Uncertainty quantification for Fisher-Kolmogorov equation on graphs with application to patient-specific Alzheimer’s disease^*

¹ MOX-Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milan 20133, Italy
² Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Station 8, Av. Piccard, Lausanne 1015, Switzerland

^** Corresponding author: mattia.corti@polimi.it

Received: 5 May 2023
Accepted: 17 November 2023

Abstract

The Fisher-Kolmogorov equation is a diffusion-reaction PDE that models the accumulation of prionic proteins, which are responsible for many different neurological disorders. The most important and studied misfolded protein in literature is the Amyloid-β, responsible for the onset of Alzheimer’s disease. Moving from medical images we construct a reduced-order model based on a graph brain connectome. The reaction coefficient of the proteins which can hardly be measured is modeled as a stochastic random field, taking into account all the many different underlying physical processes. 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.