On random sectioning of geometrically-similar bodies for morphological applications

In 1962 Weibel and Gomez published their method for estimating quantities such as the number and volume of identical biological structures by raising to the three-halves power the mean of areas measured from random cross-sections. Subsequently, Knight, Weibel and Gomez (1963), modified the formula for use when the bodies were not of the same size but remained geometrically similar. This present paper uses computer simulation to investigate the efficiencies of eight unbiased estimators for the volume of a sphere and shows that the WG formula is inferior to one based on the mean cube of transected lengths. Analysis further shows that the KWG expression cannot properly be used with varying body-size but may be recast to give Wicksell's (1925) formulation, involving the mean square of the intersected length divided by its harmonic mean. A Monte Carlo investigation for spheres with normally-distributed radii shows that, though now biased, the optimum formula for the constant size situation continues giving generally better results than the unbiased but less efficient Wicksell formula provided that the coefficient of variation is less than about 10%. A formula to estimate this coefficient is given. The results suggest the desirability of computer simulation before undertaking actual experiments.