Fig 1.
Schematic overview of the model.
We developed a mathematical model to investigate the effects of changing mutation rates during treatment on the evolution of resistance. When treated with targeted or traditional cytotoxic chemotherapy, a sensitive cancer cell (left) might give rise to a resistant cell at a rate that increases with the drug dose administered (case 1), is independent of the dose (case 2), or decreases with the dose (case 3). For simplicity, we show linear relationships between drug dose and mutation rate; however, more complex relationships can easily be considered using our general framework.
Fig 2.
Evolutionary dynamics of sensitive and resistant clones during continuous therapy with different mutation dynamics.
(A) Expected number of resistant cancer cells as a function of time during continuous therapy. Blue line: mutation rate is constant during treatment; black line: mutation rate increases with drug dose; green line: mutation rate decreases with drug dose. Red circles: simulation results. Grey shaded area indicates one analytic standard deviation from the analytic mean. (B) Probability of resistance as a function of time during continuous therapy. (C) Variance of resistant cancer cells as a function of time during continuous therapy. The following parametrizations were used for both simulation and analytic approximations: As = 0.05, Bs = 0.1, Cs = 0.005, Ar = 0.05, Br = 0.12, Cr = 0.002, and θ = 0.10. The values for Au and Bu are denoted in the panels for each corresponding scenario.
Fig 3.
Evolutionary dynamics of sensitive and resistant clones during continuous therapy with pre-existing resistant cells.
(A), (B) and (C) are examples for sine wave functional forms of birth, death and mutation rates. (A) Expected number of resistant cancer cells as a function of time during continuous therapy. Blue line: mutation rate is constant during treatment; black line: mutation rate increases with drug dose; green line: mutation rate decreases with drug dose. Red circles: simulation results for no pre-existing resistant clones; orange circle: simulation result for 3% proportion of pre-existing resistant clones; purple circle: simulation result for 5% proportion of pre-existing clones. (B) Probability of resistance as a function of time during continuous therapy. (C) Variance of resistant cancer cells as a function of time during continuous therapy. (D), (E), and (F) are examples for piecewise functional forms of birth, death and mutation rates. (D) Expected number of resistant cancer cells as a function of time during continuous therapy. (E) Probability of resistance as a function of time during continuous therapy. (F) Variance of resistant cancer cells as a function of time during continuous therapy. The death rates in Fig 2 were used here as the death rates for both upper and lower panels. The birth rates in the continuous therapy in Fig 2 were also used in this figure. The birth rates in the piecewise strategy were λXs(t) = 0.15⋅I(t / 14 mod 2 = 0) + 0.05⋅I(t / 14 mod 2 ≠ 0), λXr(t) = 0.17⋅I(t / 14 mod 2 = 0) + 0.07⋅I(t / 14 mod 2 ≠ 0).
Fig 4.
Pharmacokinetic effects influence the evolutionary dynamics of resistance.
Here we consider a baseline mutation rate per cell division of 10−8 and the effect of drug concentration on the mutation rate β = 10−9. (A) The drug concentration in vivo based on the pharmacokinetic model over time for a 100mg per day continuous dosing regime. Dotted line: loading dose; solid line: no loading dose. (B) The birth rates as a function of time t. Red line: the birth rate of the resistant cells; black line: the birth rate of the sensitive cells. (C) The mutation rate of sensitive cells as a function of time t. Blue line: constant mutation rate; green line: mutation rate monotonically decreases with the drug concentration; black line: mutation rate monotonically increases with the drug concentration. (D) The probability of resistance as a function of time t. Values for birth and death rates: μXs(t) ≈ 0.005 hour−1, μXr(t) ≈ 0.002 hour−1, λXs(t) ≈ exp(−4.4 ⋅ C(t)−3.17) hour−1, and λXr(t) ≈ −0.001 ⋅ C(t) + 0.03 hour−1.
Fig 5.
Effects of varying dosing regimes on the evolution of resistance.
Here we consider β = 10−10. (A) The dosing regimes with no loading dose for 100mg/day, 150mg/day, 1600mg/week, 1600mg/week combined with 100mg/day during the week, and 1600mg/week combined with 75mg/day during the week. (B) The dosing regimes with loading dose for 100mg/day, 150mg/day, 1600mg/week, 1600mg/week combined with 100mg/day during the week, and 1600mg/week combined with 75mg/day during the week. (C) Mutation rate as a function of treatment concentration under different assumptions: blue: independent with treatment concentration; black: increasing with treatment concentration; green: decreasing with treatment concentration. (D)-(F) Without pre-existing resistance, the probability of resistance monitored up to one month under (D) constant mutation rate, (E) mutation rate increasing with the drug concentration, and (F) mutation rate decreasing with the drug concentration. (G)-(I) With pre-existing resistance, the expected number of resistant cells monitored up to one month under (G) constant mutation rate, (H) mutation rate increasing with the drug concentration, and (I) mutation rate decreasing with the drug concentration. Dotted line: with loading dose; solid line: without loading dose. Values for birth and death rates: μXs(t) ≈ 0.005 hour−1, μXr(t) ≈ 0.002 hour−1, λXs(t) ≈ exp(−4.4 ⋅ C(t)−3.17) hour−1, and λXr(t) ≈ −0.001 ⋅ C(t) + 0.03 hour−1
Fig 6.
The optimum dosing strategy varies with dose-dependent mutation rates.
Here we consider β = 10−10. (A)–(C): (A) The area under the curve (AUC) of the probability of resistance over three months of treatment, (B) the probability of resistance after one month of treatment, and (C) the probability of resistance after three months of treatment when there are no pre-existing resistant cells for different treatment schedules (separated by different colors) and different mutation rate assumptions (independent; increasing with drug concentration; decreasing with drug concentration) indicated by the x-axis label. (D)–(F): (D) The AUC of the expected number of resistant cells over three months of treatment, (E) the expected number of resistant cells after one month of treatment, and (F) the expected number of resistant cells after three months of treatment when there are pre-existing resistant clones for different treatment schedules (separated by different colors) and different mutation rate assumptions (independent; increasing with drug concentration; decreasing with drug concentration) indicated by the x-axis label. Values for birth and death rates: μXs(t) ≈ 0.005 hour−1, μXr(t) ≈ 0.002 hour−1, λXs(t) ≈ exp(−4.4 ⋅ C(t)−3.17) hour−1, and λXr(t) ≈ −0.001 ⋅ C(t) + 0.03 hour−1.