The Impact of Novel Treatments on Aβ Burden in Alzheimer’s Disease: Insights from A Mathematical Model

Abstract

Motivated by recent therapeutic initiatives for Alzheimer's disease, we developed a mathematical model of the accumulation of amyloid β-protein (Aβ) in the brain. The model incorporates the production and clearance of Aβ monomers, and the elongation and fragmentation of polymers by monomer aggregation and break-off, respectively. Our analysis suggests that Aβ dynamics are dictated by a single unitless measure referred to as the polymerization ratio, which is the product of the production and elongation rates divided by the product of the clearance and fragmentation rates. Cerebral Aβ burden (i.e., the total number of Aβ molecules, whether they exist as monomers or polymers) attains a finite steady-state level if this ratio is less than one, and undergoes sustained growth if this ratio is greater than one. The highly nonlinear relationship between the polymerization ratio and the steady-state Aβ burden implies that a modest reduction in the polymerization ratio achieves a significant decrease in the Aβ burden. Our model also predicts that after initiation or discontinuation of treatment, it may take months to reach a new steady-state Aβ burden. Taken together, our findings suggest that the research community should focus on developing agents that provide a modest reduction of the polymerization ratio while avoiding long-term toxicity. Finally, our model can be used to indirectly estimate several crucial parameters that are difficult to measure directly: the production rate, the fragmentation rate and the strength of treatment.