Re-entrant activity and its control in a model of mammalian ventricular tissue

Response function framework for the dynamics of meandering or large-core spiral waves and modulated traveling waves

In many oscillatory or excitable systems, dynamical patterns emerge which are stationary or periodic in a moving frame of reference. Examples include traveling waves or spiral waves in chemical systems or cardiac tissue. We present a unified theoretical framework for the drift of such patterns under small external perturbations, in terms of overlap integrals between the perturbation and the adjoint critical eigenfunctions of the linearized operator (i.e., response functions). For spiral waves, the finite radius of the spiral tip trajectory and spiral wave meander are taken into account. Different coordinate systems can be chosen, depending on whether one wants to predict the motion of the spiral-wave tip, the time-averaged tip path, or the center of the meander flower. The framework is applied to analyze the drift of a meandering spiral wave in a constant external field in different regimes.

Performance evaluation of GPU parallelization, space-time adaptive algorithms, and their combination for simulating cardiac electrophysiology

The use of computer models as a tool for the study and understanding of the complex phenomena of cardiac electrophysiology has attained increased importance nowadays. At the same time, the increased complexity of the biophysical processes translates into complex computational and mathematical models. To speed up cardiac simulations and to allow more precise and realistic uses, 2 different techniques have been traditionally exploited: parallel computing and sophisticated numerical methods. In this work, we combine a modern parallel computing technique based on multicore and graphics processing units (GPUs) and a sophisticated numerical method based on a new space-time adaptive algorithm. We evaluate each technique alone and in different combinations: multicore and GPU, multicore and GPU and space adaptivity, multicore and GPU and space adaptivity and time adaptivity. All the techniques and combinations were evaluated under different scenarios: 3D simulations on slabs, 3D simulations on a ventricular mouse mesh, ie, complex geometry, sinus-rhythm, and arrhythmic conditions. Our results suggest that multicore and GPU accelerate the simulations by an approximate factor of 33×, whereas the speedups attained by the space-time adaptive algorithms were approximately 48. Nevertheless, by combining all the techniques, we obtained speedups that ranged between 165 and 498. The tested methods were able to reduce the execution time of a simulation by more than 498× for a complex cellular model in a slab geometry and by 165× in a realistic heart geometry simulating spiral waves. The proposed methods will allow faster and more realistic simulations in a feasible time with no significant loss of accuracy.

Filament Tension and Phase Locking of Meandering Scroll Waves

Meandering spiral waves are often observed in excitable media such as the Belousov-Zhabotinsky reaction and cardiac tissue. We derive a theory for drift dynamics of meandering rotors in general reaction-diffusion systems and apply it to two types of external disturbances: an external field and curvature-induced drift in three dimensions. We find two distinct regimes: with small filament curvature, meandering scroll waves exhibit filament tension, whose sign determines the stability and drift direction. In the regimes of strong external fields or meandering motion close to resonance, however, phase locking of the meander pattern is predicted and observed.

Effects of pacing magnitudes and forms on bistability width in a modeled ventricular tissue

Bistability in periodically paced cardiac tissue is relevant to cardiac arrhythmias and its control. In the present paper, one-dimensional tissue of the phase I Luo-Rudy model is numerically investigated. The effects of various parameters of pacing signals on bistability width are studied. The following conclusions are obtained: (i) Pacing can be classified into two types: pulsatile and sinusoidal types. Pulsatile pacing reduces bistability width as its magnitude is increased. Sinusoidal pacing increases the width as its amplitude is increased. (ii) In a pacing period the hyperpolarizing part plays a more important role than the depolarizing part. Variations of the hyperpolarizing ratio in a period evidently change the width of bistability and its variation tendency. (iii) A dynamical mechanism is proposed to qualitatively explain the phenomena, which reveals the reason for the different effects of pulsatile and sinusoidal pacing on bistability. The methods for changing bistability width by external pacing may help control arrhythmias in cardiology.

Resonant drift of spiral waves in the complex ginzburg-landau equation

Weak periodic external perturbations of an autowave medium can cause large-distance directed motion of the spiral wave. This happens when the period of the perturbation coincides with, or is close to the rotation period of a spiral wave, or its multiple. Such motion is called resonant or parametric drift. It may be used for low-voltage defibrillation of heart tissue. Theory of the resonant drift exists, but so far was used only qualitatively. In this paper, we show good quantitative agreement of the theory with direct numerical simulations. This is done for Complex Ginzburg-Landau Equation, one of the simplest autowave models.

