Modelling cerebrovascular pathology and the spread of amyloid beta in Alzheimer’s disease

Alzheimer’s disease (AD) is characterized in part by the accumulation and spread of amyloid beta proteins in the brain. Recent experiments have revealed that amyloid beta oligomers induce microvascular mural cells to contract, thereby constricting capillaries and increasing resistance to blood flow. Conversely, hypoperfusion promotes amyloid beta production and hinders its clearance, hence creating a pathogenic positive feedback loop. Here, we develop a mathematical model that combines protein–capillary interaction with the prion-like behaviour of amyloid beta. For sufficiently strong interaction, we find that healthy and diseased steady states, both stable, can exist simultaneously, implying that pathogenic protein seeds must exceed a critical threshold in order to trigger disease outbreak. We explore the consequences of this bistability for disease propagation through the brain’s structural connectome network. Finally, in a first attempt to model the AD two-hit vascular hypothesis mathematically, we describe how spatially localized deficits in blood supply, e.g. due to embolic stroke or atherosclerosis of the leptomeningeal vessels, may trigger disease outbreak and propagation.