The volume conductor effects of anisotropic muscle on body surfacepotentials using an eccentric spheres model

The purpose of this paper is to determine the volume conductor effects of muscle anisotropy on body surface potentials using an eccentric spheres model with a uniform double layer source configuration. Previous eccentric spheres work assumed that cardiac muscle anisotropy was small and that skeletal muscle effects could be accounted for by boundary extension, i.e., by scaling the conductivities and dimensions. However, in this paper, anisotropy for both the myocardium and the skeletal muscle is explicitly incorporated into the eccentric spheres volume conductor model. The anisotropy is treated as having uniform orthogonal components in the radial and tangential directions for both the skeletal muscle and myocardium. The solution for Laplace's equation is written in a series expansion of appropriate basis functions for each region. In the isotropic regions spherical harmonics with integer radial dependence and Legendre polynomial azimuthal dependence are utilized. For the anisotropic regions, Legendre polynomials are still appropriate for the azimuthal dependence, but noninteger powers of radial dependence are required. The approximate representation for anisotropy, i.e., the boundary extension method for the skeletal muscle and a scaled homogeneous conductivity without boundary extension for the myocardium are compared with explicit representations for the two regions. Two basic conclusions are drawn from the results. First, the treatment of skeletal muscle anisotropy by the boundary extension method is a valid and useful simplification which yields errors of 2% for the peak body surface potential. The second conclusion drawn from this study is that myocardial anisotropy has a significant effect on the magnitude of body surface potentials.(ABSTRACT TRUNCATED AT 250 WORDS)     

A bidomain model for the extracellular potential and magnetic field of cardiac tissue

In this brief communication, a bidomain volume conductor model is developed to represent cardiac muscle strands, enabling the magnetic field and extracellular potential to be calculated from the cardiac transmembrane potential. The model accounts for all action currents, including the interstitial current between the cardiac cells, and thereby allows quantitative interpretation of magnetic measurements of cardiac muscle.     

On the theory of the electrocardiogram

The biophysical basis for understanding the electrocardiogram is set forth. Bioelectric sources arise from electrical activity in the heart at the cellular level. The relation of these sources, which can be formally represented as impressed currents, to potentials involves solution of the volume conductor problem. This solution is based on Green's theorem. Sources are related to the transmembrane action potential through a bidomain model of heart muscle. Microscopic and macroscopic aspects of the bidomain model are developed. Various transformations of the source are considered, including multipoles, multiple dipoles, and replacement of the volume distribution with distributions on the heart surface. Time integrals of the waveform are related to excitation time and action potential duration. The theoretical results form the basis of a computer model of the electrocardiogram that relates skin potentials to the spatial and temporal distribution of action potentials in the heart     

Potential distribution in three-dimensional periodic myocardium.II. Application to extracellular stimulation

Modeling potential distribution in the myocardium treated as a periodic structure implies that activation from high-current stimulation with extracellular electrodes is caused by the spatially oscillating components of the transmembrane potential. This hypothesis is tested by comparing the results of the model with experimental data. The conductivity, fiber orientation, the extent of the region, the location of the pacing site, and the stimulus strength determined from experiments are components of the model used to predict the distributions of potential, potential gradient, and the transmembrane potential throughout the region. Next, assuming that a specific value of the transmembrane potential is necessary and sufficient to activate fully repolarized myocardium, the model provides an analytical relation between large-scale field parameters, such as gradient and current density, and small-scale parameters, such as transmembrane potential. This relation is used to express the stimulation threshold in terms of gradient or current density components and to explain its dependence upon fiber orientation. The concept of stimulation threshold is generalized to three dimensions, and an excitability surface is constructed, which for cardiac muscle is approximately conical in shape. The numerical values of transmembrane potential and stimulation thresholds calculated using asymptotic analysis are in agreement with the results of animal experiments, confirming the validity of this approach to study the electrophysiology of periodic cardiac muscle.     

