Structural complexity effects on transverse propagation in a two-dimensional model of myocardium

Simulation of diffuse and stringy fibrosis in a bilayer interconnected cable model of the left atrium

AIMS

The aim of this study is to design a computer model of the left atrium for investigating fibre-orientation-dependent microstructure such as stringy fibrosis.

METHODS AND RESULTS

We developed an approach for automatic construction of bilayer interconnected cable models from left atrial geometry and epi- and endocardial fibre orientation. The model consisted of two layers (epi- and endocardium) of longitudinal and transverse cables intertwined-like fabric threads, with a spatial discretization of 100 µm. Model validation was performed by comparison with cubic volumetric models in normal conditions. Then, diffuse (n = 2904), stringy (n = 3600), and mixed fibrosis patterns (n = 6840) were randomly generated by uncoupling longitudinal and transverse connections in the interconnected cable model. Fibrosis density was varied from 0% to 40% and mean stringy obstacle length from 0.1 to 2 mm. Total activation time, apparent anisotropy ratio, and local activation time jitter were computed during normal rhythm in each pattern. Non-linear regression formulas were identified for expressing measured propagation parameters as a function of fibrosis density and obstacle length (stringy and mixed patterns). Longer obstacles (even below tissue space constant) were independently associated with prolonged activation times, increased anisotropy, and local fluctuations in activation times. This effect was increased by endo-epicardial dissociation and mitigated when fibrosis was limited to the epicardium.

CONCLUSION

Interconnected cable models enable the study of microstructure in organ-size models despite limitations in the description of transmural structures.

BOUBAKER POLYNOMIALS EXPRESSION TO THE MAGNETIC PHASE-SHIFT INDUCED IN LEON–VIGMOND 3D-MODEL OF THE HUMAN HEART

The heart is simulated as an electrically controlled fluid pump which operates by mechanical contraction. Human heart modeling is undoubtedly a computationally daunting task which must incorporate magnetic, plastic, mechanic and electric patterns. In this paper, a polynomial mathematical formulation for the magnetic field phase shift in a particular heart 3D-model is proposed. The developed model can give an insight into the local and global complex dynamics of the heart magnetic field under controlled excitation as guides to monitoring i.e., the transition from normal to abnormal myocardial activity or collateral dysfunctions. The used protocol allowed handling the boundary conditions in a smooth way.

DYNAMICS OF VENTRICULAR FIBRILLATION: A MODEL AND EXPERIMENTAL STUDY

Wiggers initially described that after electrical induction of ventricular fibrillation (VF) the rhythm so induced progresses in complexity [14]. Previously Witkowski et al. [15] have described two distinct phases of perfused VF. The acute form was characterized by relatively well organized activity and was observed in the first few minutes after induction. The chronic form developed after this initial reentry and was more disorganized in nature. In this paper we describe computer simulations carried out using a complex three-dimensional model of cardiac tissue. Depending on the time constant of the dynamics of the calcium current, activity ranged from organized scroll waves, to highly disorganized chaotic activity in which it was possible to identify up to 10 different vortex filaments which continually drifted and split. The model data was compared to experimental measurements via the maximum cross-correlation function described previously by Witkowski et al. [15]. We found that the max cross-correlation function for a single scroll wave ranged between 0.6 and 0.9, and is similar to what was experimentally observed from optical transmembrane potential mapping during early VF. For the highly disorganized activity the maximum cross-correlation function was in the range of 0.2, which similarly was in same range reported experimentally for fully developed VF.

