Simultaneous analysis of the cardiac electric and magnetic fields using the scalar multipole expansion

A theoretical analysis of electric and magnetic fields of the heart, based solely upon the scalar multipole expansion, is carried out in order to gain an insight into the interrelation of the data contained in electro- and magnetocardiological measurements. The usual multipole expansion is applied for the electric field, however corresp[nding equivalent multipoles are formulated as idealized generators, having not only flow sources, but also vortex sources of the field. Furthermore, the magnetic field in a homogeneous infinite volume conductor is expressed as a sum of two series, the first being the usual multipole expansion of the nonvortex component of the magnetic field, and the second being a sequence of magnetic fields set up by the aforementioned electric multipole generators reduced to axial form. The former term is uniquely defined by the electric multipole components, but the latter reflects properties of the cardiogenerator that can be revealed only by means of magnetic measurements. Features of the electric and magnetic multipole components as integral characteristics of the cardiogenerator are discussed and concepts of the magnetic centre and magnetic axis of the cardiogenerator are proposed. The analysis is illustrated by examples of simple generator configurations.     

On Bioelectric Potentials in an Inhomogeneous Volume Conductor

Green's theorem is used to derive two sets of expressions for the quasi-static potential distribution in an inhomogeneous volume conductor. The current density in passive regions is assumed to be linearly related instantaneously to the electric field. Two equations are derived relating potentials to an arbitrary distribution of impressed currents. In one, surfaces of discontinuity in electrical conductivity are replaced by double layers and in the other, by surface charges. A multipole equivalent generator is defined and related both to the potential distribution on the outer surface of the volume conductor and to the current sources. An alternative result involves the electric field at the outer surface rather than the potential. Finally, the impressed currents are related to electrical activity at the membranes of active cells. The normal component of membrane current density is assumed to be equal at both membrane surfaces. One expression is obtained involving the potentials at the inner and outer surfaces of the membrane. A second expression involves the transmembrane potential and the normal component of membrane current.     

Active muscle fiber as an equivalent cardiac current generator

Equations are derived describing potentials due to an active muscle fiber in an infinite medium in terms of two surface integrals—one of the propagated action potential and the other of the membrane current density, both integrals being taken over the surface of the muscle. These equations are incorporated into an equivalent cardiac current generator in which the left ventricle (i.e. the current source) is represented by a three-dimensional wedge and the thorax (i.e. the volume conductor), by a homogeneous circular cylinder. Since this current generator expresses the body surface potentials in terms of the membrane current density and the membrane potential at any point on the surface of the electrically active muscle fiber, the calculated ECG can be correlated with theactual sources within the heart. This equivalent cardiac generator possesses many of the physical and physiological properties of cardiac muscle. The equations were evaluated numerically on a digital computer. The results indicate that equivalent cardiac current generators of this type can yield clinically significant results and that further research is necessary to investigate their properties fully.     

Biophysical models of the heart electrical activity

Based on the fundamental knowledge of the space-temporal organization of extracellular electrical fields of the myocardium, a system for 3-D computer modeling of the cardiac electrical activity at different structural levels of the object is being developed at the Institute of Theoretical and Experimental Biophysics. The system is based on the earlier proposed and modified biophysical model of the electrocardiosignal genesis represented by a double electrical layer along the surface of the electrically active myocardium. The system combines the model for activation and repolarization of the heart ventricles; the advanced model for the evaluation of parameters of the cardiac electric field, which makes it possible to derive model electrocardiosignals both in the direct regime of calculation of the potentials and in the regime of calculation of electrocardiosignals from preliminarily determined components of the multipole equivalent electrical heart generator; a database for the model parameters and their combinations in the form of cards of simulated "patients", and a database of modeled electrocardiosignals. In the present paper (first from three within the framework of the problem), simulation methods in electrocardiology are briefly described and a biophysical model of the heart electrical activity is presented, which has made up the basis of the system for computer modeling of forward and inverse problems of the cardiac electric field. The parameters of the model are electrophysiological, anatomical, and biophysical characteristics of the heart.     

Studies into equine electrocardiography and vectorcardiography: I. Cardiac electric forces and the dipole vector theory

Theoretical consideration has been given in two horses to the properties of the electric field created by the equine heart acting as a simple electric generator. The principles of the vectorial theory have been applied to test the validity of application of the dipole concept. The cardiac electric forces, althrough complex in the immediate region of the heart, appear at the body surface in a similar form to those arising from a relatively immobile, single equivalent dipole. The potential value of the technique of vectorcardiography in cardiological investigations is briefly discussed.     

