Analytical description of transmembrane voltage induced by electric fields on spheroidal cells

An analytical description of transmembrane voltage induced on spherical cells was determined in the 1950s, and the tools for numerical assessment of transmembrane voltage induced on spheroidal cells were developed in the 1970s. However, it has often been claimed that an analytical description is unattainable for spheroidal cells, while others have asserted that even if attainable, it does not befit the reality due to the nonuniform membrane thickness, which is unrealistic but inevitable in spheroidal geometry. In this paper we show that for all spheroidal cells, membrane thickness is irrelevant to the induced transmembrane voltage under the assumption of a nonconductive membrane, which was also applied in the derivation of Schwan's equation. We then derive the analytical description of transmembrane voltage induced on prolate and oblate spheroidal cells. The final result, which we cast from spheroidal into more familiar spherical coordinates, represents a generalization of Schwan's equation to all spheroidal cells (of which spherical cells are a special case). The obtained expression is easy to apply, and we give a simple example of such application. We conclude the study by analyzing the variation of induced transmembrane voltage as a spheroidal cell is stretched by the field, performing one study at a constant membrane surface area, and another at a constant cell volume.