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 computer model of engineered cardiac monolayers

Engineered monolayers created using microabrasion and micropatterning methods have provided a simplified in vitro system to study the effects of anisotropy and fiber direction on electrical propagation. Interpreting the behavior in these culture systems has often been performed using classical computer models with continuous properties. However, such models do not account for the effects of random cell shapes, cell orientations, and cleft spaces inherent in these monolayers on the resulting wavefront conduction. This work presents a novel methodology for modeling a monolayer of cardiac tissue in which the factors governing cell shape, cell-to-cell coupling, and degree of cleft space are not constant but rather are treated as spatially random with assigned distributions. This modeling approach makes it possible to simulate wavefront propagation in a manner analogous to performing experiments on engineered monolayer tissues. Simulated results are compared to previously published measured data from monolayers used to investigate the role of cellular architecture on conduction velocities and anisotropy ratios. We also present an estimate for obtaining the electrical properties from these networks and demonstrate how variations in the discrete cellular architecture affect the macroscopic conductivities. The simulations support the common assumption that under normal ranges of coupling strength, tissues with relatively uniform distributions of cell shapes and connectivity can be represented using continuous models with conductivities derived from random discrete cellular architecture using either global or local estimates. The results also reveal that in the presence of abrupt changes in cell orientation, local estimates of tissue properties predict smoother changes in conductivity that may not adequately predict the discrete nature of propagation at the transition sites.

Mechanisms of atrial fibrillation termination by rapidly unbinding Na+ channel blockers: insights from mathematical models and experimental correlates

Atrial fibrillation (AF) is the most common sustained clinical arrhythmia and is a problem of growing proportions. Recent studies have increased interest in fast-unbinding Na(+) channel blockers like vernakalant (RSD1235) and ranolazine for AF therapy, but the mechanism of efficacy is poorly understood. To study how fast-unbinding I(Na) blockers affect AF, we developed realistic mathematical models of state-dependent Na(+) channel block, using a lidocaine model as a prototype, and studied the effects on simulated cholinergic AF in two- and three-dimensional atrial substrates. We then compared the results with in vivo effects of lidocaine on vagotonic AF in dogs. Lidocaine action was modeled with the Hondeghem-Katzung modulated-receptor theory and maximum affinity for activated Na(+) channels. Lidocaine produced frequency-dependent Na(+) channel blocking and conduction slowing effects and terminated AF in both two- and three-dimensional models with concentration-dependent efficacy (maximum approximately 89% at 60 microM). AF termination was not related to increases in wavelength, which tended to decrease with the drug, but rather to decreased source Na(+) current in the face of large ACh-sensitive K(+) current-related sinks, leading to the destabilization of primary generator rotors and a great reduction in wavebreak, which caused primary rotor annihilations in the absence of secondary rotors to resume generator activity. Lidocaine also reduced the variability and maximum values of the dominant frequency distribution during AF. Qualitatively similar results were obtained in vivo for lidocaine effects on vagal AF in dogs, with an efficacy of 86% at 2 mg/kg iv, as well as with simulations using the guarded-receptor model of lidocaine action. These results provide new insights into the mechanisms by which rapidly unbinding class I antiarrhythmic agents, a class including several novel compounds of considerable promise, terminate AF.

Effect of bundle branch block on cardiac output: a whole heart simulation study

The heart is an electrically controlled fluid pump which operates by mechanical contraction. Whole heart modelling is a computationally daunting task which must incorporate several subsystems: mechanical, electrical, and fluidic. Numerous feedback mechanisms on many levels, and operating at different scales, exist to finely control behaviour. Understanding these interactions is necessary to understand heart operation, as well as pathologies and therapies. A review of the components in such a model is given. The authors then present a framework for their electro-mechano-fluidic whole heart model based on cable methods. The model incorporates atria and ventricles, and has functioning valves with papillary muscles. The effect of altered propagation due to left and right bundle branch block on cardiac output is examined using the cable-based model. Results are compared to clinically observed phenomena. Good agreement was obtained, but tighter coupling of mechanical and electrical events is needed to fully account for behaviour. Cable-based models offer an alternative to continuum models.

