Tensor-Based Morphometry

Tensor-Based Morphometry J Ashburner and GR Ridgway, UCL Institute of Neurology, London, UK ã 2015 Elsevier Inc. All rights reserved.

Glossary Adjoint transport Involves transporting geometric information via its tangent vector (velocity fields) (http://en. wikipedia.org/wiki/Adjoint_representation). Allometry Is the study of the relationship between organism size and its shape, anatomy, physiology, etc. (http://en. wikipedia.org/wiki/Allometry). Artifact Is an error in the representation of information introduced by the measuring equipment or techniques (http://en.wikipedia.org/wiki/Artifact_(error)). Coadjoint transport Involves transporting geometric information via its cotangent vector (momentum), such that its structure is conserved (http://en.wikipedia.org/wiki/ Coadjoint_representation). Deformation Is the transformation of an object from one configuration to another (http://en.wikipedia.org/wiki/ Deformation_(mechanics)). Diffeomorphic Satisfies the requirements of a diffeomorphism (http://en.wikipedia.org/wiki/ Diffeomorphic). Displacement Is the difference between final and initial positions, where the actual path is irrelevant (http://en. wikipedia.org/wiki/Displacement_field_(mechanics)). Divergence Is an operation on a vector field that measures the magnitude of the field’s source or sink (http://en. wikipedia.org/wiki/Divergence). Exploratory analysis Is an approach to analyzing datasets to summarize their main characteristics (http://en.wikipedia. org/wiki/Exploratory_analysis). Feature Is an individual measurable property of an observed phenomenon (http://en.wikipedia.org/wiki/Features_ (pattern_recognition)). General linear model Is a statistical linear model, which may be either univariate or multivariate (http://en. wikipedia.org/wiki/General_linear_model). Generalized linear model Is a generalization of multiple regression that allows response variables to be drawn from distributions other than Gaussians (http://en.wikipedia.org/ wiki/Generalized_linear_model). Generative model Is a model encoding a probability distribution from which observable data are treated as a sample (http://en.wikipedia.org/wiki/Generative_model). Gradient of a scalar field Is a vector field pointing in the direction of greatest rate of increase and whose magnitude is that rate of increase (http://en.wikipedia.org/wiki/ Gradient). Jacobian Usually refers to the matrix of all first-order partial derivatives of a vector-valued function (http://en.wikipedia. org/wiki/Jacobian). Jacobian determinant Is the determinant of a Jacobian matrix, which encodes the factor by which a function expands or shrinks volumes (http://en.wikipedia.org/wiki/ Jacobian).

Brain Mapping: An Encyclopedic Reference

Landmark Is usually a biologically meaningful point, which defines homologous parts of an organism across some population (http://en.wikipedia.org/wiki/ Landmark_point). Mapping Is a synonym for function or denotes a particular kind of function (http://en.wikipedia.org/wiki/Map_ (mathematics)). Mass-univariate statistics Concerns the analysis of multivariate datasets, but with an assumption that dependent variables are independent from each other. Matrix logarithm Is a generalization of a scalar logarithm, which is (in a sense) the inverse of a matrix exponential (http://en.wikipedia.org/wiki/Matrix_logarithm). Measure Is a generalization of the concepts of length, area, and volume (http://en.wikipedia.org/wiki/Measure_ (mathematics)). Model Is a description of a system using mathematical concepts and language (http://en.wikipedia.org/wiki/ Mathematical_model). Morphometric Is a quantitative analysis of form, a concept that encompasses size and shape (http://en.wikipedia.org/ wiki/Morphometrics). Multivariate statistics Concerns the analysis of datasets where simultaneous observations of multiple dependent variables (e.g., voxels in an image) are made (http://en. wikipedia.org/wiki/Multivariate_statistics). Nonparametric statistics Assumes that the observations are not drawn from a probability distribution with a characteristic structure or parameters (http://en.wikipedia. org/wiki/Non-parametric_statistics). Null hypothesis Is the general or default position (used by frequentist statisticians) that there is no relationship between two phenomena (http://en.wikipedia.org/wiki/ Null_hypothesis). Objective function Is a function that maps values of one or more variables onto a real number, which intuitively represents some ‘cost’ (http://en.wikipedia.org/wiki/ Objective_function). Parallel transport Is a way of transporting geometric information along smooth curves in a manifold (http://en. wikipedia.org/wiki/Parallel_transport). Parametric statistics Assumes that observations are drawn from a type of probability distribution, such that inferences may be made about the parameters of the distribution (http://en.wikipedia.org/wiki/Parametric_statistics). Pose Is an object’s position and orientation relative to some coordinate system (http://en.wikipedia.org/wiki/Pose_ (computer_vision)). Principal component analysis Is a procedure that uses orthogonal transformation to convert a set of possibly correlated variables into a set of linearly uncorrelated variables (http://en.wikipedia.org/wiki/ Principal_component_analysis).

