## Probability Maps

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FIGURE 18 Brain distortions induced by tumor tissue: probability maps for major sulci in both hemispheres. Color-coded probability maps (right) quantify the impact of two focal metastatic tumors (illustrated in red; see cryosection blockface, left) on the supracallosal, parieto-occipital, and anterior and posterior calcarine sulci in both brain hemispheres. See also Plate 94.

FIGURE 19 Pathology detection in Alzheimer's disease. A color-coded probability map (a), shown on a 3D graphical surface model of an Alzheimer's patient's cortex, provides probability statements about the deviation of cortical regions from the norm. The inherent variability in normal cortical anatomy is encoded in the form of a surface-based probability field, known as an anisotropic lattice process, or a random vector field (see Section 4). Adaptive to their biological context, these algorithms use contextual information on anatomic variations before assigning a probability value to each cortical point. The resulting map exhibits regions of severely depressed probability values (p<0.00001), particularly in inferior frontal cortex. A probability map is also shown (b) for a normal control subject, age-matched to the AD patient and to subjects in the reference archive. 12.8% and 19.5% of the left and right inferior frontal cortex were severely abnormal (p< 0.00001) in the Alzheimer's patient, while only 0.21% and 0.37% of the same areas were indicated as abnormal in the control subject. The system is refined as the underlying database of subjects increases in size and content. See also Plate 95.

FIGURE 19 Pathology detection in Alzheimer's disease. A color-coded probability map (a), shown on a 3D graphical surface model of an Alzheimer's patient's cortex, provides probability statements about the deviation of cortical regions from the norm. The inherent variability in normal cortical anatomy is encoded in the form of a surface-based probability field, known as an anisotropic lattice process, or a random vector field (see Section 4). Adaptive to their biological context, these algorithms use contextual information on anatomic variations before assigning a probability value to each cortical point. The resulting map exhibits regions of severely depressed probability values (p<0.00001), particularly in inferior frontal cortex. A probability map is also shown (b) for a normal control subject, age-matched to the AD patient and to subjects in the reference archive. 12.8% and 19.5% of the left and right inferior frontal cortex were severely abnormal (p< 0.00001) in the Alzheimer's patient, while only 0.21% and 0.37% of the same areas were indicated as abnormal in the control subject. The system is refined as the underlying database of subjects increases in size and content. See also Plate 95.

with a Procrustes mean shape. Residual deformations that reflect individual change or anatomic difference are then expressed in terms of an orthogonal system of principal deformations derived from the bending energy matrix of the operator that governs the deformation [11]. A number of modes of variation, based on the eigenvectors of the covariance matrix of landmark positions, can be determined to describe the main factors by which the instance shapes tend to deform from the generic shape. Of particular relevance are methods used to define a mean shape in such a way that departures from this mean shape can be treated as a linear process. Linearization of the pathology detection problem, by constructing Riemannian shape manifolds and their associated tangent spaces, allows the use of conventional statistics and linear decomposition of departures from the mean to characterize shape change. These approaches have been applied to detect structural anomalies in schizophrenia [11,33].

### 4.10 Pattern-Theoretic Approaches

In a related approach based on pattern theory [59], a spectral approach to representing anatomic variation is developed. This was the first approach to build on the framework of deformable atlases by representing variation in terms of probabilistic transformations applied to deformable neuroanatomic templates. Deformation maps expressing variations in normal anatomies are calculated with a nonlinear registration procedure based on continuum mechanics [15,82]. As noted in Section 2, the deformational behavior of each subject's anatomy, driven into correspondence with other anatomies, is expressed as a system of partial differential equations. The equations are governed by a differential operator (such as the Laplacian V2, or Cauchy-Navier operator (1 + ^)V(V^) + ^V2) that controls the way in which one anatomy is deformed into the other. The properties of this operator can be used to make the deformation reflect the mechanical properties of deformable elastic or fluid media. Each deformation map is then expanded in terms of the eigenfunctions of the governing operator, and Gaussian probability measures are defined on the resulting sequences of expansion coefficients [1,59]. In Grenander's formalism, the distribution of the random deformation fields u(x) is assumed to satisfy the stochastic differential equation

Here L is the operator governing the deformation and e(x) is a 3 x 1 random noise vector field, whose coefficients in L's eigenbasis are zero-mean independent Gaussian variables with variances ak 2. If the differential operator L has eigenbasis {fk} with eigenvalues {1k}, a probability density can be defined directly on the deformation field's expansion coefficients

then p(Z1, •••, Zn) = exp —(1/2) = to n log{2^k2/1k2}

4.11 Learning Information on Anatomic Variability

Essentially this spectral formulation is a model of anatomic variability. The parameters of this model are learned from an anatomic image database. Model parameters include the ak, but even the eigenelements {1k, 9k} can be learned, if L is treated as a parameterized operator, in much the same way as the Green's functions' parameters were learned in the neural net registration model discussed earlier. Once the model parameters are learned, every subject's anatomy can be represented by a feature vector (z1; •••, zn), whose elements are just the coefficients of the deformation field required to match their particular anatomy with a mean anatomical template (e.g., Fig. 16). The probability of abnormality in a new subject can therefore be estimated from (26).

