Minimal principle for rotor filaments

Scroll Waves and Filaments in Excitable Media of Higher Spatial Dimension

Excitable media are ubiquitous in nature, and in such systems the local excitation tends to self-organize in traveling waves, or in rotating spiral-shaped patterns in two or three spatial dimensions. Examples include waves during a pandemic or electrical scroll waves in the heart. Here we show that such phenomena can be extended to a space of four or more dimensions and propose that connections of excitable elements in a network setting can be regarded as additional spatial dimensions. Numerical simulations are performed in four dimensions using the FitzHugh-Nagumo model, showing that the vortices rotate around a two-dimensional surface which we define as the superfilament. Evolution equations are derived for general superfilaments of codimension two in an N-dimensional space, and their equilibrium configurations are proven to be minimal surfaces. We suggest that biological excitable systems, such as the heart or brain which have nonlocal connections can be regarded, at least partially, as multidimensional excitable media and discuss further possible studies in this direction.

Impact of electrode orientation, myocardial wall thickness, and myofiber direction on intracardiac electrograms: numerical modeling and analytical solutions

Intracardiac electrograms (iEGMs) are time traces of the electrical potential recorded close to the heart muscle. We calculate unipolar and bipolar iEGMs analytically for a myocardial slab with parallel myofibers and validate them against numerical bidomain simulations. The analytical solution obtained via the method of mirrors is an infinite series of arctangents. It goes beyond the solid angle theory and is in good agreement with the simulations, even though bath loading effects were not accounted for in the analytical calculation. At a large distance from the myocardium, iEGMs decay as 1/R (unipolar), 1/R ² (bipolar and parallel), and 1/R ³ (bipolar and perpendicular to the endocardium). At the endocardial surface, there is a mathematical branch cut. Here, we show how a thicker myocardium generates iEGMs with larger amplitudes and how anisotropy affects the iEGM width and amplitude. If only the leading-order term of our expansion is retained, it can be determined how the conductivities of the bath, torso, myocardium, and myofiber direction together determine the iEGM amplitude. Our results will be useful in the quantitative interpretation of iEGMs, the selection of thresholds to characterize viable tissues, and for future inferences of tissue parameters.

Rotor mechanism and its mapping in atrial fibrillation

Treatment of atrial fibrillation (AF) remains challenging despite significant progress in understanding its underlying mechanisms. The first detailed, quantitative theory of functional re-entry, the 'leading circle' model, was developed more than 40 years ago. Subsequently, in decades of study, an alternative paradigm based on spiral waves has long been postulated to drive AF. The rotor as a 'spiral wave generator' is a curved 'vortex' formed by spin motion in the two-dimensional plane, identified using advanced mapping methods in experimental and clinical AF. However, it is challenging to achieve complementary results between experimental results and clinical studies due to the limitation in research methods and the complexity of the rotor mechanism. Here, we review knowledge garnered over decades on generation, electrophysiological properties, and three-dimensional (3D) structure diversity of the rotor mechanism and make a comparison among recent clinical approaches to identify rotors. Although initial studies of rotor ablation at many independent centres have achieved promising results, some inconclusive outcomes exist in others. We propose that the clinical rotor identification might be substantially influenced by (i) non-identical surface activation patterns, which resulted from a diverse 3D form of scroll wave, and (ii) inadequate resolution of mapping techniques. With rapidly advancing theoretical and technological developments, future work is required to resolve clinically relevant limitations in current basic and clinical research methodology, translate from one to the other, and resolve available mapping techniques.

Scroll wave with negative filament tension in a model of the left ventricle of the human heart and its overdrive pacing

Nonlinear waves of electrical excitation initiate cardiac contraction. Abnormal wave propagation in the heart, e.g., spiral waves, can lead to sudden cardiac arrest. This study analyzed the dynamics of spiral waves under the influence of an instability called negative filament tension, and examined how the spiral waves can be eliminated through high-frequency pacing. A generic anatomical model of the left ventricle of the human heart and the Aliev-Panfilov model for cardiac tissue were used. The study showed that the source of such arrhythmia is elongated filaments with lengths that can be 10-20 times greater than the characteristic thickness of the heart wall. In anisotropic tissue, the filament elongated before it was annihilated at the base of the heart. The spiral waves were eliminated through overdrive pacing with stimulation periods from 0.8 to 0.95 relative to the spiral wave period. The minimum time for the expulsion was about 10 s.

A Phase Defect Framework for the Analysis of Cardiac Arrhythmia Patterns

During cardiac arrhythmias, dynamical patterns of electrical activation form and evolve, which are of interest to understand and cure heart rhythm disorders. The analysis of these patterns is commonly performed by calculating the local activation phase and searching for phase singularities (PSs), i.e., points around which all phases are present. Here we propose an alternative framework, which focuses on phase defect lines (PDLs) and surfaces (PDSs) as more general mechanisms, which include PSs as a specific case. The proposed framework enables two conceptual unifications: between the local activation time and phase description, and between conduction block lines and the central regions of linear-core rotors. A simple PDL detection method is proposed and applied to data from simulations and optical mapping experiments. Our analysis of ventricular tachycardia in rabbit hearts (n = 6) shows that nearly all detected PSs were found on PDLs, but the PDLs had a significantly longer lifespan than the detected PSs. Since the proposed framework revisits basic building blocks of cardiac activation patterns, it can become a useful tool for further theory development and experimental analysis.

