H. J. G. Gundersen

The efficiency of systematic sampling in stereology and its prediction

The superior efficiency of systematic sampling at all levels in stereological studies is emphasized and various commonly used ways of implementing it are briefly described. Summarizing recent theoretical and experimental studies a set of very simple estimators of efficiency are presented and illustrated with a variety of biological examples. In particular, a nomogram for predicting the necessary number of points when performing point counting is provided. The very efficient and simple unbiased estimator of the volume of an arbitrary object based on Cavalieris principle is dealt with in some detail. The efficiency of the systematic fractionating of an object is also illustrated.

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.

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:

The efficiency of systematic sampling in stereology — reconsidered

In the present paper, we summarize and further develop recent research in the estimation of the variance of stereological estimators based on systematic sampling. In particular, it is emphasized that the relevant estimation procedure depends on the sampling density. The validity of the variance estimation is examined in a collection of data sets, obtained by systematic sampling. Practical recommendations are also provided in a separate section.

Stereological estimation of the volume‐weighted mean volume of arbitrary particles observed on random sections*

A stereological estimator of the weighted mean volume of particles of arbitrary shape is described. This unbiased estimator is based on simple point‐sampling of linear intercept lengths. The complete absence of shape assumptions effectively breaks the long‐standing ‘convexity‐barrier’: the only requirement here is that individual particles can be unambiguously identified by their profiles on random sections. Practical details of the simple estimation procedure and an example with very irregular particles are reported. Finally, an estimator of the variance of the weighted distribution of particle volume is discussed. This estimator is also valid for particles of arbitrary shape. For any mixture of ellipsoids (spheres, oblates, prolates and triaxial ellipsoids) the estimator is reduced to a simple function of measurements of diameters in the section plane.

The total number of neurons in the human neocortex unbiasedly estimated using optical disectors

An efficient method is presented for obtaining, in under 4h, an unbiased estimate of the total number of neurons in the human neocortex, with a coefficient of error on the estimate of ∼ 5%. The novel sampling scheme used in this study is unbiased and was designed so that only a small amount of neocortical grey matter had to be removed. Hence, the majority of the cerebral grey matter and all the internal grey matter was left intact for further resampling and analysis. Each cerebral hemisphere was subdivided into the four major neocortical regions, sliced coronally at 7‐mm intervals and the volume of the neocortex determined using Cavalieris principle. Uniform sampling of neocortex was performed in the hemisphere followed by regional subsampling with a varying sampling fraction being taken from each region. Neuronal density estimates were made in thick plastic sections using optical disectors. Shrinkage estimates were made in parallel with the number estimates and found to be negligible. The total number of neocortical neurons in the right hemisphere of five normal 80‐year‐old men was found to be 13·7 × 109 with an inter‐individual coefficient of variation of 12%.

The smooth fractionator

A modification of the general fractionator sampling technique called the smooth fractionator is presented. It may be used in almost every situation in which sampling is performed from distinct items that are uniquely defined, often they are physically separated items or clusters like pieces, blocks, slabs, sections, etc. To each item is associated a ‘guesstimate’ or an associated variable with a more‐or‐less close – and possibly biased − relationship to the content of the item. The smooth fractionator is systematic sampling among the items arranged according to the guesstimates in a unique, symmetric sequence with one peak and minimal jumps. The smooth fractionator is both very simple to implement and so efficient that it should probably always be used unless the natural sequence of the sampling items is equally smooth.

Estimation of structural anisotropy based on volume orientation: a new concept

The quantification of anisotropy—its main direction and the degree of dispersion around it—is desirable in numerous research fields dealing with physical structures. Conventional methods are based on the orientation of interface elements. The results of these methods do not always agree with perceived anisotropy, and anisotropic structures do not necessarily turn out to be ‘anisotropic’ using these methods. In the present paper, we propose an alternative to curve and surface orientation, namely volume orientation. Using trabecular bone as an example of a two‐phase anisotropic structure, the new concept is studied in some detail. In particular, a parametric method of estimating volume orientation from sections is presented and discussed.

THE OPTICAL ROTATOR

The optical rotator is an unbiased, local stereological principle for estimation of cell volume and cell surface area in thick, transparent slabs. The underlying principle was first described in 1993 by Kiêu &38; Jensen ( J. Microsc170, 45–51) who also derived an estimator of length. In this study we further discuss the methods derived from this principle and present two new local volume estimators.

Efficient estimation of cell volume and number using the nucleator and the disector

The nucleator allows the unbiased estimation of absolute structural quantities of suitably sampled, arbitrarily shaped structures from observations made from arbitrary points using isotropic probes. A number of time‐saving modifications using the nucleator and the consequences of the modifications are studied in terms of their bias and efficiency. Using rat neocortex as an example, a description is given of how to estimate mean neuronal volume and total neuron number efficiently from only a few pairs of plastic sections with a thickness of about 3 μm.

Particle sizes and their distributions estimated from line- and point-sampled intercepts. Including graphical unfolding

Information about particle size is currently obtained almost exclusively by the use of stereological methods which lead to estimates of the number distribution of linear particle size. The main point of this presentation is to stress the freedom to choose more appropriate parameters for size among a host of options, including particle surface area and volume. Moreover, particle size information may often be considered advantageously in terms of particle distributions based on structural characteristics rather than number distributions. Some of these other distribution types are correctly represented in samples of intercept lengths obtained by line‐ and point‐sampling, respectively. The known and quite simple theory of sampling intercepts is summarized and developed further in several different directions, including a derivation of the distribution of intercept length in ellipsoids, graphical unfolding procedures, and mean size estimators. The potential of the approach is illustrated—but not exhausted—by the existence of a general mean size estimator based on minimal assumptions regarding particle shape.