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Principal geodesic analysis Is a generalization of principal component analysis to non-Euclidean settings (http://en. wikipedia.org/wiki/Principal_geodesic_analysis). Random field Is a generalization of a stochastic process to multiple dimensions, such that the underlying parameters can be vectors or points on a manifold http://en.wikipedia. org/wiki/Random_field). Registration Is the process of transforming different sets of data into one coordinate system (http://en.wikipedia.org/ wiki/Image_registration). Regularization Is a process of introducing additional information in order to solve ill-posed problems or prevent overfitting (http://en.wikipedia.org/wiki/Regularization_ (mathematics)). Shear Is displacement of points in a fixed direction by an amount proportional to their signed distances from a line parallel to that direction (http://en.wikipedia.org/wiki/ Shear_mapping). Spatial normalization Is a term used by neuroimagers that refers to warping images of different individuals to a common coordinate system (http://en.wikipedia.org/wiki/ Spatial_normalization). Statistical hypothesis testing Is a method for making decisions using data from a scientific study (http:// en.wikipedia.org/wiki/Statistical_hypothesis_testing). Statistical parametric map Is a statistical technique, created by Karl Friston, for localizing statistically significant differences among populations of images (http:// en.wikipedia.org/wiki/Statistical_parametric_mapping).

Introduction Morphometrics refers to the quantitative analysis of form, which is a concept that encompasses both the size and shape of an organism or organ. In neuroimaging, morphometric approaches are typically used to characterize differences among populations of subjects or to identify features that correlate with some measurement of interest. These measurements may be clinical scores, test score results, genetic measurements, or anything else of interest to the investigator. The usual approaches involve extracting anatomical features or descriptors from MRI data of the subjects and performing some form of statistical analysis on them. This article concerns tensor-based morphometric techniques, which involve analyzing features that principally relate to the relative volumes of structures, as estimated by image registration. The mathematics involved in morphometrics can be quite complicated, but we try to keep it relatively simple in this article. Morphometrics has a long history throughout many areas of biology. Most applications do not have the benefit of imaging devices that enable 3-D volumetric scans to be collected, so generally focus on working with things that can easily be measured from the organ or organism itself. Traditional approaches were limited to measures such as lengths, widths, angles, and distances, which were subjected to statistical analysis. When technological advances made it easier to record

Strain tensor Is a description of stretching and shearing due to shape changes, ignoring changes due to pose differences (http://en.wikipedia.org/wiki/Deformation_(mechanics)). Template Is some form of reference image, or set of images, that serves as a model or standard for alignment with scans of individual subjects. Tensor field Has a tensor at each point in space (http://en. wikipedia.org/wiki/Tensor_field). Tensor-based morphometry Is a term used in neuroimaging to refer to characterizing anatomical differences among populations of subjects via Jacobians of deformations (or similar). Thin-plate spline Is a form of radial basis-function representation of displacement fields that gives a smooth representation (http://en.wikipedia.org/wiki/ Thin_plate_spline). Univariate statistics Concerns the analysis of datasets where a single dependent variable is measured (http://en. wikipedia.org/wiki/Univariate). Velocity field Is a vector field used to mathematically describe the motion of a fluid (http://en.wikipedia.org/wiki/ Velocity_field). Voxel Is a volume element in 3-D images, analogous to a pixel (picture element) in 2-D images (http://en.wikipedia. org/wiki/Voxel). Voxel-based morphometry Is a term used in neuroimaging to refer to characterizing anatomical differences among populations of subjects via spatially blurred tissue maps (or similar) (http://en.wikipedia.org/wiki/Voxelbased_morphometry).

