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

Curvature-mediated source and sink effects on the genesis of premature ventricular complexes in long QT syndrome

Premature ventricular complexes (PVCs) are spontaneous excitations occurring in the ventricles of the heart that are associated with ventricular arrhythmias and sudden cardiac death. Under long QT conditions, PVCs can be mediated by repolarization gradient (RG) and early afterdepolarizations (EADs), yet the effects of heterogeneities or geometry of the RG or EAD regions on PVC genesis remain incompletely understood. In this study, we use computer simulation to systematically investigate the effects of the curvature of the RG border region on PVC genesis under long QT conditions. We show that PVCs can be either promoted or suppressed by negative or positive RG border curvature depending on the source and sink conditions. When the origin of oscillation is in the source region and the source is too strong, a positive RG border curvature can promote PVCs by causing the source area to oscillate. When the origin of oscillation is in the sink region, a negative RG border curvature can promote PVCs by causing the sink area to oscillate. Furthermore, EAD-mediated PVCs are also promoted by negative border curvature. We also investigate the effects of wavefront curvature and show that PVCs are promoted by convex but suppressed by concave wavefronts; however, the effect of wavefront curvature is much smaller than that of RG border curvature. In conclusion, besides the increase of RG and occurrence of EADs caused by QT prolongation, the geometry of the RG border plays important roles in PVC genesis, which can greatly increase the risk of arrhythmias in cardiac diseases.NEW & NOTEWORTHY The effects of the curvature or geometry of the repolarization gradient region and wavefront curvature on the genesis of premature ventricular complexes are systematically investigated using computer modeling and simulation. Premature ventricular complexes can be promoted by either positive or negative curvature of the gradient region depending on the source and sink conditions. The underlying mechanisms of the curvature effects are revealed, which provides mechanistic insights into arrhythmogenesis in cardiac diseases.

The transient outward potassium current plays a key role in spiral wave breakup in ventricular tissue

Spiral wave reentry as a mechanism of lethal ventricular arrhythmias has been widely demonstrated in animal experiments and recordings from human hearts. It has been shown that in structurally normal hearts spiral waves are unstable, breaking up into multiple wavelets via dynamical instabilities. However, many of the second-generation action potential models give rise only to stable spiral waves, raising issues regarding the underlying mechanisms of spiral wave breakup. In this study, we carried out computer simulations of two-dimensional homogeneous tissues using five ventricular action potential models. We show that the transient outward potassium current (Ito), although it is not required, plays a key role in promoting spiral wave breakup in all five models. As the maximum conductance of Ito increases, it first promotes spiral wave breakup and then stabilizes the spiral waves. In the absence of Ito, speeding up the L-type calcium kinetics can prevent spiral wave breakup, however, with the same speedup kinetics, spiral wave breakup can be promoted by increasing Ito. Increasing Ito promotes single-cell dynamical instabilities, including action potential duration alternans and chaos, and increasing Ito further suppresses these action potential dynamics. These cellular properties agree with the observation that increasing Ito first promotes spiral wave breakup and then stabilizes spiral waves in tissue. Implications of our observations to spiral wave dynamics in the real hearts and action potential model improvements are discussed.NEW & NOTEWORTHY Spiral wave breakup manifesting as multiple wavelets is a mechanism of ventricular fibrillation. It has been known that spiral wave breakup in cardiac tissue can be caused by a steeply sloped action potential duration restitution curve, a property mainly determined by the recovery of L-type calcium current. Here, we show that the transient outward potassium current (Ito) is another current that plays a key role in spiral wave breakup, that is, spiral waves can be stable for low and high maximum Ito conductance but breakup occurs for intermediate maximum Ito conductance. Since Ito is present in normal hearts of many species and required for Brugada syndrome, it may play an important role in the spiral wave stability and arrhythmogenesis under both normal condition and Brugada syndrome.

