Development of a computer algorithm for the detection of phase singularities and initial application to analyze simulations of atrial fibrillation

Rotor Localization and Phase Mapping of Cardiac Excitation Waves Using Deep Neural Networks

The analysis of electrical impulse phenomena in cardiac muscle tissue is important for the diagnosis of heart rhythm disorders and other cardiac pathophysiology. Cardiac mapping techniques acquire local temporal measurements and combine them to visualize the spread of electrophysiological wave phenomena across the heart surface. However, low spatial resolution, sparse measurement locations, noise and other artifacts make it challenging to accurately visualize spatio-temporal activity. For instance, electro-anatomical catheter mapping is severely limited by the sparsity of the measurements, and optical mapping is prone to noise and motion artifacts. In the past, several approaches have been proposed to create more reliable maps from noisy or sparse mapping data. Here, we demonstrate that deep learning can be used to compute phase maps and detect phase singularities in optical mapping videos of ventricular fibrillation, as well as in very noisy, low-resolution and extremely sparse simulated data of reentrant wave chaos mimicking catheter mapping data. The self-supervised deep learning approach is fundamentally different from classical phase mapping techniques. Rather than encoding a phase signal from time-series data, a deep neural network instead learns to directly associate phase maps and the positions of phase singularities with short spatio-temporal sequences of electrical data. We tested several neural network architectures, based on a convolutional neural network (CNN) with an encoding and decoding structure, to predict phase maps or rotor core positions either directly or indirectly via the prediction of phase maps and a subsequent classical calculation of phase singularities. Predictions can be performed across different data, with models being trained on one species and then successfully applied to another, or being trained solely on simulated data and then applied to experimental data. Neural networks provide a promising alternative to conventional phase mapping and rotor core localization methods. Future uses may include the analysis of optical mapping studies in basic cardiovascular research, as well as the mapping of atrial fibrillation in the clinical setting.

Topological charge-density method of identifying phase singularities in cardiac fibrillation

Spiral waves represent the key motifs of typical self-sustained dynamical patterns in excitable systems such as cardiac tissue. The motion of phase singularities (PSs) that lies at the center of spiral waves captures many qualitative and, in some cases, quantitative features of their complex dynamics. Recent clinical studies suggested that ablating the tissue at PS locations may cure atrial fibrillation. Here, we propose a different method to determine the location of PSs. Starting from the definition of the topological charge of spiral waves, we obtain the expression of the topological charge density in a discrete case. With this expression, we can calculate the topological charge at each grid in the space directly, so as to accurately identify the position of PSs.

Spatial phase discontinuity at the center of moving cardiac spiral waves

BACKGROUND

Precise analysis of cardiac spiral wave (SW) dynamics is essential for effective arrhythmia treatment. Although the phase singularity (PS) point in the spatial phase map has been used to determine the cardiac SW center for decades, quantitative detection algorithms that assume PS as a point fail to trace complex and rapid PS dynamics. Through a detailed analysis of numerical simulations, we examined our hypothesis that a boundary of spatial phase discontinuity induced by a focal conduction block exists around the moving SW center in the phase map.

METHOD

In a numerical simulation model of a 2D cardiac sheet, three different types of SWs (short wavelength; long wavelength; and low excitability) were induced by regulating ion channels. Discontinuities of all boundaries among adjacent cells at each instance were evaluated by calculating the phase bipolarity (PB). The total amount of phase transition (PTA) in each cell during the study period was evaluated.

RESULTS

Pivoting, drifting, and shifting SWs were observed in the short-wavelength, low-excitability, and long-wavelength models, respectively. For both the drifting and shifting cases, long high-PB edges were observed on the SW trajectories. In all cases, the conduction block (CB) was observed at the same boundaries. These were also identical to the boundaries in the PTA maps.

CONCLUSIONS

The analysis of the simulations revealed that the conduction block at the center of a moving SW induces discontinuous boundaries in spatial phase maps that represent a more appropriate model of the SW center than the PS point.

