Dynamics of Scroll Wave in a Three-Dimensional System with Changing Gradient

The dynamics of a scroll wave in an excitable medium with gradient excitability is studied in detail. Three parameter regimes can be distinguished by the degree of gradient. For a small gradient, the system reaches a simple rotating synchronization. In this regime, the rigid rotating velocity of spiral waves is maximal in the layers with the highest filament twist. As the excitability gradient increases, the scroll wave evolutes into a meandering synchronous state. This transition is accompanied by a variation in twisting rate. Filament twisting may prevent the breakup of spiral waves in the bottom layers with a low excitability with which a spiral breaks in a 2D medium. When the gradient is large enough, the twisted filament breaks up, which results in a semi-turbulent state where the lower part is turbulent while the upper part contains a scroll wave with a low twisting filament.     

Negative Tension of Scroll Wave Filaments and Turbulence in Three-Dimensional Excitable Media and Application in Cardiac Dynamics

Scroll waves are vortices that occur in three-dimensional excitable media. Scroll waves have been observed in a variety of systems including cardiac tissue, where they are associated with cardiac arrhythmias. The disorganization of scroll waves into chaotic behavior is thought to be the mechanism of ventricular fibrillation, whose lethality is widely known. One possible mechanism for this process of scroll wave instability is negative filament tension. It was discovered in 1987 in a simple two variables model of an excitable medium. Since that time, negative filament tension of scroll waves and the resulting complex, often turbulent dynamics was studied in many generic models of excitable media as well as in physiologically realistic models of cardiac tissue. In this article, we review the work in this area from the first simulations in FitzHugh–Nagumo type models to recent studies involving detailed ionic models of cardiac tissue. We discuss the relation of negative filament tension and tissue excitability and the effects of discreteness in the tissue on the filament tension. Finally, we consider the application of the negative tension mechanism to computational cardiology, where it may be regarded as a fundamental mechanism that explains differences in the onset of arrhythmias in thin and thick tissue.     

Is monomorphic arrhythmia monomorphic?

It is known that the properties of myocardium affect ECG. We studied the dependence of some ECG characteristics on the threshold of excitation by means of mathematical modeling of monomorphic arrhythmias in a homogeneous two-dimensional excitable medium. We found that monomorphic arrhythmias appear at both decreased and increased values of the excitation threshold, but there are some differences between ECG characteristics for such cases.     

Is monomorphic tachycardia indeed monomorphic?

The dependence of some ECG characteristics on the excitation threshold was studied by mathematical modeling of cardiac arrhythmia in a 2D homogeneous excitable medium. It is shown that monomorphic tachycardias can arise both at elevated and at lowered excitability but the ECG characteristics in these cases are different.     

Quantitative analysis of variability of electrocardiograms typical for polymorphic arrhythmias

A new approach to the analysis of variability of electrocardiograms (ECGs) typical of polymorphic arrhythmias is developed. In these ECGs, separate QRS complexes can be often hardly identified. As a result, the mathematical methods that have been elaborated hitherto are not suitable for such arrhythmias. The approach presented here is based on the quantitative estimation of the variability of neighboring parts of the ECG. In this case, the necessity of the identification of separate QRS complexes ceases to be significant. Based on this approach, the analysis of normalized ECG variability is developed in the framework of which two indices that characterize the oscillation variability and its changes in time are related to a part of the ECG and/or the ECG as a whole. Variations of these indices allow both the polymorphism of a separate ECG to be estimated and different ECGs to be compared with each other. The method presented may be useful in studies of the mechanisms and in the diagnosis of polymorphic arrhythmias.     

Instability of a three-dimnsional linear vortex in a simple model of a heterogenous excitable medium

We studied the dynamics of three-dimensional linear vortex in a heterogeneous excitable medium. Using the Aliev-Panfilov model we showed that even a small heterogeneity in excitation threshold can lead to drastic deformations of the vortex filament and an instability in its behavior.     

