Generation of reentry in anisotropic myocardium

Overdrive pacing of spiral waves in a model of human ventricular tissue

High-voltage electrical defibrillation remains the only reliable method of quickly controlling life-threatening cardiac arrhythmias. This paper is devoted to studying an alternative approach, low-voltage cardioversion (LVC), which is based on ideas from non-linear dynamics and aims to remove sources of cardiac arrhythmias by applying high-frequency stimulation to cardiac tissue. We perform a detailed in-silico study of the elimination of arrhythmias caused by rotating spiral waves in a TP06 model of human cardiac tissue. We consider three parameter sets with slopes of the APD restitution curve of 0.7, 1.1 and 1.4, and we study LVC at the baseline and under the blocking of INa and ICaL and under the application of the drugs verapamil and amiodarone. We show that pacing can remove spiral waves; however, its efficiency can be substantially reduced by dynamic instabilities. We classify these instabilities and show that the blocking of INa and the application of amiodarone increase the efficiency of the method, while the blocking of ICaL and the application of verapamil decrease the efficiency. We discuss the mechanisms and the possible clinical applications resulting from our study.

Use of an automaton model to suggest methods for cessation of intractable fibrillatory activity

BACKGROUND

Atrial fibrillation (AF) is the most common heart arrhythmia, and permanent AF is an intractable medical problem. If cessation of permanent AF were possible, via extensive substrate ablation or multisite stimulation, it could significantly improve the public health.

METHOD

A cellular automaton composed of 576 × 576 computerized grid nodes, described in detail previously, was used to test hypotheses concerning the cessation of fibrillatory electrical activity. A refractory period gradient across the grid, and addition of randomly located nonconducting fibers, were utilized as conditions leading to fibrillatory activity. A premature S1-S2 stimulus was applied to one grid corner, resulting in unidirectional conduction block at some locations, followed by rotational activity and random propagation of activation wavelets throughout the grid, none of which terminated spontaneously. Simulated ablation lesions of dimension 20 × 20 grid nodes, imparted at core locations of rotational activity, and multisite electrode stimulation (MES) applied at nodes where recovery of excitability had occurred, were used in attempts to terminate fibrillatory activity. Six impositions of random fiber location were utilized in separate trials.

RESULTS

Simulated ablation lesions eliminated the targeted swirling vortices; however, additional vortices then often appeared at other locations. After ablating approximately one third of the grid area, localized vortices were eliminated, but individual wavelets continued to propagate about longer viable pathways forming at ablation lesions. Thus extensive ablation was unsuccessful in terminating arrhythmia. However, MES applied uniformly throughout the grid, with a coupling interval slightly longer than the maximum refractory period, terminated fibrillatory activity in some trials. More efficaciously, application of MES with a coupling interval half the maximum refractory period of the grid succeeded in capture of activation at all nodes, and when followed by a doubling of the MES coupling interval, resulted in cessation of all fibrillatory activity.

CONCLUSIONS

It is possible to terminate simulated fibrillatory activity in a computerized grid that would otherwise be intractable, using multisite stimulation with a coupling interval related to the maximum refractory period of the substrate. If each MES stimulating electrode could be individually controlled, it would be possible to apply a stimulation pattern mimicking the normal heart activation sequence.

Effects of refractory gradients and ablation on fibrillatory activity

BACKGROUND

The mechanisms involved in onset, maintenance, and termination of atrial fibrillation are not well understood. A biophysical model could be useful to determine how the events unfold.

METHOD

A two-dimensional cellular automaton consisting of 576 × 576 grid nodes was implemented to demonstrate the types of electrical activity that may occur in compromised atrial substrate. Electrical activation between nodes was made anisotropic (2:1), and the refractory period (RP) was adjusted from 74 to 192 ms in the spatial domain. Presence of collagen fibers were simulated as short lines of conduction block at many random grid sites, while ablation lesions were delineated as longer lines of block. An S1-S2 pulse from one grid corner was utilized to initiate simulated electrical activity. Simulations were done in which 1. no ablation lines, 2. random ablation lines, and 3. parallel ablation lines were added to the grid to determine how this affected the formation and annihilation of rotational activity after S1-S2 stimulation.

