Parallel transport in diffeomorphisms distinguishes the time-dependent pattern of hippocampal surface deformation due to healthy aging and the dementia of the Alzheimer's type

Abstract

Hippocampal surface structure was assessed at twice 2 years apart in 26 nondemented subjects (CDR 0), in 18 subjects with early dementia of Alzheimer type (DAT, CDR 0.5), and in 9 subjects who converted from the nondemented (CDR 0) to the demented (CDR 0.5) state using magnetic resonance (MR) imaging. We used parallel transport in diffeomorphisms under the large deformation diffeomorphic metric mapping framework to translate within-subject deformation of the hippocampal surface as represented in the MR images between the two time points in a global template coordinate system. We then performed hypothesis testing on the longitudinal variation of hippocampal shape in the global template. Both subjects with early DAT and converters showed greater rates of hippocampal deformation across time than nondemented controls within every subfield of the hippocampus. In a random field analysis, inward surface deformation across time occurred in a non-uniform manner across the hippocampal surface in subjects with early DAT relative to the nondemented controls. Also, compared to the controls, the lateral aspect of the left hippocampal tail showed inward surface deformation in the converters. Using surface deformation patterns as features in a linear discriminant analysis, we were able to respectively distinguish converters and patients with early DAT from healthy nondemented controls at classification rates of 0.77 and 0.87, which were obtained in the same training set using the leave-one-out cross validation approach.

Fig. 1

Schematics of transport in diffeomorphisms for studying longitudinal shape variation. There are three levels of analysis. In the first level, the initial momentum ${\alpha }_0^{\left(j\right)}$ encoding the deformation of the hippocampus between the baseline and follow-up within subject j is computed via LDDMM surface mapping. The geodesic $x_t^{\left(j\right)}$ connecting subject j at the baseline to the global template is also computed via LDDMM surface mapping in the second level of the analysis. Finally, ${\alpha }_0^{\left(j\right)}$ is parallel transported to the global template along $x_t^{\left(j\right)}$ via the technique described in Methods.

Fig.2

Panel (A) shows the hippocampus of a subject at the baseline. The hippocampus surface of this subject at the follow-up (green) is superimposed with one at the baseline in Panel (B). Panel (C) shows the global hippocampus template. Panel (D) shows the hippocampus of this subject at the follow-up (green) represented in the global template coordinates (gray).

Fig. 3

The global left and right hippocampal templates were manually divided into three subfields, subiculum in green, cornu ammonis 1 (CA1) in red, and the rest (CA2,3,4, and gyrus dentatus) in blue.

Fig. 4

Figure shows Jacobian determinant of the deformation between the baseline and its follow-up surfaces averaged over each clinical population in the global template coordinates. Panels (A, C, E) illustrate those for the left hippocampus, while panels (B, D, F) show those for the right hippocampus. The bottom and top views of the hippocampus are respectively shown in the left and right sides of each panel (see the reference of subfield divisions in Figure 3).

Fig. 5

Panels (A, B) show average Jacobian determinant within each subfield of the hippocampus. Each data point represents one measurement from one subfield of the left or right hippocampus of 53 subjects.

Fig. 6

Statistically significant shape difference between diagnostic groups is constructed using the eigenfunctions of the Laplace-Beltrami operator in the global template coordinates. The left and right columns respectively correspond to left and right hippocampi. Color encodes the Jacobian difference between two groups. The warm color denotes that the outward surface deformation in the former group relative to the later group, while the cool color corresponds to inward surface deformation.