Filament instability and rotational tissue anisotropy: A numerical study using detailed cardiac models

The physics of heart rhythm disorders

The global burden caused by cardiovascular disease is substantial, with heart disease representing the most common cause of death around the world. There remains a need to develop better mechanistic models of cardiac function in order to combat this health concern. Heart rhythm disorders, or arrhythmias, are one particular type of disease which has been amenable to quantitative investigation. Here we review the application of quantitative methodologies to explore dynamical questions pertaining to arrhythmias. We begin by describing single-cell models of cardiac myocytes, from which two and three dimensional models can be constructed. Special focus is placed on results relating to pattern formation across these spatially-distributed systems, especially the formation of spiral waves of activation. Next, we discuss mechanisms which can lead to the initiation of arrhythmias, focusing on the dynamical state of spatially discordant alternans, and outline proposed mechanisms perpetuating arrhythmias such as fibrillation. We then review experimental and clinical results related to the spatio-temporal mapping of heart rhythm disorders. Finally, we describe treatment options for heart rhythm disorders and demonstrate how statistical physics tools can provide insights into the dynamics of heart rhythm disorders.

Sensitivity of a data-assimilation system for reconstructing three-dimensional cardiac electrical dynamics

Modelling of cardiac electrical behaviour has led to important mechanistic insights, but important challenges, including uncertainty in model formulations and parameter values, make it difficult to obtain quantitatively accurate results. An alternative approach is combining models with observations from experiments to produce a data-informed reconstruction of system states over time. Here, we extend our earlier data-assimilation studies using an ensemble Kalman filter to reconstruct a three-dimensional time series of states with complex spatio-temporal dynamics using only surface observations of voltage. We consider the effects of several algorithmic and model parameters on the accuracy of reconstructions of known scroll-wave truth states using synthetic observations. In particular, we study the algorithm's sensitivity to parameters governing different parts of the process and its robustness to several model-error conditions. We find that the algorithm can achieve an acceptable level of error in many cases, with the weakest performance occurring for model-error cases and more extreme parameter regimes with more complex dynamics. Analysis of the poorest-performing cases indicates an initial decrease in error followed by an increase when the ensemble spread is reduced. Our results suggest avenues for further improvement through increasing ensemble spread by incorporating additive inflation or using a parameter or multi-model ensemble. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

Slow Recovery of Excitability Increases Ventricular Fibrillation Risk as Identified by Emulation

Purpose: Rotor stability and meandering are key mechanisms determining and sustaining cardiac fibrillation, with important implications for anti-arrhythmic drug development. However, little is yet known on how rotor dynamics are modulated by variability in cellular electrophysiology, particularly on kinetic properties of ion channel recovery.

Methods: We propose a novel emulation approach, based on Gaussian process regression augmented with machine learning, for data enrichment, automatic detection, classification, and analysis of re-entrant biomarkers in cardiac tissue. More than 5,000 monodomain simulations of long-lasting arrhythmic episodes with Fenton-Karma ionic dynamics, further enriched by emulation to 80 million electrophysiological scenarios, were conducted to investigate the role of variability in ion channel densities and kinetics in modulating rotor-driven arrhythmic behavior.

Results: Our methods predicted the class of excitation behavior with classification accuracy up to 96%, and emulation effectively predicted frequency, stability, and spatial biomarkers of functional re-entry. We demonstrate that the excitation wavelength interpretation of re-entrant behavior hides critical information about rotor persistence and devolution into fibrillation. In particular, whereas action potential duration directly modulates rotor frequency and meandering, critical windows of excitability are identified as the main determinants of breakup. Further novel electrophysiological insights of particular relevance for ventricular arrhythmias arise from our multivariate analysis, including the role of incomplete activation of slow inward currents in mediating tissue rate-dependence and dispersion of repolarization, and the emergence of slow recovery of excitability as a significant promoter of this mechanism of dispersion and increased arrhythmic risk.