the locations of landmarks, a number of new morphometric approaches appeared, which were largely inspired by the work of Thompson (1917, 1942). Instead of analyzing lengths, widths, etc., the new geometric morphometrics (Adams, Rohlf, & Slice, 2004; Rohlf & Marcus, 1993) involved analyzing landmark coordinates in space, after first correcting for pose (and possibly size). Instead of treating data in a feature by feature way, multivariate analyses of landmark positions, or of thin-plate spline coefficients, preserved geometric relationships among all the points. Most areas of biology are limited to making measurements on the outside surface of whatever organ or organism they chose to study, whereas neuroimagers have the advantage of being able to measure a much wider variety of things inside the brain. Some brain morphometric studies involve volumes obtained by manually tracing regions in scans, although these are mostly limited to a handful of structures with clear boundaries. While there may be a wealth of findings pertaining to these particular structures (e.g., ventricles and hippocampi), other brain regions can easily be neglected. It would be a mistake to assume that neurological disorders only affect those structures we can manually outline. Manual tracing of a structure also ignores the potential variability within that structure, for example, one hippocampal subfield could be relatively smaller and another relatively larger with no detectable change in the overall volume.

INTRODUCTION TO METHODS AND MODELING | Tensor-Based Morphometry

Neuroimagers tend not to use much landmark data. In part, this is because there are few discrete and readily identifiable points within the brain. Instead, the field relies on correspondences estimated by automatic or semiautomatic image registration approaches. Providing the same software and settings are used, such approaches should lead to fully reproducible results, irrespective of who runs it. The following section briefly describes some of the statistical testing procedures that may be applied to morphometric features. This is followed by a section about the types of features that are typically extracted for tensor-based morphometric (TBM) studies.

Statistical Analysis TBM usually involves the framework of statistical parametric mapping (SPM) (Friston et al., 1995), which localizes statistically significant regional differences. Extensive details of the procedures involved are described in other articles of this book, so only a brief summary will be outlined here. Essentially, image data from a number of subjects are preprocessed by aligning them to a common anatomical frame of reference, which gives a feature representation that is more amenable to voxel-wise statistical testing. SPMs of voxel-wise univariate measures allow simple questions to be addressed, such as where does the chosen morphometric feature correlate with a particular regressor of interest? Typically, parametric statistical procedures (t-tests and F-tests) are used within the frequentist framework. Hypotheses can be formulated within the framework of a univariate general linear model (GLM), whereby a vector of observations is modeled by a linear combination of user-specified regressors (Friston et al., 1995). The GLM is a flexible framework that allows many different tests to be applied, ranging from group comparisons and identification of differences that are related to specified covariates such as disease severity or age to complex interactions between different effects of interest. Because the pattern of difference to be determined is not specified a priori, SPM analyses are a hybrid between statistical hypothesis testing and exploratory analyses. Rather than test a single hypothesis, the approach involves testing hypotheses at each voxel of the preprocessed data. A Bonferroni correction could be used to correct for the multiple comparisons if the tests were independent, but this is not normally the case because of the inherent spatial smoothness of the data. In practice, a correction for the multitude of tests is usually obtained via random field theory (Friston, Holmes, Poline, Price, & Frith, 1996; Worsley et al., 1996), thus allowing a correction for multiple dependent comparisons that controls the rate of false-positive results. Alternatively, SPMs may be corrected for the rate of false discoveries (Chumbley, Worsley, Flandin, & Friston, 2010; Genovese, Lazar, & Nichols, 2002). There are also a variety of other statistical analytic methods that may be applied to the feature data. For example, nonparametric approaches (Nichols & Holmes, 2002) may be applied in situations where parametric modeling assumptions do not hold. In recent years, there has been a rediscovery of Bayesian approaches by the neuroimaging community, leading

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to various Bayesian inference procedures for localizing differences (Friston & Penny, 2003; Penny & Ridgway, 2013).

