Stereology
Encyclopedia
Stereology was originally defined as "the spatial interpretation of sections". It is an interdisciplinary field that is largely concerned with the three-dimensional interpretation of planar 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 (see examples below). 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 dimension
al 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
.
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.
Stereologists have helped to
detect many fundamental scientific errors arising from the misinterpretation of plane sections. Such errors are
surprisingly common. For example:
.
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).
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.
In order 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:
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.
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 (including numbers of crystals). However, they are extremely sensitive to departures from the assumptions.
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 (without the need to delineate a reference volume). 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.
discovers connections between geometry
and probability, that ultimately lay the foundations
for stereology.
1843 Mining geologist A.E. Delesse
invents the
first practical technique (Delesse's principle)
for determining volume fraction in 3D from area fraction on sections
1885 mathematician M.W. Crofton
publishes theory
of `geometrical probability' including stereological methods.
1895 first known description of a correct method
for counting cells in microscopy.
1898 geologist A. Rosiwal explains how to determine
volume fraction from length fraction on linear transects
1916 S.J. Shand builds the first integrating linear accumulator to
automate stereological work
1919 committee of ASTM (American Society for Testing and
Materials) established to standardise measurement of grain size.
1923 statistician S.D. Wicksell formulates the general problem of
particle size - inferring the distribution of sizes of 3-D particles
from the observed distribution of sizes of their 2-D profiles - and
solves it for spherical particles.
1929 mathematician H. Steinhaus develops stereological principles
for measuring length of curves in 2D.
1930-33 geologists A.A. Glagolev and E. Thomson independently
propose techniques for determining volume fraction from the
ratio of counts of test points.
1940's cancer researcher H. Chalkley publishes methods for
determining surface area from plane sections.
1944 mathematician P.A.P. Moran describes a method for measuring
the surface area of a convex object from the area of projected images.
1946 anatomist Abercrombie shows that many current methods for
counting cells are erroneous, and proposes a correct method
1946–58 materials scientist S.A. Saltykov publishes methods for
determining surface area and length from plane sections.
1948 biologist H. Elias uncovers a one-hundred-year-old
misunderstanding of the structure of mammalian liver.
1952 Tomkeieff and Campbell calculate the internal surface area
of a human lung.
1961 word 'stereology' coined.
Foundation of the International Society of Stereology
1961 materials scientists Rhines and De Hoff develop a method
for estimating the number of objects e.g. grains, particles, cells
of convex shape.
1966 Weibel and Elias calculate the efficiency of
stereological sampling techniques.
1972 E. Underwood describes stereological techniques for
projected images.
1975-80 statisticians R.E. Miles and P.J. Davy show that
stereology can be formulated as a survey sampling technique,
and develop design-based methods.
1983 R.E Miles and (independently) E.B. Jensen and H.J.G. Gundersen
develop point-sampled intercept methods for inferring the
mean volume of arbitrarily-shaped particles from plane sections
1984 D.C Sterio describes the `disector'
counting method
1985 stereologist H. Haug criticises the dogma
that the normal human brain progressively loses neurons
with age. He shows that the existing evidence is invalid.
1985 statistician A. Baddeley introduces the method of
vertical sections.
1986 Gundersen proposes the `fractionator' sampling technique
1988-92 Gundersen and Jensen propose the `nucleator'
and `rotator' techniques for estimating particle volume.
1998 Kubinova introduces the first virtual probe that estimates
surface area in preferential slices.
1999 Larsen and Gundersen introduce global spatial sampling
for estimation of total length in preferential slices.
2002-05 Mouton and Gokhale introduce "space balls" and "virtual cycloid"
for estimation of total length and total surface area, respectively,
in preferential slices.
2008 Gundersen, Gardi, Nyengaard introduce the proportionator
the most efficient stereological method known.
The primary scientific journals for stereology are "Journal of Microscopy" and "Image Analysis & Stereology" (ex Acta Stereologica). The timeline for stereology-related book titles, chapters, and editorials:
Cross section (geometry)
In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc...
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 (see examples below). 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
Microscopy
Microscopy is the technical field of using microscopes to view samples and objects that cannot be seen with the unaided eye...
(such as petrography
Petrography
Petrography is a branch of petrology that focuses on detailed descriptions of rocks. Someone who studies petrography is called a petrographer. The mineral content and the textural relationships within the rock are described in detail. Petrographic descriptions start with the field notes at the...
