Biophysics-Based Data Assimilation of Longitudinal Tau and Amyloid- \(\beta \) PET Scans

Misfolded tau and amyloid- \(\beta \) ( \(\text {A}\beta \) ) are hallmark proteins of Alzheimer’s Disease (AD). Due to their clinical significance, rich datasets that track their temporal evolution have been created. For example, ADNI has hundreds of subjects with PET imaging of both these two proteins. Interpreting and combining this data beyond statistical correlations remains a challenge. Biophysical models offer a complementary avenue to assimilating such complex data and eventually helping us better understand disease progression. To this end, we introduce a mathematical model that tracks the dynamics of four species (normal and abnormal tau and \(\text {A}\beta \) ) and uses a graph to approximate their spatial coupling. The graph nodes represent gray matter regions of interest (ROI), and the edges represent tractography-based connectivity between ROIs. We model interspecies interactions, migration, proliferation, and clearance. Our biophysical model has seven unknown scalar parameters plus unknown initial conditions for tau and \(\text {A}\beta \) . Using imaging scans, we can calibrate these parameters by solving an inverse problem. The scans comprise longitudinal tau and \(\text {A}\beta \) PET scans, along with MRI for subject-specific anatomy. We propose an inversion algorithm that stably reconstructs the unknown parameters. We verify and test its numerical stability in the presence of noise using synthetic data. We discovered that the inversion is more stable when using multiple scans. Finally, we apply the overall methodology on 334 subjects from the ADNI dataset and compare it to a commonly used tau-only model calibrated by a single PET scan. We report the \(R^2\) and relative fitting error metrics. The proposed method achieves \(R^2=0.82\) compared to \(R^2=0.64\) of the tau-only single-scan reconstruction.