  Cognitive decline due to Alzheimer's disease (AD) is closely associated with
brain structure alterations captured by structural magnetic resonance imaging
(sMRI). It supports the validity to develop sMRI-based univariate
neurodegeneration biomarkers (UNB). However, existing UNB work either fails to
model large group variances or does not capture AD dementia (ADD) induced
changes. We propose a novel low-rank and sparse subspace decomposition method
capable of stably quantifying the morphological changes induced by ADD.
Specifically, we propose a numerically efficient rank minimization mechanism to
extract group common structure and impose regularization constraints to encode
the original 3D morphometry connectivity. Further, we generate
regions-of-interest (ROI) with group difference study between common subspaces
of $A\beta+$ AD and $A\beta-$ cognitively unimpaired (CU) groups. A univariate
morphometry index (UMI) is constructed from these ROIs by summarizing
individual morphological characteristics weighted by normalized difference
between $A\beta+$ AD and $A\beta-$ CU groups. We use hippocampal surface radial
distance feature to compute the UMIs and validate our work in the Alzheimer's
Disease Neuroimaging Initiative (ADNI) cohort. With hippocampal UMIs, the
estimated minimum sample sizes needed to detect a 25$\%$ reduction in the mean
annual change with 80$\%$ power and two-tailed $P=0.05$ are 116, 279 and 387
for the longitudinal $A\beta+$ AD, $A\beta+$ mild cognitive impairment (MCI)
and $A\beta+$ CU groups, respectively. Additionally, for MCI patients, UMIs
well correlate with hazard ratio of conversion to AD ($4.3$, $95\%$
CI=$2.3-8.2$) within 18 months. Our experimental results outperform traditional
hippocampal volume measures and suggest the application of UMI as a potential
UNB.
