We present a method of directly optimizing on deviations in clinical goal values in radiation therapy treatment planning. Using a new mathematical framework in which metrics derived from the dose–volume histogram are regarded as functionals of an auxiliary random variable, we are able to obtain volume-at-dose and dose-at-volume as infinitely differentiable functions of the dose distribution with easily evaluable function values and gradients. Motivated by the connection to risk measures in finance, which is formalized in this framework, we also derive closed-form formulas for mean-tail-dose and demonstrate its capability of reducing extreme dose values in tail distributions. Numerical experiments performed on a prostate and a head-and-neck patient case show that the direct optimization of dose–volume histogram metrics produced marginally better results than or outperformed conventional planning objectives in terms of clinical goal fulfilment, control of low- and high-dose tails of target distributions and general plan quality defined by a pre-specified evaluation measure. The proposed framework eliminates the disconnect between optimization functions and evaluation metrics and may thus reduce the need for repetitive user interaction associated with conventional treatment planning. The method also has the potential of enhancing plan optimization in other settings such as multicriteria optimization and automated treatment planning.

我们提出了一种直接优化放射治疗计划中临床目标值偏差的方法。使用新的数学框架，其中从剂量-体积直方图导出的度量被视为辅助随机变量的函数，我们能够获得剂量体积和剂量体积作为剂量分布的无限微分函数易于评估的函数值和梯度。受此框架中形式化的金融风险措施的影响，我们还推导出了平均尾剂量的封闭式公式，并证明了其减少尾分布中极端剂量值的能力。在前列腺和头颈部患者病例上进行的数值实验表明，剂量-体积直方图指标的直接优化在临床目标实现、低和高控制方面产生了略好于或优于传统规划目标的结果。目标分布的剂量尾部和由预先指定的评估措施定义的总体计划质量。所提出的框架消除了优化功能和评估指标之间的脱节，因此可以减少与传统治疗计划相关的重复用户交互的需要。该方法还具有在其他设置中增强计划优化的潜力，例如多标准优化和自动治疗计划。