The Fisher-Kolmogorov equation is a diffusion-reaction PDE that is used to model the accumulation of prionic proteins, which are responsible for many different neurological disorders. Likely, the most important and studied misfolded protein in literature is the Amyloid-$\beta$, responsible for the onset of Alzheimer disease. Starting from medical images we construct a reduced-order model based on a graph brain connectome. The reaction coefficient of the proteins is modelled as a stochastic random field, taking into account all the many different underlying physical processes, which can hardly be measured. Its probability distribution is inferred by means of the Monte Carlo Markov Chain method applied to clinical data. The resulting model is patient-specific and can be employed for predicting the disease's future development. Forward uncertainty quantification techniques (Monte Carlo and sparse grid stochastic collocation) are applied with the aim of quantifying the impact of the variability of the reaction coefficient on the progression of protein accumulation within the next 20 years.

翻译：Fisher-Kolmogorov方程是一种扩散-反应型偏微分方程，用于模拟引发多种神经疾病的朊病毒蛋白积累过程。在文献中，最受关注且研究最深入的错误折叠蛋白当属β-淀粉样蛋白（Amyloid-β），它是导致阿尔茨海默病的元凶。我们从医学影像出发，基于大脑连接组图构建了降阶模型。考虑到众多难以测量的潜在物理过程，将蛋白质反应系数建模为随机场，并通过蒙特卡罗马尔可夫链方法结合临床数据推断其概率分布。该模型具有患者特异性，可用于预测疾病未来进展。采用前向不确定性量化技术（蒙特卡洛法与稀疏网格随机配置法），旨在量化反应系数变异性对未来20年内蛋白质积累进程的影响。