Multiscale modeling of calcium cycling in cardiac ventricular myocyte: macroscopic consequences of microscopic dyadic function

Role of ryanodine receptor cooperativity in Ca2+-wave-mediated triggered activity in cardiomyocytes

Ca2+ waves are known to trigger delayed after-depolarizations that can cause malignant cardiac arrhythmias. However, modelling Ca2+ waves using physiologically realistic models has remained a major challenge. Existing models with low Ca2+ sensitivity of ryanodine receptors (RyRs) necessitate large release currents, leading to an unrealistically large Ca2+ transient amplitude incompatible with the experimental observations. Consequently, current physiologically detailed models of delayed after-depolarizations resort to unrealistic cell architectures to produce Ca2+ waves with a normal Ca2+ transient amplitude. Here, we address these challenges by incorporating RyR cooperativity into a physiologically detailed model with a realistic cell architecture. We represent RyR cooperativity phenomenologically through a Hill coefficient within the sigmoid function of RyR open probability. Simulations in permeabilized myocytes with high Ca2+ sensitivity reveal that a sufficiently large Hill coefficient is required for Ca2+ wave propagation via the fire-diffuse-fire mechanism. In intact myocytes, propagating Ca2+ waves can occur only within an intermediate Hill coefficient range. Within this range, the spark rate is neither too low, enabling Ca2+ wave propagation, nor too high, allowing for the maintenance of a high sarcoplasmic reticulum load during diastole of the action potential. Moreover, this model successfully replicates other experimentally observed manifestations of Ca2+-wave-mediated triggered activity, including phase 2 and phase 3 early after-depolarizations and high-frequency voltage-Ca2+ oscillations. These oscillations feature an elevated take-off potential with depolarization mediated by the L-type Ca2+ current. The model also sheds light on the roles of luminal gating of RyRs and the mobile buffer ATP in the genesis of these arrhythmogenic phenomena. KEY POINTS: Existing mathematical models of Ca2+ waves use an excessively large Ca2+-release current or unrealistic diffusive coupling between release units. Our physiologically realistic model, using a Hill coefficient in the ryanodine receptor (RyR) gating function to represent RyR cooperativity, addresses these limitations and generates organized Ca2+ waves at Hill coefficients ranging from ∼5 to 10, as opposed to the traditional value of 2. This range of Hill coefficients gives a spark rate neither too low, thereby enabling Ca2+ wave propagation, nor too high, allowing for the maintenance of a high sarcoplasmic reticulum load during the plateau phase of the action potential. Additionally, the model generates Ca2+-wave-mediated phase 2 and phase 3 early after-depolarizations, and coupled membrane voltage with Ca2+ oscillations mediated by the L-type Ca2+ current. This study suggests that pharmacologically targeting RyR cooperativity could be a promising strategy for treating cardiac arrhythmias linked to Ca2+-wave-mediated triggered activity.

Diversity of cells and signals in the cardiovascular system

This white paper is the outcome of the seventh UC Davis Cardiovascular Research Symposium on Systems Approach to Understanding Cardiovascular Disease and Arrhythmia. This biannual meeting aims to bring together leading experts in subfields of cardiovascular biomedicine to focus on topics of importance to the field. The theme of the 2022 Symposium was 'Cell Diversity in the Cardiovascular System, cell-autonomous and cell-cell signalling'. Experts in the field contributed their experimental and mathematical modelling perspectives and discussed emerging questions, controversies, and challenges in examining cell and signal diversity, co-ordination and interrelationships involved in cardiovascular function. This paper originates from the topics of formal presentations and informal discussions from the Symposium, which aimed to develop a holistic view of how the multiple cell types in the cardiovascular system integrate to influence cardiovascular function, disease progression and therapeutic strategies. The first section describes the major cell types (e.g. cardiomyocytes, vascular smooth muscle and endothelial cells, fibroblasts, neurons, immune cells, etc.) and the signals involved in cardiovascular function. The second section emphasizes the complexity at the subcellular, cellular and system levels in the context of cardiovascular development, ageing and disease. Finally, the third section surveys the technological innovations that allow the interrogation of this diversity and advancing our understanding of the integrated cardiovascular function and dysfunction.

Modeling Calcium Cycling in the Heart: Progress, Pitfalls, and Challenges

Intracellular calcium (Ca) cycling in the heart plays key roles in excitation-contraction coupling and arrhythmogenesis. In cardiac myocytes, the Ca release channels, i.e., the ryanodine receptors (RyRs), are clustered in the sarcoplasmic reticulum membrane, forming Ca release units (CRUs). The RyRs in a CRU act collectively to give rise to discrete Ca release events, called Ca sparks. A cell contains hundreds to thousands of CRUs, diffusively coupled via Ca to form a CRU network. A rich spectrum of spatiotemporal Ca dynamics is observed in cardiac myocytes, including Ca sparks, spark clusters, mini-waves, persistent whole-cell waves, and oscillations. Models of different temporal and spatial scales have been developed to investigate these dynamics. Due to the complexities of the CRU network and the spatiotemporal Ca dynamics, it is challenging to model the Ca cycling dynamics in the cardiac system, particularly at the tissue sales. In this article, we review the progress of modeling of Ca cycling in cardiac systems from single RyRs to the tissue scale, the pros and cons of the current models and different modeling approaches, and the challenges to be tackled in the future.

