Although several metrics have been proposed to measure brain atrophy, cortical thickness, cortical volume, and gray matter density are the most popular measures due to their intuitive interpretations.

Gray matter density maps are derived from anatomical MRI scans after normalized images are segmented into three different tissue types, specifically gray matter, white matter, and cerebrospinal fluid (CSF). The gray matter density maps are then generated by computing the concentration or amount of gray matter within each voxel of the brain. Hence, higher gray matter density is associated with a greater concentration of neurons and synapses, while lower density indicates potential neuronal loss or atrophy. While gray matter density maps can be thought of as the relative concentration of gray matter to other tissue types, and their modulation with the relative voxel volumes, as measured by the Jacobian determinants of the deformation field, can be considered as the absolute amount of gray matter.

After brain image segmentation, two computational models for surface representation of the white matter and pial surfaces are constructed. Here, the white matter surface is defined as the boundary between the gray matter and white matter, while the pial surface is the outer boundary of the cortex, separating gray matter from the cerebrospinal fluid. Thus, the cortical thickness maps are calculated as the shortest distance between the white matter surface and the pial surface representations at each vertex. Similarly, cortical volume maps are computed as the local volumes between the white matter and pial surfaces.

Since the dimensions of the large-scale vertices-by-voxels cross-correlation matrix between thickness and tau is typically larger than its rank, its exploration for revealing spatial patterns becomes impractical.

Therefore, dimensionality reduction techniques based on matrix decompositions, like SVD, are in order. In practice, the full cross-correlation matrix is approximated by the first few components, ordered according to their corresponding percentage of the explained co-variability. By algebraic properties of the SVD techniques, one can easily obtain spatial eigenimages for each imaging modality, which can be interpreted as snapshots or simplified views of the most relevant cross-correlation patterns. Also, the individual scores associated with such eigenimages can be submitted to statistical inference by assuming linear relationship and interactions within the framework of a General Linear Model (GLM).