Homogeneous Volume Conductor Research Articles

Eccentric dipole in a spherical medium: generalized expression for surface potentials.

A new formulation is presented for surface potentials produced on a homogeneous spherical volume conductor by an eccentric current dipole contained therein. The formulation is free of interminancies for all relevant dipole locations and leads directly to the solution of surface potentials due to an eccentric quadripole.

A new mathematical model of electrical cardiac activity

A mathematical model of electrical heart activity is constructed from thirteen electrical dipoles of known and fixed location and direction, and of variable moment. The moment of each is proportional to the monophasic action potential of a heart component and is found from the solution of a differential equation with discontinuous right-hand side. These equations differ according to whether the component is, or is not, pacemaker tissue. The conduction between components is modeled on biophysical principles, and a twelve-lead electrocardiograph is modeled on elementary dipole theory with the body regarded as a homogeneous volume conductor of infinite extent. High fidelity normal twelve-lead electrocardiograms, and some arrhythmic ones are generated by this model.

The electrical field produced by the eccentric current dipole in the nonhomogeneous conductor

T he problem investigated has important implications which bear on the electrical field which is produced by the heartbeat. For nearly 30 years1 the basic approach to human and animal electrocardiography has regarded the “heart generator” as a source-sink distribution which emanates from the heart muscle fibers. The most general viewpoint is that of treating the “heart generator” as a source-sink multipole in which the net pole strength is zero. In the general theory of multipoles the expansion for the potential includes a first term (tensor of rank zero) which vanishes when the net pole strength is zero. The second term of the expansion is the potential of a dipole or tensor of rank one. Additional terms are potentials of the quadripole, the octapole, and tensors of higher rank. At distances from the multipole large in comparison with its “circumference,” the cube and the higher powers of these distances enter the denominators of the terms which are components of tensors of rank two and higher, and the magnitudes of these terms may be neglected. Consequently, the potential field throughout the greater part of the body trunk is essentially equivalent to that produced by a dipole, or a tensor of rank one. Her5 the variations of body surface potentials are due to changes in strength of dipole moment &I and to rotations of the moment M-axis with each heartbeat. For the past 30 years the heart generator has been treated physically as though it produced the electrical field of a dipole in a homogeneous volume conductor.l This last simplification is, of course, a mere approximation of the truth. Nevertheless, this approximation has offered great advantages to both teachers and investigators. Recent investigations2-4 have naturally become more exacting and have introduced the general idea of different specific resistivities making up a nonhomogeneous conductor. It is the object of this article to help remove the foundations of electrocardiography from the approximate foundations of the homogeneous volume conductor to that of the more realistic nonhomogeneous volume conductor.

Epicardial excitation pattern as observed in the isolated revived and perfused fetal human heart.

The resuscitated fetal human heart can be used as an experimental tool for the investigation of the excitatory process in the human heart. During perfusion the configuration of the epicardial electrocardiograms does not change appreciably. For accurate recording permitting a detailed analysis, the use of a high-fidelity oscillograph is absolutely necessary, otherwise no identification of the rapid portion of the intrinsic deflection can be made. The heart is suspended in an homogenous volume conductor at 37 C. and perfused with fluid matching as closely as possible the composition of the extracellular fluid. The morphology of the unipolar complexes is difficult to describe adequately. At the anterior side of the right and left ventricles rS complexes are found. Some parts of the left ventricular wall and posteroapical region of the left ventricle show QR complexes. In the area bordering the A-V groove one-third the distance from apex to basis, deep Q waves and even QS complexes are found. The relatively late intrinsic deflection here points to late excitation of this region. In regions neighboring the A-V groove, the Q wave diminishes in size and the R increases. The left posterior part of the left atrium is activated latest in the atrial cycle. The epicardial excitation pattern is surprisingly simple. The excitation wave reaches the region of attachment of the right anterior papillary muscle first, then spreads radially with varying velocity across the left ventricle towards the posterobasal region, while another wave spreads simultaneously across the right ventricle towards the same region. There is, therefore, a double envelopment of the epicardial surface. A comparison of the complexes having an intrinsic deflection occurring at the same time shows conclusively that, even over the same ventricle, the morphology of these complexes may be completely different. Thus the excitation time does not determine the morphology of the complexes. There is no typical right or left ventricular pattern. There appears to be no relationship between thickness of the heart wall and height of the R wave.

A method for applying approximately ideal lead connections to homogeneous volume conductors of irregular shape

1. 1. Approximately ideal lead connections may be applied to irregularly shaped, electrically homogeneous volume conductors on the basis of well-defined theoretical principles rather than by means of trial and error methods. 2. 2. These underlying principles were tested and amply confirmed by studies conducted on a series of laminar models. 3. 3. This study confirms both theoretically and experimentally the relationship that the electrical moment due to a number of sources and sinks within a homogeneous volume conductor is equivalent to a surface integral involving unit normals and surface potentials. 4. 4. The method of ideal lead connection proposed here would undoubtedly be cumbersome, but it might profitably be applied to scalar and vector electrocardiography at an investigative level.

Vectorcardiography as studied on the isolated mammalian heart suspended in a homogeneous volume conductor.

Vectorcardiography as studied on the isolated mammalian heart suspended in a homogeneous volume conductor.

The zero of potential of the electric field produced by the heart beat; the problem with reference to homogenous volume conductors.

