization (EAD)-mediated triggered activity (Hopf) that may cause ventricular tachycardia and ventricular fibrillation (VT/VF respectively). The self-termination of the triggered activity is compatible with the homoclinic bifurcation. Ionic and intracellular calcium dynamics underlying these dynamics are discussed using available experimental and simulation data. The dynamic analysis provides novel insights into the mechanisms of VT/VF, a major cause of sudden cardiac death in the US.

Dependency of calcium alternans on ryanodine receptor refractoriness

Background

Rapid pacing rates induce alternations in the cytosolic calcium concentration caused by fluctuations in calcium released from the sarcoplasmic reticulum (SR). However, the relationship between calcium alternans and refractoriness of the SR calcium release channel (RyR2) remains elusive.

Methodology/Principal Findings

To investigate how ryanodine receptor (RyR2) refractoriness modulates calcium handling on a beat-to-beat basis using a numerical rabbit cardiomyocyte model. We used a mathematical rabbit cardiomyocyte model to study the beat-to-beat calcium response as a function of RyR2 activation and inactivation. Bi-dimensional maps were constructed depicting the beat-to-beat response. When alternans was observed, a novel numerical clamping protocol was used to determine whether alternans was caused by oscillations in SR calcium loading or by RyR2 refractoriness. Using this protocol, we identified regions of RyR2 gating parameters where SR calcium loading or RyR2 refractoriness underlie the induction of calcium alternans, and we found that at the onset of alternans both mechanisms contribute. At low inactivation rates of the RyR2, calcium alternans was caused by alternation in SR calcium loading, while at low activation rates it was caused by alternation in the level of available RyR2s.

Conclusions/Significance

We have mapped cardiomyocyte beat-to-beat responses as a function of RyR2 activation and inactivation, identifying domains where SR calcium load or RyR2 refractoriness underlie the induction of calcium alternans. A corollary of this work is that RyR2 refractoriness due to slow recovery from inactivation can be the cause of calcium alternans even when alternation in SR calcium load is present.

Decoding myocardial Ca²⁺ signals across multiple spatial scales: a role for sensitivity analysis

Numerous studies have employed mathematical modeling to quantitatively understand release of Ca(2+) from the sarcoplasmic reticulum (SR) in the heart. Models have been used to investigate physiologically important phenomena such as triggering of SR Ca(2+) release by Ca(2+) entry across the cell membrane and spontaneous leak of Ca(2+) from the SR in quiescent heart cells. In this review we summarize studies that have modeled myocardial Ca(2+) at different spatial scales: the sub-cellular level, the cellular level, and the multicellular level. We discuss each category of models from the standpoint of parameter sensitivity analysis, a common simulation procedure that can generate quantitative, comprehensive predictions about how changes in conditions influence model output. We propose that this is a useful perspective for conceptualizing models, in part because a sensitivity analysis requires the investigator to define the relevant parameters and model outputs. This procedure therefore helps to illustrate the capabilities and limitations of each model. We further suggest that in future studies, sensitivity analyses will aid in simplifying complex models and in suggesting experiments to differentiate between competing models built with different assumptions. We conclude with a discussion of unresolved questions that are likely to be addressed over the next several years.

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.

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.

Calcium alternans in a couplon network model of ventricular myocytes: role of sarcoplasmic reticulum load

Intracellular calcium (Ca) alternans in cardiac myocytes have been shown in many experimental studies, and the mechanisms remain incompletely understood. We recently developed a "3R theory" that links Ca sparks to whole cell Ca alternans through three critical properties: randomness of Ca sparks; recruitment of a Ca spark by neighboring Ca sparks; and refractoriness of Ca release units. In this study, we used computer simulation of a physiologically detailed mathematical model of a ventricular myocyte couplon network to study how sarcoplasmic reticulum (SR) Ca load and other physiological parameters, such as ryanodine receptor sensitivity, SR uptake rate, Na-Ca exchange strength, and Ca buffer levels affect Ca alternans in the context of 3R theory. We developed a method to calculate the parameters used in the 3R theory (i.e., the primary spark rate and the recruitment rate) from the physiologically detailed Ca cycling model and paced the model periodically to elicit Ca alternans. We show that alternans only occurs for an intermediate range of the SR Ca load, and the underlying mechanism can be explained via its effects on the 3Rs. Furthermore, we show that altering the physiological parameters not only directly changes the 3Rs but also alters the SR Ca load, having an indirect effect on the 3Rs as well. Therefore, our present study links the SR Ca load and other physiological parameters to whole cell Ca alternans through the framework of the 3R theory, providing a general mechanistic understanding of Ca alternans in ventricular myocytes.

Non-linear dynamics of cardiac alternans: subcellular to tissue-level mechanisms of arrhythmia

Cardiac repolarization alternans is a rhythm disturbance of the heart in which rapid stimulation elicits a beat-to-beat alternation in the duration of action potentials and magnitude of intracellular calcium transients in individual cardiac myocytes. Although this phenomenon has been identified as a potential precursor to dangerous reentrant arrhythmias and sudden cardiac death, significant uncertainty remains regarding its mechanism and no clinically practical means of halting its occurrence or progression currently exists. Cardiac alternans has well-characterized tissue, cellular, and subcellular manifestations, the mechanisms and interplay of which are an active area of research.