Conclusions: Our results mechanistically explain pro-arrhythmic effects of class Ic anti-arrhythmics in the ventricles despite their established role in the pharmacological management of atrial fibrillation. This is mediated by their slow recovery of excitability mode of action, promoting incomplete activation of slow inward currents and therefore increased dispersion of repolarization, given the larger influence of these currents in modulating the action potential in the ventricles compared to the atria. These results exemplify the potential of emulation techniques in elucidating novel mechanisms of arrhythmia and further application to cardiac electrophysiology.

Systematic reduction of a detailed atrial myocyte model

Cardiac arrhythmias are a major health concern and often involve poorly understood mechanisms. Mathematical modeling is able to provide insights into these mechanisms which might result in better treatment options. A key element of this modeling is a description of the electrophysiological properties of cardiac cells. A number of electrophysiological models have been developed, ranging from highly detailed and complex models, containing numerous parameters and variables, to simplified models in which variables and parameters no longer directly correspond to electrophysiological quantities. In this study, we present a systematic reduction of the complexity of the detailed model of Koivumaki et al. using the recently developed manifold boundary approximation method. We reduce the original model, containing 42 variables and 37 parameters, to a model with only 11 variables and 5 parameters and show that this reduced model can accurately reproduce the action potential shape and restitution curve of the original model. The reduced model contains only five currents and all variables and parameters can be directly linked to electrophysiological quantities. Due to its reduction in complexity, simulation times of our model are decreased more than three-fold. Furthermore, fitting the reduced model to clinical data is much more efficient, a potentially important step towards patient-specific modeling.

Modelling far field pacing for terminating spiral waves pinned to ischaemic heterogeneities in cardiac tissue

In cardiac tissue, electrical spiral waves pinned to a heterogeneity can be unpinned (and eventually terminated) using electric far field pulses and recruiting the heterogeneity as a virtual electrode. While for isotropic media the process of unpinning is much better understood, the case of an anisotropic substrate with different conductivities in different directions still needs intensive investigation. To study the impact of anisotropy on the unpinning process, we present numerical simulations based on the bidomain formulation of the phase I of the Luo and Rudy action potential model modified due to the occurrence of acute myocardial ischaemia. Simulating a rotating spiral wave pinned to an ischaemic heterogeneity, we compare the success of sequences of far field pulses in the isotropic and the anisotropic case for spirals still in transient or in steady rotation states. Our results clearly indicate that the range of pacing parameters resulting in successful termination of pinned spiral waves is larger in anisotropic tissue than in an isotropic medium.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

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.

Mechanistically based mapping of human cardiac fibrillation

The mechanisms underpinning human cardiac fibrillation remain elusive. In his 1913 paper ‘On dynamic equilibrium in the heart’, Mines proposed that an activation wave front could propagate repeatedly in a circle, initiated by a stimulus in the vulnerable period. While the dynamics of activation and recovery are central to cardiac fibrillation, these physiological data are rarely used in clinical mapping. Fibrillation is a rapid irregular rhythm with spatiotemporal disorder resulting from two fundamental mechanisms – sources in preferred cardiac regions or spatially diffuse self‐sustaining activity, i.e. with no preferred source. On close inspection, however, this debate may also reflect mapping technique. Fibrillation is initiated from triggers by regional dispersion in repolarization, slow conduction and wavebreak, then sustained by non‐uniform interactions of these mechanisms. Notably, optical mapping of action potentials in atrial fibrillation (AF) show spiral wave sources (rotors) in nearly all studies including humans, while most traditional electrogram analyses of AF do not. Techniques may diverge in fibrillation because electrograms summate non‐coherent waves within an undefined field whereas optical maps define waves with a visually defined field. Also fibrillation operates at the limits of activation and recovery, which are well represented by action potentials while fibrillatory electrograms poorly represent repolarization. We conclude by suggesting areas for study that may be used, until such time as optical mapping is clinically feasible, to improve mechanistic understanding and therapy of human cardiac fibrillation.