Multi-Scale Computational Modeling of Spatial Calcium Handling From Nanodomain to Whole-Heart: Overview and Perspectives

Regulation of intracellular calcium is a critical component of cardiac electrophysiology and excitation-contraction coupling. The calcium spark, the fundamental element of the intracellular calcium transient, is initiated in specialized nanodomains which co-locate the ryanodine receptors and L-type calcium channels. However, calcium homeostasis is ultimately regulated at the cellular scale, by the interaction of spatially separated but diffusively coupled nanodomains with other sub-cellular and surface-membrane calcium transport channels with strong non-linear interactions; and cardiac electrophysiology and arrhythmia mechanisms are ultimately tissue-scale phenomena, regulated by the interaction of a heterogeneous population of coupled myocytes. Recent advances in imaging modalities and image-analysis are enabling the super-resolution reconstruction of the structures responsible for regulating calcium homeostasis, including the internal structure of nanodomains themselves. Extrapolating functional and imaging data from the nanodomain to the whole-heart is non-trivial, yet essential for translational insight into disease mechanisms. Computational modeling has important roles to play in relating structural and functional data at the sub-cellular scale and translating data across the scales. This review covers recent methodological advances that enable image-based modeling of the single nanodomain and whole cardiomyocyte, as well as the development of multi-scale simulation approaches to integrate data from nanometer to whole-heart. Firstly, methods to overcome the computational challenges of simulating spatial calcium dynamics in the nanodomain are discussed, including image-based modeling at this scale. Then, recent whole-cell models, capable of capturing a range of different structures (such as the T-system and mitochondria) and cellular heterogeneity/variability are discussed at two different levels of discretization. Novel methods to integrate the models and data across the scales and simulate stochastic dynamics in tissue-scale models are then discussed, enabling elucidation of the mechanisms by which nanodomain remodeling underlies arrhythmia and contractile dysfunction. Perspectives on model differences and future directions are provided throughout.

Ca2+ Release via IP3 Receptors Shapes the Cardiac Ca2+ Transient for Hypertrophic Signaling

Calcium (Ca²⁺) plays a central role in mediating both contractile function and hypertrophic signaling in ventricular cardiomyocytes. L-type Ca²⁺ channels trigger release of Ca²⁺ from ryanodine receptors for cellular contraction, whereas signaling downstream of G-protein-coupled receptors stimulates Ca²⁺ release via inositol 1,4,5-trisphosphate receptors (IP₃Rs), engaging hypertrophic signaling pathways. Modulation of the amplitude, duration, and duty cycle of the cytosolic Ca²⁺ contraction signal and spatial localization have all been proposed to encode this hypertrophic signal. Given current knowledge of IP₃Rs, we develop a model describing the effect of functional interaction (cross talk) between ryanodine receptor and IP₃R channels on the Ca²⁺ transient and examine the sensitivity of the Ca²⁺ transient shape to properties of IP₃R activation. A key result of our study is that IP₃R activation increases Ca²⁺ transient duration for a broad range of IP₃R properties, but the effect of IP₃R activation on Ca²⁺ transient amplitude is dependent on IP₃ concentration. Furthermore we demonstrate that IP₃-mediated Ca²⁺ release in the cytosol increases the duty cycle of the Ca²⁺ transient, the fraction of the cycle for which [Ca²⁺] is elevated, across a broad range of parameter values and IP₃ concentrations. When coupled to a model of downstream transcription factor (NFAT) activation, we demonstrate that there is a high correspondence between the Ca²⁺ transient duty cycle and the proportion of activated NFAT in the nucleus. These findings suggest increased cytosolic Ca²⁺ duty cycle as a plausible mechanism for IP₃-dependent hypertrophic signaling via Ca²⁺-sensitive transcription factors such as NFAT in ventricular cardiomyocytes.

Multi-scale approaches for the simulation of cardiac electrophysiology: I - Sub-cellular and stochastic calcium dynamics from cell to organ

Computational models of the heart at multiple spatial scales, from sub-cellular nanodomains to the whole-organ, are a powerful tool for the simulation of cardiac electrophysiology. Application of these models has provided remarkable insight into the normal and pathological functioning of the heart. In these two articles, we present methods for modelling cardiac electrophysiology at all of these spatial scales. In part one, presented here, we discuss methods and approaches for modelling sub-cellular calcium dynamics at the whole-cell and organ scales, valuable for modelling excitation-contraction coupling and mechanisms of arrhythmia triggers.

