Large deformation diffeomorphism and momentum based hippocampal shape discrimination in dementia of the Alzheimer type

Abstract

In large-deformation diffeomorphic metric mapping (LDDMM), the diffeomorphic matching of images are modeled as evolution in time, or a flow, of an associated smooth velocity vector field v controlling the evolution. The initial momentum parameterizes the whole geodesic and encodes the shape and form of the target image. Thus, methods such as principal component analysis (PCA) of the initial momentum leads to analysis of anatomical shape and form in target images without being restricted to small-deformation assumption in the analysis of linear displacements. We apply this approach to a study of dementia of the Alzheimer type (DAT). The left hippocampus in the DAT group shows significant shape abnormality while the right hippocampus shows similar pattern of abnormality. Further, PCA of the initial momentum leads to correct classification of 12 out of 18 DAT subjects and 22 out of 26 control subjects.

Fig. 1

Diffeomorphic flow.

Fig. 2

Correlations between displacement vector fields (u) and initial velocity vector fields (v₀). Only the surface vertices that show significant correlation were colored according to the correlation, others were colored as yellow-green. Top, middle and bottom row shows correlation between v₀ and u along the x, y, and z axes, respectively. Column a shows dorsal view (from the top) and column b shows ventral view (from the bottom) of the hippocampus.

Fig. 3

PCA of Lv₀. Row (1) shows the distribution of eigenvalues for all eigenfunctions. Row (2) shows the distribution of mean coefficients (CDR 0 and CDR 0.5 groups) associated with the first 20 principal components. Row (3) shows the permutation tests for group differences using the first 20 principal components. The p values shown are calculated from [see (12)]. Also shown are: 1) $\widehat{F}\left(T^2\right)$ value (solid blue line) of the Control-versus-DAT group comparison; 2) theoretical F-distribution (solid red curve) with (20,23) degrees of freedom superimposed on the empirical distribution; 3) p = .05 (red dotted line) and p = .01 (red dot-dash line) for reference. Column (a) is for the left hippocampus where the first 20 principal components account for 82.9% of the total variance. Column (b) is for the right hippocampus where the first 20 principal components account for 80.5% of the total variance.

Fig. 4

Visualization of the pattern of hippocampal deformities in subjects with very mild DAT (CDR 0.5) compared with nondemented subjects (CDR 0) (Data taken from [3]). The flame coloring represents the z -scores between the two groups of subjects. Inward variation of the hippocampal surface is represented by cooler colors (i.e., blue to purple), while outward variation is represented by warmer colors (i.e., orange to red). In (1a) the pair of hippocampal surfaces are shown from above, with the head of the hippocampus pointing toward the bottom edge of the figure panel, and the left hippocampus is on the right-hand side of the panel. In (1b) the hippocampal surfaces are shown from below, with the head of the hippocampus pointing toward the top edge of the figure panel, and the left hippocampus is on the right-hand side of the panel.