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.

本研究使用深度学习和人工智能基于反应扩散方程发现了一种数学模型用于描述阿尔茨海默病中 Tau 蛋白的异常析聚和扩散，可能促进对神经退行性疾病的早期诊断和治疗。

结合 PINNs 与符号回归发现阿尔茨海默病的反应扩散模型

Discovering a reaction-diffusion model for Alzheimer's disease by  combining PINNs with symbolic regression

I show stable, localized, single and multi-spot patterns of three classes -
stationary, moving, and rotating - that exist within a limited range of
parameter values in the two-dimensional Gray-Scott reaction-diffusion model
with ${\sigma} = D_u/D_v = 2$. These patterns exist in domains of any size, and
appear to derive their stability from a constructive reinforcement effect of
the standing waves that surround any feature. There are several common elements
- including a spot that behaves as a quasiparticle, a U-shaped stripe, and a
ring or annulus, or a portion thereof - which combine to form a great variety
of stable structures. These patterns interact with each other in a variety of
ways. There are similarities to other reaction-diffusion systems and to
physical experiments; I offer several suggestions for further research.

该研究使用二维 Gray-Scott 反应扩散模型，展示出单个和多个稳定的、本地化的、静态、移动和旋转的模式。这些模式存在于特定参数范围内，并从包围任何特征的驻波的建设性加固效应中获得稳定性。这些模式相互作用，并与其他反应扩散系统和物理实验相似，并提出了进一步研究的若干建议。