Drug resistance is a key impediment to cancer treatment, however, the resistance mechanism remains controversial. Experiment evidence indicated that persister cells, a subpopulation in a transient pseudo-dormant state, are posited to play a potential role in the emergence of resistance. In this study, we propose a novel mathematical model for describing the interactions among sensitive, persister, and resistant cells to qualitatively and quantitatively analyze how persister cells affect cancer evolution and unravel the underlying resistance mechanisms. The proposed model is validated by theoretical analysis and fitting of actual human CT scan and mice data. Theoretical analysis demonstrates the existence and stability of the multiple steady states, as well as the global asymptotical stability of a unique steady state. Furthermore, bifurcation analysis reveals that the key factors, including the apoptotic rate of sensitive cells and the transformation rate between sensitive and persister cells, induce bistable phenomenon and are characterized by double saddle–node and transcritical bifurcations. The bistable region indicates that persister cells under high stable steady state as a reservoir could promote the emergence of resistant cells conferring a proliferative advantage and conversely when they are in lower stable steady state, the source of resistant cells and resistance process will be controlled. Therefore, the first transcritical bifurcation point can be viewed as an early indicator to detect critical transitions from low resistance to high resistance. Together, the proposed model and quantitative results would provide new insights for searching for strategies in modulating and decreasing the risk of drug resistance.

中文翻译：

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结合持久细胞的抗癌数学建模和动态分析

耐药性是癌症治疗的一个关键障碍，然而，耐药机制仍存在争议。实验证据表明，持久细胞（处于短暂假休眠状态的亚群）被认为在耐药性的出现中发挥潜在作用。在这项研究中，我们提出了一种新的数学模型来描述敏感细胞、持续细胞和耐药细胞之间的相互作用，以定性和定量分析持续细胞如何影响癌症进化并揭示潜在的耐药机制。通过理论分析以及实际人体 CT 扫描和小鼠数据的拟合，验证了所提出的模型。理论分析证明了多重稳态的存在和稳定性，以及唯一稳态的全局渐近稳定性。此外，分叉分析表明，敏感细胞的凋亡率以及敏感细胞和持久细胞之间的转化率等关键因素引起双稳态现象，并以双鞍结和跨临界分叉为特征。双稳态区域表明，处于高稳态的存留细胞作为储存库，可以促进耐药细胞的出现，从而具有增殖优势；反之，当它们处于较低的稳态时，耐药细胞的来源和耐药过程将受到控制。因此，第一个跨临界分岔点可以被视为检测从低电阻到高电阻的关键转变的早期指标。总之，所提出的模型和定量结果将为寻找调节和降低耐药风险的策略提供新的见解。