Simple model for identifying critical regions in atrial fibrillation

Implementation of a Cellular Automaton for efficient simulations of atrial arrhythmias

In silico models offer a promising advancement for studying cardiac arrhythmias and their clinical implications. However, existing detailed mathematical models often suffer from prolonged computational time compared to diagnostic needs. This study introduces a Cellular Automaton (CA) model tailored to replicate atrial electrophysiology in different stages of Atrial Fibrillation (AF), including persistent AF (PsAF). The CA, using a finite set of states, has been trained using biophysical simulations on a reduced domain for a large set of pacing conditions. Fine-tuning included tissue heterogeneity and anisotropic propagation through pacing simulations. Characterized by Action Potential Duration (APD), Diastolic Interval (DI) and Conduction Velocity (CV) for varying levels of electrical remodeling, the biophysical simulations introduced restitution curves or surfaces into the CA. Validation involved a comprehensive comparison with realistic 2D and 3D atrial models, evaluating healthy and pro-arrhythmic behaviors. Comparisons between CA and biophysical solver revealed striking proximity, with a Cycle Length difference of <10 ms in self-sustained re-entry and a 4.66±0.57 ms difference in depolarization times across the complete atrial geometry. Notably, the CA model exhibited a 80% accuracy, 96% specificity and 45% sensitivity in predicting AF inducibility under different pacing sites and substrate conditions. Additionally, the CA allowed for a 64-fold decrease in computing time compared to the biophysical solver. CA emerges as an efficient and valid model for simulation of atrial electrophysiology across different stages of AF, with potential as a general screening tool for rapid tests. While biophysical tests are recommended for investigating specific mechanisms, CA proves valuable in clinical applications for personalized therapy planning through digital twin simulations.

The physics of heart rhythm disorders

The global burden caused by cardiovascular disease is substantial, with heart disease representing the most common cause of death around the world. There remains a need to develop better mechanistic models of cardiac function in order to combat this health concern. Heart rhythm disorders, or arrhythmias, are one particular type of disease which has been amenable to quantitative investigation. Here we review the application of quantitative methodologies to explore dynamical questions pertaining to arrhythmias. We begin by describing single-cell models of cardiac myocytes, from which two and three dimensional models can be constructed. Special focus is placed on results relating to pattern formation across these spatially-distributed systems, especially the formation of spiral waves of activation. Next, we discuss mechanisms which can lead to the initiation of arrhythmias, focusing on the dynamical state of spatially discordant alternans, and outline proposed mechanisms perpetuating arrhythmias such as fibrillation. We then review experimental and clinical results related to the spatio-temporal mapping of heart rhythm disorders. Finally, we describe treatment options for heart rhythm disorders and demonstrate how statistical physics tools can provide insights into the dynamics of heart rhythm disorders.

Identifying locations susceptible to micro-anatomical reentry using a spatial network representation of atrial fibre maps

Micro-anatomical reentry has been identified as a potential driver of atrial fibrillation (AF). In this paper, we introduce a novel computational method which aims to identify which atrial regions are most susceptible to micro-reentry. The approach, which considers the structural basis for micro-reentry only, is based on the premise that the accumulation of electrically insulating interstitial fibrosis can be modelled by simulating percolation-like phenomena on spatial networks. Our results suggest that at high coupling, where micro-reentry is rare, the micro-reentrant substrate is highly clustered in areas where the atrial walls are thin and have convex wall morphology, likely facilitating localised treatment via ablation. However, as transverse connections between fibres are removed, mimicking the accumulation of interstitial fibrosis, the substrate becomes less spatially clustered, and the bias to forming in thin, convex regions of the atria is reduced, possibly restricting the efficacy of localised ablation. Comparing our algorithm on image-based models with and without atrial fibre structure, we find that strong longitudinal fibre coupling can suppress the micro-reentrant substrate, whereas regions with disordered fibre orientations have an enhanced risk of micro-reentry. With further development, these methods may be useful for modelling the temporal development of the fibrotic substrate on an individualised basis.

