Three-dimensional Scroll Waves Research Articles

Reconstruction of three-dimensional scroll waves in excitable media from two-dimensional observations using deep neural networks.

Scroll wave dynamics are thought to underlie life-threatening ventricular fibrillation. However, direct observations of three-dimensional electrical scroll waves remain elusive, as there is no direct way to measure action potential wave patterns transmurally throughout the thick ventricular heart muscle. Here we study whether it is possible to reconstruct simulated scroll waves and scroll wave chaos using deep learning. We trained encoding-decoding convolutional neural networks to predict three-dimensional scroll wave dynamics inside bulk-shaped excitable media from two-dimensional observations of the wave dynamics on the bulk's surface. We tested whether observations from one or two opposing surfaces would be sufficient and whether transparency or measurements of surface deformations enhances the reconstruction. Further, we evaluated the approach's robustness against noise and tested the feasibility of predicting the bulk's thickness. We distinguished isotropic and anisotropic, as well as opaque and transparent, excitable media as models for cardiac tissue and the Belousov-Zhabotinsky chemical reaction, respectively. While we demonstrate that it is possible to reconstruct three-dimensional scroll wave dynamics, we also show that it is challenging to reconstruct complicated scroll wave chaos and that prediction outcomes depend on various factors such as transparency, anisotropy, and ultimately the thickness of the medium compared to the size of the scroll waves. In particular, we found that anisotropy provides crucial information for neural networks to decode depth, which facilitates the reconstructions. In the future, deep neural networks could be used to visualize intramural action potential wave patterns from epi- or endocardial measurements.

Novel approaches to mechanism-based atrial fibrillation ablation.

Modern cardiac electrophysiology has reported significant advances in the understanding of mechanisms underlying complex wave propagation patterns during atrial fibrillation (AF), although disagreements remain. One school of thought adheres to the long-held postulate that AF is the result of randomly propagating wavelets that wonder throughout the atria. Another school supports the notion that AF is deterministic in that it depends on a small number of high-frequency rotors generating three-dimensional scroll waves that propagate throughout the atria. The spiralling waves are thought to interact with anatomic and functional obstacles, leading to fragmentation and new wavelet formation associated with the irregular activation patterns documented on AF tracings. The deterministic hypothesis is consistent with demonstrable hierarchical gradients of activation frequency and AF termination on ablation at specific (non-random) atrial regions. During the last decade, data from realistic animal models and pilot clinical series have triggered a new era of novel methodologies to identify and ablate AF drivers outside the pulmonary veins. New generation electroanatomical mapping systems and multielectrode mapping catheters, complimented by powerful mathematical analyses, have generated the necessary platforms and tools for moving these approaches into clinical procedures. Recent clinical data using such platforms have provided encouraging evidence supporting the feasibility of targeting and effectively ablating driver regions in addition to pulmonary vein isolation in persistent AF. Here, we review state-of-the-art technologies and provide a comprehensive historical perspective, characterization, classification, and expected outcomes of current mechanism-based methods for AF ablation. We discuss also the challenges and expected future directions that scientists and clinicians will face in their efforts to understand AF dynamics and successfully implement any novel method into regular clinical practice.

Induced drift of scroll waves in the Aliev–Panfilov model and in an axisymmetric heart left ventricle

AbstractThe low-voltage cardioversion-defibrillation is a modern sparing electrotherapy method for such dangerous heart arrhythmias as paroxysmal tachycardia and fibrillation. In an excitable medium, such arrhythmias relate to appearance of spiral waves of electrical excitation, and the spiral waves are superseded to the electric boundary of the medium in the process of treatment due to high-frequency stimulation from the electrode. In this paper we consider the Aliev–Panfilov myocardial model, which provides a positive tension of three-dimensional scroll waves, and an axisymmetric model of the left ventricle of the human heart. Two relations of anisotropy are considered, namely, isotropy and physiological anisotropy. The periods of stimulation with an apical electrode are found so that the electrode successfully entrains its rhythm in the medium, the spiral wave is superseded to the base of the ventricle, and disappears. The results are compared in two-dimensional and three-dimensional media. The intervals of effective stimulation periods are sufficiently close to each other in the two-dimensional case and in the anatomical model. However, the use of the anatomical model is essential in determination of the time of superseding.

