Conditions for propagation and block of excitation in an asymptotic model of atrial tissue

Reconstructing cardiac electrical excitations from optical mapping recordings

The reconstruction of electrical excitation patterns through the unobserved depth of the tissue is essential to realizing the potential of computational models in cardiac medicine. We have utilized experimental optical-mapping recordings of cardiac electrical excitation on the epicardial and endocardial surfaces of a canine ventricle as observations directing a local ensemble transform Kalman filter data assimilation scheme. We demonstrate that the inclusion of explicit information about the stimulation protocol can marginally improve the confidence of the ensemble reconstruction and the reliability of the assimilation over time. Likewise, we consider the efficacy of stochastic modeling additions to the assimilation scheme in the context of experimentally derived observation sets. Approximation error is addressed at both the observation and modeling stages through the uncertainty of observations and the specification of the model used in the assimilation ensemble. We find that perturbative modifications to the observations have marginal to deleterious effects on the accuracy and robustness of the state reconstruction. Furthermore, we find that incorporating additional information from the observations into the model itself (in the case of stimulus and stochastic currents) has a marginal improvement on the reconstruction accuracy over a fully autonomous model, while complicating the model itself and thus introducing potential for new types of model errors. That the inclusion of explicit modeling information has negligible to negative effects on the reconstruction implies the need for new avenues for optimization of data assimilation schemes applied to cardiac electrical excitation.

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.

Addendum: Action potential propagation and block in a model of atrial tissue with myocyte-fibroblast coupling

The analytical theory of our earlier study (Mortensen et al., 2021, Math. Med. Biol., 38, 106-131) is extended to address the outstanding cases of fibroblast barrier distribution and myocyte strait distribution. In particular, closed-form approximations to the resting membrane potential and to the critical parameter values for propagation are derived for these two non-uniform fibroblast distributions and are in good agreement with numerical estimates.

Action potential propagation and block in a model of atrial tissue with myocyte-fibroblast coupling

The electrical coupling between myocytes and fibroblasts and the spacial distribution of fibroblasts within myocardial tissues are significant factors in triggering and sustaining cardiac arrhythmias, but their roles are poorly understood. This article describes both direct numerical simulations and an asymptotic theory of propagation and block of electrical excitation in a model of atrial tissue with myocyte-fibroblast coupling. In particular, three idealized fibroblast distributions are introduced: uniform distribution, fibroblast barrier and myocyte strait-all believed to be constituent blocks of realistic fibroblast distributions. Primary action potential biomarkers including conduction velocity, peak potential and triangulation index are estimated from direct simulations in all cases. Propagation block is found to occur at certain critical values of the parameters defining each idealized fibroblast distribution, and these critical values are accurately determined. An asymptotic theory proposed earlier is extended and applied to the case of a uniform fibroblast distribution. Biomarker values are obtained from hybrid analytical-numerical solutions of coupled fast-time and slow-time periodic boundary value problems and compare well to direct numerical simulations. The boundary of absolute refractoriness is determined solely by the fast-time problem and is found to depend on the values of the myocyte potential and on the slow inactivation variable of the sodium current ahead of the propagating pulse. In turn, these quantities are estimated from the slow-time problem using a regular perturbation expansion to find the steady state of the coupled myocyte-fibroblast kinetics. The asymptotic theory gives a simple analytical expression that captures with remarkable accuracy the block of propagation in the presence of fibroblasts.

Mechanism for Mechanical Wave Break in the Heart Muscle

Using a reaction-diffusion-mechanics model we identify a mechanism for mechanical wave break in the heart muscle. For a wide range of strengths and durations an external mechanical load causes wave front dissipation leading to formation and breakup of spiral waves. We explain the mechanism, and discuss under which conditions it can cause or abolish cardiac arrhythmias.

Detection of the propagating direction of electrical wavefront in atrial fibrillation

Atrial Fibrillation (AF) is one of the most common sustained arrhythmia, which can increase the risk of heart failure and stroke. Understanding the complex electrical dynamics of AF and correctly targeting AF sources for ablation therapies remain challenging in clinical practice. This is due to the incapability to reconstruct the electrical dynamic of AF, and lack of efficient approach for AF source identification. This paper builds a multi-sale framework for modeling of the abnormal electrical propagation in AF initiated by triggers from Pulmonary Veins (PVs). A new algorithm is developed to detect the propagating direction of electrical wavefronts. The detection algorithm is further validated using modeling results. The developed multi-scale framework and the detection algorithm will contribute to AF diagnosis and potentially improve the treatment outcomes of AF ablations.

