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Morphometrics (from
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
μορϕή ''morphe'', "shape, form", and -μετρία ''metria'', "measurement") or morphometry refers to the quantitative analysis of ''form'', a concept that encompasses size and shape. Morphometric analyses are commonly performed on organisms, and are useful in analyzing their fossil record, the impact of mutations on shape, developmental changes in form, covariances between ecological factors and shape, as well for estimating quantitative-genetic parameters of shape. Morphometrics can be used to quantify a trait of evolutionary significance, and by detecting changes in the shape, deduce something of their
ontogeny Ontogeny (also ontogenesis) is the origination and development of an organism (both physical and psychological, e.g., moral development), usually from the time of fertilization of the egg to adult. The term can also be used to refer to the s ...
, function or evolutionary relationships. A major objective of morphometrics is to statistically test hypotheses about the factors that affect shape. "Morphometrics", in the broader sense, is also used to precisely locate certain areas of organs such as the brain, and in describing the shapes of other things.


Forms

Three general approaches to form are usually distinguished: traditional morphometrics, landmark-based morphometrics and outline-based morphometrics.


"Traditional" morphometrics

Traditional morphometrics analyzes lengths, widths, masses, angles, ratios and areas.Marcus, L. F. (1990). Chapter 4. Traditional morphometrics. In Proceedings of the Michigan Morphometric Workshop. Special Publication No. 2. F. J. Rohlf and F. L. Bookstein. Ann Arbor MI, The University of Michigan Museum of Zoology: 77–122. In general, traditional morphometric data are measurements of size. A drawback of using many measurements of size is that most will be highly correlated; as a result, there are few independent variables despite the many measurements. For instance, tibia length will vary with femur length and also with humerus and ulna length and even with measurements of the head. Traditional morphometric data are nonetheless useful when either absolute or relative sizes are of particular interest, such as in studies of growth. These data are also useful when size measurements are of theoretical importance such as body mass and limb cross-sectional area and length in studies of functional morphology. However, these measurements have one important limitation: they contain little information about the spatial distribution of shape changes across the organism. They are also useful when determining the extent to which certain pollutants have affected an individual. These indices include the hepatosomatic index, gonadosomatic index and also the condition factors (shakumbila, 2014).


Landmark-based geometric morphometrics

In landmark-based geometric morphometrics, the spatial information missing from traditional morphometrics is contained in the data, because the data are coordinates of
landmarks A landmark is a recognizable natural or artificial feature used for navigation, a feature that stands out from its near environment and is often visible from long distances. In modern use, the term can also be applied to smaller structures or f ...
: discrete anatomical loci that are arguably ''homologous'' in all individuals in the analysis (i.e. they can be regarded as the "same" point in each specimens in the study). For example, where two specific sutures intersect is a landmark, as are intersections between veins on an insect wing or leaf, or
foramina In anatomy and osteology, a foramen (;Entry "foramen"
in
, small holes through which veins and blood vessels pass. Landmark-based studies have traditionally analyzed 2D data, but with the increasing availability of 3D imaging techniques, 3D analyses are becoming more feasible even for small structures such as teeth. Finding enough landmarks to provide a comprehensive description of shape can be difficult when working with fossils or easily damaged specimens. That is because all landmarks must be present in all specimens, although coordinates of missing landmarks can be estimated. The data for each individual consists of a ''configuration'' of landmarks. There are three recognized categories of landmarks. ''Type 1 landmarks'' are defined locally, i.e. in terms of structures close to that point; for example, an intersection between three sutures, or intersections between veins on an insect wing are locally defined and surrounded by tissue on all sides. ''Type 3 landmarks'', in contrast, are defined in terms of points far away from the landmark, and are often defined in terms of a point "furthest away" from another point. ''Type 2 landmarks'' are intermediate; this category includes points such as the tip structure, or local minima and maxima of curvature. They are defined in terms of local features, but they are not surrounded on all sides. In addition to landmarks, there are ''semilandmarks'', points whose position along a curve is arbitrary but which provide information about curvature in two or three dimensions.