Reconstructing three-dimensional reentrant cardiac electrical wave dynamics using data assimilation

For many years, reentrant scroll waves have been predicted and studied as an underlying mechanism for cardiac arrhythmias using numerical techniques, and high-resolution mapping studies using fluorescence recordings from the surfaces of cardiac tissue preparations have confirmed the presence of visible spiral waves. However, assessing the three-dimensional dynamics of these reentrant waves using experimental techniques has been limited to verifying stable scroll-wave dynamics in relatively thin preparations. We propose a different approach to recovering the three-dimensional dynamics of reentrant waves in the heart. By applying techniques commonly used in weather forecasting, we combine dual-surface observations from a particular experiment with predictions from a numerical model to reconstruct the full three-dimensional time series of the experiment. Here, we use model-generated surrogate observations from a numerical experiment to evaluate the performance of the ensemble Kalman filter in reconstructing such time series for a discordant alternans state in one spatial dimension and for scroll waves in three dimensions. We show that our approach is able to recover time series of both observed and unobserved variables matching the truth. Where nearby observations are available, the error is reduced below the synthetic observation error, with a smaller reduction with increased distance from observations. Our findings demonstrate that state reconstruction for spatiotemporally complex cardiac electrical dynamics is possible and will lead naturally to applications using real experimental data.

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.

Scroll-wave dynamics in human cardiac tissue: lessons from a mathematical model with inhomogeneities and fiber architecture

Cardiac arrhythmias, such as ventricular tachycardia (VT) and ventricular fibrillation (VF), are among the leading causes of death in the industrialized world. These are associated with the formation of spiral and scroll waves of electrical activation in cardiac tissue; single spiral and scroll waves are believed to be associated with VT whereas their turbulent analogs are associated with VF. Thus, the study of these waves is an important biophysical problem. We present a systematic study of the combined effects of muscle-fiber rotation and inhomogeneities on scroll-wave dynamics in the TNNP (ten Tusscher Noble Noble Panfilov) model for human cardiac tissue. In particular, we use the three-dimensional TNNP model with fiber rotation and consider both conduction and ionic inhomogeneities. We find that, in addition to displaying a sensitive dependence on the positions, sizes, and types of inhomogeneities, scroll-wave dynamics also depends delicately upon the degree of fiber rotation. We find that the tendency of scroll waves to anchor to cylindrical conduction inhomogeneities increases with the radius of the inhomogeneity. Furthermore, the filament of the scroll wave can exhibit drift or meandering, transmural bending, twisting, and break-up. If the scroll-wave filament exhibits weak meandering, then there is a fine balance between the anchoring of this wave at the inhomogeneity and a disruption of wave-pinning by fiber rotation. If this filament displays strong meandering, then again the anchoring is suppressed by fiber rotation; also, the scroll wave can be eliminated from most of the layers only to be regenerated by a seed wave. Ionic inhomogeneities can also lead to an anchoring of the scroll wave; scroll waves can now enter the region inside an ionic inhomogeneity and can display a coexistence of spatiotemporal chaos and quasi-periodic behavior in different parts of the simulation domain. We discuss the experimental implications of our study.

Vortex filament dynamics in computational models of ventricular fibrillation in the heart

In three-dimensional cardiac tissue, the re-entrant waves that sustain ventricular fibrillation rotate around a line of phase singularity or vortex filament. The aim of this study was to investigate how the behavior of these vortex filaments is influenced by membrane kinetics, initial conditions, and tissue geometry in computational models of excitable tissue. A monodomain model of cardiac tissue was used, with kinetics described by a three-variable simplified ionic model (3V-SIM). Two versions of 3V-SIM were used, one with steep action potential duration restitution, and one with reduced excitability. Re-entrant fibrillation was then simulated in three tissue geometries: a cube, a slab, and an anatomically detailed model of rabbit ventricles. Filaments were identified using a phase-based method, and the number, size, origin, and orientation of filaments was tracked throughout each simulation. The main finding of this study is that kinetics, initial conditions, geometry, and anisotropy all affected the number, proliferation, and orientation of vortex filaments in re-entrant fibrillation. An important finding of this study was that the behavior of vortex filaments in simplified slab geometry representing part of the ventricular wall did not necessarily predict behavior in an anatomically detailed model of the rabbit ventricles.

