Electrical Wave Propagation in an Anisotropic Model of the Left Ventricle Based on Analytical Description of Cardiac Architecture

We develop a numerical approach based on our recent analytical model of fiber structure in the left ventricle of the human heart. A special curvilinear coordinate system is proposed to analytically include realistic ventricular shape and myofiber directions. With this anatomical model, electrophysiological simulations can be performed on a rectangular coordinate grid. We apply our method to study the effect of fiber rotation and electrical anisotropy of cardiac tissue (i.e., the ratio of the conductivity coefficients along and across the myocardial fibers) on wave propagation using the ten Tusscher–Panfilov (2006) ionic model for human ventricular cells. We show that fiber rotation increases the speed of cardiac activation and attenuates the effects of anisotropy. Our results show that the fiber rotation in the heart is an important factor underlying cardiac excitation. We also study scroll wave dynamics in our model and show the drift of a scroll wave filament whose velocity depends non-monotonically on the fiber rotation angle; the period of scroll wave rotation decreases with an increase of the fiber rotation angle; an increase in anisotropy may cause the breakup of a scroll wave, similar to the mother rotor mechanism of ventricular fibrillation.     

Effects of early afterdepolarizations on reentry in cardiac tissue: a simulation study

Early afterdepolarizations (EADs) are classically generated at slow heart rates when repolarization reserve is reduced by genetic diseases or drugs. However, EADs may also occur at rapid heart rates if repolarization reserve is sufficiently reduced. In this setting, spontaneous diastolic sarcoplasmic reticulum (SR) Ca release can facilitate cellular EAD formation by augmenting inward currents during the action potential plateau, allowing reactivation of the window L-type Ca current to reverse repolarization. Here, we investigated the effects of spontaneous SR Ca release-induced EADs on reentrant wave propagation in simulated one-, two-, and three-dimensional homogeneous cardiac tissue using a version of the Luo-Rudy dynamic ventricular action potential model modified to increase the likelihood of these EADs. We found: 1) during reentry, nonuniformity in spontaneous SR Ca release related to subtle differences in excitation history throughout the tissue created adjacent regions with and without EADs. This allowed EADs to initiate new wavefronts propagating into repolarized tissue; 2) EAD-generated wavefronts could propagate in either the original or opposite direction, as a single new wave or two new waves, depending on the refractoriness of tissue bordering the EAD region; 3) by suddenly prolonging local refractoriness, EADs caused rapid rotor displacement, shifting the electrical axis; and 4) rapid rotor displacement promoted self-termination by collision with tissue borders, but persistent EADs could regenerate single or multiple focal excitations that reinitiated reentry. These findings may explain many features of Torsades des pointes, such as perpetuation by focal excitations, rapidly changing electrical axis, frequent self-termination, and occasional degeneration to fibrillation.     

Wavefront propagation in an activation model of the anisotropic cardiac tissue: asymptotic analysis and numerical simulations

In this paper we present a macroscopic model of the excitation process in the myocardium. The composite and anisotropic structure of the cardiac tissue is represented by a bidomain, i.e. a set of two coupled anisotropic media. The model is characterized by a non linear system of two partial differential equations of parabolic and elliptic type. A singular perturbation analysis is carried out to investigate the cardiac potential field and the structure of the moving excitation wavefront. As a consequence the cardiac current sources are approximated by an oblique dipole layer structure and the motion of the wavefront is described by eikonal equations. Finally numerical simulations are carried out in order to analyze some complex phenomena related to the spreading of the wavefront, like the front-front or front-wall collision. The results yielded by the excitation model and the eikonal equations are compared.     

Spiral waves in two-dimensional models of ventricular muscle: formation of a stationary core.

