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-反应偏微分方程，用于模拟朊蛋白的累积过程，这类蛋白是多种神经退行性疾病的诱因。在文献中，最重要且研究最广泛的错误折叠蛋白很可能是淀粉样蛋白-β，它是阿尔茨海默病的发病根源。基于医学图像，我们构建了一个以脑连接组图为基的降阶模型。蛋白质的反应系数被建模为随机场，以涵盖所有难以测量的复杂底层物理过程。通过蒙特卡洛马尔可夫链方法对临床数据进行推断，可获得该系数的概率分布。该模型具有患者特异性，可用于预测疾病的未来发展。我们应用前向不确定性量化技术（蒙特卡洛法和稀疏网格随机配置法），旨在评估反应系数变异性对未来20年内蛋白质累积进程的影响。