Theoretical and experimental derivation of the function relating the size distribution of spherical particles in a volume element to that obtained from images of sections taken through it

Abstract

Summary. The mathematical formulation of three classes of theories con cerning the problem outlined in the title of this study was critically assessed and experimentally evaluated. It was found that only one class of the theories describes the relation between the distribution of particles in a volume element to that observed in sections through this volume in a satisfactory way. The errors in the remaining two classes could be traced and the corrections made after a mathematically rigorous analysis brought these theories to comply with a model which, as we could prove experimentally, deserves a high level of confidence.A statistical analysis of the dimensions of particles contained at random in a volume element from images of sections through the specimen meets with considerable difficulties. It seems intuitively obvious that the function relating the size distribution observed in the projected images to that in the volume element depends on a variable into which enter the orientation of the particles as well as a term associating mean particle diameter with section thickness. If this latter factor is not taken into account, variations in size even of particles not giving the orientation problem as, e.g., of spherical organelles under anisotonic conditions, could be seriously misinterpreted.In this study, the mathematical approach of several authors to this statistical problem was critically appraised and the domain of validity of each of the theories determined experimentally. The fact that the available theoretical treatments can be classified in three groups greatly facilitated this task.The first class of models is based on the assumption that the section thickness be negligible as compared to the diameter of the elements contained in the sample [9,6,4,3]. A second class is represented by the work of Coup]and [2] who recently proposed a theory in which the section thickness is considered finite. Though the thickness is introduced in the probability considerations of cutting a spherical element in a tissue, this theory is regarded incomplete since the thickness has to be introduced also in the probability term relating the apparent to the real