Origin choice and petal loss in the flower garden of spiral wave tip trajectories

Rotating spiral waves have been observed in numerous biological and physical systems. These spiral waves can be stationary, meander, or even degenerate into multiple unstable rotating waves. The spatiotemporal behavior of spiral waves has been extensively quantified by tracking spiral wave tip trajectories. However, the precise methodology of identifying the spiral wave tip and its influence on the specific patterns of behavior remains a largely unexplored topic of research. Here we use a two-state variable FitzHugh-Nagumo model to simulate stationary and meandering spiral waves and examine the spatiotemporal representation of the system's state variables in both the real (i.e., physical) and state spaces. We show that mapping between these two spaces provides a method to demarcate the spiral wave tip as the center of rotation of the solution to the underlying nonlinear partial differential equations. This approach leads to the simplest tip trajectories by eliminating portions resulting from the rotational component of the spiral wave.

Low energy defibrillation in human cardiac tissue: a simulation study

We aim to assess the effectiveness of feedback-controlled resonant drift pacing as a method for low energy defibrillation. Antitachycardia pacing is the only low energy defibrillation approach to have gained clinical significance, but it is still suboptimal. Low energy defibrillation would avoid adverse side effects associated with high voltage shocks and allow the application of implantable cardioverter defibrillator (ICD) therapy, in cases where such therapy is not tolerated today. We present results of computer simulations of a bidomain model of cardiac tissue with human atrial ionic kinetics. Reentry was initiated and low energy shocks were applied with the same period as the reentry, using feedback to maintain resonance. We demonstrate that such stimulation can move the core of reentrant patterns, in the direction that depends on the location of the electrodes and the time delay in the feedback. Termination of reentry is achieved with shock strength one-order-of-magnitude weaker than in conventional single-shock defibrillation. We conclude that resonant drift pacing can terminate reentry at a fraction of the shock strength currently used for defibrillation and can potentially work where antitachycardia pacing fails, due to the feedback mechanisms. Success depends on a number of details that these numerical simulations have uncovered.

Mechanisms of the acute ischemia-induced arrhythmogenesis – A simulation study

The underlying ionic mechanisms of ischemic-induced arrhythmia were studied by the computer simulation method. To approximate the real situation, ischemic cells were simulated by considering the three major component conditions of acute ischemia (elevated extracellular K(+) concentration, acidosis and anoxia) at the level of ionic currents and ionic concentrations, and a round ischemic zone was introduced into a homogeneous healthy sheet to avoid sharp angle of the ischemic tissue. The constructed models were solved using the operator splitting and adaptive time step methods, and the perturbation finite difference (PFD) scheme was first used to integrate the partial differential equations (PDEs) in the model. The numerical experiments showed that the action potential durations (APDs) of ischemic cells did not exhibited rate adaptation characteristic, resulting in flattening of the APD restitution curve. With reduction of sodium channel availability and long recovery of excitability, refractory period of the ischemic tissue was significantly prolonged, and could no longer be considered as same as APD. Slope of the conduction velocity (CV) restitution curve increased both in normal and ischemic region when pacing cycle length (PCL) was short, and refractory period dispersion increased with shortening of PCL as well. Therefore, dynamic changes of CV and dispersion of refractory period rather than APD were suggested to be the fundamental mechanisms of arrhythmia in regional ischemic myocardium.

Structure-Related Initiation of Reentry by Rapid Pacing in Monolayers of Cardiac Cells

This study examines how a zigzag pattern of conduction, a form of structural heterogeneity frequently found in old or diseased hearts, affects the vulnerability to reentry during rapid pacing. A central rectangular island (8×4 mm) containing a predefined zigzag pattern was created in cultured isotropic monolayers of neonatal rat ventricular myocytes. Impulse propagation was optically mapped from 253 sites using voltage-sensitive dye and was anisotropic within the zigzag island. With increasing interval between neighboring transverse connections ( a ), relative to the distance between longitudinal strands ( b ), transverse conduction velocity (CV) decreased to 66±6%, 20±2%, and 15±2% of CV in the surrounding isotropic region, whereas longitudinal CV increased to 102±8%, 113±12%, and 131±23% for a : b ratios of 1:1, 1:5, and 1:9, respectively. During rapid pacing, propagation distal to the island was steered from the side of the island with more transverse connections (“dominant” side) toward the side with fewer connections (“weak” side). Increased asymmetry in the pattern accentuated this effect, and resulted in increased rate-dependent differences in CV on the 2 sides. Consequently, a functional obstacle formed on the weak side, followed by development of single loop reentry. The reentrant wave revolved around a line of block defined by the border of the island. Reentry chirality was determined by the weak side location, and the pacing rate needed to initiate reentry decreased with increased asymmetry in the pattern. In conclusion, reentry is readily induced by rapid pacing in confluent cardiac cell monolayers containing a central and asymmetric island of zigzag conduction.

