This study constructs a breast cancer model based on threshold-triggered pulse therapy, aiming to maintain tumor burden below a specified threshold level. This dynamic cancer management regimen, which integrates threshold strategy and pulse therapy, effectively curbs tumor overproliferation and avoids dormancy associated with continuous medication. The model exhibits a class of complex \((L+m)T\) -periodic solution, where L is a any non-negative integer and m is a any positive integer. To investigate the complex dynamics of cancer recurrence, we focus on analyzing periodic solution in the dormant cancer cell-free state. For order-1 periodic solution, we establish the existence conditions of \((L+1)T\) -periodic solution, analyze its local stability by Floquet theory, and prove its global stability by the comparison theorem. For order- m periodic solution, we confirm the existence of \((1+m)T\) -periodic solution. In addition, the threshold-controlled system precludes the existence of active cancer cell-free periodic solution. This indicates that strategies aimed at complete eradication are suboptimal compared to those aimed at containment below a critical threshold. Furthermore, bifurcation analysis near \((L+1)T\) -periodic solution reveals the existence of positive periodic solution. Numerical simulations validate the existence and stability of \((L+m)T\) -periodic solution in dormant cancer cell-free state and illustrate the impact of monitoring period, threshold strategy, pulse therapy and activation rate of dormant cancer cells on breast cancer recurrence dynamics. Variations in monitoring period T can drive transitions from a semi-trivial periodic solution characterizing the absence of dormant cancer cells to a positive periodic solution indicating their presence. The threshold level directly affects the non-treatment frequency L and treatment frequency m . An appropriate combination of anti-estrogen therapy and targeted therapy may prevent the emergence of acquired resistance and tumor dormancy. Enhancing treatment efficacy may also be achieved by increasing the activation rate. The main results indicate that dynamically adjusting the monitoring period and threshold level combined with appropriate pulse interventions can effectively suppress tumor overgrowth and prevent active cancer cells from entering dormancy. This dynamic management scheme provides novel insights for preventing early-stage breast cancer progression and improving patient survival rates.
Su et al. (Fri,) studied this question.