, materials science
Materials science
Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. This scientific field investigates the relationship between the structure of materials at atomic or molecular scales and their macroscopic properties. It incorporates...
, and biosciences including histology
Histology
Histology is the study of the microscopic anatomy of cells and tissues of plants and animals. It is performed by examining cells and tissues commonly by sectioning and staining; followed by examination under a light microscope or electron microscope...
, bone
Bone
Bones are rigid organs that constitute part of the endoskeleton of vertebrates. They support, and protect the various organs of the body, produce red and white blood cells and store minerals. Bone tissue is a type of dense connective tissue...
and neuroanatomy
Neuroanatomy
Neuroanatomy is the study of the anatomy and organization of the nervous system. In contrast to animals with radial symmetry, whose nervous system consists of a distributed network of cells, animals with bilateral symmetry have segregated, defined nervous systems, and thus we can begin to speak of...
). Stereology is a developing science with many important innovations being developed mainly in Europe. New innovations such as the proportionator
Proportionator
The proportionator is the most efficient unbiased stereological method used to estimate population size. A typical application is counting the number of cells in an organ. The proportionator is related to the optical fractionator and physical disector methods that also estimate population...
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 dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al information from lower dimensional samples.
Stereology is based on fundamental principles of geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
(e.g. Cavalieri's principle
Cavalieri's principle
In geometry, Cavalieri's principle, sometimes called the method of indivisibles, named after Bonaventura Cavalieri, is as follows:* 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane...
) and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
(mainly survey sampling
Survey sampling
In statistics, survey sampling describes the process of selecting a sample of elements from a target population in order to conduct a survey.A survey may refer to many different types or techniques of observation, but in the context of survey sampling it most often involves a questionnaire used to...
inference). It is a completely different approach from computed tomography
Computed tomography
X-ray computed tomography or Computer tomography , is a medical imaging method employing tomography created by computer processing...
.
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");
- 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
); - 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).
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 tomographyComputed tomography
X-ray computed tomography or Computer tomography , is a medical imaging method employing tomography created by computer processing...
.
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 statisticalStatistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
principles to extrapolate three-dimensional shapes from plane section(s) of a material. The statistical principles are the same as those of survey sampling
Survey sampling
In statistics, survey sampling describes the process of selecting a sample of elements from a target population in order to conduct a survey.A survey may refer to many different types or techniques of observation, but in the context of survey sampling it most often involves a questionnaire used to...
(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.
In order 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:
- We assume that any plane section is typical (e.g. assume that the material is completely homogeneous); or
- We select plane sections at random, according to a specified random sampling protocol
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 (including numbers of crystals). 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
Systematic sampling
Systematic sampling is a statistical method involving the selection of elements from an ordered sampling frame. The most common form of systematic sampling is an equal-probability method, in which every kth element in the frame is selected, where k, the sampling interval , is calculated as:k =...
and cluster sampling
Cluster sampling
Cluster Sampling is a sampling technique used when "natural" groupings are evident in a statistical population. It is often used in marketing research. In this technique, the total population is divided into these groups and a sample of the groups is selected. Then the required information is...
we can effectively sample a fixed fraction of the entire material (without the need to delineate a reference volume). 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.
Stereology timeline
1733 G. BuffonGeorges-Louis Leclerc, Comte de Buffon
Georges-Louis Leclerc, Comte de Buffon was a French naturalist, mathematician, cosmologist, and encyclopedic author.His works influenced the next two generations of naturalists, including Jean-Baptiste Lamarck and Georges Cuvier...
discovers connections between geometry
and probability, that ultimately lay the foundations
for stereology.
1843 Mining geologist A.E. Delesse
Achille Ernest Oscar Joseph Delesse
Achille Ernest Oscar Joseph Delesse was a French geologist and mineralogist.He was born at Metz. At the age of twenty he entered the Ecole Polytechnique, and subsequently passed through the Ecole des Mines...
invents the
first practical technique (Delesse's principle)
for determining volume fraction in 3D from area fraction on sections
1885 mathematician M.W. Crofton
Morgan Crofton
Morgan Crofton was a mathematician who contributed to the field of geometric probability theory. He also worked with James Joseph Sylvester and contributed an article on probability to the 9th edition of the Encyclopædia Britannica...
publishes theory
of `geometrical probability' including stereological methods.
1895 first known description of a correct method
for counting cells in microscopy.
1898 geologist A. Rosiwal explains how to determine
volume fraction from length fraction on linear transects
1916 S.J. Shand builds the first integrating linear accumulator to
automate stereological work
1919 committee of ASTM (American Society for Testing and
Materials) established to standardise measurement of grain size.
