Stereology of arbitrary particles. A review of unbiased number and size estimators and the presentation of some new ones, in memory of William R. Thompson

J Microsc. 1986 Jul;143(Pt 1):3-45.


This paper deals with isolated, countable items, often termed particles, in three-dimensional space. Its substance is the unbiased stereological estimation of the number, height, surface and volume of such particles without any assumptions about their shape. The full range of estimators is described, some of them for the first time, some in an improved form, several in more than one version, and all of them under the single, absolute requirement that one can in fact identify what one is quantifying on sections. In terms of the minimal number of sections for the analysis, the estimators may be classified as follows: On a single section it is possible to estimate vV, the mean volume of particles in the volume-weighted or 'sieving'-distribution. On two parallel sections, separated by a known distance, estimators exist of particle number and of all mean sizes (height, surface and volume) in the ordinary number distribution, as well as of SDN(v), the standard deviation in the number distribution of particle volumes. If the containing space is relatively transparent the sections may be two optical sections within one thick physical section. On a stack of parallel sections, at least as high as the largest particle, and separated by known distances, one can get twelve mean sizes and twelve distributions of individual sizes: all combinations of three sizes: height, surface and volume in four different types of distributions: number, height, surface and volume. Fulfilling the sampling requirements of the above two estimation principles it has been shown very recently that by combining them one may even estimate mean sizes and number of arbitrary particles in a stack of sections with constant but unknown separation. Finally, a unique, unbiased estimator of the total number of items in a specimen is described for the use of which one need not measure the distance between sections, nor their thickness, nor the volume of the specimen, nor assume anything about shrinkage/swelling, sectioning compression or lost caps. It is the fractionator.

MeSH terms

  • Animals
  • Mathematics
  • Microscopy, Electron / methods*
  • Models, Biological*
  • Surface Properties