Validating cortical surface analysis of medial prefrontal cortex

Morphometry of superior temporal gyrus and planum temporale in schizophrenia and psychotic bipolar disorder

Structural abnormalities in temporal lobe, including the superior temporal gyrus (STG) and planum temporale (PT), have been reported in schizophrenia (SCZ) and bipolar disorder (BPD) patients. While most MRI studies have suggested gray matter volume and surface area reduction in temporal lobe regions, few have explored changes in laminar thickness in PT and STG in SCZ and BPD. ROI subvolumes of the STG from 94 subjects were used to yield gray matter volume, gray/white surface area and laminar thickness for STG and PT cortical regions. Morphometric analysis suggests that there may be gender and laterality effects on the size and shape of the PT in BPD (n=36) and SCZ (n=31) with reduced laterality in PT in subjects with SCZ but not in BPD. In addition, PT surface area was seen to be larger in males, and asymmetry in PT surface area was larger in BPD. Subjects with SCZ had reduced thickness and smaller asymmetry in PT volume. Thus, the PT probably plays a more sensitive role than the STG in structural abnormalities seen in SCZ.

Censoring distances based on labeled cortical distance maps in cortical morphometry

It has been demonstrated that shape differences in cortical structures may be manifested in neuropsychiatric disorders. Such morphometric differences can be measured by labeled cortical distance mapping (LCDM) which characterizes the morphometry of the laminar cortical mantle of cortical structures. LCDM data consist of signed/labeled distances of gray matter (GM) voxels with respect to GM/white matter (WM) surface. Volumes and other summary measures for each subject and the pooled distances can help determine the morphometric differences between diagnostic groups, however they do not reveal all the morphometric information contained in LCDM distances. To extract more information from LCDM data, censoring of the pooled distances is introduced for each diagnostic group where the range of LCDM distances is partitioned at a fixed increment size; and at each censoring step, the distances not exceeding the censoring distance are kept. Censored LCDM distances inherit the advantages of the pooled distances but also provide information about the location of morphometric differences which cannot be obtained from the pooled distances. However, at each step, the censored distances aggregate, which might confound the results. The influence of data aggregation is investigated with an extensive Monte Carlo simulation analysis and it is demonstrated that this influence is negligible. As an illustrative example, GM of ventral medial prefrontal cortices (VMPFCs) of subjects with major depressive disorder (MDD), subjects at high risk (HR) of MDD, and healthy control (Ctrl) subjects are used. A significant reduction in laminar thickness of the VMPFC in MDD and HR subjects is observed compared to Ctrl subjects. Moreover, the GM LCDM distances (i.e., locations with respect to the GM/WM surface) for which these differences start to occur are determined. The methodology is also applicable to LCDM-based morphometric measures of other cortical structures affected by disease.

STATISTICAL ANALYSIS OF CORTICAL MORPHOMETRICS USING POOLED DISTANCES BASED ON LABELED CORTICAL DISTANCE MAPS

Neuropsychiatric disorders have been demonstrated to manifest shape differences in cortical structures. Labeled Cortical Distance Mapping (LCDM) is a powerful tool in quantifying such morphometric differences and characterizes the morphometry of the laminar cortical mantle of cortical structures. Specifically, LCDM data are distances of labeled gray matter (GM) voxels with respect to the gray/white matter cortical surface. Volumes and descriptive measures (such as means and variances for each subject) based on LCDM distances provide descriptive summary information on some of the shape characteristics. However, additional morphometrics are contained in the data and their analysis may provide additional clues to underlying differences in cortical characteristics. To use more of this information, we pool (merge) LCDM distances from subjects in the same group. These pooled distances can help detect morphometric differences between groups, but do not provide information about the locations of such differences in the tissue in question. In this article, we check for the influence of the assumption violations on the analysis of pooled LCDM distances. We demonstrate that the classical parametric tests are robust to the non-normality and within sample dependence of LCDM distances and nonparametric tests are robust to within sample dependence of LCDM distances. We specify the types of alternatives for which the tests are more sensitive. We also show that the pooled LCDM distances provide powerful results for group differences in distribution of LCDM distances. As an illustrative example, we use GM in the ventral medial prefrontal cortex (VMPFC) in subjects with major depressive disorder (MDD), subjects at high risk (HR) of MDD, and healthy subjects. Significant morphometric differences were found in VMPFC due to MDD or being at HR. In particular, the analysis indicated that distances in left and right VMPFCs tend to decrease due to MDD or being at HR, possibly as a result of thinning. The methodology can also be applied to other cortical structures.

