The use of mathematical and computational models for reliable predictions of tumor growth and decline in living organisms is one of the foremost challenges in modern predictive science, as it must cope with uncertainties in observational data, model selection, model parameters, and model inadequacy, all for very complex physical and biological systems. In this paper, large classes of parametric models of tumor growth in vascular tissue are discussed including models for radiation therapy. Observational data is obtained from MRI of a murine model of glioma and observed over a period of about three weeks, with X-ray radiation administered 14.5 days into the experimental program. Parametric models of tumor proliferation and decline are presented based on the balance laws of continuum mixture theory, particularly mass balance, and from accepted biological hypotheses on tumor growth. Among these are new model classes that include characterizations of effects of radiation and simple models of mechanical deformation of tumors. The Occam Plausibility Algorithm (OPAL) is implemented to provide a Bayesian statistical calibration of the model classes, 39 models in all, as well as the determination of the most plausible models in these classes relative to the observational data, and to assess model inadequacy through statistical validation processes. Discussions of the numerical analysis of finite element approximations of the system of stochastic, nonlinear partial differential equations characterizing the model classes, as well as the sampling algorithms for Monte Carlo and Markov chain Monte Carlo (MCMC) methods employed in solving the forward stochastic problem, and in computing posterior distributions of parameters and model plausibilities are provided. The results of the analyses described suggest that the general framework developed can provide a useful approach for predicting tumor growth and the effects of radiation.

中文翻译：

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基于无创影像数据的辐射对肿瘤生长影响预测模型的选择与验证

使用数学和计算模型对活生物体的肿瘤生长和衰退进行可靠预测是现代预测科学的首要挑战之一，因为它必须应对观察数据、模型选择、模型参数和模型不足的不确定性，所有用于非常复杂的物理和生物系统。在本文中，讨论了血管组织中肿瘤生长的大量参数模型，包括放射治疗模型。观察数据是从小鼠神经胶质瘤模型的 MRI 中获得的，并在大约三周的时间内观察到，在实验程序中进行了 14.5 天的 X 射线辐射。基于连续混合理论的平衡定律，特别是质量平衡，提出了肿瘤增殖和衰退的参数模型，以及来自公认的关于肿瘤生长的生物学假设。其中包括新的模型类，包括辐射效应的表征和肿瘤机械变形的简单模型。实施奥卡姆可信度算法 (OPAL) 以提供模型类的贝叶斯统计校准，总共 39 个模型，以及确定这些类中相对于观测数据最可信的模型，并通过以下方式评估模型的不足统计验证过程。讨论表征模型类的随机、非线性偏微分方程系统的有限元近似值的数值分析，以及用于求解正向随机问题的蒙特卡罗和马尔可夫链蒙特卡罗 (MCMC) 方法的采样算法，并在计算参数和模型合理性的后验分布时提供。所描述的分析结果表明，所开发的一般框架可以提供一种有用的方法来预测肿瘤生长和辐射的影响。