RESULTS

As the premature (S2) wavefront traversed the grid, rotational activity formed near boundaries where wavefronts propagated from shorter to longer refractory regions, causing unidirectional block, and were anchored by fiber clusters. Multiple wavelets appeared when wavefronts originating from different driving rotational features collided, and/or by their encounter with RP discontinuities. With the addition of randomly orientated simulated ablation lesions, followed by reinduction of fibrillatory activity, mean activation interval (AI) prolonged from a baseline level of 144.2 ms-160.3 ms (p < 0.001 in most comparisons). During fibrillatory activity, when parallel ablation lines were added to short RP regions, AI prolonged to 150.4 ms (p < 0.001), and when added to long RP regions, AI prolonged to 185.3 ms (p < 0.001). In all cases, AI prolongation after simulated ablation resulted from reduced number and/or from the isolation of local drivers, so that distant drivers in short RP regions activated long RP regions N:1, while distant drivers in long RP regions activated short RP regions at a relatively slow rate.

CONCLUSIONS

An automaton model was found useful to generate and test hypotheses concerning fibrillatory activity, which can then be validated in the clinical electrophysiology laboratory.

Development of an automaton model of rotational activity driving atrial fibrillation

BACKGROUND

Atrial fibrillation (AF) is difficult to treat effectively, owing to uncertainty in where to best ablate to eliminate arrhythmogenic substrate. A model providing insight into the electrical activation events would be useful to guide catheter ablation strategy. Method A two-dimensional, 576×576 node automaton was developed to simulate atrial electrical activity. The substrate field was altered by the presence of differing refractory period at varying locations. Fibrosis was added in the form of short, randomly positioned lines of conduction block. Larger areas of block were used to simulate ablation lesions. Anisotropy was imposed in a 2:1 ratio. A premature electrical impulse from one of four grid corners was utilized to initiate activation.

RESULTS

Rotational activity was uninducible when refractory patch dimensions were less than 20×20mm. For larger refractory regions, a single premature stimulus was capable of inducing an average of 1.19±1.10 rotors, which often formed near the patch edges. A maximum of 5 rotors formed when refractory patch dimensions approached the size of the entire left atrial virtual field. Rotors formed along a refractory patch edge, after wavefront arrival was delayed at turning points or due to the presence of a fiber cluster of sufficient size. However, rotational activity could also occur around a large fiber cluster without the need of spatially variable refractoriness. When obstacles to conduction were lacking in size, nascent rotors drifted and either extinguished, or stabilized upon anchoring at a sufficiently large fiber cluster elsewhere in the field. Transient rotors terminated when traversing a region with differing refractory periods, if no obstacle to conduction was present to sufficiently delay wavefront arrival beyond the longest refractory period. Other rotors were annihilated when a nearby rotor with faster spin rate gradually interrupted the activation pathway. Elimination of anchors by removal, or by simulated ablation over a sufficient region, prevented rotor onset at a particular location where it would otherwise form.

CONCLUSIONS

The presence of obstacles to conduction and spatial differences in refractory period are important parameters for initiating and maintaining rotational activity in this simulation of an atrial substrate.

Asymptotic wave propagation in excitable media

Wave shape and velocity are important issues in reaction-diffusion systems, and are often the result of competition in media with heterogeneous conduction properties. Asymptotic wave front propagation at maximal conduction velocity has been previously reported in the context of anisotropic cardiac tissue, but it is unknown whether this is a universal property of excitable tissues where conduction velocity can be locally modulated by mechanisms other than anisotropy. Here, we investigate the impact of conduction heterogeneities and boundary effects on wave propagation in excitable media. Following a theoretical analysis, we find that wave-front cusps occur where local velocity is reduced and that asymptotic wave fronts propagate at the maximal translational conduction velocity. Simulations performed in different reaction-diffusion systems, including cardiac tissue, confirm our theoretical findings. We conclude that this property can be found in a wide range of reaction-diffusion systems with excitable dynamics and that asymptotic wave-front shapes can be predicted.

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.

