Simulation of propagation along a cylindrical bundle of cardiac tissue--II: Results of simulation

Bidomain modeling of electrical and mechanical properties of cardiac tissue

Throughout the history of cardiac research, there has been a clear need to establish mathematical models to complement experimental studies. In an effort to create a more complete picture of cardiac phenomena, the bidomain model was established in the late 1970s to better understand pacing and defibrillation in the heart. This mathematical model has seen ongoing use in cardiac research, offering mechanistic insight that could not be obtained from experimental pursuits. Introduced from a historical perspective, the origins of the bidomain model are reviewed to provide a foundation for researchers new to the field and those conducting interdisciplinary research. The interplay of theory and experiment with the bidomain model is explored, and the contributions of this model to cardiac biophysics are critically evaluated. Also discussed is the mechanical bidomain model, which is employed to describe mechanotransduction. Current challenges and outstanding questions in the use of the bidomain model are addressed to give a forward-facing perspective of the model in future studies.

Analytic modeling of conductively anisotropic neural tissue

The abdominal ganglion of the Aplysia californica is an established in vitro model for studying neuroelectric behavior in the presence of an applied electrical current and recently used in studies of magnetic resonance electrical impedance tomography (MREIT) which allows for quantitative visualization of spatially distributed current and magnetic flux densities. Understanding the impact the Aplysia geometry and anisotropic conductivity have on applied electromagnetic fields is central to intepreting and refining MREIT data and protocols, respectively. Here we present a simplified bidomain model of an in vitro experimental preparation of the Aplysia abdominal ganglion, describing the tissue as a radially anisotropic sphere with equal anisotropy ratios, i.e., where radial conductivities in both intra- and extra-cellular regions are ten times that of their polar and azimuthal conductivities. The fully three dimensional problem is validated through comparisons with limiting examples of 2D isotropic analyses. Results may be useful in validating finite element models of MREIT experiments and have broader relevance to analysis of MREIT data obtained from complex neural architecture in the human brain.

Analytic Modeling of Neural Tissue: I. A Spherical Bidomain

Presented here is a model of neural tissue in a conductive medium stimulated by externally injected currents. The tissue is described as a conductively isotropic bidomain, i.e. comprised of intra and extracellular regions that occupy the same space, as well as the membrane that divides them, and the injection currents are described as a pair of source and sink points. The problem is solved in three spatial dimensions and defined in spherical coordinates [Formula: see text]. The system of coupled partial differential equations is solved by recasting the problem to be in terms of the membrane and a monodomain, interpreted as a weighted average of the intra and extracellular domains. The membrane and monodomain are defined by the scalar Helmholtz and Laplace equations, respectively, which are both separable in spherical coordinates. Product solutions are thus assumed and given through certain transcendental functions. From these electrical potentials, analytic expressions for current density are derived and from those fields the magnetic flux density is calculated. Numerical examples are considered wherein the interstitial conductivity is varied, as well as the limiting case of the problem simplifying to two dimensions due to azimuthal independence. Finally, future modeling work is discussed.

A brief history of tissue models for cardiac electrophysiology

The last four decades have produced a number of significant advances in the developments of computer models to simulate and investigate the electrical activity of cardiac tissue. The tissue descriptions that underlie these simulations have been built from a combination of clever insight and careful comparison with measured data at multiple scales. Tissue models have not only led to greater insights into the mechanisms of life-threatening arrhythmias but have been used to engineer new therapies to treat the consequences of cardiac disease. This paper is a look back at the early years in the cardiac modeling and the challenges facing the field as models move toward the clinic.

Models of cardiac tissue electrophysiology: progress, challenges and open questions

Models of cardiac tissue electrophysiology are an important component of the Cardiac Physiome Project, which is an international effort to build biophysically based multi-scale mathematical models of the heart. Models of tissue electrophysiology can provide a bridge between electrophysiological cell models at smaller scales, and tissue mechanics, metabolism and blood flow at larger scales. This paper is a critical review of cardiac tissue electrophysiology models, focussing on the micro-structure of cardiac tissue, generic behaviours of action potential propagation, different models of cardiac tissue electrophysiology, the choice of parameter values and tissue geometry, emergent properties in tissue models, numerical techniques and computational issues. We propose a tentative list of information that could be included in published descriptions of tissue electrophysiology models, and used to support interpretation and evaluation of simulation results. We conclude with a discussion of challenges and open questions.

