Slow–Fast Model and Therapy Optimization for Oncolytic Treatment of Tumors

The present work studies models of oncolytic virotherapy without space variable in which virus replication occurs at a faster time scale than tumor growth. We address the questions of the modeling of virus injection in this slow–fast system and of the optimal timing for different treatment strategies. To this aim, we first derive the asymptotic of a three-species slow–fast model and obtain a two-species dynamical system, where the variables are tumor cells and infected tumor cells. We fully characterize the behavior of this system depending on the various biological parameters. In the second part, we address the modeling of virus injection and its expression in the two-species system, where the amount of virus does not appear explicitly. We prove that the injection can be described by an instantaneous jump in the phase plane, where a certain amount of tumors cells are transformed instantly into infected tumor cells. This description allows discussing qualitatively the timing of different injections in the frame of successive treatment strategies. This work is illustrated by numerical simulations. The timing and amount of injected virus may have counterintuitive optimal values; nevertheless, the understanding is clear from the phase space analysis.