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

Alexander D. Gilbert, Frances Y. Kuo, Dirk Nuyens, Graham Pash, Ian H. Sloan, Karen E. Willcox

TL;DR

This paper develops quasi-Monte Carlo methods to quantify uncertainty in tumor growth modeled by a semi-linear parabolic PDE with random diffusion and proliferation coefficients. It proves well-posedness and explicit a priori bounds, then derives parametric regularity for uniform random fields via a reparameterization that yields a nonlinearity amenable to QMC error analysis. Using Banach-space QMC theory, the authors establish dimension-truncation and QMC error bounds, and validate these results with numerical experiments showing linear convergence for uniform fields and promising faster-than-MC convergence for lognormal fields. The work advances uncertainty quantification for nonlinear PDEs in computational oncology and motivates further theoretical development for lognormal models and discretization effects.

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.

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

TL;DR

This paper develops quasi-Monte Carlo methods to quantify uncertainty in tumor growth modeled by a semi-linear parabolic PDE with random diffusion and proliferation coefficients. It proves well-posedness and explicit a priori bounds, then derives parametric regularity for uniform random fields via a reparameterization that yields a nonlinearity amenable to QMC error analysis. Using Banach-space QMC theory, the authors establish dimension-truncation and QMC error bounds, and validate these results with numerical experiments showing linear convergence for uniform fields and promising faster-than-MC convergence for lognormal fields. The work advances uncertainty quantification for nonlinear PDEs in computational oncology and motivates further theoretical development for lognormal models and discretization effects.

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.

Paper Structure

This paper contains 15 sections, 5 theorems, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Governing equation
  4. Functional framework
  5. Reparameterization
  6. Well-posedness
  7. A priori bound
  8. Parametric regularity
  9. Uniform random field model
  10. Parametric regularity for uniform random fields
  11. QMC error analysis for uniform random fields
  12. Numerical experiments
  13. Experiments with uniform random fields
  14. Experiments with lognormal random fields
  15. Conclusions

Key Result

Theorem 2.1

For every ${\boldsymbol{y}}\in U$ and $\lambda^{\boldsymbol{y}} > \kappa_{\max}^{\boldsymbol{y}}\ge 0$, consider pde II eq:pde2 with bounded initial condition $w_0\in L^2(\Omega)$, $0\le w_0({\boldsymbol{x}})\le 1$ for all ${\boldsymbol{x}}\in\Omega$, and bounded coefficients eq:pde2-sub--eq:pde2-ab

Figures (3)

  • Figure 1: The approximate root-mean-square error \ref{['eqn:rms']} of the randomized qmc approximation and the standard error of the mc estimator of the integral ${\mathbb{E}}[G(u_s)]$ for different numbers of total pde evaluations.
  • Figure 2: Left: Visualization of anatomy and slice of the left-hemisphere. Center: The computational mesh used for simulations, with $9,995$ vertices. Right: A representative sample of the diffusion coefficient.
  • Figure 3: Left: The empirical spectral decay of the two Gaussian random fields compared to the theoretical rate accompanied. Right: The approximate root-mean-square error \ref{['eqn:rms']} for the qmc approximation and the standard error for the mc estimator of the integral ${\mathbb{E}}[G(u_s)]$.

Theorems & Definitions (5)

  • Theorem 2.1: existence, uniqueness, boundedness
  • Theorem 2.2: a priori bound
  • Theorem 3.1: parametric regularity
  • Theorem 4.1: dimension truncation error
  • Theorem 4.2: QMC error bound