Stochastic spontaneous calcium release events and sodium channelopathies promote ventricular arrhythmias

Premature ventricular complexes (PVCs), the first initiating beats of a variety of cardiac arrhythmias, have been associated with spontaneous calcium release (SCR) events at the cell level. However, the mechanisms underlying the degeneration of such PVCs into arrhythmias are not fully understood. The objective of this study was to investigate the conditions under which SCR-mediated PVCs can lead to ventricular arrhythmias. In particular, we sought to determine whether sodium (Na⁺) current loss-of-function in the structurally normal ventricles provides a substrate for unidirectional conduction block and reentry initiated by SCR-mediated PVCs. To achieve this goal, a stochastic model of SCR was incorporated into an anatomically accurate compute model of the rabbit ventricles with the His-Purkinje system (HPS). Simulations with reduced Na⁺ current due to a negative-shift in the steady-state channel inactivation showed that SCR-mediated delayed afterdepolarizations led to PVC formation in the HPS, where the electrotonic load was lower, conduction block, and reentry in the 3D myocardium. Moreover, arrhythmia initiation was only possible when intrinsic electrophysiological heterogeneity in action potential within the ventricles was present. In conclusion, while benign in healthy individuals SCR-mediated PVCs can lead to life-threatening ventricular arrhythmias when combined with Na⁺ channelopathies.

A computational model of spatio-temporal cardiac intracellular calcium handling with realistic structure and spatial flux distribution from sarcoplasmic reticulum and t-tubule reconstructions

Intracellular calcium cycling is a vital component of cardiac excitation-contraction coupling. The key structures responsible for controlling calcium dynamics are the cell membrane (comprising the surface sarcolemma and transverse-tubules), the intracellular calcium store (the sarcoplasmic reticulum), and the co-localisation of these two structures to form dyads within which calcium-induced-calcium-release occurs. The organisation of these structures tightly controls intracellular calcium dynamics. In this study, we present a computational model of intracellular calcium cycling in three-dimensions (3-D), which incorporates high resolution reconstructions of these key regulatory structures, attained through imaging of tissue taken from the sheep left ventricle using serial block face scanning electron microscopy. An approach was developed to model the sarcoplasmic reticulum structure at the whole-cell scale, by reducing its full 3-D structure to a 3-D network of one-dimensional strands. The model reproduces intracellular calcium dynamics during control pacing and reveals the high-resolution 3-D spatial structure of calcium gradients and intracellular fluxes in both the cytoplasm and sarcoplasmic reticulum. We also demonstrated the capability of the model to reproduce potentially pro-arrhythmic dynamics under perturbed conditions, pertaining to calcium-transient alternans and spontaneous release events. Comparison with idealised cell models emphasised the importance of structure in determining calcium gradients and controlling the spatial dynamics associated with calcium-transient alternans, wherein the probabilistic nature of dyad activation and recruitment was constrained. The model was further used to highlight the criticality in calcium spark propagation in relation to inter-dyad distances. The model presented provides a powerful tool for future investigation of structure-function relationships underlying physiological and pathophysiological intracellular calcium handling phenomena at the whole-cell. The approach allows for the first time direct integration of high-resolution images of 3-D intracellular structures with models of calcium cycling, presenting the possibility to directly assess the functional impact of structural remodelling at the cellular scale.

The organisation of the membrane and sub-cellular structures of cells in the heart closely controls the coupling between its electrical and mechanical function. Computational models of the cellular calcium handling system, which is responsible for this electro-mechanical coupling, have been developed in recent years to study underlying structure-function relationships. Previous models have been largely idealised in structure; we present a new model which incorporates experimental data describing the high-resolution organisation of the primary structures involved in calcium dynamics. Significantly, the structure of the intracellular calcium store is modelled for the first time. The model is shown to reproduce calcium dynamics in control cells in both normal and abnormal conditions, demonstrating its suitability for future investigation of structure-function relationships. Thus, the model presented provides a powerful tool for the direct integration of experimentally acquired structural data in healthy and diseased cells and assessment of the role of structure in regulating normal and abnormal calcium dynamics.