Previous experimental studies have clearly demonstrated the existence of drifting and stationary electrical spiral waves in cardiac muscle and their involvement in cardiac arrhythmias. Here we present results of a study of reentrant excitation in computer simulations based on a membrane model of the ventricular cell. We have explored in detail the parameter space of the model, using tools derived from previous numerical studies in excitation-dynamics models. We have found appropriate parametric conditions for sustained stable spiral wave dynamics (1 s of activity or approximately 10 rotations) in simulations of an anisotropic (ratio in velocity 4:1) cardiac sheet of 2 cm x 2 cm. Initially, we used a model that reproduced well the characteristics of planar electrical waves exhibited by thin sheets of sheep ventricular epicardial muscle during rapid pacing at a cycle length of 300 ms. Under these conditions, the refractory period was 147 ms; the action potential duration (APD) was 120 ms; the propagation velocity along fibers was 33 cm/s; and the wavelength along fibers was 4.85 cm. Using cross-field stimulation in this model, we obtained a stable self-sustaining spiral wave rotating around an unexcited core of 1.75 mm x 7 mm at a period of 115 ms, which reproduced well the experimental results. Thus the data demonstrate that stable spiral wave activity can occur in small cardiac sheets whose wavelength during planar wave excitation in the longitudinal direction is larger than the size of the sheet. Analysis of the mechanism of this observation demonstrates that, during rotating activity, the core exerts a strong electrotonic influence that effectively abbreviates APD (and thus wavelength) in its immediate surroundings and is responsible for the stabilization and perpetuation of the activity. We conclude that appropriate adjustments in the kinetics of the activation front (i.e., threshold for activation and upstroke velocity of the initiating beat) of currently available models of the cardiac cell allow accurate reproduction of experimentally observed self-sustaining spiral wave activity. As such, the results set the stage for an understanding of functional reentry in terms of ionic mechanisms.     

Electrophysiological and Structural Remodeling in Heart Failure Modulate Arrhythmogenesis. 2D Simulation Study

BackgroundHeart failure is operationally defined as the inability of the heart to maintain blood flow to meet the needs of the body and it is the final common pathway of various cardiac pathologies. Electrophysiological remodeling, intercellular uncoupling and a pro-fibrotic response have been identified as major arrhythmogenic factors in heart failure.ObjectiveIn this study we investigate vulnerability to reentry under heart failure conditions by incorporating established electrophysiological and anatomical remodeling using computer simulations.MethodsThe electrical activity of human transmural ventricular tissue (5 cm×5 cm) was simulated using the human ventricular action potential model Grandi et al. under control and heart failure conditions. The MacCannell et al. model was used to model fibroblast electrical activity, and their electrotonic interactions with myocytes. Selected degrees of diffuse fibrosis and variations in intercellular coupling were considered and the vulnerable window (VW) for reentry was evaluated following cross-field stimulation.ResultsNo reentry was observed in normal conditions or in the presence of HF ionic remodeling. However, defined amount of fibrosis and/or cellular uncoupling were sufficient to elicit reentrant activity. Under conditions where reentry was generated, HF electrophysiological remodeling did not alter the width of the VW. However, intermediate fibrosis and cellular uncoupling significantly widened the VW. In addition, biphasic behavior was observed, as very high fibrotic content or very low tissue conductivity hampered the development of reentry. Detailed phase analysis of reentry dynamics revealed an increase of phase singularities with progressive fibrotic components.ConclusionStructural remodeling is a key factor in the genesis of vulnerability to reentry. A range of intermediate levels of fibrosis and intercellular uncoupling can combine to favor reentrant activity.     

An improved numerical method for strong coupling of excitation and contraction models in the heart

Quantifying the interactions between excitation and contraction is fundamental to furthering our understanding of cardiac physiology. To date simulating these effects in strongly coupled excitation and contraction tissue models has proved computationally challenging. This is in part due to the numerical methods implemented to maintain numerical stability in previous simulations, which produced computationally intensive problems. In this study, we analytically identify and quantify the velocity and length dependent sources of instability in the current established coupling method and propose a new method which addresses these issues. Specifically, we account for the length and velocity dependence of active tension within the finite deformation equations, such that the active tension is updated at each intermediate Newton iteration, within the mechanics solution step. We then demonstrate that the model is stable and converges in a three-dimensional rod under isometric contraction. Subsequently, we show that the coupling method can produce stable solutions in a cube with many of the attributes present in the heart, including asymmetrical activation, an inhomogeneous fibre field and a nonlinear constitutive law. The results show no instabilities and quantify the error introduced by discrete length updates. This validates our proposed coupling framework, demonstrating significant improvement in the stability of excitation and contraction simulations.     