Computational modeling and numerical methods for spatiotemporal calcium cycling in ventricular myocytes

Intracellular calcium (Ca) cycling dynamics in cardiac myocytes is regulated by a complex network of spatially distributed organelles, such as sarcoplasmic reticulum (SR), mitochondria, and myofibrils. In this study, we present a mathematical model of intracellular Ca cycling and numerical and computational methods for computer simulations. The model consists of a coupled Ca release unit (CRU) network, which includes a SR domain and a myoplasm domain. Each CRU contains 10 L-type Ca channels and 100 ryanodine receptor channels, with individual channels simulated stochastically using a variant of Gillespie’s method, modified here to handle time-dependent transition rates. Both the SR domain and the myoplasm domain in each CRU are modeled by 5 × 5 × 5 voxels to maintain proper Ca diffusion. Advanced numerical algorithms implemented on graphical processing units were used for fast computational simulations. For a myocyte containing 100 × 20 × 10 CRUs, a 1-s heart time simulation takes about 10 min of machine time on a single NVIDIA Tesla C2050. Examples of simulated Ca cycling dynamics, such as Ca sparks, Ca waves, and Ca alternans, are shown.

Data-based theoretical identification of subcellular calcium compartments and estimation of calcium dynamics in cardiac myocytes

In cardiac cells, Ca(2+) release flux (J(rel)) via ryanodine receptors (RyRs) from the sarcoplasmic reticulum (SR) has a complex effect on the action potential (AP). Coupling between J(rel) and the AP occurs via L-type Ca(2+) channels (I(Ca)) and the Na(+)/Ca(2+) exchanger (I(NCX)). We used a combined experimental and modelling approach to study interactions between J(rel), I(Ca) and I(NCX) in porcine ventricular myocytes.We tested the hypothesis that during normal uniform J(rel), the interaction between these fluxes can be represented as occurring in two myoplasmic subcompartments for Ca(2+) distribution, one (T-space) associated with RyR and enclosed by the junctional portion of the SR membrane and corresponding T-tubular portion of the sarcolemma, the other (M-space) encompassing the rest of the myoplasm. I(Ca) and I(NCX) were partitioned into subpopulations in the T-space and M-space sarcolemma. We denoted free Ca(2+) concentrations in T-space and M-space Ca(t) and Ca(m), respectively. Experiments were designed to allow separate measurements of I(Ca) and I(NCX) as a function of J(rel). Inclusion of T-space in themodel allowed us to reproduce in silico the following important experimental results: (1) hysteresis of I(NCX) dependence on Ca(m); (2) delay between peak I(NCX) and peak Ca(m) during caffeine application protocol; (3) delay between I(NCX) and Ca(m) during Ca(2+)-induced-Ca(2+)-release; (4) rapid I(Ca) inactivation (within 2 ms) due to J(rel), with magnitude graded as a function of the SR Ca(2+) content; (5) time delay between I(Ca) inactivation due to J(rel) and Ca(m). Partition of 25% NCX in T-space and 75% in M-space provided the best fit to the experimental data. Measured Ca(m) and I(Ca) or I(NCX) were used as input to the model for estimating Ca(t). The actual model-computed Ca(t), obtained by simulating specific experimental protocols, was used as a gold standard for comparison. The model predicted peak Ca(t) in the range of 6–25 μM, with time to equilibrium of Ca(t) with Ca(m) of ~350 ms. These Ca(t) values are in the range of LCC and RyR sensitivity to Ca(2+). An increase of the SR Ca(2+) load increased the time to equilibrium. The I(Ca)-based estimation method was most accurate during the ascending phase of Ca(t). The I(NCX)-based method provided a good estimate for the descending phase of Ca(t). Thus, application of both methods in combination provides the best estimate of the entire Ca(t) time course.

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.

Cardiac cellular coupling and the spread of early instabilities in intracellular Ca2+

Recent experimental and modeling studies demonstrate the fine spatial scale, complex nature, and independent contribution of Ca(2+) dynamics as a proarrhythmic factor in the heart. The mechanism of progression of cell-level Ca(2+) instabilities, known as alternans, to tissue-level arrhythmias is not well understood. Because gap junction coupling dictates cardiac syncytial properties, we set out to elucidate its role in the spatiotemporal evolution of Ca(2+) instabilities. We experimentally perturbed cellular coupling in cardiac syncytium in vitro. Coupling was quantified by fluorescence recovery after photobleaching, and related to function, including subtle fine-scale Ca(2+) alternans, captured by optical mapping. Conduction velocity and threshold for alternans monotonically increased with coupling. Lower coupling enhanced Ca(2+) alternans amplitude, but the spatial spread of early (<2 Hz) alternation was the greatest under intermediate (not low) coupling. This nonmonotonic relationship was closely matched by the percent of samples exhibiting large-scale alternans at higher pacing rates. Computer modeling corroborated these experimental findings for strong but not weak electromechanical (voltage-Ca(2+)) coupling, and offered mechanistic insight. In conclusion, using experimental and modeling approaches, we reveal a general mechanism for the spatial spread of subtle cellular Ca(2+) alternans that relies on a combination of gap-junctional and voltage-Ca(2+) coupling.