Procrustes-based geometric morphometrics

Shape analysis begins by removing the information that is not about shape. By definition, shape is not altered by translation, scaling or rotation. Thus, to compare shapes, the non-shape information is removed from the coordinates of landmarks. There is more than one way to do these three operations. One method is to fix the coordinates of two points to (0,0) and (0,1), which are the two ends of a baseline. In one step, the shapes are translated to the same position (the same two coordinates are fixed to those values), the shapes are scaled (to unit baseline length) and the shapes are rotated. An alternative, and preferred method, is
Procrustes superimposition In statistics, Procrustes analysis is a form of statistical shape analysis used to analyse the distribution of a set of shapes. The name ''Procrustes'' ( el, Προκρούστης) refers to a bandit from Greek mythology who made his victims fi ...
. This method translates the centroid of the shapes to (0,0); the ''x'' coordinate of the centroid is the average of the ''x'' coordinates of the landmarks, and the ''y'' coordinate of the centroid is the average of the ''y''-coordinates. Shapes are scaled to unit centroid size, which is the square root of the summed squared distances of each landmark to the centroid. The configuration is rotated to minimize the deviation between it and a reference, typically the mean shape. In the case of semi-landmarks, variation in position along the curve is also removed. Because shape space is curved, analyses are done by projecting shapes onto a space tangent to shape space. Within the tangent space, conventional multivariate statistical methods such as multivariate analysis of variance and multivariate regression, can be used to test statistical hypotheses about shape. Procrustes-based analyses have some limitations. One is that the Procrustes superimposition uses a least-squares criterion to find the optimal rotation; consequently, variation that is localized to a single landmark will be smeared out across many. This is called the 'Pinocchio effect'. Another is that the superimposition may itself impose a pattern of covariation on the landmarks. Additionally, any information that cannot be captured by landmarks and semilandmarks cannot be analyzed, including classical measurements like "greatest skull breadth". Moreover, there are criticisms of Procrustes-based methods that motivate an alternative approach to analyzing landmark data.


Euclidean distance matrix analysis


Diffeomorphometry

Diffeomorphometry is the focus on comparison of shapes and forms with a metric structure based on diffeomorphisms, and is central to the field of
computational anatomy Computational anatomy is an interdisciplinary field of biology focused on quantitative investigation and modelling of anatomical shapes variability. It involves the development and application of mathematical, statistical and data-analytical metho ...
. Diffeomorphic registration, introduced in the 90s, is now an important player with existing code bases organized around ANTS, DARTEL, DEMONS, LDDMM, StationaryLDDMM are examples of actively used computational codes for constructing correspondences between coordinate systems based on sparse features and dense images.
Voxel-based morphometry Voxel-based morphometry is a computational approach to neuroanatomy that measures differences in local concentrations of brain tissue, through a voxel-wise comparison of multiple brain images. In traditional morphometry, volume of the whole br ...
(VBM) is an important technology built on many of these principles. Methods based on diffeomorphic flows are used in For example, deformations could be diffeomorphisms of the ambient space, resulting in the LDDMM (
Large Deformation Diffeomorphic Metric Mapping Large deformation diffeomorphic metric mapping (LDDMM) is a specific suite of algorithms used for diffeomorphic mapping and manipulating dense imagery based on diffeomorphic metric mapping within the academic discipline of computational anatomy, ...
) framework for shape comparison. On such deformations is the right invariant metric of
Computational Anatomy Computational anatomy is an interdisciplinary field of biology focused on quantitative investigation and modelling of anatomical shapes variability. It involves the development and application of mathematical, statistical and data-analytical metho ...
which generalizes the metric of non-compressible Eulerian flows but to include the Sobolev norm ensuring smoothness of the flows, metrics have now been defined associated to Hamiltonian controls of diffeomorphic flows.


