Numerical modeling of senile plaque development under conditions of limited diffusivity of amyloid-β monomers

This paper introduces a new model to simulate the progression of senile plaques, focusing on scenarios where concentrations of amyloid beta (Aβ) monomers and aggregates vary between neurons. Extracellular variations in these concentrations may arise due to limited diffusivity of Aβ monomers and a high rate of Aβ monomer production at lipid membranes, requiring a substantial concentration gradient for diffusion-driven transport of Aβ monomers. The dimensionless formulation of the model is presented, which identifies four key dimensionless parameters governing the solutions for Aβ monomer and aggregate concentrations, as well as the radius of a growing Aβ plaque within the control volume. These parameters include the dimensionless diffusivity of Aβ monomers, the dimensionless rate of Aβ monomer production, and the dimensionless half-lives of Aβ monomers and aggregates. A dimensionless parameter is then introduced to evaluate the validity of the lumped capacitance approximation. An approximate solution is derived for the scenario involving large diffusivity of Aβ monomers and dysfunctional protein degradation machinery, resulting in infinitely long half-lives for Aβ monomers and aggregates. In this scenario, the concentrations of Aβ aggregates and the radius of the Aβ plaque depend solely on a single dimensionless parameter that characterizes the rate of Aβ monomer production. According to the approximate solution, the concentration of Aβ aggregates is linearly dependent on the rate of monomer production, and the radius of an Aβ plaque is directly proportional to the cube root of the rate of monomer production. However, when departing from the conditions of the approximate solution (e.g., finite half-lives), the concentrations of Aβ monomers and aggregates, along with the plaque radius, exhibit complex dependencies on all four dimensionless parameters. For instance, under physiological half-life conditions, the plaque radius reaches a maximum value and stabilizes thereafter.