Simulation of the effect of rogue ryanodine receptors on a calcium wave in ventricular myocytes with heart failure

Changes in cellular Ca2+ and Na+ regulation during the progression towards heart failure

In adapting to disease and loss of tissue, the heart shows great phenotypic plasticity that involves changes to its structure, composition and electrophysiology. Together with parallel whole body cardiovascular adaptations, the initial decline in cardiac function resulting from the insult is compensated. However, in the long term, the heart muscle begins to fail and patients with this condition have a very poor prognosis, with many dying from disturbances of rhythm. The surviving myocytes of these hearts gain Na+ , which is positively inotropic because of alterations to Ca2+ fluxes mediated by the Na+ /Ca2+ exchange, but compromises Ca2+ -dependent energy metabolism in mitochondria. Uptake of Ca2+ into the sarcoplasmic reticulum (SR) is reduced because of diminished function of SR Ca2+ ATPases. The result of increased Ca2+ influx and reduced SR Ca2+ uptake is an increase in the diastolic cytosolic Ca2+ concentration, which promotes spontaneous SR Ca2+ release and induces delayed afterdepolarisations. Action potential duration prolongs because of increased late Na+ current and changes in expression and function of other ion channels and transporters increasing the probability of the formation of early afterdepolarisations. There is a reduction in T-tubule density and so the normal spatial arrangements required for efficient excitation-contraction coupling are compromised and lead to temporal delays in Ca2+ release from the SR. Therefore, the structural and electrophysiological responses that occur to provide compensation do so at the expense of (1) increasing the likelihood of arrhythmogenesis; (2) activating hypertrophic, apoptotic and Ca2+ signalling pathways; and (3) decreasing the efficiency of SR Ca2+ release.

Estimation of stochastic behaviour in cardiac myocytes: I. Ca2+ movements inside the cytosol and sarcoplasmic reticulum on curvilinear domains

Since the discovery of Ca2+ sparks and their stochastic behaviour in cardiac myocytes, models have focused on the inclusion of stochasticity in their studies. While most models pay much attention to the stochastic modelling of cytosolic Ca2+ concentration the coupling of Ca2+ sparks and blinks in a stochastic model has not been explored fully. The cell morphology in in silico studies in the past is assumed to be Cartesian, spherical or cylindrical. The application on curvilinear grids can easily address certain restrictions posed by such grid set up and provide more realistic cell morphology. In this paper, we present a stochastic reaction-diffusion model that couples Ca2+ sparks and blinks in realistic shapes of cells in curvilinear domains. Methodology: Transformation of the model was performed to the curvilinear coordinate system. The set of equations is used to produce Ca2+ waves initiated from sparks and blinks. A non-buffered and non-dyed version as well as a buffered and dyed version of these equations were studied in light of observing the dynamics on the two different systems. For comparison, results for both the Cartesian and curvilinear grids are provided. Results and conclusions: A successful demonstration of the application of curvilinear grids serving as basis for future developments.

Effects of rogue ryanodine receptors on Ca2+ sparks in cardiac myocytes

Ca2+ sparks and Ca2+ quarks, arising from clustered and rogue ryanodine receptors (RyRs), are significant Ca2+ release events from the junctional sarcoplasmic reticulum (JSR). Based on the anomalous subdiffusion of Ca2+ in the cytoplasm, a mathematical model was developed to investigate the effects of rogue RyRs on Ca2+ sparks in cardiac myocytes. Ca2+ quarks and sparks from the stochastic opening of rogue and clustered RyRs are numerically reproduced and agree with experimental measurements. It is found that the stochastic opening Ca2+ release units (CRUs) of clustered RyRs are regulated by free Ca2+ concentration in the JSR lumen (i.e. [Ca2+]lumen). The frequency of spontaneous Ca2+ sparks is remarkably increased by the rogue RyRs opening at high [Ca2+]lumen, but not at low [Ca2+]lumen. Hence, the opening of rogue RyRs contributes to the formation of Ca2+ sparks at high [Ca2+]lumen. The interplay of Ca2+ sparks and Ca2+ quarks has been discussed in detail. This work is of significance to provide insight into understanding Ca2+ release mechanisms in cardiac myocytes.

