Panfilov Model Research Articles

Бифуркационное пятно на параметрическом портрете двумерной версии модели Алиева—Панфилова

For the first time, a parametric portrait of a two-dimensional version of the Aliev—Panfilov model is presented, indicating the position of the bifurcation boundary and the bifurcation spot on it. The difference between this model and the "classical" models of autowave processes is demonstrated. Some special cases of the behavior of an autowave vortex are presented, which have not been described in the scientific literature before. The publication is intended primarily for specialists in the fields of mathematical biology, mathematical physics of biological objects and biophysics.

A fast computational model for the electrophysiology of the whole human heart

In this study we present a novel computational model for unprecedented simulations of the whole cardiac electrophysiology. According to the heterogeneous electrophysiologic properties of the heart, the whole cardiac geometry is decomposed into a set of coupled conductive media having different topology and electrical conductivities: (i) a network of slender bundles comprising a fast conduction atrial network, the AV–node and the ventricular bundles; (ii) the Purkinje network; and (iii) the atrial and ventricular myocardium. The propagation of the action potential in these conductive media is governed by the bidomain/monodomain equations, which are discretized in space using an in–house finite volume method and coupled to three different cell models, the Courtemanche model [1] for the atrial myocytes, the Stewart model [2] for the Purkinje Network and the ten Tusscher–Panfilov model [3] for the ventricular myocytes. The developed numerical model correctly reproduces the cardiac electrophysiology of the whole human heart in healthy and pathologic conditions and it can be tailored to study and optimize resynchronization therapies or invasive surgical procedures. Importantly, the whole solver is GPU–accelerated using CUDA Fortran providing an unprecedented speedup, thus opening the way for systematic parametric studies and uncertainty quantification analyses.

Induced drift of scroll waves in the Aliev–Panfilov model and in an axisymmetric heart left ventricle

AbstractThe low-voltage cardioversion-defibrillation is a modern sparing electrotherapy method for such dangerous heart arrhythmias as paroxysmal tachycardia and fibrillation. In an excitable medium, such arrhythmias relate to appearance of spiral waves of electrical excitation, and the spiral waves are superseded to the electric boundary of the medium in the process of treatment due to high-frequency stimulation from the electrode. In this paper we consider the Aliev–Panfilov myocardial model, which provides a positive tension of three-dimensional scroll waves, and an axisymmetric model of the left ventricle of the human heart. Two relations of anisotropy are considered, namely, isotropy and physiological anisotropy. The periods of stimulation with an apical electrode are found so that the electrode successfully entrains its rhythm in the medium, the spiral wave is superseded to the base of the ventricle, and disappears. The results are compared in two-dimensional and three-dimensional media. The intervals of effective stimulation periods are sufficiently close to each other in the two-dimensional case and in the anatomical model. However, the use of the anatomical model is essential in determination of the time of superseding.

On the existence–uniqueness and computation of solution of a class of cardiac electric activity models

In this study, we consider a system of coupled PDEs–ODEs which govern the electric activity in heart with a diffusion term modeling the potential in the surrounding tissue and the nonlinear ionic model proposed by the different reduced cell level models. We prove the existence and uniqueness of the bidomain model with the Morris and Lecar ionic model. The global existence of solution is established based on regularization argument using Fedo–Galerkin/compactness approach. The uniqueness of the solution is shown based on Gronwell’s lemma upon some special treatment of nonlinear terms. The computational realization of the monodomain model with different reduced ionic test models is presented. Also, the coupled parabolic PDE and ODE system modeling the cardiac electric activity in a monodomain system representation of cardiac tissue with different ionic models (at cell level), viz. FitzHugh–Nagumo model, Roger–McCulloch model (RMM), Aliev–Panfilov model and Mitchell–Schaeffer model (MSM), is numerically analyzed by Galerkin linear finite elements in space and the backward Euler method in time. Simulations are done using FreeFem++ open source, and the evolution pattern of the solution has been investigated. MSM and RMM models are unconditionally stable with respect to applied stimulus and initial state, although other models are conditionally stable.

