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.

翻译：本研究探讨拟蒙特卡洛（QMC）方法在一类用于模拟肿瘤生长的半线性抛物反应-扩散偏微分方程中的应用。肿瘤生长的数学模型本质上是现象学模型，主要描述肿瘤向周围健康组织的浸润过程、现有肿瘤的增殖行为以及患者对化疗和放疗等治疗方式的反应。显著的个体间差异、疾病固有的异质性、稀疏且含噪声的数据采集以及模型本身的局限性，共同导致模型参数存在显著的不确定性。将这些不确定性高效地通过模型传递以计算目标量（QoIs）至关重要，这些计算结果可为临床决策提供依据。我们证明QMC方法能有效计算具有实际意义的目标量的期望值。研究建立了模型的适定性理论，并以此推导了均匀随机场情形下的理论误差界。数值实验验证了理论上的线性误差收敛速率，该速率优于标准蒙特卡洛方法。针对对数正态随机场的计算结果亦展现出良好前景，为后续理论发展提供了动力。