Drift laws for spiral waves on curved anisotropic surfaces

Rotating spiral waves organize spatial patterns in chemical, physical, and biological excitable systems. Factors affecting their dynamics, such as spatiotemporal drift, are of great interest for particular applications. Here, we propose a quantitative description for spiral wave dynamics on curved surfaces which shows that for a wide class of systems, including the Belousov-Zhabotinsky reaction and anisotropic cardiac tissue, the Ricci curvature scalar of the surface is the main determinant of spiral wave drift. The theory provides explicit equations for spiral wave drift direction, drift velocity, and the period of rotation. Depending on the parameters, the drift can be directed to the regions of either maximal or minimal Ricci scalar curvature, which was verified by direct numerical simulations.

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.

A theory of biological relativity: no privileged level of causation

Must higher level biological processes always be derivable from lower level data and mechanisms, as assumed by the idea that an organism is completely defined by its genome? Or are higher level properties necessarily also causes of lower level behaviour, involving actions and interactions both ways? This article uses modelling of the heart, and its experimental basis, to show that downward causation is necessary and that this form of causation can be represented as the influences of initial and boundary conditions on the solutions of the differential equations used to represent the lower level processes. These insights are then generalized. A priori, there is no privileged level of causation. The relations between this form of 'biological relativity' and forms of relativity in physics are discussed. Biological relativity can be seen as an extension of the relativity principle by avoiding the assumption that there is a privileged scale at which biological functions are determined.

Reflection and attachment of spirals at obstacles for the Fitzhugh-Nagumo and Beeler-Reuter models

In this paper, the Fitzhugh-Nagumo (FHN) equations and a modified FHN (MFHN) are considered. For the modified version, the recovery variable v has three different time scales. By considering different parameters in the local dynamics of the MFHN equations, it is observed that the phenomenon of reflection and annihilation at an impermeable boundary is observed just as in the Beeler-Reuter model. The interaction of spirals obtained with the FHN, MFHN, and Beeler-Reuter model, and an obstacle is also considered. The phenomenon of reflection of the spiral wave at a boundary changes when the boundary becomes an obstacle. Four properties for attachment of a spiral wave to an obstacle are presented in this work.

Nondestructive optical determination of fiber organization in intact myocardial wall

Mapping the myocardial fiber organization is important for assessing the electrical and mechanical properties of normal and diseased hearts. Current methods to determine the fiber organization have several limitations: histological sectioning mechanically distorts the tissue and is labor-intensive, while diffusion tensor imaging has low spatial resolution and requires expensive MRI scanners. Here, we utilized optical clearing, a fluorescent dye, and confocal microscopy to create three-dimensional reconstructions of the myocardial fiber organization of guinea pig and mouse hearts. We have optimized the staining and clearing procedure to allow for the nondestructive imaging of whole hearts with a thickness up to 3.5 mm. Myocardial fibers could clearly be identified at all depths in all preparations. We determined the change of fiber orientation across strips of guinea pig left ventricular wall. Our study confirms the qualitative result that there is a steady counterclockwise fiber rotation across the ventricular wall. Quantitatively, we found a total fiber rotation of 105.7+/-14.9 degrees (mean+/-standard error of the mean); this value lies within the range reported by previous studies. These results show that optical clearing, in combination with a fluorescent dye and confocal microscopy, is a practical and accurate method for determining myocardial fiber organization.

Intra-myocardial cusp waves and their manifestation in optical mapping signals

The rotating fiber orientation within the cardiac wall substantially affects the electrical propagation and can cause intra-myocardial cusp waves. Numerical simulations have shown that the cusps form in layers where propagation is perpendicular to the fiber orientation and lead to complex wave front morphologies. They can travel across layers and break through at the epi- or endocardial surfaces where they cause apparent accelerations of propagation. The validation of these results remains a major experimental challenge. In the present study, we investigate both computationally and experimentally how intramural cusp waves can be detected using optical imaging. Our simulations show that cusps alter the optical upstroke morphology and can be detected well before they reach the surface (up to 1 mm deep). Experiments in Langendorff-perfused guinea pig hearts are consistent with our numerical findings.

