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 3-dimensional space with measurements on a random test plane are Nvo = Nao/2.hivin.r and .**GRAPHIC**. for the diameter .delta. of circular sections spheres; and Nvo = N.lambda.o/.pi..hivin.r2Q2 and .**GRAPHIC**. for the length .lambda. of chords delivered by intersection of a random test line, where Nvo, Nao and N.lambda.o 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 0 or a positive integer; .hivin.r the arithmetical mean of r; (.delta.n) and (.lambda.n) the means of the n-th powers of .delta. and .lambda., respectively; and Qn a quotient defined by Qn = (rn)/.hivin.rn. The ratio of measured (.delta.2)/.hivin..delta.2 or (.lambda.2)/.hivin..lambda.2 is used for calculating 1 of the parameters of assumed theoretical distribution functions. A 2nd parameter is then estimated from .hivin..delta. or .hivin..lambda.. The method was applied to the normal [human] 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.