Multiscale modeling methods in biomechanics

More and more frequently, computational biomechanics deals with problems where the portion of physical reality to be modeled spans over such a large range of spatial and temporal dimensions, that it is impossible to represent it as a single space-time continuum. We are forced to consider multiple space-time continua, each representing the phenomenon of interest at a characteristic space-time scale. Multiscale models describe a complex process across multiple scales, and account for how quantities transform as we move from one scale to another. This review offers a set of definitions for this emerging field, and provides a brief summary of the most recent developments on multiscale modeling in biomechanics. Of all possible perspectives, we chose that of the modeling intent, which vastly affect the nature and the structure of each research activity. To the purpose we organized all papers reviewed in three categories: 'causal confirmation,' where multiscale models are used as materializations of the causation theories; 'predictive accuracy,' where multiscale modeling is aimed to improve the predictive accuracy; and 'determination of effect,' where multiscale modeling is used to model how a change at one scale manifests in an effect at another radically different space-time scale. Consistent with how the volume of computational biomechanics research is distributed across application targets, we extensively reviewed papers targeting the musculoskeletal and the cardiovascular systems, and covered only a few exemplary papers targeting other organ systems. The review shows a research subdomain still in its infancy, where causal confirmation papers remain the most common. WIREs Syst Biol Med 2017, 9:e1375. doi: 10.1002/wsbm.1375 For further resources related to this article, please visit the WIREs website.

Studying dyadic structure-function relationships: a review of current modeling approaches and new insights into Ca2+ (mis)handling

Excitation–contraction coupling in cardiac myocytes requires calcium influx through L-type calcium channels in the sarcolemma, which gates calcium release through sarcoplasmic reticulum ryanodine receptors in a process known as calcium-induced calcium release, producing a myoplasmic calcium transient and enabling cardiomyocyte contraction. The spatio-temporal dynamics of calcium release, buffering, and reuptake into the sarcoplasmic reticulum play a central role in excitation–contraction coupling in both normal and diseased cardiac myocytes. However, further quantitative understanding of these cells’ calcium machinery and the study of mechanisms that underlie both normal cardiac function and calcium-dependent etiologies in heart disease requires accurate knowledge of cardiac ultrastructure, protein distribution and subcellular function. As current imaging techniques are limited in spatial resolution, limiting insight into changes in calcium handling, computational models of excitation–contraction coupling have been increasingly employed to probe these structure–function relationships. This review will focus on the development of structural models of cardiac calcium dynamics at the subcellular level, orienting the reader broadly towards the development of models of subcellular calcium handling in cardiomyocytes. Specific focus will be given to progress in recent years in terms of multi-scale modeling employing resolved spatial models of subcellular calcium machinery. A review of the state-of-the-art will be followed by a review of emergent insights into calcium-dependent etiologies in heart disease and, finally, we will offer a perspective on future directions for related computational modeling and simulation efforts.

Computational cardiology and risk stratification for sudden cardiac death: one of the grand challenges for cardiology in the 21st century

Risk stratification in the context of sudden cardiac death has been acknowledged as one of the major challenges facing cardiology for the past four decades. In recent years, the advent of high performance computing has facilitated organ-level simulation of the heart, meaning we can now examine the causes, mechanisms and impact of cardiac dysfunction in silico. As a result, computational cardiology, largely driven by the Physiome project, now stands at the threshold of clinical utility in regards to risk stratification and treatment of patients at risk of sudden cardiac death. In this white paper, we outline a roadmap of what needs to be done to make this translational step, using the relatively well-developed case of acquired or drug-induced long QT syndrome as an exemplar case.

Superresolution modeling of calcium release in the heart

Stable calcium-induced calcium release (CICR) is critical for maintaining normal cellular contraction during cardiac excitation-contraction coupling. The fundamental element of CICR in the heart is the calcium (Ca²⁺) spark, which arises from a cluster of ryanodine receptors (RyR). Opening of these RyR clusters is triggered to produce a local, regenerative release of Ca²⁺ from the sarcoplasmic reticulum (SR). The Ca²⁺ leak out of the SR is an important process for cellular Ca²⁺ management, and it is critically influenced by spark fidelity, i.e., the probability that a spontaneous RyR opening triggers a Ca²⁺ spark. Here, we present a detailed, three-dimensional model of a cardiac Ca²⁺ release unit that incorporates diffusion, intracellular buffering systems, and stochastically gated ion channels. The model exhibits realistic Ca²⁺ sparks and robust Ca²⁺ spark termination across a wide range of geometries and conditions. Furthermore, the model captures the details of Ca²⁺ spark and nonspark-based SR Ca²⁺ leak, and it produces normal excitation-contraction coupling gain. We show that SR luminal Ca²⁺-dependent regulation of the RyR is not critical for spark termination, but it can explain the exponential rise in the SR Ca²⁺ leak-load relationship demonstrated in previous experimental work. Perturbations to subspace dimensions, which have been observed in experimental models of disease, strongly alter Ca²⁺ spark dynamics. In addition, we find that the structure of RyR clusters also influences Ca²⁺ release properties due to variations in inter-RyR coupling via local subspace Ca²⁺ concentration ([Ca²⁺]_ss). These results are illustrated for RyR clusters based on super-resolution stimulated emission depletion microscopy. Finally, we present a believed-novel approach by which the spark fidelity of a RyR cluster can be predicted from structural information of the cluster using the maximum eigenvalue of its adjacency matrix. These results provide critical insights into CICR dynamics in heart, under normal and pathological conditions.

