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

Adaptive therapy is a promising paradigm for treating cancers, that exploits competitive interactions between drug-sensitive and drug-resistant cells, thereby avoiding or delaying treatment failure due to evolution of drug resistance within the tumor. Previous studies have shown the mathematical possibility of building cyclic schemes of drug administration which restore tumor composition to its exact initial value in deterministic models. However, algorithms for cycle design, the conditions on which such algorithms are certain to work, as well as conditions for cycle stability remain elusive. Here, we state biologically motivated hypotheses that guarantee existence of such cycles in two deterministic classes of mathematical models already considered in the literature: Lotka-Volterra and adjusted replicator dynamics. We stress that not only existence of cyclic schemes, but also stability of such cycles is a relevant feature for applications in real clinical scenarios. We also analyze stochastic versions of the above deterministic models, a necessary step if we want to take into account that real tumors are composed by a finite population of cells subject to randomness, a relevant feature in the context of low tumor burden. We argue that the stability of the deterministic cycles is also relevant for the stochastic version of the models. In fact, Dua, Ma and Newton [Cancers (2021)] and Park and Newton [Phys. Rev. E (2023)] observed breakdown of deterministic cycles in a stochastic model (Moran process) for a tumor. Our findings indicate that the breakdown phenomenon is not due to stochasticity itself, but to the deterministic instability inherent in the cycles of the referenced papers. We then illustrate how stable deterministic cycles avoid for very large times the breakdown of cyclic treatments in stochastic tumor models.