Boundary-induced reentry in homogeneous excitable tissue

Heterogeneity of cardiac electrical properties can lead to heart rhythm disorders. Numerical studies have shown that stimuli chosen to maximize dynamic heterogeneity terminate wave propagation. However, experimental investigations suggest that similar sequences induce fragmentation of the wave fronts, rather than complete wave block. In this paper we show that an insulating boundary in an otherwise homogeneous medium can disrupt dynamically induced wave block by breaking a symmetry in the spatial pattern of action potential duration, leading to unidirectional block and reentrant activation.

Minimal model for human ventricular action potentials in tissue

Modeling the dynamics of wave propagation in human ventricular tissue and studying wave stability require models that reproduce realistic characteristics in tissue. We present a minimal ventricular (MV) human model that is designed to reproduce important tissue-level characteristics of epicardial, endocardial and midmyocardial cells, including action potential (AP) amplitudes and morphologies, upstroke velocities, steady-state action potential duration (APD) and conduction velocity (CV) restitution curves, minimum APD, and minimum diastolic interval. The model is then compared with three previously published human ventricular cell models, the Priebe and Beuckelmann (PB), the Ten Tusscher-Noble-Noble-Panfilov (TNNP), and the Iyer-Mazhari-Winslow (IMW). For the first time, the stability of reentrant waves for all four models is analyzed, and quantitative comparisons are made among the models in single cells and in tissue. The PB, TNNP, and IMW models exhibit quantitative differences in APD and CV rate adaptation, as well as completely different reentrant wave dynamics of quasi-breakup, stability, and breakup, respectively. All the models have dominant frequencies comparable to clinical values except for the IMW model, which has a large range of frequencies extending beyond the clinical range for both ventricular tachycardia (VT) and ventricular fibrillation (VF). The TNNP and IMW models possess a large degree of short-term memory and we show for the first time the existence of memory in CV restitution. The MV model also can be fitted to reproduce the dynamics of other models and is computationally more efficient: the times required to simulate the MV, TNNP, PB and IMW models follow the ratio 1:31:50:8084.

Enhanced transmural fiber rotation and connexin 43 heterogeneity are associated with an increased upper limit of vulnerability in a transgenic rabbit model of human hypertrophic cardiomyopathy

Human hypertrophic cardiomyopathy, characterized by cardiac hypertrophy and myocyte disarray, is the most common cause of sudden cardiac death in the young. Hypertrophic cardiomyopathy is often caused by mutations in sarcomeric genes. We sought to determine arrhythmia propensity and underlying mechanisms contributing to arrhythmia in a transgenic (TG) rabbit model (beta-myosin heavy chain-Q403) of human hypertrophic cardiomyopathy. Langendorff-perfused hearts from TG (n=6) and wild-type (WT) rabbits (n=6) were optically mapped. The upper and lower limits of vulnerability, action potential duration (APD) restitution, and conduction velocity were measured. The transmural fiber angle shift was determined using diffusion tensor MRI. The transmural distribution of connexin 43 was quantified with immunohistochemistry. The upper limit of vulnerability was significantly increased in TG versus WT hearts (13.3+/-2.1 versus 7.4+/-2.3 V/cm; P=3.2e(-5)), whereas the lower limits of vulnerability were similar. APD restitution, conduction velocities, and anisotropy were also similar. Left ventricular transmural fiber rotation was significantly higher in TG versus WT hearts (95.6+/-10.9 degrees versus 79.2+/-7.8 degrees; P=0.039). The connexin 43 density was significantly increased in the mid-myocardium of TG hearts compared with WT (5.46+/-2.44% versus 2.68+/-0.77%; P=0.024), and similar densities were observed in the endo- and epicardium. Because a nearly 2-fold increase in upper limit of vulnerability was observed in the TG hearts without significant changes in APD restitution, conduction velocity, or the anisotropy ratio, we conclude that structural remodeling may underlie the elevated upper limit of vulnerability in human hypertrophic cardiomyopathy.

