  Longitudinal magnetic resonance imaging data is used to model trajectories of
change in brain regions of interest to identify areas susceptible to atrophy in
those with neurodegenerative conditions like Alzheimer's disease. Most methods
for extracting brain regions are applied to scans from study participants
independently, resulting in wide variability in shape and volume estimates of
these regions over time in longitudinal studies. To address this problem, we
propose a longitudinal principal manifold estimation method, which seeks to
recover smooth, longitudinally meaningful manifold estimates of shapes over
time. The proposed approach uses a smoothing spline to smooth over the
coefficients of principal manifold embedding functions estimated at each time
point. This mitigates the effects of random disturbances to the manifold
between time points. Additionally, we propose a novel data augmentation
approach to enable principal manifold estimation on self-intersecting
manifolds. Simulation studies demonstrate performance improvements over naive
applications of principal manifold estimation and principal curve/surface
methods. The proposed method improves the estimation of surfaces of
hippocampuses and thalamuses using data from participants of the Alzheimer's
Disease Neuroimaging Initiative. An analysis of magnetic resonance imaging data
from 236 individuals shows the advantages of our proposed methods that leverage
regional longitudinal trends for segmentation.