Simplifying the Process of Going From Cells to Tissues Using Statistical Mechanics

The value of digital twins for prototyping controllers or interventions in a sandbox environment are well-established in engineering and physics. However, this is challenging for biophysics trying to seamlessly compose models of multiple spatial and temporal scale behavior into the digital twin. Two challenges stand out as constraining progress: (i) ensuring physical consistency of conservation laws across composite models and (ii) drawing useful and timely clinical and scientific information from conceptually and computationally complex models. Challenge (i) can be robustly addressed with bondgraphs. However, challenge (ii) is exacerbated using this approach. The complexity question can be looked at from multiple angles. First from the perspective of discretizations that reflect underlying biophysics (functional tissue units) and secondly by exploring maximum entropy as the principle guiding multicellular biophysics. Statistical mechanics, long applied to understanding emergent phenomena from atomic physics, coupled with the observation that cellular architecture in tissue is orchestrated by biophysical constraints on metabolism and communication, shows conceptual promise. This architecture along with cell specific properties can be used to define tissue specific network motifs associated with energetic contributions. Complexity can be addressed based on energy considerations and finding mean measures of dependent variables. A probability distribution of the tissue's network motif can be approximated with exponential random graph models. A prototype problem shows how these approaches could be implemented in practice and the type of information that could be extracted.

A Review on Atrial Fibrillation (Computer Simulation and Clinical Perspectives)

Atrial fibrillation (AF), a heart condition, has been a well-researched topic for the past few decades. This multidisciplinary field of study deals with signal processing, finite element analysis, mathematical modeling, optimization, and clinical procedure. This article is focused on a comprehensive review of journal articles published in the field of AF. Topics from the age-old fundamental concepts to specialized modern techniques involved in today’s AF research are discussed. It was found that a lot of research articles have already been published in modeling and simulation of AF. In comparison to that, the diagnosis and post-operative procedures for AF patients have not yet been totally understood or explored by the researchers. The simulation and modeling of AF have been investigated by many researchers in this field. Cellular model, tissue model, and geometric model among others have been used to simulate AF. Due to a very complex nature, the causes of AF have not been fully perceived to date, but the simulated results are validated with real-life patient data. Many algorithms have been proposed to detect the source of AF in human atria. There are many ablation strategies for AF patients, but the search for more efficient ablation strategies is still going on. AF management for patients with different stages of AF has been discussed in the literature as well but is somehow limited mostly to the patients with persistent AF. The authors hope that this study helps to find existing research gaps in the analysis and the diagnosis of AF.

Understanding the transition from paroxysmal to persistent atrial fibrillation

Atrial fibrillation (AF) is the most common cardiac arrhytmia, characterized by the chaotic motion of electrical wavefronts in the atria. In clinical practice, AF is classified under two primary categories: paroxysmal AF, short intermittent episodes separated by periods of normal electrical activity; and persistent AF, longer uninterrupted episodes of chaotic electrical activity. However, the precise reasons why AF in a given patient is paroxysmal or persistent is poorly understood. Recently, we have introduced the percolation-based Christensen-Manani-Peters (CMP) model of AF which naturally exhibits both paroxysmal and persistent AF, but precisely how these differences emerge in the model is unclear. In this paper, we dissect the CMP model to identify the cause of these different AF classifications. Starting from a mean-field model where we describe AF as a simple birth-death process, we add layers of complexity to the model and show that persistent AF arises from reentrant circuits which exhibit an asymmetry in their probability of activation relative to deactivation. As a result, different simulations generated at identical model parameters can exhibit fibrillatory episodes spanning several orders of magnitude from a few seconds to months. These findings demonstrate that diverse, complex fibrillatory dynamics can emerge from very simple dynamics in models of AF.

Generation of ECG signals from a reaction-diffusion model spatially discretized

We propose a model to generate electrocardiogram signals based on a discretized reaction-diffusion system to produce a set of three nonlinear oscillators that simulate the main pacemakers in the heart. The model reproduces electrocardiograms from healthy hearts and from patients suffering various well-known rhythm disorders. In particular, it is shown that under ventricular fibrillation, the electrocardiogram signal is chaotic and the transition from sinus rhythm to chaos is consistent with the Ruelle-Takens-Newhouse route to chaos, as experimental studies indicate. The proposed model constitutes a useful tool for research, medical education, and clinical testing purposes. An electronic device based on the model was built for these purposes

Unified mechanism of local drivers in a percolation model of atrial fibrillation

The mechanisms of atrial fibrillation (AF) are poorly understood, resulting in disappointing success rates of ablative treatment. Different mechanisms defined largely by different atrial activation patterns have been proposed and, arguably, this dispute has slowed the progress of AF research. Recent clinical evidence suggests a unifying mechanism of local drivers based on sustained reentrant circuits in the complex atrial architecture. Here, we present a percolation inspired computational model showing spontaneous emergence of AF that strongly supports, and gives a theoretical explanation for, the clinically observed diversity of activation. We show that the difference in surface activation patterns is a direct consequence of the thickness of the discrete network of heart muscle cells through which electrical signals percolate to reach the imaged surface. The model naturally follows the clinical spectrum of AF spanning sinus rhythm, paroxysmal AF, and persistent AF as the decoupling of myocardial cells results in the lattice approaching the percolation threshold. This allows the model to make the prediction that, for paroxysmal AF, reentrant circuits emerge near the endocardium, but in persistent AF they emerge deeper in the bulk of the atrial wall. If experimentally verified, this may go towards explaining the lowering ablation success rate as AF becomes more persistent.

