  Alzheimer's disease (AD) is characterized by complex and largely unknown
progression dynamics affecting the brain's morphology. Although the disease
evolution spans decades, to date we cannot rely on long-term data to model the
pathological progression, since most of the available measures are on a
short-term scale. It is therefore difficult to understand and quantify the
temporal progression patterns affecting the brain regions across the AD
evolution. In this work, we tackle this problem by presenting a generative
model based on probabilistic matrix factorization across temporal and spatial
sources. The proposed method addresses the problem of disease progression
modelling by introducing clinically-inspired statistical priors. To promote
smoothness in time and model plausible pathological evolutions, the temporal
sources are defined as monotonic and independent Gaussian Processes. We also
estimate an individual time-shift parameter for each patient to automatically
position him/her along the sources time-axis. To encode the spatial continuity
of the brain sub-structures, the spatial sources are modeled as Gaussian random
fields. We test our algorithm on grey matter maps extracted from brain
structural images. The experiments highlight differential temporal progression
patterns mapping brain regions key to the AD pathology, and reveal a
disease-specific time scale associated with the decline of volumetric
biomarkers across clinical stages.
