Effect normalisation and pharmacokinetics

The integral of f_v (Eqs. 1–3) over some period of time from t₀ to t₁

can be used as a measurement of how effective a treatment is at inhibiting growth. If f_v(t) is periodic, i.e. there exists a certain period τ=t₁−t₀ for which f_v(t)=f_v(t+τ), it can be shown that if the integrals of f_v over time (Eq. 5) are equal for two different treatments, their effects are also equal (Additional file 1). Note that at steady state the TKI blood plasma concentration profile, and by extension f_v(t), is periodic with period τ – the dosing interval. The steady state concentration profile is given by [45]:

The dose, fractional bioavailability, salt factor and volume of distribution (D, F, S, V_d respectively) serve to determine the magnitude of the plasma concentration, while the absorption and elimination rates (k_a, k_e) and the dosing interval τ determine the shape of the concentration curve over time. The parameters were determined numerically to reproduce half-times and peak plasma concentrations from clinical studies given dosages and dosing intervals (vide infra).

A combination of drugs can lead to a lower total amount of drugs required while maintaining a treatment effect comparable to either drug alone. For two drugs normalised to the same effectiveness we can define a dose reduction factor ζ of a combination of x parts drug A and (1−x) parts drug B as (for an exclusive interaction)

For a nonexclusive interaction:

Note that if ζ=0.5 and x=0.5 (an equal amount of each drug, effect–wise) then the actual amount required of either drug is only 25% of what would be required of the respective drugs in a monodrug therapy.

It follows that we can calculate an effective IC₅₀ value (IC50eff) for a drug combination (defined by the dose ratio x) in terms of a combined pseudo-concentration C by rearranging Eqs. 2 and 3 when f_v=0.5 (exclusive interaction):

or (nonexclusive interaction)

Solutions to these equations are provided in the supplementary material, Eqs. S16 and S17.

The IC50eff falls somewhere between the IC₅₀ values of each of the two drugs, but there is a separate effect which can produce an extra drug sensitivity for combinations. If two drugs are used to approximately the same degree and are approximately equally sensitive to a mutation, their effect will be less than expected, due to a difference in the shape of the dose-response curve as illustrated in Fig. 1. A demonstration of this effect is available in the supplementary material, Fig. S1.

Dose-effect curves for a monodrug or exclusive combination and a nonexclusive combination (Eqs. 2 and 3). Given an effect target (f_v=0.2 in the figure) the nonexclusive combination achieves it with a lower total drug concentration (a). The increased IC₅₀ of a mutation has the same effect as reducing the drug concentration (Eqs. 2 and 3). At this lowered concentration, the monodrug or exclusive combination achieves a greater degree of inhibition (i.e. lower f_v, b)