Multi-scale computational modelling in biology and physiology

Recent advances in biotechnology and the availability of ever more powerful computers have led to the formulation of increasingly complex models at all levels of biology. One of the main aims of systems biology is to couple these together to produce integrated models across multiple spatial scales and physical processes. In this review, we formulate a definition of multi-scale in terms of levels of biological organisation and describe the types of model that are found at each level. Key issues that arise in trying to formulate and solve multi-scale and multi-physics models are considered and examples of how these issues have been addressed are given for two of the more mature fields in computational biology: the molecular dynamics of ion channels and cardiac modelling. As even more complex models are developed over the coming few years, it will be necessary to develop new methods to model them (in particular in coupling across the interface between stochastic and deterministic processes) and new techniques will be required to compute their solutions efficiently on massively parallel computers. We outline how we envisage these developments occurring.

A guide to modelling cardiac electrical activity in anatomically detailed ventricles

One of the most recent trends in cardiac electrophysiology is the development of integrative anatomically accurate models of the heart, which include description of cardiac activity from sub-cellular and cellular level to the level of the whole organ. In order to construct this type of model, a researcher needs to collect a wide range of information from books and journal articles on various aspects of biology, physiology, electrophysiology, numerical mathematics and computer programming. The aim of this methodological article is to survey recent developments in integrative modelling of electrical activity in the ventricles of the heart, and to provide a practical guide to the resources and tools that are available for work in this exciting and challenging area.

Continuous and discontinuous propagation in heart muscle

Although cardiac muscle has been considered as an electrical syncytium for a long time, several aspects of conduction of the electrical impulse in the heart could not be explained by a continuous approach. Even the simple anisotropic nature of cardiac muscle gives rise to a number of conduction characteristics that contradicts with results from continuous models. It is, however, evident that cardiac conduction must be discontinuous in nature owing to recurrent resistive discontinuities caused by the cellular interconnections. In this review, the role of discontinuities at the cellular and macroscopic level on cardiac propagation will be discussed.

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.

Epicardial fiber organization in swine right ventricle and its impact on propagation

Fiber organization is important for myocardial excitation and contraction. It can be a major factor in arrhythmogenesis and current distribution during defibrillation shocks. In this study, we report the discovery of a previously undetected thin epicardial layer in swine right ventricle (RV) with distinctly different fiber orientation, which significantly affects epicardial propagation. Experiments were conducted in isolated coronary-perfused right ventricular free wall preparations (n=8) stained with the voltage-sensitive dye di-4-ANEPPS. Optical signals were recorded from the epicardium with a CCD video camera at 800 fps. Preparations were sectioned parallel to the epicardial surface with a resolution of 50 mum or better. To link the histological data with the observed activation patterns, resulting fiber angles were introduced into a 3D computer model to simulate the electrical activation and voltage-dependent optical signals. In all preparations, we detected a thin epicardial layer with almost no depth-dependent fiber rotation. The thickness of this layer (z(0)) varied from 110 to 930 microm. At the boundary of this layer, we observed an abrupt change in fiber angle by 64+/-13 degrees followed by a gradual fiber rotation in the underlying layers. In preparations with z(0) <700 microm, optical mapping during epicardial stimulation revealed unusual diamond- and rectangular-shaped activation fronts with two axes of fast conduction. Computer simulations accurately predicted the features of the experimentally recorded activation fronts. The free wall of swine RV has a thin epicardial layer with distinctly different fiber orientation, which can significantly affect propagation and give rise to unusually shaped activation fronts. This is important for understanding electrical propagation in the heart, and further refines the existing knowledge of myocardial fiber architecture.

Intramural wave propagation in cardiac tissue: asymptotic solutions and cusp waves

The cardiac muscle is well known to conduct electric impulses anisotropically, showing a larger conduction velocity along than across fibers. Fiber orientation is not uniform within the cardiac wall, but rotates by as much as 180 degrees throughout the wall thickness. Numerical simulations and experiments have indicated that this rotational anisotropy considerably affects the spread of excitation in cardiac tissue: the wave front shows a complex intramural shape with trailing cusps. The cusps can travel across layers and reach the epicardial and endocardial surfaces where they cause apparent accelerations of propagation. In the present study we provide an analytical description of the asymptotic wave front, as well as of cusp waves. We investigate the motion of cusp waves, based on the assumption that they occur at the intersection of asymptotic solutions, and we show that our theoretical analysis is in close agreement with numerical simulations. The asymptotic solutions are found to be determined purely by the fiber organization within the cardiac wall, independent of the excitable properties of cardiac tissue.