Microdomain [Ca(2+)] Fluctuations Alter Temporal Dynamics in Models of Ca(2+)-Dependent Signaling Cascades and Synaptic Vesicle Release

Ca(2+)-dependent signaling is often localized in spatially restricted microdomains and may involve only 1 to 100 Ca(2+) ions. Fluctuations in the microdomain Ca(2+) concentration (Ca(2+)) can arise from a wide range of elementary processes, including diffusion, Ca(2+) influx, and association/dissociation with Ca(2+) binding proteins or buffers. However, it is unclear to what extent these fluctuations alter Ca(2+)-dependent signaling. We construct Markov models of a general Ca(2+)-dependent signaling cascade and Ca(2+)-triggered synaptic vesicle release. We compare the hitting (release) time distribution and statistics for models that account for [Ca(2+)] fluctuations with the corresponding models that neglect these fluctuations. In general, when Ca(2+) fluctuations are much faster than the characteristic time for the signaling event, the hitting time distributions and statistics for the models with and without Ca(2+) fluctuation are similar. However, when the timescale of Ca(2+) fluctuations is on the same order as the signaling cascade or slower, the hitting time mean and variability are typically increased, in particular when the average number of microdomain Ca(2+) ions is small, a consequence of a long-tailed hitting time distribution. In a model of Ca(2+)-triggered synaptic vesicle release, we demonstrate the conditions for which [Ca(2+)] fluctuations do and do not alter the distribution, mean, and variability of release timing. We find that both the release time mean and variability can be increased, demonstrating that Ca(2+) fluctuations are an important aspect of microdomain Ca(2+) signaling and further suggesting that Ca(2+) fluctuations in the presynaptic terminal may contribute to variability in synaptic vesicle release and thus variability in neuronal spiking.

Long-Lasting Sparks: Multi-Metastability and Release Competition in the Calcium Release Unit Network

Calcium (Ca) sparks are elementary events of biological Ca signaling. A normal Ca spark has a brief duration in the range of 10 to 100 ms, but long-lasting sparks with durations of several hundred milliseconds to seconds are also widely observed. Experiments have shown that the transition from normal to long-lasting sparks can occur when ryanodine receptor (RyR) open probability is either increased or decreased. Here, we demonstrate theoretically and computationally that long-lasting sparks emerge as a collective dynamical behavior of the network of diffusively coupled Ca release units (CRUs). We show that normal sparks occur when the CRU network is monostable and excitable, while long-lasting sparks occur when the network dynamics possesses multiple metastable attractors, each attractor corresponding to a different spatial firing pattern of sparks. We further highlight the mechanisms and conditions that produce long-lasting sparks, demonstrating the existence of an optimal range of RyR open probability favoring long-lasting sparks. We find that when CRU firings are sparse and sarcoplasmic reticulum (SR) Ca load is high, increasing RyR open probability promotes long-lasting sparks by potentiating Ca-induced Ca release (CICR). In contrast, when CICR is already strong enough to produce frequent firings, decreasing RyR open probability counter-intuitively promotes long-lasting sparks by decreasing spark frequency. The decrease in spark frequency promotes intra-SR Ca diffusion from neighboring non-firing CRUs to the firing CRUs, which helps to maintain the local SR Ca concentration of the firing CRUs above a critical level to sustain firing. In this setting, decreasing RyR open probability further suppresses long-lasting sparks by weakening CICR. Since a long-lasting spark terminates via the Kramers’ escape process over a potential barrier, its duration exhibits an exponential distribution determined by the barrier height and noise strength, which is modulated differently by different ways of altering the Ca release flux strength.

Calcium (Ca) sparks, resulting from Ca-induced Ca release, are elementary events of biological Ca signaling. Sparks are normally brief, but long-lasting sparks have been widely observed experimentally under various conditions. The underlying mechanisms of spark duration or termination and the corresponding determinants remain a topic of debate. In this study, we demonstrate theoretically and computationally that normal brief sparks are excitable transients, while long-lasting sparks are multiple metastable states emerging in the diffusively coupled Ca release unit network, as a result of cooperativity and release competition among the Ca release units. Termination of a long-lasting spark is a Kramers’ escape process over a potential barrier, and the spark duration is the first-passage time, exhibiting an exponential distribution.