Unidirectional pinning and hysteresis of spatially discordant alternans in cardiac tissue

Spatially discordant alternans is a widely observed pattern of voltage and calcium signals in cardiac tissue that can precipitate lethal cardiac arrhythmia. Using spatially coupled iterative maps of the beat-to-beat dynamics, we explore this pattern's dynamics in the regime of a calcium-dominated period-doubling instability at the single-cell level. We find a novel nonlinear bifurcation associated with the formation of a discontinuous jump in the amplitude of calcium alternans at nodes separating discordant regions. We show that this jump unidirectionally pins nodes by preventing their motion away from the pacing site following a pacing rate decrease but permitting motion towards this site following a rate increase. This unidirectional pinning leads to strongly history-dependent node motion that is strongly arrhythmogenic.

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.

Mathematical modeling mechanisms of arrhythmias in transgenic mouse heart overexpressing TNF-α

Transgenic mice overexpressing tumor necrosis factor-α (TNF-α mice) possess many of the features of human heart failure, such as dilated cardiomyopathy, impaired Ca(2+) handling, arrhythmias, and decreased survival. Although TNF-α mice have been studied extensively with a number of experimental methods, the mechanisms of heart failure are not completely understood. We created a mathematical model that reproduced experimentally observed changes in the action potential (AP) and Ca(2+) handling of isolated TNF-α mice ventricular myocytes. To study the contribution of the differences in ion currents, AP, Ca(2+) handling, and intercellular coupling to the development of arrhythmias in TNF-α mice, we further created several multicellular model tissues with combinations of wild-type (WT)/reduced gap junction conductance, WT/prolonged AP, and WT/decreased Na(+) current (I(Na)) amplitude. All model tissues were examined for susceptibility to Ca(2+) alternans, AP propagation block, and reentry. Our modeling results demonstrated that, similar to experimental data in TNF-α mice, Ca(2+) alternans in TNF-α tissues developed at longer basic cycle lengths. The greater susceptibility to Ca(2+) alternans was attributed to the prolonged AP, resulting in larger inactivation of I(Na), and to the decreased SR Ca(2+) uptake and corresponding smaller SR Ca(2+) load. Simulations demonstrated that AP prolongation induces an increased susceptibility to AP propagation block. Programmed stimulation of the model tissues with a premature impulse showed that reduced gap junction conduction increased the vulnerable window for initiation reentry, supporting the idea that reduced intercellular coupling is the major factor for reentrant arrhythmias in TNF-α mice.

Multi-scale modeling in biology: how to bridge the gaps between scales?

Human physiological functions are regulated across many orders of magnitude in space and time. Integrating the information and dynamics from one scale to another is critical for the understanding of human physiology and the treatment of diseases. Multi-scale modeling, as a computational approach, has been widely adopted by researchers in computational and systems biology. A key unsolved issue is how to represent appropriately the dynamical behaviors of a high-dimensional model of a lower scale by a low-dimensional model of a higher scale, so that it can be used to investigate complex dynamical behaviors at even higher scales of integration. In the article, we first review the widely-used different modeling methodologies and their applications at different scales. We then discuss the gaps between different modeling methodologies and between scales, and discuss potential methods for bridging the gaps between scales.

A mathematical model of spontaneous calcium release in cardiac myocytes

In cardiac myocytes, calcium (Ca) can be released from the sarcoplasmic reticulum independently of Ca influx from voltage-dependent membrane channels. This efflux of Ca, referred to as spontaneous Ca release (SCR), is due to Ryanodine receptor fluctuations, which can induce spontaneous Ca sparks, which propagate to form Ca waves. This release of Ca can then induce delayed after-depolarizations (DADs), which can lead to arrhythmogenic-triggered activity in the heart. However, despite its importance, to date there is no mathematical model of SCR that accounts for experimentally observed features of subcellular Ca. In this article, we present an experimentally based model of SCR that reproduces the timing distribution of spontaneous Ca sparks and key features of the propagation of Ca waves emanating from these spontaneous sparks. We have coupled this model to an ionic model for the rabbit ventricular action potential to simulate SCR within several thousand cells in cardiac tissue. We implement this model to study the formation of an ectopic beat on a cable of cells that exhibit SCR-induced DADs.

