Optimal control for anti-abeta treatment in Alzheimer's disease using a reaction-diffusion model

Alzheimer's disease (AD) is a progressive neurodegenerative disorder that severely impairs survival and quality of life. While anti-amyloid beta (Aβ) therapies can slow disease progression, their efficacy depends on personalized dosing that maximizes benefits and minimizes risks, such as amyloid-related imaging abnormalities (ARIA). Mathematical modelling offers a powerful tool for understanding AD dynamics and optimizing treatment, yet most models focus solely on temporal behaviour, overlooking spatial heterogeneity within the brain. In this study, we propose a spatially explicit reaction-diffusion model to describe Aβ plaque dynamics. We formulate an optimal control problem to minimize plaque concentration while balancing therapeutic efficacy and treatment risk. Under reasonable assumptions, we establish well-posedness and uniqueness of the optimal solution. A finite element method (FEM)-based numerical framework is developed to compute personalized treatment strategies. Our model is calibrated using longitudinal Aβ positron emission tomography (PET) data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), enabling estimation of patient-specific parameters, such as growth rate and effective diffusivity. Results show that optimized treatment strategies consistently outperform constant dosing regimens across patient groups, achieving substantial reductions in cumulative amyloid burden while minimizing side effects. This integrated, data-driven framework advances personalized, spatially informed therapeutic optimization for AD.