Validation of alternating Kernel mixture method: application to tissue segmentation of cortical and subcortical structures

This paper describes the application of the alternating Kernel mixture (AKM) segmentation algorithm to high resolution MRI subvolumes acquired from a 1.5T scanner (hippocampus, n = 10 and prefrontal cortex, n = 9) and a 3T scanner (hippocampus, n = 10 and occipital lobe, n = 10). Segmentation of the subvolumes into cerebrospinal fluid, gray matter, and white matter tissue is validated by comparison with manual segmentation. When compared with other segmentation methods that use traditional Bayesian segmentation, AKM yields smaller errors (P < .005, exact Wilcoxon signed rank test) demonstrating the robustness and wide applicability of AKM across different structures. By generating multiple mixtures for each tissue compartment, AKM mimics the increased variation of manual segmentation in partial volumes due to the highly folded tissues. AKM's superior performance makes it useful for tissue segmentation of subcortical and cortical structures in large-scale neuroimaging studies.

Mapping progressive brain structural changes in early Alzheimer's disease and mild cognitive impairment

Alzheimer's disease (AD), the most common neurodegenerative disorder of the elderly, ranks third in health care cost after heart disease and cancer. Given the disproportionate aging of the population in all developed countries, the socio-economic impact of AD will continue to rise. Mild cognitive impairment (MCI), a transitional state between normal aging and dementia, carries a four- to sixfold increased risk of future diagnosis of dementia. As complete drug-induced reversal of AD symptoms seems unlikely, researchers are now focusing on the earliest stages of AD where a therapeutic intervention is likely to realize the greatest impact. Recently neuroimaging has received significant scientific consideration as a promising in vivo disease-tracking modality that can also provide potential surrogate biomarkers for therapeutic trials. While several volumetric techniques laid the foundation of the neuroimaging research in AD and MCI, more precise computational anatomy techniques have recently become available. This new technology detects and visualizes discrete changes in cortical and hippocampal integrity and tracks the spread of AD pathology throughout the living brain. Related methods can visualize regionally specific correlations between brain atrophy and important proxy measures of disease such as neuropsychological tests, age of onset or factors that may influence disease progression. We describe extensively validated cortical and hippocampal mapping techniques that are sensitive to clinically relevant changes even in the single individual, and can identify group differences in epidemiological studies or clinical treatment trials. We give an overview of some recent neuroimaging advances in AD and MCI and discuss strengths and weaknesses of the various analytic approaches.

Segmentation of arteries in MPRAGE images of the ventral medial prefrontal cortex

A method for removing arteries that appear bright with intensities similar to white matter in Magnetized Prepared Rapid Gradient Echo images of the ventral medial prefrontal cortex is described. The Fast Marching method is used to generate a curve within the artery. Then, the largest connected component is selected to segment the artery which is used to mask the image. The surface reconstructed from the masked image yielded cortical thickness maps similar to those generated by manually pruning the arteries from surfaces reconstructed from the original image. The method may be useful in masking vasculature in other cortical regions.

Abnormalities of cingulate gyrus neuroanatomy in schizophrenia

OBJECTIVE AND METHODS

Abnormalities of the neuroanatomy of the gray matter of the cingulate gyrus, especially its anterior segment, have been suggested to be an important characteristic of schizophrenia. In this study, T1-weighted magnetic resonance scans were collected in 53 individuals with schizophrenia and 68 comparison subjects matched for age, gender, race and parental socioeconomic status. We applied Labeled Cortical Mantle Distance Mapping to assess the volume, surface area and thickness of the cortical mantle within the anterior (AC) and posterior (PC) segments of the cingulate gyrus, excluding the paracingulate gyrus, and related these anatomical measures to measures of psychopathology and illness duration.

RESULTS

After covarying for total cerebral volume, individuals with schizophrenia showed smaller AC gray matter volume (p=0.024), thickness (trend, p=0.081), but not surface area (p=0.16), than comparison subjects. Similar group differences were found for PC gray matter volume (p=0.0005) and thickness (trend, p=0.055), but not surface area (p=0.15). Across both groups, there was a significant L>R asymmetry in thickness of the AC, and a significant L>R asymmetry in the surface area of the PC. However, there were no significant group-by-hemisphere interactions. In the individuals with schizophrenia, thinning of the AC, but not the PC, was correlated with a longer duration of illness and a greater severity of psychotic symptoms.