Understanding cardiac alternans: a piecewise linear modeling framework

Cardiac alternans is a beat-to-beat alternation in action potential duration (APD) and intracellular calcium (Ca(2+)) cycling seen in cardiac myocytes under rapid pacing that is believed to be a precursor to fibrillation. The cellular mechanisms of these rhythms and the coupling between cellular Ca(2+) and voltage dynamics have been extensively studied leading to the development of a class of physiologically detailed models. These have been shown numerically to reproduce many of the features of myocyte response to pacing, including alternans, and have been analyzed mathematically using various approximation techniques that allow for the formulation of a low dimensional map to describe the evolution of APDs. The seminal work by Shiferaw and Karma is of particular interest in this regard [Shiferaw, Y. and Karma, A., "Turing instability mediated by voltage and calcium diffusion in paced cardiac cells," Proc. Natl. Acad. Sci. U.S.A. 103, 5670-5675 (2006)]. Here, we establish that the key dynamical behaviors of the Shiferaw-Karma model are arranged around a set of switches. These are shown to be the main elements for organizing the nonlinear behavior of the model. Exploiting this observation, we show that a piecewise linear caricature of the Shiferaw-Karma model, with a set of appropriate switching manifolds, can be constructed that preserves the physiological interpretation of the original model while being amenable to a systematic mathematical analysis. In illustration of this point, we formulate the dynamics of Ca(2+) cycling (in response to pacing) and compute the properties of periodic orbits in terms of a stroboscopic map that can be constructed without approximation. Using this, we show that alternans emerge via a period-doubling instability and track this bifurcation in terms of physiologically important parameters. We also show that when coupled to a spatially extended model for Ca(2+) transport, the model supports spatially varying patterns of alternans. We analyze the onset of this instability with a generalization of the master stability approach to accommodate the nonsmooth nature of our system.

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.

Modeling CICR in rat ventricular myocytes: voltage clamp studies

Background

The past thirty-five years have seen an intense search for the molecular mechanisms underlying calcium-induced calcium-release (CICR) in cardiac myocytes, with voltage clamp (VC) studies being the leading tool employed. Several VC protocols including lowering of extracellular calcium to affect Ca²⁺ loading of the sarcoplasmic reticulum (SR), and administration of blockers caffeine and thapsigargin have been utilized to probe the phenomena surrounding SR Ca²⁺ release. Here, we develop a deterministic mathematical model of a rat ventricular myocyte under VC conditions, to better understand mechanisms underlying the response of an isolated cell to calcium perturbation. Motivation for the study was to pinpoint key control variables influencing CICR and examine the role of CICR in the context of a physiological control system regulating cytosolic Ca²⁺ concentration ([Ca²⁺]_myo).

Methods

The cell model consists of an electrical-equivalent model for the cell membrane and a fluid-compartment model describing the flux of ionic species between the extracellular and several intracellular compartments (cell cytosol, SR and the dyadic coupling unit (DCU), in which resides the mechanistic basis of CICR). The DCU is described as a controller-actuator mechanism, internally stabilized by negative feedback control of the unit's two diametrically-opposed Ca²⁺ channels (trigger-channel and release-channel). It releases Ca²⁺ flux into the cyto-plasm and is in turn enclosed within a negative feedback loop involving the SERCA pump, regulating[Ca²⁺]_myo.

Results

Our model reproduces measured VC data published by several laboratories, and generates graded Ca²⁺ release at high Ca²⁺ gain in a homeostatically-controlled environment where [Ca²⁺]_myo is precisely regulated. We elucidate the importance of the DCU elements in this process, particularly the role of the ryanodine receptor in controlling SR Ca²⁺ release, its activation by trigger Ca²⁺, and its refractory characteristics mediated by the luminal SR Ca²⁺ sensor. Proper functioning of the DCU, sodium-calcium exchangers and SERCA pump are important in achieving negative feedback control and hence Ca²⁺ homeostasis.

Conclusions

We examine the role of the above Ca²⁺ regulating mechanisms in handling various types of induced disturbances in Ca²⁺ levels by quantifying cellular Ca²⁺ balance. Our model provides biophysically-based explanations of phenomena associated with CICR generating useful and testable hypotheses.

The role of mitochondria for the regulation of cardiac alternans

Electro-mechanical and Ca alternans is a beat-to-beat alternation of action potential duration, contraction strength and Ca transient amplitude observed in cardiac myocytes at regular stimulation frequency. Ca alternans is a multifactorial process that is causally linked to cardiac arrhythmias. At the cellular level, conditions that increase fractional release from the sarcoplasmic reticulum or reduce diastolic Ca sequestration favor the occurrence of alternans. Mitochondria play a significant role in cardiac excitation–contraction coupling and Ca signaling by providing the energy for contraction and ATP-dependent processes and possibly by serving as Ca buffering organelles. Here we tested the hypothesis that impairment of mitochondrial function generates conditions that favor the occurrence of Ca alternans. Alternans were elicited by electrical pacing (>1 Hz) in single cat atrial myocytes and intracellular Ca ([Ca]_i) was measured with the fluorescent Ca indicator Indo-1. The degree of alternans was quantified as the alternans ratio (AR = 1 − S/L, where S/L is the ratio of the small to the large amplitude of a pair of alternating Ca transients). Dissipation of mitochondrial membrane potential (with FCCP) as well as inhibition of mitochondrial F₁/F₀-ATP synthase (oligomycin), electron transport chain (rotenone, antimycin, CN⁻), Ca-dependent dehydrogenases and mitochondrial Ca uptake or extrusion, all enhanced AR and lowered the threshold for the occurrence of Ca alternans. The data indicate that impairment of mitochondrial function adversely affects cardiac Ca cycling leading to proarrhythmic Ca alternans.