Heart failure-induced atrial remodelling promotes electrical and conduction alternans

Heart failure (HF) is associated with an increased propensity for atrial fibrillation (AF), causing higher mortality than AF or HF alone. It is hypothesized that HF-induced remodelling of atrial cellular and tissue properties promotes the genesis of atrial action potential (AP) alternans and conduction alternans that perpetuate AF. However, the mechanism underlying the increased susceptibility to atrial alternans in HF remains incompletely elucidated. In this study, we investigated the effects of how HF-induced atrial cellular electrophysiological (with prolonged AP duration) and tissue structural (reduced cell-to-cell coupling caused by atrial fibrosis) remodelling can have an effect on the generation of atrial AP alternans and their conduction at the cellular and one-dimensional (1D) tissue levels. Simulation results showed that HF-induced atrial electrical remodelling prolonged AP duration, which was accompanied by an increased sarcoplasmic reticulum (SR) Ca²⁺ content and Ca²⁺ transient amplitude. Further analysis demonstrated that HF-induced atrial electrical remodelling increased susceptibility to atrial alternans mainly due to the increased sarcoplasmic reticulum Ca²⁺-ATPase (SERCA) Ca²⁺ reuptake, modulated by increased phospholamban (PLB) phosphorylation, and the decreased transient outward K⁺ current (I_to). The underlying mechanism has been suggested that the increased SR Ca²⁺ content and prolonged AP did not fully recover to their previous levels at the end of diastole, resulting in a smaller SR Ca²⁺ release and AP in the next beat. These produced Ca²⁺ transient alternans and AP alternans, and further caused AP alternans and Ca²⁺ transient alternans through Ca²⁺→AP coupling and AP→Ca²⁺ coupling, respectively. Simulation of a 1D tissue model showed that the combined action of HF-induced ion channel remodelling and a decrease in cell-to-cell coupling due to fibrosis increased the heart tissue’s susceptibility to the formation of spatially discordant alternans, resulting in an increased functional AP propagation dispersion, which is pro-arrhythmic. These findings provide insights into how HF promotes atrial arrhythmia in association with atrial alternans.

Atrial Fibrillation (AF) is the most common arrhythmia in adults, especially in the elderly, with the increased incidence of stroke being a major complication that increases morbidity and mortality. The occurrence of AF is often accompanied by heart failure (HF). AF and HF are also known to have the bidirectional relationship that AF worsens HF and HF promotes AF. HF can induce atrial remodelling, including electrical remodelling, atrial fibrosis, stretch and dilatation, and oxidative stress, in which many factors are associated with arrhythmogenic atrial alternans. HF-induced atrial remodelling varies during various stages and complications of HF, but possible mechanisms underlying their pro-susceptibility to alternans have not been completely elucidated. In this study, we investigated the effects of HF-induced atrial remodelling with prolonged action potential duration (APD) and decreased cell-to-cell coupling on susceptibility to atrial alternans. Simulation results showed that HF-induced an increase in sarcoplasmic reticulum Ca²⁺-ATPase (SERCA) Ca²⁺ reuptake caused by increased phospholamban phosphorylation and a decrease in transient outward K⁺ current played significant roles in the genesis of Ca²⁺ transient alternans and action potential alternans at the single-cell level. The HF-induced decline of cell-to-cell coupling and APD prolongation promoted the genesis of spatially discordant alternans in atrial tissue. This provides insights into how HF facilitates atrial arrhythmia in relation to atrial alternans.

Altered calcium handling produces reentry-promoting action potential alternans in atrial fibrillation-remodeled hearts

Atrial fibrillation (AF) alters atrial cardiomyocyte (ACM) Ca2+ handling, promoting ectopic beat formation. We examined the effects of AF-associated remodeling on Ca2+-related action potential dynamics and consequences for AF susceptibility. AF was maintained electrically in dogs by right atrial (RA) tachypacing. ACMs isolated from AF dogs showed increased Ca2+ release refractoriness, spontaneous Ca2+ spark frequency, and cycle length (CL) threshold for Ca2+ and action potential duration (APD) alternans versus controls. AF increased the in situ CL threshold for Ca2+/APD alternans and spatial dispersion in Ca2+ release recovery kinetics, leading to spatially discordant alternans associated with reentrant rotor formation and susceptibility to AF induction/maintenance. The clinically available agent dantrolene reduced Ca2+ leak and CL threshold for Ca2+/APD alternans in ACMs and AF dog right atrium, while suppressing AF susceptibility; caffeine increased Ca2+ leak and CL threshold for Ca2+/APD alternans in control dog ACMs and RA tissues. In vivo, the atrial repolarization alternans CL threshold was increased in AF versus control, as was AF vulnerability. Intravenous dantrolene restored repolarization alternans threshold and reduced AF vulnerability. Immunoblots showed reduced expression of total and phosphorylated ryanodine receptors and calsequestrin in AF and unchanged phospholamban/SERCA expression. Thus, along with promoting spontaneous ectopy, AF-induced Ca2+ handling abnormalities favor AF by enhancing vulnerability to repolarization alternans, promoting initiation and maintenance of reentrant activity; dantrolene provides a lead molecule to target this mechanism.

