On the design and stability of cancer adaptive therapy cycles: Deterministic and stochastic models

Adaptive therapy is a promising paradigm for cancer treatment exploiting competition between drug-sensitive and drug-resistant cells to delay evolution of drug resistance. Previous studies demonstrated that cyclic drug administration can restore tumor composition to its initial value in deterministic models. However, conditions and methods for designing such cycles deserve better investigation. We present biologically motivated conditions to construct such cycles in two well-known deterministic frameworks, Lotka-Volterra and adjusted replicator dynamics, and provide algorithms for building cycles using two drugs and a period with no drugs. Moreover, we analyze stability of these cycles, an essential consideration for their clinical applicability. We conjecture that a cycle is stable whenever the averaged treatment is stable, conversely it is unstable when the averaged treatment is also unstable. We further investigate stochastic counterparts of both models to account for the finite cell population and randomness inherent to real tumors. Our results reveal that the breakdown of cyclic behavior in stochastic settings, see Dua et al. (2021) and Park and Newton (2023), is not caused by stochasticity per se, but by instability of the corresponding deterministic cycles used as examples. In contrast, we demonstrate that stable deterministic cycles give rise to stable cyclic behavior despite stochastic fluctuations, highlighting the importance of stability in adaptive therapy. We illustrate how stable deterministic cycles avoid for large times the breakdown of cyclic treatments in stochastic models. These findings establish a coherent framework linking deterministic cycle stability to stochastic robustness, offering theoretical support for the design of clinically resilient adaptive cancer therapies.