Increased interstitial loading reduces the effect of microstructural variations in cardiac tissue

Electrical propagation in diseased and aging hearts is strongly influenced by structural changes that occur in both the intracellular and interstitial spaces of cardiac tissue; however, very few studies have investigated how interactions between the two spaces affect propagation at the microscale. In this study, we used one-dimensional microstructural computer models of interconnected ventricular myocytes to systematically investigate how increasing the effective interstitial resistivity (rho(oeff)) influences action potential propagation in fibers with variations in intracellular properties such as cell coupling and cell length. Changes in rho(oeff) were incorporated into a monodomain model using a correction to the intracellular properties that was based on bidomain simulations. The results showed that increasing rho(oeff) in poorly coupled one-dimensional fibers alters the distribution of electrical load at the microscale and causes propagation to become more continuous. In the poorly coupled fiber, this continuous state is characterized by decreased gap junction delay, sustained conduction velocity, increased sodium current, reduced maximum upstroke velocity, and increased safety factor. Long, poorly coupled cells experience greater loading effects than short cells and show the greatest initial response to changes in rho(oeff). In inhomogeneous fibers with adjacent well-coupled and poorly coupled regions, increasing rho(oeff) in the poorly coupled region also reduces source-load mismatch, which delays the onset of conduction block and reduces the dispersion of repolarization at the transition between the two regions. Increasing the rho(oeff) minimizes the effect of cell-to-cell variations and may influence the pattern of activation in critical regimes characterized by low intercellular coupling, microstructural heterogeneity, and reduced or abnormal membrane excitability.

Effect of nonuniform interstitial space properties on impulse propagation: a discrete multidomain model

This work presents a discrete multidomain model that describes ionic diffusion pathways between connected cells and within the interstitium. Unlike classical models of impulse propagation, the intracellular and extracellular spaces are represented as spatially distinct volumes with dynamic/static boundary conditions that electrically couple neighboring spaces. The model is used to investigate the impact of nonuniform geometrical and electrical properties of the interstitial space surrounding a fiber on conduction velocity and action potential waveshape. Comparison of the multidomain and bidomain models shows that although the conduction velocity is relatively insensitive to cases that confine 50% of the membrane surface by narrow extracellular depths (> or =2 nm), the action potential morphology varies greatly around the fiber perimeter, resulting in changes in the magnitude of extracellular potential in the tight spaces. Results also show that when the conductivity of the tight spaces is sufficiently reduced, the membrane adjacent to the tight space is eliminated from participating in propagation, and the conduction velocity increases. Owing to its ability to describe the spatial discontinuity of cardiac microstructure, the discrete multidomain can be used to determine appropriate tissue properties for use in classical macroscopic models such as the bidomain during normal and pathophysiological conditions.

Propagation velocity profile in a cross-section of a cardiac muscle bundle from PSpice simulation

BACKGROUND

The effect of depth on propagation velocity within a bundle of cardiac muscle fibers is likely to be an important factor in the genesis of some heart arrhythmias. MODEL AND METHODS: The velocity profile of simulated action potentials propagated down a bundle of parallel cardiac muscle fibers was examined in a cross-section of the bundle using a PSpice model. The model (20 x 10) consisted of 20 chains in parallel, each chain being 10 cells in length. All 20 chains were stimulated simultaneously at the left end of the bundle using rectangular current pulses (0.25 nA, 0.25 ms duration) applied intracellularly. The simulated bundle was symmetrical at the top and bottom (including two grounds), and voltage markers were placed intracellularly only in cells 1, 5 and 10 of each chain to limit the total number of traces to 60. All electrical parameters were standard values; the variables were (1) the number of longitudinal gap-junction (G-j) channels (0, 1, 10, 100), (2) the longitudinal resistance between the parallel chains (Rol2) (reflecting the closeness of the packing of the chains), and (3) the bundle termination resistance at the two ends of the bundle (RBT). The standard values for Rol2 and RBT were 200 KOmega.

RESULTS

The velocity profile was bell-shaped when there was 0 or only 1 gj-channel. With standard Rol2 and RBT values, the velocity at the surface of the bundle (theta1 and theta20) was more than double (2.15 x) that at the core of the bundle (theta10, theta11). This surface:core ratio of velocities was dependent on the values of Rol2 and RBT. When Rol2 was lowered 10-fold, theta1 increased slightly and theta2decreased slightly. When there were 100 gj-channels, the velocity profile was flat, i.e. the velocity at the core was about the same as that at the surface. Both velocities were more than 10-fold higher than in the absence of gj-channels. Varying Rol2 and RBT had almost no effect. When there were 10 gj-channels, the cross-sectional velocity profile was bullet-shaped, but with a low surface/core ratio, with standard Rol2 and RBT values.

CONCLUSION

When there were no or few gj-channels (0 or 1), the profile was bell-shaped with the core velocity less than half that at the surface. In contrast, when there were many gj-channels (100), the profile was flat. Therefore, when some gj-channels close under pathophysiological conditions, this marked velocity profile could contribute to the genesis of arrhythmias.