Spatially Discordant Repolarization Alternans in the Absence of Conduction Velocity Restitution

Spatially discordant alternans (SDA) of action potential duration (APD) has been widely observed in cardiac tissue and is linked to cardiac arrhythmogenesis. Theoretical studies have shown that conduction velocity restitution (CVR) is required for the formation of SDA. However, this theory is not completely supported by experiments, indicating that other mechanisms may exist. In this study, we carried out computer simulations using mathematical models of action potentials to investigate the mechanisms of SDA in cardiac tissue. We show that when CVR is present and engaged, such as fast pacing from one side of the tissue, the spatial pattern of APD in the tissue undergoes either spatially concordant alternans or SDA, independent of initial conditions or tissue heterogeneities. When CVR is not engaged, such as simultaneous pacing of the whole tissue or under normal/slow heart rates, the spatial pattern of APD in the tissue can have multiple solutions, including spatially concordant alternans and different SDA patterns, depending on heterogeneous initial conditions or pre-existing repolarization heterogeneities. In homogeneous tissue, curved nodal lines are not stable, which either evolve into straight lines or disappear. However, in heterogeneous itssue, curved nodal lines can be stable, depending on their initial locations and shapes relative to the structure of the heterogeneity. Therefore, CVR-induced SDA and non-CVR-induced SDA exhibit different dynamical properties, which may be responsible for the different SDA properties observed in experimental studies and arrhythmogenesis in different clinical settings.

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

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

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.

Chaos in the genesis and maintenance of cardiac arrhythmias

Dynamical chaos, an irregular behavior of deterministic systems, has been widely shown in nature. It also has been demonstrated in cardiac myocytes in many studies, including rapid pacing-induced irregular beat-to-beat action potential alterations and slow pacing-induced irregular early afterdepolarizations, etc. Here we review the roles of chaos in the genesis of cardiac arrhythmias, the transition to ventricular fibrillation, and the spontaneous termination of fibrillation, based on evidence from computer simulation of mathematical models and experiments of animal models.

Effect of activation sequence on transmural patterns of repolarization and action potential duration in rabbit ventricular myocardium

Although transmural heterogeneity of action potential duration (APD) is established in single cells isolated from different tissue layers, the extent to which it produces transmural gradients of repolarization in electrotonically coupled ventricular myocardium remains controversial. The purpose of this study was to examine the relative contribution of intrinsic cellular gradients of APD and electrotonic influences to transmural repolarization in rabbit ventricular myocardium. Transmural optical mapping was performed in left ventricular wedge preparations from eight rabbits. Transmural patterns of activation, repolarization, and APD were recorded during endocardial and epicardial stimulation. Experimental results were compared with modeled data during variations in electrotonic coupling. A transmural gradient of APD was evident during endocardial stimulation, which reflected differences previously seen in isolated cells, with the longest APD at the endocardium and the shortest at the epicardium (endo: 165 ± 5 vs. epi: 147 ± 4 ms; P < 0.05). During epicardial stimulation, this gradient reversed (epi: 162 ± 4 vs. endo: 148 ± 6 ms; P < 0.05). In both activation sequences, transmural repolarization followed activation and APD shortened along the activation path such that significant transmural gradients of repolarization did not occur. This correlation between transmural activation time and APD was recapitulated in simulations and varied with changes in intercellular coupling, confirming that it is mediated by electrotonic current flow between cells. These data suggest that electrotonic influences are important in determining the transmural repolarization sequence in rabbit ventricular myocardium and that they are sufficient to overcome intrinsic differences in the electrophysiological properties of the cells across the ventricular wall.

Vulnerable windows define susceptibility to alternans and spatial discordance

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

Electrotonic effects on action potential duration in perfused rat hearts

Electrotonic interactions in cardiac tissue have been shown previously to modulate dynamical properties of the myocardium such as action potential duration (APD) and action potential duration restitution. A recent computational study indicated that these electrotonic effects may be strongest in small murine hearts. In the present study, we investigate experimentally how APD is modulated by activation sequence and pacing rate using optical mapping in Langendorff perfused rat hearts. Our results show that following an epicardial point stimulus, a strong correlation exists between epicardial APD and activation time, with decreasing APD for increasing activation time. This effect is preserved for all pacing frequencies (6-14 Hz) investigated in this study. Our experimental results are validated by detailed three-dimensional computer simulations. These simulations also demonstrate a strong transmural APD dependence on activation sequence, which, near the pacing site, is sufficient to mask the intrinsic transmural gradient.