The acceptable zeros of the potential of the electrical fields produced by certain dipole distributions in homogeneous volume conductors is discussed. A bridge circuit is described by which a solution of the three-arm and the four-arm central terminals of Wilson may be solved for a zero of potential of the field produced by an arbitrary distribution of dipoles in a homogeneous volume conductor. An acceptable zero of potential for evaluation of the potentials in a locus on the "body" surface distant from the heart is described.

Intrabronchial electrocardiography.

The Einthoven hypothesis implies that differences in the conductivity of body tissues are not sufficiently great to invalidate the concept that the body is a homogeneous volume conductor. Some investigators believe the lungs are sufficiently poor conductors to make questionable the concept that the body can be regarded as a homogeneous volume conductor. Intrabronchial electrocardiography is presented as a possible method for studying this problem. In ten individuals without manifest cardiac disease the distribution of potential variations in the lungs as manifested by the QRS complexes corresponded approximately and qualitatively with the distribution of the QRS complexes recorded from the body surfaces.

Electrical Instrumentation in Medical Research

RECENTLY there has been a heartening trend in biological research toward closer collaboration between the physical and the biological sciences. This article is a resume of a program of this nature being pursued at the University of Pennsylvania. In order that such a program be well balanced, research, education, and instrumentation are all required. These three topics are discussed and new technical developments are described. Work done in the field of electrocardiography may be cited as one example of research. In explanation of the distribution of potentials on the body surface arising from heart action, an ingenious theory has evolved which is widely accepted. However, some of the fundamental postulates on which this theory is based are open to question. Chief among these is the assumption that for electrocardiographic purposes the body behaves as a homogenous volume conductor. Through the years, numerous attempts have been made to check this assumption both by measuring directly the resistivities of the various tissues within the body and by more indirect means. Investigators have reached widely differing conclusions from their experiments so that there is still no definitive answer to the question. This past work has been reviewed and evaluated and a series of experiments is under way which, it is hoped, will throw further light on the subject.

An interpretation of the injury and the ischemic effects of myocardial infarction in accordance with the laws which determine the flow of electric currents in homogeneous volume conductors, and in accordance with relevant pathologic changes

1. 1. There is a current difference of opinion regarding the interpretation to be placed upon the RS-T junction displacements caused by myocardial infarction. 2. 2. When an electrical situation similar to that which exists with certain varieties of direct subepicardial damage is applied to the ordinary varieties of infarction, according to the laws which determine the flow of electric currencs in homogeneous volume conductors, the predicted electrocardiographic effects prove to be the inverse of those which are known to occur clinically and experimentally in association with infarction. 3. 3. Inasmuch as these laws have been shown to apply to problems of the kind herein considered with sufficient accuracy to be useful, it appeared highly probable that the electrical situations caused by direct damage and by infarction, although fundamentally similar, must differ considerably in pattern detail. 4. 4. A summary of the pathologic changes is presented, together with the requirements deduced from a critical analysis of the electrocardiographic changes associated with infarction. The data led to a reasonable definition of the source and sink distribution of the electrical field of injury and of the neighboring zone of ischemia which are associated with infarction. Moreover, an explanation for the occurrence in sequence of the RS-T junction displacements and the primary T-wave changes is regarded as an integral part of the problem. This sequential relationship, together with heretofore unexplained QRS changes, is thus accounted for incidentally. The RS-T junction displacements in the extremity leads are regarded as the result of the development and subsequent decay of the spatial axis of injury-î. The primary T-wave changes have been ascribed elsewhere to a diversion and subsequent reversion of the spatial gradient of duration of the excited ventricular state. In general, the injured zone caused by recent infarction is perifocal or semiperifocal with respect to the dead zone, depending upon whether the infarct is essentially subendocardial or transmural. The ischemic zone is at first small and semiperifocal with respect to the injured zone. Later, as the injured zone diminishes in the direction of the dead zone and finally disappears, the ischemic zone necessarily undergoes an inward expansion and becomes directly perifocal with respect to the dead zone of a subendocardial infarct, or directly semiperifocal with respect to the dead zone of a transmural infarct. The changes are ascribed chiefly to improvement in the local circulation. 5. 5. The explanation offered for the RS-T junction displacements and the sequential occurrence of T-wave changes as a result of infarction has the additional advantage of being directly applicable to similar but more transient serial changes which are observed in certain patients during and immediately after an attack of angina pectoris. 6. 6. A summary of certain electrocardiographic interpretations is presented which is regarded as tentatively acceptable for curves that reflect the complex combination of infarction and pericarditis.

The Potential Produced by Cardiac Muscle. A General and a Particular Solution

Let us consider a mass of cardiac muscle immersed in an extensive homogeneous volume conductor. The value of the potential at a point in the conductor, produced by any given distribution of depolarization or repolarization may be obtained theoretically in the following way:Let v2 denote the particular region or volume of the muscle mass which is undergoing depolarization at a given instant. Let us choose any point O as the origin of a rectangular coordinate system X, Y, Z. Let (X2 Y2, Z2) be any convenient point within the region v2. Let dv2 be an element of volume of v2 at the point (X2, Y2, Z2). Let the magnitude of the vector O represent the intensity of depolarization of the element dv2, and let the direction of O be that of a line drawn from the effective negative toward the effective positive ionic charge within the element dv2. The vector quantity Odv2 is then the electric moment of depolarization.Let us choose next any other (fixed) point (X1, Y1, Z1) within the volume conductor, in the vicinity o...