Dynamics of waste proteins in brain tissue: Numerical insights into Alzheimer's risk factors

Over the past few decades, research has indicated that the buildup of waste proteins, like amyloid-$\beta$ ($\mathrm{A}\beta$), in the brain's interstitial spaces is linked to neurodegenerative diseases like Alzheimer's, but the details of how such proteins are removed from the brain are not well understood. We have developed a numerical model to investigate the aggregation and clearance mechanisms of $\mathrm{A}\beta$ in the interstitial spaces of the brain. The model describes the volume-averaged transport of $\mathrm{A}\beta$ in a segment of the brain interstitium modeled as a porous medium, oriented between the perivascular space (fluid-filled channel surrounding a blood vessel) of a penetrating arteriole and that of a venule. Our numerical approach solves $N$ coupled advection-diffusion-aggregation equations that model the production, aggregation, fragmentation, and clearance of $N$ species of $\mathrm{A}\beta$. We simulate $N=50$ species to investigate the oligomer-size dependence of clearance and aggregation. We introduce a timescale plot that helps predict $\mathrm{A}\beta$ buildup for different neurological conditions. We show that a sudden increase in monomer concentration, as occurs in conditions like traumatic brain injury, leads to significant plaque formation, which can qualitatively be predicted using the timescale plot. Our results also indicate that impaired protein clearance (as occurs with aging) and fragmentation are both mechanisms that sustain large intermediate oligomer concentrations. Our results provide novel insight into several known risk factors for Alzheimer's disease and cognitive decline, and we introduce a unique framing of $\mathrm{A}\beta$ dynamics as a competition between different timescales associated with production rates, aggregation rates, and clearance conditions.