Action potential duration gradient protects the right atrium from fibrillating

Atrial fibrillation (AF) is the most common cardiac arrhythmia. It is characterized by rapid and disorganized electrical activity in the atria. Atrial arrhythmias can be triggered from an ectopic focus, i.e., an abnormal impulse originating in an area other than the sinus node, generating reentrant waves. The regional ionic heterogeneities found in the atria cause a gradual shortening of the action potential duration (APD) with increased distance from the sinoatrial node. It is generally thought that the only electrophysiological consequence of the spatial dispersion of cardiac action potentials (AP) is the enhancement of reentry. This paper investigates the effect of a gradient in APD on arrhythmogenesis via computer simulations. A gradient of ionic properties was introduced into a computationally efficient computer model of the canine atria to produce a smooth distribution of APDs. The window of vulnerability for ectopic beat-induction of reentry was determined for both left atrium (LA) and the right atrium (RA) stimulation, with and without an APD gradient. The shortened windows of vulnerability in the RA, due to the addition of the APD gradient, suggests a protective mechanism against AF. The left atrial window of vulnerability was slightly longer from ionic dispersion.

The effect of vagally induced dispersion of action potential duration on atrial arrhythmogenesis

OBJECTIVE

The purpose of this study is to ascertain the effects of spatially variable ACh distributions on arrhythmogenesis in a morphologically realistic computer model of canine atria.

BACKGROUND

Vagal stimulation releases acetylcholine (ACh), which causes a dose-dependent reduction in action potential duration (APD) in the atria. Due to the nonuniform distribution of nerve endings, APD dispersion may result, which has been shown to play a role in the breakup of activity.

METHODS

Reentry was initiated in a computationally efficient, morphologically realistic computer model of the atria. Discrete regions corresponding to ACh release sites, referred to as islands, were assigned shortened APDs in an ACh-dependent fashion. Island APD was varied as well as the basal APD. The window of vulnerability for ectopic beat-induction of sustained reentry was determined for both left atrial(LA) and right atrial (RA) stimulation. The resulting reentries were categorized based on type and location.

RESULTS

1) Atrial geometry severely restricts the formation of reentrant circuits. 2) Wave fractionation only occurred for large differences between island and basal APD. 3) Small ACh concentration differences produced stable figure-of-8 reentrant patterns. 4) Large islands displayed more wave breakup but could sometimes anchor reentries.

CONCLUSIONS

Large APD gradients produced by ACh heterogeneity can lead to a breakdown of organized activity.

Computer simulations of successful defibrillation in decoupled and non-uniform cardiac tissue

Aim

The aim of the present study is to investigate the origin and effect of virtual electrode polarization in uniform, decoupled and non-uniform cardiac tissue during field stimulation.

Methods

A discrete bidomain model with active membrane behaviour was used to simulate normal cardiac tissue as well as cardiac tissue that is decoupled due to fibrosis and gap junction remodelling. Various uniform and non-uniform electric fields were applied to the external domain of uniform, decoupled and non-uniform resting cardiac tissue as well as cardiac tissue in which spiral waves were induced.

Results

Field stimulation applied on non-uniform tissue results in more virtual electrodes compared with uniform tissue. The spiral waves were terminated in decoupled tissue, but not in uniform, homogeneous tissue. By gradually increasing local differences in intracellular conductivities, the amount and spread of virtual electrodes increased and the spiral waves were terminated.

Conclusion

Fast depolarization of the tissue after field stimulation may be explained by intracellular decoupling and spatial heterogeneity present in normal and pathological cardiac tissue. We demonstrated that termination of spiral waves by means of field stimulation can be achieved when the tissue is modelled as a non-uniform, anisotropic bidomain with active membrane behaviour.

Computational tools for modeling electrical activity in cardiac tissue

Computer models offer many attractive benefits. However, the modeling of cardiac tissue is computationally expensive due to several physical constraints which result in fine spatiotemporal discretization over large spatiotemporal regions. Our laboratory has been actively trying to develop new techniques to make large scale cardiac simulations tractable over the past 15 years. This paper describes the latest modeling software that our group has developed, called Carp (Cardiac arrhythmias research package). It is designed to run in both shared memory and clustered computing environments. Carp aims to be modular and flexible by following a plug-in framework. This allows the latest models and most efficient solvers to be incorporated as well as enabling run-time selection of techniques. Performance results are given for a large-scale simulation which utilized a comprehensive membrane ionic current description.