Recent developments in using mechanistic cardiac modelling for drug safety evaluation

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Modelling and simulation can streamline decision making in drug safety testing.

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Computational cardiac electrophysiology is a mature technology with a long heritage.

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There are many challenges and opportunities in using in silico techniques in future.

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We discuss how models can be used at different stages of drug discovery.

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CiPA will combine screening platforms, human cell assays and in silico predictions.

On the tenth anniversary of two key International Conference on Harmonisation (ICH) guidelines relating to cardiac proarrhythmic safety, an initiative aims to consider the implementation of a new paradigm that combines in vitro and in silico technologies to improve risk assessment. The Comprehensive In Vitro Proarrhythmia Assay (CiPA) initiative (co-sponsored by the Cardiac Safety Research Consortium, Health and Environmental Sciences Institute, Safety Pharmacology Society and FDA) is a bold and welcome step in using computational tools for regulatory decision making. This review compares and contrasts the state-of-the-art tools from empirical to mechanistic models of cardiac electrophysiology, and how they can and should be used in combination with experimental tests for compound decision making.

Myokit: A simple interface to cardiac cellular electrophysiology

Myokit is a new powerful and versatile software tool for modeling and simulation of cardiac cellular electrophysiology. Myokit consists of an easy-to-read modeling language, a graphical user interface, single and multi-cell simulation engines and a library of advanced analysis tools accessible through a Python interface. Models can be loaded from Myokit's native file format or imported from CellML. Model export is provided to C, MATLAB, CellML, CUDA and OpenCL. Patch-clamp data can be imported and used to estimate model parameters. In this paper, we review existing tools to simulate the cardiac cellular action potential to find that current tools do not cater specifically to model development and that there is a gap between easy-to-use but limited software and powerful tools that require strong programming skills from their users. We then describe Myokit's capabilities, focusing on its model description language, simulation engines and import/export facilities in detail. Using three examples, we show how Myokit can be used for clinically relevant investigations, multi-model testing and parameter estimation in Markov models, all with minimal programming effort from the user. This way, Myokit bridges a gap between performance, versatility and user-friendliness.

A Genetic Algorithm Optimization Method for Mapping Non-Conducting Atrial Regions: A Theoretical Feasibility Study

Atrial ablation has been recently utilized for curing atrial fibrillation. The success rate of empirical ablation is relatively low as often the exact locations of the arrhythmogenic sources remain elusive. Guided ablation has been proposed to improve ablation technique by providing guidance regarding the potential localization of the sources; yet to date no main technological solution has been widely adopted as an integral part of the ablation process. Here we propose a genetic algorithm optimization technique to map a major arrhythmogenic substance-non-conducting regions (NCRs). Excitation delays in a set of electrodes of known locations are measured following external tissue stimulation, and the spatial distribution of obstacles that is most likely to yield the measured delays is reconstructed. A forward problem module was solved to provide synthetic time delay measurements using a 2D human atrial model with known NCR distribution. An inverse genetic algorithm module was implemented to optimally reconstruct the locations of the now unknown obstacle distribution using the synthetic measurements. The performance of the algorithm was demonstrated for several distributions varying in NCR number and shape. The proposed algorithm was found robust to measurements with a signal-to-noise ratio of at least -20 dB, and for measuring electrodes separated by up to 3.2 mm. Our results support the feasibility of the proposed algorithm in mapping NCRs; nevertheless, further research is required prior to clinical implementation for incorporating more complex atrial tissue geometrical configurations as well as for testing the algorithm with experimental data.

Semianalytical approach to criteria for ignition of excitation waves

We consider the problem of ignition of propagating waves in one-dimensional bistable or excitable systems by an instantaneous spatially extended stimulus. Earlier we proposed a method [I. Idris and V. N. Biktashev, Phys. Rev. Lett. 101, 244101 (2008)] for analytical description of the threshold conditions based on an approximation of the (center-)stable manifold of a certain critical solution. Here we generalize this method to address a wider class of excitable systems, such as multicomponent reaction-diffusion systems and systems with non-self-adjoint linearized operators, including systems with moving critical fronts and pulses. We also explore an extension of this method from a linear to a quadratic approximation of the (center-)stable manifold, resulting in some cases in a significant increase in accuracy. The applicability of the approach is demonstrated on five test problems ranging from archetypal examples such as the Zeldovich-Frank-Kamenetsky equation to near realistic examples such as the Beeler-Reuter model of cardiac excitation. While the method is analytical in nature, it is recognized that essential ingredients of the theory can be calculated explicitly only in exceptional cases, so we also describe methods suitable for calculating these ingredients numerically.