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.     

Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias

We introduce the concept of a contracting excitable medium that is capable of conducting non-linear waves of excitation that in turn initiate contraction. Furthermore, these kinematic deformations have a feedback effect on the excitation properties of the medium. Electrical characteristics resemble basic models of cardiac excitation that have been used to successfully study mechanisms of reentrant cardiac arrhythmias in electrophysiology. We present a computational framework that employs electromechanical and mechanoelectric feedback to couple a three-variable FitzHugh–Nagumo-type excitation-tension model to the non-linear stress equilibrium equations, which govern large deformation hyperelasticity. Numerically, the coupled electromechanical model combines a finite difference method approach to integrate the excitation equations, with a Galerkin finite element method to solve the equations governing tissue mechanics. We present example computations demonstrating various effects of contraction on stationary rotating spiral waves and spiral wave break. We show that tissue mechanics significantly contributes to the dynamics of electrical propagation, and that a coupled electromechanical approach should be pursued in future electrophysiological modelling studies.     

The emergence of wave emitting centres in an excitable medium

The response of an excitable biological medium to a double local stimulus is considered within the context of a mathematical model for a layer of starving cells of Dictyostelium discoideum, with both spatially one- and two-dimensional (1D and 2D) system being investigated. In contrast to the response usually seen in excitable media, whereby each superthreshold stimulus delivered to the relaxed medium results in the initiation of just one travelling wave, a source emitting a sequence of waves can develop in the present excitable medium after the second stimulus. In a 1D system, only transient wave sources forming a limited number of waves are found. In 2D systems, a permanent wave sources consisting in a pair of spirals are observed as well as the transient wave sources forming circular wave patterns. The general features of the medium dynamics that underlie the observed responses to the double stimulus are discussed.     

Pulsed low-energy stimulation initiates electric turbulence in cardiac tissue

Interruptions in nonlinear wave propagation, commonly referred to as wave breaks, are typical of many complex excitable systems. In the heart they lead to lethal rhythm disorders, the so-called arrhythmias, which are one of the main causes of sudden death in the industrialized world. Progress in the treatment and therapy of cardiac arrhythmias requires a detailed understanding of the triggers and dynamics of these wave breaks. In particular, two very important questions are: 1) What determines the potential of a wave break to initiate re-entry? and 2) How do these breaks evolve such that the system is able to maintain spatiotemporally chaotic electrical activity? Here we approach these questions numerically using optogenetics in an in silico model of human atrial tissue that has undergone chronic atrial fibrillation (cAF) remodelling. In the lesser studied sub-threshold illumination régime, we discover a new mechanism of wave break initiation in cardiac tissue that occurs for gentle slopes of the restitution characteristics. This mechanism involves the creation of conduction blocks through a combination of wavefront-waveback interaction, reshaping of the wave profile and heterogeneous recovery from the excitation of the spatially extended medium, leading to the creation of re-excitable windows for sustained re-entry. This finding is an important contribution to cardiac arrhythmia research as it identifies scenarios in which low-energy perturbations to cardiac rhythm can be potentially life-threatening.     

Dynamics of intramural scroll waves in three-dimensional continuous myocardium with rotational anisotropy.

It has been suggested that reentrant activity in three-dimensional cardiac muscle may be organized as a scroll wave rotating around a singularity line called the filament. Experimental studies indicate that filaments are often concealed inside the ventricular wall and consequently, scroll waves do not manifest reentrant activity on the surface. Here we analyse how such concealed scroll waves are affected by a twisted anisotropy resulting from rotation of layers of muscle fibers inside the ventricular wall. We used a computer model of a ventricular slab (15x15x15 mm(3)) with a fiber twist of 120 degrees from endocardium to epicardium. The action potential was simulated using FitzHugh-Nagumo equations. Scroll waves with rectilinear filaments were initiated at various depths of the slab and at different angles with respect to fiber orientation. The analysis shows that independent of initial conditions, after a certain transitional period, the filament aligns with the local fiber orientation. The alignment of the filament is determined by the directional variations in cell coupling due to fiber rotation and by boundary conditions. Our findings provide a mechanistic explanation for the prevalence of intramural reentry over transmural reentry during polymorphic ventricular tachycardia and fibrillation.     

