Department of Mathematics, COMSATS University Islamabad, 45550 Islamabad, Pakistan.
Department of Computer Sciences and Mathematics, Lebanese American University, 1102-2801 Beirut, Lebanon.
Front Biosci (Landmark Ed). 2023 Aug 18;28(8):174. doi: 10.31083/j.fbl2808174.
Cancer is the biggest cause of mortality globally, with approximately 10 million fatalities expected by 2020, or about one in every six deaths. Breast, lung, colon, rectum, and prostate cancers are the most prevalent types of cancer.
In this work, fractional modeling is presented which describes the dynamics of cancer treatment with mixed therapies (immunotherapy and chemotherapy). Mathematical models of cancer treatment are important to understand the dynamical behavior of the disease. Fractional models are studied considering immunotherapy and chemotherapy to control cancer growth at the level of cell populations. The models consist of the system of fractional differential equations (FDEs). Fractional term is defined by Caputo fractional derivative. The models are solved numerically by using Adams-Bashforth-Moulton method.
For all fractional models the reasonable range of fractional order is between β = 0.6 and β = 0.9. The equilibrium points and stability analysis are presented. Moreover, positivity and boundedness of the solution are proved. Furthermore, a graphical representation of cancerous cells, immunotherapy and chemotherapy is presented to understand the behaviour of cancer treatment.
At the end, a curve fitting procedure is presented which may help medical practitioners to treat cancer patients.
癌症是全球最大的死亡原因，预计到 2020 年将有大约 1000 万人死亡，约占每六例死亡中的一例。乳腺癌、肺癌、结肠癌、直肠癌和前列腺癌是最常见的癌症类型。
本工作提出了分数建模，用于描述混合疗法（免疫疗法和化疗）治疗癌症的动力学。癌症治疗的数学模型对于理解疾病的动力学行为很重要。考虑到免疫疗法和化疗可以控制细胞群体水平的癌症生长，研究了分数模型。模型由分数微分方程（FDE）系统组成。分数项由 Caputo 分数导数定义。使用 Adams-Bashforth-Moulton 方法对模型进行数值求解。
对于所有分数模型，分数阶的合理范围在β=0.6 和β=0.9 之间。给出了平衡点和稳定性分析。此外，证明了解的正定性和有界性。进一步，提出了癌症细胞、免疫疗法和化疗的图形表示，以了解癌症治疗的行为。
最后，提出了曲线拟合程序，这可能有助于医生治疗癌症患者。
