Samanta Sudip, Sarkar Sandip, Kayal Kaushik
Department of Mathematics, Bankura University, Bankura 722155, India.
Sri Aurobindo Vidyamandir, Chandannagar, Hooghly, West Bengal 712136, India.
MethodsX. 2024 Aug 13;13:102895. doi: 10.1016/j.mex.2024.102895. eCollection 2024 Dec.
This paper investigates the dynamics of a three-dimensional nonlinear cancer model involving interactions among cancer cells, normal cells, and immune cells. By performing a linear stability analysis of the equilibria and investigating the Hopf bifurcation in relation to the immune cell growth rate, we reveal the possibility of chaotic behavior when radiation is absent. However, with the appropriate implementation of radiotherapy, the cancer model demonstrates stable solutions, transitioning from chaotic oscillations through period-halving bifurcation. Additionally, we propose and examine an indirect Z-control mechanism within the cancer model. Our findings indicate that using the indirect Z-controller on the immune population successfully manages chaos and adjusts the cancer cell density to a desired level. Through extensive investigation, we demonstrate the robustness of the Z-controller in managing oscillations and provide insights into determining the minimum number of immune cells needed to achieve a predetermined cancer cell density. This study underscores the importance of control mechanisms in mitigating cancer progression and highlights the potential of Z-control for therapeutic intervention strategies.
本文研究了一个三维非线性癌症模型的动力学，该模型涉及癌细胞、正常细胞和免疫细胞之间的相互作用。通过对平衡点进行线性稳定性分析，并研究与免疫细胞生长速率相关的霍普夫分岔，我们揭示了在无辐射情况下出现混沌行为的可能性。然而，通过适当实施放射治疗，癌症模型呈现出稳定解，从混沌振荡通过倍周期分岔转变而来。此外，我们在癌症模型中提出并研究了一种间接Z控制机制。我们的研究结果表明，对免疫群体使用间接Z控制器能够成功地控制混沌，并将癌细胞密度调整到期望水平。通过广泛研究，我们证明了Z控制器在控制振荡方面的鲁棒性，并为确定实现预定癌细胞密度所需的最小免疫细胞数量提供了见解。本研究强调了控制机制在减轻癌症进展中的重要性，并突出了Z控制在治疗干预策略中的潜力。