Modeling and simulation of cardiac electric activity in a human cardiac tissue with multiple ischemic zones.

In this work, a human ventricular model (ten Tusscher and Panfilov model) coupled with the tissue level monodomain model is used to analyze the influence of multiple myocardial ischemia on the human cardiac tissue. The existence and uniqueness of the ischemic model comprising the monodomain model with a discontinuous ionic model for the human cardiac tissue is discussed. The coupled system of partial differential equation and ordinary differential equations are solved numerically using [Formula: see text] finite elements in space and Backward Euler finite difference scheme in time. The apriori finite element error estimate for the numerical scheme has been shown to be of [Formula: see text]. Essentially, we evaluate the impact of the increasing size of the ischemic region and the presence of the multiple ischemic regions having equal or different intensities on the neighboring healthy part of the cardiac tissue. We examine both the individual and the combined influence of two types of ischemia, Hyperkalemia (with the variation of the extracellular potassium ion concentration, [Formula: see text]) and Hypoxia (with the variation of intracellular Adenosine triphosphate (ATP) concentration via parameter [Formula: see text]) on the cardiac electrical activity of cardiac tissue. We observe that with the increase in the ischemic region size by a factor five times, there is an additional almost 10% drop in the action potential duration (APD) in the neighboring healthy regions. The combined effect of Hyperkalemia and Hypoxia brings an additional 12% drop in APD in the ischemic subregions and an additional 5% drop in APD in the neighboring healthy part of the cardic tissue in comparison to the only Hyperkalemic ischemia. When the Hyperkalemic and/or Hypoxic degeneracy of a ischemic zone is non-uniform then innercore degeneracy has greater influence on resting potential and APD of outercore of variable intensity ischemic zone than the other way. Also, increasing the number of ischemic subregions from 2 to 4 leads to a 4% drop in APD.

Analytical properties and exact solution of the Aliev-Panfilov model

Some analytical properties of the Aliyev – Panfilov model describing the propagation of impulses through the cardiac tissue are studied. It is shown that the Aliyev – Panfilov system generally does not pass the Painleve test. The relations between the parameters of the model for which there is an exact solution in the form of a kink corresponding to the process of switching the system from one state of stable equilibrium to another are established.

Gating-enhanced IMEX splitting methods for cardiac monodomain simulation

The electrical activity in excitable cardiac tissue can be simulated using the so-called monodomain model. The monodomain model is a continuum-based multi-scale model that consists of non-linear ordinary differential equations describing the electrical activity at the cellular scale along with a semi-linear parabolic partial differential equation describing electrical propagation at the tissue scale. The standard “scale-based” splitting method for simulating the monodomain model is to split the tissue and cell models, applying different integrators to each. Typically, the tissue model is simulated with an implicit time-integration method, and the cell model is simulated with an explicit or explicit-exponential one. We demonstrate that the application of implicit-explicit (IMEX) linear multistep or Runge–Kutta methods to this splitting can have poor stability properties when the cell model is stiff. We propose a novel “gating-enhanced” IMEX splitting that treats the tissue variable and the (typically stiff) cell model gating variables together implicitly. The performance of 14 different IMEX methods using both splittings is measured in a variety of one- and two-dimensional experiments. The low incremental overhead combined with the substantially improved stability of the gating-enhanced splitting is shown to result in a performance increase of approximately a factor of four for simulations of the monodomain model with the stiff ten Tusscher–Panfilov model of human endocardial cells.