The effect of simplifying assumptions in the bidomain model of cardiac tissue: application to ST segment shifts during partial ischaemia

In this study various electrical conductivity approximations used in bidomain models of cardiac tissue are considered. Comparisons are based on epicardial surface potential distributions arising from regions of subendocardial ischaemia situated within the cardiac tissue. Approximations studied are a single conductivity bidomain model, an isotropic bidomain model and equal and reciprocal anisotropy ratios both with and without fibre rotation. It is demonstrated both analytically and numerically that the approximations involving a single conductivity bidomain, an isotropic bidomain or equal anisotropy ratios (ignoring fibre rotation) results in identical epicardial potential distributions for all degrees of subendocardial ischaemia. This result is contrary to experimental observations. It is further shown that by assuming reciprocal anisotropy ratios, epicardial potential distributions vary with the degree of subendocardial ischaemia. However, it is concluded that unequal anisotropy ratios must be used to obtain the true character of experimental observations.

The importance of anisotropy in modeling ST segment shift in subendocardial ischaemia

In this paper, a simple mathematical model of a slab of cardiac tissue is presented in an attempt to better understand the relationship between subendocardial ischaemia and the resulting epicardial potential distributions. The cardiac tissue is represented by the bidomain model where tissue anisotropy and fiber rotation have been incorporated with a view to predicting the epicardial surface potential distribution. The source of electric potential in this steady-state problem is the difference between plateau potentials in normal and ischaemic tissue, where it is assumed that ischaemic tissue has a lower plateau potential. Simulations with tissue anisotropy and no fiber rotation are also considered. Simulations are performed for various thicknesses of the transition region between normal and ischaemic tissue and for various sizes of the ischaemic region. The simulated epicardial potential distributions, based on an anisotropic model of the cardiac tissue, show that there are large potential gradients above the border of the ischaemic region and that there are dips in the potential distribution above the region of ischaemia. It could be concluded from the simulations that it would be possible to predict the region of subendocardial ischaemia from the epicardial potential distribution, a conclusion contrary to observed experimental data. Possible reasons for this discrepancy are discussed. In the interests of mathematical simplicity, isotropic models of the cardiac tissue are also considered, but results from these simulations predict epicardial potential distributions vastly different from experimental observations. A major conclusion from this work is that tissue anisotropy and fiber rotation must be included to obtain meaningful and realistic epicardial potential distributions.

Influence of a perfusing bath on the foot of the cardiac action potential

Recently, Spach et al (Circ Res. 1998;83:1144-1164) measured the transmembrane action potential 150 to 200 microm below the tissue surface during longitudinal and transverse propagation. They found that "during longitudinal propagation there was initial slowing of V(m) [action potential] foot that resulted in deviations from a simple exponential. " (p 1144). They attributed this behavior to the effects of capillaries on propagation. The purpose of this commentary is to show that the perfusing bath plays an important role in determining the time course of the action potential foot, even when the transmembrane potential is measured 150 microm below the tissue surface. Using numerical simulations based on the bidomain model, we find that the action potential foot for transverse propagation is nearly exponential (tau(foot)=314 micros). For longitudinal propagation, the action potential foot is not exponential because of an initial slowing (best-fit tau(foot)=483 micros). We conclude that the perfusing bath must be taken into account when interpreting data showing differences in the shape of the action potential foot with propagation direction, even if the transmembrane potential is measured 150 microm below the tissue surface.

Effects of spatial segmentation in the continuous model of excitation propagation in cardiac muscle

INTRODUCTION

Spatial segmentation is essential for the numerical simulation of excitation propagation in cardiac muscle.

METHODS AND RESULTS

This study evaluated the effects of spatial segmentation on action potential and on the velocity of propagation in a continuous one-dimensional model of cardiac muscle [intracellular and extracellular resistivities along (L) and transverse (T) to the muscle fibers: 402 omega(cm) (R(e), L), 3,620 omega(cm) (R(e), T), 48 omega(cm) (R(e), L), and 126 omega(cm) (R(e), T), J of Physiol 255:335-346, 1976) and either Luo-Rudy (L-R, Circ Res 68:1501-1526, 1991) or Beeler-Reuter (B-R, J Physiol 268:177-210, 1977) ionic currents. Related cable equations for active membrane are derived. Spatial segmentations of < 31.2 microm (L, L-R), < 11.5 microm (T, L-R), < 44.7 microm (L, B-R), and < 16.5 microm (T, B-R) were required for < 1% errors in the characteristic parameters of action potential. Similarly, spatial segmentations of < 54.5 microm (L, L-R), < 20.1 microm (T, L-R), < 84.3 microm (L, B-R), and < 31.2 microm (T, B-R) were required for < 1 % errors in the velocity of conduction.

CONCLUSION

In general, spatial segmentations of < 26.9% and < 50.8% of the space constant of a fully activated membrane gave < 1.0% errors in the characteristic parameters of action potential and in the velocity of propagation, respectively, for both membranes. The action potential duration was relatively insensitive to the spatial segmentation. Our analysis suggests that lambda(full is a better criterion for the selection of spatial segmentation in numerical simulation than the space constant of the resting membrane.

A numerical method for the solution of the bidomain equations in cardiac tissue

A numerical scheme for efficient integration of the bidomain model of action potential propagation in cardiac tissue is presented. The scheme is a mixed implicit-explicit scheme with no stability time step restrictions and requires that only linear systems of equations be solved at each time step. The method is faster than a fully explicit scheme and there is no increase in algorithmic complexity to use this method instead of a fully explicit method. The speedup factor depends on the timestep size, which can be set solely on the basis of the demands for accuracy. (c) 1998 American Institute of Physics.