Lessons learned from multi-scale modeling of the failing heart

Heart failure constitutes a major public health problem worldwide. Affected patients experience a number of changes in the electrical function of the heart that predispose to potentially lethal cardiac arrhythmias. Due to the multitude of electrophysiological changes that may occur during heart failure, the scientific literature is complex and sometimes ambiguous, perhaps because these findings are highly dependent on the etiology, the stage of heart failure, and the experimental model used to study these changes. Nevertheless, a number of common features of failing hearts have been documented. Prolongation of the action potential (AP) involving ion channel remodeling and alterations in calcium handling have been established as the hallmark characteristics of myocytes isolated from failing hearts. Intercellular uncoupling and fibrosis are identified as major arrhythmogenic factors. Multi-scale computational simulations are a powerful tool that complements experimental and clinical research. The development of biophysically detailed computer models of single myocytes and cardiac tissues has contributed greatly to our understanding of processes underlying excitation and repolarization in the heart. The electrical, structural, and metabolic remodeling that arises in cardiac tissues during heart failure has been addressed from different computational perspectives to further understand the arrhythmogenic substrate. This review summarizes the contributions from computational modeling and simulation to predict the underlying mechanisms of heart failure phenotypes and their implications for arrhythmogenesis, ranging from the cellular level to whole-heart simulations. The main aspects of heart failure are presented in several related sections. An overview of the main electrophysiological and structural changes that have been observed experimentally in failing hearts is followed by the description and discussion of the simulation work in this field at the cellular level, and then in 2D and 3D cardiac structures. The implications for arrhythmogenesis in heart failure are also discussed including therapeutic measures, such as drug effects and cardiac resynchronization therapy. Finally, the future challenges in heart failure modeling and simulation will be discussed.

Stochastic spontaneous calcium release events trigger premature ventricular complexes by overcoming electrotonic load

AIMS

Premature ventricular complexes (PVCs) due to spontaneous calcium (Ca) release (SCR) events at the cell level can precipitate ventricular arrhythmias. However, the mechanistic link between SCRs and PVC formation remains incompletely understood. The aim of this study was to investigate the conditions under which delayed afterdepolarizations resulting from stochastic subcellular SCR events can overcome electrotonic source-sink mismatch, leading to PVC initiation.

METHODS AND RESULTS

A stochastic subcellular-scale mathematical model of SCR was incorporated in a realistic model of the rabbit ventricles and Purkinje system (PS). Elevated levels of diastolic sarcoplasmic reticulum Ca(2+) (CaSR) were imposed until triggered activity was observed, allowing us to compile statistics on probability, timing, and location of PVCs. At CaSR≥ 1500 µmol/L PVCs originated in the PS. When SCR was incapacitated in the PS, PVCs also emerged in the ventricles, but at a higher CaSR (≥1550 µmol/L) and with longer waiting times. For each model configuration tested, the probability of PVC occurrence increased from 0 to 100% within a well-defined critical CaSR range; this transition was much more abrupt in organ-scale models (∼50 µmol/L CaSR range) than in the tissue strand (∼100 µmol/L) or single-cell (∼450 µmol/L) models. Among PVCs originating in the PS, ∼68% were located near Purkinje-ventricular junctions (<1 mm).

CONCLUSION

SCR events overcome source-sink mismatch to trigger PVCs at a critical CaSR threshold. Above this threshold, PVCs emerge due to increased probability and reduced variability in timing of SCR events, leading to significant diastolic depolarization. Sites of lower electronic load, such as the PS, are preferential locations for triggering.

Images as drivers of progress in cardiac computational modelling

Computational models have become a fundamental tool in cardiac research. Models are evolving to cover multiple scales and physical mechanisms. They are moving towards mechanistic descriptions of personalised structure and function, including effects of natural variability. These developments are underpinned to a large extent by advances in imaging technologies. This article reviews how novel imaging technologies, or the innovative use and extension of established ones, integrate with computational models and drive novel insights into cardiac biophysics. In terms of structural characterization, we discuss how imaging is allowing a wide range of scales to be considered, from cellular levels to whole organs. We analyse how the evolution from structural to functional imaging is opening new avenues for computational models, and in this respect we review methods for measurement of electrical activity, mechanics and flow. Finally, we consider ways in which combined imaging and modelling research is likely to continue advancing cardiac research, and identify some of the main challenges that remain to be solved.

Determinants of beat-to-beat variability of repolarization duration in the canine ventricular myocyte: a computational analysis