Voxel-wise Multivariate Analyses Univariate GLMs are a special case of the more general multivariate GLM. In principle, SPMs can also be obtained from the results of voxel-wise multivariate tests. Instead of one variable per voxel of a subject, tests within a multivariate GLM could effectively involve two or more variables. Following voxel-wise multivariate tests, similar corrections based on random field theory can be applied as in the univariate case (Cao & Worsley, 1999; Carbonell, Worsley, & Galan, 2011; Worsley, Taylor, Tomaiuolo, & Lerch, 2004). Readers without a mathematical background can safely skip the remainder of this subsection. In a univariate GLM, y ¼ Xb þ e, t- and F-contrasts can be seen as special cases of a likelihood ratio test comparing a restricted model (X0, under the null hypothesis) to the unrestricted full model (X) F¼

y T ðXX þ  X 0 X 0 þ Þy rankðX Þ  rankðX0 Þ = yT ðI  XX þ Þy rankðX Þ SSH DFH ¼ = SSE DFE

where Xþ denotes the pseudo-inverse of X (which can be computed with, e.g., MATLAB’sppinv ffiffiffi command). The t-contrast ^ see is just the signed version of F (with the sign of CT b; succeeding text). The multivariate equivalent is based on Wilks’ lambda (or related test statistics, such as Roy’s greatest root), which has a closely related form as it is also derived from the likelihood ratio:  T  Y ðI  XX þ ÞY  jSSEj ¼ L ¼  T Y ðI  X 0 X 0 þ ÞY  jSSH þ SSEj In both univariate and multivariate cases, the sums of squares (or sums of squares and cross products matrix) for the hypothesis, SSH, can be expressed in terms of a contrast C in the parameters b^ as follows: SSH ¼ Y T ðXX þ  X 0 X 0 þ ÞY    þ ^T C CT X T X þ C CT b^ ¼b which allows one to test a general linear null hypothesis CT b^ ¼ 0 without explicitly determining the implied reduced model X0.

Whole-Brain Multivariate Analyses An alternative to the SPM framework involves whole-brain multivariate modeling. Within the frequentist setting, this may be achieved by reducing the dimensionality of the data, using a method such as principal component analysis, followed by performing statistical tests using the multivariate linear model (e.g., based on Wilks’ lambda, described in the preceding text) (Ashburner et al., 1998). This type of approach is similar to that used by conventional geometric morphometrics (Adams et al., 2004; Klingenberg, 2011; Mitteroecker & Gunz, 2009; Rohlf & Marcus, 1993; Slice, 2007), but with features derived

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from deformations instead of just a few landmark coordinates (as in Bookstein, 1997, 1999; Dryden & Mardia, 1998). When applied on a global scale, this approach simply identifies whether there are significant differences between overall shapes among the brains of different populations. More recently, Bayesian multivariate approaches (Bishop, 2006; Friston et al., 2008; Rasmussen & Williams, 2006) are being applied, allowing more elegant approaches to be used to circumvent the curse of dimensionality (see Ashburner and Klo¨ppel (2011) for more on this subject). Measures of statistical significance could be assessed using the principles underlying Bayesian model selection, although currently, it is more common to see cross validation used.

Template

Image

Warped template

Warped image

Feature Representations There are many ways of characterizing anatomical differences among populations or finding correlations between anatomy and, for example, disease severity. Over the years, there has been a proliferation in the types of features that can be tested, although most neuroimaging comparisons are made using the ‘voxel-based morphometric’ approach. However, a number of other data representations may also be subjected to statistical analysis. One of the challenges for morphometry is to identify shape modeling features that best differentiate among populations. Where there are differences between the cortical thickness in one population and that of another, then cortical thickness would be the most discriminative shape feature to use. An analysis of regional gray matter volumes may partially reveal those differences, but it would not be as accurate as looking at thickness itself. Similarly, if the difference between groups is best characterized by cortical surface areas, then an analysis of cortical thickness is unlikely to show much of interest. In general, determining the most accurate representations of differences among populations of subjects is something to be done empirically.

Deformation Fields Currently, most morphometric studies in neuroimaging are based on T1-weighted scans. MRI scans contain a variety of artifacts, many of which will impact on any kind of morphometric analysis. These are especially important for studies that combine scans from multiple scanners (Jovicich et al., 2009). Spatial distortions arising from gradient nonlinearities impact any kind of morphometric analysis, although there are a variety of correction methods for these (Janke, Zhao, Cowin, Galloway, & Doddrell, 2004; Jovicich et al., 2006). Further information about optimizing image acquisition parameters, artifact correction, etc., for large morphometric studies may be found in Jack et al. (2008). TBM requires that the images of all subjects in the study to be aligned together by some form of ‘spatial normalization.’ In neuroimaging, the primary result of spatially normalizing a series of images is that they all conform to the same space, enabling region-by-region comparisons to be performed. However, for TBM, the main objective is to obtain a set of parameterizations of the spatial transformations required to

