Subfigure (A) shows the dosing schedule specified by the growth rate of sensitive cancer cells as a function of time. The dashed line represents a dosing schedule in which the drug is administered at the maximum dose tolerated without treatment breaks. The solid line represents a schedule in which the drug is administered in pulses followed by drug holidays. Subfigures (B) and (C) show sample paths for the sensitive cell population during pulsed and continuous therapy, respectively. During treatment pulses as well as during continuous therapy, the sensitive cancer cell population declines while it expands during treatment breaks. Parameters are M = 10000, r ₁ = 1.75, d ₁ = 1.0, c ₁ = 1.0, T_on = 3.0, and T_off = 1.0.

(A) The probability of developing resistance during continuous therapy as a function of the growth rate of sensitive cells during treatment, q ₁. The probability of resistance increases with the growth rate. Solid lines represent the analytical approximations and Monte Carlo simulation results are plotted as dots. Parameters are M = 10000, q₂ = 1.25, u = 10⁻⁵, and c ₁ = c ₂ = 1.0. (B) We show the probability of developing resistance during continuous therapy as a function of the number of sensitive cancer cells at diagnosis, M. The probability of resistance increases linearly with M. Parameters are q ₁ = 0.75 and all others as in (A). (C) We show the expected number of resistant cells as a function of time during continuous therapy for different values of the growth rate of sensitive cells during treatment, q ₁. The expected number of resistant cells increases with time and with q ₁. Parameters are M = 1000, q ₂ = 1.2, u = 10⁻⁴, and c ₁ = c₂ = 1.0. (D) We show the expected number of resistant cells as a function of time during continuous therapy for different values of the growth rate of resistant cells during treatment, q ₂. The expected number of resistant cells increases with time and with q ₂. Parameters are q ₁ = 0.5 and all others as in (C).

(A) We show the probability of resistance during pulsed therapy as a function of the growth rate of sensitive cancer cells during each treatment phase, q ₁. The risk of resistance increases as the growth rate is enhanced. Lines represent the analytical predictions and dots show the Monte Carlo simulation results. Parameters are M = 1000, r ₁ = 1.3, r ₂ = q ₂ = 1.1, u = 10⁻⁴, and d ₁ = d ₂ = c ₁ = c ₂ = 1.0. (B) We show the probability of resistance during pulsed therapy as a function of the growth rate of resistant cancer cells during a treatment phase, q ₂. The risk of resistance increases as the growth rate is enhanced. Parameters are q ₁ = 0.5 and all others as in (A). (C) We show the probability of resistance during pulsed therapy as a function of the duration of each treatment phase, T_on. The risk of resistance decreases as the pulse becomes longer. Parameters are as above. (D) We show the probability of resistance during pulsed therapy as a function of the initial size of the sensitive cancer cell population, M. The risk of resistance increases with M. Parameters are as above. (E) and (F) We show the expected number of resistant cells as a function of time during pulsed therapy when q ₁, (E), and q ₂, (F), are varied. The expected number of resistant cells increases with both quantities. Parameters are as above.

Subfigure (A) shows an example toxicity constraint in the form of a function defining the maximum tolerated duration of treatment, T_on, versus the death rate of sensitive cancer cells during therapy, c ₁, for a 28-day treatment cycle. This type of toxicity constraint can be derived from experimental and clinical data in the form shown in subfigures (B) and (C). In subfigure (B), we display an example of the type of data available from clinical trials that relate T_on to the drug concentration. Hence for a given drug concentration, a treatment pulse of T_on days was tolerated. In (C), we show an example of how the drug concentration can be related to the death rate of sensitive cancer cells during treatment, c ₁, through in vitro experiments.

Subfigure (A) shows the relationship between dose and plasma C_max concentration obtained after a single dose (µM). These data were reported in ,. Subfigure (B) shows points depicting clinically tolerated schedules, as reported in –. The doses administered are converted to plasma concentration using the linear relationship found in subfigure (A). We also plot a conservative toxicity constraint where we assume that any concentration or length of pulse increased to levels beyond those tolerated in the trials would not be admissible.

(A) We show examples for toxicity constraints which vary the endpoint representing the maximum dose that can be tolerated for a full treatment cycle of K = T_on days. The treatment cycle is K = 28 days, and the endpoint representing the maximum dose that is tolerated for just one day is specified by the death rate c ₁ = 6. Black dots represent different continuous-dose endpoints. (B) We show examples for toxicity constraints which vary the endpoint representing the maximum dose that can be tolerated for a single day only, T_on = 1. Parameters are as above and black points represent the high-dose endpoints. (C) We show examples for toxicity constraints which vary the degree of the convexity of the curve. Parameters as above. (D) We show an example for a toxicity constraint (equation (16)) in which the range of c ₁ is restricted to satisfy the viable treatment option constraint, γ<0. The dotted line shows the full toxicity constraint curve, T₁¹, while the solid line shows the portion of the curve in the range in which γ<0. Parameters are r ₁ = 1.3, d ₁ = 1, and q ₁ = 1.3.

(A) We show the probability of resistance, P, over the range of treatment schedules specified by the restricted (solid) constraint curve from . The probability of resistance is minimized when c ₁ = 3.15. We can then use the toxicity constraint curve to identify all other specifications of the dosing schedule, such as T_on, T_off, the concentration etc. Parameters are r ₁ = 1.3, d ₁ = 1, q ₁ = 1.3, r ₂ = 1.3, d ₂ = 1, M = 10⁹, and u = 5·10⁻¹⁰. (B) We show the time-independent term of the equation describing the number of resistant cells, C_R, over the same constraint curve as above. This term is minimized when c ₁ = 3.4, which does not coincide with the optimum dosing strategy identified in (A). Therefore the strategy minimizing the probability of resistance may not coincide with the strategy maximizing the time until detection of resistant cells. (C) We show the value proportional to the conditional number of resistant cells, C_R/P, over the same constraint curve. Parameters are as above.

In column (A), the color at each point represents the value of c ₁ that corresponds to the dosing schedule which minimizes the probability of resistance. In column (B), the color at each point represents the value of c ₁ that corresponds to the dosing schedule which minimizes the conditional expected number of resistant cells. In column (C), the minimal probability of resistance corresponding to the optimal schedule from column (A) is plotted. Rows 1,2, and 3 show results for constraints T ₁ ¹, T ₁ ², and T ₁ ³. Parameters are d ₁ = 1, q ₁ = 1.3, M = 10⁹, and u = 5·10⁻¹⁰.

In column (A), the color at each point represents the value of c ₁ that corresponds to the dosing schedule which minimizes the probability of resistance. In column (B), the color at each point represents the value of c ₁ that corresponds to the dosing schedule which minimizes the conditional expected number of resistant cells. In column (C), the minimal probability of resistance corresponding to the optimal schedule from column (A) is plotted. Rows 1,2, and 3 show results for constraints T ₁ ¹, T ₁ ², and T ₁ ³. Parameters are d ₁ = 1, q ₁ = 1.3, M = 10⁹, and u = 5·10⁻¹⁰.

In column (A), the color at each point represents the value of c ₁ that corresponds to the dosing schedule which minimizes the probability of resistance. In column (B), the color at each point represents the value of c ₁ that corresponds to the dosing schedule which minimizes the conditional expected number of resistant cells. In column (C), the minimal probability of resistance corresponding to the optimal schedule from column (A) is plotted. Rows 1,2, and 3 show results for constraints T ₁ ¹, T ₁ ², and T ₁ ³. Parameters are d ₁ = 1, q ₁ = 1.3, M = 10⁹, and u = 5·10⁻¹⁰.