Analytical description of the transmembrane voltage induced on arbitrarily oriented ellipsoidal and cylindrical cells

We present an analytical equation for the transmembrane voltage (Deltaphi) induced by a homogeneous AC field on arbitrarily oriented cells of the general ellipsoidal shape. The equation generalizes the Schwan equation for spherical cells and describes the dependence of Deltaphi on field frequency, cell size and shape, membrane capacitance, conductivities of cytoplasm, membrane and external medium, the location of the membrane site under consideration, and on the orientation of the cell with respect to the field. The derivation is based on the fact that the cytoplasm and the Maxwellian equivalent body of the whole cell are both of a general ellipsoidal shape and must thus exhibit constant local fields. The constant fields allow for a relatively simple description of the potentials on the internal and external membrane sides, leading to Deltaphi. For this, the properties of cytoplasm, membrane, and external medium have been introduced into a special, finite element model. We found that Deltaphi can be unambiguously defined for non-spherical cells, provided that the membrane thickness is thin in comparison to the cell dimensions.