The response of a spherical heart to a uniform electric field: abidomain 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     

Quantitative Formulations of Electrophysiological Sources of Potential Fields in Volume Conductors

The noninvasive measurement of electrical potentials at the surface of the body (e.g., the electrocardiogram) has long been considered an important tool in clinical diagnosis. Electrophysiological modeling and simulation is valuable as an aid in the interpretation of such potential recordings. In all cases, the potential field can be considered to arise from bioelectric sources operating in a volume conductor. This paper concentrates on the appropriate quantitative formulation for these sources. Such sources arise from excitable cells undergoing action potentials (primary sources) or at passive boundaries between regions of different conductivity (secondary sources). These sources are described and discussed for arbitrary cell shapes, circular cylindrical cells, conductive media with piecewise constant conductivity, and for syncytial tissue whose macroscopic properties are anisotropic.     

Analytic solution of the anisotropic bidomain equations for myocardial tissue: the effect of adjoining conductive regions

The anisotropic bidomain model for the propagation of electrical activation in the human myocardium H consists of coupled elliptic-parabolic partial differential equations for the transmembrane potential Vm, intracellular potential phi(i), and extracellular potential phi(e) in H, together with quasi-static equations for the potential distribution phiB in the surrounding (passive) isotropic extracardiac regions B. Four local parameters sigma((i,e) (l,t)) specify the conductivities in the longitudinal (l) and transverse (t) directions with respect to cardiac muscle fibers. Continuous current flow is required at the interface S(H) between H and B. We derive analytic formulas for Vm, phi(e), phi(i), and phiB for plane wave propagation in a uniformly anisotropic slab surmounted by a homogeneous region of conductivity sigmaB. No assumptions are required regarding the anisotropy ratios of the conductivity coefficients. The properties of these solutions are examined with a view to providing insight into the effect of the passive region B on the propagation of Vm and phi(e) in H. We show that for a suitably chosen boundary condition, the problem can be reduced to solving the bidomain equations in H alone.     

A survey of the near-field far-field inverse scattering inverse source integral equation

The Kirchhoff direct integration of the scalar wave equation is reviewed, and some properties of the Kirchhoff surface integral are discussed, from the perspective of the inverse scattering inverse source problem. A modified Kirchhoff surface integral is introduced, leading to a Fredholm integral equation of the first kind for the unknown sources (induced by the incident field) inside a volume in terms of the (scattered) fields on the surface enclosing this volume. The properties and physical meaning of this integral equation are discussed. A generalization of this integral equation for the vector electromagnetic wave equations is presented.     

Potential distribution in three-dimensional periodic myocardium. I.Solution with two-scale asymptotic analysis

The use of two-scale asymptotic analysis allows development of a model of the steady-state potential distribution in three-dimensional cardiac muscle preserving the underlying cellular network. The myocardium is modeled as a periodic structure consisting of cylindrical cells embedded in extracellular fluid and connected by longitudinal and side junctions. The method is applicable to cardiac muscle of arbitrary extent since the periodicity of the tissue is dealt with analytically, and thus numerical computations require no more resources than a continuous volume conductor problem. The asymptotic analysis approach reveals that the potential in a periodic myocardium consists of two components. The large-scale component provides the baseline for the total solution and can be determined from the anisotropic monodomain model associated with the original periodic problem. The method provides the formula for calculating the conductivity of the equivalent monodomain model on the basis of cell geometry and conductivity distribution in the cardiac tissue. The small-scale component reflects the periodicity of the underlying structure and oscillates with periods determined by the dimensions of cardiac cells. The magnitude of these oscillations depends upon the gradient of the large-scale component. During stimulation with extracellular electrodes, the small-scale component determines both the shape and the magnitude of the transmembrane potential, while the influence of the large-scale component is negligible. Hence, the small-scale component merits closer attention in pacing and defibrillation studies, especially since the model based on two-scale asymptotic analysis provides an effective means of its computation.     