Mapping Technologies for Catheter Ablation of Atrial Fibrillation Beyond Pulmonary Vein Isolation

Catheter ablation remains the most effective and relatively minimally invasive therapy for rhythm control in patients with AF. Ablation has consistently shown a reduction of arrhythmia-related symptoms and significant improvement in patients’ quality of life compared with medical treatment. The ablation strategy relies on a well-established anatomical approach of effective pulmonary vein isolation. Additional anatomical targets have been reported with the aim of increasing procedure success in complex substrates. However, larger ablated areas with uncertainty of targeting relevant regions for AF initiation or maintenance are not exempt from the potential risk of complications and pro-arrhythmia. Recent developments in mapping tools and computational methods for advanced signal processing during AF have reported novel strategies to identify atrial regions associated with AF maintenance. These novel tools – although mainly limited to research series – represent a significant step forward towards the understanding of complex patterns of propagation during AF and the potential achievement of patient-tailored AF ablation strategies for the near future.

Rotor Drivers in Induction and Maintenance of Atrial Fibrillation

Atrial fibrillation is the most common arrhythmia in clinical practice. It is associated with an increased risk of stroke, chronic heart failure, and sudden cardiac death. Our options of restoring and maintaining sinus rhythm have a very limited effect, both in the case of antiarrhythmic and catheter treatment. Catheter ablation has proven to be a more effective approach than antiarrhythmic therapy. The success rate of the procedure reaches 70%. However, radiofrequency ablation is associated with a risk of complications, with 4.5% of patients likely to develop major complications, including tamponade (1.31%), femoral pseudoaneurysm (0.71%), and death (0.15%). Given the generally recognized dominant role of the pulmonary veins in the induction of atrial fibrillation, their electrical isolation has become the recommended tactic of the catheter approach. In the case of patients with paroxysmal form of atrial fibrillation, the success rate of the procedure reaches 87%. Unfortunately, in the case of persistent forms of atrial fibrillation, the effectiveness of the primary procedure decreases to 28% and reaches 51% with repeated interventions. In addition to the anatomically oriented isolation of the pulmonary veins, a number of strategies have been proposed to reach the secondary zones of atrial fibrillation induction. The results of recent studies on the effectiveness of strategies for ablation of rotor regions and their role in the induction and maintenance of AF may lead to the further development of catheter ablation techniques and an individual radiofrequency ablation approach in a particular patient.

Optical Mapping

Optical mapping of electrical activity in the heart is based on voltage-sensitive and lipophilic fluorescence dyes. Optical signals recorded from cardiac cells correlate well with their transmembrane potentials. High spatiotemporal resolution, wide field mapping, and high sensitivity to transmembrane potential enable detailed characterization of action potential initiation and propagation. Optical mapping is used to study complex patterns of excitation propagation, including propagation across the sinoatrial and atrioventricular nodes and during atrial and ventricular arrhythmias.Optical mapping is used to study the role of reentrant activity in atrial and ventricular fibrillation.

Conduction in the Heart Wall: Helicoidal Fibers Minimize Diffusion Bias

The mammalian heart must function as an efficient pump while simultaneously conducting electrical signals to drive the contraction process. In the ventricles, electrical activation begins at the insertion points of the Purkinje network in the endocardium. How does the diffusion component of the subsequent excitation wave propagate from the endocardium in a healthy heart wall without creating directional biases? We show that this is a consequence of the particular geometric organization of myocytes in the heart wall. Using a generalized helicoid to model fiber orientation, we treat the myocardium as a curved space via Riemannian geometry, and then use stochastic calculus to model local signal diffusion. Our analysis shows that the helicoidal arrangement of myocytes minimizes the directional biases that could lead to aberrant propagation, thereby explaining how electrophysiological principles are consistent with local measurements of cardiac fiber geometry. We discuss our results in the context of the need to balance electrical and mechanical requirements for heart function.

Hidden structures of information transport underlying spiral wave dynamics

A spiral wave is a macroscopic dynamics of excitable media that plays an important role in several distinct systems, including the Belousov-Zhabotinsky reaction, seizures in the brain, and lethal arrhythmia in the heart. Because the spiral wave dynamics can exhibit a wide spectrum of behaviors, its precise quantification can be challenging. Here we present a hybrid geometric and information-theoretic approach to quantifying the spiral wave dynamics. We demonstrate the effectiveness of our approach by applying it to numerical simulations of a two-dimensional excitable medium with different numbers and spatial patterns of spiral waves. We show that, by defining the information flow over the excitable medium, hidden coherent structures emerge that effectively quantify the information transport underlying the spiral wave dynamics. Most importantly, we find that some coherent structures become more clearly defined over a longer observation period. These findings provide validity with our approach to quantitatively characterize the spiral wave dynamics by focusing on information transport. Our approach is computationally efficient and is applicable to many excitable media of interest in distinct physical, chemical, and biological systems. Our approach could ultimately contribute to an improved therapy of clinical conditions such as seizures and cardiac arrhythmia by identifying potential targets of interventional therapies.