Figure 1 Illustration of warping some synthetic images. Note that it shows an inexact matching between them, which is typically what would be expected when aligning real MRI across subjects.

match the different shaped brains to the same template (see Figures 1 and 2). For morphometric studies, these deformations must be mappings from the template to each of the individuals in the study, so these need to be inverted if the registration algorithm generates mappings from each individual to the template. Encoded within each deformation is information about the individual image shapes, which may be further characterized using a variety of statistical procedures. In theory, the choice of reference template used for spatial normalization will influence the findings of a study. Basic common sense tells us that intensity properties of the template should match those of the study data. For example, less accurate findings would be obtained from a study where a meansquares difference matching term was used to align a set of T2-weighted images to a template based on T1-weighted data. In addition, registration errors can be reduced by having the shape of the template brain as similar as possible to those of the subjects in the study. Providing certain objective functions are used to drive the alignment, this is often best achieved by ‘groupwise’ registration approaches, whereby the template is computed as some form of shape and intensity average from the brains in the study (Ashburner, Andersson, & Friston, 2000; Ashburner & Friston, 2009; Joshi, Davis, Jomier, & Gerig, 2004). An alternative approach involves aligning the images in a study with multiple templates, which are single-subject images of different subjects (Koikkalainen et al., 2011; Lepore´ et al., 2008), that is followed by some form of feature averaging procedure. Groupwise registration is often suboptimal when it is driven by certain information–theoretic objective functions, so these multitemplate (also known as multiatlas)

INTRODUCTION TO METHODS AND MODELING | Tensor-Based Morphometry

Deformation

Horizontal component

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Vertical component

Identity

Displacements

Figure 2 The components of a 2-D deformation. Top row: The deformation (from Figure 1), with its horizontal and vertical components. Middle row: An identity transform, with its horizontal and vertical components. Bottom row: Displacements obtained by subtracting the identity transform from the deformation.

procedures are intended to reduce the bias incurred by selecting a particular individual’s image as a template. In general, one should not expect to obtain exactly the same findings from morphometric studies using different image registration software. Different algorithms use different models and assumptions, and in the absence of clear theoretical preferences, the optimal ones can only be determined empirically. Image registration algorithms have a number of settings, and changes to these will generally lead to changes in the findings of a study. For example, Figure 3 shows a simulated image aligned using a variety of regularization settings (but the same algorithm), each giving different maps of relative volumes. The more accurately the registration model is specified, the more accurately the findings from a study will reflect real underlying biological differences.

The Jacobian Tensors A simple morphometric approach would be to examine the deformations themselves by treating them as vector fields representing displacements. These may be analyzed within a

multivariate framework (Ashburner et al., 1998) after appropriate corrections to factor out pose. Some previous works have applied voxel-wise Hotelling’s T2 tests on the displacements at each and every voxel (Gaser, Volz, Kiebel, Riehemann, & Sauer, 1999; Thompson & Toga, 1999), with statistical significance assessed by random field corrections (Cao & Worsley, 1999). However, this approach does not directly localize differences that are intrinsic to the brains themselves. Rather, it identifies those brain structures that are in different locations in space, which depends upon how the poses and possibly sizes of the brains are factored out of the estimated deformations (Klingenberg, 2013). The objective of TBM is usually to localize regions of shape differences among groups of brains, based on deformations that map points in a template (x1,x2,x3) to equivalent points in individual source images (y1,y2,y3). In principle, the Jacobian matrices of the deformations (a second-order tensor field given by the spatial derivatives of the transformation; see Figure 4) should be more reliable indicators of local brain shape than displacements.

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Figure 3 Effects of different regularization settings on estimates of Jacobian determinants. Left column: Image aligned with different regularization. Middle column: Estimated deformations. Right column: Estimated Jacobian determinants (all shown on the same color scale).

A Jacobian matrix contains information about the local stretching, shearing, and rotation involved in the deformation and is defined at each point by 2

3 @y1 =@x1 @y1 =@x2 @y1 =@x3 J ¼ 4 @y2 =@x1 @y2 =@x2 @y2 =@x3 5 @y3 =@x1 @y3 =@x2 @y3 =@x3

determinants at each point (Davatzikos et al., 1996; Freeborough & Fox, 1998; Machado, Gee, & Campos, 1998; Studholme et al., 2004). This type of morphometry is useful for studies that have specific questions about whether growth or volume loss has occurred. The field obtained by taking the determinants at each point gives a map of structural volumes relative to those of a reference image.