Beat-to-beat variability of repolarization duration (BVR) is an intrinsic characteristic of cardiac function and a better marker of proarrhythmia than repolarization prolongation alone. The ionic mechanisms underlying baseline BVR in physiological conditions, its rate dependence, and the factors contributing to increased BVR in pathologies remain incompletely understood. Here, we employed computer modeling to provide novel insights into the subcellular mechanisms of BVR under physiological conditions and during simulated drug-induced repolarization prolongation, mimicking long-QT syndromes type 1, 2, and 3. We developed stochastic implementations of 13 major ionic currents and fluxes in a model of canine ventricular-myocyte electrophysiology. Combined stochastic gating of these components resulted in short- and long-term variability, consistent with experimental data from isolated canine ventricular myocytes. The model indicated that the magnitude of stochastic fluctuations is rate dependent due to the rate dependence of action-potential (AP) duration (APD). This process (the “active” component) and the intrinsic nonlinear relationship between membrane current and APD (“intrinsic component”) contribute to the rate dependence of BVR. We identified a major role in physiological BVR for stochastic gating of the persistent Na⁺ current (I_Na) and rapidly activating delayed-rectifier K⁺ current (I_Kr). Inhibition of I_Kr or augmentation of I_Na significantly increased BVR, whereas subsequent β-adrenergic receptor stimulation reduced it, similar to experimental findings in isolated myocytes. In contrast, β-adrenergic stimulation increased BVR in simulated long-QT syndrome type 1. In addition to stochastic channel gating, AP morphology, APD, and beat-to-beat variations in Ca²⁺ were found to modulate single-cell BVR. Cell-to-cell coupling decreased BVR and this was more pronounced when a model cell with increased BVR was coupled to a model cell with normal BVR. In conclusion, our results provide new insights into the ionic mechanisms underlying BVR and suggest that BVR reflects multiple potentially proarrhythmic parameters, including increased ion-channel stochasticity, prolonged APD, and abnormal Ca²⁺ handling.

Every heartbeat has an electrical recovery (repolarization) interval that varies in duration from beat to beat. Excessive beat-to-beat variability of repolarization duration has been shown to be a risk marker of potentially fatal heart-rhythm disorders, but the contributing subcellular mechanisms remain incompletely understood. Computational models have greatly enhanced our understanding of several basic electrophysiological mechanisms. We developed a detailed computer model of the ventricular myocyte that can simulate beat-to-beat changes in repolarization duration by taking into account stochastic changes in the opening and closing of individual ion channels responsible for all main ion currents. The model accurately reproduced experimental data from isolated myocytes under both physiological and pathological conditions. Using the model, we identified several mechanisms contributing to repolarization variability, including stochastic gating of ion channels, duration and morphology of the repolarization phase, and intracellular calcium handling, thereby providing insights into its basis as a proarrhythmic marker. Our computer model provides a detailed framework to study the dynamics of cardiac electrophysiology and arrhythmias.

Parameter sensitivity analysis of stochastic models provides insights into cardiac calcium sparks

We present a parameter sensitivity analysis method that is appropriate for stochastic models, and we demonstrate how this analysis generates experimentally testable predictions about the factors that influence local Ca(2+) release in heart cells. The method involves randomly varying all parameters, running a single simulation with each set of parameters, running simulations with hundreds of model variants, then statistically relating the parameters to the simulation results using regression methods. We tested this method on a stochastic model, containing 18 parameters, of the cardiac Ca(2+) spark. Results show that multivariable linear regression can successfully relate parameters to continuous model outputs such as Ca(2+) spark amplitude and duration, and multivariable logistic regression can provide insight into how parameters affect Ca(2+) spark triggering (a probabilistic process that is all-or-none in a single simulation). Benchmark studies demonstrate that this method is less computationally intensive than standard methods by a factor of 16. Importantly, predictions were tested experimentally by measuring Ca(2+) sparks in mice with knockout of the sarcoplasmic reticulum protein triadin. These mice exhibit multiple changes in Ca(2+) release unit structures, and the regression model both accurately predicts changes in Ca(2+) spark amplitude (30% decrease in model, 29% decrease in experiments) and provides an intuitive and quantitative understanding of how much each alteration contributes to the result. This approach is therefore an effective, efficient, and predictive method for analyzing stochastic mathematical models to gain biological insight.

Alterations in T-tubule and dyad structure in heart disease: challenges and opportunities for computational analyses

Compelling recent experimental results make clear that sub-cellular structures are altered in ventricular myocytes during the development of heart failure, in both human samples and diverse experimental models. These alterations can include, but are not limited to, changes in the clusters of sarcoplasmic reticulum (SR) Ca(2+)-release channels, ryanodine receptors, and changes in the average distance between the cell membrane and ryanodine receptor clusters. In this review, we discuss the potential consequences of these structural alterations on the triggering of SR Ca(2+) release during excitation-contraction coupling. In particular, we describe how mathematical models of local SR Ca(2+) release can be used to predict functional changes resulting from diverse modifications that occur in disease states. We review recent studies that have used simulations to understand the consequences of sub-cellular structural changes, and we discuss modifications that will allow for future modelling studies to address unresolved questions. We conclude with a discussion of improvements in both experimental and mathematical modelling techniques that will be required to provide a stronger quantitative understanding of the functional consequences of changes in sub-cellular structure in heart disease.

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.