CONCLUSIONS

Individuals with schizophrenia showed smaller gray matter volumes across the entire cingulate gyrus, mostly due to a reduction in cortical mantle thickness. However, structural measures of the AC were more closely related to clinical features of the illness.

Diffeomorphic metric surface mapping in subregion of the superior temporal gyrus

This paper describes the application of large deformation diffeomorphic metric mapping to cortical surfaces based on the shape and geometric properties of subregions of the superior temporal gyrus in the human brain. The anatomical surfaces of the cortex are represented as triangulated meshes. The diffeomorphic matching algorithm is implemented by defining a norm between the triangulated meshes, based on the algorithms of Vaillant and Glaunès. The diffeomorphic correspondence is defined as a flow of the extrinsic three dimensional coordinates containing the cortical surface that registers the initial and target geometry by minimizing the norm. The methods are demonstrated in 40 high-resolution MRI cortical surfaces of planum temporale (PT) constructed from subsets of the superior temporal gyrus (STG). The effectiveness of the algorithm is demonstrated via the Euclidean positional distance, distance of normal vectors, and curvature before and after the surface matching as well as the comparison with a landmark matching algorithm. The results demonstrate that both the positional and shape variability of the anatomical configurations are being represented by the diffeomorphic maps.

Gender effects on cortical thickness and the influence of scaling

Using magnetic resonance imaging and well-validated computational cortical pattern matching methods in a large and well-matched sample of healthy subjects (n = 60), we analyzed the regional specificity of gender-related cortical thickness differences across the lateral and medial cortices at submillimeter resolution. To establish the influences of brain size correction on gender effects, comparisons were performed with and without applying affine transformations to scale each image volume to a template. We revealed significantly greater cortical thickness in women compared to men, after correcting for individual differences in brain size, while no significant regional thickness increases were observed in males. The pattern and direction of the results were similar without brain size correction, although effects were less pronounced and a small cortical region in the lateral temporal lobes showed greater thickness in males. Our gender-specific findings support a dimorphic organization in male and female brains that appears to involve the architecture of the cortical mantle and that manifests as increased thickness in female brains. This sexual dimorphism favoring women, even without correcting for brain size, may have functional significance and possibly account for gender-specific abilities and/or behavioral differences between sexes.

Estimating linear cortical magnification in human primary visual cortex via dynamic programming

Human primary visual cortex is organized retinotopically, with adjacent locations in cortex representing adjacent locations on the retina. The spatial sampling in cortex is highly nonuniform: the amount of cortex devoted to a unit area of retina decreases with increasing retinal eccentricity. This sampling property can be quantified by the linear cortical magnification factor, which is expressed in terms of millimeters of cortex per degree of visual angle. In this paper, we present a new method using dynamic programming and fMRI retinotopic eccentricity mapping to estimate the linear cortical magnification factor in human primary visual cortex (V1). We localized cortical activity while subjects viewed each of seven stationary contrast- reversing radial checkerboard rings of equal thickness that tiled the visual field from 1.62 to 12.96 degrees of eccentricity. Imaging data from all epochs of each ring were contrasted with data from fixation epochs on a subject-by-subject basis. The resulting t statistic maps were then superimposed on a local coordinate system constructed from the gray/white matter boundary surface of each individual subject's occipital lobe, separately for each ring. Smoothed maps of functional activity on the cortical surface were constructed using orthonormal bases of the Laplace-Beltrami operator that incorporate the geometry of the cortical surface. This allowed us to stably track the ridge of maximum activation due to each ring via dynamic programming optimization over all possible paths on the cortical surface. We estimated the linear cortical magnification factor by calculating geodesic distances between activation ridges on the cortical surface in a population of five normal subjects. The reliability of these estimates was assessed by comparing results based on data from one quadrant to those based on data from the full hemifield along with a split-half reliability analysis.