Ca2+ alternans in a cardiac myocyte model that uses moment equations to represent heterogeneous junctional SR Ca2+

Multiscale whole-cell models that accurately represent local control of Ca2+-induced Ca2+ release in cardiac myocytes can reproduce high-gain Ca2+ release that is graded with changes in membrane potential. Using a recently introduced formalism that represents heterogeneous local Ca2+ using moment equations, we present a model of cardiac myocyte Ca2+ cycling that exhibits alternating sarcoplasmic reticulum (SR) Ca2+ release when periodically stimulated by depolarizing voltage pulses. The model predicts that the distribution of junctional SR [Ca2+] across a large population of Ca2+ release units is distinct on alternating cycles. Load-release and release-uptake functions computed from this model give insight into how Ca2+ fluxes and stimulation frequency combine to determine the presence or absence of Ca2+ alternans. Our results show that the conditions for the onset of Ca2+ alternans cannot be explained solely by the steepness of the load-release function, but that changes in the release-uptake process also play an important role. We analyze the effect of the junctional SR refilling time constant on Ca2+ alternans and conclude that physiologically realistic models of defective Ca2+ cycling must represent the dynamics of heterogeneous junctional SR [Ca2+] without assuming rapid equilibration of junctional and network SR [Ca2+].

Feedback-control induced pattern formation in cardiac myocytes: a mathematical modeling study

Cardiac alternans is a dangerous rhythm disturbance of the heart, in which rapid stimulation elicits a beat-to-beat alternation in the action potential duration (APD) and calcium (Ca) transient amplitude of individual myocytes. Recently, "subcellular alternans", in which the Ca transients of adjacent regions within individual myocytes alternate out-of-phase, has been observed. A previous theoretical study suggested that subcellular alternans may result during static pacing from a Turing-type symmetry breaking instability, but this was only predicted in a subset of cardiac myocytes (with negative Ca to voltage (Ca-->V(m)) coupling) and has never been directly verified experimentally. A recent experimental study, however, showed that subcellular alternans is dynamically induced in the remaining subset of myocytes during pacing with a simple feedback control algorithm ("alternans control"). Here we show that alternans control pacing changes the effective coupling between the APD and the Ca transient (V(m)-->Ca coupling), such that subcellular alternans is predicted to occur by a Turing instability in cells with positive Ca-->V(m) coupling. In addition to strengthening the understanding of the proposed mechanism for subcellular alternans formation, this work (in concert with previous theoretical and experimental results) illuminates subcellular alternans as a striking example of a biological Turing instability in which the diffusing morphogens can be clearly identified.

Models of cardiac excitation-contraction coupling in ventricular myocytes

Mathematical and computational modeling of cardiac excitation-contraction coupling has produced considerable insights into how the heart muscle contracts. With the increase in biophysical and physiological data available, the modeling has become more sophisticated with investigations spanning in scale from molecular components to whole cells. These modeling efforts have provided insight into cardiac excitation-contraction coupling that advanced and complemented experimental studies. One goal is to extend these detailed cellular models to model the whole heart. While this has been done with mechanical and electrophysiological models, the complexity and fast time course of calcium dynamics have made inclusion of detailed calcium dynamics in whole heart models impractical. Novel methods such as the probability density approach and moment closure technique which increase computational efficiency might make this tractable.

Spark-induced sparks as a mechanism of intracellular calcium alternans in cardiac myocytes

RATIONALE

Intracellular calcium (Ca) alternans has been widely studied in cardiac myocytes and tissue, yet the underlying mechanism remains controversial.

OBJECTIVE

In this study, we used computational modeling and simulation to study how randomly occurring Ca sparks interact collectively to result in whole-cell Ca alternans.

METHODS AND RESULTS

We developed a spatially distributed intracellular Ca cycling model in which Ca release units (CRUs) are locally coupled by Ca diffusion throughout the myoplasm and sarcoplasmic reticulum (SR) network. Ca sparks occur randomly in the CRU network when periodically paced with a clamped voltage waveform, but Ca alternans develops as the pacing speeds up. Combining computational simulation with theoretical analysis, we show that Ca alternans emerges as a collective behavior of Ca sparks, determined by 3 critical properties of the CRU network from which Ca sparks arise: "randomness" (of Ca spark activation), "refractoriness" (of a CRU after a Ca spark), and "recruitment" (Ca sparks inducing Ca sparks in adjacent CRUs). We also show that the steep nonlinear relationship between fractional SR Ca release and SR Ca load arises naturally as a collective behavior of Ca sparks, and Ca alternans can occur even when SR Ca is held constant.

CONCLUSIONS

We present a general theory for the mechanisms of intracellular Ca alternans, which mechanistically links Ca sparks to whole-cell Ca alternans, and is applicable to Ca alternans in both physiological and pathophysiological conditions.