Cardiac electrical dynamics: maximizing dynamical heterogeneity

The relationships between key features of the cardiac electrical activity, such as electrical restitution, discordant alternans, wavebreak, and reentry, and the onset of ventricular tachyarrhythmias have been characterized extensively under the condition of constant rapid pacing. However, it is unlikely that this scenario applies directly to the clinical situation, where the induction of ventricular tachycardia (VT) typically is associated with the interruption of normal cardiac rhythm by several premature beats. To address this issue, we have developed a general theory to explain why specific patterns of premature stimuli increase dynamic heterogeneity of repolarization and precipitate conduction block. The theory predicts that conduction block is caused by (1) creation of a spatial gradient in diastolic interval (DI) by waves traveling at slightly different velocities (ie, conduction velocity dispersion) and (2) amplification of the spatial gradient in DI over subsequent action potentials, secondary to a strong dependence of action potential duration on the preceding DI (ie, a steep action potential duration restitution function). Tests of this theory have been conducted in computer models of homogeneous tissue, where increased spatial dispersion of repolarization during premature stimulation can be attributed solely to the development of dynamical heterogeneity, and in a canine model exhibiting spontaneously occurring VT and sudden death. Our results thus far indicate that the probability of inducing ventricular fibrillation (VF) in the animal model is highest for those sequences predicted to cause conduction block in the computer model. An understanding of the mechanisms underlying these observations will help to identify key electrical phenomena in the onset of VT and fibrillation. Drug and electrical therapies can then be improved by targeting these specific phenomena.

Maintenance of ventricular fibrillation in heterogeneous ventricle

Although ventricular fibrillation (VF) is the prevalent cause of sudden cardiac death, the mechanisms that underlie VF remain elusive. One possible explanation is that VF is driven by a single robust rotor that is the source of wavefronts that break-up due to functional heterogeneities. Previous 2D computer simulations have proposed that a heterogeneity in background potassium current (IK1) can serve as the substrate for the formation of mother rotor activity. This study incorporates IK1 heterogeneity between the left and right ventricle in a realistic 3D rabbit ventricle model to examine its effects on the organization of VF. Computer simulations show that the IK1 heterogeneity contributes to the initiation and maintenance of VF by providing regions of different refractoriness which serves as sites of wave break and rotor formation. A single rotor that drives the fibrillatory activity in the ventricle is not found in this study. Instead, multiple sites of reentry are recorded throughout the ventricle. Calculation of dominant frequencies for each myocardial node yields no significant difference between the dominant frequency of the LV and the RV. The 3D computer simulations suggest that IK1 spatial heterogeneity alone can not lead to the formation of a stable rotor.

Dynamics of conduction blocks in a model of paced cardiac tissue

We study numerically the dynamics of conduction blocks using a detailed electrophysiological model. We find that this dynamics depends critically on the size of the paced region. Small pacing regions lead to stationary conduction blocks while larger pacing regions can lead to conduction blocks that travel periodically towards the pacing region. We show that this size-dependence dynamics can lead to a novel arrhythmogenic mechanism. Furthermore, we show that the essential phenomena can be captured in a much simpler coupled-map model.

Modeling wave propagation in realistic heart geometries using the phase-field method

We present a novel algorithm for modeling electrical wave propagation in anatomical models of the heart. The algorithm uses a phase-field approach that represents the boundaries between the heart muscle and the surrounding medium as a spatially diffuse interface of finite thickness. The chief advantage of this method is to automatically handle the boundary conditions of the voltage in complex geometries without the need to track the location of these boundaries explicitly. The algorithm is shown to converge accurately in nontrivial test geometries with no-flux (zero normal current) boundary conditions as the width of the diffuse interface becomes small compared to the width of the cardiac action potential wavefront. Moreover, the method is illustrated for anatomically realistic models of isolated rabbit and canine ventricles as well as human atria.