Cellular mechanism of cardiac alternans: an unresolved chicken or egg problem

T-wave alternans, a specific form of cardiac alternans, has been associated with the increased susceptibility to cardiac arrhythmias and sudden cardiac death (SCD). Plenty of evidence has related cardiac alternans at the tissue level to the instability of voltage kinetics or Ca(2+) handling dynamics at the cellular level. However, to date, none of the existing experiments could identify the exact cellular mechanism of cardiac alternans due to the bi-directional coupling between voltage kinetics and Ca(2+) handling dynamics. Either of these systems could be the origin of alternans and the other follows as a secondary change, therefore making the cellular mechanism of alternans a difficult chicken or egg problem. In this context, theoretical analysis combined with experimental techniques provides a possibility to explore this problem. In this review, we will summarize the experimental and theoretical advances in understanding the cellular mechanism of alternans. We focus on the roles of action potential duration (APD) restitution and Ca(2+) handling dynamics in the genesis of alternans and show how the theoretical analysis combined with experimental techniques has provided us a new insight into the cellular mechanism of alternans. We also discuss the possible reasons of increased propensity for alternans in heart failure (HF) and the new possible therapeutic targets. Finally, according to the level of electrophysiological recording techniques and theoretical strategies, we list some critical experimental or theoretical challenges which may help to determine the origin of alternans and to find more effective therapeutic targets in the future.

Calcium waves initiating from the anomalous subdiffusive calcium sparks

The objective of the study is to investigate the propagation of Ca(2+) waves in full-width cardiac myocytes and carry out sensitivity analysis to study the effects of various physiological parameters on global Ca(2+) waves. Based on the anomalous subdiffusion of Ca(2+) sparks, a mathematical model was proposed to characterize the Ca(2+) waves. The computed results were in agreement with the experimental measurements using confocal microscopy. This model includes variables of current through the Ca(2+) release unit (CRU; ICRU), duration of current flow through CRU (Topen), Ca(2+) sensitivity parameter (K), the longitudinal and transverse spatial separation of CRUs (lx and ly, where x denotes longitudinal direction (x-axis) and y denotes transverse direction (y-axis)) and Ca(2+) diffusion coefficients (Dx, Dy). The spatio-temporal mechanism of the anomalous Ca(2+) sparks led to results that were different from those based on Fick's law. The major findings were reported as: ICRU affected the dynamic properties of Ca(2+) waves more significantly than Topen; the effect of K on the properties of Ca(2+) waves was negligible; ly affected the amplitude significantly, but lx affected the longitudinal velocity significantly; in turn, the limitation and significance of the study are discussed.

Ca²⁺ waves in the heart

Ca(2+) waves were probably first observed in the early 1940s. Since then Ca(2+) waves have captured the attention of an eclectic mixture of mathematicians, neuroscientists, muscle physiologists, developmental biologists, and clinical cardiologists. This review discusses the current state of mathematical models of Ca(2+) waves, the normal physiological functions Ca(2+) waves might serve in cardiac cells, as well as how the spatial arrangement of Ca(2+) release channels shape Ca(2+) waves, and we introduce the idea of Ca(2+) phase waves that might provide a useful framework for understanding triggered arrhythmias.

Modeling calcium wave based on anomalous subdiffusion of calcium sparks in cardiac myocytes

sparks and waves play important roles in calcium release and calcium propagation during the excitation-contraction (EC) coupling process in cardiac myocytes. Although the classical Fick’s law is widely used to model sparks and waves in cardiac myocytes, it fails to reasonably explain the full-width at half maximum(FWHM) paradox. However, the anomalous subdiffusion model successfully reproduces sparks of experimental results. In this paper, in the light of anomalous subdiffusion of sparks, we develop a mathematical model of calcium wave in cardiac myocytes by using stochastic release of release units (CRUs). Our model successfully reproduces calcium waves with physiological parameters. The results reveal how concentration waves propagate from an initial firing of one CRU at a corner or in the middle of considered region, answer how large in magnitude of an anomalous spark can induce a wave. With physiological currents (2pA) through CRUs, it is shown that an initial firing of four adjacent CRUs can form a wave. Furthermore, the phenomenon of calcium waves collision is also investigated.

Simulation of arrhythmogenic effect of rogue RyRs in failing heart by using a coupled model

Cardiac cells with heart failure are usually characterized by impairment of Ca²⁺ handling with smaller SR Ca²⁺ store and high risk of triggered activities. In this study, we developed a coupled model by integrating the spatiotemporal Ca²⁺ reaction-diffusion system into the cellular electrophysiological model. With the coupled model, the subcellular Ca²⁺ dynamics and global cellular electrophysiology could be simultaneously traced. The proposed coupled model was then applied to study the effects of rogue RyRs on Ca²⁺ cycling and membrane potential in failing heart. The simulation results suggested that, in the presence of rogue RyRs, Ca²⁺ dynamics is unstable and Ca²⁺ waves are prone to be initiated spontaneously. These release events would elevate the membrane potential substantially which might induce delayed afterdepolarizations or triggered action potentials. Moreover, the variation of membrane potential depolarization is indicated to be dependent on the distribution density of rogue RyR channels. This study provides a new possible arrhythmogenic mechanism for heart failure from subcellular to cellular level.

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.