A series expression for the surface area of an ellipsoid and its application to the computation of the surface area of avian erythrocytes

Knowledge of the surface area of cells is necessary for biophysical studies in which the permeability coefficient of a cell-type for a solute is defined. Under the microscope, avian erythrocytes have the appearance of an oval discus usually with a central elongated nucleus. The dimensions of these cells have been obtained for a large range of species over the past century. However, estimates of the surface areas have been obtained using mathematically simplistic models such as flattened circular cylinders. We modelled red cells of the domestic chicken (Gallus domesticus) as ellipsoids using the previously published dimensions to obtain the three semi-axis lengths. Although the mathematical expression for the volume of an ellipsoid is well known, an expression for the surface area is not. We present a general expression for the surface area of an ellipsoid that is a power series involving elliptic integrals and functions; the latter being rapidly evaluated by computer using standard series expressions. Our estimates of area are compared with earlier ones and those obtained by numerical integration of the surface.