Multistability property in cardiac ionic models of mammalian and human ventricular cells

The underlying mechanisms of irregular cardiac rhythms are still poorly understood. Many experimental and modeling studies are aimed at identifying factors which cause cardiac arrhythmias. However, a lack of understanding of heart rhythm dynamical properties makes it difficult to uncover precise mechanisms of electrical instabilities, and hence to predict the onset of heart rhythm disorders. We review and compare the existing methods of studying cardiac dynamics, including restitution protocol (S1-S2), dynamic restitution protocol and multistability test protocol (S1-CI-S2). We focus on cardiac cell dynamics to elucidate regularities of heart rhythm. We demonstrate the advantages of our newly proposed systematic approach of analysis of cardiac cell dynamics using mammalian Luo Rudy 1991 and human ventricular Ten Tusscher 2006 single cell models under healthy and diseased conditions such as altered K(+) or Ca(2+) related currents. We investigate the role of ionic properties and the shape of an action potential on the nonlinear dynamics of electrical processes in periodically stimulated cardiac cells. We show the existence of multistability property for human ventricular cells. Moreover, the multistability is proposed to be an intrinsic property of cardiac cells, and is also suggested to be one of the mechanisms which could underlie the sudden triggering of life-threatening ventricular arrhythmias in the human heart.

Properties and ionic mechanisms of action potential adaptation, restitution, and accommodation in canine epicardium

Computational models of cardiac myocytes are important tools for understanding ionic mechanisms of arrhythmia. This work presents a new model of the canine epicardial myocyte that reproduces a wide range of experimentally observed rate-dependent behaviors in cardiac cell and tissue, including action potential (AP) duration (APD) adaptation, restitution, and accommodation. Model behavior depends on updated formulations for the 4-aminopyridine-sensitive transient outward current (I(to1)), the slow component of the delayed rectifier K(+) current (I(Ks)), the L-type Ca(2+) channel current (I(Ca,L)), and the Na(+)-K(+) pump current (I(NaK)) fit to data from canine ventricular myocytes. We found that I(to1) plays a limited role in potentiating peak I(Ca,L) and sarcoplasmic reticulum Ca(2+) release for propagated APs but modulates the time course of APD restitution. I(Ks) plays an important role in APD shortening at short diastolic intervals, despite a limited role in AP repolarization at longer cycle lengths. In addition, we found that I(Ca,L) plays a critical role in APD accommodation and rate dependence of APD restitution. Ca(2+) entry via I(Ca,L) at fast rate drives increased Na(+)-Ca(2+) exchanger Ca(2+) extrusion and Na(+) entry, which in turn increases Na(+) extrusion via outward I(NaK). APD accommodation results from this increased outward I(NaK). Our simulation results provide valuable insight into the mechanistic basis of rate-dependent phenomena important for determining the heart's response to rapid and irregular pacing rates (e.g., arrhythmia). Accurate simulation of rate-dependent phenomena and increased understanding of their mechanistic basis will lead to more realistic multicellular simulations of arrhythmia and identification of molecular therapeutic targets.

Intrinsic inhomogeneities and the coexistence of spirals with different periods of rotation

We propose a mechanism by which wave fronts emanating from a spiral may break far from the spiral core due to intrinsic spatial inhomogeneities. A series of computer simulations are presented to demonstrate how coupling domains, which on their own would not cause breakup, may cause a single spiral to break into many spirals.