Time resolution for wavefront and phase singularity tracking using activation maps in cardiac propagation models

The dynamics of cardiac fibrillation can be described by the number, the trajectory, the stability, and the lifespan of phase singularities (PSs). Accurate PS tracking is straightforward in simple uniform tissues but becomes more challenging as fibrosis, structural heterogeneity, and strong anisotropy are combined. In this paper, we derive a mathematical formulation for PS tracking in two-dimensional reaction-diffusion models. The method simultaneously tracks wavefronts and PS based on activation maps at full spatiotemporal resolution. PS tracking is formulated as a linear assignment problem solved by the Hungarian algorithm. The cost matrix incorporates information about distances between PS, chirality, and wavefronts. A graph of PS trajectories is generated to represent the creations and annihilations of PS pairs. Structure-preserving graph transformations are applied to provide a simplified description at longer observation time scales. The approach is validated in 180 simulations of fibrillation in four different types of substrates featuring, respectively, wavebreaks, ionic heterogeneities, fibrosis, and breakthrough patterns. The time step of PS tracking is studied in the range from 0.1 to 10 ms. The results show the benefits of improving time resolution from 1 to 0.1 ms. The tracking error rate decreases by an order of magnitude because the occurrence of simultaneous events becomes less likely. As observed on PS survival curves, the graph-based analysis facilitates the identification of macroscopically stable rotors despite wavefront fragmentation by fibrosis.

Interaction of phase singularities on the spiral wave tail: reconsideration of capturing the excitable gap

The action mechanism of stimulation toward spiral waves (SWs) owing to the complex excitation patterns that occur just after point stimulation has not yet been experimentally clarified. This study sought to test our hypothesis that the effect of capturing excitable gap of SWs by stimulation can also be explained as the interaction of original phase singularity (PS) and PSs induced by the stimulation on the wave tail (WT) of the original SW. Phase variance analysis was used to quantitatively analyze the postshock PS trajectories. In a two-dimensional subepicardial layer of Langendorff-perfused rabbit hearts, optical mapping was used to record the excitation pattern during stimulation. After a SW was induced by S1–S2 shock, single biphasic point stimulation S3 was applied. In 70 of the S1-S2-S3 stimulation episodes applied on 6 hearts, the original PS was clearly observed just before the S3 point stimulation in 37 episodes. Pairwise PSs were newly induced by the S3 in 20 episodes. The original PS collided with the newly induced PSs in 16 episodes; otherwise, they did not interact with the original PS. SW shift occurred most efficiently when the S3 shock was applied at the relative refractory period, and PS shifted in the direction of the WT. In conclusion, quantitative tracking of PS clarified that stimulation in desirable conditions induces pairwise PSs on WT and that the collision of PSs causes SW shift along the WT. The results of this study indicate the importance of the interaction of shock-induced excitation with the WT for effective stimulation.

NEW & NOTEWORTHY The quantitative analysis of spiral wave dynamics during stimulation clarified the action mechanism of capturing the excitable gap, i.e., the induction of pairwise phase singularities on the wave tail and spiral wave shift along the wave tail as a result of these interactions. The importance of the wave tail for effective stimulation was revealed.

Effectiveness of atrial fibrillation rotor ablation is dependent on conduction velocity: An in-silico 3-dimensional modeling study

BACKGROUND

We previously reported that stable rotors are observed in in-silico human atrial fibrillation (AF) models, and are well represented by a dominant frequency (DF). In the current study, we hypothesized that the outcome of DF ablation is affected by conduction velocity (CV) conditions and examined this hypothesis using in-silico 3D-AF modeling.

METHODS

We integrated 3D CT images of left atrium obtained from 10 patients with persistent AF (80% male, 61.8±13.5 years old) into in-silico AF model. We compared AF maintenance durations (max 300s), spatiotemporal stabilities of DF, phase singularity (PS) number, life-span of PS, and AF termination or defragmentation rates after virtual DF ablation with 5 different CV conditions (0.2, 0.3, 0.4, 0.5, and 0.6m/s).

