Intermittent self-organization of scroll wave turbulence in three-dimensional excitable media

We study the asymptotic behavior of scroll wave turbulence in large three-dimensional excitable media modeled by FitzHugh-Nagumo equations. The focus is on the type of turbulence caused by negative tension of scroll wave filaments, which is considered to be one of the mechanisms of cardiac fibrillation. We discovered that the initial increase in turbulence complexity can be followed by intermittent self-organization, when complex filament tangles are replaced by a small number of relatively stable triple filament strands. The intermittency is the result of a competition between the destabilizing effect of negative tension and mutual attraction of filaments with similar orientation.     

Vortex Turbulence due to the Interplay of Filament Tension and Rotational Anisotropy

The mechanism of scroll wave turbulence is investigated in excitable media with rotational anisotropy. We adopt the Barkley model with heterogeneity in the diffusion constants. Through comparative numerical studies, we demonstrate the vortex turbulence results from the rotational anisotropy＇s cooperation with negative filament tension or competition with positive filament tension. The presence of rotational anisotropy can enlarge the parameter region leading to negative-tension induced wave turbulence in isotropic media.     

Negative filament tension in the Luo-Rudy model of cardiac tissue

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, which lethality is widely known. One of the possible mechanisms of scroll wave instability is negative filament tension, which was studied theoretically using low-dimensional models of excitable medium. In this article we perform a numerical study of negative filament tension using the Luo-Rudy phase 1 model, which is widely used in cardiac electrophysiology. We show that this instability exists in this model, study its manifestation and discuss its relation to cardiac arrhythmogenesis.     

Alternative stable scroll waves and conversion of autowave turbulence

Rotating spiral and scroll waves (vortices) are investigated in the FitzHugh-Nagumo model of excitable media. The focus is on a parameter region in which there exists bistability between alternative stable vortices with distinct periods. Response functions are used to predict the filament tension of the alternative scrolls and it is shown that the slow-period scroll has negative filament tension, while the filament tension of the fast-period scroll changes sign within a hysteresis loop. The predictions are confirmed by direct simulations. Further investigations show that the slow-period scrolls display features similar to delayed after-depolarization and tend to develop into turbulence similar to ventricular fibrillation (VF). Scrolls with positive filament tension collapse or stabilize, similar to monomorphic ventricular tachycardia (VT). Perturbations, such as boundary interaction or shock stimulus, can convert the vortex with negative filament tension into the vortex with positive filament tension. This may correspond to transition from VF to VT unrelated to pinning.     

Suppress Winfree turbulence by local forcing excitable systems

The occurrence of Winfree turbulence is currently regarded as one of the principal mechanisms underlying cardiac fibrillation. We develop a local stimulation method that suppresses Winfree turbulence in three-dimensional excitable media. We find that Winfree turbulence can be effectively suppressed by locally injecting periodic signals to only a very small subset (around some surface region) of total space sites. Our method for the first time demonstrates the effectiveness of local low-amplitude periodic excitations in suppressing turbulence in 3D excitable media and has fundamental improvements in efficiency, convenience, and turbulence suppression speed compared with previous strategies. Therefore, it has great potential for developing into a practical low-amplitude defibrillation approach.     

Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation

Wave propagation in ventricular muscle is rendered highly anisotropic by the intramural rotation of the fiber. This rotational anisotropy is especially important because it can produce a twist of electrical vortices, which measures the rate of rotation (in degree/mm) of activation wavefronts in successive planes perpendicular to a line of phase singularity, or filament. This twist can then significantly alter the dynamics of the filament. This paper explores this dynamics via numerical simulation. After a review of the literature, we present modeling tools that include: (i) a simplified ionic model with three membrane currents that approximates well the restitution properties and spiral wave behavior of more complex ionic models of cardiac action potential (Beeler-Reuter and others), and (ii) a semi-implicit algorithm for the fast solution of monodomain cable equations with rotational anisotropy. We then discuss selected results of a simulation study of vortex dynamics in a parallelepipedal slab of ventricular muscle of varying wall thickness (S) and fiber rotation rate (theta(z)). The main finding is that rotational anisotropy generates a sufficiently large twist to destabilize a single transmural filament and cause a transition to a wave turbulent state characterized by a high density of chaotically moving filaments. This instability is manifested by the propagation of localized disturbances along the filament and has no previously known analog in isotropic excitable media. These disturbances correspond to highly twisted and distorted regions of filament, or "twistons," that create vortex rings when colliding with the natural boundaries of the ventricle. Moreover, when sufficiently twisted, these rings expand and create additional filaments by further colliding with boundaries. This instability mechanism is distinct from the commonly invoked patchy failure or wave breakup that is not observed here during the initial instability. For modified Beeler-Reuter-like kinetics with stable reentry in two dimensions, decay into turbulence occurs in the left ventricle in about one second above a critical wall thickness in the range of 4-6 mm that matches experiment. However this decay is suppressed by uniformly decreasing excitability. Specific experiments to test these results, and a method to characterize the filament density during fibrillation are discussed. Results are contrasted with other mechanisms of fibrillation and future prospects are summarized. (c)1998 American Institute of Physics.     

