An Overview of Drug Combination Analysis with Isobolograms

Abstract

Drugs given in combination may produce effects that are greater than or less than the effect predicted from their individual potencies. The historical basis for predicting the effect of a combination is based on the concept of dose equivalence; i.e., an equally effective dose (a) of one will add to the dose (b) of the other in the combination situation. For drugs with a constant relative potency, this leads to linear additive isoboles (a-b curves of constant effect), whereas a varying potency ratio produces nonlinear additive isoboles. Determination of the additive isobole is a necessary procedure for assessing both synergistic and antagonistic interactions of the combination. This review discusses both variable and constant relative potency situations and provides the mathematical formulas needed to distinguish these cases.This communication is concerned with the analysis of combinations of two drugs that produce overtly similar effects that are measurable. Therefore, each drug is an agonist that displays dose dependence. As studies of drug combinations have become more common, there has emerged an increased use of the isobologram, a graph that was introduced many years ago (Loewe, 1927(Loewe, , 1928. That graph, constructed on a coordinate system composed of the individual drug doses, commonly contains a straight "line of additivity" that is employed to distinguish additive from synergistic and antagonistic interactions. This graphical construction is based on the assumption of a constant relative potency.In a previous review, Tallarida (2001) discussed the use and construction of the common (linear) isobole, the set of points (dose pairs) that give a specified effect magnitude. A subsequent study (Grabovsky and Tallarida, 2004) considered combinations of a full and partial agonist, a situation that necessarily means a variable relative potency. That situation was shown to lead to nonlinear isoboles of additivity instead of the widely applied linear isobole and demonstrated that experimental results in this case could be mistaken for synergism or antagonism. That result (nonlinearity), which represents a departure from the common use of isobolograms, prompted further attention to other situations of variable relative potency. However, the variability condition was not explicitly connected to Loewe's concept of dose equivalence in the author's previous review. In this review, we make this explicit by showing (for the first time) that the derivation leading to curved isoboles is consistent with the same concept of dose equivalence that was employed by Loewe. A further consequence of this concept is in its application to two full agonists, with varying relative potency, a case that is shown here to lead to not just one but two nonlinear but symmetric isoboles. The demonstration (proof) of symmetry of this pair of isoboles, presented here for the first time, provides a new criterion for distinguishing between additive and nonadditive interactions.The theoretical basis for this graphical procedure, i.e., t...

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