  Misfolded tau proteins play a critical role in the progression and pathology
of Alzheimer's disease. Recent studies suggest that the spatio-temporal pattern
of misfolded tau follows a reaction-diffusion type equation. However, the
precise mathematical model and parameters that characterize the progression of
misfolded protein across the brain remain incompletely understood. Here, we use
deep learning and artificial intelligence to discover a mathematical model for
the progression of Alzheimer's disease using longitudinal tau positron emission
tomography from the Alzheimer's Disease Neuroimaging Initiative database.
Specifically, we integrate physics informed neural networks (PINNs) and
symbolic regression to discover a reaction-diffusion type partial differential
equation for tau protein misfolding and spreading. First, we demonstrate the
potential of our model and parameter discovery on synthetic data. Then, we
apply our method to discover the best model and parameters to explain tau
imaging data from 46 individuals who are likely to develop Alzheimer's disease
and 30 healthy controls. Our symbolic regression discovers different misfolding
models $f(c)$ for two groups, with a faster misfolding for the Alzheimer's
group, $f(c) = 0.23c^3 - 1.34c^2 + 1.11c$, than for the healthy control group,
$f(c) = -c^3 +0.62c^2 + 0.39c$. Our results suggest that PINNs, supplemented by
symbolic regression, can discover a reaction-diffusion type model to explain
misfolded tau protein concentrations in Alzheimer's disease. We expect our
study to be the starting point for a more holistic analysis to provide
image-based technologies for early diagnosis, and ideally early treatment of
neurodegeneration in Alzheimer's disease and possibly other misfolding-protein
based neurodegenerative disorders.
