Quasi-Monte Carlo methods for uncertainty quantification of tumor growth modeled by a parametric semi-linear parabolic reaction-diffusion equation
For researchers in computational medicine, it demonstrates that QMC can efficiently propagate parametric uncertainty in tumor growth models, potentially improving clinical decision-making.
The paper applies quasi-Monte Carlo (QMC) methods to quantify uncertainty in a tumor growth model governed by a semi-linear parabolic PDE, achieving a theoretical linear error rate superior to standard Monte Carlo, which is verified numerically.
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.