Riemannian Shape Analysis of the Corpus Callosum in Kendall Space: Aging and Alzheimer's Disease

The corpus callosum (CC) is a major white-matter structure and a well-established marker of brain aging, but most studies quantify it using scalar summaries that discard its boundary geometry. We present a Riemannian shape-space framework for analyzing age-related morphological change in the midsagittal CC, applied to the OASIS-1 cohort. Each contour is represented by $128$ landmarks and embedded into Kendall shape space, where translation, rotation, and scale are removed. We derive a multivariate geodesic regression with exact Riemannian gradients and use the fitted age-velocity field to localize age-related deformation to five anatomical sub-regions. In the cognitively normal cohort ($n=252$), geodesic regression outperforms the Euclidean linear benchmark ($R^2=0.1355$ vs.\ $0.1216$). Regional energy is posterior-dominant: the Splenium carries $41.4\mathrm{\%}$ and the Isthmus $23.8\mathrm{\%}$ of total age-related shape change, together accounting for $\sim 65\mathrm{\%}$ despite comprising only $\sim 35\mathrm{\%}$ of landmarks. Signed projections confirm the ordering (Splenium $r=0.570$; Isthmus $r=0.473$). In contrast, age explains less than $0.5\mathrm{\%}$ of shape variance in Alzheimer’s disease ($n=88$), indicating that the disease disrupts the healthy aging trajectory. A tangent-space classifier achieves an age-group AUC of $0.791$ from the 2D contour alone, exceeding a recent volumetric benchmark ($0.67$).