Computational techniques for solving the bidomain equations in three dimensions

The bidomain equations are the most complete description of cardiac electrical activity. Their numerical solution is, however, computationally demanding, especially in three dimensions, because of the fine temporal and spatial sampling required. This paper methodically examines computational performance when solving the bidomain equations. Several techniques to speed up this computation are examined in this paper. The first step was to recast the equations into a parabolic part and an elliptic part. The parabolic part was solved by either the finite-element method (FEM) or the interconnected cable model model (ICCM). The elliptic equation was solved by FEM on a coarser grid than the parabolic problem and at a reduced frequency. The performance of iterative and direct linear equation system solvers was analyzed as well as the scalability and parallelizability of each method. Results indicate that the ICCM was twice as fast as the FEM for solving the parabolic problem, but when the total problem was considered, this resulted in only a 20% decrease in computation time. The elliptic problem could be solved on a coarser grid at one-quarter of the frequency at which the parabolic problem was solved and still maintain reasonable accuracy. Direct methods were faster than iterative methods by at least 50% when a good estimate of the extracellular potential was required. Parallelization over four processors was efficient only when the model comprised at least 500,000 nodes. Thus, it was possible to speed up solution of the bidomain equations by an order of magnitude with a slight decrease in accuracy.

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

Atrial fibrillation (AF) is a common cardiac arrhythmia, but its mechanisms are incompletely understood. The identification of phase singularities (PSs) has been used to define spiral waves involved in maintaining the arrhythmia, as well as daughter wavelets. In the past, PSs have often been identified manually. Automated PS detection algorithms have been described previously, but when we attempted to apply a previously developed algorithm we experienced problems with false positives that made the results difficult to use directly. We therefore developed a tool for PS identification that uses multiple strategies incorporating both image analysis and mathematical convolution for automated detection with optimized sensitivity and specificity, followed by manual verification. The tool was then applied to analyze PS behavior in simulations of AF maintained in the presence of spatially distributed acetylcholine effects in cell grids of varying size. These analyses indicated that in almost all cases, a single PS lasted throughout the simulation, corresponding to the central-core tip of a single spiral wave that maintained AF. The sustained PS always localized to an area of low acetylcholine concentration. When the grid became very small and no area of low acetylcholine concentration was surrounded by zones of higher concentration, AF could not be sustained. The behavior of PSs and the mechanisms of AF were qualitatively constant over an 11.1-fold range of atrial grid size, suggesting that the classical emphasis on tissue size as a primary determinant of fibrillatory behavior may be overstated. (c) 2002 American Institute of Physics.

Cholinergic atrial fibrillation in a computer model of a two-dimensional sheet of canine atrial cells with realistic ionic properties

Classical concepts of atrial fibrillation (AF) have been rooted in Moe's multiple-wavelet hypothesis and simple cellular-automaton computer model. Recent experimental work has raised questions about the multiple-wavelet mechanism, suggesting a discrete "driver region" underlying AF. We reexplored the theoretical basis for AF with a 2-dimensional computer model of a 5x10-cm sheet of atrial cells with realistic ionic and coupling properties. Vagal actions were formulated based on patch-clamp studies of acetylcholine (ACh) effects. In control, a single extrastimulus resulted in a highly meandering unstable spiral wave. Simulated electrograms showed fibrillatory activity, with a dominant frequency (DF, 6.5 Hz) that correlated with the mean rate. Uniform ACh reduced core meander of the spiral wave by approximately 70% (as measured by the standard deviation of spiral-wave tip position) and accelerated the DF to 17.0 Hz. Simulated vagally induced refractoriness heterogeneity caused wavefront breakup as accelerated reentrant activity in regions of short refractoriness impinged on regions unable to respond in a 1:1 fashion because of longer refractoriness. In 7 simulations spanning the range of conditions giving sustained AF, 5 were maintained by single dominant spiral waves. On average, 3.0+/-1.3 wavelets were present (range, 1 to 7). Most wavelets were short-lived and did not contribute to AF maintenance. In contrast to predictions of the multiple-wavelet hypothesis, but in agreement with recent experimental evidence, our model indicates that AF can result from relatively stable primary spiral-wave generators and is significantly organized. Our results suggest that vagal AF may arise from ACh-induced stabilization of the primary spiral-wave generator and disorganization of the heterogeneous tissue response. The full text of this article is available at http://www.circresaha.org.