Preventing alternans-induced spiral wave breakup in cardiac tissue: an ion-channel-based approach

The detailed processes involved in spiral wave breakup, believed to be one major mechanism by which tachycardia evolves into fibrillation, are still poorly understood. This has rendered difficult the proper design of an efficient and practical control stimulus protocol to eliminate such events. In order to gain new insights into the underlying electrophysiological and dynamical mechanisms of breakup, we applied linear perturbation theory to a steadily rotating spiral wave in two spatial dimensions. The tissue was composed of cells modeled using the Fenton-Karma equations whose parameters were chosen to emphasize alternans as a primary mechanism for breakup. Along with one meandering mode, not just one but several unstable alternans modes were found with differing growth rates, frequencies, and spatial structures. As the conductance of the fast inward current was increased, the instability of the modes increased, consistent with increased meandering and propensity for spiral breakup in simulations. We also explored a promising new approach, based on the theory, for the design of an energy efficient electrical stimulus protocol to control spiral wave breakup. The novelty lies in addressing the problem directly at the ion channel level and taking advantage of the inherent two dimensional nature of the rotating wave. With the help of the eigenmode method, we were able to calculate the exact timing and amplitude of the stimulus, and locate it optimally to maximize efficiency. The analysis led to a special-case example that demonstrated that a single, properly timed stimulus can have a global effect, suppressing all growing alternans modes over the entire tissue, thus inhibiting spiral wave breakup.

Examination of optical depth effects on fluorescence imaging of cardiac propagation

Optical mapping with voltage-sensitive dyes provides a high-resolution technique to observe cardiac electrodynamic behavior. Although most studies assume that the fluorescent signal is emitted from the surface layer of cells, the effects of signal attenuation with depth on signal interpretation are still unclear. This simulation study examines the effects of a depth-weighted signal on epicardial activation patterns and filament localization. We simulated filament behavior using a detailed cardiac model, and compared the signal obtained from the top (epicardial) layer of the spatial domain with the calculated weighted signal. General observations included a prolongation of the action upstroke duration, early upstroke initiation, and reduction in signal amplitude in the weighted signal. A shallow filament was found to produce a dual-humped action potential morphology consistent with previously reported observations. Simulated scroll wave breakup exhibited effects such as the false appearance of graded potentials, apparent supramaximal conduction velocities, and a spatially blurred signal with the local amplitude dependent upon the immediate subepicardial activity; the combination of these effects produced a corresponding change in the accuracy of filament localization. Our results indicate that the depth-dependent optical signal has significant consequences on the interpretation of epicardial activation dynamics.

Suppression of alternans and conduction blocks despite steep APD restitution: electrotonic, memory, and conduction velocity restitution effects

We examine the utility of the action potential (AP) duration (APD) restitution curve slope in predicting the onset of electrical alternans when electrotonic and memory effects are considered. We develop and use two ionic cell models without memory that have the same restitution curve with slope >1 but different AP shapes and, therefore, different electrotonic effects. We also study a third cell model that incorporates short-term memory of previous cycle lengths, so that it has a family of S1-S2 restitution curves as well as a dynamic restitution curve with slope >1. Our results indicate that both electrotonic and memory effects can suppress alternans, even when the APD restitution curve is steep. In the absence of memory, electrotonic currents related to the shape of the AP, as well as conduction velocity restitution, can affect how alternans develops in tissue and, in some cases, can prevent its induction entirely, even when isolated cells exhibit alternans. When short-term memory is included, alternans may not occur in isolated cells, despite a steep APD restitution curve, and may or may not occur in tissue, depending on conduction velocity restitution. We show for the first time that electrotonic and memory effects can prevent conduction blocks and stabilize reentrant waves in two and three dimensions. Thus we find that the slope of the APD restitution curve alone does not always well predict the onset of alternans and that incorporating electrotonic and memory effects may provide a more useful alternans criterion. A Data Supplement containing movies and JAVA applets is available online at http://ajpheart.physiology.org/cgi/content/full/00747.2003/DC1 .

A simulation study of the effects of cardiac anatomy in ventricular fibrillation

In ventricular fibrillation (VF), the principal cause of sudden cardiac death, waves of electrical excitation break up into turbulent and incoherent fragments. The causes of this breakup have been intensely debated. Breakup can be caused by fixed anatomical properties of the tissue, such as the biventricular geometry and the inherent anisotropy of cardiac conduction. However, wavebreak can also be caused purely by instabilities in wave conduction that arise from ion channel dynamics, which represent potential targets for drug action. To study the interaction between these two wave-breaking mechanisms, we used a physiologically based mathematical model of the ventricular cell, together with a realistic three-dimensional computer model of cardiac anatomy, including the distribution of fiber angles throughout the myocardium. We find that dynamical instabilities remain a major cause of the wavebreak that drives VF, even in an anatomically realistic heart. With cell physiology in its usual operating regime, dynamics and anatomical features interact to promote wavebreak and VF. However, if dynamical instability is reduced, for example by modeling of certain pharmacologic interventions, electrical waves do not break up into fibrillation, despite anatomical complexity. Thus, interventions that promote dynamical wave stability show promise as an antifibrillatory strategy in this more realistic setting.

