Estimation of the wear volume after total hip replacement. A simple access to geometrical concepts

Various formulas have been proposed to calculate the volume of prosthetic wear from the penetration depth of the head as assessed on plain radiographs, based on idealized, three-dimensional geometrical models of a prosthetic hip. However, for most published formulas no (or no simple) derivation is available and not all of them are correct. We describe a simple geometrical model that allows for transparent derivation of equations for various components of prosthetic wear volume and compare the calculated volumes with those obtained from published equations. These components are: (1) a right generalized cylinder resulting from a linear shift of a half spherical part of the prosthetic head into the hemispherical cup, (2) an additional wedge that is "cut" from the cup and (3) the wear from an optional additional cylindrical portion of the cup. We emphasize that calculation of a three-dimensional wear volume from linear penetration depth should be based on a geometrical concept that is transparent and simple enough for clinical research, such as the one presented. The incorrect formula of Kabo et al. should be completely abandoned.