Outline analysis

Outline analysis is another approach to analyzing shape. What distinguishes outline analysis is that coefficients of mathematical functions are fitted to points sampled along the outline. There are a number of ways of quantifying an outline. Older techniques such as the "fit to a polynomial curve" and Principal components quantitative analysis have been superseded by the two main modern approaches: eigenshape analysis, and
elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
Fourier analysis (EFA), using hand- or computer-traced outlines. The former involves fitting a preset number of semilandmarks at equal intervals around the outline of a shape, recording the deviation of each step from semilandmark to semilandmark from what the angle of that step would be were the object a simple circle.For an example "in use", see The latter defines the outline as the sum of the minimum number of ellipses required to mimic the shape.e.g. Both methods have their weaknesses; the most dangerous (and easily overcome) is their susceptibility to noise in the outline. Likewise, neither compares homologous points, and global change is always given more weight than local variation (which may have large biological consequences). Eigenshape analysis requires an equivalent starting point to be set for each specimen, which can be a source of error EFA also suffers from redundancy in that not all variables are independent. On the other hand, it is possible to apply them to complex curves without having to define a centroid; this makes removing the effect of location, size and rotation much simpler. The perceived failings of outline morphometrics are that it doesn't compare points of a homologous origin, and that it oversimplifies complex shapes by restricting itself to considering the outline and not internal changes. Also, since it works by approximating the outline by a series of ellipses, it deals poorly with pointed shapes. One criticism of outline-based methods is that they disregard homology – a famous example of this disregard being the ability of outline-based methods to compare a
scapula The scapula (plural scapulae or scapulas), also known as the shoulder blade, is the bone that connects the humerus (upper arm bone) with the clavicle (collar bone). Like their connected bones, the scapulae are paired, with each scapula on eith ...
to a potato chip. Such a comparison which would not be possible if the data were restricted to biologically homologous points. An argument against that critique is that, if landmark approaches to morphometrics can be used to test biological hypotheses in the absence of homology data, it is inappropriate to fault outline-based approaches for enabling the same types of studies.


Analyzing data

Multivariate statistical methods can be used to test statistical hypotheses about factors that affect shape and to visualize their effects. To visualize the patterns of variation in the data, the data need to be reduced to a comprehensible (low-dimensional) form. Principal component analysis (PCA) is a commonly employed tool to summarize the variation. Simply put, the technique projects as much of the overall variation as possible into a few dimensions. See the figure at the right for an example. Each axis on a PCA plot is an
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of the covariance matrix of shape variables. The first axis accounts for maximum variation in the sample, with further axes representing further ways in which the samples vary. The pattern of clustering of samples in this morphospace represents similarities and differences in shapes, which can reflect
phylogenetic relationship In biology, phylogenetics (; from Greek φυλή/ φῦλον [] "tribe, clan, race", and wikt:γενετικός, γενετικός [] "origin, source, birth") is the study of the evolutionary history and relationships among or within groups o ...
s. As well as exploring patterns of variation, Multivariate statistical methods can be used to test statistical hypotheses about factors that affect shape and to visualize their effects, although PCA is not needed for this purpose unless the method requires inverting the variance-covariance matrix. Landmark data allow the difference between population means, or the deviation an individual from its population mean, to be visualized in at least two ways. One depicts vectors at landmarks that show the magnitude and direction in which that landmark is displaced relative to the others. The second depicts the difference via the
thin plate splines Thin plate splines (TPS) are a spline-based technique for data interpolation and smoothing. They were introduced to geometric design by Duchon. They are an important special case of a polyharmonic spline. Robust Point Matching (RPM) is a common ex ...
, an interpolation function that models change ''between'' landmarks from the data of changes in coordinates ''of'' landmarks. This function produces what look like deformed grids; where regions that relatively elongated, the grid will look stretched and where those regions are relatively shortened, the grid will look compressed.