Nonlinear physics of electrical wave propagation in the heart: a review

The beating of the heart is a synchronized contraction of muscle cells (myocytes) that is triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media with applications to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact for cardiac arrhythmias.

Presence and stability of rotors in atrial fibrillation: evidence and therapeutic implications

Rotor-guided ablation has opened new perspectives into the therapy of atrial fibrillation (AF). Analysis of the spatio-temporal cardiac excitation patterns in the frequency and phase domains has demonstrated the importance of rotors in research models of AF, however, the dynamics and role of rotors in human AF are still controversial. In this review, the current knowledge gained through research models and patient data that support the notion that rotors are key players in AF maintenance is summarized. We report and discuss discrepancies regarding rotor prevalence and stability in various studies, which can be attributed in part to methodological differences among mapping systems. Future research for validation and improvement of current clinical electrophysiology mapping technologies will be crucial for developing mechanistic-based selection and application of the best therapeutic strategy for individual AF patient, being it, pharmaceutical, ablative, or other approach.

Variational principle for nonlinear wave propagation in dissipative systems

The dynamics of many natural systems is dominated by nonlinear waves propagating through the medium. We show that in any extended system that supports nonlinear wave fronts with positive surface tension, the asymptotic wave-front dynamics can be formulated as a gradient system, even when the underlying evolution equations for the field variables cannot be written as a gradient system. The variational potential is simply given by a linear combination of the occupied volume and surface area of the wave front and changes monotonically over time.

Filament Dynamics during Simulated Ventricular Fibrillation in a High-Resolution Rabbit Heart

The mechanisms underlying ventricular fibrillation (VF) are not well understood. The electrical activity on the heart surface during VF has been recorded extensively in the experimental setting and in some cases clinically; however, corresponding transmural activation patterns are prohibitively difficult to measure. In this paper, we use a high-resolution biventricular heart model to study three-dimensional electrical activity during fibrillation, focusing on the driving sources of VF: “filaments,” the organising centres of unstable reentrant scroll waves. We show, for the first time, specific 3D filament dynamics during simulated VF in a whole heart geometry that includes fine-scale anatomical structures. Our results suggest that transmural activity is much more complex than what would be expected from surface observations alone. We present examples of complex intramural activity, including filament breakup and reattachment, anchoring to the thin right ventricular apex; rapid transitions among various filament shapes; and filament lengths much greater than wall thickness. We also present evidence for anatomy playing a major role in VF development and coronary vessels and trabeculae influencing filament dynamics. Overall, our results indicate that intramural activity during simulated VF is extraordinarily complex and suggest that further investigation of 3D filaments is necessary to fully comprehend recorded surface patterns.

Modifying Ventricular Fibrillation by Targeted Rotor Substrate Ablation: Proof-of-Concept from Experimental Studies to Clinical VF

INTRODUCTION

Recent work has suggested a role for organized sources in sustaining ventricular fibrillation (VF). We assessed whether ablation of rotor substrate could modulate VF inducibility in canines, and used this proof-of-concept as a foundation to suppress antiarrhythmic drug-refractory clinical VF in a patient with structural heart disease.

METHODS AND RESULTS

In 9 dogs, we introduced 64-electrode basket catheters into one or both ventricles, used rapid pacing at a recorded induction threshold to initiate VF, and then defibrillated after 18±8 seconds. Endocardial rotor sites were identified from basket recordings using phase mapping, and ablation was performed at nonrotor (sham) locations (7 ± 2 minutes) and then at rotor sites (8 ± 2 minutes, P = 0.10 vs. sham); the induction threshold was remeasured after each. Sham ablation did not alter canine VF induction threshold (preablation 150 ± 16 milliseconds, postablation 144 ± 16 milliseconds, P = 0.54). However, rotor site ablation rendered VF noninducible in 6/9 animals (P = 0.041), and increased VF induction threshold in the remaining 3. Clinical proof-of-concept was performed in a patient with repetitive ICD shocks due to VF refractory to antiarrhythmic drugs. Following biventricular basket insertion, VF was induced and then defibrillated. Mapping identified 4 rotors localized at borderzone tissue, and rotor site ablation (6.3 ± 1.5 minutes/site) rendered VF noninducible. The VF burden fell from 7 ICD shocks in 8 months preablation to zero ICD therapies at 1 year, without antiarrhythmic medications.

CONCLUSIONS

Targeted rotor substrate ablation suppressed VF in an experimental model and a patient with refractory VF. Further studies are warranted on the efficacy of VF source modulation.

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.