The Jacobian Determinants

Logarithms and Exponentials

Determinants of square matrices play an important role in computational anatomy. The most straightforward form of TBM involves comparing relative volumes of different brain structures, where the volumes are derived from the Jacobian

There are a number of nonlinearities to consider when analyzing the shapes and sizes of structures. If a structure stays the same shape but is doubled in length or width, its surface area will be scaled by a factor of 4 and its volume by a factor of 8.

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Figure 4 For 2-D deformations, a Jacobian tensor field encodes a 2  2 matrix at each point. Top: Horizontal and vertical gradients of the horizontal component of the deformation in Figure 2. Bottom: Horizontal and vertical gradients of the vertical component.

Similarly, if the length is scaled by a factor of 3, its surface will be scaled by nine and its volume by 27. A linear relationship between the score and length (1, 2, 3) will mean that there is some nonlinear relationship with area (1, 4, 9) and volume (1, 8, 27). When relating size measurements with, for example, a clinical score, should the analysis be set up so as to identify correlations in length, area, or volume? A reasonable solution is to work with the logarithms of the measures. Growth is a process of self-multiplication. One cell divides into two, then four, eight, etc. This means that for an organ growing at a constant rate, the rate at which more tissue is generated will be proportional to the amount of tissue in existence. This leads to an exponential increase in size. Similarly, atrophy at a constant rate will lead to an exponential decrease in size. Log-transforming Jacobian determinants are only possible if they are >0. This means that the deformations must be oneto-one mappings, such that there is no folding present (see Figure 5). Many nonlinear image registration algorithms parameterize deformations in terms of displacement fields, which do not necessarily enforce well-behaved Jacobian determinants. In contrast, diffeomorphic registration algorithms use a different way of encoding deformations, which involves building up deformations by composing a number of much

smaller displacement fields together. Providing that the constituent deformations are sufficiently small to be one-to-one, the result from composing them should also be a one-to-one mapping (Christensen et al., 1995). Even so, the discrete nature of the actual implementations means that care needs to be taken when computing Jacobians to ensure that they have positive determinants. Some diffeomorphic approaches (Ashburner & Friston, 2011; Beg, Miller, Trouve´, & Younes, 2005; Vialard, Risser, Rueckert, & Cotter, 2012) generate a vector field referred to as the initial velocity. Rather than analyze the features of the deformations, it is possible to work instead with features extracted from this initial velocity field. In particular, the divergence of the initial velocity provides a feature that is numerically similar to the logarithm of the Jacobian determinants. These divergences encode the volumetric growth rates required to achieve alignment of the images, according to the diffeomorphic registration model. If they are integrated over some brain region, this gives the rate at which tissue flows into the region (this is known as the ‘divergence theorem,’ or ‘Gauss’s theorem’). The divergence is computed by summing the diagonal elements of the Jacobian matrices of the velocity field. One advantage of working with divergences, rather than the logs of the Jacobian determinants, is that they are linear

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Folded deformation

Jacobian determinants

Detail of folding

Figure 5 A deformation with folding. Left: Full deformation, generated by doubling the displacement in Figure 2. Center: Jacobian determinants, containing two regions of negative values. Right: Detail of the folded region.

functions of the velocity fields and can often be better behaved numerically. This brings TBM approaches a step closer to procedures such as ‘principal geodesic analysis’ (Fletcher, Joshi, Lu, & Pizer, 2003; Fletcher, Lu, Pizer, & Joshi, 2004). Divergences of displacement fields may also be used for morphometry (Chung et al., 2001; Thirion & Calmon, 1999).

