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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...

Two drugs used in combination may produce enhanced or reduced effects. The degree of enhancement or reduction is measured from the interaction index (gamma), a quantity that indicates the changed potency of the combination. The index is therefore a quantitative marker for the drug combination and effect metric used. Methodology for measuring the interaction index utilizes the combination and individual drug dose-effect data suitably modeled by regression techniques that most often produce linear plots of effect on log dose from which isobolar analysis is employed. The isobologram provides a simple and convenient graphical assessment of the interaction index but an independent statistical analysis is needed to assess its precision. In some cases, the relative potency of the constituent drugs is the same at all effect levels. When this is so, it is shown that the interaction index can be measured by either an isobolar or an alternate method that is illustrated here. These calculations demonstrate that these different methods of analysis yield the same value of gamma, and do so with comparable precision.

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.

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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., ED 50 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.When two drugs produce overtly similar effects, e.g., two analgesics or two antihypertensives, their presence together may produce an effect whose magnitude is different from that predicted by the individual drug potencies. Combinations that achieve predictable effects based on individual potencies are said to be additive. Determining the additive effect for a drug combination is the first step in detecting synergism. That determination is straightforward when the drugs act through different receptors and have a constant relative potency R, i.e., when equally effective doses have the same ratio over the range of effects being studied. The constancy of R is implicit in most studies of combinations that use isobolographic analysis. In such studies the dose-response curves of the individual agents allow a determination of each drug's dose that gives a specified effect such as the half-maximal effect. That dose, the D 50 (or ED 50 if the dose-response data are quantal), for each drug is plotted as an intercept on a Cartesian coordinate system in which the axes represent the individual drug doses, and the two intercepts define a straight line called the line of additivity. All points (a,b) on this line are dose pairs, called additive isoboles, that give the half-maximal effect if the joint action is in accord with their individual potencies. Although an effect level ϭ 50% E max is the most common, any other effect level up to the ma...

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