The pinwheel experiment re-revisited.

Virtual electrode induced phase singularity hypothesis explains the origin of cardiac arrhythmias caused by artificial electrical induction of rotors, i.e. vortex-like self-sustained sources of activity. This mechanism is thought to underlie both stimulus-induced arrhythmias and shock defibrillation therapy. In this paper, we extend this hypothesis to three dimensions using the bidomain model of cardiac tissue. We predict that virtual electrode polarization can produce three topologically distinct types of rotors anchored to: (1) transmural I-shaped scroll wave filaments; (2) near-surface U-shaped scroll wave filaments; and (3) intramural O-shaped scroll wave filaments.     

Scroll wave dynamics in a three-dimensional cardiac tissue model: roles of restitution, thickness, and fiber rotation

Scroll wave (vortex) breakup is hypothesized to underlie ventricular fibrillation, the leading cause of sudden cardiac death. We simulated scroll wave behaviors in a three-dimensional cardiac tissue model, using phase I of the Luo-Rudy (LR1) action potential model. The effects of action potential duration (APD) restitution, tissue thickness, filament twist, and fiber rotation were studied. We found that APD restitution is the major determinant of scroll wave behavior and that instabilities arising from APD restitution are the main determinants of scroll wave breakup in this cardiac model. We did not see a "thickness-induced instability" in the LR1 model, but a minimum thickness is required for scroll breakup in the presence of fiber rotation. The major effect of fiber rotation is to maintain twist in a scroll wave, promoting filament bending and thus scroll breakup. In addition, fiber rotation induces curvature in the scroll wave, which weakens conduction and further facilitates wave break.     

Dynamics of a transmural scroll wave in ground squirrel myocardium

A study was made of tachyarrhythmia evoked by a premature stimulus (4 ms at 4–5 diastolic thresholds) following a train of rectangular pulses (4 ms at 2 diastolic thresholds, repetition rate of 0.5 or 2 s^?1). The spatiotemporal distribution of the potential over the endo- and epicardial surfaces of a thin (～1 mm) specimen of ground squirrel ventricular myocardium was monitored with two arrays of 32 unipolar electrodes each. The electrographic data were processed into isochrone maps reflecting the spread of activation over the surfaces. These maps were further analyzed to infer the 3D structure and dynamics of the vortex (scroll) wave. During the evolution of a transmural scroll, (i) the filament could be normal to the myocardial surfaces as well as oblique at varying angles (down to 12°); (ii) the scroll could drift as a whole, whereby the filament remained self-parallel or changed its inclination; in other cases, one (endocardial) core was anchored while the other changed its position (precession of the filament); (iii) the vortex cores on both or only one surface changed in size and shape; (iv) the filament could be repeatedly twisted through various angles and untwisted. Scroll rotation was attended with excitation breakthroughs that might have originated from filament bending as well as from focal sources.     