Quantal currents and potential in the three-dimensional anisotropic bidomain model of smooth muscle

The potential generated in the smooth muscle of the vas deferens on release of a quantum of transmitter from a varicosity was analyzed using a three-dimensional bidomain continuum model. Current was injected at the origin of the bidomain; this current had the temporal characteristics of the junctional current. The membrane potential, intracellular potential, and extracellular potential, as well as the extracellular current, were then calculated throughout the bidomain at different times. Calculations were performed to show the effect of changing the anisotropy ratios of the intracellular and extracellular conductivities on the spread of current and potential in each of the three dimensions. These results provide a theoretical framework for ascertaining the time course of transmitter interaction at a varicosity following the secretion of a quantum of transmitter.

A generalized activating function for predicting virtual electrodes in cardiac tissue

To fully understand the mechanisms of defibrillation, it is critical to know how a given electrical stimulus causes membrane polarizations in cardiac tissue. We have extended the concept of the activating function, originally used to describe neuronal stimulation, to derive a new expression that identifies the sources that drive changes in transmembrane potential. Source terms, or virtual electrodes, consist of either second derivatives of extracellular potential weighted by intracellular conductivity or extracellular potential gradients weighted by derivatives of intracellular conductivity. The full response of passive tissue can be considered, in simple cases, to be a convolution of this "generalized activating function" with the impulse response of the tissue. Computer simulations of a two-dimensional sheet of passive myocardium under steady-state conditions demonstrate that this source term is useful for estimating the effects of applied electrical stimuli. The generalized activating function predicts oppositely polarized regions of tissue when unequally anisotropic tissue is point stimulated and a monopolar response when a point stimulus is applied to isotropic tissue. In the bulk of the myocardium, this new expression is helpful for understanding mechanisms by which virtual electrodes can be produced, such as the hypothetical "sawtooth" pattern of polarization, as well as polarization owing to regions of depressed conductivity, missing cells or clefts, changes in fiber diameter, or fiber curvature. In comparing solutions obtained with an assumed extracellular potential distribution to those with fully coupled intra- and extracellular domains, we find that the former provides a reliable estimate of the total solution. Thus the generalized activating function that we have derived provides a useful way of understanding virtual electrode effects in cardiac tissue.

Autonomic neuromuscular transmission at a varicosity

This review attempts to clarify the definition of what constitutes an autonomic neuromuscular function formed by a varicosity. Ultrastructural studies of serial sections through varicosities, partly or wholly bare of Schwann cell covering, show that areas of close apposition occur between varicosities and muscle cell membrane that vary between 20 and 150 nm, depending on the muscle considered. Consideration of the diffusion of purine transmitters and their receptor kinetics after secretion in a packet show that the number of purinergic receptor channels opened at a site of 150 nm apposition by a varicosity is about 15% of that at a site of 50 nm apposition. These results, together with the analysis of the stochastic fast component and the deterministic slow components of the rising phase of the EJP suggest that the stochastic fast component is due to varicosities that form especially close appositions (20-50 nm), whereas the deterministic slow component is due to the large number of varicosities at distances up to about 150 nm. Varicosities forming appositions of 20-150 nm with muscle cells several hundred micrometers long possess junctional receptor types distinct from extrajunctional receptors. According to this argument, then, there are two different classes of varicosities: one that gives rise to a relatively large junctional current and another that is responsible for a very small junctional current. Present evidence suggests that two subclasses of varicosities can be discerned amongst the varicosities that generate large junctional currents. One of these subclasses of varicosity possesses relatively few post-junctional receptors compared with the amount of transmitter reaching the receptors from the varicosity, so that the junctional current generated is determined by the size of the receptor population; in this case, the size of the transmitter packages released from these varicosities is unknown and the size of the junctional current is relatively constant. The other subclass of varicosity possesses large receptor patches, sufficient to accommodate the largest amounts of transmitter released from the varicosities: in this case, the size of the transmitter packages is shown to be highly non-uniform. These speculations await confirmation by direct labelling of the receptor patches beneath varicosities, a possibility that is likely to be realized in the near future.

Effect of a perfusing bath on the rate of rise of an action potential propagating through a slab of cardiac tissue

Experiments show that the rate of rise of the action potential depends on the direction of propagation in cardiac tissue. Two interpretations of these experiments have been presented: (i) the data are evidence of discrete propagation in cardiac tissue, and (ii) the data are an effect of the perfusing bath. In this paper we present a mathematical model that supports the second interpretation. We use the bidomain model to simulate action potential propagation through a slab of cardiac tissue perfused by a bath. We assume an intracellular potential distribution and solve the bidomain equations analytically for the transmembrane and extracellular potentials. The key assumption in our model is that the intracellular potential is independent of depth within the tissue. This assumption ensures that all three boundary conditions at the surface of a bidomain are satisfied simultaneously. One advantage of this model over previous numerical calculations is that we obtain an analytical solution for the transmembrane potential. The model predicts that the bath reduces the rate of rise of the transmembrane action potential at the tissue surface, and that this reduction depends on the direction of propagation. The model is consistent with the hypothesis that the perfusing bath causes the observed dependence of the action-potential rate of rise on the direction of propagation, and that this dependence has nothing to do with discrete properties of cardiac tissue.