Filament behavior in a computational model of ventricular fibrillation in the canine heart

The aim of this paper was to quantify the behavior of filaments in a computational model of re-entrant ventricular fibrillation. We simulated cardiac activation in an anisotropic monodomain with excitation described by the Fenton-Karma model with Beeler-Reuter restitution, and geometry by the Auckland canine ventricle. We initiated re-entry in the left and right ventricular free walls, as well as the septum. The number of filaments increased during the first 1.5 s before reaching a plateau with a mean value of about 36 in each simulation. Most re-entrant filaments were between 10 and 20 mm long. The proportion of filaments touching the epicardial surface was 65%, but most of these were visible for much less than one period of re-entry. This paper shows that useful information about filament dynamics can be gleaned from models of fibrillation in complex geometries, and suggests that the interplay of filament creation and destruction may offer a target for antifibrillatory therapy.

Effect of regional differences in cardiac cellular electrophysiology on the stability of ventricular arrhythmias: a computational study

Re-entry is an important mechanism of cardiac arrhythmias. During re-entry a wave of electrical activation repeatedly propagates into recovered tissue, rotating around a rod-like filament. Breakdown of a single re-entrant wave into multiple waves is believed to underlie the transition from ventricular tachycardia to ventricular fibrillation. Several mechanisms of breakup have been identified including the effect of anisotropic conduction in the ventricular wall. Cells in the inner and outer layers of the ventricular wall have different action potential durations (APD), and support re-entrant waves with different periods. The aim of this study was to use a computational approach to study twisting and breakdown in a transmural re-entrant wave spanning these regions, and examine the relative role of this effect and anisotropic conduction. We used a simplified model of action potential conduction in the ventricular wall that we modified so that it supported stable re-entry in an anisotropic model with uniform APD. We first examined the effect of regional differences on breakdown in an isotropic model with transmural differences in APD, and found that twisting of the re-entrant filament resulted in buckling and breakdown during the second cycle of re-entry. We found that breakdown was amplified in the anisotropic model, resulting in complex activation in the region of longest APD. This study shows that regional differences in cardiac electrophysiology are a potentially important mechanism for destabilizing re-entry and may act synergistically with other mechanisms to mediate the transition from ventricular tachycardia to ventricular fibrillation.

Minimal principle for rotor filaments

Three-dimensional rotors, or scroll waves, provide essential insight into the activity of excitable media. They also are a suspected cause in the formation and maintenance of ventricular fibrillation, whose lethality is well known. It is therefore of considerable interest to find out what configurations can be adopted by such pathologies. A scroll's behavior is embodied in its organizing center or filament, a largely quiescent tube about which the scroll rotates. Predicting filament shape has normally required computer-intensive simulations of the whole scroll in time. We have found a fast and robust principle that yields the prediction for stationary filaments on a purely geometrical basis, blind to the reaction parameters of the medium. The procedure is to calculate the filament shape as a minimal path. We work in singly diffusive media whose diffusivity tensor--and no other feature--varies spatially. Mathematical and numerical evidence is presented for the proposition that a stable filament is a geodesic in a three-dimensional space whose metric is given by the inverse diffusivity tensor of the medium. Away from the boundaries, a stable filament is unaffected by the reaction parameters. The algorithmic aspects of this work are subsidiary to our main purpose of drawing attention to the universal and unexpectedly exact fit of an elementary geodesic principle within reaction-diffusion theories.