The inconsistencies of present-day mathematical-physical methods pertaining to current generators in volume conductors used in electrocardiography

The present-day practices of electrocardiography and vectorardiography are based upon the theory that the surface potential differences can be assumed to be due to a single dipole inside the body. It is shown in this paper that a dipole cannot account for all the surface potentials due to realistic current generators, and hence the determination of the current generator from surface potential measurements based upon such a theory will lead to inconsistent representations of the heart for one and the same subject. To demonstrate this point two eccentric dipoles of different strengths and locations representing two muscle fibers are taken to be the current generator in a homogeneous spherical conductor. The exact surface potentials are then expressed by means of the “interior sphere theorem” of the authors. With these expressions the magnitude, direction, and location of the resultant dipole are determined by the method of D. Gabor and C. V. Nelson (J. App. Physics,25, 413–16, 1954). The surface potentials due to this resultant dipole are again exactly expressed by means of the “interior sphere theorem” and compared with those due to the eccentric dipoles assumed. It can be seen that the differences can be considerable. It is suggested that the multipole model of the authors (Bull. Math. Biophysics,20, 203–16, 1958) be used as a more accurate and the only unique representation of the heart. This investigation was supported by the National Heart Institute under a research grant H-2263(c).     

Use of translucent indium tin oxide to measure stimulatory effects of a passive conductor during field stimulation of rabbit hearts

Biomathematical models and experiments have indicated that passive extracellular conductors influence field stimulation. Because metallic conductors prevent optical mapping under the conductor, we have evaluated a passive translucent indium tin oxide (ITO) thin-film conductor to allow mapping of transmembrane potential (V(m)) and stimulatory current under the conductor. A 1-cm ITO disk was patterned photolithographically and positioned between 0.3-cm(2) mesh shock electrodes on the ventricular epicardium of isolated perfused rabbit hearts stained with 4-{2-[6-(dibutylamino)-2-naphthylenal]ethenyl}-1-(3-sulfopropyl)-, hydroxide, inner salt (di-4-ANEPPS). For a 1-A, 10-ms shock during the action potential plateau, optical maps from fluorescence collected using emission ratiometry (excitation at 488 nm and emissions at 510-570 and >590 nm) indicated that the disk altered V(m) by as much as the height of an action potential. DeltaV(m) became more positive near the edge of the disk, where the ITO conductance gradient was parallel to applied current, and more negative near the opposite edge, where the gradient was not parallel to current. For diastolic shocks, the disk expedited membrane excitation at the sites of positive DeltaV(m) in the heart and in a cardiac model with realistic ITO disk surface and interfacial conductances. Optical maps of ITO transmittance and the model indicated that the disk introduced anodal and cathodal stimulatory current at opposite edges of the disk. Thus ITO allows study of the stimulatory effects of a passive conductor in an electric field.     

Integral characteristics of the cardiac electrical excitation wave]

A set of characterisitics of the cardiac electrical generator is described which expresses in an integral form some important properties of the electrical excitation wave, in particular its summary intensity, average spatial localization and distinction from a uniform double layer with planar rim. Relation of the proposed model to the multipole equivalent generator is discussed, and procedures for computing its characteristics are given. Calculation results for these characteristics on the basis of experimentally measured electrical field potentials of isolated dog hearts are presented.     

心肌电兴奋传导速度对心律失常影响的仿真研究

心电场是由心肌的电活动产生的。心肌细胞的电特性及心肌细胞间的传导关系决定了体表电位的分布及心电图的变化。心肌电兴奋传导速度则是影响心肌间兴奋传导关系的重要参数之一。由于很难通过实验方法来人为改变电兴奋传导速度,因而临床上有关该参数对心律影响的定量知识相当缺乏。本文采用真实三雏躯干模型及心脏模型,对心肌电兴奋传导速度与心律变化的关系进行定量仿真研究。结果表明,兴奋传导速度决定了整个心电图的变化,而局部普通心肌的传导速度在相当范围内变化似乎对心电图影响不明显,但传导速度超过一定范围后可能产生突变。     

Measurement of the electric field of the heart in a homogeneous volume conductor]

The paper describes a technique and some results of experimental measurements of electrical potentials generated by an isolated dog heart in homogeneous conductor, drawing equipotential maps of the field, and calculating the characteristics of the dipole equivalent generator of the heart. The form of potential distribution on a spherical surface around the heart and its ideal orthogonal vectorcardiograms are discussed.     