Reentrant waves in a ring of embryonic chick ventricular cells imaged with a Ca2+ sensitive dye

According to the classic model initially formulated by Mines, reentrant cardiac arrhythmias may be associated with waves circulating in a ring geometry. This study was designed to study the dynamics of reentry in a ring geometry of cardiac tissue culture. Reentrant calcium waves in rings of cultured embryonic chick cardiac myocytes were imaged using a macroscope to monitor the fluorescence of intracellular Calcium Green-1 dye. The rings displayed a variety of stable rhythms including pacemaker activity and spontaneous reentry. Waves originating from a localized pacemaker could lead to reentry as a consequence of unidirectional block. In addition, more complex patterns were observed due to the interactions between reentrant and pacemaker rhythms. These rhythms included instances in which pacemakers accelerated the reentrant rhythm, and instances in which the excitation was blocked in the vicinity of pacemakers. During reentrant activity an appropriately timed electrical stimulus could induce resetting of activity or cause complete annihilation of the propagating waves. This experimental preparation reveals many spontaneously occuring complex rhythms. These complex rhythms are hypothesized to reflect interactions between spontaneous pacemakers, wave propagation, refractory period, and overdrive suppression. This preparation may serve as a useful model system to further investigate complex dynamics arising during reentrant rhythms in cardiac tissue.     

Theoretical analysis of the magnetocardiographic pattern for reentry wave propagation in a three-dimensional human heart model

We present a computational study of reentry wave propagation using electrophysiological models of human cardiac cells and the associated magnetic field map of a human heart. We examined the details of magnetic field variation and related physiological parameters for reentry waves in two-dimensional (2-D) human atrial tissue and a three-dimensional (3-D) human ventricle model. A 3-D mesh system representing the human ventricle was reconstructed from the surface geometry of a human heart. We used existing human cardiac cell models to simulate action potential (AP) propagation in atrial tissue and 3-D ventricular geometry, and a finite element method and the Galerkin approximation to discretize the 3-D domain spatially. The reentry wave was generated using an S1-S2 protocol. The calculations of the magnetic field pattern assumed a horizontally layered conductor for reentry wave propagation in the 3-D ventricle. We also compared the AP and magnetocardiograph (MCG) magnitudes during reentry wave propagation to those during normal wave propagation. The temporal changes in the reentry wave motion and magnetic field map patterns were also analyzed using two well-known MCG parameters: the current dipole direction and strength. The current vector in a reentry wave forms a rotating spiral. We delineated the magnetic field using the changes in the vector angle during a reentry wave, demonstrating that the MCG pattern can be helpful for theoretical analysis of reentry waves.     

Soft Tissue Modelling of Cardiac Fibres for Use in Coupled Mechano-Electric Simulations

The numerical solution of the coupled system of partial differential and ordinary differential equations that model the whole heart in three dimensions is a considerable computational challenge. As a consequence, it is not computationally practical—either in terms of memory or time—to repeat simulations on a finer computational mesh to ensure that convergence of the solution has been attained. In an attempt to avoid this problem while retaining mathematical rigour, we derive a one dimensional model of a cardiac fibre that takes account of elasticity properties in three structurally defined axes within the myocardial tissue. This model of a cardiac fibre is then coupled with an electrophysiological cell model and a model of cellular electromechanics to allow us to simulate the coupling of the electrical and mechanical activity of the heart. We demonstrate that currently used numerical methods for coupling electrical and mechanical activity do not work in this case, and identify appropriate numerical techniques that may be used when solving the governing equations. This allows us to perform a series of simulations that: (i) investigate the effect of some of the assumptions inherent in other models; and (ii) reproduce qualitatively some experimental observations.     

Soliton-Like Regimes and Excitation Pulse Reflection (Echo) in Homogeneous Cardiac Purkinje Fibers: Results of Numerical Simulations

On the basis of numerical simulations of the partial McAllister-Noble-Tsien equations quantitatively describing the dynamics of electrical processes in conductive cardiac Purkinje fibers we reveal unusual – soliton-like – regimes of interaction of nonlinear excitation pulses governing the heart contraction rhythm: reflection of colliding pulses instead of their annihilation. The phenomenological mechanism of the reflection effects is that in a narrow (but finite) range of the system parameters the traveling pulse presents a doublet consisting of a high-amplitude leader followed by a low-amplitude subthreshold wave. Upon collisions of pulses the leaders are annihilated, but subthreshold waves summarize becoming superthreshold and initiating two novel echo-pulses traveling in opposite directions. The phenomenon revealed presents an analogy to the effect of reflection of colliding nerve pulses, predicted recently, and can be of use in getting insight into the mechanisms of heart rhythm disturbances.     