3D Finite Element Electrical Model of Larval Zebrafish ECG Signals

Assessment of heart function in zebrafish larvae using electrocardiography (ECG) is a potentially useful tool in developing cardiac treatments and the assessment of drug therapies. In order to better understand how a measured ECG waveform is related to the structure of the heart, its position within the larva and the position of the electrodes, a 3D model of a 3 days post fertilisation (dpf) larval zebrafish was developed to simulate cardiac electrical activity and investigate the voltage distribution throughout the body. The geometry consisted of two main components; the zebrafish body was modelled as a homogeneous volume, while the heart was split into five distinct regions (sinoatrial region, atrial wall, atrioventricular band, ventricular wall and heart chambers). Similarly, the electrical model consisted of two parts with the body described by Laplace’s equation and the heart using a bidomain ionic model based upon the Fitzhugh-Nagumo equations. Each region of the heart was differentiated by action potential (AP) parameters and activation wave conduction velocities, which were fitted and scaled based on previously published experimental results. ECG measurements in vivo at different electrode recording positions were then compared to the model results. The model was able to simulate action potentials, wave propagation and all the major features (P wave, R wave, T wave) of the ECG, as well as polarity of the peaks observed at each position. This model was based upon our current understanding of the structure of the normal zebrafish larval heart. Further development would enable us to incorporate features associated with the diseased heart and hence assist in the interpretation of larval zebrafish ECGs in these conditions.     

Dynamics of virtual electrode-induced scroll-wave reentry in a 3D bidomain model

Functional reentry in the heart can be caused by a wave front of excitation rotating around its edge. Previous simulations on the basis of monodomain cable equations predicted the existence of self-sustained, vortex-like wave fronts (scroll waves) rotating around a filament in three dimensions. In our simulations, we used the more accurate bidomain model with modified Beeler-Reuter ionic kinetics to study the dynamics of scroll-wave filaments in a 16 x 8 x 1.5-mm slab of ventricular tissue with straight fibers. Wave fronts were identified as the areas with inward current. Their edges represented the filaments. Both transmural and intramural reentries with I- and U-shaped filaments, respectively, were obtained by the S1-S2 point stimulation protocol through the virtual electrode-induced phase singularity mechanism. The filaments meandered along elongated trajectories and tended to attach to the tissue boundaries exposed to air (no current flow) rather than to the bath (zero extracellular potential). They completely detached from electroporated (zero transmembrane potential) boundaries. In our simulations, the presence of the bath led to generation of only U-shaped filaments, which survived for the 1.5-mm-thick slab but not for the slabs of 0.5- or 3-mm thicknesses. Thus boundary conditions may be another determinant of the type and dynamics of reentry.     

A condition for setting off ectopic waves in computational models of excitable cells

The purpose of this paper is to study the stability of steady state solutions of the Monodomain model equipped with Luo-Rudy I kinetics. It is well established that re-entrant arrhythmias can be created in computational models of excitable cells. Such arrhythmias can be initiated by applying an external stimulus that interacts with a partially refractory region, and spawn breaking waves that can eventually generate extremely complex wave patterns commonly referred to as fibrillation. An ectopic wave is one possible stimulus that may initiate fibrillation. Physiologically, it is well known that ectopic waves exist, but the mechanism for initiating ectopic waves in a large collection of cells is poorly understood. In the present paper we consider computational models of collections of excitable cells in one and two spatial dimensions. The cells are modeled by Luo-Rudy I kinetics, and we assume that the spatial dynamics is governed by the Monodomain model. The mathematical analysis is carried out for a reduced model that is known to provide good approximations of the initial phase of solutions of the Luo-Rudy I model. A further simplification is also introduced to motivate and explain the results for the more complicated models. In the analysis the cells are divided into two regions; one region (N) consists of normal cells as model by the standard Luo-Rudy I model, and another region (A) where the cells are automatic in the sense that they would act as pacemaker cells if they where isolated from their surroundings. We let delta denote the spatial diffusion and a denote a characteristic length of the automatic region. It has previously been shown that reducing diffusion or increasing the automatic region enhances ectopic activity. Here we derive a condition for the transition from stable resting state to ectopic wave spread. Under suitable assumptions on the model we provide mathematical and computational arguments indicating that there is a constant eta such that a steady state solution of this system is stable whenever delta approximately > etaa(2), and unstable whenever delta approximately < etaa(2).     