Control of cardiac alternans in an electromechanical model of cardiac tissue

Electrical alternations in cardiac action potential duration have been shown to be a precursor to arrhythmias and sudden cardiac death. Through the mechanism of excitation–contraction coupling, the presence of electrical alternans induces alternations in the heart muscle contractile activity. Also, contraction of cardiac tissue affects the process of cardiac electric wave propagation through the mechanism of the so-called mechanoelectrical feedback. Electrical excitation and contraction of cardiac tissue can be linked by an electromechanical model such as the Nash–Panfilov model. In this work, we explore the feasibility of suppressing cardiac alternans in the Nash-Panfilov model which is employed for small and large deformations. Several electrical pacing and mechanical perturbation feedback strategies are considered to demonstrate successful suppression of alternans on a one-dimensional cable. This is the first attempt to combine electrophysiologically relevant cardiac models of electrical wave propagation and contractility of cardiac tissue in a synergistic effort to suppress cardiac alternans. Numerical examples are provided to illustrate the feasibility and the effects of the proposed algorithms to suppress cardiac alternans.

Entropy Rate Maps of Complex Excitable Dynamics in Cardiac Monolayers

The characterization of spatiotemporal complexity remains a challenging task. This holds in particular for the analysis of data from fluorescence imaging (optical mapping), which allows for the measurement of membrane potential and intracellular calcium at high spatial and temporal resolutions and, therefore, allows for an investigation of cardiac dynamics. Dominant frequency maps and the analysis of phase singularities are frequently used for this type of excitable media. These methods address some important aspects of cardiac dynamics; however, they only consider very specific properties of excitable media. To extend the scope of the analysis, we present a measure based on entropy rates for determining spatiotemporal complexity patterns of excitable media. Simulated data generated by the Aliev–Panfilov model and the cubic Barkley model are used to validate this method. Then, we apply it to optical mapping data from monolayers of cardiac cells from chicken embryos and compare our findings with dominant frequency maps and the analysis of phase singularities. The studies indicate that entropy rate maps provide additional information about local complexity, the origins of wave breakup and the development of patterns governing unstable wave propagation.

Instability of periodic traveling wave solutions in a modified FitzHugh–Nagumo model for excitable media

We introduce a two-variable system of reaction–diffusion equations for excitable media. We numerically investigate the existence and stability of periodic traveling wave solutions in a two-dimensional parameter plane. Our results based on the method of continuation show a stability change of Eckhaus type. There are two families of periodic traveling waves: fast and slow. The fast family is stable in the case of standard FitzHugh–Nagumo excitable system. However, we observe that the fast family becomes unstable in our model. Consequently, it bifurcates to an oscillating wave. We explain this phenomenon by numerically calculating the essential spectra of the periodic traveling wave solutions. Moreover, we study the stability of the periodic traveling wave solutions for the Aliev–Panfilov excitable system and compare its results with the proposed model. We show that the stability boundary of our model agrees qualitatively with that of the Aliev–Panfilov model, whereas it is different in the FitzHugh–Nagumo model. In addition, our results based on the stability boundary of the models show oscillating periodic traveling waves, except for the FitzHugh–Nagumo model.

Numerical Method of Determining a Localized Initial Cardiac Excitation for the Aliev–Panfilov Model from Measurements on the Inner Boundary

We consider a numerical method of determining a localized initial excitation for cardiac excitation mathematical models. In the direct problem we model the variation of fields in the region corresponding to a heart section. The variation of the potential is described by the Aliev---Panfilov model for regions from R 2. The inverse problem involves determining the coordinates of a localized initial excitation in the heart from potential values at several point on the inner boundary of the ventricles. Results of computer experiments are reported.

Numerical solution of an inverse problem for the modified aliev–panfilov model

We pose the inverse problem for the modified Aliev–Panfilov model, which involves determining the coefficient of a system of partial differential equations dependent on spatial variables from supplementary observations of the solution on the boundary. This inverse problem may be interpreted as a problem to find the shape and location of the cardiac region damaged by myocardial infarct. A numerical method is proposed for solving the problem and computer experiments illustrating its implementation are reported.

