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

Mattia Corti, Francesca Bonizzoni, Paola F. Antonietti, Alfio M. Quarteroni

TL;DR

The study advances patient-specific Alzheimer’s progression modeling by coupling a graph-based Fisher-Kolmogorov framework with uncertainty quantification. It calibrates a piecewise-constant reaction parameter from PET data using MCMC (inverse UQ) and propagates this uncertainty through forward UQ via Monte Carlo and sparse-grid stochastic collocation to predict Amyloid-$\beta$ accumulation over $t\in[0,20]$ years on a brain connectome. The results show regionally heterogeneous parameter distributions and demonstrate that sparse-grid collocation provides higher accuracy than Monte Carlo for a given number of PDE solves, while revealing increasing uncertainty over time. This approach yields quantified projections that could inform clinical decisions and motivates extensions to 3D geometries and multi-patient studies.

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-$β$, 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.

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

TL;DR

The study advances patient-specific Alzheimer’s progression modeling by coupling a graph-based Fisher-Kolmogorov framework with uncertainty quantification. It calibrates a piecewise-constant reaction parameter from PET data using MCMC (inverse UQ) and propagates this uncertainty through forward UQ via Monte Carlo and sparse-grid stochastic collocation to predict Amyloid- accumulation over years on a brain connectome. The results show regionally heterogeneous parameter distributions and demonstrate that sparse-grid collocation provides higher accuracy than Monte Carlo for a given number of PDE solves, while revealing increasing uncertainty over time. This approach yields quantified projections that could inform clinical decisions and motivates extensions to 3D geometries and multi-patient studies.

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-, 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.
Paper Structure (13 sections, 19 equations, 7 figures, 3 tables, 2 algorithms)

This paper contains 13 sections, 19 equations, 7 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Discrete formulation
  4. UQ methods for FK with stochastic reaction on graphs
  5. Inverse Uncertainty Quantification
  6. Forward Uncertainty Quantification
  7. MC method
  8. Sparse grid SC method
  9. Numerical results
  10. Graph reconstruction and PET projection
  11. Estimation of parameter distributions
  12. Prediction of disease development
  13. Conclusions

Figures (7)

  • Figure 1: Schematic representation of a weighted undirected graph.
  • Figure 2: Graph discretization of the brain (each colour in Figures (c) and (d) denotes a different brain's region).
  • Figure 3: Local graphs of the seven regions of the brain and brain connectogram between different regions (excluding the connections with weight lower than $5\%$ of the principal one)
  • Figure 4: Data extracted from PET medical images. The first PET is acquired at 61 years, and the second one is acquired at 68 years.
  • Figure 5: Results of the MCMC algorithm. Histogram and Gaussian distribution associated with each lobe of the brain and comparison between the medical data and the numerical results ($T=20$).
  • ...and 2 more figures