Abstract This article presents a fractional-order mathematical model of the biological phenomena that occur in cancer therapy. The formulation generalizes the one proposed by Soto-Ortiza and Finley that consists of eighteen integer order differential equations and intends to serve as a platform for cancer treatment design. The fractional model is used to test the hypothesis that a combination of anti-Vascular Endothelial Growth Factor (VEGF) treatment with immunotherapy, involving injections of unlicensed dendritic cells (DC), can lead to the tumor eradication by concealing the suppressor VEGF. The new approach adopts derivatives defined in the Caputo sense and implements an optimal control strategy. Two control variables, one for immunotherapy and another for anti-angiogenic therapy, are considered for reducing the number of the cancer cells. Moreover, two numerical methods are introduced and their stability analysis is studied. Numerical simulations illustrate the proposed concepts.

中文翻译：

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用于延迟癌症治疗的可变阶分数模型的优化控制

摘要 本文介绍了癌症治疗中发生的生物学现象的分数阶数学模型。该公式概括了 Soto-Ortiza 和 Finley 提出的公式，该公式由 18 个整数阶微分方程组成，旨在作为癌症治疗设计的平台。分数模型用于检验以下假设：抗血管内皮生长因子 (VEGF) 治疗与免疫疗法的组合，包括注射未经许可的树突细胞 (DC)，可以通过隐藏抑制因子 VEGF 来根除肿瘤。新方法采用在 Caputo 意义上定义的导数并实施最佳控制策略。两个控制变量，一个用于免疫疗法，另一个用于抗血管生成疗法，被认为用于减少癌细胞的数量。此外，介绍了两种数值方法并研究了它们的稳定性分析。数值模拟说明了所提出的概念。