放射治疗剂量计划的鲁棒概率优化
Robust and probabilistic optimization of dose schedules in radiotherapy
摘要：
在线性二次模型的背景下，我们考虑了参数不确定性对最优辐射计划的影响。我们的兴趣来自于以下观察结果：如果在放射计划期间未考虑OAR和肿瘤对放射的敏感性或OAR的保留因子的患者间差异，则治疗的性能可能会严重降低，或者OAR可能会接收比最大阈值大得多的剂量。本文提出了两个放射调度的概念，将患者间的变异性到调度优化问题。第一种方法是一个强大的配方，制定了一个保守的模型，优化可能发生的最坏情况剂量调度的问题。第二种方法是概率方法，其中模型参数由一组随机变量给出。这个公式确保我们的约束满足给定的概率，我们的目标函数达到所需的水平与规定的概率。我们使用了一个转换，以减少由此产生的优化问题，以两个维度。我们证明了最优解位于可行域的边界上，并使用分支定界算法找到全局最优解。我们观察到，如果在最佳的常规时间表的分数的数量是相同的鲁棒性和随机的解决方案，它是最好的管理相等或更小的总剂量。此外，如果与常规时间表相比，概率或鲁棒解决方案中存在更多（更少）治疗会话，则预计总剂量平方（总剂量）会减少。最后，我们进行了数值实验，在设置头颈部肿瘤，以揭示最佳时间表的参数不确定性的影响，并评估模型的敏感性，关键模型参数的选择。
英文摘要：
We consider the effects of parameter uncertainty on the optimal radiation schedule in the context of the linear-quadratic model. Our interest arises from the observation that if inter-patient variations in OAR and tumor sensitivities to radiation or sparing factor of the OAR are not accounted for during radiation scheduling, the performance of the therapy may be strongly degraded or the OAR may receive a substantially larger dose than the maximum threshold. This paper proposes two radiation scheduling concepts to incorporate inter-patient variability into the scheduling optimization problem. The first approach is a robust formulation that formulates the problem as a conservative model that optimizes the worst case dose scheduling that may occur. The second method is a probabilistic approach, where the model parameters are given by a set of random variables. This formulation insures that our constraints are satisfied with a given probability, and that our objective function achieves a desired level with a stated probability. We used a transformation to reduce the resulting optimization problem to two dimensions. We showed that the optimal solution lies on the boundary of the feasible region and we used a branch and bound algorithm to find the global optimal solution. We observed that if the number of fractions in the optimal conventional schedule is the same as the robust and stochastic solutions, it is preferable to administer equal or smaller total dose. In addition if there exist more (fewer) treatment sessions in the probabilistic or robust solution compared to the conventional schedule, a reduction in total dose squared (total dose) will be expected. Finally, we performed numerical experiments in the setting of head-and-neck tumors to reveal the effect of parameter uncertainty on optimal schedules and to evaluate the sensitivity of the model to the choice of key model parameters.
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