RESULTS

1. AF maintenance duration (p<0.001), spatiotemporal mean variance of DF (p<0.001), and the number of PS (p = 0.023) showed CV dependent bimodal patterns (highest at CV0.4m/s and lowest at CV0.6m/s) consistently. 2. After 10% highest DF ablation, AF defragmentation rates were the lowest at CV0.4m/s (37.8%), but highest at CV0.5 and 0.6m/s (all 100%, p<0.001). 3. In the episodes with AF termination or defragmentation followed by 10% highest DF ablation, baseline AF maintenance duration was shorter (p<0.001), spatiotemporal mean variance of DF was lower (p = 0.014), and the number of PS was lower (p = 0.004) than those with failed AF defragmentation after DF ablation.

CONCLUSION

Virtual ablation of DF, which may indicate AF driver, was more likely to terminate or defragment AF with spatiotemporally stable DF, but not likely to do so in long-lasting and sustained AF conditions, depending on CV.

A statistical model of false negative and false positive detection of phase singularities

The complexity of cardiac fibrillation dynamics can be assessed by analyzing the distribution of phase singularities (PSs) observed using mapping systems. Interelectrode distance, however, limits the accuracy of PS detection. To investigate in a theoretical framework the PS false negative and false positive rates in relation to the characteristics of the mapping system and fibrillation dynamics, we propose a statistical model of phase maps with controllable number and locations of PSs. In this model, phase maps are generated from randomly distributed PSs with physiologically-plausible directions of rotation. Noise and distortion of the phase are added. PSs are detected using topological charge contour integrals on regular grids of varying resolutions. Over 100 × 10⁶ realizations of the random field process are used to estimate average false negative and false positive rates using a Monte-Carlo approach. The false detection rates are shown to depend on the average distance between neighboring PSs expressed in units of interelectrode distance, following approximately a power law with exponents in the range of 1.14 to 2 for false negatives and around 2.8 for false positives. In the presence of noise or distortion of phase, false detection rates at high resolution tend to a non-zero noise-dependent lower bound. This model provides an easy-to-implement tool for benchmarking PS detection algorithms over a broad range of configurations with multiple PSs.

A New Efficient Method for Detecting Phase Singularity in Cardiac Fibrillation

Background

The point of phase singularity (PS) is considered to represent a spiral wave core or a rotor in cardiac fibrillation. Computational efficiency is important for detection of PS in clinical electrophysiology. We developed a novel algorithm for highly efficient and robust detection of PS.

Methods

In contrast to the conventional method, which calculates PS based on the line integral of the phase around a PS point equal to ±2π (the Iyer-Gray method), the proposed algorithm (the location-centric method) looks for the phase discontinuity point at which PS actually occurs. We tested the efficiency and robustness of these two methods in a two-dimensional mathematical model of atrial fibrillation (AF), with and without remodeling of ionic currents.

Results

1. There was a significant association, in terms of the Hausdorff distance (3.30 ± 0.0 mm), between the PS points measured using the Iyer-Gray and location-centric methods, with almost identical PS trajectories generated by the two methods. 2. For the condition of electrical remodeling of AF (0.3 × I_CaL), the PS points calculated by the two methods were satisfactorily co-localized (with the Hausdorff distance of 1.64 ± 0.09 mm). 3. The proposed location-centric method was substantially more efficient than the Iyer-Gray method, with a 28.6-fold and 28.2-fold shorter run times for the control and remodeling scenarios, respectively.

Conclusion

We propose a new location-centric method for calculating PS, which is robust and more efficient compared with the conventionally used method.

Anti-arrhythmic strategies for atrial fibrillation: The role of computational modeling in discovery, development, and optimization

Atrial fibrillation (AF), the most common cardiac arrhythmia, is associated with increased risk of cerebrovascular stroke, and with several other pathologies, including heart failure. Current therapies for AF are targeted at reducing risk of stroke (anticoagulation) and tachycardia-induced cardiomyopathy (rate or rhythm control). Rate control, typically achieved by atrioventricular nodal blocking drugs, is often insufficient to alleviate symptoms. Rhythm control approaches include antiarrhythmic drugs, electrical cardioversion, and ablation strategies. Here, we offer several examples of how computational modeling can provide a quantitative framework for integrating multiscale data to: (a) gain insight into multiscale mechanisms of AF; (b) identify and test pharmacological and electrical therapy and interventions; and (c) support clinical decisions. We review how modeling approaches have evolved and contributed to the research pipeline and preclinical development and discuss future directions and challenges in the field.

