Mathematical Sciences: Markov Models for the Time Development of DNA Damage Repair

辐射生物学中与时间相关的问题将用数学方法进行研究。重点将放在使用解析解、边界或近似来理解数值结果，反之亦然。将数学结果与目前发表的实验结果进行详细的比较。主要项目是利用连续时间马尔可夫链模拟电离辐射对哺乳动物细胞的损伤。具体使用的发电机将是对最近论文中介绍的发电机的修改（Albright 1989; Tobias et al. 1989; Curtis 1989; Sachs et al. 1990; Hlatky et al. 1990; Hahnfeldt et al. 1991）。将考虑以下生物学终点：(A)辐照期间DNA双链断裂和其他染色体损伤的产率；(B)辐照前后DNA双链断裂通过细胞酶修复和错误修复过程产生的染色体畸变的产率和变异；(C)辐照后克隆生存活细胞的比例。在(B)和(C)中，主要的数学技术将是使用嵌入的离散时间马尔可夫链以适合于数值评估和与实验比较的形式计算出长时间渐近行为，特别是在人类淋巴细胞上的实验。(A)主要涉及使用微扰理论和Perron-Frobenius理论对发生器的特征值。许多适用于这些端点之一的旧模型可以被视为马尔可夫模型的确定性近似，并且将系统地研究这种近似的有效性域。第二个项目将是使用完整的、时变的反应扩散方程和真实的反应项来分析球形肿瘤模型。电离辐射对DNA损伤的数学建模对于将中等辐射剂量的实验结果与极低剂量和剂量率的结果联系起来尤其重要，而极低剂量和剂量率与公共卫生问题最为相关。与大多数旧模型不同，这里使用的模型系统地考虑了剂量的波动和细胞间的反应。这些结果将有助于对电离辐射后染色体畸变（如双中心）如何发展的基本理解。这种认识反过来应该改进生物剂量学、放射性暴露风险评估和放射治疗的方法。

Time-dependent problems of interest in radiation biology will be investigated mathematically. Emphasis will be on using analytic solutions, bounds or approximations to understand numerical results and vice-versa. Detailed comparisons of the mathematical results with current published experiments will be made. The main project is to use continuous time Markov chains for modeling ionizing radiation damage to mammalian cells. The specific generators used will be modifications of those introduced in recent papers (Albright 1989; Tobias et al. 1989; Curtis 1989; Sachs et al. 1990; Hlatky et al. 1990; Hahnfeldt et al. 1991). The following biological endpoints will be considered: (A) the yield of DNA double strand breaks and other chromosome damage during irradiation; (B) the yield and variance of chromosome aberrations evolving from DNA double strand breaks via enzymatic cellular repair and misrepair processes during and after irradiation; (C) the fraction of cells clonogenically viable after irradiation. In (B) and (C) the main mathematical technique will be to use the embedded discrete time Markov chain to work out the long time asymptotic behavior in a form suitable for numerical evaluation and comparison to experiments, especially experiments on human lymphocytes. (A) mainly involves use of perturbation theory and of Perron-Frobenius theory for the eigenvalues of the generator. Many older models applicable to one of these endpoints can be regarded as deterministic approximations to the Markov models and the domain of validity of such approximations will be systematically studied. A secondary project will be to use the full, time-dependent reaction-diffusion equations with realistic reaction terms for analyzing spheroid tumor models. Mathematical modeling of ionizing radiation damage to DNA is particularly important in relating experimental results for moderate radiation doses to results at the very low doses and dose rates which are most relevant in public health concerns. Unlike most older models, the models to be used here systematically take into account fluctuations in dose and response from cell to cell. The results will contribute to a basic understanding of how chromosome aberrations such as dicentrics develop following ionizing radiation. Such understanding in turn should improve methods of biological dosimetry, of risk assessment for exposures to radioactivity, and of radiotherapy.