Optimal Treatment Planning in Radiotherapy based on Boltzmann Transport Equations

我们计划研究应用于临床外束放疗的最优控制问题。本文的出发点是将处理计划问题表述为玻尔兹曼连续减速模型的最优控制问题。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成。除了分析研究解的存在性和规律性以及最优控制外，我们还研究了高维相空间的矩逼近，以便设计有效的数值格式。优化和矩逼近之间的关系也将被研究。我们将解析和数值结果应用于实际CT -患者数据，并与标准蒙特卡罗方法求解控制问题得到的最优治疗方案进行比较。

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.