Dynamics and interaction of filaments in a computational model of re-entrant ventricular fibrillation

Ventricular fibrillation (VF) is a lethal cardiac arrhythmia. Re-entry, in which action potential wavefronts rotate around filaments, is believed to sustain VF. In this study we used a computational model of multiple wavelet fibrillation in the thin-walled right ventricle (10 mm thick) and the thicker walled left ventricle (16 mm thick) to investigate the effect of tissue thickness and initiation protocol on re-entry, and to examine whether filament dynamics and interaction in the model could explain why re-entry is both rarely observed and short-lived in experimental studies that map electrical activation on the heart surface. We found (i) that the density of filaments, the proportion of transmural filaments and the proportion of filaments visible on the model surface were all higher in the 10 mm simulation, (ii) that the initiation protocol influences the rate of filament breakdown but not the number of filaments present after 1 s, and (iii) that although many filaments are visible on the surface of the model, the majority are visible for less than one rotation. This study shows that tissue thickness, geometry and initiation protocol influence electrical activation during VF, and that the rapid motion and interaction of filaments result in transient appearance of surface re-entry.

Scroll waves in isotropic excitable media: linear instabilities, bifurcations, and restabilized states

Scroll waves are three-dimensional analogs of spiral waves. The linear stability spectrum of untwisted and twisted scroll waves is computed for a two-variable reaction-diffusion model of an excitable medium. Different bands of modes are seen to be unstable in different regions of parameter space. The corresponding bifurcations and bifurcated states are characterized by performing direct numerical simulations. In addition, computations of the adjoint linear stability operator eigenmodes are also performed and serve to obtain a number of matrix elements characterizing the long-wavelength deformations of scroll waves.

A method to quantify the dynamics and complexity of re-entry in computational models of ventricular fibrillation

Ventricular fibrillation is a deadly cardiac arrhythmia. There is evidence that electrical activity in cardiac tissue is sustained during fibrillation by re-entrant waves that rotate around filaments. In this paper we develop a method for identifying and tracking filaments in a computational model of ventricular fibrillation. This method identifies the birth, death, bifurcation and amalgamation of filaments and these events are summarized on a directed graph. The approach described in this study provides ways to quantify the complex patterns of electrical activity seen in computational models of fibrillation, to relate the behaviour of computational models to experimental data and thus to gain insights into the underlying mechanisms of this dangerous arrhythmia.

Ionic charge conservation and long-term steady state in the Luo-Rudy dynamic cell model

It has been postulated that cardiac cell models accounting for changes in intracellular ion concentrations violate a conservation principle, and, as a result, computed parameters (e.g., ion concentrations and transmembrane potential, V(m)) drift in time, never attaining steady state. To address this issue, models have been proposed that invoke the charge conservation principle to calculate V(m) from ion concentrations ("algebraic" method), rather than from transmembrane current ("differential" method). The aims of this study are to compare model behavior during prolonged periods of pacing using the algebraic and differential methods, and to address the issue of model drift. We pace the Luo-Rudy dynamic model of a cardiac ventricular cell and compare the time-dependent behavior of computed parameters using the algebraic and differential methods. When ions carried by the stimulus current are taken into account, the algebraic and differential methods yield identical results and neither shows drift in computed parameters. The present study establishes the proper pacing protocol for simulation studies of cellular behavior during long periods of rapid pacing. Such studies are essential for mechanistic understanding of arrhythmogenesis, since cells are subjected to rapid periodic stimulation during many arrhythmias.

Scroll waves in spherical shell geometries

The evolution of scroll waves in excitable media with spherical shell geometries is studied as a function of shell thickness and outer radius. The motion of scroll wave filaments that are the locii of phaseless points in the medium and organize the wave pattern is investigated. When the inner radius is sufficiently large the filaments remain attached to both the inner and outer surfaces. The minimum size of the sphere that supports spiral waves and the maximum number of spiral waves that can be sustained on a sphere of given size are determined for both regular and random initial distributions. When the inner radius is too small to support spiral waves the filaments detach from the inner surface and form a curved filament connecting the two spiral tips in the surface. In certain parameter domains the filament is an arc of a circle that shrinks with constant shape. For parameter values close to the meandering border, the filament grows and collisions with the sphere walls lead to turbulent filament dynamics. (c) 2001 American Institute of Physics.