Abnormal cortical complexity and thickness profiles mapped in Williams syndrome

We identified and mapped an anatomically localized failure of cortical maturation in Williams syndrome (WS), a genetic condition associated with deletion of approximately 20 contiguous genes on chromosome 7. Detailed three-dimensional (3D) maps of cortical thickness, based on magnetic resonance imaging (MRI) scans of 164 brain hemispheres, identified a delimited zone of right hemisphere perisylvian cortex that was thicker in WS than in matched controls, despite pervasive gray and white matter deficits and reduced total cerebral volumes. 3D cortical surface models were extracted from 82 T1-weighted brain MRI scans (256 x 192 x 124 volumes) of 42 subjects with genetically confirmed WS (mean +/- SD, 29.2 +/- 9.0 years of age; 19 males, 23 females) and 40 age-matched healthy controls (27.5 +/- 7.4 years of age; 16 males, 24 females). A cortical pattern-matching technique used 72 sulcal landmarks traced on each brain as anchors to align cortical thickness maps across subjects, build group average maps, and identify regions with altered cortical thickness in WS. Cortical models were remeshed in frequency space to compute their fractal dimension (surface complexity) for each hemisphere and lobe. Surface complexity was significantly increased in WS (p < 0.0015 and p < 0.0014 for left and right hemispheres, respectively) and correlated with temporoparietal gyrification differences, classified via Steinmetz criteria. In WS, cortical thickness was increased by 5-10% in a circumscribed right hemisphere perisylvian and inferior temporal zone (p < 0.002). Spatially extended cortical regions were identified with increased complexity and thickness; cortical thickness and complexity were also positively correlated in controls (p < 0.03). These findings visualize cortical zones with altered anatomy in WS, which merit additional study with techniques to assess function and connectivity.

Segmenting magnetic resonance images via hierarchical mixture modelling

We present a statistically innovative as well as scientifically and practically relevant method for automatically segmenting magnetic resonance images using hierarchical mixture models. Our method is a general tool for automated cortical analysis which promises to contribute substantially to the science of neuropsychiatry. We demonstrate that our method has advantages over competing approaches on a magnetic resonance brain imagery segmentation task.

A stochastic model for studying the laminar structure of cortex from MRI

The human cerebral cortex is a laminar structure about 3 mm thick, and is easily visualized with current magnetic resonance (MR) technology. The thickness of the cortex varies locally by region, and is likely to be influenced by such factors as development, disease and aging. Thus, accurate measurements of local cortical thickness are likely to be of interest to other researchers. We develop a parametric stochastic model relating the laminar structure of local regions of the cerebral cortex to MR image data. Parameters of the model include local thickness, and statistics describing white, gray and cerebrospinal fluid (CSF) image intensity values as a function of the normal distance from the center of a voxel to a local coordinate system anchored at the gray/white matter interface. Our fundamental data object, the intensity-distance histogram (IDH), is a two-dimensional (2-D) generalization of the conventional 1-D image intensity histogram, which indexes voxels not only by their intensity value, but also by their normal distance to the gray/white interface. We model the IDH empirically as a marked Poisson process with marking process a Gaussian random field model of image intensity indexed against normal distance. In this paper, we relate the parameters of the IDH model to the local geometry of the cortex. A maximum-likelihood framework estimates the parameters of the model from the data. Here, we show estimates of these parameters for 10 volumes in the posterior cingulate, and 6 volumes in the anterior and posterior banks of the central sulcus. The accuracy of the estimates is quantified via Cramer-Rao bounds. We believe that this relatively crude model can be extended in a straightforward fashion to other biologically and theoretically interesting problems such as segmentation, surface area estimation, and estimating the thickness distribution in a variety of biologically relevant contexts.

Program for Assisted Labeling of Sulcal Regions (PALS): description and reliability

With the improvements in techniques for generating surface models from magnetic resonance (MR) images, it has recently become feasible to study the morphological characteristics of the human brain cortex in vivo. Studies of the entire surface are important for measuring global features, but analysis of specific cortical regions of interest provides a more detailed understanding of structure. We have previously developed a method for automatically segmenting regions of interest from the cortical surface using a watershed transform. Each segmented region corresponds to a cortical sulcus and is thus termed a "sulcal region." In this work, we describe two important augmentations of this methodology. First, we describe a user interface that allows for the efficient labeling of the segmented sulcal regions called the Program for Assisted Labeling of Sulcal Regions (PALS). An additional augmentation allows for even finer divisions on the cortex with a methodology that employs the fast marching technique to track a curve on the cortical surface that is then used to separate segmented regions. After regions of interest have been identified, we compute both the cortical surface area and gray matter volume. Reliability experiments are performed to assess both the long-term stability and short-term repeatability of the proposed techniques. These experiments indicate the proposed methodology gives both highly stable and repeatable results.

Computational anatomy: shape, growth, and atrophy comparison via diffeomorphisms

Computational anatomy (CA) is the mathematical study of anatomy I in I = I(alpha) o G, an orbit under groups of diffeomorphisms (i.e., smooth invertible mappings) g in G of anatomical exemplars I(alpha) in I. The observable images are the output of medical imaging devices. There are three components that CA examines: (i) constructions of the anatomical submanifolds, (ii) comparison of the anatomical manifolds via estimation of the underlying diffeomorphisms g in G defining the shape or geometry of the anatomical manifolds, and (iii) generation of probability laws of anatomical variation P(.) on the images I for inference and disease testing within anatomical models. This paper reviews recent advances in these three areas applied to shape, growth, and atrophy.