Termination of Scroll Waves by Surface Impacts.

Three-dimensional scroll waves direct cell movement and gene expression, and induce chaos in the brain and heart. We found an approach to terminate multiple three-dimensional scrolls. A pulse of a properly configured electric field detaches scroll filaments from the surface. They shrink due to filament tension and disappear. Since wave emission from small heterogeneities is not used, this approach requires a much lower electric field. It is not sensitive to the details of the excitable medium. It may affect future studies of low-energy chaos termination in the heart.

Electromechanical vortex filaments during cardiac fibrillation.

The self-organized dynamics of vortex-like rotating waves, which are also known as scroll waves, are the basis of the formation of complex spatiotemporal patterns in many excitable chemical and biological systems. In the heart, filament-like phase singularities that are associated with three-dimensional scroll waves are considered to be the organizing centres of life-threatening cardiac arrhythmias. The mechanisms that underlie the onset, maintenance and control of electromechanical turbulence in the heart are inherently three-dimensional phenomena. However, it has not previously been possible to visualize the three-dimensional spatiotemporal dynamics of scroll waves inside cardiac tissues. Here we show that three-dimensional mechanical scroll waves and filament-like phase singularities can be observed deep inside the contracting heart wall using high-resolution four-dimensional ultrasound-based strain imaging. We found that mechanical phase singularities co-exist with electrical phase singularities during cardiac fibrillation. We investigated the dynamics of electrical and mechanical phase singularities by simultaneously measuring the membrane potential, intracellular calcium concentration and mechanical contractions of the heart. We show that cardiac fibrillation can be characterized using the three-dimensional spatiotemporal dynamics of mechanical phase singularities, which arise inside the fibrillating contracting ventricular wall. We demonstrate that electrical and mechanical phase singularities show complex interactions and we characterize their dynamics in terms of trajectories, topological charge and lifetime. We anticipate that our findings will provide novel perspectives for non-invasive diagnostic imaging and therapeutic applications.

Stabilization of three-dimensional scroll waves and suppression of spatiotemporal chaos by heterogeneities.

Scroll waves in a three-dimensional medium with negative filament tension may break up and display spatiotemporal chaos. The presence of heterogeneities can influence the evolution of the medium, in particular scroll waves may pin to such heterogeneities. We show that as a result the medium may be stabilized by heterogeneities of a suitably chosen geometry. Thin rodlike heterogeneities suppress otherwise developing spatiotemporal chaos and additionally clear out already existing chaotic excitation patterns.

Unpinning of scroll waves under the influence of a thermal gradient.

Three-dimensional scroll waves of electrical activity are among the mechanisms believed to be responsible for the rapid, unsynchronized contraction of cardiac ventricles, thereby reducing the heart's ability to pump blood. Scroll waves can attach themselves to unexcitable obstacles, and this sometimes highly elongates their life span. Hence, the unpinning and annihilation of these vortices has attracted much attention in recent decades. In this work, we study the influence of a thermal gradient on scroll waves pinned to inert obstacles, in the Belousov-Zhabotinsky reaction. Under a temperature gradient, scroll rings were seen to unpin from these obstacles, thus strikingly reducing their lifetimes. These results were also reproduced by numerical simulations using the Barkley model.

Analysis of anchor-size effects on pinned scroll waves and measurement of filament rigidity.

Inert, spherical heterogeneities can pin three-dimensional scroll waves in the excitable Belousov-Zhabotinsky reaction. Three pinning sites cause initially circular rotation backbones to approach equilateral triangles. The resulting stationary shapes show convex deviations that increase with decreasing anchor radii. This dependence is interpreted as a transition between filament termination at large surfaces and true, local pinning of a continuous curve. The shapes of the filament segments are described by a hyperbolic cosine function which is predicted by kinematic theory that considers filament tension and rigidity. The latter value is measured as (1.0±0.7)×10-6 cm4/s.