Molecular mechanisms and global dynamics of fibrillation: an integrative approach to the underlying basis of vortex-like reentry

Art Winfree's scientific legacy has been particularly important to our laboratory whose major goal is to understand the mechanisms of ventricular fibrillation (VF). Here, we take an integrative approach to review recent studies on the manner in which nonlinear electrical waves organize to result in VF. We describe the contribution of specific potassium channel proteins and of the myocardial fiber structure to such organization. The discussion centers on data derived from a model of stable VF in the Langendorff-perfused guinea pig heart that demonstrates distinct patterns of organization in the left (LV) and right (RV) ventricles. Analysis of optical mapping data reveals that VF excitation frequencies are distributed throughout the ventricles in clearly demarcated domains. The highest frequency domains are found on the anterior wall of the LV at a location where sustained reentrant activity is present. The optical data suggest that a high frequency rotor that remains stationary in the LV is the mechanism that sustains VF in this model. Computer simulations predict that the inward rectifying potassium current (IK1) is an essential determinant of rotor stability and frequency, and patch-clamp results strongly suggest that the outward component of IK1 of cells in the LV is significantly larger than in the RV. Additional computer simulations and analytical procedures predict that the filaments of the reentrant activity (scroll waves) adopt a non-random configuration depending on fiber organization within the ventricular wall. Using the minimal principle we have concluded that filaments align with the trajectory of least resistance (i.e. the geodesic) between their endpoints. Overall, the data discussed have opened new and potentially exciting avenues of research on the possible role played by inward rectifier channels in the mechanism of VF, as well as the organization of its reentrant sources in three-dimensional cardiac muscle. Such an integrative approach may lead us toward an understanding of the molecular and structural basis of VF and hopefully to new preventative approaches.

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.

Modelling the heart: insights, failures and progress

Mathematical models of the heart have developed over a period of about 40 years. Cell types in all regions of the heart have been modelled and they are now being incorporated into anatomically detailed models of the whole organ. This combination is leading to the creation of the first 'virtual organ,' which is being used in drug discovery and testing, and in simulating the action of devices, such as cardiac defibrillators. Simulation is a necessary tool of analysis in attempting to understand biological complexity. We often learn as much from the failures as from the successes of mathematical models. It is the iterative interaction between experiment and simulation that is important. Examples are given where this process has been instrumental in some of the major advances in the field.

Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity

It has become widely accepted that the most dangerous cardiac arrhythmias are due to reentrant waves, i.e., electrical wave(s) that recirculate repeatedly throughout the tissue at a higher frequency than the waves produced by the heart's natural pacemaker (sinoatrial node). However, the complicated structure of cardiac tissue, as well as the complex ionic currents in the cell, have made it extremely difficult to pinpoint the detailed dynamics of these life-threatening reentrant arrhythmias. A simplified ionic model of the cardiac action potential (AP), which can be fitted to a wide variety of experimentally and numerically obtained mesoscopic characteristics of cardiac tissue such as AP shape and restitution of AP duration and conduction velocity, is used to explain many different mechanisms of spiral wave breakup which in principle can occur in cardiac tissue. Some, but not all, of these mechanisms have been observed before using other models; therefore, the purpose of this paper is to demonstrate them using just one framework model and to explain the different parameter regimes or physiological properties necessary for each mechanism (such as high or low excitability, corresponding to normal or ischemic tissue, spiral tip trajectory types, and tissue structures such as rotational anisotropy and periodic boundary conditions). Each mechanism is compared with data from other ionic models or experiments to illustrate that they are not model-specific phenomena. Movies showing all the breakup mechanisms are available at http://arrhythmia.hofstra.edu/breakup and at ftp://ftp.aip.org/epaps/chaos/E-CHAOEH-12-039203/ INDEX.html. The fact that many different breakup mechanisms exist has important implications for antiarrhythmic drug design and for comparisons of fibrillation experiments using different species, electromechanical uncoupling drugs, and initiation protocols. (c) 2002 American Institute of Physics.

