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FIGURE 13 Average cortex in Alzheimer's disease. The average cortical surface for a group of subjects (N = 9, Alzheimer's patients) is shown as a graphically rendered surface model. If sulcal position vectors are averaged without aligning the intervening gyral patterns (top), sulcal features are not reinforced across subjects, and a smooth average cortex is produced. By matching gyral patterns across subjects before averaging, a crisper average cortex is produced (bottom). Sulcal features that consistently occur across all subjects appear in their average geometric configuration. See also Plate 90.

FIGURE 14 3D cortical variability in Talairach stereotaxic space. The profile of variability across the cortex is shown (N = 9 Alzheimer's patients), after differences in brain orientation and size are removed by transforming individual data into Talairach stereotaxic space. The following views are shown: oblique frontal, frontal, right, left, top, bottom. Extreme variability in posterior perisylvian zones and superior frontal association cortex (16-18 mm; red colors) contrasts sharply with the comparative invariance of primary sensory, motor, and orbitofrontal cortex (2-5 mm, blue colors). See also Plate 91.

FIGURE 13 Average cortex in Alzheimer's disease. The average cortical surface for a group of subjects (N = 9, Alzheimer's patients) is shown as a graphically rendered surface model. If sulcal position vectors are averaged without aligning the intervening gyral patterns (top), sulcal features are not reinforced across subjects, and a smooth average cortex is produced. By matching gyral patterns across subjects before averaging, a crisper average cortex is produced (bottom). Sulcal features that consistently occur across all subjects appear in their average geometric configuration. See also Plate 90.

This map shows the anatomic variability, due to differences in gyral patterning, that remains after aligning MR data into the Talairach stereotaxic space, using the transforms prescribed in the atlas [104],

4.4 Variability and Asymmetry

First, variability values rose sharply (Fig. 14) from 4-5 mm in primary motor cortex to localized peaks of maximum variability in posterior perisylvian zones and superior frontal association cortex (16-18 mm). Temporal lobe variability rose from 2-3 mm in the depths of the Sylvian fissure to 18 mm at the posterior limit of the inferior temporal sulcus in both brain hemispheres. Peak variability occurs in the vicinity of functional area MT [126] and extends into the posterior heteromodal association cortex of the parietal lobe (1418 mm). Second, there is a marked anatomic asymmetry in posterior perisylvian cortex ([52]; up to 10 mm). This asymmetry is not clearly apparent individually, but appears clearly in the average representation. It also contrasts sharply with negligible asymmetry in frontal, parietal, and occipital cortex (1-2mm). Similar studies of deep sulcal cortex found this asymmetry to be greater in Alzheimer's patients than in controls matched for age, gender, handedness, and educational level, corroborating earlier reports of an asymmetric progression of the disease [115]. The improved ability to localize asymmetry and encode its variability in a disease-specific atlas has encouraged us to develop a probabilistic atlas ofthe brain in schizophrenia [85] where cortical organization and functional lateralization are also thought to be altered ([68]; cf. [25]).

FIGURE 14 3D cortical variability in Talairach stereotaxic space. The profile of variability across the cortex is shown (N = 9 Alzheimer's patients), after differences in brain orientation and size are removed by transforming individual data into Talairach stereotaxic space. The following views are shown: oblique frontal, frontal, right, left, top, bottom. Extreme variability in posterior perisylvian zones and superior frontal association cortex (16-18 mm; red colors) contrasts sharply with the comparative invariance of primary sensory, motor, and orbitofrontal cortex (2-5 mm, blue colors). See also Plate 91.

4.5 Comparing Registration Methods

Elastic registration approaches all produce two types of output: first, they measure anatomic variability by transforming brain data to match another brain, or a common digital template; second, by creating registered images, they can reduce the confounding effects of anatomic variation when comparing other types of brain data. The relative performance of different algorithms should therefore be compared. The availability of large numbers of digital anatomical models stored in population-based brain atlases [117] presents opportunities to explore the strengths and limitations of different registration algorithms and measure their accuracy quantitatively.

In a recent project to generate an Alzheimer's disease brain atlas [79,117,131], parametric surface meshes [110] were created to model 84 structures per brain in 3D MRI scans of 9 Alzheimer's patients. The ability of different registration approaches to reduce anatomic variability was investigated. 3D patterns of residual structural variability were visualized after digitally mapping all structures from all subjects into several different coordinate systems, using the transformations described in the Talairach atlas [104], as well as automated affine and polynomial transformations of increasing com plexity [128, 130]. For all major anatomic systems (Fig. 15), automated affine registration reduced variability to a far greater degree than the Talairach system, with added benefit obtained by using higher-order polynomial mappings. As expected, the highest registration accuracy was achieved for brain structures with lowest complexity and topological variability across subjects (Fig. 15, top row). With increasing polynomial order, significantly better registration was observed across deep cortical sulci, and anatomical variation could be further reduced using model-driven algorithms that explicitly match gyral patterns.

4.6 Population-Based Brain Image Templates

Interestingly, the automated registration approaches were able to reduce anatomic variability to a slightly greater degree if a specially prepared image template was used as a registration target [117, 131]. Brain templates that have the mean global shape for a group of subjects can be created using Procrustes averaging [11] or transformation matrix averaging [130]. Brain templates that have mean shape and intensity characteristics at a local level can also be generated. Recent methods have included Bayesian pattern theory and variational calculus [59, 62], variable diffusion generators [117], or hierarchical registration of curves, surfaces, and volumes to a set of mean anatomical models [113, 115].

