N igilbglj idrf bgjbr dgjjbr

[(X*(i) V(V*)+tiV2]%(rt) +F(rrii(rl)M;u(r)=0 if&S,

FIGURE 11 Covariant tensor approach to cortical matching. Current approaches for deforming one cortex into the shape of another typically simplify the problem by first representing cortical features on a 2D plane, sphere, or ellipsoid, where the matching procedure (i.e., finding u(r2), above) is subsequently performed. Although these simple two-parameter surfaces serve as proxies for the cortex, different amounts of local dilation and contraction (g^ (r)) are required to transform the cortex into a simpler two-parameter surface. These variations complicate the direct application of 2D regularization equations for matching their features. A covariant tensor approach is introduced to address this difficulty. The regularization operator L is replaced by its covariant form L*, in which correction terms rjt compensate for fluctuations in the metric tensor of the flattening procedure. See also Plate 88.

0.2 million parameters (Fig. 12). This high-dimensional parameterization of the transformation is required to accommodate fine anatomical variations (cf. [16]).

4 Pathology Detection 4.1 Encoding Brain Variation

When applied to two different 3D brain scans, a nonlinear registration or warping algorithm calculates a deformation map (Figs 7, 12) defining the spatial relationship between them. The deformation map describes the 3D patterns of anatomic differences. Several probabilistic approaches have been proposed for encoding variation based on deformation maps, both for brain image analysis [1, 58,113, 114] and in the engineering literature on deformable templates (e.g. [135]). By defining probability distributions on the space of deformation transformations applied to a prototypical template [1,57,58], statistical parameters of these distributions can be estimated from databased anatomic data to determine the magnitude and directional biases of anatomic variation. Encoding of local variation can then be used to assess the severity of structural variants outside of the normal range, which may be related to disease [112].

4.2 Emerging Patterns

Cortical patterns are altered in schizophrenia [68], Alzheimer's disease [115], and a wide variety of developmental disorders. By using specialized strategies for group averaging of anatomy, specific features of anatomy emerge that are not observed in individual representations because of their considerable variability. Group-specific patterns of cortical organization or asymmetry can then be mapped out and visualized [85, 118].

In one study [117], mappings that deform one cortex into gyral correspondence with another were used to create an average cortex for patients with mild to moderate Alzheimer's disease (AD). Thirty-six gyral curves for nine AD patients were transferred to the cortical parameter space (as in Fig. 9b) and uniformly reparameterized, and a set of36 average gyral curves for the group was created by vector averaging of point locations on each curve. Each individual cortical pattern was then aligned with the average curve set using a spherical flow field (as in Fig. 9c). These nine flow fields were then used to create an average cortex in 3D space, as follows. If a code (that indexes 3D locations) is carried along with the flow that aligns each individual with the average folding pattern (cf. Fig. 9c), information can then be recovered at a particular location in the average folding pattern, specifying the 3D cortical points that map to it in each subject. By ruling a regular grid over the warped coded map, and reading off 3D position values for each subject, cortical positions in any subject's original 3D anatomy can be recovered. This produces a new coordinate grid on a given subject's cortex, in which particular grid points appear in the same location relative to the primary gyral pattern across all subjects (see [41] for a similar approach). By averaging these 3D positions across subjects, an average 3D cortical model was constructed for the group (Fig. 13). The resulting mapping is guaranteed to average together all points falling on the same cortical locations across the set of brains, and ensures that corresponding cortical features are averaged together.

4.3 Mapping Cortical Variability

By using the code to identify original cortical locations in 3D space, displacement maps can be recovered mapping each patient into gyrus-by-gyrus correspondence with the average cortex (Fig. 12). Anatomic variability can thus be defined at each point on the average mesh as the root mean square magnitude of the 3D displacement vectors, assigned to each point, in the surface maps from individual to average. This variability pattern is visualized as a color-coded map (Fig. 14).

FIGURE 12 Matching an individual's cortex to the average cortex. 3D variability patterns across the cortex are measured by driving individual cortical patterns into local correspondence with the average cortical model. (a) shows how the anatomy of one subject (shown as a brown surface mesh) deviates from the average cortex (shown in white), after affine alignment of the individual data. (b) shows the deformation vector field required to reconfigure the gyral pattern of the subject into the exact configuration of the average cortex. The transformation is shown as a flow field that takes the individual's anatomy onto the right hemisphere of the average cortex (blue surface mesh). The largest amount of deformation is required in the temporal and parietal cortex (pink colors, large deformation). Details of the 3D vector deformation field ((b), inset) show the local complexity of the mapping. (c) Mapping a patient into the group average configuration. Instead of matching just the cortex, this figure shows the complex transformation required to match 84 different surface models in a given patient, after affine alignment, into the configuration of an average surface set derived for the group (see Thompson et al., [117] for details). The effects of several anatomic surfaces driving the transformation are indicated, including the cingulate sulcus (CING), hippocampal surface (HPCP), superior ventricular horn (VTS), parieto-occipital sulcus, and the anterior calcarine fissure (CALCa). This surface-based vector field is extended to a full volumetric transformation field (0.1 billion degrees of freedom) that reconfigures the anatomy of the patient into correspondence with the average configuration for the group. Storage of these mappings allows quantification of local anatomic variability. See also Plate 89.

FIGURE 12 Matching an individual's cortex to the average cortex. 3D variability patterns across the cortex are measured by driving individual cortical patterns into local correspondence with the average cortical model. (a) shows how the anatomy of one subject (shown as a brown surface mesh) deviates from the average cortex (shown in white), after affine alignment of the individual data. (b) shows the deformation vector field required to reconfigure the gyral pattern of the subject into the exact configuration of the average cortex. The transformation is shown as a flow field that takes the individual's anatomy onto the right hemisphere of the average cortex (blue surface mesh). The largest amount of deformation is required in the temporal and parietal cortex (pink colors, large deformation). Details of the 3D vector deformation field ((b), inset) show the local complexity of the mapping. (c) Mapping a patient into the group average configuration. Instead of matching just the cortex, this figure shows the complex transformation required to match 84 different surface models in a given patient, after affine alignment, into the configuration of an average surface set derived for the group (see Thompson et al., [117] for details). The effects of several anatomic surfaces driving the transformation are indicated, including the cingulate sulcus (CING), hippocampal surface (HPCP), superior ventricular horn (VTS), parieto-occipital sulcus, and the anterior calcarine fissure (CALCa). This surface-based vector field is extended to a full volumetric transformation field (0.1 billion degrees of freedom) that reconfigures the anatomy of the patient into correspondence with the average configuration for the group. Storage of these mappings allows quantification of local anatomic variability. See also Plate 89.

Sulcal Pattern Matching

3D Average Anatomy

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