Identifying Potential Re-Entrant Circuit Locations From Atrial Fibre Maps

Re-entrant circuits have been identified as potential drivers of atrial fibrillation (AF). In this paper, we develop a novel computational framework for finding the locations of re-entrant circuits from high resolution fibre orientation data. The technique follows a statistical approach whereby we generate continuous fibre tracts across the tissue and couple adjacent fibres stochastically if they are within a given distance of each other. By varying the connection distance, we identify which regions are most susceptible to forming re-entrant circuits if muscle fibres are uncoupled, through the action of fibrosis or otherwise. Our results highlight the sleeves of the pulmonary veins, the posterior left atrium and the left atrial appendage as the regions most susceptible to re-entrant circuit formation. This is consistent with known risk locations in clinical AF. If the model can be personalised for individual patients undergoing ablation, future versions may be able to suggest suitable ablation targets.

Addressing challenges of quantitative methodologies and event interpretation in the study of atrial fibrillation

Atrial fibrillation (AF) is the commonest arrhythmia, yet the mechanisms of its onset and persistence are incompletely known. Although techniques for quantitative assessment have been investigated, there have been few attempts to integrate this information to advance disease treatment protocols. In this review, key quantitative methods for AF analysis are described, and suggestions are provided for the coordination of the available information, and to develop foci and directions for future research efforts. Quantitative biologists may have an interest in this topic in order to develop machine learning and tools for arrhythmia characterization, but they may perhaps have a minimal background in the clinical methodology and in the types of observed events and mechanistic hypotheses that have thus far been developed. We attempt to address these issues via exploration of the published literature. Although no new data is presented in this review, examples are shown of current lines of investigation, and in particular, how electrogram analysis and whole-chamber quantitative modeling of the left atrium may be useful to characterize fibrillatory patterns of activity, so as to propose avenues for more efficacious acquisition and interpretation of AF data.

Heart Rhythm Insights Into Structural Remodeling in Atrial Tissue: Timed Automata Approach

The heart rhythm of a person following heart transplantation (HTX) is assumed to display an intrinsic cardiac rhythm because it is significantly less influenced by the autonomic nervous system-the main source of heart rate variability in healthy people. Therefore, such a rhythm provides evidence for arrhythmogenic processes developing, usually silently, in the cardiac tissue. A model is proposed to simulate alterations in the cardiac tissue and to observe the effects of these changes on the resulting heart rhythm. The hybrid automata framework used makes it possible to represent reliably and simulate efficiently both the electrophysiology of a cardiac cell and the tissue organization. The curve fitting method used in the design of the hybrid automaton cycle follows the well-recognized physiological phases of the atrial myocyte membrane excitation. Moreover, knowledge of the complex architecture of the right atrium, the ability of the almost free design of intercellular connections makes the automata approach the only one possible. Two particular aspects are investigated: impairment of the impulse transmission between cells and structural changes in intercellular connections. The first aspect models the observed fatigue of cells due to specific cardiac tissue diseases. The second aspect simulates the increase in collagen deposition with aging. Finally, heart rhythms arising from the model are validated with the sinus heart rhythms recorded in HTX patients. The modulation in the impairment of the impulse transmission between cells reveals qualitatively the abnormally high heart rate variability observed in patients living long after HTX.

Ectopic beats arise from micro-reentries near infarct regions in simulations of a patient-specific heart model

