Postgraduate Program in Engineering Sciences, BioMath Research Group, Tijuana Institute of Technology, Tijuana, Mexico.
Department of Electric and Electronic Engineering, Durango Institute of Technology, Durango, Mexico.
Bull Math Biol. 2019 Oct;81(10):4144-4173. doi: 10.1007/s11538-019-00636-7. Epub 2019 Jul 1.
Mathematical models may allow us to improve our knowledge on tumor evolution and to better comprehend the dynamics between cancer, the immune system and the application of treatments such as chemotherapy and immunotherapy in both short and long term. In this paper, we solve the tumor clearance problem for a six-dimensional mathematical model that describes tumor evolution under immune response and chemo-immunotherapy treatments. First, by means of the localization of compact invariant sets method, we determine lower and upper bounds for all cells populations considered by the model and we use these results to establish sufficient conditions for the existence of a bounded positively invariant domain in the nonnegative orthant by applying LaSalle's invariance principle. Then, by exploiting a candidate Lyapunov function we determine sufficient conditions on the chemotherapy treatment to ensure tumor clearance. Further, we investigate the local stability of the tumor-free equilibrium point and compute conditions for asymptotic stability and tumor persistence. All conditions are given by inequalities in terms of the system parameters, and we perform numerical simulations with different values on the chemotherapy treatment to illustrate our results. Finally, we discuss the biological implications of our work.
数学模型可以帮助我们提高对肿瘤进化的认识，并更好地理解癌症、免疫系统以及化疗和免疫治疗等治疗方法在短期和长期内的相互作用。在本文中，我们解决了一个六维数学模型的肿瘤清除问题，该模型描述了免疫反应和化疗免疫治疗下的肿瘤进化。首先，通过紧不变集定位方法，我们确定了模型所考虑的所有细胞群体的上下界，并利用这些结果，通过应用 Lasalle 不变原理，在非负半轴上建立了有界正向不变域存在的充分条件。然后，通过利用候选 Lyapunov 函数，我们确定了化疗治疗以确保肿瘤清除的充分条件。此外，我们还研究了无肿瘤平衡点的局部稳定性，并计算了渐近稳定性和肿瘤持续存在的条件。所有条件都是用系统参数的不等式给出的，我们还对化疗治疗的不同值进行了数值模拟，以说明我们的结果。最后，我们讨论了我们工作的生物学意义。