Identification of Rotors during Human Atrial Fibrillation Using Contact Mapping and Phase Singularity Detection: Technical Considerations

OBJECTIVE

To explore technical challenges of phase singularity (PS) mapping during atrial fibrillation (AF) using direct contact electrograms.

METHODS

AF mapping was performed in high-density epicardial recordings of human paroxysmal (PAF) or persistent (PersAF) (N = 20 pts) AF with an array of 16 × 16 electrodes placed on atrial epicardium. PS points were detected using subsets of electrodes forming rings of varying sizes.

RESULTS

PS detection using a 2 × 2 electrode ring identified 0.88 ± 1.00 PS/s in PAF group and 3.91 ± 2.51 per s in PersAF group (p < 0.001) in 2.4 × 2.4 cm mapping area. All detected PS had a short lifespan with the longest being 1100 ms (6.8 rotations). Exploration of the PS detection in a numerical model demonstrated that at least eight electrodes are required to avoid frequent false positive PS detection due to chance. Application of a detection grid consisting a double ring of electrodes (2 × 2 and 4 × 4 rings) decreased the number of false positive detections. The double ring was more resilient to electrode swapping (with just three instances of false positives versus 4380 false positives using 2 × 2 ring).

CONCLUSIONS

The number of detected rotors critically depends upon the parameters of the detection algorithm, especially the number of electrodes used to detect PS. Based on our results, we recommend double ring comprised of 2 × 2 and 4 × 4 grid of electrodes for robust rotor detection.

SIGNIFICANCE

Great methodological care has to be taken before equating detected PS with rotating waves and using PS detection algorithms to guide catheter ablation of AF.

Regulation of Spatiotemporal Patterns by Biological Variability: General Principles and Applications to Dictyostelium discoideum

Spatiotemporal patterns often emerge from local interactions in a self-organizing fashion. In biology, the resulting patterns are also subject to the influence of the systematic differences between the system’s constituents (biological variability). This regulation of spatiotemporal patterns by biological variability is the topic of our review. We discuss several examples of correlations between cell properties and the self-organized spatiotemporal patterns, together with their relevance for biology. Our guiding, illustrative example will be spiral waves of cAMP in a colony of Dictyostelium discoideum cells. Analogous processes take place in diverse situations (such as cardiac tissue, where spiral waves occur in potentially fatal ventricular fibrillation) so a deeper understanding of this additional layer of self-organized pattern formation would be beneficial to a wide range of applications. One of the most striking differences between pattern-forming systems in physics or chemistry and those in biology is the potential importance of variability. In the former, system components are essentially identical with random fluctuations determining the details of the self-organization process and the resulting patterns. In biology, due to variability, the properties of potentially very few cells can have a driving influence on the resulting asymptotic collective state of the colony. Variability is one means of implementing a few-element control on the collective mode. Regulatory architectures, parameters of signaling cascades, and properties of structure formation processes can be "reverse-engineered" from observed spatiotemporal patterns, as different types of regulation and forms of interactions between the constituents can lead to markedly different correlations. The power of this biology-inspired view of pattern formation lies in building a bridge between two scales: the patterns as a collective state of a very large number of cells on the one hand, and the internal parameters of the single cells on the other.