Ectopic beats are known to be involved in the initiation of a variety of cardiac arrhythmias. Although their location may vary, ectopic excitations have been found to originate from infarct areas, regions of micro-fibrosis and other heterogeneous tissues. However, the underlying mechanisms that link ectopic foci to heterogeneous tissues have yet to be fully understood. In this work, we investigate the mechanism of micro-reentry that leads to the generation of ectopic beats near infarct areas using a patient-specific heart model. The patient-specific geometrical model of the heart, including scar and peri-infarct zones, is obtained through magnetic resonance imaging (MRI). The infarct region is composed of ischemic myocytes and non-conducting cells (fibrosis, for instance). Electrophysiology is captured using an established cardiac myocyte model of the human ventricle modified to describe ischemia. The simulation results clearly reveal that ectopic beats emerge from micro-reentries that are sustained by the heterogeneous structure of the infarct regions. Because microscopic information about the heterogeneous structure of the infarct regions is not available, Monte-Carlo simulations are used to identify the probabilities of an infarct region to behave as an ectopic focus for different levels of ischemia and different percentages of non-conducting cells. From the proposed model, it is observed that ectopic beats are generated when a percentage of non-conducting cells is near a topological metric known as the percolation threshold. Although the mechanism for micro-reentries was proposed half a century ago to be a source of ectopic beats or premature ventricular contractions during myocardial infarction, the present study is the first to reproduce this mechanism in-silico using patient-specific data.

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

BACKGROUND

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

METHOD

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

RESULTS

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

CONCLUSIONS

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

Loss of Side-to-Side Connections Affects the Relative Contributions of the Sodium and Calcium Current to Transverse Propagation Between Strands of Atrial Myocytes

Background: Atrial fibrillation (AF) leads to a loss of transverse connections between myocyte strands that is associated with an increased complexity and stability of AF. We have explored the interaction between longitudinal and transverse coupling, and the relative contribution of the sodium (I_Na) and calcium (I_Ca) current to propagation, both in healthy tissue and under diseased conditions using computer simulations.

Methods: Two parallel strands of atrial myocytes were modeled (Courtemanche et al. ionic model). As a control condition, every single cell was connected both transversely and longitudinally. To simulate a loss of transverse connectivity, this number was reduced to 1 in 4, 8, 12, or 16 transversely. To study the interaction with longitudinal coupling, anisotropy ratios of 3, 9, 16, and 25:1 were used. All simulations were repeated for varying degrees of I_Na and I_Ca block and the transverse activation delay (TAD) between the paced and non-paced strands was calculated for all cases.

Results: The TAD was highly sensitive to the transverse connectivity, increasing from 1 ms at 1 in 1, to 25 ms at 1 in 4, and 100 ms at 1 in 12 connectivity. The TAD also increased when longitudinal coupling was increased. Both decreasing transverse connectivity and increasing longitudinal coupling enhanced the synchronicity of activation of the non-paced strand and increased the propensity for transverse conduction block. Even after long TADs, the action potential upstroke in the non-paced strand was still mainly dependent on the I_Na. Nevertheless, I_Ca in the paced strand was essential to provide depolarizing current to the non-paced strand. Loss of transverse connections increased the sensitivity to both I_Na and I_Ca block. However, when longitudinal coupling was relatively high, transverse propagation was more sensitive to I_Ca block than to I_Na block.

Conclusions: Although transverse propagation depends on both I_Na and I_Ca, their relative contribution, and sensitivity to channel blockade, depends on the distribution of transverse connections and the axial conductivity. This simple two-strand model helps to explain the nature of atrial discontinuous conduction during structural remodeling and provides an opportunity for more effective drug development.

Effects of refractory gradients and ablation on fibrillatory activity

BACKGROUND

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

METHOD

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

RESULTS

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

CONCLUSIONS

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

Machine learning methods for locating re-entrant drivers from electrograms in a model of atrial fibrillation

Mapping resolution has recently been identified as a key limitation in successfully locating the drivers of atrial fibrillation (AF). Using a simple cellular automata model of AF, we demonstrate a method by which re-entrant drivers can be located quickly and accurately using a collection of indirect electrogram measurements. The method proposed employs simple, out-of-the-box machine learning algorithms to correlate characteristic electrogram gradients with the displacement of an electrogram recording from a re-entrant driver. Such a method is less sensitive to local fluctuations in electrical activity. As a result, the method successfully locates 95.4% of drivers in tissues containing a single driver, and 95.1% (92.6%) for the first (second) driver in tissues containing two drivers of AF. Additionally, we demonstrate how the technique can be applied to tissues with an arbitrary number of drivers. In its current form, the techniques presented are not refined enough for a clinical setting. However, the methods proposed offer a promising path for future investigations aimed at improving targeted ablation for AF.

