Aggregates of misfolded tau proteins (or just "tau" for brevity) play a crucial role in the progression of Alzheimer's disease (AD) as they correlate with cell death and accelerated tissue atrophy. Longitudinal positron emission tomography (PET) is used to measure the extent of tau inclusions; and PET-based image biomarkers are a promising technology for AD diagnosis and prognosis. Here, we propose to combine URLan-scale biophysical mathematical model and longitudinal PET to extract characteristic growth patterns and spreading of tau. The biophysical model is a reaction-advection-diffusion partial differential equation (PDE) with only two unknown parameters representing the spreading (the diffusion part of the PDE) and  growth of tau (the reaction part of the PDE). The advection term captures tissue atrophy and is obtained from diffeomorphic registration of longitudinal magnetic resonance imaging (MRI) scans. We describe the method, present a numerical scheme for the calibration of the growth and spreading parameters, perform a sensitivity study using synthetic data, and apply it to clinical scans from the ADNI dataset. Despite having only two parameters, the model can reconstruct clinical scans quite accurately  .
Aggregates of misfolded tau proteins (or just 'tau' for brevity) play a crucial role in the progression of Alzheimer's disease (AD) as they correlate with cell death and accelerated tissue atrophy. Longitudinal positron emission tomography (PET) scans can be used quantify the extend of abnormal tau spread. Such PET-based image biomarkers are a promising technology for AD diagnosis and prognosis. Here, we propose to calibrate URLan-scale biophysical mathematical model using longitudinal PET scans to extract characteristic growth patterns and spreading of tau. The biophysical model is a reaction-advection-diffusion partial differential equation (PDE) with only two scalar unknown parameters, one representing the spreading (the diffusion part of the PDE) and the other one the growth of tau (the reaction part of the PDE). The advection term captures tissue atrophy and is obtained from diffeomorphic registration of longitudinal magnetic resonance imaging (MRI) scans. We describe the method, present a numerical scheme for the calibration of the growth and spreading parameters, perform a sensitivity study using synthetic data, and we perform a preliminary evaluation on clinical scans from the ADNI dataset. We study whether such model calibration is possible and investigate the sensitivity of such calibration to the time between consecutive scans and the presence of atrophy. Our findings show that despite using only two calibration parameters, the model can reconstruct clinical scans quite accurately . We discovered that small time intervals between scans and the presence of background noise create difficulties. Our reconstructed model fits the data well, yet the study on clinical data also reveals shortcomings of the simplistic model. Interestingly, the parameters show significant variability across patients, an indication that these parameters could be useful biomarkers .
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