Mark J. West

Introduction to Stereology

Just as astrology became astronomy and alchemy became chemistry through the application of mathematics, descriptive anatomy can be expected to become more and more quantitative in nature. This article describes the basics of stereology, which provides meaningful quantitative descriptions of the geometry of three-dimensional (3D) structures from measurements that are made on two-dimensional (2D) images. With precise mathematical descriptions such as those that can be obtained with unbiased stereological techniques, it will be possible to make concise descriptions of the relationships between structure and function, of the dynamics of structure, and to reassert the importance of quantitative morphology as an essential part of the evaluation of biological tissues.

Estimating Volume in Biological Structures

Estimates of volume can be useful in comparative and experimental studies of tissues and organs. In this article, a simple procedure for making unbiased estimates of total volume using Cavalieris principle is described. This is followed by a description of how point probes can be used to make unbiased estimates of the areas of sectional profiles and how, when combined with Cavalieris principle, point counting can be used to make an unbiased estimate of the volume of a structural feature.

The Precision of Estimates in Stereological Analyses

When quantifying the total volume, number, length, or surface of morphological features in complex structures such as the brain, it is neither desirable nor feasible to make absolute determinations of these parameters. It would take a lifetime to count all of the neurons or measure the total length of dendrites in many parts of the brain. Estimates based on a relatively small number of samples of even the most complex structures will suffice, if the estimates are unbiased and have a sufficient amount of precision. It is important to have an understanding of what the precision of an estimate actually means and how it can be calculated and altered, to optimize studies with regard to the number of individuals, sections, and probes used in an analysis. An optimal sampling scheme is one that involves a minimum of sampling but still provides enough precision to draw a conclusion with a predetermined level of confidence.

Optimizing the Sampling Scheme for a Stereological Study: How Many Individuals, Sections, and Probes Should Be Used

Stereology provides meaningful quantitative descriptions of the geometry of three-dimensional (3D) structures from measurements that are made on two-dimensional (2D) images. A pilot stereological study will provide information that can be used to rationalize how many individuals, sections, and probes should be used to ensure that one is sampling enough, but not too much, to achieve the goal of a study. This general approach is exemplified in the thought experiment described here, which involves a comparison of the means of estimates of the total number of neurons N in two groups, using the simplest of statistical tests, the Students t-test. It is also applicable to studies involving other estimates of total quantities such as volume, surface, and length obtained from a parallel series of sections.

Getting Started in Stereology

Stereology involves sampling structural features in sections of tissue with geometrical probes. This article discusses some practical issues that must be dealt with when getting started in stereology, including tissue preparation methods and determining how many tissue sections and probes are needed to make a stereological estimate.

Tissue shrinkage and stereological studies.

Shrinkage often takes place in biological tissues during the different phases of preparation for microscopy. This can have detrimental effects on the stereological estimates, even when unbiased procedures are used. There are different types of shrinkage, and an awareness of them is essential when designing stereological studies. Some forms of shrinkage can be taken into account to ensure the unbiasedness of an estimator, but some cannot and should be avoided. Dimensional changes that take place during fixation, embedding, sectioning, mounting, and staining can seriously compromise ones ability to make assumption-free estimates of length and surface, but there are steps that can be taken to reduce the impact of these changes on estimates of object number and size. This article describes types of shrinkage and the effects of shrinkage on estimators of object number. It gives an example of how to make a number-weighted correction of section thickness and also discusses the consequences of shrinkage for the validity of estimates of object size.

Estimating Object Number in Biological Structures

The number of cells and subcellular structures can often be readily related to quantitative evaluations of organ and tissue function. Neurons and synapses, for example, are directly involved in the integration and transfer of information in neural systems. Their numbers are consequently important parameters in the evaluations of the functional capacity of neural systems. Only information regarding the total number of objects, such as synapses and neurons, can be used to draw conclusions regarding changes or differences in the number of these structural entities. The large numbers of neurons and synapses in the vast majority of neural systems preclude absolute determinations of their total number, that is, counting each and every neuron or synapse. However, estimates or approximations based on limited sampling can be useful if the estimates are unbiased and if the individual estimates have an acceptable amount of precision. This article discusses the estimation of object number, including sampling, indirect and direct counting techniques, sources and types of bias, and the disector counting technique. An example is also given.

Estimating length in biological structures.

Length estimates of particular features of biological tissues can be useful in evaluating function. Such estimates have been notoriously difficult to obtain because of the requirement for an isotropic interaction between the area probes and the linear features of cells and tissues, which are most likely not isotropically oriented. For complex embedded structures, such as subdivisions of the brain, the turning of the tissue before sectioning that is needed to ensure an isotropic interaction has made it difficult to delineate many regions of interest and limited the number of unbiased stereological studies of length. The recent development of a virtual isotropic spherical probe, the spaceball, makes it relatively easy for the isotropic interaction between probe and structure to be realized. This article describes the use of the spaceball probe to estimate length, and gives an example of estimating total capillary length in CA1 stratum radiatum of the human hippocampus.

Systematic versus Random Sampling in Stereological Studies

The sampling that takes place at all levels of an experimental design must be random if the estimate is to be unbiased in a statistical sense. There are two fundamental ways by which one can make a random sample of the sections and positions to be probed on the sections. Using a card-sampling analogy, one can pick any card at all out of a deck of cards. This is referred to as independent random sampling because the sampling of any one card is made without reference to the position of the other cards. The other approach to obtaining a random sample would be to pick a card within a set number of cards and others at equal intervals within the deck. Systematic sampling along one axis of many biological structures is more efficient than random sampling, because most biological structures are not randomly organized. This article discusses the merits of systematic versus random sampling in stereological studies.

Isotropy, iSectors, and Vertical Sections in Stereology

Stereology provides meaningful quantitative descriptions of the geometry of three-dimensional (3D) structures from measurements that are made on two-dimensional (2D) images. One important consideration when designing such studies is that both length and surface features may have preferred orientations in 3D space; that is, they may not be isotropic. To fully understand the global estimators of length L and surface area S, it is essential that one understand the problems that the inherent anisotropy in the structure of most living organisms generates when designing unbiased sampling schemes for estimating length and surface. These same issues also apply to the use of local estimators of object volume v and surface s, in that they use line and surface probes. These problems and the various solutions to them, which involve sectioning and probing tissue in particular ways, are presented in this article.