The response of a spherical heart to a uniform electric field: a bidomain analysis of cardiac stimulation

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Abstract
A mathematical model describing electrical stimulation of the heart is developed, in which a uniform electric field is applied to a spherical shell of cardiac tissue. The electrical properties of the tissue are characterized using the bidomain model. Analytical expressions for the induced transmembrane potential are derived for the cases of equal anisotropy ratios in the intracellular and interstitial (extracellular) spaces, and no transverse coupling between fibers. Numerical calculations of the transmembrane potential are also performed using realistic electrical conductivities. The model illustrates several mechanisms for polarization of the cell membrane, which can be divided into two categories, depending on if they polarize fibers at the heart surface only or if they polarize fibers both at the surface and within the bulk of the tissue. The latter mechanisms can be classified further according to whether they originate from continuous or discrete properties of cardiac tissue. If cardiac tissue had equal anisotropy ratios, a large membrane polarization would be induced at the heart surface that would become negligible a few length constants into the tissue. If cardiac tissue were continuous and had no transverse coupling between fibers, a membrane polarization would be induced throughout the bulk that would arise from an "activating function" similar to the one used to describe neural stimulation. Polarization would occur if the fibers were curving, if the cross-sectional area of the tissue were changing (fiber branching), or both. The numerically calculated transmembrane potential is intermediate between those predicted using the assumptions of equal anisotropy ratios and no transverse coupling between fibers. Although discrete properties of cardiac tissue are not incorporated into this model, an estimate of their effect indicates that the amplitude of the polarization caused by the resistance of the cellular junctions is similar to that caused by fiber curvature and branching. The spatial distribution of the polarization, however, is quite different.