Effect of fibre rotation on the initiation of re-entry in cardiac tissue

Transmural rotation of cardiac fibres can have a big influence on the initiation of re-entry in the heart. However, owing to computational demands, this has not been fully explored in a three-dimensional model of cardiac tissue that has a microscopic description of membrane currents, such as the Luo-Rudy model. Using a previously described model that is computationally fast, re-entry in three-dimensional blocks of cardiac tissue is induced by a cross-shock protocol, and the activity is examined. In the study, the effect of the transmural fibre rotation is ascertained by examining differences between a tissue block with no rotation and ones with 1, 2 and 3 degrees of rotation per fibre layer. The direction of the re-entry is significant in establishing whether or not re-entry can be induced, with clockwise re-entry being easier to initiate. Owing to the rotating anisotropy that results in preferential propagation along the fibre axis, the timing of the second stimulus in the cross-shock protocol has to be changed for different rates of fibre rotation. The fibre rotation either increases or decreases the window of opportunity for re-entry, depending on whether the activation front is perpendicular or parallel to the fibre direction. By varying the transmural extent of the S2, it is found that a deeper stimulus has to be applied to the blocks with fibre rotation to create re-entry. Increasing the transmural resistance also tends to reduce the extent of the S2 required to induce re-entry. Results suggest that increasing fibre rotation reduces the susceptibility of the tissue to re-entry, but that more complex spatiotemporal patterns are possible, e.g. stable figure-of-eight re-entries and transient rotors. Three mechanisms of re-entry annihilation are identified: front catchup, filling of the excitable gap and core wander.

Direct histological validation of diffusion tensor MRI in formaldehyde-fixed myocardium

Diffusion tensor MRI is emerging as a rapid, nondestructive method to map myocardial fiber organization. It accurately measures myofiber orientation in hearts bathed in or perfused with cardioplegic solution. This study shows it also accurately maps the fibrous architecture of formalin-fixed hearts. Fiber orientations obtained by MRI and histology at the same locations in an excised portion of rabbit ventricle differed on average by 3.7 degrees (SD = 6.4 degrees, N = 70), a closer correspondence than achieved with previous preparations. The longer acquisition times afforded by fixed-heart imaging provides better accuracy, and should enable high-resolution reconstruction of the entire ventricular architecture. Magn Reson Med 44:157-161, 2000.

Predicting filament drift in twisted anisotropy

Excitable media with twisted anisotropy have recently been attracting significant interest because of their applicability to wave propagation in heart tissue. Here we consider the dynamics of an intramural scroll wave whose filament lies initially within an arbitrary layer of mutually parallel cardiac fibers, and drifts parallel to itself from layer to layer. Earlier simulations have demonstrated that such a filament stabilizes in a layer whose fiber direction is the same as its own. In the present paper we analytically derive the trajectory of the filament, and obtain good agreement with earlier numerical data. For sufficiently sparse scrolls, our analysis predicts an equilibrium alignment perpendicular rather than parallel to the fibers.

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.

Quatrefoil reentry in myocardium: an optical imaging study of the induction mechanism

INTRODUCTION

The "critical point hypothesis" for induction of ventricular fibrillation has previously been extended to infer the coexistence of four critical points, and hence four simultaneous spiral reentries or a quatrefoil reentry, resulting from only one premature stimulus delivered to the same location as the pacing stimulus. An optical imaging technique was used to explore its existence and to study the induction mechanism of this peculiar reentry pattern.

METHODS AND RESULTS

In 16 isolated, Langendorff-perfused rabbit hearts, high-speed optical imaging at 133 or 267 frames/sec was performed to observe the induced response with a unipolar point electrode. A novel quatrefoil-shaped reentry pattern consisting of two pairs of opposing rotors was created by delivering long stimuli during the vulnerable phase. Successful induction occurred in a narrow range of coupling intervals. A dogbone pattern of virtual electrodes was established during the premature stimulus. Propagating wavefronts launched from the virtual anodes immediately after the termination of S2. The alternating blocking and conducting effects of the virtual electrodes, as well as the boundary between virtual cathode and virtual anode, provided the necessary pathways for quatrefoil reentry. Propagation directions of the reentrant spiral wavefronts reversed with a reversal in S2 polarity. Quatrefoil reentries were not sustained and lasted 1 to 4 complete cycles.

