Analytical description of transmembrane voltage induced by electric fieldson spheroidal cells

An analytical description of transmembrane voltage induced on spherical cel ls was determined in the 1950s, and the tools for numerical assessment of t ransmembrane voltage induced on spheroidal cells were developed in the 1970 s. However, it has often been claimed that an analytical description is una ttainable for spheroidal cells, while others have asserted that even if att ainable, it does not befit the reality due to the nonuniform membrane thick ness, which is unrealistic but inevitable in spheroidal geometry. In this p aper we show that for all spheroidal cells, membrane thickness is irrelevan t to the induced transmembrane voltage under the assumption of a nonconduct ive membrane, which was also applied in the derivation of Schwan's equation . We then derive the analytical description of transmembrane voltage induce d on prolate and oblate spheroidal cells. The final result, which we cast f rom spheroidal into more familiar spherical coordinates, represents a gener alization 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 analyz ing 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.