Effect of fibre rotation on the initiation of re-entry in cardiac tissue

Transmural rotation of cardiac fibres can have a big influence on the initiation of re-entry in the heart. However, owing to computational demands, this has not been fully explored in a three-dimensional model of cardiac tissue that has a microscopic description of membrane currents, such as the Luo-Rudy model. Using a previously described model that is computationally fast, re-entry in three-dimensional blocks of cardiac tissue is induced by a cross-shock protocol, and the activity is examined. In the study, the effect of the transmural fibre rotation is ascertained by examining differences between a tissue block with no rotation and ones with 1, 2 and 3 degrees of rotation per fibre layer. The direction of the re-entry is significant in establishing whether or not re-entry can be induced, with clockwise re-entry being easier to initiate. Owing to the rotating anisotropy that results in preferential propagation along the fibre axis, the timing of the second stimulus in the cross-shock protocol has to be changed for different rates of fibre rotation. The fibre rotation either increases or decreases the window of opportunity for re-entry, depending on whether the activation front is perpendicular or parallel to the fibre direction. By varying the transmural extent of the S2, it is found that a deeper stimulus has to be applied to the blocks with fibre rotation to create re-entry. Increasing the transmural resistance also tends to reduce the extent of the S2 required to induce re-entry. Results suggest that increasing fibre rotation reduces the susceptibility of the tissue to re-entry, but that more complex spatiotemporal patterns are possible, e.g. stable figure-of-eight re-entries and transient rotors. Three mechanisms of re-entry annihilation are identified: front catchup, filling of the excitable gap and core wander.

A flexible method for simulating cardiac conduction in three-dimensional complex geometries

In this article we present a method for creating a membrane-based computer model capable of representing a three-dimensional irregular domain. The spatial discretization of our model is based on the Finite Volume Method. In combination with a robust meshmaking tool, our method may be used to simulate conduction in an arbitrarily shaped, complex region. In this way, conduction in specific subregions of cardiac anatomy may be examined. The capabilities of this methodology are demonstrated through 3 examples. The first shows the influence of abrupt changes in tissue geometry on conduction parameters. The second highlights the ability of the model to incorporate interior boundaries and altered membrane properties. The final example shows the inclusion of a full description of the fiber architecture in a portion of the canine left ventricle. Future applications will make use of the model's capabilities to conduct investigations in complex regions including the atria.

Efficient integration of a realistic two-dimensional cardiac tissue model by domain decomposition

The size of realistic cardiac tissue models has been limited by their high computational demands. In particular, the Luo-Rudy phase II membrane model, used to simulate a thin sheet of ventricular tissue with arrays of coupled ventricular myocytes, is usually limited to 100 x 100 arrays. We introduce a new numerical method based on domain decomposition and a priority queue integration scheme which reduces the computational cost by a factor of 3-17. In the standard algorithm all the nodes advance with the same time step delta t, whose size is limited by the time scale of activation. However, at any given time, many regions may be inactive and do not require the same small delta t and consequent extensive computations. Hence, adjusting delta t locally is a key factor in improving computational efficiency, since most of the computing time is spent calculating ionic currents. This paper proposes an efficient adaptive numerical scheme for integrating a two-dimensional (2-D) propagation model, by incorporating local adjustments of delta t. In this method, alternating direction Cooley-Dodge and Rush-Larsen methods were used for numerical integration. Between consecutive integrations over the whole domain using an implicit method, the model was spatially decomposed into many subdomains, and delta t adjusted locally. The Euler method was used for numerical integration in the subdomains. Local boundary values were determined from the boundary mesh elements of the neighboring subdomains using linear interpolation. Because delta t was defined locally, a priority queue was used to store and order next update times for each subdomain. The subdomain with the earliest update time was given the highest priority and advanced first. This new method yielded stable solutions with relative errors less than 1% and reduced computation time by a factor of 3-17 and will allow much larger (e.g., 500 x 500) models based on realistic membrane kinetics and realistic dimensions to simulate reentry, triggered activity, and their interactions.

