Stereology


Stereology 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. 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, one-dimensional probes, 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 and statistics. It is a completely different approach from computed tomography.

Classical examples

Classical applications of stereology include:
The popular science fact that the human lungs have a surface area equivalent to a tennis court, 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:
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.
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..

Sampling principles

In addition to using geometrical facts, stereology applies statistical principles to extrapolate three-dimensional shapes from plane section of a material. The statistical principles are the same as those of survey sampling.
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:
or
The first approach is the one that was used in classical stereology.
Extrapolation from the sample to the 3-D material depends on the assumption that the material is homogeneous. This effectively postulates a statistical model of the material. This method of sampling is referred to as model-based sampling inference.
The second approach is the one typically used in modern stereology.
Instead of relying on model assumptions about the three-dimensional material, we take our sample of plane sections by following a randomized sampling design, for example, choosing a random position at which to start cutting the material. Extrapolation from the sample to the 3-D material is valid because of the randomness of the sampling design, so this is called design-based sampling inference.
Design-based stereological methods can be applied to materials which are inhomogeneous or cannot be assumed to be homogeneous. These methods have gained increasing popularity in the biomedical sciences, especially in lung-, kidney-, bone-, cancer- and neuro-science. Many of these applications are directed toward determining the number of elements in a particular structure, e.g. the total number of neurons in the brain.

Geometrical models

Many classical stereological techniques, in addition to assuming homogeneity, also involved mathematical modeling of the geometry of the structures under investigation.
These methods are still popular in materials science, metallurgy and petrology where shapes of e.g. crystals may be modelled as simple geometrical objects. Such geometrical models make it possible to extract additional information. However, they are extremely sensitive to departures from the assumptions.

Total quantities

In the classical examples listed above, the target quantities were relative densities: volume fraction, surface area per unit volume, and length per unit volume. Often we are more interested in total quantities such as the total surface area of the lung's gas exchange surface, or the total length of capillaries in the brain. relative densities are also problematic because, unless the material is homogeneous, they depend on the unambiguous definition of the reference volume.
Sampling principles also make it possible to estimate total quantities such as the total surface area of lung. Using techniques such as systematic sampling and cluster sampling we can effectively sample a fixed fraction of the entire material. This allows us to extrapolate from the sample to the entire material, to obtain estimates of total quantities such as the absolute surface area of lung and the absolute number of cells in the brain.

Timeline

The primary scientific journals for stereology are Journal of Microscopy and Image Analysis & Stereology.