Biophysical models of cardiac electrical activity

A system for 3D simulation of heart electrical activity at different structural levels based on fundamental knowledge on the spatiotemporal organization of extracellular electric fields in the myocardium is being developed at the Institute of Theoretical and Experimental Biophysics of the Russian Academy of Sciences. The system is based on a biophysical model of the genesis of electrocardiosignals (ECSs) in the form of a double electric layer on the surface of the electrically active myocardium, which was proposed earlier and then modified. The system combines a model of the activation and repolarization of the heart ventricles, an advanced model for determining the parameters of the heart electric field, which makes it possible to obtain model ECSs both by direct calculation of the potentials and calculation of ECSs from preliminarily determined components of a multipole equivalent heart generator, a database of model parameters and their combinations in the form of cards of simulated “patients,” and a database of simulated ECSs. This paper (the first in a series of three on the subject) briefly describes simulation methods used in electrocardiology and the biophysical model of heart electrical activity that forms the basis of the system for computer simulation of direct and inverse problems concerning the heart electric field. Electrophysiological, anatomical, and biophysical characteristics of the heart are the parameters of the model.     

An electrodiffusive neuron-extracellular-glia model for exploring the genesis of slow potentials in the brain

Within the computational neuroscience community, there has been a focus on simulating the electrical activity of neurons, while other components of brain tissue, such as glia cells and the extracellular space, are often neglected. Standard models of extracellular potentials are based on a combination of multicompartmental models describing neural electrodynamics and volume conductor theory. Such models cannot be used to simulate the slow components of extracellular potentials, which depend on ion concentration dynamics, and the effect that this has on extracellular diffusion potentials and glial buffering currents. We here present the electrodiffusive neuron-extracellular-glia (edNEG) model, which we believe is the first model to combine compartmental neuron modeling with an electrodiffusive framework for intra- and extracellular ion concentration dynamics in a local piece of neuro-glial brain tissue. The edNEG model (i) keeps track of all intraneuronal, intraglial, and extracellular ion concentrations and electrical potentials, (ii) accounts for action potentials and dendritic calcium spikes in neurons, (iii) contains a neuronal and glial homeostatic machinery that gives physiologically realistic ion concentration dynamics, (iv) accounts for electrodiffusive transmembrane, intracellular, and extracellular ionic movements, and (v) accounts for glial and neuronal swelling caused by osmotic transmembrane pressure gradients. The edNEG model accounts for the concentration-dependent effects on ECS potentials that the standard models neglect. Using the edNEG model, we analyze these effects by splitting the extracellular potential into three components: one due to neural sink/source configurations, one due to glial sink/source configurations, and one due to extracellular diffusive currents. Through a series of simulations, we analyze the roles played by the various components and how they interact in generating the total slow potential. We conclude that the three components are of comparable magnitude and that the stimulus conditions determine which of the components that dominate.     

The Application of Electromagnetic Theory to Electrocardiology: II. Numerical Solution of the Integral Equations

In an earlier paper exact integral equations were derived for the surface potentials resulting from sources within an irregularly shaped inhomogeneous body. These exact equations cannot usually be solved. In this paper a discrete analogue is constructed which is not straightforward to solve, but which can be treated by careful mathematical methods. In particular a deflation procedure greatly facilitates the iterative solution of the problem and overcomes the divergence encountered by other authors. Numerical solutions obtained for simple geometries are compared to the exact analytic solutions available in such cases. The necessary convergence of the solutions of the discrete analog towards the solution of the continuous problem is shown to occur only if the coefficients of the discrete analogue are carefully evaluated. Calculations are then presented for realistic thoracic geometries, typical results being presented as surface potential maps. Finally the important effect of the internal regional inhomogeneities, particularly a realistic cardiac blood mass, is demonstrated by obtaining vector loops with and without these effects.     

A cardiac electric field on the body surface in rats with left ventricular hypertrophy caused by experimental renovascular hypertension

The dynamics of potential distribution of cardiac electric field on the body surface was studied in renovascular hypertensive rats (Goldblatt type) during the ventricular activity. Three inversions of the mutual location of positive and negative areas of the cardiac electric field on the body surface were found in normotensive and hypertensive rats during the QRS-T period. Left ventricular hypertrophy of the heart in rats caused by renovascular hypertension results in changes of temporal and amplitude characteristics of the body surface potential distribution during the initial and terminal ventricular activity. The shifting trajectory of the positive and negative areas and their extremal ranges on the body surface does not change during the ventricular activity in rats with left ventricular hypertrophy of the heart as compared to the initial normotensive state.     

A Mathematical-Physical Model of the Genesis of the Electrocardiogram

A fundamental problem of cardiac electrophysiology is that of relating quantitatively the electrical activity within the heart to the complete timevarying potential distribution at the body surface. A new numerical method is described for the calculation of the surface potential on an irregularly shaped closed external surface due to an arbitrary source distribution in a medium containing regions of different conductivity, subject to the appropriate boundary conditions. The method is intended to provide an exact theoretical analysis of the experimental data acquired by A. M. Scher and others who have been mapping the pathways of ventricular depolarization in dogs and other animals. In anticipation of the above research program, a number of exploratory computations are reported. For example, the surface potential distribution has been calculated for a cylinder of human torso cross-section with a hemispherical dipole layer current source in approximate heart position and orientation and containing “lungs” of conductivity different from that of the surrounding medium. Under certain conditions, when lung-like inhomogeneities are introduced, a simple dipole source can generate a potential distribution having the multiple maxima and minima characteristic of higher multipole sources.     

