Morphometrical method to estimate the parameters of distribution functions assumed for spherical bodies from measurements on a random section.

Abstract

The general equations to correlate the distribution of the radius r of spheres randomly dispersed in the three-dimensional space with measurements on a random test plane are (see article) and (see article) for the diameter delta of circular sections of spheres; and (see article) and (see article) for the length lambda of chords delivered by intersection of a random test line. In the above expressions Nvo, Nao and Nlambdao are the numbers of spheres in a unit volume, of circles on a unit surface area and of chords per unit length of a test line, respectively; n is O or a positive integer; r the arithmetical mean of r; (deltan) and (lambdan) the means of the n-th powers of delta and lambda, respectively; and Qn a quotient defined by Qn = (rn)/rn. The ratio of measured (delta2/delta2 or (lambda2/lambda2 is used for calculating one of the parameters of assumed theoretical distribution functions. A second parameter is then estimated from delta or lambda. The method was applied to the normal pancreatic islets, and the use of chord length lambda was preferred to that of diameter delta, because the error due to the failure in identifying very small islet sections was minimized in the former.