Uniqueness and stability of action potential models during rest, pacing, and conduction using problem-solving environment

Development and application of physiologically detailed dynamic models of the action potential (AP) and Ca2+ cycling in cardiac cells is a rapidly growing aspect of computational cardiac electrophysiology. Given the large scale of the nonlinear system involved, questions were recently raised regarding reproducibility, numerical stability, and uniqueness of model solutions, as well as ability of the model to simulate AP propagation in multicellular configurations. To address these issues, we reexamined ventricular models of myocyte AP developed in our laboratory with the following results. 1), Recognizing that the model involves a system of differential-algebraic equations, a procedure is developed for estimating consistent initial conditions that insure uniqueness and stability of the solution. 2), Model parameters that can be used to modify these initial conditions according to experimental values are identified. 3), A convergence criterion for steady-state solution is defined based on tracking the incremental contribution of each ion species to the membrane voltage. 4), Singularities in state variable formulations are removed analytically. 5), A biphasic current stimulus is implemented to completely eliminate stimulus artifact during long-term pacing over a broad range of frequencies. 6), Using the AP computed based on 1-5 above, an efficient scheme is developed for computing propagation in multicellular models.

Spatiotemporal intracellular calcium dynamics during cardiac alternans

Cellular calcium transient alternans are beat-to-beat alternations in the peak cytosolic calcium concentration exhibited by cardiac cells during rapid electrical stimulation or under pathological conditions. Calcium transient alternans promote action potential duration alternans, which have been linked to the onset of life-threatening ventricular arrhythmias. Here we use a recently developed physiologically detailed mathematical model of ventricular myocytes to investigate both stochastic and deterministic aspects of intracellular calcium dynamics during alternans. The model combines a spatially distributed description of intracellular calcium cycling, where a large number of calcium release units are spatially distributed throughout the cell, with a full set of ionic membrane currents. The results demonstrate that ion channel stochasticity at the level of single calcium release units can influence the whole-cell alternans dynamics by causing phase reversals over many beats during fixed frequency pacing close to the alternans bifurcation. They also demonstrate the existence of a wide range of dynamical states. Depending on the sign and magnitude of calcium-voltage coupling, calcium alternans can be spatially synchronized or desynchronized, in or out of phase with action potential duration alternans, and the node separating out-of-phase regions of calcium alternans can be expelled from or trapped inside the cell. This range of states is found to be larger than previously anticipated by including a robust global attractor where calcium alternans can be spatially synchronized but out of phase with action potential duration alternans. The results are explained by a combined theoretical analysis of alternans stability and node motion using general iterative maps of the beat-to-beat dynamics and amplitude equations.

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

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

K+ current changes account for the rate dependence of the action potential in the human atrial myocyte

Ongoing investigation of the electrophysiology and pathophysiology of the human atria requires an accurate representation of the membrane dynamics of the human atrial myocyte. However, existing models of the human atrial myocyte action potential do not accurately reproduce experimental observations with respect to the kinetics of key repolarizing currents or rate dependence of the action potential and fail to properly enforce charge conservation, an essential characteristic in any model of the cardiac membrane. In addition, recent advances in experimental methods have resulted in new data regarding the kinetics of repolarizing currents in the human atria. The goal of this study was to develop a new model of the human atrial action potential, based on the Nygren et al. model of the human atrial myocyte and newly available experimental data, that ensures an accurate representation of repolarization processes and reproduction of action potential rate dependence and enforces charge conservation. Specifically, the transient outward K(+) current (I(t)) and ultrarapid rectifier K(+) current (I(Kur)) were newly formulated. The inwardly recitifying K(+) current (I(K1)) was also reanalyzed and implemented appropriately. Simulations of the human atrial myocyte action potential with this new model demonstrated that early repolarization is dependent on the relative conductances of I(t) and I(Kur), whereas densities of both I(Kur) and I(K1) underlie later repolarization. In addition, this model reproduces experimental measurements of rate dependence of I(t), I(Kur), and action potential duration. This new model constitutes an improved representation of excitability and repolarization reserve in the human atrial myocyte and, therefore, provides a useful computational tool for future studies involving the human atrium in both health and disease.

Redox modification of ryanodine receptors underlies calcium alternans in a canine model of sudden cardiac death

AIMS

Although cardiac alternans is a known predictor of lethal arrhythmias, its underlying causes remain largely undefined in disease settings. The potential role of, and mechanisms responsible for, beat-to-beat alternations in the amplitude of systolic Ca(2+) transients (Ca(2+) alternans) was investigated in a canine post-myocardial infarction (MI) model of sudden cardiac death (SCD).

METHODS AND RESULTS

Post-MI dogs had preserved left ventricular (LV) function and susceptibility to ventricular fibrillation (VF) during exercise. LV wedge preparations from VF dogs were more susceptible to action potential (AP) alternans and the frequency-dependence of Ca(2+) alternans was shifted towards slower rates in myocytes isolated from VF dogs relative to controls. In both groups of cells, cytosolic Ca(2+) transients ([Ca(2+)](c)) alternated in phase with changes in diastolic Ca(2+) in sarcoplasmic reticulum ([Ca(2+)](SR)), but the dependence of [Ca(2+)](c) amplitude on [Ca(2+)](SR) was steeper in VF cells. Abnormal ryanodine receptor (RyR) function in VF cells was indicated by increased fractional Ca(2+) release for a given amplitude of Ca(2+) current and elevated diastolic RyR-mediated SR Ca(2+) leak. SR Ca(2+) uptake activity did not differ between VF and control cells. VF myocytes had an increased rate of reactive oxygen species production and increased RyR oxidation. Treatment of VF myocytes with reducing agents normalized parameters of Ca(2+) handling and shifted the threshold of Ca(2+) alternans to higher frequencies.