Mechanistic Comparison of "Nearly Missed" Versus "On-Target" Rotor Ablation

OBJECTIVES

This study used advanced optical mapping techniques to examine atrial fibrillation (AF) dynamics before and after 2 distinct electrogram-based ablation strategies: complex fractionated atrial electrograms (CFAEs) and DFmax/rotor ablation.

BACKGROUND

Among the electrogram analytical features proposed to unravel the atrial regions that perpetuate AF, CFAEs, highest dominant frequency sites (DFmax), and, more recently, phase analysis-enabled rotor mapping have received the largest attention. Still, the mechanisms by which these approaches modulate AF dynamics and lead to AF termination are unknown.

METHODS

In Langendorff-perfused sheep hearts, AF was maintained by the continuous perfusion of acetylcholine and high-resolution endocardial-epicardial optical videos were recorded from the left atrial free wall and the posterior left atrium. Then, DFmax/rotor regions (n = 7), or CFAE regions harboring the highest wavebreak density (HWD) (n = 5), were targeted with a 4F ablation catheter (5 to 15 W, 30 to 60 s/point). Thereafter, we examined the changes in AF dynamics and whether AF terminated.

RESULTS

DFmax/rotor point ablation resulted in a significant decrease in DFmax values. In 2 animals AF terminated, whereas in the remaining 5 animals the post-ablation DFmax domain remained in the vicinity of its pre-ablation location. However, after HWD/CFAEs density ablation, DFmax values did not change, AF did not terminate, and post-ablation DFmax domains relocated from the left atrial free wall to the pulmonary vein-posterior left atrium region. In another group of hearts (n = 12), we observed that upon a progressive increase in acetylcholine concentration-mimicking the acute electrophysiological changes occurring after ablation-3-dimensional rotors drifted from one atrial region to another along large gradients of myocardial thickness.

CONCLUSIONS

"On-target" DFmax/rotor ablation leads to the annihilation of the fibrillation-driving rotor. This translates into large decreases in AF frequency or AF termination. In contrast, "nearly missed" HWD/CFAEs ablation spares the fibrillation-driving rotor, and set the stage for rotor drift along large myocardial thickness gradients.

Generalized minimal principle for rotor filaments

To a reaction-diffusion medium with an inhomogeneous anisotropic diffusion tensor D, we add a fourth spatial dimension such that the determinant of the diffusion tensor is constant in four dimensions. We propose a generalized minimal principle for rotor filaments, stating that the scroll wave filament strives to minimize its surface area in the higher-dimensional space. As a consequence, stationary scroll wave filaments in the original 3D medium are geodesic curves with respect to the metric tensor G=det(D)D(-1). The theory is confirmed by numerical simulations for positive and negative filament tension and a model with a non-stationary spiral core. We conclude that filaments in cardiac tissue with positive tension preferentially reside or anchor in regions where cardiac cells are less interconnected, such as portions of the cardiac wall with a large number of cleavage planes.

Refraction of scroll-wave filaments at the boundary between two reaction-diffusion media

We explore the shape and the dynamics of scroll-wave filaments in excitable media with an abruptly changing diffusion tensor, important for cardiac applications. We show that, similar to a beam of light, the filament refracts at the boundary separating domains with different diffusion. We derive the laws of filament refraction and test their validity in computational experiments. We discovered that at small angles to the interface, the filament can become unstable and develop oscillations. The nature of the observed instabilities, as well as overall theoretical and experimental significance of the findings, is discussed.

Rotation of a scroll wave reveals the metric tensor for the associated geodesic filament

This study considers a reaction-diffusion medium that supports a rotating scroll wave with stationary filament and diffusivity matrix D. The filament is a geodesic curve in three-space. The metric tensor g associated with that geodesic is obtained in terms of D, resulting in g=(detD)D-1. Here detD (the determinant of D) is not necessarily constant in space. The detD factor, although strongly indicated previously, has not yet been derived deductively, as it is here. We take the opportunity to outline, starting from the reaction-diffusion equations, a sequence of mathematical steps leading to both geodesic and metric.

Optical Mapping of Ventricular Fibrillation Dynamics

There is very limited information regarding the dynamic patterns of the electrical activity during ventricular fibrillation (VF) in humans. Most of the data used to generate and test hypotheses regarding the mechanisms of VF come from animal models and computer simulations and the quantification of VF patterns is non-trivial. Many of the experimental recordings of the dynamic spatial patterns of VF have been obtained from mammals using "optical mapping" or "video imaging" technology in which "phase maps" are derived from high-resolution transmembrane recordings from the heart surface. The surface manifestation of the unstable reentrant waves sustaining VF can be identified as "phase singularities" and their number and location provide one measure of VF complexity. After providing a brief history of optical mapping of VF, we compare and contrast a quantitative analysis of VF patterns from the heart surface for four different animal models, hence providing physiological insight into the variety of VF dynamics among species. We found that in all four animal models the action potential duration restitution slope was actually negative during VF and that the spatial dispersion of electrophysiological parameters were not different during the first second of VF compared to pacing immediately before VF initiation. Surprisingly, our results suggest that APD restitution and spatial dispersion may not be essential causes of VF dynamics. Analyses of electrophysiological quantities in the four animal models are consistent with the idea that VF is essentially a two-dimensional phenomenon in small rabbit hearts whose size are near the boundary of the "critical mass" required to sustain VF, while VF in large pig hearts is three-dimensional and exhibits the maximal theoretical phase singularity density, and thus will not terminate spontaneously.