Periodic forcing of scroll rings and control of Winfree turbulence in excitable media

By simulations of the Barkley model, action of uniform periodic nonresonant forcing on scroll rings and wave turbulence in three-dimensional excitable media is investigated. Sufficiently strong rapid forcing converts expanding scroll rings into the collapsing ones and suppresses the Winfree turbulence caused by the negative tension of wave filaments. Slow strong forcing has an opposite effect, leading to expansion of scroll rings and induction of the turbulence. These effects are explained in the framework of the phenomenological kinematic theory of scroll waves.     

A geometric theory for scroll wave filaments in anisotropic excitable media

Scroll waves are an important example of self-organisation in excitable media. In cardiac tissue, scroll waves of electrical activity underlie lethal ventricular arrhythmias and fibrillation. They rotate around a topological line defect which has been termed the filament. Numerical investigation has shown that anisotropy can substantially affect the dynamics of scroll waves. It has recently been hypothesised that stationary scroll wave filaments in cardiac tissue describe geodesics in a space whose metric is the inverse diffusion tensor. Several computational studies have validated this hypothesis, but until now no quantitative theory has been provided to study the effects of anisotropy on scroll wave filaments. Here, we review in detail the recently developed covariant formalism for scroll wave dynamics in general anisotropy and derive the equations of motion of filaments. These equations are fully covariant under general spatial coordinate transformations and describe the motion of filaments in a curved space whose metric tensor is the inverse diffusion tensor. Our dynamic equations are valid for thin filaments and for general anisotropy and we show that stationary filaments obey the geodesic equation. We extend previous work by allowing spatial variations in the determinant of the diffusion tensor and the reaction parameters, leading to drift of the filament.     

Effects of reduced discrete coupling on filament tension in excitable media

Wave propagation in the heart has a discrete nature, because it is mediated by discrete intercellular connections via gap junctions. Although effects of discreteness on wave propagation have been studied for planar traveling waves and vortexes (spiral waves) in two dimensions, its possible effects on vortexes (scroll waves) in three dimensions are not yet explored. In this article, we study the effect of discrete cell coupling on the filament dynamics in a generic model of an excitable medium. We find that reduced cell coupling decreases the line tension of scroll wave filaments and may induce negative filament tension instability in three-dimensional excitable lattices.     

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.     

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.     

Covariant stringlike dynamics of scroll wave filaments in anisotropic cardiac tissue

It has been hypothesized that stationary scroll wave filaments in cardiac tissue describe a geodesic in a curved space whose metric is the inverse diffusion tensor. Several numerical studies support this hypothesis, but no analytical proof has been provided yet for general anisotropy. In this Letter, we derive dynamic equations for the filament in the case of general anisotropy. These equations are covariant under general spatial coordinate transformations and describe the motion of a stringlike object in a curved space whose metric tensor is the inverse diffusion tensor. Therefore the behavior of scroll wave filaments in excitable media with anisotropy is similar to the one of cosmic strings in a curved universe. Our dynamic equations are valid for thin filaments and for general anisotropy. We show that stationary filaments obey the geodesic equation.     

On one-eighth law in unforced two-dimensional turbulence

We critically revisit the various attempts to prove one-eighth law in two-dimensional (2D) turbulence and reconcile them. Herein, the one-eighth law has been proved for unforced 2D incompressible high Reynolds number turbulence. An exact expression of the time derivative of two-point second order velocity correlation function is also derived for the enstrophy cascade dominated regime.     

Suppression of the Spiral Wave and Turbulence in the Excitability-Modulated Media

Periodical forcing is used to control the spiral wave and turbulence in the modified Fithzhugh-Nagumo equation (MFHNe) when excitability is changed. The decisive parameter ε of (MFHNe), which describes the ratio of time scales of the fast activator u and the slow inhibitor variable v, is supposed to increase linearly to simulate the excitability modulation in the media. In the numerical simulation, a local periodical stimulus is imposed on the left border of the media and the periods of external forcing are adjusted according to the approximate formula ω ∝1/ε ^1/3 so that using the most appropriate frequency for the external forcing can approach a shorter transient period. It is found that the spiral wave and turbulence can be removed successfully by using an appropriate periodical forcing on the left border of the media. The mean activator and distribution of frequency of all the sites are also used to analyze the transition of spiral wave.     

Eikonal formulation of the minimal principle for scroll wave filaments

Recently, Wellner et al. [Proc. Natl. Acad. Sci. U.S.A. 99, 8015 (2002)]] proposed a principle for predicting a stable scroll wave filament shape as a geodesic in a 3D space with a metric determined by the inverse diffusivity tensor of the medium. Using the Hamilton-Jacobi theory we show that this geodesic is the shortest path for a wave propagating through the medium. This allows the use of shortest path algorithms to predict filament shapes, which we confirm numerically for a medium with orthotropic anisotropy. Our method can be used in cardiac tissue experiments since it does not require knowledge of the tissue anisotropy.     

