Stereology is a branch of applied mathematics that is the three-dimensional interpretation of two-dimensional cross sections of materials or tissues. It provides practical techniques for extracting quantitative information about a three-dimensional material from measurements made on two-dimensional planar sections of the material. Stereology is a method that utilizes random, systematic sampling to provide unbiased and quantitative data. It is an important and efficient tool in many applications of microscopy (such as petrography, materials science, and biosciences including histology, bone and neuroanatomy). Stereology is a developing science with many important innovations being developed mainly in Europe. New innovations such as the proportionator continue to make important improvements in the efficiency of stereological procedures. In addition to two-dimensional plane sections, stereology also applies to three-dimensional slabs (e.g. 3D microscope images), one-dimensional probes (e.g. needle biopsy), projected images, and other kinds of 'sampling'. It is especially useful when the sample has a lower spatial dimension than the original material. Hence, stereology is often defined as the science of estimating higher-dimensional information from lower-dimensional samples. Stereology is based on fundamental principles of geometry (e.g. Cavalieri's principle) and statistics (mainly survey sampling inference). It is a completely different approach from computed tomography.
Classical examples Classical applications of stereology include:
calculating the volume fraction of quartz in a rock by measuring the area fraction of quartz on a typical polished plane section of rock ("Delesse principle" from Achille Delesse); calculating the surface area of pores per unit volume in a ceramic, by measuring the length of profiles of pore boundary per unit area on a typical plane section of the ceramic (multiplied by 4 / π {\displaystyle 4/\pi } ); calculating the total length of capillaries per unit volume of a biological tissue, by counting the number of profiles of capillaries per unit area on a typical histological section of the tissue (multiplied by 2). Find the parameters such as Bone Volume, Trabecular thickness and trabecular number in a given sample of bone. The popular science fact that the human lungs have a surface area (of gas exchange surface) equivalent to a tennis court (75 square meters), was obtained by stereological methods. Similarly for statements about the total length of nerve fibres, capillaries etc. in the human body.
Errors in spatial interpretation The word Stereology was coined in 1961 and defined as `the spatial interpretation of sections'. This reflects the founders' idea that stereology also offers insights and rules for the qualitative interpretation of sections. Stereologists have helped to detect many fundamental scientific errors arising from the misinterpretation of plane sections. Such errors are surprisingly common. For example:
plane sections of quenched steel contain thin linear streaks of Martensite. For many years this was interpreted as demonstrating that the Martensite inclusions are "needle-like". But if every plane section shows linear profiles, then the Martensite inclusions must be plate-like, rather than needle-like. (Length on sections is related to area in 3D). the internal structure of mammalian liver was misunderstood for 100 years (1848–1948) because of a similar error. a biological tissue containing capillaries is sectioned. Researchers count the number of profiles of capillaries that are visible in a microscope field, and report the "number of capillaries" or "number of capillaries per unit area". This is an error because the number of capillary profiles on a plane section is related to the length of capillaries, not to their number (which may not even be well-defined). (Number in 2D is related to length in 3D). researchers compare plane sections of normal and diseased tissue from an organ. They find that a certain type of cell is seen more frequently in the diseased tissue. They conclude that the disease involves proliferation of these cells. However, the number of cell profiles seen on a section depends both on the number of cells and on their sizes. So it is possible that the disease process simply involves an increase in the size of cells, without any proliferation. (Number in 2D is related to length or height in 3D). the construction of historic Tabby buildings in the Carolinas was assumed to be done with sand obtained from sand pits. Stereological studies demonstrated that the sand was obtained from dunes facing the bays. This has caused the method of construction to be rethought as well as methods of restoration.
Stereology is not tomography Stereology is a completely different enterprise from computed tomography. A computed tomography algorithm effectively reconstructs the complete internal three-dimensional geometry of an object, given a complete set of all plane sections through it (or equivalent X-ray data). On the contrary, stereological techniques require only a few 'representative' plane sections, from which they statistically extrapolate the three-dimensional material. Stereology exploits the fact that some 3-D quantities can be determined without 3-D reconstruction: for example, the 3-D volume of any object can be determined from the 2-D areas of its plane sections, without reconstructing the object. (This means that stereology only works for certain quantities like volume, and not for other quantities).
Sampling principles In addition to using geometrical facts, stereology applies statistical principles to extrapolate three-dimensional shapes from plane section(s) of a material. The statistical principles are the same as those of survey sampling (used to draw inferences about a human population from an opinion poll, etc.). Statisticians regard stereology as a form of sampling theory for spatial populations. To extrapolate from a few plane sections to the three-dimensional material, essentially the sections must be 'typical' or 'representative' of the entire material. There are basically two ways to ensure this:
It is assumed that any plane section is typical (e.g. assume that the material is completely homogeneous); or
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