Rihan F A, Abdelrahman D H, Al-Maskari F, Ibrahim F, Abdeen M A
Department of Mathematical Sciences, College of Science, UAE University, P.O. Box 15551, Al-Ain, UAE ; Department of Mathematics, Faculty of Science, Helwan University, Cairo 11795, Egypt.
Department of Mathematical Sciences, College of Science, UAE University, P.O. Box 15551, Al-Ain, UAE.
Comput Math Methods Med. 2014;2014:982978. doi: 10.1155/2014/982978. Epub 2014 Aug 14.
We present a delay differential model with optimal control that describes the interactions of the tumour cells and immune response cells with external therapy. The intracellular delay is incorporated into the model to justify the time required to stimulate the effector cells. The optimal control variables are incorporated to identify the best treatment strategy with minimum side effects by blocking the production of new tumour cells and keeping the number of normal cells above 75% of its carrying capacity. Existence of the optimal control pair and optimality system are established. Pontryagin's maximum principle is applicable to characterize the optimal controls. The model displays a tumour-free steady state and up to three coexisting steady states. The numerical results show that the optimal treatment strategies reduce the tumour cells load and increase the effector cells after a few days of therapy. The performance of combination therapy protocol of immunochemotherapy is better than the standard protocol of chemotherapy alone.
我们提出了一个带有最优控制的时滞微分模型，该模型描述了肿瘤细胞和免疫反应细胞与外部治疗之间的相互作用。将细胞内时滞纳入模型，以说明刺激效应细胞所需的时间。引入最优控制变量，通过阻断新肿瘤细胞的产生并使正常细胞数量保持在其承载能力的75%以上，来确定副作用最小的最佳治疗策略。建立了最优控制对的存在性和最优性系统。庞特里亚金极大值原理适用于刻画最优控制。该模型显示出一个无肿瘤稳态和多达三个共存稳态。数值结果表明，最优治疗策略在治疗几天后可降低肿瘤细胞负荷并增加效应细胞。免疫化疗联合治疗方案的性能优于单纯化疗的标准方案。