Ecology and evolutionary biology

D'Arcy Thompson Sir D'Arcy Wentworth Thompson CB FRS FRSE (2 May 1860 – 21 June 1948) was a Scottish biologist, mathematician and classics scholar. He was a pioneer of mathematical and theoretical biology, travelled on expeditions to the Bering Strait a ...
in 1917 suggested that shapes in many different species could also be related in this way. In the case of shells and horns he gave a fairly precise analysis… But he also drew various pictures of fishes and skulls, and argued that they were related by deformations of coordinates. Shape analysis is widely used in ecology and evolutionary biology to study plasticity, evolutionary changes in shape and in evolutionary developmental biology to study the evolution of the ontogeny of shape, as well as the developmental origins of developmental stability, canalization and modularity. Many other applications of shape analysis in ecology and evolutionary biology can be found in the introductory text:


Neuroimaging

In
neuroimaging Neuroimaging is the use of quantitative (computational) techniques to study the structure and function of the central nervous system, developed as an objective way of scientifically studying the healthy human brain in a non-invasive manner. Incr ...
, the most common variants are
voxel-based morphometry Voxel-based morphometry is a computational approach to neuroanatomy that measures differences in local concentrations of brain tissue, through a voxel-wise comparison of multiple brain images. In traditional morphometry, volume of the whole br ...
, deformation-based morphometry and
surface-based morphometry Brain morphometry is a subfield of both morphometry and the brain sciences, concerned with the measurement of brain structures and changes thereof during development, aging, learning, disease and evolution. Since autopsy-like dissection is gener ...
of the
brain A brain is an organ that serves as the center of the nervous system in all vertebrate and most invertebrate animals. It is located in the head, usually close to the sensory organs for senses such as vision. It is the most complex organ in a ve ...
.


Bone histomorphometry

Histomorphometry of bone involves obtaining a bone biopsy specimen and processing of bone specimens in the laboratory, obtaining estimates of the proportional volumes and surfaces occupied by different components of bone. First the bone is broken down by baths in highly concentrated
ethanol Ethanol (abbr. EtOH; also called ethyl alcohol, grain alcohol, drinking alcohol, or simply alcohol) is an organic compound. It is an alcohol with the chemical formula . Its formula can be also written as or (an ethyl group linked to a ...
and
acetone Acetone (2-propanone or dimethyl ketone), is an organic compound with the formula . It is the simplest and smallest ketone (). It is a colorless, highly volatile and flammable liquid with a characteristic pungent odour. Acetone is miscib ...
. The bone is then embedded and stained so that it can be visualized/analyzed under a
microscope A microscope () is a laboratory instrument used to examine objects that are too small to be seen by the naked eye. Microscopy is the science of investigating small objects and structures using a microscope. Microscopic means being invisi ...
. Obtaining a bone biopsy is accomplished by using a bone biopsy trephine.


See also

*
Allometry Allometry is the study of the relationship of body size to shape, anatomy, physiology and finally behaviour, first outlined by Otto Snell in 1892, by D'Arcy Thompson in 1917 in '' On Growth and Form'' and by Julian Huxley in 1932. Overview Allom ...
* Allometric engineering *
Brain morphometry Brain morphometry is a subfield of both morphometry and the brain sciences, concerned with the measurement of brain structures and changes thereof during development, aging, learning, disease and evolution. Since autopsy-like dissection is gener ...
*
D'Arcy Wentworth Thompson Sir D'Arcy Wentworth Thompson CB FRS FRSE (2 May 1860 – 21 June 1948) was a Scottish biologist, mathematician and classics scholar. He was a pioneer of mathematical and theoretical biology Mathematical and theoretical biology, or biomat ...
* Geometric morphometrics in anthropology * Geomorphometrics *
Meristics Meristics is an area of ichthyology and herpetology which relates to counting quantitative features of fish and reptiles, such as the number of fins or scales. A meristic (countable trait) can be used to describe a particular species of fish, or us ...
*
Phylogenetic comparative methods Phylogenetic comparative methods (PCMs) use information on the historical relationships of lineages ( phylogenies) to test evolutionary hypotheses. The comparative method has a long history in evolutionary biology; indeed, Charles Darwin used diff ...


Notes

from Greek: "morph," meaning shape or form, and "metron”, measurement


References


Bibliography

* * * * * * *


External links

*
PASTSHAPE
– Elliptic Fourier Descriptors
Morphometric software
– Archive of many different types of software for use in morphometrics - especially geometric morphometrics. {{Authority control Comparative anatomy Bioinformatics Computational anatomy