In this work we present a new stochastic framework for obtaining optimal treatment regimes in prostate cancer. We model the realistic scenario of randomized clinical trials for incorporating randomness related to interaction between a prostate cancer cell and androgen cell quota, due to cancer heterogeneities, across different patients in a given group, using a Liouville partial differential equation. We then solve two optimization problems: one for determining the model parameters to fit the measured data and the second to determine the optimal androgen deprivation therapy. The optimization problems are implemented using a positive, stable, and conservative finite volume solver for the Liouville equations and the projected non-linear conjugate gradient method. Several numerical results, including comparison with ordinary differential equations modeling framework, demonstrate the robustness and accuracy of our proposed framework to obtain optimal treatment regimes in real time.

中文翻译：

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前列腺癌的刘维尔最优控制框架


在这项工作中，我们提出了一个新的随机框架，用于获得前列腺癌的最佳治疗方案。我们使用刘维尔偏微分方程对随机临床试验的现实场景进行建模，以将由于癌症异质性而导致的前列腺癌细胞和雄激素细胞配额之间相互作用相关的随机性纳入给定组中不同患者的模型。然后，我们解决两个优化问题：一是确定模型参数以适应测量数据，二是确定最佳雄激素剥夺疗法。优化问题是使用刘维尔方程的正、稳定、保守的有限体积求解器和投影非线性共轭梯度法来实现的。一些数值结果，包括与常微分方程建模框架的比较，证明了我们提出的框架实时获得最佳治疗方案的鲁棒性和准确性。