Fig. 2: Optimal dosing under linear induction of tolerance.
From: Optimal dosing of anti-cancer treatment under drug-induced plasticity
a In our investigation of linear induction of tolerance, we consider three cases. In Case I, the drug only increases the transition rate from sensitivity to tolerance (μ(c) = μ0; + kc and ν(c) = ν0), in Case II, the drug only decreases the transition rate from tolerance to sensitivity (μ(c) = μ0 and ν(c) = ν0 − mc), and in Case III, the drug does both (μ(c) = μ0 + kc and ν(c) = ν0 − mc). b Optimal dosing strategies under linear induction of tolerance computed using the forward-backward sweep method. In all cases, the optimal dosing strategy involves both a transient phase and an equilibrium phase. During the equilibrium phase, a constant dose c* is applied, which minimizes the long-run tumor growth rate σ(c) under a constant dose c, as indicated by the dotted lines. The grayed-out part of each dosing strategy is a boundary effect which arises due to the optimization goal of minimizing the tumor size at a specific time T, with no regard for what is best in the long run. If we were to extend the treatment horizon beyond time T, the period applying the constant dose c* would be extended (Supplementary Fig. 1). We therefore ignore this boundary effect and focus on what is optimal over a longer time horizon. c Proportion of sensitive cells f0(t) over time under the optimal dosing strategy. The constant equilibrium dose c* maintains the tumor at a fixed proportion between sensitive and tolerant cells. d Tumor size evolution under the optimal strategy compared with the tumor size evolution under the constant-dose strategy applying c* throughout. For Case I, the two strategies are the same. For Cases II/III, the optimal strategy results in 85.6%/55.4% greater tumor reduction during the initial stages of treatment, and 59.4%/24.2% greater reduction long-term. e If the proportion of sensitive cells f0(0) at the start of treatment is changed, only the transient phase of the optimal treatment is affected, whereas the optimal equilibrium dose c* remains the same. For Case I, the transient phase is independent of the initial condition.