Electrical Wave Propagation in an Anisotropic Model of the Left Ventricle Based on Analytical Description of Cardiac Architecture

We develop a numerical approach based on our recent analytical model of fiber structure in the left ventricle of the human heart. A special curvilinear coordinate system is proposed to analytically include realistic ventricular shape and myofiber directions. With this anatomical model, electrophysiological simulations can be performed on a rectangular coordinate grid. We apply our method to study the effect of fiber rotation and electrical anisotropy of cardiac tissue (i.e., the ratio of the conductivity coefficients along and across the myocardial fibers) on wave propagation using the ten Tusscher–Panfilov (2006) ionic model for human ventricular cells. We show that fiber rotation increases the speed of cardiac activation and attenuates the effects of anisotropy. Our results show that the fiber rotation in the heart is an important factor underlying cardiac excitation. We also study scroll wave dynamics in our model and show the drift of a scroll wave filament whose velocity depends non-monotonically on the fiber rotation angle; the period of scroll wave rotation decreases with an increase of the fiber rotation angle; an increase in anisotropy may cause the breakup of a scroll wave, similar to the mother rotor mechanism of ventricular fibrillation.     

Dynamics of a vortex with the U-shaped filament (ends “anchored”) in the heart of the ground squirrel

The dynamics of an electrical scroll wave with the U-shaped filament with both ends of the filament being “anchored” on the endocardial surface and the dependence of the structure of pseudoECG on the dynamics of the vortex during the development of polymorphic tachysystolia have been studied by applying premature stimuli to the “target phase” with subsequent registration of the spatial and temporal distribution of electrical potential throughout the surface (endocardial and epicardial) of a thin (≈1 mm) preparation. It was found that (1) the pseudoECG of the polymorphic form during the tachysystolia attack can be observed in the case that the position of the filament ends on the surfaces of the preparation does not practically change from turn to turn (filament ends are “anchored”); (2) the thread of a scroll wave during this attack can twist and untwin (twisted filament), just as it was the case for scroll waves with a straight filament; (3) in the case of pseudoECG of polymorphic form, the twisting and untwining of the filament were stronger (the angle of maximal twisting was 120 degrees and more), and the angle of twisting changed by a substantially greater value from turn to turn as compared with the pseudoECG of monomorphic form; (4) in the case of pseudoECG of polymorphic form, the time interval between the appearance of waves on the surfaces of the preparation (T _epi-endo) was substantially greater and changed to a greater extent from turn to turn of the vortex; and (5) simultaneously with the appearance of pseudoECG of polymorphic form and the onset of changes in the twisting of the scroll and the T _epi-endo interval indicated in (2–4), significant changes in the patterns of coverage of the surface by excitation occurred. Based on the results obtained, an explanation of the reasons for the appearance of excitation breakdown patterns on the surface of the myocardium was proposed, which differs from the traditional viewpoint. These patterns may be the result of reflection on myocardial surfaces of the activity of not different simultaneously occurring sources of initiation of excitation but of a single three-dimensional vortex whose filament twists when passing through the thickness of the myocardium and can closely approach one or the other surface.     

ECG Changes in Subarachnoid Haemorrhage: A Synopsis

Subarachnoid haemorrhage (SAH) is a neurological emergency with high mortality rates. It is mainly caused by rupture of an aneurysm (congenital/infectious/traumatic) or rupture of an arteriovenous malformation. Electrocardiograms (ECGs) done in patients with SAH have shown morphological changes as well as arrhythmias. Subarachnoid haemorrhage (SAH) patients have often been misdiagnosed to have cardiac abnormalities based on their ECGs when in many of those instances the ECG change had been the result of the SAH itself. They have led to unnecessary and wasteful investigations and therapies in many occasions. Hence the current article is an effort at consolidating the information available in an attempt to avoid possible errors in diagnosis by house staff and internists. There are two mechanisms that might mediate ECG changes in patients with SAH, i.e. autonomic neural stimulation from the hypothalamus or elevated levels of circulating catecholamine. Hypothalamic stimulation may cause ECG changes without associated myocardial damage whereas elevated catecholamine levels have been correlated with QT-interval prolongation and myocardial damage.