Geodesic distance on a Grassmannian for monitoring the progression of Alzheimer's disease

We propose a geodesic distance on a Grassmannian manifold that can be used to quantify the shape progression patterns of the bilateral hippocampi, amygdalas, and lateral ventricles in healthy control (HC), mild cognitive impairment (MCI), and Alzheimer's disease (AD). Longitudinal magnetic resonance imaging (MRI) scans of 754 subjects (3092 scans in total) were used in this study. Longitudinally, the geodesic distance was found to be proportional to the elapsed time separating the two scans in question. Cross-sectionally, utilizing a linear mixed-effects statistical model, we found that each structure's annualized rate of change in the geodesic distance followed the order of AD>MCI>HC, with statistical significance being reached in every case. In addition, for each of the six structures of interest, within the same time interval (e.g., from baseline to the 6th month), we observed significant correlations between the geodesic distance and the cognitive deterioration as quantified by the ADAS-cog increase and the MMSE decrease. Furthermore, as the disease progresses over time, this linkage between the inter-shape geodesic distance and the cognitive decline becomes considerably stronger and more significant.