Loading effect of fibroblast-myocyte coupling on resting potential, impulse propagation, and repolarization: insights from a microstructure model

The numerous nonmyocytes present within the myocardium may establish electrical connections with myocytes through gap junctions, formed naturally or as a result of a cell therapy. The strength of the coupling and its potential impact on action potential characteristics and conduction are not well understood. This study used computer simulation to investigate the load-induced electrophysiological consequences of the coupling of myocytes with fibroblasts, where the fibroblast resting potential, density, distribution, and coupling strength were varied. Conduction velocity (CV), upstroke velocity, and action potential duration (APD) were analyzed for longitudinal and transverse impulse propagation in a two-dimensional microstructure tissue model, developed to represent a monolayer culture of cardiac cells covered by a layer of fibroblasts. The results show that 1) at weak coupling (<0.25 nS), the myocyte resting potential was elevated, leading to CV up to 5% faster than control; 2) at intermediate coupling, the myocyte resting potential elevation saturated, whereas the current flowing from the myocyte to the fibroblast progressively slowed down both CV and upstroke velocity; 3) at strong couplings (>8 nS), all of the effects saturated; and 4) APD at 90% repolarization was usually prolonged by 0-20 ms (up to 60-80 ms for high fibroblast density and coupling) by the coupling to fibroblasts. The changes in APD depended on the fibroblast resting potential. This complex, coupling-dependent interaction of fibroblast and myocytes also has relevance to the integration of other nonmyocytes in the heart, such as those used in cellular therapies.

Multistability as intrinsic property of a single cardiac cell: a simulation study

Multistability mechanism or the appearance of coexisting rhythms at periodical stimulation of single cardiac cells is explored. A computation analysis is carried out using the phase 1 Luo and Rudy action potential model. Using double stage protocol of stimulation we have changed continuously the moment of application of the premature or delayed stimulus or the so-called 'coupling interval' and investigated the stable states of action potential duration. A multistability zone is established implying a stimulation frequency range where different initial conditions (premature or delayed stimulus application moments) lead to different stable rhythms. Basins of attraction are discovered on the transmembrane potential curve. Each basin is characterized by a set of initial conditions leading to a certain stable rhythm. Basins of attraction are shown to coexist for both simple and complex entrainment modes, which results in multistability. Obtained basins of attraction of complex rhythms on the curves of transmembrane potential are expected to be linked with the occurrence of cardiac vulnerable windows on ECG records during which extra stimuli can induce life threatening arrhythmias.

Monophasic action potentials generated by bidomain modeling as a tool for detecting cardiac repolarization times

Unipolar electrograms (EGs) and hybrid (or unorthodox or unipolar) monophasic action potentials (HMAPs) are currently the only proposed extracellular electrical recording techniques for obtaining cardiac recovery maps with high spatial resolution in exposed and isolated hearts. Estimates of the repolarization times from the HMAP downstroke phase have been the subject of recent controversies. The goal of this paper is to computationally address the controversies concerning the HMAP information content, in particular the reliability of estimating the repolarization time from the HMAP downstroke phase. Three-dimensional numerical simulations were performed by using the anisotropic bidomain model with a region of short action potential durations. EGs, transmembrane action potentials (TAPs), and HMAPs elicited by an epicardial stimulation close or away from a permanently depolarized site were computed. The repolarization time was computed as the moment of EG fastest upstroke (RTeg) during the T wave, of HMAP fastest downstroke (RTHMAP), and of TAP fastest downstroke (RTtap). The latter was taken as the gold standard for repolarization time. We also compared the times (RT90HMAP, RT90tap) when the HMAP and TAP first reach 90% of their resting value during the downstroke. For all explored sites, the HMAP downstroke closely followed the TAP downstroke, which is the expression of local repolarization activity. Results show that HMAP and TAP markers are highly correlated, and both markers RTHMAP and RTeg (RT90HMAP) are reliable estimates of the TAP reference marker RTtap (RT90tap). Therefore, the downstroke phase of the HMAP contains valuable information for assessing repolarization times.

