Optimal Control For Anti-Abeta Treatment in Alzheimer's Disease using a Reaction-Diffusion Model

TL;DR

This work tackles the problem of optimizing anti-Abeta therapy by incorporating spatial heterogeneity of amyloid plaques through a reaction-diffusion PDE of Fisher–KPP type. It formulates a PDE-constrained optimal control problem with a cost that balances plaque reduction and treatment-side effects, proves well-posedness and local convexity for large $\alpha$, and solves the problem with a FEM-based Linear Combination Adjoint Method. The approach is calibrated on longitudinal ABeta PET data from ADNI to obtain patient-specific diffusion and growth parameters, enabling personalized treatment planning. Numerical results in 1D, 2D PET-informed, and 3D brain-surface settings show that the optimized dosing outperforms constant dosing in reducing cumulative amyloid burden while mitigating risks. The framework thus offers a data-driven, spatially informed pathway toward safer and more effective individualized AD therapies, with future work including additional biomarkers and multi-objective optimization.

Abstract

Alzheimer's disease (AD) is a progressive neurodegenerative disorder that severely impairs survival and quality of life. While anti-amyloid beta (Abeta) 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 modeling offers a powerful tool for understanding AD dynamics and optimizing treatment, yet most models focus solely on temporal behavior, overlooking spatial heterogeneity within the brain. In this study, we propose a spatially explicit reaction-diffusion model to describe Abeta 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 Abeta 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 Alzheimer's disease.

Figures (9)

• Figure 1: The difference defined as $\delta u(\mathbf{x}, t) = u(\mathbf{x}, t) - u^*(\mathbf{x}, t)$, where $u$ denotes the solution under the constant treatment $C$, and $u^*$ under the optimal treatment $C^*$. We set $T = 42$ and initialized $u_0$ using the PET images with $\alpha = 100$, $\rho = 0.012$, and $D = 0.002$ for Eq. \ref{['eq:model']} and Eq. \ref{['eq:objective']}.
• Figure 2: Left: The original PET scan of the brain for the AD patient. Middle: The mesh was generated based on the PET scan data. Right: The initial condition $u_0(x,y)$ generated from the PET scan data using the formula in Eq. \ref{['IC']}.
• Figure 3: The top row shows the temporal accumulation of the constant treatment, represented by $\int_0^t u(\mathbf{x},s)\,ds$, while the bottom row illustrates the accumulation under the optimal treatment, $\int_0^t u^*(\mathbf{x},s)\,ds$. Both quantities are displayed at selected time points ($t = 10, 20, 30, 40$), allowing visual comparison of the cumulative effects of the constant and optimal treatments over time.
• Figure 4: The difference defined as $\delta u(\mathbf{x}, t) = u(\mathbf{x}, t) - u^*(\mathbf{x}, t)$, where $u$ denotes the solution under the constant treatment $C$, and $u^*$ under the optimal treatment $C^*$. For five subjects across five different patient groups: Cognitively Normal (CN), Significant Memory Complaint (SMC) but clinically normal, Early Mild Cognitive Impairment (EMCI), Late Mild Cognitive Impairment (LMCI), and Alzheimer's Disease (AD). We set $T = 42$ and initialized $u_0$ using the PET images with $\alpha = 10^6$, $\rho = 0.012$, and $D = 0.002$ for Eq. \ref{['eq:model']} and Eq. \ref{['eq:objective']}.
• Figure 5: Visualization of the PET-based initial condition. Left: Original PET node data visualized as colored scatter points. Middle: Surface mesh generated from the brain geometry. Right: Initial condition $u_0$ interpolated over the mesh elements from the PET values.

Theorems & Definitions (17)

• Definition 2.1
• Lemma 2.2: Existence and Uniqueness of the weak solution
• Lemma 2.3: Uniform Bounds
• Lemma 3.1: Sensitivity Equation
• Lemma 3.2
• Lemma 3.3: Necessary Condition for Optimality
• proof
• Theorem 3.4: Local Strict Convexity of the Objective Functional
• proof
• Theorem 3.5: Existence and Uniqueness of the Optimal Control