Several approaches, many of them based on high-dimensional image transformations, are under active development to create average brain templates. Average templates have been made for the Macaque brain [59], and for individual structures such as the corpus callosum [26, 51], central sulcus [75], cingulate and paracingulate sulci [88,114], and hippocampus

FIGURE 15 Comparing different registration approaches. The ability of different registration algorithms to reduce anatomic variability in a group of subjects (N = 9, Alzheimer's patients) is shown here. Digital anatomic models for each subject were mapped into common coordinate spaces using the transformations specified in the Talairach atlas (Tal; [104]), as well as automated affine (or first-order) and eighth-order polynomial mappings as implemented in the Automated Image Registration package [128, 130, 131]. After applying each type of mapping to the models from all subjects, the residual variability of ventricular (top row) and deep cortical surfaces (middle row) and superficial sulci (bottom row) is shown as a color-coded map across each structure. The color represents the 3D root mean square distance from individual models to an average model for each structure, where distance is measured to the near-point on the average mesh model [117]. As expected, polynomial transformations reduce variability more effectively than affine transformations, and both outperform the Talairach system. At the cortex, model-driven registration can be used to explicitly match gyral patterns, improving registration still further. See also Plate 92.

[25,62,66,117], as well as for transformed representations of the human and Macaque cortex [36,41,59,117]. Under various metrics, incoming subjects deviate least from these mean brain templates in terms of both image intensity and anatomy (Fig. 16). Registration of new data to these templates therefore requires minimal image distortion, allows faster algorithm convergence, and may help to avoid nonglobal stationary points of the registration measure as the parameter space is searched for an optimal match. For these reasons, templates that reflect the mean geometry and intensity of a group are a topic of active research [59,117,131].

4.7 Random Tensor Field Models

Analysis of variance in 3D deformation fields that match different subjects' anatomies shows considerable promise in being able to differentiate intrasubject (between hemisphere), intersubject, and intergroup contributions to brain variation in human populations [115]. In Thompson et al. [112], we developed an approach to detect structural abnormalities in individual subjects. Based on a reference image archive of brain scans from normal subjects, a family of volumetric warps was generated to encode statistical properties and directional biases of local anatomical variation throughout the brain (see Fig. 17). To identify differences in brain structure between two groups, or between a patient and the normal database, the following steps were followed. After affine components of the deformation fields are factored out, we defined Wj (x) as the deformation vector required to match the structure at position x in an atlas template with its counterpart in subject i of group j, and modeled the deformations as

FIGURE 16 Average brain templates and 3D cortical variability. Axial, sagittal, and coronal images are shown from a variety of population-based brain image templates. For comparison purposes, the left column shows a widely used average intensity dataset (ICBM305) based on 305 young normal subjects, created by the International Consortium for Brain Mapping [39]; by contrast, the middle and right columns show average brain templates created from high-resolution 3D MRI scans of Alzheimer's disease patients. (middle column): Affine Brain Template, constructed by averaging normalized MR intensities on a voxel-by-voxel basis data after automated affine registration; (right column): Continuum-Mechanical Brain Template, based on intensity averaging after continuum-mechanical transformation. By using spatial transformations of increasing complexity, each patient's anatomy can increasingly be reconfigured into the average anatomical configuration for the group. After intensity correction and normalization, the reconfigured scans are then averaged on a pixel-by-pixel basis to produce a group image template with the average geometry and average image intensity for the group. Anatomical features are highly resolved, even at the cortex. Transformations of extremely high spatial dimension are required to match cortical features with sufficient accuracy to resolve them after scans are averaged together. See also Plate 93.

Here ^(x) is the mean deformation for group j; £(x) is a nonstationary, anisotropic covariance tensor field, which relaxes the confidence threshold for detecting abnormal structure in regions where normal variability is extreme; E(x)1/2 is the upper triangular Cholesky factor tensor field; and £ij(x) is a trivariate random vector field whose components are independent stationary Gaussian random fields. Deviations from a mean deformation field based on normal subjects were modeled, for small N, as a Hotelling's T2-distributed random field, or as N^ro, as a y2 (chi-squared) distributed random field with three degrees of freedom, defined at nodes (u, v) in parametric mesh models of the anatomy of new subjects [112]. A T2 or F statistic that indicates evidence of significant difference in deformations between two groups of subjects, or between a single subject and the normal database, can be calculated. By calculating these statistics at each lattice location in a 3D image, or across a parameterized 3D anatomical surface, a statistic image is formed. The global maximum of the random deformation field, or derived tensor fields [115], can be used to identify the presence of structural change in disease [14,132,133]. Random field approaches, some of which are now widely used in packages for analyzing functional brain images [45], use the Euler characteristic (EC) of the excursion sets of a random field as an estimator of the number of local nonzero signal components, above a given threshold in a statistic image. They also use the expected value of the EC as an approximate p-value for the local maximum [132,133]. This probability value allows a definitive statement of whether a statistically abnormal structural difference has been detected or not.

Probabilistic atlases based on random deformation fields have been used to assess gender-specific differences in the brain [14] and to detect structural abnormalities induced by tumor

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