Optimal control approach to termination of re-entry waves in cardiac electrophysiology

This work proposes an optimal control approach for the termination of re-entry waves in cardiac electrophysiology. The control enters as an extracellular current density into the bidomain equations which are well established model equations in the literature to describe the electrical behavior of the cardiac tissue. The optimal control formulation is inspired, in part, by the dynamical systems behavior of the underlying system of differential equations. Existence of optimal controls is established and the optimality system is derived formally. The numerical realization is described in detail and numerical experiments, which demonstrate the capability of influencing and terminating reentry phenomena, are presented.     

A tale of two dogs: analyzing two models of canine ventricular electrophysiology

The extensive development of detailed mathematical models of cardiac myocyte electrophysiology in recent years has led to a proliferation of models, including many that model the same animal species and specific region of the heart and thus would be expected to have similar properties. In this paper we review and compare two recently developed mathematical models of the electrophysiology of canine ventricular myocytes. To clarify their similarities and differences, we also present studies using them in a range of preparations from single cells to two-dimensional tissue. The models are compared with each other and with new and previously published experimental results in terms of a number of their properties, including action potential morphologies; transmembrane currents during normal heart rates and during alternans; alternans onsets, magnitudes, and cessations; and reentry dynamics of spiral waves. Action potential applets and spiral wave movies for the two canine ventricular models are available online as supplemental material. We find a number of differences between the models, including their rate dependence, alternans dynamics, and reentry stability, and a number of differences compared with experiments. Differences between models of the same species and region of the heart are not unique to these canine models. Similar differences can be found in the behavior of two models of human ventricular myocytes and of human atrial myocytes. We provide several possible explanations for the differences observed in models of the same species and region of the heart and discuss the implications for the applicability of models in addressing questions of mechanism in cardiac electrophysiology.     

Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias

We introduce the concept of a contracting excitable medium that is capable of conducting non-linear waves of excitation that in turn initiate contraction. Furthermore, these kinematic deformations have a feedback effect on the excitation properties of the medium. Electrical characteristics resemble basic models of cardiac excitation that have been used to successfully study mechanisms of reentrant cardiac arrhythmias in electrophysiology. We present a computational framework that employs electromechanical and mechanoelectric feedback to couple a three-variable FitzHugh–Nagumo-type excitation-tension model to the non-linear stress equilibrium equations, which govern large deformation hyperelasticity. Numerically, the coupled electromechanical model combines a finite difference method approach to integrate the excitation equations, with a Galerkin finite element method to solve the equations governing tissue mechanics. We present example computations demonstrating various effects of contraction on stationary rotating spiral waves and spiral wave break. We show that tissue mechanics significantly contributes to the dynamics of electrical propagation, and that a coupled electromechanical approach should be pursued in future electrophysiological modelling studies.     

Dynamics of action potential head-tail interaction during reentry in cardiac tissue: ionic mechanisms

In a sufficiently short reentry pathway, the excitation wave front (head) propagates into tissue that is partially refractory (tail) from the previous action potential (AP). We incorporate a detailed mathematical model of the ventricular myocyte into a one-dimensional closed pathway to investigate the effects of head-tail interaction and ion accumulation on the dynamics of reentry. The results were the following: 1) a high degree of head-tail interaction produces oscillations in several AP properties; 2) Ca(2+)-transient oscillations are in phase with AP duration oscillations and are often of greater magnitude; 3) as the wave front propagates around the pathway, AP properties undergo periodic spatial oscillations that produce complicated temporal oscillations at a single site; 4) depending on the degree of head-tail interaction, intracellular [Na(+)] accumulation during reentry either stabilizes or destabilizes reentry; and 5) elevated extracellular [K(+)] destabilizes reentry by prolonging the tail of postrepolarization refractoriness.     