Eccentric multipole representations of current generators in a spherical volume conductor

In Yeh, Martinek and de Beaumont (Bull. Math. Biophysics,20, 203–16, 1958), a method is presented for determining successively better central multipole representations of the current generators in a homogeneous conducting sphere by measuring surface potentials at a successively increasing number of points. This paper generalizes the method such that the multipoles may be located at any chosen point in the conductor. The spherical harmonic expansion is advantageously used and the “interior sphere theorem” of Ludford, Martinek and Yeh (Proc. Cambridge Philos. Soc.,51, 389–93, 1955) makes possible disturbance potential expressions in closed forms. A method for approximate determination of the eccentricity is also presented. In the theory of electrocardiography, the eccentric multipoles can more accurately represent the heart as a current generator with fewer surface potential measurements than the central multipoles. This investigation was supported by The National Heart Institute under Research Grant H-2263(c-4).     

Phase and amplitude maps of the electric organ discharge of the weakly electric fish,Apteronotus leptorhynchus

Summary The electric organ discharge (EOD) potential was mapped on the skin and midplane of several Apteronotus leptorhynchus. The frequency components of the EOD on the surface of the fish have extremely stable amplitude and phase. However, the waveform varies considerably with different positions on the body surface. Peaks and zero crossings of the potential propagate along the fish's body, and there is no point where the potential is always zero. The EOD differs significantly from a sinusoid over at least one third of the body and tail. A qualitative comparison between fish showed that each individual had a unique spatiotemporal pattern of the EOD potential on its body.The potential waveforms have been assembled into high temporal and spatial resolution maps which show the dynamics of the EOD. Animation sequences and Macintosh software are available by anonymous ftp (mordor.cns.caltech.edu; cd/pub/ElectricFish).We interpret the EOD maps in terms of ramifications on electric organ control and electroreception. The electrocytes comprising the electric organ do not all fire in unison, indicating that the command pathway is not synchronized overall. The maps suggest that electroreceptors in different regions fulfill different computational roles in electroreception. Receptor mechanisms may exist to make use of the phase information or harmonic content of the EOD, so that both spatial and temporal patterns could contribute information useful for electrolocation and communication.Abbreviations EOD electric organ discharge - EO electric organ - CV coefficient of variance     

On the calculation of magnetic fields based on multipole modeling of focal biological current sources.

Spatially restricted biological current distributions, like the primary neuronal response in the human somatosensory cortex evoked by electric nerve stimulation, can be described adequately by a current multipole expansion. Here analytic formulas are derived for computing magnetic fields induced by current multipoles in terms of an nth-order derivative of the dipole field. The required differential operators are given in closed form for arbitrary order. The concept is realized in different forms for an expansion of the scalar as well as the dyadic Green's function, the latter allowing for separation of those multipolar source components that are electrically silent but magnetically detectable. The resulting formulas are generally applicable for current sources embedded in arbitrarily shaped volume conductors. By using neurophysiologically relevant source parameters, examples are provided for a spherical volume conductor with an analytically given dipole field. An analysis of the signal-to-noise ratio for multipole coefficients up to the octapolar term indicates that the lateral extent of cortical current sources can be detected by magnetoencephalographic recordings.     

Theoretical analysis of pre-receptor image conditioning in weakly electric fish

Electroreceptive fish detect nearby objects by processing the information contained in the pattern of electric currents through the skin. The distribution of local transepidermal voltage or current density on the sensory surface of the fish's skin is the electric image of the surrounding environment. This article reports a model study of the quantitative effect of the conductance of the internal tissues and the skin on electric image generation in Gnathonemus petersii (Günther 1862). Using realistic modelling, we calculated the electric image of a metal object on a simulated fish having different combinations of internal tissues and skin conductances. An object perturbs an electric field as if it were a distribution of electric sources. The equivalent distribution of electric sources is referred to as an object's imprimence. The high conductivity of the fish body lowers the load resistance of a given object's imprimence, increasing the electric image. It also funnels the current generated by the electric organ in such a way that the field and the imprimence of objects in the vicinity of the rostral electric fovea are enhanced. Regarding skin conductance, our results show that the actual value is in the optimal range for transcutaneous voltage modulation by nearby objects. This result suggests that "voltage" is the answer to the long-standing question as to whether current or voltage is the effective stimulus for electroreceptors. Our analysis shows that the fish body should be conceived as an object that interacts with nearby objects, conditioning the electric image. The concept of imprimence can be extended to other sensory systems, facilitating the identification of features common to different perceptual systems.