Interactions between adjacent fibers in a cardiac muscle bundle

A strand of cardiac muscle was modeled as a small bundle of individual fibers surrounded by a large volume conductor. The bundle is a uniform assembly of small identical cylindrical fibers, arranged as a series of concentric layers, and its behavior is examined in the presence (coupled bundle) or absence (uncoupled bundle) of transverse resistive coupling between adjacent fibers. Individual fibers are continuous cables of excitable membrane, with circumferential segmentation into 12 equal patches to make the membrane potential changes dependent upon the local interstitial potential. The minimum spacing (d) between adjacent fibers is used to modify the interstitial microstructural organization and the intracellular volume fraction (fi). When d is small enough (d < 0.01 micron), fi remains unchanged at its maximum of about 90%, the interstitial potential is large, the transverse interstitial resistance is high, and the proximity effect arising from the close juxtaposition of adjacent fibers is important. A surface fiber of the uncoupled bundle exhibits little sensitivity to changes in the interstitial microstructure, owing to the dominant influence of the external volume conductor, whereas the central fiber shows a large decrease in velocity, substantial waveshape modifications, and a large increase in interstitial potential as d is reduced. In the coupled bundle, all fibers adopt the same velocity during uniform propagation, owing to the strong transverse resistive coupling; when d is reduced in the range of d < 0.01 micron, the velocity and interstitial potential changes are less pronounced than in the uncoupled bundle. When d is large enough (d > 0.01 micron), the bundle behavior (coupled and uncoupled) approaches that obtained with a bidomain formulation.

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.

Propagation on a central fiber surrounded by inactive fibers in a multifibered bundle model

We studied uniform propagation on a central active fiber surrounded by inactive fibers in a multifibered bundle model lying in a large volume conductor. The behavior of a fully active bundle is considered in a companion paper. The bundle is formed by concentric layers of small cylindrical fibers (radius 5 microns), with a uniform minimum distance (d) between any two adjacent fibers, to yield a bundle radius of about 72 microns. Individual fibers are identical continuous cables of excitable membrane based on a modified Beeler-Reuter model. The intracellular volume fraction (fi) increases to a maximum of about 90% as d is reduced and remains unchanged for d < 0.01 micron. In the range of d < 0.01 micron, the central fiber is effectively shielded from external effects by the first concentric layer of inactive fibers, and a large capacitive load current flows across the surrounding inactive membranes. In addition, the fiber proximity produces a circumferentially nonuniform current density (proximity effect) that is equivalent to an increased average longitudinal interstitial resistance. The conduction velocity is reduced as d becomes smaller in the range of d < 0.1 micron, the interstitial potential becomes larger, and both the maximum rate of rise and time constant of the foot of the upstroke are increased. On the other hand, for d > 0.1 micron, there are negligible changes in the shape of the upstroke, and the behavior of the central fiber is close to that of a uniform cable in a restricted volume conductor. For d larger than about 1.2 microns, the active fiber environment is close to an unbounded isotropic volume conductor.

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.

Computer simulation of epicardial potentials using a heart-torso model with realistic geometry

Previous cardiac simulation studies have focused on simulating the activation isochrones and subsequently the body surface potentials. Epicardial potentials, which are important for clinical application as well as for electrocardiographic inverse problems studies, however, have usually been neglected. This paper describes a procedure of simulating epicardial potentials using a microcomputer-based heart-torso model with realistic geometry. Our heart model developed earlier is composed of approximately 65,000 cell units which are arranged in a cubic close-packed structure. An action potential waveform with variable in duration is assigned to each unit. The heart model, together with the epicardial surface model constructed recently, are mounted in an inhomogeneous human torso model. Electric dipoles, which are proportional to the spatial gradient of the action potential, are generated in all the cell units. These dipoles give rise to a potential distribution on the epicardial surface, which is calculated by means of the boundary element method. The simulated epicardial potential maps during a normal heart beat and in a preexcited beat to mimic Wolff-Parkinson-White (WPW) syndrome are in close agreement with those reported in the literature.