Effects of bath resistance on action potentials in the squid giant axon: myocardial implications

This study presents a simplified version of the quasi-one-dimensional theory (Wu, J., E. A. Johnson, and J. M. Kootsey. 1996. A quasi-one-dimensional theory for anisotropic propagation of excitation in cardiac muscle. Biophys. J. 71:2427-2439) with two components of the extracellular current, along and perpendicular to the axis, and a simulation and its experimental confirmation for the giant axon of the squid. By extending the one-dimensional core conductor cable equations, this theory predicts, as confirmed by the experiment, that the shapes of the intracellular and the extracellular action potentials are related to the resistance of the bath. Such a result was previously only expected by the field theories. The correlation between the shapes of the intracellular and the extracellular potentials of the giant axon of the squid resembles that observed during the anisotropic propagation of excitation in cardiac muscle. Therefore, this study not only develops a quasi-one-dimensional theory for a squid axon, but also provides one possible factor contributing to the anisotropic propagation of action potentials in cardiac muscle.

A quasi-one-dimensional theory for anisotropic propagation of excitation in cardiac muscle

It has been shown that propagation of excitation in cardiac muscle is anisotropic. Compared to propagation at right angles to the long axes of the fibers, propagation along the long axis is faster, the extracellular action potential (AP) is larger in amplitude, and the intracellular AP has a lower maximum rate of depolarization, a larger time constant of the foot, and a lower peak amplitude. These observations are contrary to the predictions of classical one-dimensional (1-D) cable theory and, thus far, no satisfactory theory for them has been reported. As an alternative description of propagation in cardiac muscle, this study provides a quasi-1-D theory that includes a simplified description of the effects of action currents in extracellular space as well as resistive coupling between surface and deeper fibers in cardiac muscle. In terms of classical 1-D theory, this quasi-1-D theory reveals that the anisotropies in the wave form of the AP arise from modifications in the effective membrane ionic current and capacitance. The theory also shows that it is propagation in the longitudinal, not in the transverse direction that deviates from classical 1-D cable theory.

Anisotropy, fiber curvature, and bath loading effects on activation in thin and thick cardiac tissue preparations: simulations in a three-dimensional bidomain model

INTRODUCTION

A modeling study is presented to explore the effects of tissue conductivity, fiber orientation, and presence of an adjoining extracellular volume conductor on electrical conduction in cardiac muscle. Simulated results are compared with those of classical in vitro experiments on superfused thin layer preparations and on whole hearts.

METHODS AND RESULTS

The tissue is modeled as a three-dimensional bidomain block adjoining an isotropic bath. In the thin layer model, the fibers are assumed parallel. In the thick block model, fiber rotation, curvature, and tipping are incorporated. Results from the thin layer model explain experimental observations that the rate of rise of the entire action potential upstroke is faster and the magnitude of the extracellular potential is smaller across fibers than along fibers in a uniformly propagating front. The simulation identified that this behavior only arises in tissue with unequal anisotropy in the two spaces and adjoining an extracellular bath. Simulated conduction and potential distributions in the thick block model are shown to well approximate experimental maps. The potentials are sensitive to changes in the fiber orientations. A slight 5 degrees tipping of intramural fibers out of the planes parallel to the epicardium and endocardium will lead to an asymmetry of the magnitudes of the positive regions. In addition, the introduction of fiber curvature leads to more realistic isochrone and extracellular potential distributions. The orientation of the central negative region of the extracellular potential is shown to be determined by the average of the fiber direction at the plane of pacing and the plane of recording.