Existence and Stability of Traveling Pulses in a Reaction–Diffusion-Mechanics System

In this article, we analyze traveling waves in a reaction–diffusion-mechanics (RDM) system. The system consists of a modified FitzHugh–Nagumo equation, also known as the Aliev–Panfilov model, coupled bidirectionally with an elasticity equation for a deformable medium. In one direction, contraction and expansion of the elastic medium decreases and increases, respectively, the ionic currents and also alters the diffusivity. In the other direction, the dynamics of the R–D components directly influence the deformation of the medium. We demonstrate the existence of traveling waves on the real line using geometric singular perturbation theory. We also establish the linear stability of these traveling waves using the theory of exponential dichotomies.

Elimination of Breathing Spiral Waves in the Aliev–Panfilov Model

We study spiral waves in an annulus with the Aliev–Panfilov model. The spiral pattern exhibits a regular breathing motion in a certain parameter range. A time-periodic force is applied for the elimination of spiral waves. The defibrillation becomes most effective if the period of the sinusoidal force is suitably chosen near the resonance condition. Next, we investigate a delayed feedback method for the elimination of spiral waves. The feedback method is effective because the feedback force oscillates with the breathing period and the resonance effect works naturally.

Atrial and ventricular fibrillation: computational simulation of spiral waves in cardiac tissue

This manuscript proposes a novel, efficient finite element solution technique for the computational simulation of cardiac electrophysiology. We apply a two-parameter model that is characterized through a fast action potential and a slow recovery variable. The former is introduced globally as a nodal degree of freedom, whereas the latter is treated locally as internal variable on the integration point level. This particular discretization is extremely efficient and highly modular since different cardiac cell models can be incorporated straightforwardly through only minor local modifications on the integration point level. In this manuscript, we illustrate the algorithm in terms of the Aliev–Panfilov model for cardiomyocytes. To ensure unconditional stability, a backward Euler scheme is applied to discretize the evolution equation for both the action potential and the recovery variable in time. To increase robustness and guarantee optimal quadratic convergence, we suggest an incremental iterative Newton–Raphson scheme and illustrate the consistent linearization of the weak form of the excitation problem. The proposed algorithm is illustrated by means of two- and three-dimensional examples of re-entrant spiral and scroll waves characteristic of cardiac arrhythmias in atrial and ventricular fibrillation.

From in vivo plasma composition to in vitro cardiac electrophysiology and in silico virtual heart: the extracellular calcium enigma

In spite of its potential impact on simulation results, the problem of setting the appropriate Ca(2+) concentration ([Ca(2+)](o)) in computational cardiac models has not yet been properly considered. Usually [Ca(2+)](o) values are derived from in vitro electrophysiology. Unfortunately, [Ca(2+)](o) in the experiments is set significantly far (1.8 or 2 mM) from the physiological [Ca(2+)] in blood (1.0-1.3 mM). We analysed the inconsistency of [Ca(2+)](o) among in vivo, in vitro and in silico studies and the dependence of cardiac action potential (AP) duration (APD) on [Ca(2+)](o). Laboratory measurements confirmed the difference between standard extracellular solutions and normal blood [Ca(2+)]. Experimental data on human atrial cardiomyocytes confirmed literature data, demonstrating an inverse relationship between APD and [Ca(2+)](o). Sensitivity analysis of APD on [Ca(2+)](o) for five of the most used cardiac cell models was performed. Most of the models responded with AP prolongation to increases in [Ca(2+)](o), i.e. opposite to the AP shortening observed in vitro and in vivo. Modifications to the Ten Tusscher-Panfilov model were implemented to demonstrate that qualitative consistency among in vivo, in vitro and in silico studies can be achieved. The Courtemanche atrial model was used to test the effect of changing [Ca(2+)](o) on quantitative predictions about the effect of K(+) current blockade. The present analysis suggests that (i) [Ca(2+)](o) in cardiac AP models should be changed from 1.8 to 2 mM to approximately 1.15 mM in order to reproduce in vivo conditions, (ii) the sensitivity to [Ca(2+)](o) of ventricular AP models should be improved in order to simulate real conditions, (iii) modifications to the formulation of Ca(2+)-dependent I(CaL) inactivation can make models more suitable to analyse AP when [Ca(2+)](o) is set to lower physiological values, and (iv) it could be misleading to use non-physiological high [Ca(2+)](o) when the quantitative analysis of in vivo pathophysiological mechanisms is the ultimate aim of simulation.