Enhanced self-termination of re-entrant arrhythmias as a pharmacological strategy for antiarrhythmic action

Ventricular tachycardia and fibrillation are potentially lethal cardiac arrhythmias generated by high frequency, irregular spatio-temporal electrical activity. Re-entrant propagation has been demonstrated as a mechanism generating these arrhythmias in computational and in vitro animal models of these arrhythmias. Re-entry can be idealised in homogenous isotropic virtual cardiac tissues as spiral and scroll wave solutions of reaction-diffusion equations. A spiral wave in a bounded medium can be terminated if its core reaches a boundary. Ventricular tachyarrhythmias in patients are sometimes observed to spontaneously self-terminate. One possible mechanism for self-termination of a spiral wave is meander of its core to an inexcitable boundary. We have previously proposed the hypothesis that the spatial extent of meander of a re-entrant wave in the heart can be directly related to its probability of self-termination, and so inversely related to its lethality. Meander in two-dimensional virtual ventricular tissues based on the Oxsoft family of cell models, with membrane excitation parameters simulating the inherited long Q-T syndromes has been shown to be consistent with this hypothesis: the largest meander is seen in the syndrome with the lowest probability of death per arrhythmic episode. Here we extend our previous results to virtual tissues based on the Luo-Rudy family of models. Consistent with our hypothesis, for both families of models, whose different ionic mechanisms produce different patterns of meander, the LQT virtual tissue with the larger meander simulates the syndrome with the lower probability of death per episode. Further, we search the parameter space of the repolarizing currents to find their conductance parameter values that give increased meander of spiral waves. These parameters may provide targets for antiarrhythmic drugs designed to act by increasing the likelihood of self-termination of re-entrant arrhythmias. (c) 2002 American Institute of Physics.

A model for glucose-induced wave propagation in pancreatic islets of Langerhans

A reaction-diffusion type model is constructed, describing the spatio-temporal dynamics of the basic intracellular variables assumed to be involved in the initiation of the insulin secretion process by beta -cells in the pancreatic islets of Langerhans. The model includes equations for the electric membrane potential of the cells, with respective kinetics for ionic currents, for concentrations of both free and stored intracellular calcium, and for the intra- and extracellular concentrations of glucose. An empirical expression connecting the equation for the intracellular glucose concentration to the electrical equation is introduced. The model reproduces the events observed in experiments in vitro upon external glucose application to the islets of Langerhans, such as usual bursting oscillations of the membrane potential and corresponding oscillations of the intracellular calcium concentration. It also allows simulation of electric wave propagation through the islet, initiated by the spatial gradient of glucose concentration within the islet. The gradient emerges due to glucose diffusing into the islets from the external medium, being high at the edges. The latter results show that glucose diffusion presents a means for wave initiation in the islets, which supports our previous assumption (Aslanidi et al., 2001).

Characterization of patterned irregularity in locally interacting, spatially extended systems: Ventricular fibrillation

The re-entrant ventricular arrhythmias of monomorphic ventricular tachycardia and fibrillation are produced by abnormal spatio-temporal patterns of propagation in the ventricular myocardium. These behaviors can be described by solutions of reaction-diffusion equation excitable medium models. The direct comparison of such solutions with existing experimental observations is virtually impossible as there are too many factors to be taken into account, including not only the complicated dynamics of the re-entrant waves of excitation in the tissue, but also the way the appearance of these waves on the surface is modified by the inhomogeneity, anisotropy and three-dimensional nature of heart tissue. One way of indirect comparison is to compare characteristics of the complexity of the model and the real data, that are invariant under these modifications of the signal. Karhunen-Loeve decomposition is a standard tool for evaluating the complexity of multidimensional signals. A comparison of the separate and conjoint complexities of the signals on the opposite sides of the preparation can be considered as an indicator how much three-dimensional effects are essential in the preparation behavior. (c) 2001 American Institute of Physics.

Computational models of normal and abnormal action potential propagation in cardiac tissue: linking experimental and clinical cardiology

Computational models have the potential to make a huge impact on our understanding of normal and abnormal cardiac function. The aim of this article is to review tools that have been developed to simulate the electrophysiology of cardiac cells and tissue, and to show how computational models have been used to gain insight into normal and abnormal action potential propagation. Some of the practical problems experienced in the development and application of these models are described, and examples are given.