Dynamical effects of diffusive cell coupling on cardiac excitation and propagation: a simulation study

Cell coupling is considered to be important for cardiac action potential propagation and arrhythmogenesis. We carried out computer simulations to investigate the effects of stimulation strength and cell-to-cell coupling on action potential duration (APD) restitution, APD alternans, and stability of reentry in models of isolated cell, one-dimensional cable, and two-dimensional tissue. Phase I formulation of the Luo and Rudy action potential model was used. We found that stronger stimulation resulted in a shallower APD restitution curve and onset of APD alternans at a faster pacing rate. Reducing diffusive coupling between cells prolonged APD. Weaker diffusive currents along the direction of propagation steepened APD restitution and caused APD alternans to occur at a slower pacing rate in tissue. Diffusive current due to curvature changed APD but had little effect on APD restitution slope and onset of instability. Heterogeneous cell coupling caused APD inhomogeneities in space. Reduction in coupling strength either uniformly or randomly had little effect on the rotation period and stability of a reentry, but random cell decoupling slowed the rotation period and, thus, stabilized the reentry, preventing it from breaking up into multiple waves. Therefore, in addition to its effects on action potential conduction velocity, diffusive cell coupling also affects APD in a rate-dependent manner, causes electrophysiological heterogeneities, and thus modulates the dynamics of cardiac excitation. These effects are brought about by the modulation of ionic current activation and inactivation.     

Electrophysiological basis for antiarrhythmic drug action

Arrhythmias result from abnormalities of impulse initiation or impulse conduction or a combination of both. Abnormal impulse initiation results from either automaticity or triggered activity. Automaticity can further be subdivided into (1) automaticity caused by the normal automatic mechanism (a normal property of cardiac cells in the sinus node, in some parts of the atria, in the atrioventricular junctional region, and in the His-Purkinje system) and (2) automaticity caused by an abnormal mechanism (resulting from a decrease in membrane potential of cardiac fibers, which normally have a high level of membrane potential). Triggered activity is caused by afterdepolarizations, which are second depolarizations that occur either during repolarization (referred to as early afterdepolarizations) or after repolarization is complete or nearly complete (referred to as delayed afterdepolarizations). Abnormal impulse conduction results in reentrant excitation. Usually a combination of slowed conduction and unidirectional conduction block provides the conditions necessary for reentry to occur.     

An inverse modeling approach for stress estimation in mitral valve anterior leaflet valvuloplasty for in-vivo valvular biomaterial assessment

Estimation of regional tissue stresses in the functioning heart valve remains an important goal in our understanding of normal valve function and in developing novel engineered tissue strategies for valvular repair and replacement. Methods to accurately estimate regional tissue stresses are thus needed for this purpose, and in particular to develop accurate, statistically informed means to validate computational models of valve function. Moreover, there exists no currently accepted method to evaluate engineered heart valve tissues and replacement heart valve biomaterials undergoing valvular stresses in blood contact. While we have utilized mitral valve anterior leaflet valvuloplasty as an experimental approach to address this limitation, robust computational techniques to estimate implant stresses are required. In the present study, we developed a novel numerical analysis approach for estimation of the in-vivo stresses of the central region of the mitral valve anterior leaflet (MVAL) delimited by a sonocrystal transducer array. The in-vivo material properties of the MVAL were simulated using an inverse FE modeling approach based on three pseudo-hyperelastic constitutive models: the neo-Hookean, exponential-type isotropic, and full collagen–fiber mapped transversely isotropic models. A series of numerical replications with varying structural configurations were developed by incorporating measured statistical variations in MVAL local preferred fiber directions and fiber splay. These model replications were then used to investigate how known variations in the valve tissue microstructure influence the estimated ROI stresses and its variation at each time point during a cardiac cycle. Simulations were also able to include estimates of the variation in tissue stresses for an individual specimen dataset over the cardiac cycle. Of the three material models, the transversely anisotropic model produced the most accurate results, with ROI averaged stresses at the fully-loaded state of  432.6±46.5 kPa and 241.4±40.5 kPa in the radial and circumferential directions, respectively. We conclude that the present approach can provide robust instantaneous mean and variation estimates of tissue stresses of the central regions of the MVAL.