Supplementary Figure 9 from Ultrasensitive Response Explains the Benefit of Combination Chemotherapy Despite Drug Antagonism

Supplemental Figure 9. Model including a distribution of growth rates. The model presented in Figures 4 and 5 assigns cells to one of two outcomes after drug treatment: growing at a consistent rate, or dying at a consistent rate. In reality cells may experience more varied responses to treatment including various growth and death rates as well as cytostatic responses. To confirm that our experimental observations and conclusions about concurrent versus sequential therapy are robust to the possibility of graded drug responses, we here present a more complex model in which increasing drug doses causes, at first, a gradual decline in growth rate, then growth arrest, and then an increasing death rate. A) Each cell is assigned a unique growth rate (GR) where GR = 1 represents living cells with a doubling time of 2. 5 days, GR = 0 represents cytostatic cells, and GR = -1 represents dying cells with a half-life of 2. 5 days. GR is modelled as increasing linearly with drug dose, d, such that Growth Rate (d) =1- (1+Maximum Killing) *d where Maximum Killing = -1. Maximum rate of killing could in principle be faster, but experimental measurements shows that the rate of death after high doses of doxorubicin and 4-H-cyclophosphamide reaches a plateau that is similar to the uninhibited rate of growth (but for being a negative rate instead of a positive rate), which corresponds to GR = -1 (Supplemental Figure 8A). Here d represents effective combined dose accounting for drug interactions, ultrasensitivity, and cell-to-cell heterogeneity in sensitivity. Specifically, as a function of single-drug doses a and b, drug interaction i, LQ model parameters α and β, and relative drug sensitivity x, the combined effective drug dose is d = 10ˣ (alpha (a + b + (i a b) / (a+b) ) + beta (a + b + (i a b) / (a+b) ) ²) Parameters defining ultrasensitivity (α and β), drug-drug interaction (i) and cellular heterogeneity (x) are unchanged from Figures 4 and 5. Recall that in a population of tumor cells, x is normally distributed around 0; therefore 10ˣ is log-normally distributed around 1 and describes cell-to-cell heterogeneity in drug sensitivity, rendering the combination therapy more or less potent against individual cells. The outcome of this model is that given doses of two drugs a and b, cancer cells in a population experience a variety of GR values spanning from 1 to -1, meaning that some cells are partially growth inhibited, some are arrested, and some are dying at various rates. After an initial drug treatment, an individual cell’s GR value (rate of growth or death) is constant until day 9, unless a sequential treatment on day 6 exerts a stronger effect that results in faster death or more severe growth inhibition. That is, the second dose can intensify response but not weaken it. B) This model with more diverse cellular responses to treatment also qualitatively reproduces the experimentally observed superiority of concurrent treatment over sequential treatment, when it includes ultrasensitivity and cellular heterogeneity. Yellow triangles mark simulated treatment times (concurrent: C and H both on day 0; sequential: C on day 0 and H on day 6).