Periodic Conductivity as a Mechanism for Cardiac Stimulation and Defibrillation

This study examines the distribution of the transmembrane potential in the periodic strand of cardiac muscle established by configurations of sources similar to those arising during extracellular stimulation and defibrillation, during intracellular stimulation, and during propagation of action potential. The closed-form solution indicates that during extracellular stimulation with large current and during defibrillation, the periodic component of the transmembrane potential is very important. We postulate that this periodic component causes the depolarization or defibrillation in cardiac muscle, which is different from the depolarization mechanism for a continuous fiber. On the other hand, during propagation and intracellular stimulation, the periodic component only slightly modifies the monotonic decrease of the transmembrane potential, which suggests that the mechanism of propagation in discrete structures may be similar to that of the continuous fiber.     

Patterns of and mechanisms for shock-induced polarization in the heart: a bidomain analysis

This paper examines the combined action of cardiac fiber curvature and transmural fiber rotation in polarizing the myocardium under the conditions of a strong electrical shock. The study utilizes a three-dimensional finite element model and the continuous bidomain representation of cardiac tissue to model steady-state polarization resulting from a defibrillation-strength uniform applied field. Fiber architecture is incorporated in the model via the shape of the heart, an ellipsoid of variable ellipticity index, and via an analytical function, linear or nonlinear, describing the transmural fiber rotation. Analytical estimates and numerical results are provided for the location and shape of the "bulk" polarization (polarization away from the tissue boundaries) as a function of the fiber field, or more specifically, of the conductivity changes in axial and radial direction with respect to the applied electrical field lines. Polarization in the tissue "bulk" is shown to exist only under the condition of unequal anisotropy ratios in the extra- and intracellular spaces. Variations in heart geometry and, thus, fiber curvature, are found to lead to change in location of the zones of significant membrane polarization. The transmural fiber rotation function modulates the transmembrane potential profile in the radial direction. A higher gradient of the transmural transmembrane potential is observed in the presence of fiber rotation as compared to the no rotation case. The analysis presented here is a step forward in understanding the interaction between tissue structure and applied electric field in establishing the pattern of membrane polarization during the initial phase of the defibrillation shock.     

A complete linear discretization for calculating the magnetic fieldusing the boundary element method

An analytic solution is derived for the magnetic field generated by current sources located in a piecewise homogeneous volume conductor. A linear discretization approach is used, whereby the surface potential is assumed to be a piecewise linear function over each tessellated surface defining the regions of differing conductivity. The magnetic field is shown to be a linear combination of vector functions which are strictly dependent on the geometry of the problem, the surface tesselation, and the observation point     

An equivalent current source model and laplacian weighted minimum norm current estimates of brain electrical activity

We have developed a method for estimating the three-dimensional distribution of equivalent current sources inside the brain from scalp potentials. Laplacian weighted minimum norm algorithm has been used in the present study to estimate the inverse solutions. A three-concentric-sphere inhomogeneous head model was used to represent the head volume conductor. A closed-form solution of the electrical potential over the scalp and inside the brain due to a point current source was developed for the three-concentric-sphere inhomogeneous head model. Computer simulation studies were conducted to validate the proposed equivalent current source imaging. Assuming source configurations as either multiple dipoles or point current sources/sinks, in computer simulations we used our method to reconstruct these sources, and compared with the equivalent dipole source imaging. Human experimental studies were also conducted and the equivalent current source imaging was performed on the visual evoked potential data. These results highlight the advantages of the equivalent current source imaging and suggest that it may become an alternative approach to imaging spatially distributed current sources-sinks in the brain and other organ systems.     

Discrete versus syncytial tissue behavior in a model of cardiacstimulation. II. Results of simulation

For pt. I see ibid., vol. 43, no. 12, p. 1129-40 (1996). The research presented here combines mathematical modeling and computer simulation in developing a new model of the membrane polarization induced in the myocardium by the applied electric field. Employing this new model termed the “periodic” bidomain model, the steady-state distribution of the transmembrane potential is calculated on a slice of cardiac tissue composed of abutting myocytes and subjected to two point-source extracellular current stimuli. The goal of this study is to examine the relative contribution of cellular discreteness and macroscopic syncytial tissue behavior in the mechanism by which the applied electric field alters the transmembrane potential in cardiac muscle. The results showed the existence of oscillatory changes in the transmembrane potential at cell ends owing to the local resistive inhomogeneities (gap-junctions). This low-magnitude sawtooth component in the transmembrane potential is superimposed over large-scale transmembrane potential excursions associated with the syncytial (collective) fiber behavior. The character of the cardiac response to stimulation is determined primarily by the large-scale syncytial tissue behavior. The sawtooth contributes to the overall tissue response only in regions where the large-scale transmembrane potential component is small     

