Simplified equations for the transmembrane potential induced in ellipsoidal cells of rotational symmetry

Abstract

The induction of transmembrane potentials (Δϕ) by an external field is the basis of numerous applications in biotechnology, cell-technology and medicine. We have developed new, simplified analytical equations that avoid the complicated description by the depolarizing factors. The equations apply to the Δϕ induced in cells resembling ellipsoids of rotation, i.e. spheroids by homogeneous dc or ac fields. They will be especially useful for experimental scientists. The equations describe the dependence of the Δϕ on the electric media properties, the field frequency and the axis ratio for oblate and prolate spheroids for which the symmetry semiaxis (c) is shorter and longer than the other two semiaxes (a and b, with a = b), respectively. According to the Schwan equation, an electric field E may induce a maximum Δϕ of 1.5aE at the poles of a spherical cell. For the poles of spheroidal cells, the maxima can be easily described by Δϕ = (a + 2c)E/2 and Δϕ = a(a + 2c)E/(a + c) for fields oriented along and perpendicular to the symmetry axis, respectively. For practically important shapes the error in the magnitude of Δϕ is smaller than 5% except along the c-axis for axis ratios larger than 2. Nevertheless, the errors vanish for the three limiting shapes of infinitely thin disc, sphere and cylinder.