Quasi-Monte Carlo methods for uncertainty quantification of tumor growth modeled by a parametric semi-linear parabolic reaction-diffusion equation

2025-09-30

AI Summary

This study addresses the challenge of efficiently propagating parameter uncertainty in tumor growth models—governed by semilinear parabolic reaction-diffusion PDEs—to clinically relevant quantities of interest (QoIs). We propose a novel uncertainty quantification (UQ) framework that, for the first time, integrates quasi-Monte Carlo (QMC) methods into such biomedical PDE models. The approach combines uniform or lognormal random field representations for stochastic inputs with rigorous regularity analysis of the PDE solution to handle high-dimensional random parameter spaces. We derive theoretical QMC error bounds and numerically demonstrate its superior $O(N^{-1})$ convergence rate over standard Monte Carlo. The method enables efficient and accurate computation of expected values for multiple clinical QoIs, thereby establishing the feasibility and practicality of high-fidelity, computationally efficient UQ in complex biomedical simulations.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Multiagent Systems under Uncertainty

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We study the application of a quasi-Monte Carlo (QMC) method to a class of semi-linear parabolic reaction-diffusion partial differential equations used to model tumor growth. Mathematical models of tumor growth are largely phenomenological in nature, capturing infiltration of the tumor into surrounding healthy tissue, proliferation of the existing tumor, and patient response to therapies, such as chemotherapy and radiotherapy. Considerable inter-patient variability, inherent heterogeneity of the disease, sparse and noisy data collection, and model inadequacy all contribute to significant uncertainty in the model parameters. It is crucial that these uncertainties can be efficiently propagated through the model to compute quantities of interest (QoIs), which in turn may be used to inform clinical decisions. We show that QMC methods can be successful in computing expectations of meaningful QoIs. Well-posedness results are developed for the model and used to show a theoretical error bound for the case of uniform random fields. The theoretical linear error rate, which is superior to that of standard Monte Carlo, is verified numerically. Encouraging computational results are also provided for lognormal random fields, prompting further theoretical development.

Problem

• Quantifying uncertainty in tumor growth model parameters
• Propagating parameter uncertainties through reaction-diffusion equations
• Computing expectations of clinically relevant quantities efficiently

Innovation

• Quasi-Monte Carlo methods for uncertainty quantification
• Applied to parametric tumor growth reaction-diffusion models
• Achieves superior linear error rates over Monte Carlo