Nonequilibrium Arrhythmic States and Transitions in a Mathematical Model for Diffuse Fibrosis in Human Cardiac Tissue

We present a comprehensive numerical study of spiral-and scroll-wave dynamics in a state-of-the-art mathematical model for human ventricular tissue with fiber rotation, transmural heterogeneity, myocytes, and fibroblasts. Our mathematical model introduces fibroblasts randomly, to mimic diffuse fibrosis, in the ten Tusscher-Noble-Noble-Panfilov (TNNP) model for human ventricular tissue; the passive fibroblasts in our model do not exhibit an action potential in the absence of coupling with myocytes; and we allow for a coupling between nearby myocytes and fibroblasts. Our study of a single myocyte-fibroblast (MF) composite, with a single myocyte coupled to fibroblasts via a gap-junctional conductance , reveals five qualitatively different responses for this composite. Our investigations of two-dimensional domains with a random distribution of fibroblasts in a myocyte background reveal that, as the percentage of fibroblasts increases, the conduction velocity of a plane wave decreases until there is conduction failure. If we consider spiral-wave dynamics in such a medium we find, in two dimensions, a variety of nonequilibrium states, temporally periodic, quasiperiodic, chaotic, and quiescent, and an intricate sequence of transitions between them; we also study the analogous sequence of transitions for three-dimensional scroll waves in a three-dimensional version of our mathematical model that includes both fiber rotation and transmural heterogeneity. We thus elucidate random-fibrosis-induced nonequilibrium transitions, which lead to conduction block for spiral waves in two dimensions and scroll waves in three dimensions. We explore possible experimental implications of our mathematical and numerical studies for plane-, spiral-, and scroll-wave dynamics in cardiac tissue with fibrosis.

New Resolution Strategy for Multiscale Reaction Waves using Time Operator Splitting, Space Adaptive Multiresolution, and Dedicated High Order Implicit/Explicit Time Integrators

We tackle the numerical simulation of reaction-diffusion equations modeling multi-scale reaction waves. This type of problem induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of steep spatial gradients in the reaction fronts, spatially very localized. In this paper, we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts. The paper considers a high order implicit time integration of the reaction and an explicit one for the diffusion term in order to build a time operator splitting scheme that exploits efficiently the special features of each problem. Based on recent theoretical studies of numerical analysis such a strategy leads to a splitting time step which is restricted by neither the fastest scales in the source term nor by stability constraints of the diffusive steps but only by the physics of the phenomenon. We aim thus at solving complete models including all time and space scales within a prescribed accuracy, considering large simulation domains with conventional computing resources. The efficiency is evaluated through the numerical simulation of configurations which were so far out of reach of standard methods in the field of nonlinear chemical dynamics for two-dimensional spiral waves and three-dimensional scroll waves, as an illustration. Future extensions of the proposed strategy to more complex configurations involving other physical phenomena as well as optimization capability on new computer architectures are discussed.

Topologically Mismatched Pinning of Scroll Waves

Three-dimensional scroll waves in excitable reaction−diffusion systems rotate around one-dimensional phase singularities. Although these filaments are dynamic objects, their motion can be arrested if pinned to unexcitable obstacles. We study this vortex pinning for a case in which the topology of the initial filament (circular loop) differs from the topology of the obstacle (double torus). In experiments with the Belousov−Zhabotinsky reaction and numerical simulations, we show that pinning is possible and that it induces periodic rotation patterns. Continuous rotation extends around either one or both halves of the double torus. The latter case is doubly degenerate because waves can move through the two holes in the same or opposite directions. Opposite rotation is organized by a two-armed vortex segment that emits waves at a period approximately two-times larger than the other pinned vortex states.

