Neural operator-based digital twins for modeling amyloid-$β$ and tau propagation and treatment optimization in Alzheimer's disease

Accurately predicting the spatiotemporal evolution of amyloid-$β$ and tau proteins at the individual level is critical for improving the diagnosis and treatment of Alzheimer's disease. We consider the problem of constructing patient-specific digital twins that model the propagation of these biomarkers on the cortical surface using reaction--diffusion dynamics. A major challenge is that the underlying nonlinear aggregation mechanisms are unknown and must be inferred from sparse, noisy, and heterogeneous longitudinal PET imaging data. To address this, we develop a data-driven framework that learns biomarker dynamics directly from clinical observations. The approach combines operator learning with reduced-order representations to infer governing equations of disease progression from data. Using this framework, we achieve predictive accuracies of 87\% for amyloid-$β$ and 81\% for tau. Building on the learned dynamics, we further formulate a PDE-constrained optimal control problem to design personalized therapeutic strategies that regulate pathological protein propagation. By integrating data-driven dynamical modeling with treatment optimization, the proposed digital twin framework provides an interpretable and predictive platform for understanding disease progression and enabling precision interventions in neurodegenerative disorders.