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

Abstract

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 coördinate 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 Ø represent the intensity of depolarization of the element dv₂, and let the direction of Ø be that of a line drawn from the effective negative toward the effective positive ionic charge within the element dv₂. The vector quantity Ødv₂ 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 of the muscle mass, at which it is desired to know the value of the potential V due to the distribution of depolarization v₂. Let r₁ and r₂ be radius vectors drawn from O to the points (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂) respectively. Let r be a vector drawn from the latter to the former point so that r = r₁ —r₂. Since the elementary potential dV at (X₁, Y₁, Z₁) due to the elementary distribution dv₂ varies inversely with the square of the distance r and directly with the cosine of the angle (r,⊘), we have dV = r.⊘dv₂/r³. Consequently, where the triple integral is to be taken over the whole of the volume v₂.