Forward and Inverse Potential Field Solutions for Cardiac Strands of Cylindrical Geometry

Abstract

This paper deals with the classical forward and inverse volume conductor field problems associated with the isolated active cardiac muscle preparations of cylindrical geometry. Specifically, these are the Purkinje fiber, the atrial trabeculum, and the (idealized) single atrial cell. The electrical behavior of the multicellular preparations (Purkinje strand and atrial trabeculum) is modeled in terms of the electrical activity of an equivalent single cell, with a representative membrane that separates an anisotropic intracellular medium from an isotropic extracellular medium. The isolated single atrial fiber is considered an interesting special case anid is modeled in an idealized sense as a long cylindrical cell with an isotropic internal medium. A model based on potential theory is developed for the equivalent cardiac cell; it is based on a solution of Laplace's equation in the media of interest, subject to appropriate boundary conditions. The solution for potential at an arbitrary point in the extracellular medium is in the form of a Fourier integral; the equation is subsequently reformulated into a more convenient computational form using a discrete Fourier transform (DFT) method. Implementation of this method, using a fast Fourier transform (FFT) technique, results in a fast and efficient numerical algorithm for the calculation of volume conductor potentials. A benefit of this approach is that the classical forward and inverse problems in electrophysiology may be viewed as equivalent filtering problems.