慢性粒细胞白血病是一种血癌，酪氨酸激酶抑制剂的标准治疗对大多数患者来说是成功的。停止治疗后，一半反应良好的患者要么长时间出现检测不到的肿瘤细胞水平，要么表现出肿瘤负荷持续波动在非常低的水平。受随之而来的问题的启发，观察到的动力学是否反映了肿瘤-免疫相互作用中出现的周期性振荡，在这项工作中，我们分析了一个描述 CML 免疫反应的常微分方程组，其中针对白血病的功能反应和免疫招募都表现出最佳激活窗口。除了研究平衡点的稳定性之外，我们还提供了严格的证明，证明该模型至少表现出两种类型的分岔：围绕无肿瘤平衡点的跨临界分岔和围绕生物学上合理的平衡点的 Hopf 分岔，为以下问题提供了肯定的答案：我们最初的问题。我们将注意力集中在 Hopf 分岔上，研究极限环的出现并通过计算 Lyapunov 系数来分析其稳定性。然后我们通过基于临床相关参数的数值模拟来说明我们的理论结果。除了数学兴趣之外，我们的结果表明，在 CML 患者中观察到的低肿瘤负荷水平的波动可能是捕食者-猎物相互作用产生的周期性轨道的结果。© 2024。作者，获得 Springer 独家许可-Verlag GmbH 德国，施普林格自然集团的一部分。

Chronic Myeloid Leukemia is a blood cancer for which standard therapy with Tyrosine-Kinase Inhibitors is successful in the majority of patients. After discontinuation of treatment half of the well-responding patients either present undetectable levels of tumor cells for a long time or exhibit sustained fluctuations of tumor load oscillating at very low levels. Motivated by the consequent question of whether the observed kinetics reflect periodic oscillations emerging from tumor-immune interactions, in this work, we analyze a system of ordinary differential equations describing the immune response to CML where both the functional response against leukemia and the immune recruitment exhibit optimal activation windows. Besides investigating the stability of the equilibrium points, we provide rigorous proofs that the model exhibits at least two types of bifurcations: a transcritical bifurcation around the tumor-free equilibrium point and a Hopf bifurcation around a biologically plausible equilibrium point, providing an affirmative answer to our initial question. Focusing our attention on the Hopf bifurcation, we examine the emergence of limit cycles and analyze their stability through the calculation of Lyapunov coefficients. Then we illustrate our theoretical results with numerical simulations based on clinically relevant parameters. Besides the mathematical interest, our results suggest that the fluctuating levels of low tumor load observed in CML patients may be a consequence of periodic orbits arising from predator-prey-like interactions.© 2024. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.