Computational modeling of cardiac electrophysiology: A novel finite element approach

AbstractThe key objective of this work is the design of an unconditionally stable, robust, efficient, modular, and easily expandable finite element‐based simulation tool for cardiac electrophysiology. In contrast to existing formulations, we propose a global–local split of the system of equations in which the global variable is the fast action potential that is introduced as a nodal degree of freedom, whereas the local variable is the slow recovery variable introduced as an internal variable on the integration point level. Cell‐specific excitation characteristics are thus strictly local and only affect the constitutive level. We illustrate the modular character of the model in terms of the FitzHugh–Nagumo model for oscillatory pacemaker cells and the Aliev–Panfilov model for non‐oscillatory ventricular muscle cells. We apply an implicit Euler backward finite difference scheme for the temporal discretization and a finite element scheme for the spatial discretization. The resulting non‐linear system of equations is solved with an incremental iterative Newton–Raphson solution procedure. Since this framework only introduces one single scalar‐valued variable on the node level, it is extremely efficient, remarkably stable, and highly robust. The features of the general framework will be demonstrated by selected benchmark problems for cardiac physiology and a two‐dimensional patient‐specific cardiac excitation problem. Copyright © 2009 John Wiley & Sons, Ltd.

Building maps of local apparent conductivity of the epicardium with a 2-D electrophysiological model of the heart

In this paper, we address the problem of estimating the parameters of an electrophysiological model of the heart from a set of electrical recordings. The chosen model is the reaction-diffusion model on the transmembrane potential proposed by Aliev and Panfilov. For this model of the transmembrane, we estimate a local apparent two-dimensional conductivity from a measured depolarization time distribution. First, we perform an initial adjustment including the choice of initial conditions and of a set of global parameters. We then propose a local estimation by minimizing the quadratic error between the depolarization time computed by the model and the measures. As a first step we address the problem on the epicardial surface in the case of an isotropic version of the Aliev and Panfilov model. The minimization is performed using Brent method without computing the derivative of the error. The feasibility of the approach is demonstrated on synthetic electrophysiological measurements. A proof of concept is obtained on real electrophysiological measures of normal and infarcted canine hearts.

Wave Propagation in Excitable Media with Randomly Distributed Obstacles

We study the effect of small, randomly distributed obstacles on wave propagation in two-dimensional (2D) and 3D excitable media described by the Aliev--Panfilov model. We find that increasing the number of obstacles decreases the conduction velocity of plane waves and decreases the effective diffusion coefficient in the eikonal curvature equation. The presence of obstacles also increases the inducibility of wave breaks and spiral waves in 2D and 3D excitable media, but suppresses spiral breakup induced by a steep restitution curve mechanism. We discuss the mechanisms of the observed effects, the differences between 2D and 3D excitable media, as well as the relevance of our study to processes of wave propagation in cardiac tissue, including arrhythmogenesis in the presence of fibrosis in the myocardium.

VENTRICULAR FIBRILLATION IN A SIMPLE EXCITABLE MEDIUM MODEL OF CARDIAC TISSUE

Ventricular fibrillation (VF), the major reason behind sudden cardiac death, is turbulent cardiac electrical activity in which rapid, irregular disturbances in spatiotemporal electrical activation of heart make it incapable of any concerted pumping action. We give a brief overview of the simple Panfilov model for ventricular fibrillation, with emphasis on studies that have elucidated the nature of spiral turbulence which is the analog of VF here. The control of such turbulence is briefly touched upon. Preliminary results are presented for the effects of conduction inhomogeneity on spiral breakup, and the transition from functional to anatomical reentry as a function of the size and position of the inhomogeneity.