Saberian Fatemeh, Ghate Archis, Kim Minsun
Industrial and Systems Engineering, University of Washington, Box 352650, Seattle, WA 98195, USA
Industrial and Systems Engineering, University of Washington, Box 352650, Seattle, WA 98195, USA.
Math Med Biol. 2016 Jun;33(2):211-52. doi: 10.1093/imammb/dqv015. Epub 2015 May 15.
The goal in radiotherapy is to maximize the biological effect (BE) of radiation on the tumour while limiting its toxic effects on healthy anatomies. Treatment is administered over several sessions to give the normal tissue time to recover as it has better damage-repair capabilities than tumour cells. This is termed fractionation. A key problem in radiotherapy involves finding an optimal number of treatment sessions (fractions) and the corresponding dosing schedule. A major limitation of existing mathematically rigorous work on this problem is that it includes only a single normal tissue. Since essentially no anatomical region of interest includes only one normal tissue, these models may incorrectly identify the optimal number of fractions and the corresponding dosing schedule. We present a formulation of the optimal fractionation problem that includes multiple normal tissues. Our model can tackle any combination of maximum dose, mean dose and dose-volume type constraints for serial and parallel normal tissues as this is characteristic of most treatment protocols. We also allow for a spatially heterogeneous dose distribution within each normal tissue. Furthermore, we do not a priori assume that the doses are invariant across fractions. Finally, our model uses a spatially optimized treatment plan as input and hence can be seamlessly combined with any treatment planning system. Our formulation is a mixed-integer, non-convex, quadratically constrained quadratic programming problem. In order to simplify this computationally challenging problem without loss of optimality, we establish sufficient conditions under which equal-dosage or single-dosage fractionation is optimal. Based on the prevalent estimates of tumour and normal tissue model parameters, these conditions are expected to hold in many types of commonly studied tumours, such as those similar to head-and-neck and prostate cancers. This motivates a simple reformulation of our problem that leads to a closed-form formula for the dose per fraction. We then establish that the tumour-BE is quasiconcave in the number of fractions; this ultimately helps in identifying the optimal number of fractions. We perform extensive numerical experiments using 10 head-and-neck and prostate test cases to uncover several clinically relevant insights.
放射治疗的目标是在限制辐射对健康组织产生毒性作用的同时，最大化辐射对肿瘤的生物学效应（BE）。治疗分多次进行，以便正常组织有时间恢复，因为其损伤修复能力优于肿瘤细胞。这被称为分割放疗。放射治疗中的一个关键问题是找到最佳的治疗次数（分割数）和相应的给药方案。现有关于此问题的数学严谨研究的一个主要局限性在于，其仅包含单一正常组织。由于实际上没有任何感兴趣的解剖区域仅包含一种正常组织，这些模型可能会错误地确定最佳分割数和相应的给药方案。我们提出了一种包含多个正常组织的最佳分割问题的公式化表述。我们的模型可以处理串联和平行正常组织的最大剂量、平均剂量和剂量体积类型约束的任何组合，因为这是大多数治疗方案的特点。我们还考虑了每个正常组织内空间异质的剂量分布。此外，我们并不先验地假设各分割之间的剂量是不变的。最后，我们的模型使用空间优化的治疗计划作为输入，因此可以与任何治疗计划系统无缝结合。我们的公式化表述是一个混合整数、非凸、二次约束二次规划问题。为了在不损失最优性的情况下简化这个计算具有挑战性的问题，我们建立了等剂量或单剂量分割为最优的充分条件。基于肿瘤和正常组织模型参数的普遍估计，预计这些条件在许多常见研究的肿瘤类型中成立，例如类似于头颈癌和前列腺癌的肿瘤。这促使我们对问题进行简单的重新表述，从而得到每个分割剂量的闭式公式。然后我们确定肿瘤生物学效应在分割数上是拟凹的；这最终有助于确定最佳分割数。我们使用10个头颈和前列腺测试案例进行了广泛的数值实验，以揭示一些临床相关的见解。