Surface and intramural reentrant patterns during atrial fibrillation in the sheep

INTRODUCTION

This article is part of the Focus Theme of Methods of Information in Medicine on "Biosignal Interpretation: Advanced Methods for Studying Cardiovascular and Respiratory Systems".

BACKGROUND

Atrial fibrillation (AF) is the most common sustained cardiac arrhythmia in humans and is predicted to dramatically increase its prevalence in the future. High-resolution mapping data and Fourier power spectral analysis with its dominant frequency support the hypothesis that AF in the structurally normal sheep heart and in some patients often presents organized drivers in the form of periodic surface re-entries or breakthroughs. Nevertheless, the dynamics of those surface patterns of activity, as well as their intramural components are still poorly understood.

OBJECTIVE

To present data on AF waves from the surface of isolated sheep hearts and discuss the interpretation of their intramural patterns.

METHODS

We used a combination of endocardial-epicardial optical mapping with phase and spectral analysis as well as computer simulation of the re-entrant activity in the myocardial wall.

RESULTS

Analysis of the surfaces' optical mapping data in the phase domain reveals that activation of the posterior left atrium (PLA) consisted of alternating patterns of breakthroughs and reentries. The patterns on the endocardial and epicardial PLA surface at any given moment of time of the AF could be either identical or not identical, and the activity in the thickness of the PLA wall is hypothesized to conform to either ectopic discharge or reentrant scroll waves, but a definite evidence for the presence of such mechanisms is currently lacking. A universal minimal-principle theory is shown in a computer model to result in a tendency of the axis of the scroll waves to align with the myocardial fibers inside the wall.

CONCLUSION

The tendency of filaments of scroll waves to align with myocardial fibers may contribute to the variety and intermittency of surface rotors seen in AF.

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.

Effective dynamics of twisted and curved scroll waves using virtual filaments

Scroll waves are three-dimensional excitation patterns that rotate around a central filament curve; they occur in many physical, biological, and chemical systems. We explicitly derive the equations of motion for scroll wave filaments in reaction-diffusion systems with isotropic diffusion up to third order in the filament's twist and curvature. The net drift components define at every instance of time a virtual filament which lies close to the instantaneous filament. Importantly, virtual filaments obey simpler, time-independent laws of motion which we analytically derive here and illustrate with numerical examples. Stability analysis of scroll waves is performed using virtual filaments, showing that filament curvature and twist add as quadratic terms to the nominal filament tension. Applications to oscillating chemical reactions and cardiac tissue are discussed.

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.

Twists of opposite handedness on a scroll wave

The dynamic interaction of scroll waves in the Belousov-Zhabotinsky reaction with a vertically orientated gradient of excitability is studied by optical tomography. This study focuses on scroll waves, whose filaments were oriented almost perpendicular to the gradient. Whereas scroll waves with filaments exactly perpendicular to the gradient remain unaffected, filaments with a component parallel to the gradient develop a twist. Scroll waves with U-shaped filaments exhibit twists starting from both of its ends, resulting in scroll waves whose filaments display a pair of twists of opposite handedness. These twists are separated by a nodal plane where the filament remains straight and untwisted. The experimental findings were reproduced by numerical simulations using the Oregonator model and a linear gradient of excitability almost perpendicular to the orientation of the filament.

Thermodynamic metrics and optimal paths

A fundamental problem in modern thermodynamics is how a molecular-scale machine performs useful work, while operating away from thermal equilibrium without excessive dissipation. To this end, we derive a friction tensor that induces a Riemannian manifold on the space of thermodynamic states. Within the linear-response regime, this metric structure controls the dissipation of finite-time transformations, and bestows optimal protocols with many useful properties. We discuss the connection to the existing thermodynamic length formalism, and demonstrate the utility of this metric by solving for optimal control parameter protocols in a simple nonequilibrium model.

[Basic mechanisms of the new antiarrhythmic drugs in atrial fibrillation]

Atrial fibrillation (AF) is the most common sustained arrhythmia seen in clinical practice. Despite of new technological breakthroughs and the understanding of the mechanisms underlying AF, based on animal models and ablation procedures in patients, the antiarrhythmic drugs remain the main therapeutic strategy to restore and maintain the sinus rhythm. New antiarrhythmic drugs are already available in the clinical practice and many others are under development. The new antiarrhythmic drugs have the capability to block atrial-specific ionic currents, which are involved in the maintenance of the arrhythmia. Parallel, increasing evidence supports the use of compounds to regulate the arrhythmogenic atrial substrate involved in the long-term maintenance of the arrhythmia (upstream therapies). This article reviews the new antiarrhythmic drugs and upstream therapies, based on the current knowledge of the mechanisms involved in the maintenance of AF.