Direct and indirect effects of fetal irradiation on cortical gray and white matter volume in the macaque

BACKGROUND

Schizophrenia is associated with reductions in thalamic neuronal number and cortical gray matter volume. Exposure of nonhuman primates to x-irradiation in early gestation has previously been shown to decrease thalamic volume and neuronal number. Here we examine whether early gestational irradiation also results in cortical volume reduction.

METHODS

High-resolution, T1-weighted magnetic resonance scans were collected in adult monkeys 1) exposed to irradiation during the early gestational period (E33-E42) corresponding to thalamic neurogenesis, 2) irradiated in midgestation (E70-81) during neocortical neurogenesis, and 3) not exposed to irradiation. Cortical gray matter and white matter volumes were derived via manual segmentation; frontal and nonfrontal volumes were distinguished via sulcal landmarks.

RESULTS

Monkeys irradiated in early gestation exhibited a trend reduction in nonfrontal gray matter volume (17%) and significant reductions in white matter volume in frontal (26%) and nonfrontal (36%) lobes. Monkeys irradiated in midgestation had smaller gray (frontal: 28%; nonfrontal: 22%) and white matter (frontal: 29%; nonfrontal: 38%) volumes.

CONCLUSIONS

The cortical deficits observed in midgestationally irradiated monkeys are consistent with a reduction in cortical neuronal number. Cortical volume reductions following early gestational irradiation may be secondary to reduced thalamic neuronal number and therefore model the thalamocortical pathology of schizophrenia.

Validation of semiautomated methods for quantifying cingulate cortical metrics in schizophrenia

This paper validates semiautomated methods for reconstructing cortical surfaces of the cingulate gyrus from high-resolution magnetic resonance (MR) images. Bayesian segmentation was used to delineate the image voxels into five tissue types: cerebrospinal fluid (CSF), gray matter (GM), white matter (WM), and partial volumes of CSF/GM and GM/WM; the tissues were then recalibrated as CSF, GM, and WM via the Neyman-Pearson Likelihood Ratio Test. To generate cortical surfaces at the interface of GM and WM, the thresholds between the tissue types were first used to reassign partial volume voxels to CSF, GM, and WM with minimum error (that varied from 0.06 to 0.15 for the 10 subjects). Next, topology-correct cortical surfaces were generated and validated with almost all surface vertices lying within one voxel (0.5 mm) of hand contours. Dynamic programming was used to delineate and extract the cingulate gyrus from the cortical surfaces based on its gyral and sulcal boundaries. The intraclass correlation coefficient for surface area obtained by two raters for all 10 surfaces was 0.82. In addition, by repeating the entire procedure three times in one subject, we obtained a coefficient of variation of 0.0438 for surface area.

Computational anatomy and neuropsychiatric disease: probabilistic assessment of variation and statistical inference of group difference, hemispheric asymmetry, and time-dependent change

Three components of computational anatomy (CA) are reviewed in this paper: (i) the computation of large-deformation maps, that is, for any given coordinate system representations of two anatomies, computing the diffeomorphic transformation from one to the other; (ii) the computation of empirical probability laws of anatomical variation between anatomies; and (iii) the construction of inferences regarding neuropsychiatric disease states. CA utilizes spatial-temporal vector field information obtained from large-deformation maps to assess anatomical variabilities and facilitate the detection and quantification of abnormalities of brain structure in subjects with neuropsychiatric disorders. Neuroanatomical structures are divided into two types: subcortical structures-gray matter (GM) volumes enclosed by a single surface-and cortical mantle structures-anatomically distinct portions of the cerebral cortical mantle layered between the white matter (WM) and cerebrospinal fluid (CSF). Because of fundamental differences in the geometry of these two types of structures, image-based large-deformation high-dimensional brain mapping (HDBM-LD) and large-deformation diffeomorphic metric matching (LDDMM) were developed for the study of subcortical structures and labeled cortical mantle distance mapping (LCMDM) was developed for the study of cortical mantle structures. Studies of neuropsychiatric disorders using CA usually require the testing of hypothesized group differences with relatively small numbers of subjects per group. Approaches that increase the power for testing such hypotheses include methods to quantify the shapes of individual structures, relationships between the shapes of related structures (e.g., asymmetry), and changes of shapes over time. Promising preliminary studies employing these approaches to studies of subjects with schizophrenia and very mild to mild Alzheimer's disease (AD) are presented.

