Optimal Treatment Planning in Radiotherapy based on Boltzmann Transport Equations

我们计划研究临床外照射治疗中的最优控制问题。本文的出发点是将治疗计划问题表述为Boltzmann连续减速模型的最优控制问题。在数学上，该模型由能量、空间和角度变量的积分-偏微分方程组组成。除了对解的存在性、正则性和最优控制进行分析研究外，我们还研究了对高维相空间的矩逼近，以设计有效的数值格式。还将研究最优化和矩逼近之间的关系。我们使用实际CT−患者数据的分析和数值结果，并将通过求解控制问题得到的最优治疗方案与标准蒙特卡罗方法进行了比较。

We plan to investigate optimal control problems used in clinical external beam radiotherapy. The starting point is the formulation of the treatment planning problem as an optimal control problem for the Boltzmann Continuous Slowing Down model. Mathematically, the model consists of an integro–partial differential equation in energy, spatial and angular variables. Besides analytical investigations on existence and regularity of solutions and optimal controls we study moment approximations to the high–dimensional phase space in order to devise efficient numerical schemes. The relation between optimization and moment approximation will also be investigated. We use the analytical and numerical results on real CT−patient data and compare the optimal treatment plans obtain by solving the control problem with standard Monte–Carlo methods.