Role of the Left Atrial Anatomy in the Maintenance of Three-Dimensional Scroll Waves in Acute, Stretch-Related and Cholinergic Atrial Fibrillation

The mechanisms that maintain atrial fibrillation (AF) are incompletely understood. We hypothesized that 3-D scroll waves (SWs) confined to specific left atrial (LA) anatomical structure are involved in the maintenance of stretch-related (SRAF) and cholinergic AF with stretch (ChAF). Methods: In 13 isolated sheep hearts, optical membrane potential images were recorded, by using two CCD cameras and a cardio-endoscope, simultaneously from the epicardial and endocardial surfaces of the posterior LA (PLA, n=8) or the LA appendage (LAA, n=5). SRAF was induced by burst pacing under increased LA pressure (12 cm H2O), and ChAF did so additional perfusion of 0.1 µM acetylcholine. Results: During SRAF, major activation pattern in LAA was either breakthrough (44%) or one-way propagation (30%), whereas that in PLA was SWs (30%), which were anchored at LA-PV junction. During ChAF, major activation pattern in LAA was SWs (53%), which were anchored at the periphery of pectinate muscles. Numerical simulations suggest that the thinnest region of LA will favor the anchoring of SWs via shortening of atrial wave length resulting from an increase of stretch-activated ion channel conductance. Conclusion: Transmural SWs play an important role to maintain AF induced by mechanical stretch and cholinergic stimulation. The thinnest LA region will provide a structural substrate in favor of perpetuation of these SWs.

Scroll Wave Filaments Terminate in the Back of Traveling Fronts

Experiments with the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction demonstrate that three-dimensional scroll waves can rotate around filaments that end in the wake of a traveling excitation pulse. The vortex structures nucleate during the collision of three nonrotating excitation pulses. The nucleation process and the wave-termination of filaments are direct consequences of the system's anomalous dispersion relation. Vortex filaments are found to expand with about twice the speed of their anchoring wave fronts. Filament expansion is accompanied by the build-up of phase differences in spiral rotation creating strongly twisted wave structures. Experiments employ optical tomography for the reconstruction of the three-dimensional wave patterns.

Large-excitability asymptotics for scroll waves in three-dimensional excitable media.

Three-dimensional scroll waves are considered in a reaction-diffusion model of excitable media in the large excitability limit. Coordinates based on the scroll filament are defined and shown to provide a natural extension of the coordinates used for two-dimensional spiral waves. The leading-order free-boundary equations for interface motion in three dimensions are explicitly derived in these coordinates. Three specific examples are considered: straight twisted scroll waves, axisymmetric scroll waves, and helical scroll waves. The equations for the fields at leading and first order in the core region are given.

Evidence of three-dimensional scroll waves with ribbon-shaped filament as a mechanism of ventricular tachycardia in the isolated rabbit heart.

Rotating vortices have been observed in excitable media of different nature. Vortices may sustain life or kill in different species, by underlining morphogenesis in Dictiostelium discoideum during starvation, or arrhythmias during sudden cardiac death in mammals. Investigation of vortices in the heart has been limited by two-dimensional experimental techniques. In contrast, three-dimensional (3D) Belousov-Zhabotinsky excitable medium and mathematical models have been shown to sustain scroll-shaped waves. The heart is a 3D structure; therefore, scroll waves may underlie cardiac arrhythmias. We used potentiometric dye and optical mapping techniques to study vortices during ventricular arrhythmias. The core of all observed vortices were linearly shaped and > or = 9 mm (48 episodes, six hearts). As shown by Allessie et al. in the rabbit atrium, ventricular signals recorded within 1 to 2 mm from the line of block were dual humped, suggesting there is electrotonic interaction across the line of block. We hypothesized that the line of block represents epicardial intersection of ribbon-shaped filaments, which in some cases may be oriented under an angle or parallel to the epicardium. In 14 episodes we observed dual-humped optical recordings at one side of the line of block at a distance up to 12 mm. The two humps may represent the signatures of two activation wavefronts propagating above and below the filament, which in this area is close to the epicardial surface. The activation sequence of the two waves is consistent with the idea of a scroll wave with ribbon-like filament. Our data provide new insights into the shape and dynamics of the filament of the 3D scroll wave, which underlies the mechanism of ventricular tachycardia in the rabbit heart. The filament of the scroll wave may be ribbon shaped, with a significant width > or = 9 mm and a thickness of 1 to 2 mm. No evidence of fully excitable cells in the core of vortex was observed.