### 4.12 Disease Classification and Subtyping

A second opportunity arises if a disease-specific atlas is available. This type of atlas represents a homogeneous group of patients matched for age, gender, and relevant demographic factors [118,131]. If the parameters of anatomical variation are altered in disease, a pattern classifier can readily be constructed to classify new subjects according to their statistical distance from the diseased group mean relative to the normal group mean [66, 112]. From a validation standpoint, the operating characteristics of such a system can be investigated (i.e., false positives versus false negatives; [66,112]). Currently being tested as a framework for encoding anatomic variation, pattern-theoretic and other random tensor-based approaches build on the framework of deformable atlases and show considerable promise in the automated detection of pathology [62,66].

### 4.13 Pathology Detection in Image Databases

Pattern recognition algorithms for automated identification of brain structures can benefit greatly from encoded information on anatomic variability. We have developed a Bayesian approach to identify the corpus callosum in each image in an MRI database [88a]. The shape of a deformable curve (Fig. 20, panel 7) is progressively tuned to optimize a mathematical criterion measuring how likely it is that it has found the corpus callosum. The measure includes terms that reward contours based on their agreement with a diffused edge map (panels 7— 9), their geometric regularity, and their statistical abnormality when compared with a distribution of normal shapes. By

FIGURE 20 Automated detection of structures in image databases. Here an algorithm is used to find the corpus callosum boundary (panel 9) in each image in an anatomic database (N = 104; [88a]). The output of an edge detector (panel 2) is run through a connectivity filter that suppresses the smallest connected sets of edge pixels. The filtered edge image is then diffused over time (panels 4—6) and a deformable curve (panel 7) is adapted to optimize a matching measure (panel 10). This measure penalizes curve shapes that are too bent or stretched, that fail to overlap the diffused edge image, or that are unlikely based on a statistical distribution of normal corpus callosum shapes. Given an image database, algorithm parameters (such as the size of the connectivity filter; panel 11) can be tuned based on their overall performance on an image database. Their optimal values differ depending on how noisy the images are. Boundaries were averaged from patients with Alzheimer's disease and from elderly controls matched for age, educational level, gender, and handedness. Panel 12 shows a focal shape inflection in the Alzheimer's patients relative to normal elderly subjects of the same age, and a statistically significant tissue loss in the isthmus (the second sector, when the structure is partitioned into fifths). The isthmus connects regions of temporo-parietal cortex that exhibit early neuronal loss and perfusion deficits in AD [115]. See also Plate 96.

FIGURE 20 Automated detection of structures in image databases. Here an algorithm is used to find the corpus callosum boundary (panel 9) in each image in an anatomic database (N = 104; [88a]). The output of an edge detector (panel 2) is run through a connectivity filter that suppresses the smallest connected sets of edge pixels. The filtered edge image is then diffused over time (panels 4—6) and a deformable curve (panel 7) is adapted to optimize a matching measure (panel 10). This measure penalizes curve shapes that are too bent or stretched, that fail to overlap the diffused edge image, or that are unlikely based on a statistical distribution of normal corpus callosum shapes. Given an image database, algorithm parameters (such as the size of the connectivity filter; panel 11) can be tuned based on their overall performance on an image database. Their optimal values differ depending on how noisy the images are. Boundaries were averaged from patients with Alzheimer's disease and from elderly controls matched for age, educational level, gender, and handedness. Panel 12 shows a focal shape inflection in the Alzheimer's patients relative to normal elderly subjects of the same age, and a statistically significant tissue loss in the isthmus (the second sector, when the structure is partitioned into fifths). The isthmus connects regions of temporo-parietal cortex that exhibit early neuronal loss and perfusion deficits in AD [115]. See also Plate 96.

averaging contours derived from an image database, structural Agency, under Grant G-1-00001, by a Fellowship of the abnormalities associated with Alzheimer's Disease and schizo- Howard Hughes Medical Institute, and by a research grant phrenia were identified (Fig. 20, [85,115]). Automated from the U.S-U.K. Fulbright Commission, London. Special parameterization of structures will accelerate the identification thanks go to our colleagues Roger Woods, Michael Mega, Jay and analysis of disease-specific structural patterns. Giedd, David MacDonald, Colin Holmes, Alan Evans, and

John Mazziotta, whose advice and support have been invaluable in these investigations.

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