Pattern formation is abundant in nature—from the rich ornaments of sea shells and the diversity of animal coat patterns to the myriad of fractal structures in biology and pattern-forming colonies of bacteria. Particularly fascinating are patterns changing with time, spatiotemporal patterns, like propagating waves and aggregation streams. Bacteria form large branched and nested aggregation-like patterns to immobilize themselves against water flow. The individual amoeba in Dictyostelium discoideum colonies initiates a transition to a collective multicellular state via a quorum-sensing form of communication—a cAMP signal propagating through the community in the form of spiral waves—and the subsequent chemotactic response of the cells leads to branch-like aggregation streams. The theoretical principle underlying most of these spatial and spatiotemporal patterns is self-organization, in which local interactions lead to patterns as large-scale collective”modes” of the system. Over more than half a century, these patterns have been classified and analyzed according to a”physics paradigm,” investigating such questions as how parameters regulate the transitions among patterns, which (types of) interactions lead to such large-scale patterns, and whether there are "critical parameter values" marking the sharp, spontaneous onset of patterns. A fundamental discovery has been that simple local interaction rules can lead to complex large-scale patterns. The specific pattern "layouts" (i.e., their spatial arrangement and their geometric constraints) have received less attention. However, there is a major difference between patterns in physics and chemistry on the one hand and patterns in biology on the other: in biology, patterns often have an important functional role for the biological system and can be considered to be under evolutionary selection. From this perspective, we can expect that individual biological elements exert some control on the emerging patterns. Here we explore spiral wave patterns as a prominent example to illustrate the regulation of spatiotemporal patterns by biological variability. We propose a new approach to studying spatiotemporal data in biology: analyzing the correlation between the spatial distribution of the constituents’ properties and the features of the spatiotemporal pattern. This general concept is illustrated by simulated patterns and experimental data of a model organism of biological pattern formation, the slime mold Dictyostelium discoideum. We introduce patterns starting from Turing (stripe and spot) patterns, together with target waves and spiral waves. The biological relevance of these patterns is illustrated by snapshots from real and theoretical biological systems. The principles of spiral wave formation are first explored in a stylized cellular automaton model and then reproduced in a model of Dictyostelium signaling. The shaping of spatiotemporal patterns by biological variability (i.e., by a spatial distribution of cell-to-cell differences) is demonstrated, focusing on two Dictyostelium models. Building up on this foundation, we then discuss in more detail how the nonlinearities in biological models translate the distribution of cell properties into pattern events, leaving characteristic geometric signatures.

Dynamics of atrial arrhythmias modulated by time-dependent acetylcholine concentration: a simulation study

AIM

The autonomic nervous system modulates atrial activity, notably through acetylcholine (ACh) release. This time-dependent action may alter the dynamics of atrial arrhythmia. Our aim is to investigate in a computer model the changes induced by ACh release and degradation on the dynamical regime of a reentry.

METHODS AND RESULTS

A functional reentry was simulated in a 10 × 5 cm(2) two-dimensional tissue with canine atrial membrane kinetics including an ACh-dependent K(+) current. The local ACh concentration was altered over time in a circular region following a predefined spatiotemporal profile (ACh release and degradation) characterized by its maximum ACh level, time constant of release/degradation, and diameter of the region. Phase singularities were tracked to monitor the complexity of the dynamics. Four scenarios were identified: (i) the original reentry remained stable; (ii) repolarization gradients induced by ACh release caused wavebreaks, resulting in a transient complex dynamics that spontaneously converted to a single stable reentry; (iii) the reentry self-terminated through wavebreaks and wavefront interactions; (4) wavebreaks led to a complex dynamics that converted to two or three reentries that remained stable after ACh degradation. Higher ACh level, short ACh release time constant, larger heterogeneous region, and short distance between the heterogeneous region and the spiral tip were associated with higher occurrence of ACh-induced wavebreaks.

CONCLUSION

Variation of ACh concentration over time may modulate the complexity of atrial arrhythmias.

Substrate size as a determinant of fibrillatory activity maintenance in a mathematical model of canine atrium

Tissue size has been considered an important determinant of atrial fibrillation (AF), but recent work has questioned the critical size hypothesis. Here, we use a previously developed mathematical model of the two-dimensional canine atrium with realistic action potential, ionic, and conduction properties to address substrate size effects on the maintenance of fibrillatory activity. Cholinergic AF was simulated at different acetylcholine (ACh) concentrations ([ACh]) and distributions, with substrate area varied 11.1-fold. Automated phase singularity detection was used to facilitate the analysis of arrhythmic activity. The duration of activity induced by a single extrastimulus increased with increasing substrate dimensions. Two general mechanisms underlying activity were observed and were differentially affected by substrate size. For large mean [ACh], single primary rotors anchored in low-[ACh] zones maintained activity and substrate dimensions were not critical. At lower mean [ACh], extensive spiral wave meander prevented the emergence of single stable rotors. Prolonged activity was favored when substrate size permitted a sufficiently large number of simultaneous longer-lasting rotors that extinction of all was unlikely. Thus either single dominant rotor or multiple reentrant spiral generator mechanisms could maintain fibrillatory activity in this model and were differentially dependent on substrate size. These results speak to recent debates about the role in AF of single driver rotors versus multiple reentrant circuit mechanisms by suggesting that either may maintain fibrillatory atrial activity depending on atrial size and electrophysiological properties.

