Choi Beong, Deasy Joseph O
Department of Radiation Oncology, Mallinckrodt Institute of Radiology, Washington University Medical Center, St Louis, MO 63110, USA.
Phys Med Biol. 2002 Oct 21;47(20):3579-89. doi: 10.1088/0031-9155/47/20/302.
The efficiency of intensity-modulated radiation therapy (IMRT) treatment planning depends critically on the presence or absence of multiple local minima in the feasible search space. We analyse the convexity of the generalized equivalent uniform dose equation (Niemierko A 1999 Med. Phys. 26 1100) when used either in the objective function or in the constraints. The practical importance of this analysis is that convex objective functions minimized over convex feasibility spaces do not have multiple local minima, likewise for concave objective functions maximized over convex feasibility spaces. Both of these situations are referred to as 'convex problems' and computationally efficient local search methods can be used for their solution. We also show that the Poisson-based tumour control probability objective function is strictly concave (if one neglects inter-patient heterogeneity), and hence it implies a single local minimum if maximized over a convex feasibility space. Even when including inter-patient heterogeneity, multiple local minima, although theoretically possible, are expected to be of minimal concern. The generalized equivalent uniform dose function (EUDa) is proved to be convex or concave depending on its only parameter a: when a is equal to or greater than 1, minimizing EUDa, on a convex feasibility space leads to a single minimum; when a is less than 1, maximizing EUDa, on a convex feasibility space leads to a single minimum. We also study a recently proposed practical, yet difficult, IMRT treatment planning formulation: unconstrained optimization of the objective function proposed by Wu et al (2002 Int. J. Radiat. Oncol. Biol. Phys. 52 224-35), which is expressed in terms of the EUDa for the target and normal tissues. This formulation may theoretically lead to multiple local minima. We propose a procedure for improving resulting solutions based on the convexity properties of the underlying objective function terms.
调强放射治疗（IMRT）治疗计划的效率严重依赖于可行搜索空间中是否存在多个局部最小值。我们分析了广义等效均匀剂量方程（Niemierko A 1999《医学物理》26 1100）在用于目标函数或约束条件时的凸性。该分析的实际重要性在于，在凸可行空间上最小化的凸目标函数没有多个局部最小值，同样，在凸可行空间上最大化的凹目标函数也没有多个局部最小值。这两种情况都被称为“凸问题”，并且可以使用计算效率高的局部搜索方法来求解。我们还表明，基于泊松的肿瘤控制概率目标函数是严格凹的（如果忽略患者间的异质性），因此在凸可行空间上最大化时它意味着只有一个局部最小值。即使考虑患者间的异质性，虽然理论上可能存在多个局部最小值，但预计影响极小。广义等效均匀剂量函数（EUDa）根据其唯一参数a被证明是凸的或凹的：当a等于或大于1时，在凸可行空间上最小化EUDa会得到一个单一最小值；当a小于1时，在凸可行空间上最大化EUDa会得到一个单一最小值。我们还研究了最近提出的一种实际但困难的IMRT治疗计划公式：对Wu等人（2002《国际放射肿瘤学、生物学、物理学杂志》52 224 - 35）提出的目标函数进行无约束优化，该目标函数根据靶区和正常组织的EUDa来表示。从理论上讲，这种公式可能会导致多个局部最小值。我们基于基础目标函数项的凸性属性提出了一种改进所得解的方法。