Virtual electrodes in cardiac tissue: a common mechanism for anodal and cathodal stimulation

Traditional cable analyses cannot explain complex patterns of excitation in cardiac tissue with unipolar, extracellular anodal, or cathodal stimuli. Epifluorescence imaging of the transmembrane potential during and after stimulation of both refractory and excitable tissue shows distinctive regions of simultaneous depolarization and hyperpolarization during stimulation that act as virtual cathodes and anodes. The results confirm bidomain model predictions that the onset (make) of a stimulus induces propagation from the virtual cathode, whereas stimulus termination (break) induces it from the virtual anode. In make stimulation, the virtual anode can delay activation of the underlying tissue, whereas in break stimulation this occurs under the virtual cathode. Thus make and break stimulations in cardiac tissue have a common mechanism that is the result of differences in the electrical anisotropy of the intracellular and extracellular spaces and provides clear proof of the validity of the bidomain model.

Magnetic fields from simulated cardiac action currents

A digital simulation has been performed of an idealized, thin, 2-D cardiac slice in the chi-y plane. The slice is stimulated near the center and the resulting action potential propagates outward, developing a distribution of electrical current with nonzero curl. An anisotropic bidomain model is used for the calculation, with membrane physiology based upon either just fast sodium fluxes or the more complete Beeler-Reuter myocardial model. The electrical anisotropy, expressed as the ratio of longitudinal to transverse electrical conductivity, is much greater for the inner domain than for the outer one, and this results in current loops that develop ahead of and behind the wavefront and produce a Bz magnetic field of order 10(-9) T 1 mm above the tissue, similar to recent experimental observations on canine cardiac tissue slices. The fields exhibit a quatrefoil symmetry which can be distorted by nonuniformities in the tissue. The field from repolarization currents is larger by almost an order of magnitude than might be predicted from considerations of rate of change of voltage.

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.

Linear algebraic transformations of the bidomain equations: implications for numerical methods

A mathematical framework is presented for the treatment of the bidomain equations used to model propagation in cardiac tissue. This framework is independent of the model used to represent membrane ionic currents and incorporates boundary conditions and other constraints. By representing the bidomain equations in the operator notation L phi = F, various algebraic transformations can be expressed as PLQ-1 psi = PF, where P and Q are linear operators. The authors show how previous work fits into this framework and discuss the implications of various transformation for numerical methods of solution. Although such transformations allow many choices of independent variable, these results emphasize the fundamental importance of the transmembrane potential.

A bidomain model for ring stimulation of a cardiac strand

A bidomain model is developed to represent a cardiac muscle strand undergoing stimulation from a ring electrode. Expressions are derived for the distribution of the potentials in the strand as well as in the bath perfusing the tissue, under the condition of equal anisotropy ratios in the intracellular and interstitial spaces. The present derivation provides an example of an analytical solution technique that might also prove useful for other problems in theoretical electrophysiology.

Electrical stimulation of cardiac tissue: a bidomain model with active membrane properties

Numerical calculations simulated the response of cardiac muscle to stimulation by electrical current. The bidomain model with unequal anisotropy ratios represented the tissue, and parallel leak and active sodium channels represented the membrane conductance. The speed of the wavefront was faster in the direction parallel to the myocardial fibers than in the direction perpendicular to them. However, for cathodal stimulation well above threshold, the wavefront originated farther from the cathode in the direction perpendicular to the myocardial fibers than in the direction parallel to them, consistent with observations of a dog-bone-shaped virtual cathode made by Wikswo et al., Circ. Res. 68:513-530, 1991. The model showed that the virtual cathode size and shape were dependent upon both membrane and tissue conductivities. Increasing the peak sodium conductance or reducing the transverse intracellular conductivity accentuated the dog-bone shape, while the opposite change caused the virtual cathode to become more elliptical, with the major axis of the ellipse transverse to the fiber direction. A cathodal stimulus created regions of hyperpolarization that slowed conduction of the wavefront propagating parallel to the fibers. An anodal stimulus evoked a wavefront with a complex shape; activation originated from two depolarized regions 1 to 2 mm from the stimulus site along the fiber direction. The threshold current strength (0.5 ms duration pulse) for a cathodal stimulus was 0.048 mA, and for an anodal stimulus was 0.67 mA. When the model was modified to simulate the effect of electropermeabilization, which may be present when the transmembrane potential reaches very large values near the stimulating electrode, our qualitative conclusions remained unchanged.(ABSTRACT TRUNCATED AT 250 WORDS)

The response of a spherical heart to a uniform electric field: a bidomain analysis of cardiac stimulation