Chloroquine terminates stretch-induced atrial fibrillation more effectively than flecainide in the sheep heart

BACKGROUND

Blockade of inward-rectifier K+ channels by chloroquine terminates reentry in cholinergic atrial fibrillation (AF). However, it is unknown whether inward-rectifier K+ channels and reentry are also important in maintaining stretch-induced AF (SAF). We surmised that reentry underlies SAF, and that abolishing reentry with chloroquine terminates SAF more effectively than traditional Na+-channel blockade by flecainide.

METHODS AND RESULTS

Thirty Langendorff-perfused sheep hearts were exposed to acute and continuous atrial stretch, and mapped optically and electrically. AF dynamics were studied under control and during perfusion of either chloroquine (4 µmol/L, n=7) or flecainide (2-4 µmol/L, n=5). Chloroquine increased rotor core size and decreased reentry frequency from 10.6±0.7 Hz in control to 6.3±0.7 Hz (P<0.005) just before restoring sinus rhythm (7/7). Flecainide had lesser effects on core size and reentry frequency than chloroquine and did not restore sinus rhythm (0/5). Specific IKr blockade by E-4031 (n=7) did not terminate AF when frequency values were >8 Hz. During pacing (n=11), flecainide reversibly reduced conduction velocity (≈30% at cycle length 300, 250, and 200 ms; P<0.05) to a larger extent than chloroquine (11% to 19%; cycle length, 300, 250, and 200 ms; P<0.05). Significant action potential duration prolongation was demonstrable only for chloroquine at cycle length 300 (12%) and cycle length 250 ms (9%) (P<0.05).

CONCLUSIONS

Chloroquine is more effective than flecainide in terminating SAF in isolated sheep hearts by significantly increasing core size and decreasing reentry frequency. Chloroquine's effectiveness may be explained by its inward-rectifier K+ channel blockade profile and suggest that reentry is important to maintain acute SAF.

Heterogeneous atrial wall thickness and stretch promote scroll waves anchoring during atrial fibrillation

AIMS

Atrial dilatation and myocardial stretch are strongly associated with atrial fibrillation (AF). However, the mechanisms by which the three-dimensional (3D) atrial architecture and heterogeneous stretch contribute to AF perpetuation are incompletely understood. We compared AF dynamics during stretch-related AF (pressure: 12 cmH(2)O) in normal sheep hearts (n = 5) and in persistent AF (PtAF, n = 8)-remodelled hearts subjected to prolonged atrial tachypacing. We hypothesized that, in the presence of stretch, meandering 3D atrial scroll waves (ASWs) anchor in regions of large spatial gradients in wall thickness.

METHODS AND RESULTS

We implemented a high-resolution optical mapping set-up that enabled simultaneous epicardial- and endoscopy-guided endocardial recordings of the intact atria in Langendorff-perfused normal and PtAF (AF duration: 21.3 ± 11.9 days) hearts. The numbers and lifespan of long-lasting ASWs (>3 rotations) were greater in PtAF than normal (lifespan 0.9 ± 0.5 vs. 0.4 ± 0.2 s/(3 s of AF), P< 0.05). Than normal hearts, focal breakthroughs interacted with ASWs at the posterior left atrium and left atrial appendage to maintain AF. In PtAF hearts, ASW filaments seemed to span the atrial wall from endocardium to epicardium. Numerical simulations using 3D atrial geometries (Courtemanche-Ramirez-Nattel human atrial model) predicted that, similar to experiments, filaments of meandering ASWs stabilized at locations with large gradients in myocardial thickness. Moreover, simulations predicted that ionic remodelling and heterogeneous distribution of stretch-activated channel conductances contributed to filament stabilization.

CONCLUSION

The heterogeneous atrial wall thickness and atrial stretch, together with ionic and anatomic remodelling caused by AF, are the main factors allowing ASW and AF maintenance.

Accurate eikonal-curvature relation for wave fronts in locally anisotropic reaction-diffusion systems

The dependency of wave velocity in reaction-diffusion (RD) systems on the local front curvature determines not only the stability of wave propagation, but also the fundamental properties of other spatial configurations such as vortices. This Letter gives the first derivation of a covariant eikonal-curvature relation applicable to general RD systems with spatially varying anisotropic diffusion properties, such as cardiac tissue. The theoretical prediction that waves which seem planar can nevertheless possess a nonvanishing geometrical curvature induced by local anisotropy is confirmed by numerical simulations, which reveal deviations up to 20% from the nominal plane wave speed.

