Revisiting the isobole and related quantitative methods for assessing drug synergism

Abstract

The isobole is well established and commonly used in the quantitative study of agonist drug combinations. This article reviews the isobole, its derivation from the concept of dose equivalence, and its usefulness in providing the predicted effect of an agonist drug combination, a topic not discussed in pharmacology textbooks. This review addresses that topic and also shows that an alternate method, called "Bliss independence," is inconsistent with the isobolar approach and also has a less clear conceptual basis. In its simplest application the isobole is the familiar linear plot in cartesian coordinates with intercepts representing the individual drug potencies. It is also shown that the isobole can be nonlinear, a fact recognized by its founder (Loewe) but neglected or rejected by virtually all other users. Whether its shape is linear or nonlinear the isobole is equally useful in detecting synergism and antagonism for drug combinations, and its theoretical basis leads to calculations of the expected effect of a drug combination. Numerous applications of isoboles in preclinical testing have shown that synergism or antagonism is not only a property of the two agonist drugs; the dose ratio is also important, a fact of potential importance to the design and testing of drug combinations in clinical trials.

Fig. 1.

I, dose-effect curves for agonists with a constant potency ratio (10:1) with ED₅₀ values of 200 and 20. II, the isobole for the 50% effect is linear, has intercepts equal to the respective ED₅₀ values, and obeys the equation a/200 + b/20 = 1 for doses a and b. All points (a,b) on the isobole represent dose pairs that are expected to give effect = 50% E_max when there is no interaction between the drugs. III, if the maximal effects E_A and E_B differ (say, E_A = 60 and E_B = 100), the potency ratio is not constant and the isobole, for the usual hyperbolic dose-effect curves, is nonlinear and given by the equation (Grabovsky and Tallarida, 2004): where C_A is the dose of drug A that gives effect ½ E_A. For dose-effect curves with slopes that differ this result is easily generalized as discussed in Grabovsky and Tallarida (2004) and Tallarida (2006, .

Fig. 2.

Shown here is a linear isobole with an observed synergistic point P having coordinates (a,b), the additive point S with coordinates (a*,b*), and a subadditive point Q. The broken radial line denotes the dose combinations in a fixed ratio. In this case of a linear isobole the interaction index, denoted T, is calculated from the coordinates of P as T = a/A + b/B, a value less than unity and therefore indicative of synergism. For this case (linear isobole) it also follows that T is the ratio of radial distances 0P/0S, which also equals the ratio of total doses, (a + b)/(a* + b*). Situations of suspected departures from additivity require statistical testing of the difference (observed total − additive total). The variance (square of S.E.) for the additive total Z_add is given by The S.E. of the observed total dose is obtained from the dose-effect data of the combination. These totals and their S.E. are examined as the difference as described previously (Tallarida, 2000).

Fig. 3.

Shown is a response surface for the effect of two vasoconstrictors on isolated aortic rings, where the effect is expressed as the percentage of tension produced by KCl. Three dose combinations in the fixed ratio 94:6 were tested and showed that the two highest doses produced effects above the additive surface. Based on Lamarre and Tallarida (2008) with permission.