TBM on Tensors Some shape information is lost if only the determinants of the Jacobians are considered. With many subjects in a study, a potentially more powerful form of TBM can be attained using multivariate statistics on other measures derived from the Jacobian matrices. This use of multivariate statistics not only tests for volumetric differences but also indicates whether there are any differences among lengths, areas, and the amount of shear. Because the Jacobian matrices encode both local shape (zooms and shears) and orientation, it is useful to remove the latter prior to statistical analysis. A nonsingular Jacobian matrix can be decomposed into a rotation matrix (R) and a symmetrical positive definite matrix (U), such that J ¼ RU. Matrix U (called the ‘right stretch tensor’) is derived by U ¼ (JTJ)1/2 (using matrix square roots). For a purely rigid body transformation, U ¼ I (the identity matrix). Deviations of U away from I indicate a shape change, which can be represented by a strain tensor E. For any deformation, there is a whole continuum of ways of defining strain tensors, based on a parameter m. When m is nonzero, the family of strain tensors E is given by E(m) ¼ m1 (UmI). For the special case when m is zero, the Hencky strain tensor is given by E(0) ¼ ln (U), where ln refers to a matrix logarithm. Lepore et al. (2006, 2008) showed that a voxelwise multivariate analysis of Hencky strain tensors can exhibit much greater sensitivity than a voxel-wise univariate analysis of logarithms of Jacobian determinants. Many of the concepts required for analysis of strain tensors are also found in the literature on diffusion tensor imaging (Arsigny, Fillard, Pennec, & Ayache, 2006; Pennec, 2009; Whitcher, Wisco, Hadjikhani, & Tuch, 2007).

Allometry Outside neuroimaging, biologists often consider allometry when making comparisons. The ideas behind allometry were first formulated in Sir Julian Huxley’s Problems of Relative Growth

(Huxley, 1932), whereby there is a logarithmic relationship between the sizes of the various structures or organs and the size of the entire organism. Although aware of some of the limitations of the assumptions, Huxley conceptualized growth as a process of self-multiplication of living substance, in which d d log ðyÞ ¼ k log ðxÞ dt dt where x is the magnitude (weight, volume, length, etc.) of the animal, y is the magnitude of the differentially growing organ, and b and k are constants. This gives a relation between magnitudes, which can be modeled as y ¼ bxk. He considered the meaning of the value k to be the rate of growth per unit weight, which is the growth rate at any instant, divided by the size. In neuroimaging, we generally do not consider the weight of the subjects, although corrections for measurements such as whole-brain volume or total intracranial volume are usually incorporated during statistical analyses. In general, findings are heavily dependent upon the way that this global correction is made (Barnes et al., 2010; Hu et al., 2011; Peelle, Cusack, & Henson, 2012), which can lead to widely divergent findings – particularly when comparing genders. These corrections account for some of the multivariate nature of anatomical form. If, instead of volumes, their logarithms are used, a GLM may be fit such that log(y) ¼ log(b) þ k log(x). This GLM formulation can easily be extended so that population-specific log (b) and k may be estimated. Of course, if volume estimates are not constrained to be positive, there will be problems computing their logarithms. A preferable approach, not discussed in any detail here, involves fitting a generalized linear model (GLZ – not to be confused with a GLM) or generalized additive model to the untransformed data (Schuff et al., 2012).

Longitudinal Data Often, longitudinal data are used for morphometric analyses, whereby anatomical scans of multiple subjects are collected at multiple time points. Time differences between scans vary from a few hours (Tost et al., 2010) to a few decades (Fisniku et al., 2008) with intervals of a few months being common for structural plasticity and intervals around a year being common for neurodegenerative disease. Current applications include characterizing patterns of atrophy in dementia (Freeborough & Fox, 1998) or studying brain development (see Figure 6). Other examples include studies into structural

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Figure 6 Example of atrophy measured in a single individual via longitudinal registration. Three orthogonal sections of the subject’s average image are shown above the same sections through a map of volume change. Darker regions indicate shrinkage, whereas brighter regions indicate expansion.