1923 statistician S.D. Wicksell formulates the general problem of
particle size - inferring the distribution of sizes of 3-D particles
from the observed distribution of sizes of their 2-D profiles - and
solves it for spherical particles.
1929 mathematician H. Steinhaus develops stereological principles
for measuring length of curves in 2D.
1930-33 geologists A.A. Glagolev and E. Thomson independently
propose techniques for determining volume fraction from the
ratio of counts of test points.
1940's cancer researcher H. Chalkley publishes methods for
determining surface area from plane sections.
1944 mathematician P.A.P. Moran describes a method for measuring
the surface area of a convex object from the area of projected images.
1946 anatomist Abercrombie shows that many current methods for
counting cells are erroneous, and proposes a correct method
1946–58 materials scientist S.A. Saltykov publishes methods for
determining surface area and length from plane sections.
1948 biologist H. Elias uncovers a one-hundred-year-old
misunderstanding of the structure of mammalian liver.
1952 Tomkeieff and Campbell calculate the internal surface area
of a human lung.
1961 word 'stereology' coined.
Foundation of the International Society of Stereology
1961 materials scientists Rhines and De Hoff develop a method
for estimating the number of objects e.g. grains, particles, cells
of convex shape.
1966 Weibel and Elias calculate the efficiency of
stereological sampling techniques.
1972 E. Underwood describes stereological techniques for
projected images.
1975-80 statisticians R.E. Miles and P.J. Davy show that
stereology can be formulated as a survey sampling technique,
and develop design-based methods.
1983 R.E Miles and (independently) E.B. Jensen and H.J.G. Gundersen
develop point-sampled intercept methods for inferring the
mean volume of arbitrarily-shaped particles from plane sections
1984 D.C Sterio describes the `disector'
counting method
1985 stereologist H. Haug criticises the dogma
that the normal human brain progressively loses neurons
with age. He shows that the existing evidence is invalid.
1985 statistician A. Baddeley introduces the method of
vertical sections.
1986 Gundersen proposes the `fractionator' sampling technique
1988-92 Gundersen and Jensen propose the `nucleator'
and `rotator' techniques for estimating particle volume.
1998 Kubinova introduces the first virtual probe that estimates
surface area in preferential slices.
1999 Larsen and Gundersen introduce global spatial sampling
for estimation of total length in preferential slices.
2002-05 Mouton and Gokhale introduce "space balls" and "virtual cycloid"
for estimation of total length and total surface area, respectively,
in preferential slices.
2008 Gundersen, Gardi, Nyengaard introduce the proportionator
Proportionator
The proportionator is the most efficient unbiased stereological method used to estimate population size. A typical application is counting the number of cells in an organ. The proportionator is related to the optical fractionator and physical disector methods that also estimate population...
the most efficient stereological method known.
The primary scientific journals for stereology are "Journal of Microscopy" and "Image Analysis & Stereology" (ex Acta Stereologica). The timeline for stereology-related book titles, chapters, and editorials:
- Underwood, E. (1970). Quantitative Stereology. New York, Addison-Wesley.
- Saper, C.B. (1997). Counting on our reviewers to set the standards. J. Comp. Neurol. 386(1), 1.
- Ohser, J. and F. Mücklich (2000). Statistical analysis of microstructures in materials science. New York, Wiley. | year=2009}}
- Mouton, P.R. (2009). "'Applications of Modern Stereology to Quantitative Microscopy", in Microscopy and Microanalysis, L.N. Brewer, S. McKernan, J.P. Shields, et al., (eds.), Cambridge University Press, London, Vol. 15, pp. 1526-1527.
- Mouton, P.R., Gordon, M (2010). Stereological and Image Analysis Techniques For Quantitative Assessment Of Neurotoxicology, In Neurotoxicology, 3rd Edition, Target Organ Toxicology Series, G. Jean Harry, Hugh A. Tilson, (eds).
- Mouton, P.R. (2011). Applications Of Unbiased Stereology To Neurodevelopmental Toxicology, in Developmental Neurotoxicology Research: Principles, Models, Techniques, Strategies and Mechanisms, C. Wang and W. Slikker (eds), John Wiley & Sons, Hoboken, N.J. pp. 53-77.
- Mouton, P.R. (2011). Unbiased Stereology: A Concise Guide. The Johns Hopkins University Press, Baltimore, MD. ISBN 978-0801899843