Enhancement of early warning properties in the Kuramoto model and in an atrial fibrillation model due to an external perturbation of the system

When a complex dynamical system is externally disturbed, the statistical moments of signals associated to it can be affected in ways that depend on the nature and amplitude of the perturbation. In systems that exhibit phase transitions, the statistical moments can be used as Early Warnings (EW) of the transition. A natural question is thus to wonder what effect external disturbances have on the EWs of system. In this work we study the impact of external noise added to the system on the EWs, with particular focus on understanding the importance of the amplitude and complexity of the noise. We do this by analyzing the EWs of two computational models related to biology: the Kuramoto model, which is a paradigm of synchronization for biological systems, and a cellular automaton model of cardiac dynamics which has been used as a model for atrial fibrillation. For each model we first characterize the EWs. Then, we introduce external noise of varying intensity and nature to observe what effect this has on the EWs. In both cases we find that the introduction of noise amplified the EWs, with more complex noise having a greater effect. This both offers a way to improve the chance of detection of EWs in real systems and suggests that natural variability in the real world does not have a detrimental effect on EWs, but the opposite.

Ageing related down-regulation of myocardial connexin-43 and up-regulation of MMP-2 may predict propensity to atrial fibrillation in experimental animals

Mechanisms underlying atrial fibrillation (AF), the most common cardiac arrhythmia, particularly in aged population, are not fully elucidated. We have previously shown an increased propensity of old guinea pigs (GPs) heart to inducible AF when comparing to young animals. This study aimed to verify our hypothesis that susceptibility of aged heart to AF may be attributed to abnormalities in myocardial connexin-43 (Cx43) and extracellular matrix that affect cardiac electrical properties. Experiments were conducted on male and female 4-week-old and 24-week-old GPs. Atrial tissue was processed for analysis of Cx43 topology using immunohistochemistry, expression of Cx43 protein using immunobloting, and expression of mRNA of Cx43 and extracellular matrix metalloproteinase-2 (MMP-2) using real time PCR. Immunohistochemistry revealed uniform Cx43 distribution predominantly on lateral sides of the cardiomyocytes of young male and female GP atria. In contrast, non-uniform distribution, mislocalization and reduced immunolabeling of Cx43 were detected in atria of old GPs. In parallel, the atrial tissue levels of Cx43 mRNA were significantly decreased, while mRNA expression of MMP-2 was significantly increased in old versus young GPs. The changes were more pronounced in old GPs males comparing to females. Findings indicate that age-related down-regulation of atrial Cx43 and up-regulation of MMP-2 as well as disordered Cx43 distribution can facilitate development of AF in old guinea pig hearts.

Models of life: epigenetics, diversity and cycles

This review emphasizes aspects of biology that can be understood through repeated applications of simple causal rules. The selected topics include perspectives on gene regulation, phage lambda development, epigenetics, microbial ecology, as well as model approaches to diversity and to punctuated equilibrium in evolution. Two outstanding features are repeatedly described. One is the minimal number of rules to sustain specific states of complex systems for a long time. The other is the collapse of such states and the subsequent dynamical cycle of situations that restitute the system to a potentially new metastable state.

Development of an automaton model of rotational activity driving atrial fibrillation

BACKGROUND

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

RESULTS

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

CONCLUSIONS

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

Fast propagation regions cause self-sustained reentry in excitable media

Self-sustained waves of electrophysiological activity can cause arrhythmia in the heart. These reentrant excitations have been associated with spiral waves circulating around either an anatomically defined weakly conducting region or a functionally determined core. Recently, an ablation procedure has been clinically introduced that stops atrial fibrillation of the heart by destroying the electrical activity at the spiral core. This is puzzling because the tissue at the anatomically defined spiral core would already be weakly conducting, and a further decrease should not improve the situation. In the case of a functionally determined core, an ablation procedure should even further stabilize the rotating wave. The efficacy of the procedure thus needs explanation. Here, we show theoretically that fundamentally in any excitable medium a region with a propagation velocity faster than its surrounding can act as a nucleation center for reentry and can anchor an induced spiral wave. Our findings demonstrate a mechanistic underpinning for the recently developed ablation procedure. Our theoretical results are based on a very general and widely used two-component model of an excitable medium. Moreover, the important control parameters used to realize conditions for the discovered phenomena are applicable to quite different multicomponent models.