CONCLUSION

The initiation of quatrefoil reentry followed anodal- or cathodal-break stimulation as a result of local symmetrical enhancement of the dispersion of tissue excitability. The "critical point hypothesis" provides the minimum topology required for this type of reentry; the "graded response hypothesis" can be viewed as providing a more detailed explanation of how this topology is actually realized. Triggering mechanisms due to the "break" mode of stimulation also posits a new mechanism for defibrillation.

Stretch-induced changes in heart rate and rhythm: clinical observations, experiments and mathematical models

Clinical and research data indicate that active and passive changes in the mechanical environment of the heart are capable of influencing both the initiation and the spread of cardiac excitation via pathways that are intrinsic to the heart. This direction of the cross-talk between cardiac electrical and mechanical activity is referred to as mechano-electric feedback (MEF). MEF is thought to be involved in the adjustment of heart rate to changes in mechanical load and would help to explain the precise beat-to-beat regulation of cardiac performance as it occurs even in the recently transplanted (and, thus, denervated) heart. Furthermore, there is clinical evidence that MEF may be involved in mechanical initiation of arrhythmias and fibrillation, as well as in the re-setting of disturbed heart rhythm by 'mechanical' first aid procedures. This review will outline the clinical relevance of cardiac MEF, describe cellular correlates to the responses observed in situ, and discuss the role that quantitative mathematical models may play in identifying the involvement of cardiac MEF in the regulation of heart rate and rhythm.

Histological validation of myocardial microstructure obtained from diffusion tensor magnetic resonance imaging

Diffusion tensor magnetic resonance imaging (MRI) is a possible new means of elucidating the anatomic structure of the myocardium. It enjoys several advantages over traditional histological approaches, including the ability to rapidly measure fiber organization in isolated, perfused, arrested hearts, thereby avoiding fixation and sectioning of artifacts. However, quantitative validation of this MRI method has been lacking. Here, fiber orientations estimated in the same locations in the same heart using both diffusion tensor MRI and histology are compared in a total of two perfused rabbit hearts. Fiber orientations were statistically similar for both methods and differed on average by 12 degrees at any single location. This is similar to the 10 degrees uncertainty in fiber orientation achieved with histology. In addition, imaging studies performed in a total of seven hearts support a level of organization beyond the myofiber, the recently described laminar organization of the ventricular myocardium.

A spatial scale factor for electrophysiological models of myocardium

In the United States among males 20-64 years old about 1/3 of deaths are classified as 'sudden cardiac death', and 1/4 had no forewarning, nor did autopsy turn up any visible cause. The heart just switches from its normally periodic pumping to an alternative mode called 'fibrillation' more resembling electrical turbulence. In normal tissue its mechanism is a geometrically re-entrant mode of normal propagation. Everything about this spatial pattern depends upon the magnitude of 'D', the one term with dimension involving space in the pertinent biophysical equations. Explicit or implicit estimates in current literature span orders of magnitude. In this article I argue from a diversity of recent experiments for a narrower range of realistic values. It has an important role in the spatio-temporal evolution of fibrillation and in defibrillation.

Three-dimensional analysis of regional cardiac function: a model of rabbit ventricular anatomy

The three-dimensional geometry and anisotropic properties of the heart give rise to nonhomogeneous distributions of stress, strain, electrical activation and repolarization. In this article we review the ventricular geometry and myofiber architecture of the heart, and the experimental and modeling studies of three-dimensional cardiac mechanics and electrophysiology. The development of a three-dimensional finite element model of the rabbit ventricular geometry and fiber architecture is described in detail. Finally, we review the experimental results, from the level of the cell to the intact organ, which motivate the development of coupled three-dimensional models of cardiac electromechanics and mechanoelectric feedback.

