Hans Jørgen G. Gundersen

SOME NEW, SIMPLE AND EFFICIENT STEREOLOGICAL METHODS AND THEIR USE IN PATHOLOGICAL RESEARCH AND DIAGNOSIS

Stereology is a set of simple and efficient methods for quantitation of three‐dimensional microscopic structures which is specifically tuned to provide reliable data from sections. Within the last few years, a number of new methods has been developed which are of special interest to pathologists. Methods for estimating the volume, surface area and length of any structure are described in this review. The principles on which stereology is based and the necessary sampling procedures are described and illustrated with examples. The necessary equipment, the measurements, and the calculations are invariably simple and easy.

The new stereological tools: Disector, fractionator, nucleator and point sampled intercepts and their use in pathological research and diagnosis

The new stereological methods for correct and efficient sampling and sizing of cells and other particles are reviewed. There is a hierarchy of methods starting from the simplest where even the microscopic magnification may be unknown to the most complex where typically both section thickness and the magnification must be known. Opticalsections in suitably modified microscopes can be used to improve the ease and speed with which even the most demanding of these methods are performed. The methods are illustrated by practical examples of applications to a wide range of histological entities including synapses, neurons and cancer cells, glomerular corpuscles and ovarian follicles.

Notes on the estimation of the numerical density of arbitrary profiles: the edge effect

A description is given of a family of test‐frames for obtaining an unbiased estimate of the numerical density of arbitrary profiles on a section. The counting rule pertaining to the test‐frame is simple and requires no corrections based on other estimated quantities.

Estimation of surface area from vertical sections

‘Vertical’ sections are plane sections longitudinal to a fixed (but arbitrary) axial direction. Examples are sections of a cylinder parallel to the central axis; and sections of a flat slab normal to the plane of the slab. Vertical sections of any object can be generated by placing the object on a table and taking sections perpendicular to the plane of the table.

Neocortical neuron number in humans: Effect of sex and age

Modern stereological methods provide precise and reliable estimates of the number of neurons in specific regions of the brain. We decided to estimate the total number of neocortical neurons in the normal human brain and to analyze it with respect to the major macro‐ and microscopical structural components, to study the internal relationships of these components, and to quantitate the influence of important physiological variables on brain structure. The 94 brains reported represent a consecutive collection of brains from the general Danish population. The average numbers of neocortical neurons were 19 billion in female brains and 23 billion in male brains, a 16% difference. In our study, which covered the age range from 20 years to 90 years, approximately 10% of all neocortical neurons are lost over the life span in both sexes. Sex and age were the main determinants of the total number of neurons in the human neocortex, whereas body size, per se, had no influence on neuron number. Some of the data presented have been analyzed by using new mathematical designs. An equation predicting the total neocortical neuron number in any individual in which sex and age are known is provided. J. Comp. Neurol. 384:312‐320, 1997.

Quantification of connectivity in cancellous bone, with special emphasis on 3-D reconstructions.

The connectivity of cancellous bone attracts increasing attention, as it has been hypothesized that a primary reason for decreasing strength and stiffness in osteoporosis is caused by a loss of trabecular elements and consequently a loss in connectivity. The Euler characteristic has in a few previous articles been used to express cancellous bone connectivity, but there are severe problems in using the Euler characteristic uncritically. The Euler characteristic of a three-dimensional structure is a topological invariant, which reports the number of particles of a structure plus the number of enclosed cavities minus the connectivity. As such, one must know the number of components and the number of enclosed cavities in order to use the Euler characteristic as an expression of connectivity. Another difficulty of using the Euler characteristic is that due to edge effects the Euler characteristic of an excised specimen provides a biased estimate of the Euler characteristic of the region from which the specimen was taken. In this article the intuitive concept of connectivity is given a precise mathematical definition, and a basic topological method for quantifying the connectivity of cancellous bone is presented. The method uses the Euler characteristic, but the above-mentioned problems are controlled. The development of the method and the practical implementation is based on a set of topological notes. It must be stressed that the method is free from assumptions concerning trabecular architecture, and that the method is unbiased. This is in contrast to previously presented methods. The unbiased and model-free method is used on a series of 3-D reconstructions of cancellous bone specimens, and it is demonstrated that the connectivity of cancellous bone is not simply related to volume fraction (density), and that biased and model-based 2-D methods aimed at determining connectivity do not have any general relationship to connectivity in cancellous bone.

Optimizing sampling efficiency of stereological studies in biology: or ‘Do more less well!‘

The aim of the sampling design for stereology is to obtain the maximal amount of quantitative structural information at a given total cost or effort. Principles of such optimal designs are discussed and methods for generating them are illustrated by a biological example. In general, the variation between different individuals—the biological variation—is the major determinant of overall efficiency, whereas the variation between single microscopic features is unimportant. It follows that the expenditure of time and/or money in order to increase the precision of the individual measurements is irrational in almost all studies where the emphasis is on the biological results.

The impact of recent stereological advances on quantitative studies of the nervous system

The usefulness of a number of new stereological principles for unbiased estimation of particle number and sizes and sampling of particles is illustrated together with a novel principle for unbiased estimation of anisotropic surfaces. The examples include descriptions of estimators of neurone and synapse number and sizes, synapse gradients, neurone point patterns in three-dimensional space, capillary surface area, and perikarya volumes. A major advantage of the methods is the possibility to carry out the estimation procedures in specified, well-characterized regions or layers of the brain. Some general statistical and stereological problems are briefly discussed.

Tissue shrinkage and unbiased stereological estimation of particle number and size

This paper is a review of the stereological problems related to the unbiased estimation of particle number and size when tissue deformation is present. The deformation may occur during the histological processing of the tissue. It is especially noted that the widely used optical disector may be biased by dimensional changes in the z‐axis, i.e. the direction perpendicular to the section plane. This is often the case when frozen sections or vibratome sections are used for the stereological measurements. The present paper introduces new estimators to be used in optical fractionator and optical disector designs; the first is, as usual, the simplest and most robust. Finally, it is stated that when tissue deformation only occurs in the z‐direction, unbiased estimation of particle size with several estimators is possible.

Determination of membrane thickness distribution from orthogonal intercepts

This report deals with the reconstruction of the distribution of membrane thickness T from that of orthogonal intercept length L0, measured in random section planes. In such planes the membrane appears as a band and the linear distance from one of its boundaries perpendicular to the opposite one is the length of the orthogonal intercept. Using a membrane model, an integral equation relating the probability density functions of orthogonal intercept length f(l0) and membrane thickness g(τ) is derived. Relations between moments are derived and the analytic solution to the problem of reconstructing g(τ) from f(l0) is given. The parametric approach by which it is assumed that g(τ) has some known analytic form with unknown parameters is considered, and the use of a suggested analytic form for describing the thickness distribution of the human glomerular basement membrane is discussed.