A mathematical model describing electrical stimulation of the heart is developed, in which a uniform electric field is applied to a spherical shell of cardiac tissue. The electrical properties of the tissue are characterized using the bidomain model. Analytical expressions for the induced transmembrane potential are derived for the cases of equal anisotropy ratios in the intracellular and interstitial (extracellular) spaces, and no transverse coupling between fibers. Numerical calculations of the transmembrane potential are also performed using realistic electrical conductivities. The model illustrates several mechanisms for polarization of the cell membrane, which can be divided into two categories, depending on if they polarize fibers at the heart surface only or if they polarize fibers both at the surface and within the bulk of the tissue. The latter mechanisms can be classified further according to whether they originate from continuous or discrete properties of cardiac tissue. If cardiac tissue had equal anisotropy ratios, a large membrane polarization would be induced at the heart surface that would become negligible a few length constants into the tissue. If cardiac tissue were continuous and had no transverse coupling between fibers, a membrane polarization would be induced throughout the bulk that would arise from an "activating function" similar to the one used to describe neural stimulation. Polarization would occur if the fibers were curving, if the cross-sectional area of the tissue were changing (fiber branching), or both. The numerically calculated transmembrane potential is intermediate between those predicted using the assumptions of equal anisotropy ratios and no transverse coupling between fibers. Although discrete properties of cardiac tissue are not incorporated into this model, an estimate of their effect indicates that the amplitude of the polarization caused by the resistance of the cellular junctions is similar to that caused by fiber curvature and branching. The spatial distribution of the polarization, however, is quite different.

Generation of reentry in anisotropic myocardium

INTRODUCTION

We investigated numerically the effects of the rotation of fiber axis orientation through the myocardial wall on wave propagation.

METHODS AND RESULTS

We show that because of this rotation and inherent discrete properties of myocardium, a premature stimulus can create unidirectional conduction block leading to reentry.

CONCLUSION

The dynamics of the subsequent reentrant patterns are complicated by the presence of rotational anisotropy, as the center of reentry drifts, and the reentry terminates in finite time when it collides with the domain boundary.

Potential distributions generated by point stimulation in a myocardial volume: simulation studies in a model of anisotropic ventricular muscle

INTRODUCTION

We present simulations of extracellular potential patterns elicited by delivering ectopic stimuli to a parallelepipedal slab of ventricular tissue represented as an anisotropic bidomain incorporating epi-endocardial fiber rotation.

METHODS AND RESULTS

Simulations were based on an eikonal model that determines wavefront shapes throughout the slab at every time instant during the depolarization phase, coupled with an approximate model of the action potential profile. The endocardial face of the slab was in contact with blood and the composite volume was surrounded by an insulating medium. The effect of a simplified Purkinje network was also studied.

RESULTS

(1) For all pacing depths, except endocardial pacing, a central negative area and two potential maxima were observed at QRS onset in all intramural planes parallel to the epicardium. In all planes, the axis joining the two maxima was approximately aligned with the direction of fibers in the plane of pacing. Endocardial pacing generated a different pattern, but only when blood was present; (2) During later stages of excitation, outflowing currents (from the wavefront toward the resting tissue) were always emitted, at all intramural depths, only from those portions of the wavefront that spread along fibers. At any given instant, the position of the two potential maxima in a series of planes parallel to the epicardium and intersecting the wavefront rotated as a function of depth, following the rotating direction of intramural fibers. Purkinje involvement modified the above patterns.

CONCLUSION

Epicardial and endocardial potential maps provided information on pacing site and depth and on subsequent intramural propagation by reflecting the clockwise or counter-clockwise rotation of the deep positivity. Results may be applicable to epicardial and endocardial potential maps recorded at surgery or from endocavitary probes.

A bidomain model with periodic intracellular junctions: a one-dimensional analysis

The classical bidomain model of cardiac tissue views the intracellular and extracellular (interstitial) spaces as two coupled but separate continua. In the present study, the classical bidomain model has been extended by introducing a periodic conductivity in the intracellular space to represent the junctional discontinuity between abutting myocytes. In this model the junctional region of a myocyte is represented in a way that permits variation of junction size and conductivity profile. Employing spectral techniques, a new method was developed for solving the coupled differential equations governing the intracellular and extracellular potentials in a tissue preparation of finite dimensions. Different spectral representations are used for the aperiodic intra- and extracellular potentials (finite Fourier integral transform) and for the periodic intracellular conductivity (Fourier series). As a first application of the method, the response of a 50-cell, single interior fiber to a defibrillating current is examined under steady-state conditions. Transmembrane as well as intra- and extracellular potential distributions along the fiber were calculated.

Computer simulations of three-dimensional propagation in ventricular myocardium. Effects of intramural fiber rotation and inhomogeneous conductivity on epicardial activation

Three-dimensional membrane-based simulations of action potential propagation in ventricular myocardium were performed. Specifically, the effects of the intramural rotation of the fiber axes and inhomogeneous conductivity on the timing and pattern of epicardial activation were examined. Models were built, with approximately 400,000 microscopic elements arranged in rectangular parallelepipeds in each model. Simulations used the nonlinear Ebihara and Johnson membrane equations for the fast sodium current. Constructed models had histological features of ventricular myocardium. All models were anisotropic. In a subset of the models, an abrupt intramural rotation of the fiber axes was included. This feature was also combined with randomly distributed inhomogeneous conductivity and regions of high transverse resistance to represent nonuniform anisotropy in a further subset of the models. Epicardial stimuli were applied for each simulation. Three-dimensional activation patterns and epicardial isochron maps were constructed from the simulations. We noted that the rotation of fiber axes accelerated epicardial activation distant from the stimulus site. The inhomogeneous conductivity caused regional acceleration and deceleration of activation spread. We also noted features of epicardial activation that resulted from the fiber rotation, and the inhomogeneous conductivity corresponded to that observed in maps from experimental animals.

