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，用于模拟致病性朊蛋白的积累过程，这类蛋白是多种神经性疾病的关键诱因。在文献中，最受关注且研究最为深入的错误折叠蛋白当属β-淀粉样蛋白（Amyloid-β），该蛋白与阿尔茨海默病的发病直接相关。我们基于医学影像数据，构建了以图脑连接组为基础的低阶模型。考虑到诸多难以测量的底层物理过程，将蛋白质的反应系数建模为随机场。通过蒙特卡洛马尔可夫链方法，结合临床数据推断了该随机场的概率分布。所构建的模型具有患者特异性，可用于预测疾病的未来演变。本研究采用前向不确定性量化技术（蒙特卡洛模拟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程度。