Reentry and Ectopic Pacemakers Emerge in a Three-Dimensional Model for a Slab of Cardiac Tissue with Diffuse Microfibrosis near the Percolation Threshold

Arrhythmias in cardiac tissue are generally associated with irregular electrical wave propagation in the heart. Cardiac tissue is formed by a discrete cell network, which is often heterogeneous. Recently, it was shown in simulations of two-dimensional (2D) discrete models of cardiac tissue that a wave crossing a fibrotic, heterogeneous region may produce reentry and transient or persistent ectopic activity provided the fraction of conducting connections is just above the percolation threshold. Here, we investigate the occurrence of these phenomena in three-dimensions by simulations of a discrete model representing a thin slab of cardiac tissue. This is motivated (i) by the necessity to study the relevance and properties of the percolation-related mechanism for the emergence of microreentries in three dimensions and (ii) by the fact that atrial tissue is quite thin in comparison with ventricular tissue. Here, we simplify the model by neglecting details of tissue anatomy, e. g. geometries of atria or ventricles and the anisotropy in the conductivity. Hence, our modeling study is confined to the investigation of the effect of the tissue thickness as well as to the comparison of the dynamics of electrical excitation in a 2D layer with the one in a 3D slab. Our results indicate a strong and non-trivial effect of the thickness even for thin tissue slabs on the probability of microreentries and ectopic beat generation. The strong correlation of the occurrence of microreentry with the percolation threshold reported earlier in 2D layers persists in 3D slabs. Finally, a qualitative agreement of 3D simulated electrograms in the fibrotic region with the experimentally observed complex fractional atrial electrograms (CFAE) as well as strong difference between simulated electrograms in 2D and 3D were found for the cases where reentry and ectopic activity were triggered by the micro-fibrotic region.

Determining conduction patterns on a sparse electrode grid: Implications for the analysis of clinical arrhythmias

We present a general method of utilizing bioelectric recordings from a spatially sparse electrode grid to compute a dynamic vector field describing the underlying propagation of electrical activity. This vector field, termed the wave-front flow field, permits quantitative analysis of the magnitude of rotational activity (vorticity) and focal activity (divergence) at each spatial point. We apply this method to signals recorded during arrhythmias in human atria and ventricles using a multipolar contact catheter and show that the flow fields correlate with corresponding activation maps. Further, regions of elevated vorticity and divergence correspond to sites identified as clinically significant rotors and focal sources where therapeutic intervention can be effective. These flow fields can provide quantitative insights into the dynamics of normal and abnormal conduction in humans and could potentially be used to enhance therapies for cardiac arrhythmias.

Myocardial architecture and patient variability in clinical patterns of atrial fibrillation

Atrial fibrillation (AF) increases the risk of stroke by a factor of 4-5 and is the most common abnormal heart rhythm. The progression of AF with age, from short self-terminating episodes to persistence, varies between individuals and is poorly understood. An inability to understand and predict variation in AF progression has resulted in less patient-specific therapy. Likewise, it has been a challenge to relate the microstructural features of heart muscle tissue (myocardial architecture) with the emergent temporal clinical patterns of AF. We use a simple model of activation wave-front propagation on an anisotropic structure, mimicking heart muscle tissue, to show how variation in AF behavior arises naturally from microstructural differences between individuals. We show that the stochastic nature of progressive transversal uncoupling of muscle strands (e.g., due to fibrosis or gap junctional remodeling), as occurs with age, results in variability in AF episode onset time, frequency, duration, burden, and progression between individuals. This is consistent with clinical observations. The uncoupling of muscle strands can cause critical architectural patterns in the myocardium. These critical patterns anchor microreentrant wave fronts and thereby trigger AF. It is the number of local critical patterns of uncoupling as opposed to global uncoupling that determines AF progression. This insight may eventually lead to patient-specific therapy when it becomes possible to observe the cellular structure of a patient's heart.

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.

Emergence of Asynchronous Local Clocks in Excitable Media

Excitable media such as the myocardium or the brain consist of arrays of coupled excitable elements, in which the local excitation of a single element can propagate to its neighbors in the form of a non-linear autowave. Since each element has to pass through a refractory period immediately after excitation, the frequency of autowaves is self-limiting. In this work, we consider the case where each element is spontaneously excited at a fixed average rate and thereby initiates a new autowave. Although these spontaneous self-excitation events are modelled as independent Poisson point processes with exponentially distributed waiting times, the travelling autowaves lead collectively to a non-exponential, unimodal waiting time distribution for the individual elements. With increasing system size, a global 'clock' period T emerges as the most probable waiting time for each element, which fluctuates around T with an increasingly small but non-zero variance. This apparent synchronization between asynchronous, temporally uncorrelated point processes differs from synchronization effects between perfect oscillators interacting in a phase-aligning manner. Finally, we demonstrate that asynchronous local clocks also emerge in non-homogeneous systems in which the rates of self-excitation are different for all individuals, suggesting that this novel mechanism can occur in a wide range of excitable media.