Effects of transmural electrical heterogeneities and electrotonic interactions on the dispersion of cardiac repolarization and action potential duration: A simulation study

It has been shown in the literature that myocytes isolated from the ventricular walls at various intramural depths have different action potential durations (APDs). When these myocytes are embedded in the ventricular wall, their inhomogeneous properties affect the sequence of repolarization and the actual distribution of the APDs in the entire wall. In this article, we implement a mathematical model to simulate the combined effect of (a) the non-homogeneous intrinsic membrane properties (in particular the non-homogeneous APDs) and (b) the electrotonic currents that modulate the APDs when the myocytes are embedded in the ventricular myocardium. In particular, we study the effect of (a) and (b) on the excitation and repolarization sequences and on the distribution of APDs in the ventricles. We implement a Monodomain tissue representation that includes orthotropic anisotropy, transmural fiber rotation and homogeneous or heterogeneous transmural intrinsic membrane properties, modeled according to the phase I Luo-Rudy membrane ionic model. Three-dimensional simulations are performed in a cartesian slab with a parallel finite element solver employing structured isoparametric trilinear finite elements in space and a semi-implicit adaptive method in time. Simulations of excitation and repolarization sequences elicited by epicardial or endocardial pacing show that in a homogeneous slab the repolarization pathways approximately follow the activation sequence. Conversely, in the heterogeneous cases considered in this study, we observed two repolarization wavefronts that started from the epi and the endocardial faces respectively and collided in the thickness of the wall and in one case an additional repolarization wave starting from an intramural site. Introducing the heterogeneities along the transmural epi-endocardial direction affected both the repolarization sequence and the APD dispersion, but these effects were clearly discernible only in transmural planes. By contrast, in planes parallel to epi- and endocardium the APD distribution remained remarkably similar to that observed in the homogeneous model. Therefore, the patterns of the repolarization sequence and APD dispersion on the epicardial surface (or any other intramural surface parallel to it) do not reveal the uniform transmural heterogeneity.

Dynamics and Cardiac Arrhythmias

Cardiac arrhythmias, characterized by single or multiple reentrant circuits, represent a dynamic phenomenon in an excitable medium. In this review, we provide a brief overview of how cardiac action potential duration restitution, conduction velocity restitution, and intracellular calcium cycling regulate the dynamics of action potential excitation and wave propagation in relation to the genesis and maintenance of cardiac arrhythmias.

Vulnerable window for conduction block in a one-dimensional cable of cardiac cells, 2: multiple extrasystoles

Unidirectional conduction block of premature extrasystoles can lead to initiation of cardiac reentry, causing lethal arrhythmias including ventricular fibrillation. Multiple extrasystoles are often more effective at inducing unidirectional conduction block and reentry than a single extrasystole. Since the substrate for conduction block is spatial dispersion of refractoriness, in this study we investigate how the first extrasystole modulates this dispersion to influence the "vulnerable window" for conduction block by subsequent extrasystoles, particularly in relation to action potential duration restitution and conduction velocity restitution properties. Using a kinematic model to represent wavefront-waveback interactions and simulations with the Luo-Rudy model in a one-dimensional cable of cardiac cells, we show that in homogeneous tissue, a premature extrasystole can create a large dispersion of refractoriness leading to conduction block of a subsequent extrasystole. In heterogeneous tissue, however, a premature extrasystole can either reduce or enhance the dispersion of refractoriness depending on its propagation direction with respect to the previous beat. With multiple extrasystoles at random coupling intervals, vulnerability to conduction block is proportional to their number. In general, steep action potential duration restitution and broad conduction velocity restitution promote dispersion of refractoriness in response to multiple extrasystoles, and thus enhance vulnerability to conduction block. These restitution properties also promote spatially discordant alternans, a setting which is particularly prone to conduction block. The equivalent dispersion of refractoriness created dynamically in homogeneous tissue by spatially discordant alternans is more likely to cause conduction block than a comparable degree of preexisting dispersion in heterogeneous tissue.