Labeled cortical mantle distance maps of the cingulate quantify differences between dementia of the Alzheimer type and healthy aging

The cingulate gyri in 37 subjects with and without early dementia of the Alzheimer type (DAT) were studied by using MRI at 1.0 mm3 isotropic resolution. Groups were segregated into young controls (n = 10), age-matched normal controls (n = 10), very mild DAT (n = 8), and mild DAT (n = 9). By using automated Bayesian segmentation of the cortex and gray matter/white matter (GM/WM) isosurface generation, tissue compartments were labeled into gray, white, and cerebrospinal fluid as a function of distance from the GM/WM isosurface. Cortical mantle distance maps are generated profiling the GM volume and cortical mantle distribution as a function of distance from the cortical surface. Probabilistic tests based on generalizations of Wilcoxon-Mann-Whitney tests were applied to quantify cortical mantle distribution changes with normal and abnormal aging. We find no significant change between young controls and healthy aging as measured by the GM volume and cortical mantle distribution as a function of distance in both anterior and posterior regions of the cingulate. Significant progression of GM loss is seen in the very mild DAT and mild DAT groups in all areas of the cingulate. Posterior regions show both GM volume loss as well as significant cortical mantle distribution decrease with the onset of mild DAT. The "shape of the cortical mantle" as measured by the cortical mantle distance profiles manifests a pronounced increase in variability with mild DAT.

Dynamic programming generation of boundaries of local coordinatized submanifolds in the neocortex: application to the planum temporale

Dynamic programming is used to define boundaries of cortical submanifolds with focus on the planum temporale (PT) of the superior temporal gyrus (STG), which has been implicated in a variety of neuropsychiatric disorders. To this end, automated methods are used to generate the PT manifold from 10 high-resolution MRI subvolumes ROI masks encompassing the STG. A procedure to define the subvolume ROI masks from original MRI brain scans is developed. Bayesian segmentation is then used to segment the subvolumes into cerebrospinal fluid, gray matter (GM), and white matter (WM). 3D isocontouring using the intensity value at which there is equal probability of GM and WM is used to reconstruct the triangulated graph representing the STG cortical surface, enabling principal curvature at each point on the graph to be computed. Dynamic programming is used to delineate the PT manifold by tracking principal curves from the retro-insular end of the Heschl's gyrus (HG) to the STG, along the posterior STG up to the start of the ramus and back to the retro-insular end of the HG. A coordinate system is then defined on the PT manifold. The origin is defined by the retro-insular end of the HG and the y-axis passes through the point on the posterior STG where the ramus begins. Automated labeling of GM in the STG is robust with L(1) distances between Bayesian and manual segmentation in the range 0.001-0.12 (n = 20). PT reconstruction is also robust with 90% of the vertices of the reconstructed PT within about 1 voxel (n = 20) from semiautomated contours. Finally, the reliability index (based on interrater intraclass correlation) for the surface area derived from repeated reconstructions is 0.96 for the left PT and 0.94 for the right PT, thus demonstrating the robustness of dynamic programming in defining a coordinate system on the PT. It provides a method with potential significance in the study of neuropsychiatric disorders.

On the metrics and euler-lagrange equations of computational anatomy

This paper reviews literature, current concepts and approaches in computational anatomy (CA). The model of CA is a Grenander deformable template, an orbit generated from a template under groups of diffeomorphisms. The metric space of all anatomical images is constructed from the geodesic connecting one anatomical structure to another in the orbit. The variational problems specifying these metrics are reviewed along with their associated Euler-Lagrange equations. The Euler equations of motion derived by Arnold for the geodesics in the group of divergence-free volume-preserving diffeomorphisms of incompressible fluids are generalized for the larger group of diffeomorphisms used in CA with nonconstant Jacobians. Metrics that accommodate photometric variation are described extending the anatomical model to incorporate the construction of neoplasm. Metrics on landmarked shapes are reviewed as well as Joshi's diffeomorphism metrics, Bookstein's thin-plate spline approximate-metrics, and Kendall's affine invariant metrics. We conclude by showing recent experimental results from the Toga & Thompson group in growth, the Van Essen group in macaque and human cortex mapping, and the Csernansky group in hippocampus mapping for neuropsychiatric studies in aging and schizophrenia.