Purkinje-muscle reentry as a mechanism of polymorphic ventricular arrhythmias in a 3-dimensional model of the ventricles

Multiple electrode mapping of the ventricles during complex tachyarrhythmias has revealed focal subendocardial activation whose mechanism remains unexplained. We hypothesized that reentry involving the Purkinje-muscle junctions (PMJs) may be a mechanism for such focal excitations. We have constructed an anatomically appropriate computerized 3-dimensional model of the mammalian ventricles that includes the Purkinje conduction system and 214 PMJs distributed throughout the endocardium. Isochronal maps during normal excitation, as well as during right or left bundle branch block, resembled experimental measurements and compared well with isochronal maps of propagation in the human heart. Activity observed at both sides of a PMJ in the model showed that propagation from Purkinje fibers to muscle was slower than in the opposite direction. Under these realistic and normal conditions, the evolution of reentrant activity involving muscle and the Purkinje network was simulated. The reentry pattern was independent of the initiation site. It evolved with drifting epicardial breakthroughs and transformed on the endocardium from focal activity to figure-of-8 reentry. In addition, the ECG amplitude undulated during the evolution, and decrease in the cycle period, apparent wavelength, and propagation velocity were observed. Finally, the reentry was terminated if the Purkinje system was disconnected from the muscle before it reached a relative steady state. The simulation results suggest the following: (1) Epicardial breakthroughs and endocardial focal activity may originate at the PMJs. (2) The ECG amplitude may decrease as the reentry stabilizes and the excitation wavelength decreases. (3) The Purkinje system may have a double role in the evolution of reentry: first, it is essential to the reentry at the initial stage; second, it may lead to the establishment of intramyocardial reentry, at which time the Purkinje system becomes irrelevant.

Control of re-entrant activity in a model of mammalian atrial tissue

We evaluate the feasibility of using resonant drift under feedback driven stimulation to control re-entrant excitation in atrial muscle. We simulate a two-dimensional sheet of atrial tissue, where the local kinetics are described by the Earm-Hilgemann-Noble equations for a rabbit atrial cell, and the effects of small amplitude spatially uniform forcing of the whole sheet are computed. Repetitive forcing can induce a drift of a spiral wave in the two-dimensional model, with a drift velocity of up to 10 cm s-1. For a 4 cm x 4 cm atrial surface this resonant drift can move the re-entrant spiral to the inexcitable boundaries, eliminating re-entry in less than 10 s when the amplitude of the repetitive stimulation is 10% that of the single shock defibrillation threshold.

Electrical turbulence in three-dimensional heart muscle

Rotors or vortex action potentials with a diameter of about 1 centimeter and a rotation period of about 0.1 second occur in normal myocardium just before transition to fibrillation, a disorderly pattern of action potential propagation. Numerical models and corresponding mathematical analysis have recently suggested candidate mechanisms, all two-dimensional, for this transition from periodic electrical activity to something resembling turbulence. However, comparably recent experiments unanimously show that rotors, and the spiral waves they radiate, remain stably periodic in two-dimensional myocardium. This seeming paradox suggests a transition mediated through disorderly dynamics of the electrical vortex in three dimensions, as a "vortex filament."

Nonuniform muscle fiber orientation causes spiral wave drift in a finite element model of cardiac action potential propagation

INTRODUCTION

Reentrant tachyarrhythmias are thought to involve spiral waves of excitation and recovery that may be nonstationary. The effect of muscle fiber curvature on spiral and planar wave propagation was studied using a computational model.

METHODS AND RESULTS

Two-dimensional anisotropic cardiac propagation was modeled using a finite element method to solve a modification of the FitzHugh-Nagumo equations. Spiral waves that propagated stably about a fixed core in tissue with a uniform fiber orientation were found to drift at an oblique angle to the fibers when the fibers curved. The drift velocity was linearly related to the fiber angle gradient and was 10% of the longitudinal propagation velocity with a gradient of 4 degree/cm. Planar wavefronts were also affected by fiber curvature. The maximal upstroke rate, propagation velocity, and the action potential amplitude all increased when the fibers curved toward the wavefront and decreased when they curved away. For example, when the fibers curved toward the wavefront with a moderate gradient of 15 degree/cm, maximal upstroke rate increased 74%, transverse propagation velocity increased 65%, and action potential amplitude increased 9%. This phenomenon caused the spiral wave drift: As a spiral wave traverses a cycle, the angle between the wavefront at the wavetip and the fibers changes periodically, thus altering the propagation parameters. These periodic changes affect the instantaneous radius of curvature of the wavetip path, which results in drift.

CONCLUSION

Spiral and planar waves are affected by muscle fiber curvature. The resulting dynamics may be important in determining the lifetime and stability of reentrant arrhythmias.