Uncertainty Quantification for Fisher-Kolmogorov Equation on Graphs with Application to Patient-Specific Alzheimer Disease

TL;DR

This paper develops a patient-specific stochastic model based on the Fisher-Kolmogorov equation on brain graphs to predict Alzheimer disease progression, incorporating uncertainty quantification of reaction rates inferred from clinical data.

Contribution

It introduces a novel uncertainty quantification framework for a graph-based PDE model of Alzheimer disease, integrating stochastic reaction coefficients derived from medical imaging and clinical data.

Findings

Uncertainty in reaction coefficients significantly affects disease progression predictions.

The model successfully predicts protein accumulation over 20 years.

Monte Carlo and sparse grid methods effectively quantify variability impacts.

Abstract

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.