Computational approaches to understand cardiac electrophysiology and arrhythmias

Cardiac rhythms arise from electrical activity generated by precisely timed opening and closing of ion channels in individual cardiac myocytes. These impulses spread throughout the cardiac muscle to manifest as electrical waves in the whole heart. Regularity of electrical waves is critically important since they signal the heart muscle to contract, driving the primary function of the heart to act as a pump and deliver blood to the brain and vital organs. When electrical activity goes awry during a cardiac arrhythmia, the pump does not function, the brain does not receive oxygenated blood, and death ensues. For more than 50 years, mathematically based models of cardiac electrical activity have been used to improve understanding of basic mechanisms of normal and abnormal cardiac electrical function. Computer-based modeling approaches to understand cardiac activity are uniquely helpful because they allow for distillation of complex emergent behaviors into the key contributing components underlying them. Here we review the latest advances and novel concepts in the field as they relate to understanding the complex interplay between electrical, mechanical, structural, and genetic mechanisms during arrhythmia development at the level of ion channels, cells, and tissues. We also discuss the latest computational approaches to guiding arrhythmia therapy.

Toward an integrative computational model of the Guinea pig cardiac myocyte

The local control theory of excitation-contraction (EC) coupling asserts that regulation of calcium (Ca²⁺) release occurs at the nanodomain level, where openings of single L-type Ca²⁺ channels (LCCs) trigger openings of small clusters of ryanodine receptors (RyRs) co-localized within the dyad. A consequence of local control is that the whole-cell Ca²⁺ transient is a smooth continuous function of influx of Ca²⁺ through LCCs. While this so-called graded release property has been known for some time, its functional importance to the integrated behavior of the cardiac ventricular myocyte has not been fully appreciated. We previously formulated a biophysically based model, in which LCCs and RyRs interact via a coarse-grained representation of the dyadic space. The model captures key features of local control using a low-dimensional system of ordinary differential equations. Voltage-dependent gain and graded Ca²⁺ release are emergent properties of this model by virtue of the fact that model formulation is closely based on the sub-cellular basis of local control. In this current work, we have incorporated this graded release model into a prior model of guinea pig ventricular myocyte electrophysiology, metabolism, and isometric force production. The resulting integrative model predicts the experimentally observed causal relationship between action potential (AP) shape and timing of Ca²⁺ and force transients, a relationship that is not explained by models lacking the graded release property. Model results suggest that even relatively subtle changes in AP morphology that may result, for example, from remodeling of membrane transporter expression in disease or spatial variation in cell properties, may have major impact on the temporal waveform of Ca²⁺ transients, thus influencing tissue level electromechanical function.

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.

Activation of mitochondrial calcium-independent phospholipase A2γ (iPLA2γ) by divalent cations mediating arachidonate release and production of downstream eicosanoids

Calcium-independent phospholipase A(2)γ (iPLA(2)γ) (PNPLA8) is the predominant phospholipase activity in mammalian mitochondria. However, the chemical mechanisms that regulate its activity are unknown. Here, we utilize iPLA(2)γ gain of function and loss of function genetic models to demonstrate the robust activation of iPLA(2)γ in murine myocardial mitochondria by Ca(2+) or Mg(2+) ions. Calcium ion stimulated the production of 2-arachidonoyl-lysophosphatidylcholine (2-AA-LPC) from 1-palmitoyl-2-[(14)C]arachidonoyl-sn-glycero-3-phosphocholine during incubations with wild-type heart mitochondrial homogenates. Furthermore, incubation of mitochondrial homogenates from transgenic myocardium expressing iPLA(2)γ resulted in 13- and 25-fold increases in the initial rate of radiolabeled 2-AA-LPC and arachidonic acid (AA) production, respectively, in the presence of calcium ion. Mass spectrometric analysis of the products of calcium-activated hydrolysis of endogenous mitochondrial phospholipids in transgenic iPLA(2)γ mitochondria revealed the robust production of AA, 2-AA-LPC, and 2-docosahexaenoyl-LPC that was over 10-fold greater than wild-type mitochondria. The mechanism-based inhibitor (R)-(E)-6-(bromomethylene)-3-(1-naphthalenyl)-2H-tetrahydropyran-2-one (BEL) (iPLA(2)γ selective), but not its enantiomer, (S)-BEL (iPLA(2)β selective) or pyrrolidine (cytosolic PLA(2)α selective), markedly attenuated Ca(2+)-dependent fatty acid release and polyunsaturated LPC production. Moreover, Ca(2+)-induced iPLA(2)γ activation was accompanied by the production of downstream eicosanoid metabolites that were nearly completely ablated by (R)-BEL or by genetic ablation of iPLA(2)γ. Intriguingly, Ca(2+)-induced iPLA(2)γ activation was completely inhibited by long-chain acyl-CoA (IC(50) ∼20 μm) as well as by a nonhydrolyzable acyl-CoA thioether analog. Collectively, these results demonstrate that mitochondrial iPLA(2)γ is activated by divalent cations and inhibited by acyl-CoA modulating the generation of biologically active metabolites that regulate mitochondrial bioenergetic and signaling functions.

Imaging cardiac arrhythmias

Electrocardiographic imaging can noninvasively provide an activation map of the heart’s surface to help treat arrhythmias.