Fig 1.
Arrows indicate growth, switching and death processes of the different subpopulations. Cells grow at rates ri, switch to slower growth rates at rate pS ri or to faster growth rates at rate pF ri. Cancer cell mortality from growth rate-dependent treatment (for example by the uptake of chemotherapeutics) is assumed to be proportional to growth rate and induces a mortality δ ri. Cancer cell mortality from the growth rate-independent treatment, for example immunotherapy, is captured by the death rate m.
Table 1.
Deviations from these values are reported where applicable.
Fig 2.
Population dynamics for a typical treatment scenario.
Starting from a cancer population that reached the stable distribution before detection at t = 0 treatment is applied between t = 0 and t = 150. After t = 150 relapse is monitored until t = 300. (A) Splitting the total cancer population (thick lines) into different subpopulations (thin lines) with different growth rates (colour gradient) allows for tracking the differential selection pressure that trait-dependent and trait-independent treatment types impose on different growth rates and how this selection affects the subsequent relapse dynamics. Insets show the growth rate trait distribution at various time points. The cancer cell mortality rate in the trait-independent treatment was set such that the tumour load at the end of treatment is similar to the tumour load at the end of the trait-dependent treatment. (B) Trait diversity (measured as Shannon evenness) is affected only by the growth rate-dependent treatment.
Fig 3.
Shown are 100 replicate populations for a binary trait where the two subpopulations grow with growth rates rmin and rmax, respectively, for (A) trait-independent and (B) trait-dependent treatments. The black lines represent the deterministic solution of the ordinary differential equation for the two subpopulations (Eq 1). Note that initial conditions and treatment duration are different compared to Fig 2 to allow for relapse given the discrete number of cells in the stochastic simulations. The cancer cell mortality by trait-independent treatment was set to m = 0.722 d−1 to ensure equal tumour load at the end of treatment in the deterministic model for the shorter treatment duration.
Fig 4.
Relapse time distributions extracted from the stochastic simulations in Fig 3.
Orange represents the trait-independent and blue the trait-dependent treatment type. The vertical lines indicate the relapse times from the deterministic simulations. Relapse is defined to occur when the total tumour load of a replicate exceeds 103 cells.
Fig 5.
Comparison of minimum tumour load during treatment and the relapse time when tumour load surpasses the pre-treatment maximum for different sequential treatment schemes.
Alterations between the trait-independent and the trait-dependent treatment type are fixed in the predefined sequence scheme (red to blue colour gradient corresponds to proportion of trait-dependent treatment type). In the realistic adaptive scheme, the currently best treatment type is determined at regular intervals during the treatment phase (grey colour gradient). In the optimal adaptive scheme (black dot), the treatment type that, given the current trait distribution, would exert the highest population mortality is chosen nearly instantaneously (at every step of the numerical solver). Note that these schemes have a much stronger impact on the minimal tumour load (up to a factor of 1000) than on the relapse time (up to a factor of 1.3).