2D spectral and turbulence length scale estimation with PIV

A new method to apply spatial two-dimensional power spectral density (2D PSD) analysis to the data measured with Particle Image Velocimetry (PIV) has been introduced. Applying the method to a set of the velocity vector fields characteristic turbulence length scales can be estimated. In this method the computation of 2D PSD has been performed to two kinds of pre-processed data. In the first set, the local average has been spatially subtracted (Spatial decomposition) and in the second set the time-average has been subtracted (Reynolds decomposition). In the computation of 2D PSD the 2D FFT with the variance scaling has been used. Besides 2D spectral analysis this paper uses the distribution analysis of the various turbulence quantities and a structure analysis method to estimate the dimensions of coherent structures in the flow. Another method to analyse turbulence length scales is the estimation of the spatial 2D Auto Correlation Coefficient Function (2D ACCF). All these methods applied side by side to the PIV data increase the understanding of the turbulence, its scales and the nature of the coherent structures.     

Kinematics of rigidly rotating spiral waves

Spiral waves rigidly rotating in excitable media are studied by use of a free-boundary approach. This study reveals the selection principle which determines the shape and the rotation frequency of spiral waves in an unbounded medium with a given excitability. It is shown that a rigidly rotating spiral in a medium with a strongly reduced refractoriness is supported within a range of the medium excitability restricted by two universal limits. At the low excitability limit the spiral core radius diverges, while at the high excitability limit it vanishes. The simulations performed for the medium excitability higher than the high excitability limit reveal nonstationary rotating waves, which considerably differ from well-studied meandering spiral waves. It is shown how the proposed free-boundary approach can be extended to the case of an arbitrary refractoriness. The predictions of the free-boundary approach are in good agreement with the results from numerical simulations of the underlying reaction-diffusion model and with asymptotics derived earlier for highly and weakly excitable media.     

Potential of propagation‐based synchrotron X‐ray phase‐contrast computed tomography for cardiac tissue engineering

Hydrogel‐based cardiac tissue engineering offers great promise for myocardial infarction repair. The ability to visualize engineered systems in vivo in animal models is desired to monitor the performance of cardiac constructs. However, due to the low density and weak X‐ray attenuation of hydrogels, conventional radiography and micro‐computed tomography are unable to visualize the hydrogel cardiac constructs upon their implantation, thus limiting their use in animal systems. This paper presents a study on the optimization of synchrotron X‐ray propagation‐based phase‐contrast imaging computed tomography (PCI‐CT) for three‐dimensional (3D) visualization and assessment of the hydrogel cardiac patches. First, alginate hydrogel was 3D‐printed into cardiac patches, with the pores filled by fibrin. The hydrogel patches were then surgically implanted on rat hearts. A week after surgery, the hearts including patches were excised and embedded in a soft‐tissue‐mimicking gel for imaging by using PCI‐CT at an X‐ray energy of 25 keV. During imaging, the sample‐to‐detector distances, CT‐scan time and the region of interest (ROI) were varied and examined for their effects on both imaging quality and radiation dose. The results showed that phase‐retrieved PCI‐CT images provided edge‐enhancement fringes at a sample‐to‐detector distance of 147 cm that enabled visualization of anatomical and microstructural features of the myocardium and the implanted patch in the tissue‐mimicking gel. For visualization of these features, PCI‐CT offered a significantly higher performance than the dual absorption‐phase and clinical magnetic resonance (3 T) imaging techniques. Furthermore, by reducing the total CT‐scan time and ROI, PCI‐CT was examined for lowering the effective dose, meanwhile without much loss of imaging quality. In effect, the higher soft tissue contrast and low‐dose potential of PCI‐CT has been used along with an acceptable overall animal dose to achieve the high spatial resolution needed for cardiac implant visualization. As a result, PCI‐CT at the identified imaging parameters offers great potential for 3D assessment of microstructural features of hydrogel cardiac patches.     

Characteristics and generation of elastic turbulence in a three-dimensional parallel plate channel using direct numerical simulation

Direct numerical simulations(DNSs) of purely elastic turbulence in rectilinear shear flows in a three-dimensional(3D) parallel plate channel were carried out,by which numerical databases were established.Based on the numerical databases,the present paper analyzed the structural and statistical characteristics of the elastic turbulence including flow patterns,the wall effect on the turbulent kinetic energy spectrum,and the local relationship between the flow motion and the microstructures' behavior.Moreover,to address the underlying physical mechanism of elastic turbulence,its generation was presented in terms of the global energy budget.The results showed that the flow structures in elastic turbulence were 3D with spatial scales on the order of the geometrical characteristic length,and vortex tubes were more likely to be embedded in the regions where the polymers were strongly stretched.In addition,the patterns of microstructures' elongation behave like a filament.From the results of the turbulent kinetic energy budget,it was found that the continuous energy releasing from the polymers into the main flow was the main source of the generation and maintenance of the elastic turbulent status.