Numerical and Analytical Simulation of the Growth of Amyloid-β Plaques

Abstract

Numerical and analytical solutions were employed to calculate the radius of an amyloid-β (Aβ) plaque over time. To the author's knowledge, this study presents the first model simulating the growth of Aβ plaques. Findings indicate that the plaque can attain a diameter of 50 μm after 20 years of growth, provided the Aβ monomer degradation machinery is malfunctioning. A mathematical model incorporates nucleation and autocatalytic growth processes using the Finke–Watzky model. The resulting system of ordinary differential equations was solved numerically, and for the simplified case of infinitely long Aβ monomer half-life, an analytical solution was found. Assuming that Aβ aggregates stick together and using the distance between the plaques as an input parameter of the model, it was possible to calculate the plaque radius from the concentration of Aβ aggregates. This led to the “cube root hypothesis,” positing that Aβ plaque size increases proportionally to the cube root of time. This hypothesis helps explain why larger plaques grow more slowly. Furthermore, the obtained results suggest that the plaque size is independent of the kinetic constants governing Aβ plaque agglomeration, indicating that the kinetics of Aβ plaque agglomeration is not a limiting factor for plaque growth. Instead, the plaque growth rate is limited by the rates of Aβ monomer production and degradation.