Complex chemical reactions — A review

Complex responses have been observed in a wide range of chemical and engineering systems: In well-stirred and unstirred solution-phase, in gels, in heterogeneous catalysis and dissolution reactions, in gas- and solid-phase combustion, widely in biological systems, accompanying phase transitions, in atmospheric kinetics and even in interstellar dust clouds. Complex behaviour does not, however, imply a necessarily complex underlying chemical mechanism. The appropriate feedback mechanisms, built on chain-branching, autocatalysis or self-heating, arise quite commonly in Nature. In many instances, it is appropriate first to attempt to find reduced mechanisms giving a semi-quantitative fit to observed responses. An example of such an approach applied to a model for complex oscillations of species concentrations in the mesosphere is presented, and the reduced model used to investigate the effects of longitudinal mixing in this system. Once a general understanding of a particular reaction system has been obtained, more detailed questions can be addressed. This is exemplified by a study of the development of three-dimensional scroll waves in the Belousov–Zhabotinsky reaction. Such detailed information for a specific reaction also informs the general understanding of the class of `excitable media'.

The motion of organization center of scroll waves in excitable media with single diffusion

The motion of organization center of three-dimensional untwisted scroll waves in excitable media with single diffusion is studied by singular perturbation method in this paper. The relation of curvature and the linear law are derived for untwisted organization center. These results have explicit physical meaning and are in good agreement with experiments

Scroll waves in the Belousov–Zhabotinsky reaction: exploitation of O2-effect on the ferroin-catalysed system

The inhibiting effect of dissolved oxygen on wave propagation in the ferroin-catalysed Belousov–Zhabotinsky (BZ) reaction is exploited to obtain three-dimensional scroll waves experimentally in this system. The development of the scrolls is monitored by imaging the wave evolution via a video camera and image acquisition software. Solutions for which the initial waves are initiated under an atmosphere of pure oxygen have a subexcitable or inhibited layer of typically 1 mm depth. Once the O2 is replaced by N2, this layer regains excitability and a scroll wave structure can develop. For waves initiated under air, the depth of the inhibited layer is less and scroll waves do not develop. A simple extension of the ZBKE model for the ferroin system is proposed and shown to predict the oxygen inhibition in reasonable agreement with experiment.

Spiral Wave Formation in Three-Dimensional Excitable Media.

A mechanism for the generation of spiral waves in three-dimensional excitable media is described. Experiments using the photosensitive Belousov-Zhabotinsky reaction show that circular or spiral waves appear in the wake of a propagating wave following a local perturbation in illumination intensity. The induced waves arise from three-dimensional scroll waves striking the upper boundary of the excitable medium. Spiral waves are formed over a range of illumination intensities and perturbation times which place the induced wave in the vulnerable region of the refractory wave back. [S0031-9007(96)01342-7]

Three-dimensional scroll waves of cAMP could direct cell movement and gene expression in Dictyostelium slugs.

Complex three-dimensional waves of excitation can explain the observed cell movement pattern in Dictyostelium slugs. Here we show that these three-dimensional waves can be produced by a realistic model for the cAMP relay system [Martiel, J. L. & Goldbeter, A. (1987) Biophys J. 52, 807-828]. The conversion of scroll waves in the prestalk zone of the slug into planar wave fronts in the prespore zone can result from a smaller fraction of relaying cells in the prespore zone. Further, we show that the cAMP concentrations to which cells in a slug are exposed over time display a simple pattern, despite the complex spatial geometry of the waves. This cAMP distribution agrees well with observed patterns of cAMP-regulated cell type-specific gene expression. The core of the spiral, which is a region of low cAMP concentration, might direct expression of stalk-specific genes during culmination.