Bertuzzi A, Bruni C, Papa F, Sinisgalli C
Istituto di Analisi dei Sistemi ed Informatica A. Ruberti, CNR, Viale Manzoni 30, 00185 Roma, Italy.
J Math Biol. 2013 Jan;66(1-2):311-49. doi: 10.1007/s00285-012-0512-2.
We address the problem of finding the optimal radiotherapy fractionation scheme, representing the response to radiation of tumour and normal tissues by the LQ model including exponential repopulation and sublethal damage due to incomplete repair. We formulate the nonlinear programming problem of maximizing the overall tumour damage, while keeping the damages to the late and early responding normal tissues within a given admissible level. The optimum is searched over a single week of treatment and its possible structures are identified. In the two simpler but important cases of absence of the incomplete repair term or of prevalent late constraint, we prove the uniqueness of the optimal solution and we characterize it in terms of model parameters. The optimal solution is found to be not necessarily uniform over the week. The theoretical results are confirmed by numerical tests and comparisons with literature fractionation schemes are presented.
我们研究了寻找最佳放射治疗分割方案的问题，该方案通过包含指数再增殖以及因不完全修复导致的亚致死损伤的LQ模型来表示肿瘤和正常组织对辐射的反应。我们制定了一个非线性规划问题，即在将晚期和早期反应正常组织的损伤保持在给定允许水平内的同时，最大化总体肿瘤损伤。在一周的治疗过程中搜索最优解，并确定其可能的结构。在不存在不完全修复项或主要为晚期约束这两种更简单但重要的情况下，我们证明了最优解的唯一性，并根据模型参数对其进行了刻画。结果发现最优解在一周内不一定是均匀的。通过数值测试证实了理论结果，并与文献中的分割方案进行了比较。
