Quantitative methods for assessing drug synergism

Abstract

Two or more drugs that individually produce overtly similar effects will sometimes display greatly enhanced effects when given in combination. When the combined effect is greater than that predicted by their individual potencies, the combination is said to be synergistic. A synergistic interaction allows the use of lower doses of the combination constituents, a situation that may reduce adverse reactions. Drug combinations are quite common in the treatment of cancers, infections, pain, and many other diseases and situations. The determination of synergism is a quantitative pursuit that involves a rigorous demonstration that the combination effect is greater than that which is expected from the individual drug's potencies. The basis of that demonstration is the concept of dose equivalence, which is discussed here and applied to an experimental design and data analysis known as isobolographic analysis. That method, and a related method of analysis that also uses dose equivalence, are presented in this brief review, which provides the mathematical basis for assessing synergy and an optimization strategy for determining the dose combination.

Keywords: additivity; dose-effect relation; drug combinations; isoboles; optimal dose strategy; subadditivity; synergism.

Figure 1.

Dose pairs (a,b) that plot as points on the isobole (solid line for a specified effect with intercepts A and B) represent combination doses that are additive when the 2 drugs have a constant potency ratio. Dose pairs that preserve a constant dose ratio are indicated by the radial lines for 2 different fixed-ratio dose combinations. If achievement of the specified (e.g., 50% E_max) occurs with lesser doses, such as point P below the isobole, then the dose combination in that fixed dose ratio is superadditive (synergistic). Another fixed ratio dose combination may show a dose pair that attains this effect with a dose pair (a,b) that lies above the isobole (such as point Q), and this dose combination indicates subadditivity.

Figure 2.

(Upper) This figure illustrates 2 agonists that yield different maximum effects and which are expressed by equations E = 100 dose/(dose + 20) for Drug B and E = 70 dose/(dose + 100) for Drug A; thus, the effect 50 is attained by a = 250 and b = 20 when each acts alone. (Lower) The isobole for this drug pair’s effect 50 is shown as the solid curve with intercepts that denote the individual potencies. The incorrect linear isobole is also shown (broken straight line). The curvature of the additive isobole would be erroneously interpreted as synergism if the linear isobole is used.

Figure 3.

Illustration of how a fixed dose ratio combination of 2 agonist drugs are used in a calculation of the expected (additive effect) of a combination dose. The graphs are shown for 3 different fixed ratio dose combinations and are expressed and plotted in terms of the dose a of the lower efficacy drug. The drug parameters are for Drug A, E _A = 60, C _A = 100; for Drug B, E _B = 100, C _B = 20. Details of the calculation of the combined effect are given in the text.

Figure 4.

Illustration in which synergy for the desired effect is found to exist for dose combinations in the region between radial lines C and D, whereas the toxic effect is subadditive only between radial lines A and B. The intersection is the region between C and B, and therefore dose combinations in this intersection, with practical upper limits, are a region (shown shaded) that is optimal for selecting dose ratios.