CONCLUSIONS

The simulations demonstrate the sensitivity of spread of activation and potential time courses and distributions to the underlying electrical properties in both thick and thin slabs. The bidomain model is shown to be a useful representation of cardiac tissue for interpreting experimental data of activation.

Calculation of transmembrane current from extracellular potential recordings: a model study

INTRODUCTION

A mathematical/computer model of cardiac tissue was used to study the estimation of transmembrane current (EIm) from extracellular potential recordings.

METHODS AND RESULTS

The simulated EIm of transmembrane current was compared with the simulated transmembrane current (Im), and both simulated values were compared with experimentally derived EIm obtained during sinus rhythm and ventricular fibrillation in dogs. We found that although EIm measurements slightly overestimate the duration of the Im waveform, they provide a reasonable approximation of Im during normal conduction and during decremental conduction and conduction block.

CONCLUSIONS

There is a very clear linear correlation between the time spent at or below 25% of the peak inward transmembrane current (Im25), its corresponding estimate (EIm25), the peak inward Im and EIm, and the peak ionic current, providing some evidence that EIm25 may be a suitable in vivo measure of peak ionic current.

Simulation of two-dimensional anisotropic cardiac reentry: effects of the wavelength on the reentry characteristics

A two-dimensional sheet model was used to study the dynamics of reentry around a zone of functional block. The sheet is a set of parallel, continuous, and uniform cables, transversely interconnected by a brick-wall arrangement of fixed resistors. In accord with experimental observations on cardiac tissue, longitudinal propagation is continuous, whereas transverse propagation exhibits discontinuous features. The width and length of the sheet are 1.5 and 5 cm, respectively, and the anisotropy ratio is fixed at approximately 4:1. The membrane model is a modified Beeler-Reuter formulation incorporating faster sodium current dynamics. We fixed the basic wavelength and action potential duration of the propagating impulse by dividing the time constants of the secondary inward current by an integer K. Reentry was initiated by a standard cross-shock protocol, and the rotating activity appeared as curling patterns around the point of junction (the q-point) of the activation (A) and recovery (R) fronts. The curling R front always precedes the A front and is separated from it by the excitable gap. In addition, the R front is occasionally shifted abruptly through a merging with a slow-moving triggered secondary recovery front that is dissociated from the A front and q-point. Sustained irregular reentry associated with substantial excitable gap variations was simulated with short wavelengths (K = 8 and K = 4). Unsustained reentry was obtained with a longer wavelength (K = 2), leading to a breakup of the q-point locus and the triggering of new activation fronts.

The dynamics of sustained reentry in a ring model of cardiac tissue

This paper describes the dynamics of circus movement around a fixed obstacle, using a one-dimensional continuous and uniform ring model of cardiac tissue to simulate sustained reentry. The membrane ionic current is simulated by a modified Beeler-Reuter formulation in which the kinetics of the fast sodium current were updated using more recent voltage-clamp data. Changes in the ring length are used to modify the dynamics of reentry. Reentry is stable if the ring length (X) exceeds a critical value (Xcrit) and complete block occurs if X is below a minimum (Xmin). Irregular sustained reentry is observed at intermediate ring lengths, as a narrow range of aperiodic reentry near Xcrit, and a larger range of quasi-periodic reentry at shorter ring lengths. The basic pattern of irregular reentry is an alternation between long and short cycle length, action potential duration (APD), diastolic interval (DIA), wavelength, and excitable gap. In aperiodic reentry cycle length variations are small, APD and DIA fluctuations are of medium amplitude, and conduction velocity over the whole pathway is essentially constant during successive turns. Much larger fluctuations in these various quantities occur during quasi-periodic reentry, and they increase in size as X approaches Xmin. The complexity of quasi-periodic reentry patterns is related to three factors: the slope of the APD versus DIA relation, which is greater than 1, the existence of a zone of slow conduction on the ring when the excitable gap becomes quite short, and the occurrence of triggered waves of secondary repolarization and excitability recovery. In the present model, quasi-periodic reentry with triggered secondary recovery covers most of the range of ring lengths, giving rise to sustained irregular reentry. There is very close agreement between our simulation results and experimental data obtained on rings of cardiac tissue.