Déjà vu in the theories of atrial fibrillation dynamics

This brief review looks back to the major theoretical, experimental, and clinical work on the dynamics and mechanisms of atrial fibrillation (AF). Its goal is to highlight the most important issues, controversies, and advances that have driven the field of investigation into AF mechanisms at any given time during the last ∼100 years. It emphasizes that while the history of AF research has been full of controversies from the start, such controversies have led to new information, and individual scientists have learned from those that have preceded them. However, in the face of the most common sustained cardiac arrhythmia seen in clinical practice, we are yet to fully understand its fundamental mechanisms and learn how to treat it effectively. Future research into AF dynamics and mechanisms should focus on the development and validation of new numerical and animal models. Such models should be relevant to and accurately reproduce the important substrates associated with ageing and with diseases such as hypertension, heart failure, and ischaemic heart disease which cause AF in the vast majority of patients. Knowledge derived from such models may help to greatly advance the field and hopefully lead to more effective prevention and therapy.

Frustrated drift of an anchored scroll-wave filament and the geodesic principle

We investigate anchored scroll-wave filaments in an excitable medium whose diffusivity matrix, including its determinant, is spatially nonuniform. The study is motivated by cardiological applications where scroll-wave behavior in the presence of diffusivity gradients is believed to play an important role in the development of severe arrhythmias. A diffusivity gradient is expected to make the filament drift, unless drift is prevented ("frustrated") by anchoring to localized defects in the propagation medium. The resulting stationary filament is a geodesic curve, as demonstrated here in the case of a nonzero but constant gradient. That is, the diffusivity matrix has a determinant that varies in space, in contrast to what was assumed in earlier work. Here, we show that the filament shape results from a metric tensor of the form (det D)D{-1} , where D is the diffusivity tensor. The filament's shape is solely determined by the diffusivity tensor and is independent of the equation's reaction terms. We derive the analytic solution for the filament and determine conditions for the existence of that solution. The theory is in excellent agreement with numerical simulations.

Anisotropy of wave propagation in the heart can be modeled by a Riemannian electrophysiological metric

It is well established that wave propagation in the heart is anisotropic and that the ratio of velocities in the three principal directions may be as large as v(f)v(s)v(n) approximately 4(fibers)2(sheets)1(normal). We develop an alternative view of the heart based on this fact by considering it as a non-Euclidean manifold with an electrophysiological(el-) metric based on wave velocity. This metric is more natural than the Euclidean metric for some applications, because el-distances directly encode wave propagation. We develop a model of wave propagation based on this metric; this model ignores higher-order effects like the curvature of wavefronts and the effect of the boundary, but still gives good predictions of local activation times and replicates many of the observed features of isochrones. We characterize this model for the important case of the rotational orthotropic anisotropy seen in cardiac tissue and perform numerical simulations for a slab of cardiac tissue with rotational orthotropic anisotropy and for a model of the ventricles based on diffusion tensor MRI scans of the canine heart. Even though the metric has many slow directions, we show that the rotation of the fibers leads to fast global activation. In the diffusion tensor MRI-based model, with principal velocities 0.25051 m/s, we find examples of wavefronts that eventually reach speeds up to 0.9 m/s and average velocities of 0.7 m/s. We believe that development of this non-Euclidean approach to cardiac anatomy and electrophysiology could become an important tool for the characterization of the normal and abnormal electrophysiological activity of the heart.

Atrial septopulmonary bundle of the posterior left atrium provides a substrate for atrial fibrillation initiation in a model of vagally mediated pulmonary vein tachycardia of the structurally normal heart

BACKGROUND

The posterior left atrium (PLA) and pulmonary veins (PVs) have been shown to be critical for atrial fibrillation (AF) initiation. However, the detailed mechanisms of reentry and AF initiation by PV impulses are poorly understood. We hypothesized that PV impulses trigger reentry and AF by undergoing wavebreaks as a result of sink-to-source mismatch at specific PV-PLA transitions along the septopulmonary bundle, where there are changes in thickness and fiber direction.

METHODS AND RESULTS

In 7 Langendorff-perfused sheep hearts AF was initiated by a burst of 6 pulses (CL 80 to 150ms) delivered to the left inferior or right superior PV ostium 100 to 150 ms after the sinus impulse in the presence of 0.5 micromol/L acetylcholine. The exposed septal-PLA endocardial area was mapped with high spatio-temporal resolution (DI-4-ANEPPS, 1000-fr/s) during AF initiation. Isochronal maps for each paced beat preceding AF onset were constructed to localize areas of conduction delay and block. Phase movies allowed the determination of the wavebreak sites at the onset of AF. Thereafter, the PLA myocardial wall thickness was quantified by echocardiography, and the fiber direction in the optical field of view was determined after peeling off the endocardium. Finally, isochrone, phase and conduction velocity maps were superimposed on the corresponding anatomic pictures for each of the 28 episodes of AF initiation. The longest delays of the paced PV impulses, as well as the first wavebreak, occurred at those boundaries along the septopulmonary bundle that showed sharp changes in fiber direction and the largest and most abrupt increase in myocardial thickness.

CONCLUSION

Waves propagating from the PVs into the PLA originating from a simulated PV tachycardia triggered reentry and vagally mediated AF by breaking at boundaries along the septopulmonary bundle where abrupt changes in thickness and fiber direction resulted in sink-to-source mismatch and low safety for propagation.