A Principle for Solving a Class of Anisotropic Current Flow Problems and Applications to Electrocardiography

In studies of electrocardiographic lead performance, theoretical analyses of the influence of the anisotropic heart and skeletal muscle are particularly difficult. In this paper, the basic differential equations of static fields and steady current flow are arranged to emphasize the field and conductivity dependent charge distributions which arise in anisotropic media. The equations are applied to two types of problems of immediate interest. Firstly, the equations are used to explain how anisotropic media may be included in current digital computer studies of the heart-lead relation and to conclude that the techniques which made the computer studies possible tend to lose their advantage when applied to arbitrary anisotropic configurations. Secondly, the equations are used to develop a principle which permits exact solutions for the fields of numerous simple anisotropic configurations. Three such configurations useful for heart-lead studies are analyzed with the following results: the anisotropic skeletal muscle can be treated in special cases such as a head-foot heart-vector lead approximately as isotropic with resistivity of 280 ohm cm; the closed dipolar layer in an anisotropic, inhomogeneous heart produces the same null electric field as it does in homogeneous isotropic media; bounds on the influence of the heart's anisotropy on a heart-vector lead field are estimated at plus or minus 12 percent of the average lead field intensity.     

The effect of torso impedance on epicardial and body surface potentials: a modeling study

Experimental results have been published that report marked changes in measured epicardial potentials when the conductivity of the material surrounding the heart is altered. These reports raise a question as to the validity of the traditional two step, equivalent cardiac source approach to modeling the forward problem of electrocardiology as the equivalent source calculation occurs in what is effectively an isolated cardiac region. In the physical situation the heart is surrounded by a torso that contains many different tissue types with different conductivities and is certainly not isolated. Here, a fully coupled model of the problem is employed where the electrical pathways are continuous from a cellular level through to the body surface. This model is used to investigate the effects that torso inhomogeneities have on epicardial and body surface potentials, including comparisons with a traditional two step approach. In particular, it is shown that adding lungs changes the epicardial potentials by 17%, which is consistent with the reported experimental results. In none of the tested situations did the equivalent source approach completely reproduce the fully coupled results, supporting the notion that a fully coupled approach is required to properly solve the forward problem of electrocardiology.     

Dipole and Quadrupole Measurements of an Isolated Turtle Heart

An equivalent cardiac generator, specifically the multipole model, was evaluated for an isolated turtle heart located in a spherical volume conductor. The dipole and quadrupole contributions to the potential field were calculated using weighted integrals of the surface distributions. This procedure has the advantage of eliminating errors due to the truncation of the multipole series. The dipole and quadrupole effects were calculated for the QRS portion of the cardiac waveform. A significant quadrupole contribution was found to exist.     

Simulation of propagation along a cylindrical bundle of cardiactissue. II. Results of simulation

Previous evaluations of the cylindrical bidomain model of a bundle of cardiac tissue, have been obtained by using an analytic function for the transmembrane potential and assuming the activating wavefront through the bundle cross section is planar. In this paper, nonlinear membrane kinetics are introduced into the bidomain membrane and equal anisotropy ratios are assumed, permitting the transmembrane potential to be computed and its behavior examined at different depths in the bundle and for different values of conductivity and bundle diameters. In contrast with single fiber models, the bundle model reveals that the shape of the action potential is influenced by tissue resistivities. In addition, the steady-state activation wavefront through the cross-section perpendicular to the long axis of the bundle is not planar and propagates with a velocity that lies between that of a single fiber in an unbounded volume and a single fiber in a restricted extracellular space. In general, the bundle model is shown to be significantly better than the classical single fiber model in describing the behavior of real cardiac tissue.