Unstable spiral waves and local Euclidean symmetry in a model of cardiac tissue

This paper investigates the properties of unstable single-spiral wave solutions arising in the Karma model of two-dimensional cardiac tissue. In particular, we discuss how such solutions can be computed numerically on domains of arbitrary shape and study how their stability, rotational frequency, and spatial drift depend on the size of the domain as well as the position of the spiral core with respect to the boundaries. We also discuss how the breaking of local Euclidean symmetry due to finite size effects as well as the spatial discretization of the model is reflected in the structure and dynamics of spiral waves. This analysis allows identification of a self-sustaining process responsible for maintaining the state of spiral chaos featuring multiple interacting spirals.

How the Hodgkin-Huxley equations inspired the Cardiac Physiome Project

Early modelling of cardiac cells (1960-1980) was based on extensions of the Hodgkin-Huxley nerve axon equations with additional channels incorporated, but after 1980 it became clear that processes other than ion channel gating were also critical in generating electrical activity. This article reviews the development of models representing almost all cell types in the heart, many different species, and the software tools that have been created to facilitate the cardiac Physiome Project.

Impact of tissue geometry on simulated cholinergic atrial fibrillation: a modeling study

Atrial fibrillation (AF), arising in the cardiac atria, is a common cardiac rhythm disorder that is incompletely understood. Numerous characteristics of the atrial tissue are thought to play a role in the maintenance of AF. Most traditional theoretical models of AF have considered the atrium to be a flat two-dimensional sheet. Here, we analyzed the relationship between atrial geometry, substrate size, and AF persistence, in a mathematical model involving heterogeneity. Spatially periodic properties were created by variations in times required for reactivation due to periodic acetylcholine concentration [ACh] distribution. The differences in AF maintenance between the sheet and the cylinder geometry are found for intermediate gradients of inexcitable time (intermediate [ACh]). The maximum difference in AF maintenance between geometry decreases with increasing tissue size, down to zero for a substrate of dimensions 20 × 10 cm. Generators have the tendency to be anchored to the regions of longer inexcitable period (low [ACh]). The differences in AF maintenance between geometries correlate with situations of moderate anchoring for which rotor-core drifts between low-[ACh] regions occur, favoring generator disappearance. The drift of generators increases their probability of disappearance at the tissue borders, resulting in a decreased maintenance rate in the sheet due to the higher number of no-flux boundaries. These interactions between biological variables and the role of geometry must be considered when selecting an appropriate model for AF in intact hearts.

A patient specific electro-mechanical model of the heart

This paper presents a patient specific deformable heart model that involves the known electrical and mechanical properties of the cardiac cells and tissue. The whole heart model comprises ten Tusscher's ventricular and Nygren's atrial cell models, the anatomical and electrophysiological model descriptions of the atria (introduced by Harrild et al.) and ventricle (given by Winslow et al.), and the mechanical model of the periodical cardiac contraction and resting phenomena proposed by Moireau et al. During the propagation of the depolarization wave, the kinetic, compositional and rotational anisotropy is handled by the tissue, organ and torso model. The applied patient specific parameters were determined by an evolutionary computation method. An intensive parameter reduction was performed using the abstract formulation of the searching space. This patient specific parameter representation enables the adjustment of deformable model parameters in real-time. The validation process was performed using simultaneously measured ECG and ultrasound image records that were compared with simulated signals and shapes using an abstract, parameterized evaluation criterion.