CONCLUSION

Redox modulation of RyRs promotes generation of Ca(2+) alternans by enhancing the steepness of the Ca(2+) release-load relationship and thereby providing a substrate for post-MI arrhythmias.

Cardiac alternans induced by fibroblast-myocyte coupling: mechanistic insights from computational models

Recent experimental studies have shown that fibroblasts can electrotonically couple to myocytes via gap junctions. In this study, we investigated how this coupling affects action potential and intracellular calcium (Ca(i)) cycling dynamics in simulated fibroblast-myocyte pairs and in two-dimensional tissue with random fibroblast insertions. We show that a fibroblast coupled with a myocyte generates a gap junction current flowing to the myocyte with two main components: an early pulse of transient outward current, similar to the fast transient outward current, and a later background current during the repolarizing phase. Depending on the relative prominence of the two components, fibroblast-myoycte coupling can 1) prolong or shorten action potential duration (APD), 2) promote or suppress APD alternans due to steep APD restitution (voltage driven) and also result in a novel mechanism of APD alternans at slow heart rates, 3) promote Ca(i)-driven alternans and electromechanically discordant alternans, and 4) promote spatially discordant alternans by two mechanisms: by altering conduction velocity restitution and by heterogeneous fibroblast distribution causing electromechanically concordant and discordant alternans in different regions of the tissue. Thus, through their coupling with myocytes, fibroblasts alter repolarization and Ca(i) cycling alternans at both the cellular and tissue scales, which may play important roles in arrhythmogenesis in diseased cardiac tissue with fibrosis.

Identification of Ikr kinetics and drug binding in native myocytes

Determining the effect of a compound on I_Kr is a standard screen for drug safety. Often the effect is described using a single IC₅₀ value, which is unable to capture complex effects of a drug. Using verapamil as an example, we present a method for using recordings from native myocytes at several drug doses along with qualitative features of I_Kr from published studies of HERG current to estimate parameters in a mathematical model of the drug effect on I_Kr. I_Kr was recorded from canine left ventricular myocytes using ruptured patch techniques. A voltage command protocol was used to record tail currents at voltages from −70 to −20 mV, following activating pulses over a wide range of voltages and pulse durations. Model equations were taken from a published I_Kr Markov model and the drug was modeled as binding to the open state. Parameters were estimated using a combined global and local optimization algorithm based on collected data with two additional constraints on I_KrI–V relation and I_Kr inactivation. The method produced models that quantitatively reproduce both the control I_Kr kinetics and dose dependent changes in the current. In addition, the model exhibited use and rate dependence. The results suggest that: (1) the technique proposed here has the practical potential to develop data-driven models that quantitatively reproduce channel behavior in native myocytes; (2) the method can capture important drug effects that cannot be reproduced by the IC₅₀ method. Although the method was developed for I_Kr, the same strategy can be applied to other ion channels, once appropriate channel-specific voltage protocols and qualitative features are identified.

Mechanisms Underlying the Formation and Dynamics of Subcellular Calcium Alternans in the Intact Rat Heart

Optical mapping of intact cardiac tissue reveals that, in some cases, intracellular calcium (Ca) release can alternate from one beat to the next in a large-small-large sequence, also referred to as Ca transient (CaT) alternans. CaT alternans can also become spatially phase-mismatched within a single cell, when one part of the cell alternates in a large-small-large sequence, whereas a different part alternates in a small-large-small sequence, a phenomenon known as subcellular discordant alternans. The mechanisms for the formation and spatiotemporal evolution of these phase-mismatched patterns are not known. We used confocal Ca imaging to measure CaT alternans at the sarcomeric level within individual myocytes in the intact rat heart. After a sudden change in cycle length (CL), 2 distinct spatial patterns of CaT alternans emerge. CaTs can form spatially phase-mismatched alternans patterns after the first few beats following the change in CL. The phase mismatch persists for many beats, after which it gradually becomes phase matched via the movement of nodes, which are junctures between phase-mismatched cell regions. In other examples, phase-matched alternans gradually become phase-mismatched, via the formation and movement of nodes. In these examples, we observed large beat-to-beat variations in the cell activation times, despite constant CL pacing. Using computer simulations, we explored the underlying mechanisms for these dynamical phenomena. Our results show how heterogeneity at the sarcomeric level, in conjunction with the dynamics of Ca cycling and membrane voltage, can lead to complex spatiotemporal phenomena within myocytes of the intact heart.