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plasticity, whereby anatomical changes may be attributed to some type of intervention (Draganski et al., 2004). (This is a situation where the direction of causality may be deduced from computational anatomical studies of the human brain.) These studies usually involve within-subject longitudinal registration, followed by intersubject registration to project shape changes from each subject into a common space (spatial normalization). Physical changes over relatively short time intervals are usually small, and longitudinal registration algorithms are very sensitive to biasing effects arising from not treating all data equivalently. Subtle things, such as image interpolation or the form of the regularization, have been demonstrated to have especially troubling effects (Thompson, Holland, & Alzheimer’s Disease Neuroimaging Initiative, 2011; Yushkevich et al., 2010). Previously, these concerns were only theoretical. However, investigators pay more rather attention to them (Fox, Ridgway, & Schott, 2011) now that convincing evidence has been identified. Various solutions have been proposed, which generally involve formulating longitudinal registration in a way that involves pairwise or groupwise consistency (Ashburner & Ridgway, 2013; Hua et al., 2011; Modat et al., 2012; Reuter & Fischl, 2011; Reuter, Schmansky, Rosas, & Fischl, 2012). After intrasubject longitudinal registration has been done and the deformations that align all subjects to a common template space have been estimated, the next question concerns how best to transport the longitudinal information into the common space. Currently, there are two approaches commonly used to transport this information. One approach involves just warping the features, whereas the other involves scaling the warped features by the Jacobian determinant of the deformation with which they are warped. If the longitudinal features are Jacobian determinants themselves, then the latter option will result in spatially normalized features that encode volumes in the original scans relative to those of the template that defines the common space. In an analysis of logarithms of Jacobian determinants, whether or not Jacobians obtained from longitudinal registration (e.g., y0 and y1) are scaled by the Jacobians of the spatially normalizing transformations (e.g., j) should make no difference (because log(jy1) log (jy0) ¼ log(y1) log(y0)). If logarithms are not used, it is unclear whether or not it is generally better to rescale by the Jacobians from the spatial normalization. It may be preferable for analyses that relate longitudinally estimated changes to those estimated cross-sectionally but is perhaps less appropriate for other types of analyses. Sometimes, it is desirable to do some form of analysis of the Jacobian tensors themselves or to analyze some other form of geometric information derived from within-subject longitudinal registration. The most correct way to transport these sorts of information to a common space is probably to use a mathematical procedure known as parallel transport or parallel translation. Physicists, from Einstein onwards, make use of parallel transport, although more recently, it has emerged in the field of computational anatomy (Qiu, Younes, Miller, & Csernansky, 2008; Younes, Qiu, Winslow, & Miller, 2008). Simpler (but reasonably well-justified) approaches for transporting intrasubject geometric information also exist. These include adjoint transport and coadjoint transport (Younes et al., 2008).

Outlook Morphometric approaches used by neuroimagers tend to be substantially different from those applied in other areas of biology. Within neuroimaging, there tends to be much more focus on localizing differences via mass-univariate approaches, whereas multivariate approaches tend to be favored in other fields. This difference in viewpoint has drawn criticism in the past (Bookstein, 2001), although the neuroimaging field has now begun to embrace multivariate methods rather more. Currently, most morphometric analyses involve a purely bottom-up procedure, whereby a pipeline of processing steps is applied to the data. Although still at the early stages, we are beginning to see hierarchical generative models emerge, which combine statistical modeling with registration (Allassonnie`re, Amit, & Trouve, 2007; Fishbaugh, Durrleman, & Gerig, 2011; Niethammer, Huang, & Vialard, 2011; Prastawa, Awate, & Gerig, 2012). Instead of statistical analyses that attempt to explain how the features were generated (ignoring the fact that they came from nonlinear registration), these developments involve generative models of the original image data. Such approaches may eventually enable top-down knowledge about disease status, age, etc., to inform the registration and other image processing components. As all neuroscientists know, top-down processing is essential for making sense of the world (Mumford, 1991).

Acknowledgments Image data used in Figure 6 were part of the ‘OASIS: longitudinal MRI data in nondemented and demented older adults’ dataset (Marcus, Fotenos, Csernansky, Morris, & Buckner, 2010), funded by grant numbers P50 AG05681, P01 AG03991, R01 AG021910, P20 MH071616, and U24 RR021382.

See also: INTRODUCTION TO ANATOMY AND PHYSIOLOGY: Cortical Surface Morphometry; INTRODUCTION TO METHODS AND MODELING: Analysis of Variance (ANOVA); Bayesian Model Inference; Bayesian Model Inversion; Bayesian Multiple Atlas Deformable Templates; Computing Brain Change over Time; Contrasts and Inferences; Cortical Thickness Mapping; Crossvalidation; Diffeomorphic Image Registration; False Discovery Rate Control; Modeling Brain Growth and Development; Multi-voxel Pattern Analysis; Nonlinear Registration Via Displacement Fields; Posterior Probability Maps; Surface-Based Morphometry; The General Linear Model; Topological Inference; Variational Bayes; Voxel-Based Morphometry.

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