Vortex filament dynamics in computational models of ventricular fibrillation in the heart

In three-dimensional cardiac tissue, the re-entrant waves that sustain ventricular fibrillation rotate around a line of phase singularity or vortex filament. The aim of this study was to investigate how the behavior of these vortex filaments is influenced by membrane kinetics, initial conditions, and tissue geometry in computational models of excitable tissue. A monodomain model of cardiac tissue was used, with kinetics described by a three-variable simplified ionic model (3V-SIM). Two versions of 3V-SIM were used, one with steep action potential duration restitution, and one with reduced excitability. Re-entrant fibrillation was then simulated in three tissue geometries: a cube, a slab, and an anatomically detailed model of rabbit ventricles. Filaments were identified using a phase-based method, and the number, size, origin, and orientation of filaments was tracked throughout each simulation. The main finding of this study is that kinetics, initial conditions, geometry, and anisotropy all affected the number, proliferation, and orientation of vortex filaments in re-entrant fibrillation. An important finding of this study was that the behavior of vortex filaments in simplified slab geometry representing part of the ventricular wall did not necessarily predict behavior in an anatomically detailed model of the rabbit ventricles.

Epicardial mapping of ventricular fibrillation over the posterior descending artery and left posterior papillary muscle of the swine heart

BACKGROUND

Recent studies suggest that during ventricular fibrillation (VF) epicardial vessels may be a site of conduction block and the posterior papillary muscle (PPM) in the left ventricle (LV) may be the location of a "mother rotor." The goal of this study was to obtain evidence to support or refute these possibilities.

METHODS

Epicardial activation over the posterior LV and right ventricle (RV) was mapped during the first 20 s of electrically induced VF in six open-chest pigs with a 504 electrode plaque covering a 20 cm(2) area centered over the posterior descending artery (PDA).

RESULTS

The locations of epicardial breakthrough as well as reentry clustered in time and space during VF. Spatially, reentry occurred significantly more frequently over the LV than the RV in all 48 episodes, and breakthrough clustered near the PPM (p < 0.001). Significant temporal clustering occurred in 79% of breakthrough episodes and 100% of reentry episodes. These temporal clusters occurred at different times so that there was significantly less breakthrough when reentry was present (p < 0.0001). Conduction block occurred significantly more frequently near the PDA than elsewhere.

CONCLUSIONS

The PDA is a site of epicardial block which may contribute to VF maintenance. Epicardial breakthrough clusters near the PPM. Reentry also clusters in space but at a separate site. The fact that breakthrough and reentry cluster at different locations and at different times supports the possibility of a drifting filament at the PPM so that at times reentry is present on the surface but at other times the reentrant wavefront breaks through to the epicardium.

Modulation of spiral wave reentry by K(+) channel blockade

It is well established that spiral wave reentry is the primary mechanism of ventricular tachyarrhythmias (ventricular fibrillation/tachycardia, VF/VT), but information is still limited concerning pharmacological modification of spiral waves by ion channel blockers. In this brief review, the antiarrhythmic and proarrhythmic actions of K(+)-channel blockade (I(Kr) and I (K1)) are discussed in terms of spiral wave dynamics, primarily based on recent experimental findings in ventricular preparations perfused in vitro with the aid of high-resolution optical mapping, as well as their related theoretical studies using computer simulation.

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.

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.

Detection of intramyocardial scroll waves using absorptive transillumination imaging

Optical imaging using voltage-sensitive dyes has become an important tool for studying vortex-like electrical waves in the heart. Such waves, known as spiral or scroll waves, can spontaneously form in pathological ventricular myocardium, causing ventricular fibrillation and sudden death. Until recently, observations of scroll waves were limited to their surface manifestations, thus providing little information about the shape and location of their organizing center, the filament. We use computer modeling to assess the feasibility of visualizing filaments using dynamic transillumination imaging in conjunction with near-IR voltage-sensitive absorptive dyes (absorptive transillumination). We simulate transillumination signals produced by the intramural scroll waves in a realistic slab of ventricular tissue with trabeculated endocardial surface. The computations use a detailed ionic model of electrical excitation (LRd) coupled to a photon transport model for cardiac tissue. Our simulations show that dynamic absorptive transillumination data, with subsequent processing involving either amplitude maps, time-space plots, or power-of-the-dominant-frequency maps, can be used to reliably detect intramural scroll waves through the whole thickness (approximately 10 mm) of the ventricular wall. Neither variations in the thickness of the myocardial wall nor noise impeded the detection of intramural filaments.

Scroll waves meandering in a model of an excitable medium

We study numerically the dynamics of a scroll wave in a three-dimensional (3D) excitable medium in the presence of substantial meandering of the corresponding 2D spiral wave in the Aliev-Panfilov model. We identify three types of dynamics of the scroll wave filament--quasi-2D, periodic, and aperiodic meandering--and we study their dependence on parameter settings and thickness of the medium.

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.

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.