Mechanisms of transition from normal to reentrant electrical activity in a model of rabbit atrial tissue: interaction of tissue heterogeneity and anisotropy

Experimental evidence suggests that regional differences in action potential (AP) morphology can provide a substrate for initiation and maintenance of reentrant arrhythmias in the right atrium (RA), but the relationships between the complex electrophysiological and anatomical organization of the RA and the genesis of reentry are unclear. In this study, a biophysically detailed three-dimensional computer model of the right atrial tissue was constructed to study the role of tissue heterogeneity and anisotropy in arrhythmogenesis. The model of Lindblad et al. for a rabbit atrial cell was modified to incorporate experimental data on regional differences in several ionic currents (primarily, I(Na), I(CaL), I(K1), I(to), and I(sus)) between the crista terminalis and pectinate muscle cells. The modified model was validated by its ability to reproduce the AP properties measured experimentally. The anatomical model of the rabbit RA (including tissue geometry and fiber orientation) was based on a recent histological reconstruction. Simulations with the resultant electrophysiologically and anatomically detailed three-dimensional model show that complex organization of the RA tissue causes breakdown of regular AP conduction patterns at high pacing rates (>11.75 Hz): as the AP in the crista terminalis cells is longer, and electrotonic coupling transverse to fibers of the crista terminalis is weak, high-frequency pacing at the border between the crista terminalis and pectinate muscles results in a unidirectional conduction block toward the crista terminalis and generation of reentry. Contributions of the tissue heterogeneity and anisotropy to reentry initiation mechanisms are quantified by measuring action potential duration (APD) gradients at the border between the crista terminalis and pectinate muscles: the APD gradients are high in areas where both heterogeneity and anisotropy are high, such that intrinsic APD differences are not diminished by electrotonic interactions. Thus, our detailed computer model reconstructs complex electrical activity in the RA, and provides new insights into the mechanisms of transition from focal atrial tachycardia into reentry.

Modelling excitable cells using cycle-linear hybrid automata

Cycle-linear hybrid automata (CLHAs), a new model of excitable cells that efficiently and accurately captures action-potential morphology and other typical excitable-cell characteristics such as refractoriness and restitution, is introduced. Hybrid automata combine discrete transition graphs with continuous dynamics and emerge in a natural way during the (piecewise) approximation process of any nonlinear system. CLHAs are a new form of hybrid automata that exhibit linear behaviour on a per-cycle basis but whose overall behaviour is appropriately nonlinear. To motivate the need for this modelling formalism, first it is shown how to recast two recently proposed models of excitable cells as hybrid automata: the piecewise-linear model of Biktashev and the nonlinear model of Fenton-Karma. Both of these models were designed to efficiently approximate excitable-cell behaviour. We then show that the CLHA closely mimics the behaviour of several classical highly nonlinear models of excitable cells, thereby retaining the simplicity of Biktashev's model without sacrificing the expressiveness of Fenton-Karma. CLHAs are not restricted to excitable cells; they can be used to capture the behaviour of a wide class of dynamic systems that exhibit some level of periodicity plus adaptation.

Asymptotic analysis and analytical solutions of a model of cardiac excitation

We describe an asymptotic approach to gated ionic models of single-cell cardiac excitability. It has a form essentially different from the Tikhonov fast-slow form assumed in standard asymptotic reductions of excitable systems. This is of interest since the standard approaches have been previously found inadequate to describe phenomena such as the dissipation of cardiac wave fronts and the shape of action potential at repolarization. The proposed asymptotic description overcomes these deficiencies by allowing, among other non-Tikhonov features, that a dynamical variable may change its character from fast to slow within a single solution. The general asymptotic approach is best demonstrated on an example which should be both simple and generic. The classical model of Purkinje fibers (Noble in J. Physiol. 160:317-352, 1962) has the simplest functional form of all cardiac models but according to the current understanding it assigns a physiologically incorrect role to the Na current. This leads us to suggest an "Archetypal Model" with the simplicity of the Noble model but with a structure more typical to contemporary cardiac models. We demonstrate that the Archetypal Model admits a complete asymptotic solution in quadratures. To validate our asymptotic approach, we proceed to consider an exactly solvable "caricature" of the Archetypal Model and demonstrate that the asymptotic of its exact solution coincides with the solutions obtained by substituting the "caricature" right-hand sides into the asymptotic solution of the generic Archetypal Model. This is necessary, because, unlike in standard asymptotic descriptions, no general results exist which can guarantee the proximity of the non-Tikhonov asymptotic solutions to the solutions of the corresponding detailed ionic model.

Critical fronts in initiation of excitation waves

We consider the problem of initiation of propagating waves in a one-dimensional excitable fiber. In the FitzHugh-Nagumo theory, the key role is played by "critical nucleus" and "critical pulse" solutions whose (center-) stable manifold is the threshold surface separating initial conditions leading to propagation and those leading to decay. We present evidence that in cardiac excitation models, this role is played by "critical front" solutions.