Isobolographic Analysis for Combinations of a Full and Partial Agonist: Curved Isoboles

Combinations of drugs are frequently used therapeutically to achieve an enhanced effect without using an excess quantity of either agent. If the drugs exert overtly similar action, e.g., two analgesics, the effect of the combination may be tested for additivity, i.e., an effect level that is achieved based on the individual drug potencies. But combinations of agonists will sometimes display either superadditive (synergistic) or subadditive responses. Whether the two agonists are both drugs, or a combination of a drug and an endogenous chemical, there is interest in characterizing the interaction to determine whether it departs from additivity because quantitative information of this kind, aside from its therapeutic importance, may also illuminate mechanism. A common method for this characterization uses the isobologram. This is a plot in rectangular coordinates of dose combinations (a,b) that produce the same effect level (often taken to be 50% of the maximum). In its usual form, this plot is constructed as a straight line (of additivity) connecting intercepts that represent the individually effective doses, e.g., ED₅₀ values of each. This line is the reference for distinguishing additive from nonadditive interactions accordingly as the tested combination is on or off this line. Discussed here are the assumptions that underlie this linear plot. Specifically we show that a combination of drugs with a variable potency ratio, exemplified by a full and a partial agonist, lead to curvilinear isoboles of additivity that may erroneously be attributed to either synergism or subadditivity.