A simulation of cardiac action currents having curl

A digital simulation of a two dimensional cardiac slice has been performed. It is stimulated at the center and an action potential propagates outward. An anisotropic bidomain model is used in which fast sodium physiology connects the intracellular and extracellular domains. For cases in which the inner asymmetry (expressed as longitudinal versus transverse electrical conductivity) is greater than the outer asymmetry, a current flow pattern is observed for which there is nonzero curl. Such a result explains recent observations of nonzero Bz magnetic field detected above a slab of tissue in the x-y plane. The current loop producing this field consists of outer domain current in the longitudinal direction flowing around in space to return at the AP location in the transverse direction in the outer domain and then completing the loop in the longitudinal direction by passing distally through the AP in the inner domain where resistance is extremely low.

Electrophysiological interaction through the interstitial space between adjacent unmyelinated parallel fibers

The influence of interstitial or extracellular potentials on propagation usually has been ignored, often through assuming these potentials to be insignificantly different from zero, presumably because both measurements and calculations become much more complex when interstitial interactions are included. This study arose primarily from an interest in cardiac muscle, where it has been well established that substantial interstitial potentials occur in tightly packed structures, e.g., tens of millivolts within the ventricular wall. We analyzed the electrophysiological interaction between two adjacent unmyelinated fibers within a restricted extracellular space. Numerical evaluations made use of two linked core-conductor models and Hodgkin-Huxley membrane properties. Changes in transmembrane potentials induced in the second fiber ranged from nonexistent with large intervening volumes to large enough to initiate excitation when fibers were coupled by interstitial currents through a small interstitial space. With equal interstitial and intracellular longitudinal conductivities and close coupling, the interaction was large enough (induced Vm approximately 20 mV peak-to-peak) that action potentials from one fiber initiated excitation in the other, for the 40-microns radius evaluated. With close coupling but no change in structure, propagation velocity in the first fiber varied from 1.66 mm/ms (when both fibers were simultaneously stimulated) to 2.84 mm/ms (when the second fiber remained passive). Although normal propagation through interstitial interaction is unlikely, the magnitudes of the electrotonic interactions were large and may have a substantial modulating effect on function.

Propagation model using the DiFrancesco-Noble equations. Comparison to reported experimental results

Propagation, re-entry and the effects of stimuli within the conduction system can be studied effectively with computer models when the pertinent membrane properties can be represented accurately in mathematical form. To date, no membrane models have been shown to be accurate representations during repolarisation and recovery of excitability, although for the Purkinje membrane the DiFrancesco-Noble (DN) model has become a possibility. The paper examines the DN model, restates its equations and compares simulated waveforms in a number of propagation contexts to experimental measurements reported in the literature. The objective is to determine whether or not the DN model reproduced phenomena such as supernormality, shortening in action potential duration during pacing rate increases, alternation of duration with changes in rhythm, graded responses and 'all-or-none' repolarisation in a quantitatively realistic way, as each of these come from time and space dependencies not directly a part of the ionic current measurements on which the DN model is based. The results show that the DN equations correctly simulate these situations and support the goal of having a model that is broadly applicable to Purkinje tissue, including refractory period properties and response to electrical stimulation.

A comparison of two boundary conditions used with the bidomain model of cardiac tissue

In the bidomain model, two alternative sets of boundary conditions at the interface between cardiac tissue and a saline bath have been used. It is shown that these boundary conditions are equivalent if the length constant of the tissue in the direction transverse to the fibers is much larger than the radius of the individual cardiac cells. If this is not the case, the relative merits of the two boundary conditions are closely related to the question of the applicability of a continuum model, such as the bidomain model, to describe a discrete multicellular tissue.

Modification of a cylindrical bidomain model for cardiac tissue

Previous models based on a cylindrical bidomain assumed either that the ratio of intracellular and interstitial conductivities in the principal directions were the same or that there was no radial variation in potential (i.e., a planar front, delta Vm/delta rho = 0). This paper presents a formulation and the expressions for the intracellular, interstitial, extracellular, and transmembrane potentials arising from nonplanar propagation along a cylindrical bundle of cardiac tissue represented as a bidomain with arbitrary anisotropy. For unequal anisotropy, the transmembrane current depends not only on the local change of the transmembrane potential but also on the nature of the transmembrane potential throughout the volume.

A planar slab bidomain model for cardiac tissue

A fully three-dimensional model of the ventricular or atrial free wall will involve a planar geometry of finite thickness. The governing equations for the interstitial and extracellular potential of a planar slab of cardiac tissue comprised of parallel fibers undergoing uniform plane-wave activation are presented. A comparison with a bidomain of cylindrical geometry with the same half-thickness shows that the potentials in the planar bidomain (as a function of depth) approach core-conductor behavior more quickly.