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 present 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.