A Mathematical-Physical Model of the Genesis of the Electrocardiogram

Abstract

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.

Calculation of the electrostatic-potential distribution in ohmic elliptical cylinders fed by a dipole source

“…Finally, for the purpose of checking the validity and the precision of the results obtained by Gelernter and Swihart, 18 we compute the electric potential in the elliptical homogeneous cylinder with ϭ1.308 and l/dϭ4.57, which these authors used in their calculations. We use the same position of the dipole as in their work, i.e., xЈϭ0.1866x b , yЈϭ Ϫ0.5014y b , Јϭ0.6607.…”

“…6͑a͒ of Ref. 18, keeping in mind the difference in the units and in the choice of the zero potential, which is taken at the surface-potential minimum in Ref. 18.…”

“…18 Then, we adopt the values used by these authors in their numerical computation of the electric potential in the case of the first homogeneous elliptical cylinder that they considered. This corresponds to ϭ1.308 and l/dϭ4.57.…”

Quantitative Prediction of Body Surface Potentials from Myocardial Action Potentials Using a Summed Dipole Model

“…As shown in Appendix 2 the partial derivative of the dipole potential field (6a) at the midline ( = 90 o ) is (7) through (9).…”

“…These models have become exercises in sophisticated computer science far beyond the usual domains of physiology and medicine. They have also tended to express results for computed body surface potentials in arbitrary units 8,9 , blurring the question of exactly how many millivolts of electrical potential difference are expected betweens arbitrary points A and B on the chest wall, given known normal or abnormal cardiac electrophysiology.…”