Re-entrant cardiac arrhythmias in computational models of long QT myocardium

The long QT syndrome (LQTS) is an inherited disorder in which repolarization of cardiac ventricular cells is prolonged. Patients with the LQTS are at an increased risk of ventricular cardiac arrhythmias. Two phenotypes of the inherited LQTS are caused by defects in K(+)channels (LQT1 and LQT2) and one by defects in Na(+)channels (LQT3). Patients with LQT1 are more likely to have self-terminating arrhythmias than those with LQT3. The aim of this computational study was to propose an explanation for this finding by comparing the vulnerability of normal and LQT tissue to re-entry, and estimating the likelihood of self-termination by motion of re-entrant waves to an inexcitable boundary in simulated LQT1, LQT2 and LQT3 tissue. We modified a model of mammalian cardiac cells to simulate LQT1 by reducing maximal I(K(s))conductance, LQT2 by reducing maximal I(K(r))conductance, and LQT3 by preventing complete inactivation of I(Na)channels. Each simulated phenotype was incorporated into a computational model of action potential propagation in one- and two-dimensional homogeneous tissue. Simulated LQT tissue was no more vulnerable to re-entry than simulated normal tissue, but the motion of re-entrant waves in simulated LQT1 tissue was between 2 and 5 times greater than the motion of re-entrant waves in simulated LQT2 and LQT3 tissue. These findings suggest that LQT arrhythmias do not result from increased vulnerability to re-entry, and that re-entry once initiated is more likely to self-terminate by moving to an inexcitable tissue boundary in LQT1 than in LQT2 and LQT3. This finding is consistent with clinical observations.

Reentrant waves and their elimination in a model of mammalian ventricular tissue

The vulnerability to reentrant wave propagation, its characteristics (period, meander, and stability), the effects of rotational transmural anisotropy, and the control of reentrant waves by small amplitude perturbations and large amplitude defibrillating shocks are investigated theoretically and numerically for models based on high order, stiff biophysically derived excitation equations.

Two forms of spiral-wave reentry in an ionic model of ischemic ventricular myocardium

It is well known that there is considerable spatial inhomogeneity in the electrical properties of heart muscle, and that the many interventions that increase this initial degree of inhomogeneity all make it easier to induce certain cardiac arrhythmias. We consider here the specific example of myocardial ischemia, which greatly increases the electrical heterogeneity of ventricular tissue, and often triggers life-threatening cardiac arrhythmias such as ventricular tachycardia and ventricular fibrillation. There is growing evidence that spiral-wave activity underlies these reentrant arrhythmias. We thus investigate whether spiral waves might be induced in a realistic model of inhomogeneous ventricular myocardium. We first modify the Luo and Rudy [Circ. Res. 68, 1501-1526 (1991)] ionic model of cardiac ventricular muscle so as to obtain maintained spiral-wave activity in a two-dimensional homogeneous sheet of ventricular muscle. Regional ischemia is simulated by raising the external potassium concentration ([K(+)](o)) from its nominal value of 5.4 mM in a subsection of the sheet, thus creating a localized inhomogeneity. Spiral-wave activity is induced using a pacing protocol in which the pacing frequency is gradually increased. When [K(+)](o) is sufficiently high in the abnormal area (e.g., 20 mM), there is complete block of propagation of the action potential into that area, resulting in a free end or wave break as the activation wave front encounters the abnormal area. As pacing continues, the free end of the activation wave front traveling in the normal area increasingly separates or detaches from the border between normal and abnormal tissue, eventually resulting in the formation of a maintained spiral wave, whose core lies entirely within an area of normal tissue lying outside of the abnormal area ("type I" spiral wave). At lower [K(+)](o) (e.g., 10.5 mM) in the abnormal area, there is no longer complete block of propagation into the abnormal area; instead, there is partial entrance block into the abnormal area, as well as exit block out of that area. In this case, a different kind of spiral wave (transient "type II" spiral wave) can be evoked, whose induction involves retrograde propagation of the action potential through the abnormal area. The number of turns made by the type II spiral wave depends on several factors, including the level of [K(+)](o) within the abnormal area and its physical size. If the pacing protocol is changed by adding two additional stimuli, a type I spiral wave is instead produced at [K(+)](o)=10.5 mM. When pacing is continued beyond this point, apparently aperiodic multiple spiral-wave activity is seen during pacing. We discuss the relevance of our results for arrythmogenesis in both the ischemic and nonischemic heart. (c) 1998 American Institute of Physics.

Qualitative modeling of mechanoelectrical feedback in a ventricular cell

Mechanical changes in the heart muscle can influence its electrical properties through a process called mechanoelectrical feedback (MEF). This feedback can operate via changes in calcium dynamics during the cross-bridge cycle or via mechanosensitive (stretch-activated) channels. We present a four-variable ordinary differential equation (ODE) system that caricatures the electrical and mechanical activity of a ventricular cell and their mutual interactions. A three-variable excitable system with restitution properties of the FitzHugh-Nagumo type is coupled to a fourth equation which describes changes in cell length during a lightly loaded contraction. The resulting four-variable system models MEF in a cell and can be incorporated into spatially distributed models for mechanoelectric behavior during wave propagation in the cardiac tissue.