Mechanisms of Atrial Fibrillation Termination by Pure Sodium Channel Blockade in an Ionically-Realistic Mathematical Model

The mechanisms by which Na + -channel blocking antiarrhythmic drugs terminate atrial fibrillation (AF) remain unclear. Classical “leading-circle” theory suggests that Na + -channel blockade should, if anything, promote re-entry. We used an ionically-based mathematical model of vagotonic AF to evaluate the effects of applying pure Na + -current (I Na ) inhibition during sustained arrhythmia. Under control conditions, AF was maintained by 1 or 2 dominant spiral waves, with fibrillatory propagation at critical levels of action potential duration (APD) dispersion. I Na inhibition terminated AF increasingly with increasing block, terminating all AF at 65% block. During 1:1 conduction, I Na inhibition reduced APD (by 13% at 4 Hz and 60% block), conduction velocity (by 37%), and re-entry wavelength (by 24%). During AF, I Na inhibition increased the size of primary rotors and reduced re-entry rate (eg, dominant frequency decreased by 33% at 60% I Na inhibition) while decreasing generation of secondary wavelets by wavebreak. Three mechanisms contributed to I Na block–induced AF termination in the model: (1) enlargement of the center of rotation beyond the capacity of the computational substrate; (2) decreased anchoring to functional obstacles, increasing meander and extinction at boundaries; and (3) reduction in the number of secondary wavelets that could provide new primary rotors. Optical mapping in isolated sheep hearts confirmed that tetrodotoxin dose-dependently terminates AF while producing effects qualitatively like those of I Na inhibition in the mathematical model. We conclude that pure I Na inhibition terminates AF, producing activation changes consistent with previous clinical and experimental observations. These results provide insights into previously enigmatic mechanisms of class I antiarrhythmic drug-induced AF termination. The full text of this article is available online at http://circres.ahajournals.org

Parallel Multigrid Preconditioner for the Cardiac Bidomain Model

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue but are computationally expensive, limiting the size of the problem which can be modeled. The purpose of this study is to determine more efficient ways to solve the elliptic portion of the bidomain equations, the most computationally expensive part of the computation. Specifically, we assessed the performance of a parallel multigrid (MG) preconditioner for a conjugate gradient solver. We employed an operator splitting technique, dividing the computation in a parabolic equation, an elliptical equation, and a nonlinear system of ordinary differential equations at each time step. The elliptic equation was solved by the preconditioned conjugate gradient method, and the traditional block incomplete LU parallel preconditioner (ILU) was compared to MG. Execution time was minimized for each preconditioner by adjusting the fill-in factor for ILU, and by choosing the optimal number of levels for MG. The parallel implementation was based on the PETSc library and we report results for up to 16 nodes on a distributed cluster, for two and three dimensional simulations. A direct solver was also available to compare results for single processor runs. MG was found to solve the system in one third of the time required by ILU but required about 40% more memory. Thus, MG offered an attractive tradeoff between memory usage and speed, since its performance lay between those of the classic iterative methods (slow and low memory consumption) and direct methods (fast and high memory consumption). Results suggest the MG preconditioner is well suited for quickly and accurately solving the bidomain equations.

Introduction: Mapping and control of complex cardiac arrhythmias

This paper serves as an introduction to the Focus Issue on mapping and control of complex cardiac arrhythmias. We first introduce basic concepts of cardiac electrophysiology and describe the main clinical methods being used to treat arrhythmia. We then provide a brief summary of the main themes contained in the articles in this Focus Issue. In recent years there have been important advances in the ability to map the spread of excitation in intact hearts and in laboratory settings. This work has been combined with simulations that use increasingly realistic geometry and physiology. Waves of excitation and contraction in the heart do not always propagate with constant velocity but are often subject to instabilities that may lead to fluctuations in velocity and cycle time. Such instabilities are often treated best in the context of simple one- or two-dimensional geometries. An understanding of the mechanisms of propagation and wave stability is leading to the implementation of different stimulation protocols in an effort to modify or eliminate abnormal rhythms. (c) 2002 American Institute of Physics.