Acidosis and ischemia increase cellular Ca2+ transient alternans and repolarization alternans susceptibility in the intact rat heart

Cardiac cellular Ca(2+) transient (CaT) alternans and electrocardiographic T-wave alternans (TWA) often develop in myocardial ischemia, but the mechanisms for this relationship have not been elucidated. Acidosis is a major component of ischemia, but there is no direct evidence linking acidosis-induced cellular CaT alternans to ischemia-induced CaT alternans and TWA in whole heart. We used laser-scanning confocal microscopy to measure intracellular Ca(2+) (Ca(i)(2+)) cycling in individual myocytes of fluo-4 AM-loaded rat hearts and simultaneously recorded pseudo-ECGs to investigate changes in CaTs and late-phase repolarization, respectively, during baseline and rapid pacing under control and either globally acidic or globally ischemic conditions. Acidosis (hypercapnia; pH 6.6) increased diastolic Ca(i)(2+) levels, prolonged CaT duration, and shifted to slower heart rates both the development of pacing-induced acidosis-induced CaT alternans (both concordant and discordant) and of repolarization alternans (RPA, a measure of TWA in rat ECGs). The magnitudes of these shifts were equivalent for both CaT alternans and RPA, suggesting a close association between them. Nearly identical results were found in low-flow global ischemia. Additionally, ischemic preconditioning reduced the increased propensity for CaT alternans and RPA development and was mimicked by preconditioning by acidosis alone. Our results demonstrate that global acidosis or ischemia modifies Ca(i)(2+) cycling in myocytes such that the diastolic Ca(i)(2+) rises and the cellular CaT duration is prolonged, causing spatially concordant as well as spatially discordant cellular CaT alternans to develop at slower heart rates than in controls. Since RPA also developed at slower heart rates, our results suggest that acidosis is a major contributor to CaT alternans, which underlies the proarrhythmic state induced by myocardial ischemia and therefore may play a role in its modulation and prevention.

A computational study of mother rotor VF in the human ventricles

Sudden cardiac death is one of the major causes of death in the industrialized world. It is most often caused by a cardiac arrhythmia called ventricular fibrillation (VF). Despite its large social and economical impact, the mechanisms for VF in the human heart yet remain to be identified. Two of the most frequently discussed mechanisms observed in experiments with animal hearts are the multiple wavelet and mother rotor hypotheses. Most recordings of VF in animal hearts are consistent with the multiple wavelet mechanism. However, in animal hearts, mother rotor fibrillation has also been observed. For both multiple wavelet and mother rotor VF, cardiac heterogeneity plays an important role. Clinical data of action potential restitution measured from the surface of human hearts have been recently published. These in vivo data show a substantial degree of spatial heterogeneity. Using these clinical restitution data, we studied the dynamics of VF in the human heart using a heterogeneous computational model of human ventricles. We hypothesized that this observed heterogeneity can serve as a substrate for mother rotor fibrillation. We found that, based on these data, mother rotor VF can occur in the human heart and that ablation of the mother rotor terminates VF. Furthermore, we found that both mother rotor and multiple wavelet VF can occur in the same heart depending on the initial conditions at the onset of VF. We studied the organization of these two types of VF in terms of filament numbers, excitation periods, and frequency domains. We conclude that mother rotor fibrillation is a possible mechanism in the human heart.

Tachycardia-induced early afterdepolarizations: insights into potential ionic mechanisms from computer simulations

Although early afterdepolarizations (EADs) are classically thought to occur at slow heart rates, mounting evidence suggests that EADs may also occur at rapid heart rates produced by tachyarrhythmias, due to Ca overload of the sarcoplasmic reticulum (SR) leading to spontaneous SR Ca release. We hypothesized that the mechanism of tachycardia-induced EADs depends on the spatial and temporal morphology of spontaneous SR Ca release, and tested this hypothesis in computer simulations using a ventricular action potential mathematical model. Using two previously suggested spontaneous release morphologies, we found two distinct tachycardia-induced EAD mechanisms: one mechanistically similar to bradycardia-induced EADs, the other to delayed afterdepolarizations (DADs).

Cardiac repolarization instability during normal postnatal development

Long QT syndrome is a disease characterized by abnormal lengthening of the QT interval and by sudden cardiac death. It is a disease of development, with the incidence of a sudden event increasing during childhood. Repolarization instability during postnatal development could make the substrate susceptible to a fatal arrhythmia. Dynamic changes in repolarization that occur on a beat-to-beat basis, known as alternans, are a hallmark of electrical instability. T-wave alternans (TWA) in the electrocardiogram correlates with arrhythmia risk and long-term survival in adults. We determined TWA properties longitudinally in vivo in 7 propofol-sedated New Zealand white rabbits using transesophageal pacing weekly from 2 to 10 weeks of age. Furthermore, TWA induction after the onset of rapid pacing was characterized in vitro in 6 infant (2 weeks) and 6 adolescent (7 weeks) isolated, arterially perfused rabbit hearts. In vivo, TWA amplitude was maximum at 2 weeks and declined with age. Isoproterenol increased TWA at 8 weeks (adolescence). In vitro, large-amplitude TWA was induced with rapid pacing in both infant and adolescents but decreased to low, steady-state levels in infants. We conclude that TWA properties are age dependent in rabbit. Significant TWA is induced in rabbit at the onset of rapid pacing.