A collocation--Galerkin finite element model of cardiac action potential propagation

A new computational method was developed for modeling the effects of the geometric complexity, nonuniform muscle fiber orientation, and material inhomogeneity of the ventricular wall on cardiac impulse propagation. The method was used to solve a modification to the FitzHugh-Nagumo system of equations. The geometry, local muscle fiber orientation, and material parameters of the domain were defined using linear Lagrange or cubic Hermite finite element interpolation. Spatial variations of time-dependent excitation and recovery variables were approximated using cubic Hermite finite element interpolation, and the governing finite element equations were assembled using the collocation method. To overcome the deficiencies of conventional collocation methods on irregular domains, Galerkin equations for the no-flux boundary conditions were used instead of collocation equations for the boundary degrees-of-freedom. The resulting system was evolved using an adaptive Runge-Kutta method. Converged two-dimensional simulations of normal propagation showed that this method requires less CPU time than a traditional finite difference discretization. The model also reproduced several other physiologic phenomena known to be important in arrhythmogenesis including: Wenckebach periodicity, slowed propagation and unidirectional block due to wavefront curvature, reentry around a fixed obstacle, and spiral wave reentry. In a new result, we observed wavespeed variations and block due to nonuniform muscle fiber orientation. The findings suggest that the finite element method is suitable for studying normal and pathological cardiac activation and has significant advantages over existing techniques.

Electrical coupling and impulse propagation in anatomically modeled ventricular tissue

Computer simulations were used to study the role of resistive couplings on flat-wave action potential propagation through a thin sheet of ventricular tissue. Unlike simulations using continuous or periodic structures, this unique electrical model includes random size cells with random spaced longitudinal and lateral connections to simulate the physiologic structure of the tissue. The resolution of the electrical model is ten microns, thus providing a simulated view at the subcellular level. Flat-wave longitudinal propagation was evaluated with an electrical circuit of over 140,000 circuit elements, modeling a 0.25 mm by 5.0 mm sheet of tissue. An electrical circuit of over 84,000 circuit elements, modeling a 0.5 mm by 1.5 mm sheet was used to study flat-wave transverse propagation. Under normal cellular coupling conditions, at the macrostructure level, electrical conduction through the simulated sheets appeared continuous and directional differences in conduction velocity, action potential amplitude and Vmax were observed. However, at the subcellular level (10 microns) unequal action potential delays were measured at the longitudinal and lateral gap junctions and irregular wave-shapes were observed in the propagating signal. Furthermore, when the modeled tissue was homogeneously uncoupled at the gap junctions conduction velocities decreased as the action potential delay between modeled cells increased. The variability in the measured action potential was most significant in areas with fewer lateral gap junctions, i.e., lateral gap junctions between fibers were separated by a distance of 100 microns or more.

Propagation of activation in cardiac muscle

The hypothesis of local circuit current flow underlying propagation of activation in cardiac muscle has been extensively documented by one-dimensional and two-dimensional simulation studies. The assumptions of spatially uniform membrane capacitance and membrane ionic properties yield simulation results that are in good agreement with experimental observations in healthy cardiac muscle, thereby indicating that differences in propagation velocity and action potential upstroke between longitudinal and transverse directions can be explained solely on the basis of anisotropic intercellular coupling. Two-dimensional model studies of anisotropic propagation have also stressed the more efficient charging of the membrane capacitance and higher safety factor of propagation in the transverse direction. These conditions favor the occurrence of longitudinal unidirectional block and the initiation of reentry via transverse propagation. The authors simulated rotating waves initiated by properly phased transverse and longitudinal plane waves in a two-dimensional sheet model. Sustained propagation requires a minimum anisotropy ratio, corresponding to a velocity ratio of about 4:1. It was found, for uniform anisotropy, that the central focus wandered slightly. A higher anisotropy ratio favors a more stable rotating pattern and a more restricted movement of the central focus.