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

This paper introduces a method to simulate the progression of senile plaques, focusing on scenarios where concentrations of amyloid beta (A{beta}) monomers and aggregates vary between neurons. Extracellular variations in these concentrations may arise due to limited diffusivity of A{beta} monomers and a high rate of A{beta} monomer production at lipid membranes, requiring a substantial concentration gradient for diffusion-driven transport of A{beta} monomers. The dimensionless formulation of the model is presented, identifying four key dimensionless parameters governing the solutions for A{beta} monomer and aggregate concentrations, as well as the radius of a growing A{beta} plaque within the control volume. These parameters include the dimensionless diffusivity of A{beta} monomers, the dimensionless rate of A{beta} monomer production, and the dimensionless half-lives of A{beta} monomers and aggregates. A dimensionless parameter is introduced to assess the validity of the lumped capacitance approximation. An approximate solution is derived for the scenario involving large diffusivity of A{beta} monomers and dysfunctional protein degradation machinery, resulting in infinitely long half-lives for A{beta} monomers and aggregates. In this scenario, the concentrations of A{beta} aggregates and the radius of the A{beta} plaque depend solely on a single dimensionless parameter that characterizes the rate of A{beta} monomer production. According to the approximate solution, the concentration of A{beta} aggregates is linearly dependent on the rate of monomer production, and the radius of an A{beta} 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 A{beta} 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.