The following letter is a response to the report by Grabovsky and Tallarida (2004) entitled “Isobolographic Analysis for Combinations of a Full and Partial Agonist: Curved Isoboles” (J Pharmacol Exp Ther310:981–986). In their article, Grabovsky and Tallarida address the resultant effect of therapeutic drugs when they are administered under combination. Although their concepts relate to drug therapeutics, they can be applied in an ecotoxicological context as well. Indeed, predicting the combined response of two therapeutic drugs that have similar effects (e.g., two analgesics) and predicting the toxic effect in organisms exposed to two toxic substances that have a similar toxic mode of action are quite similar. Thus, consequences of present work may have extreme importance in clinical research and therapies, in ecotoxicological risk assessment, and even in the improvement of environmental legislation criteria.
Currently, most combinatorial exposure studies adopt either the isobolographic (Loewe, 1953) or the toxic unit concept (Sprague, 1970), of which the former is used more in pharmacology and the latter is more often applied in ecotoxicology. Both methods are actually based on the same definition of additivity: where a_{x} and b_{y} represent combinational doses of two drugs that result in the same level of effect than the individual doses A_{x} or B_{y} of each drug (Berenbaum, 1989; Tallarida, 1992). This formula has been widely used in many fields without a formal proof until Berenbaum (1989) proposed a general validation by constructing sham combinations of agents that mimic the real combination. However, there are several authors that still remain skeptical about the general validity of this formula when the log doseeffect curves of the individual agents are not parallel (Norwood et al., 2003; van der Hoeven, 2004).
Against this background, Grabovsky and Tallarida (2004) presented an isobolic concept framework that is based on individual doseresponse curves and on the use of equivalent doses of agonists in an effort to prove different cases where this general assumed additive model (eq. 1) is erroneous (i.e., nonparallelism of the individual log doseresponse curves and different drug efficacy). In this way, Grabovsky and Tallarida (2004) obtain curved isoboles for those cases instead of the generally expected linear isoboles. By extension, this study would have the ability to predict the expected effect of any combination of therapeutic drugs or toxicants as well. However, although some deficiencies of the general model are pointed out in Grabovsky and Tallarida (2004) (as the assumption of constant relative potency of drugs), some subtle assumptions still hold in their framework that may lead to erroneous conclusions.
First, Grabovsky and Tallarida (2004) provide a thoughtful way of expressing the concentration of drug A into equivalents of drug B, standardized to the same effect. This is an analogous approach to “constructing a sham combination” followed by Berenbaum (1989), but Grabovsky and Tallarida (2004) clearly show that previous transformation is only valid for the case of constant relative potency between both drugs.
Second, Grabovsky and Tallarida (2004) state the following:
Additivity is based on the concept of dose equivalence.... This equivalence is the basis of the relation derived for the additive isobole so that the combination doses, a and b, can be expressed as a dose of either one of them.... Thus, dose b plus this equivalent becomes the dose of drug B that yields the expected effect when this summed dose is used in the doseeffect equation of drug B.
Hence, the effect of the combined agonists (i.e., E_{a + b}) is assumed to be the same as the effect of the summed dose of the first agonist (which is dose a) plus the equivalent dose of A (i.e., a′) that causes the same effect as the agonist B dose (b). Obviously, this approach should not depend on the agonist that is chosen for the transformation (i.e., b → a′ or a → b′). Thus, it is assumed that:
However, that assumption gives incongruent results when applied to agonists that have different slopes in their log doseresponse curves (p ≠ q). As an example, when two full agonists are present in a mixture at their halfmaximal dose (i.e., a = A_{50} and b = B_{50}), the drug A dose equivalent to a B_{50} dose of drug B would be A_{50}, and vice versa (a′ = A_{50} and b′ = B_{50}). Therefore, E_{a + b} = E_{2a} = E_{2b}. However, if both agonists have different slopes in their doseresponse curves, E_{a + b} = E_{2a} = E_{2b} would no longer hold, because the effect would be greater in the agonist that has the highest slope.
It can be proven mathematically that the expected isoboles for each transformation differ. Consequently, the predicted isoboles from Grabovsky and Tallarida (2004) cannot be applied for testing additivity when the slopes of the doseresponse curves of both agonists differ. Assuming the case of two full agonists with variable relative potency: where A and B are the doses of drug A and B, A_{50} and B_{50} are the median effective concentrations of both drugs, and q and p are the Hill coefficients or the sigmoidicity of both doseeffect curves (analogous to the slope of the curves). The amount of drug B having the same effect of an a amount of drug A will be: and analogously: Thus, the expected effect for both combined drugs using the A drug as a reference would be: The isoboles of that given effect E_{a + b} (noted as E for simplicity) can be obtained by combining eqs. 6 and 7: Finding a:
Since isobolograms are plots in a twodimensional AB plane, the formulas of the isoboles would be in the form of b as a function of a. Therefore, the corresponding isobole could be obtained from eq. 8 as: In the same way, the expected effect of both drugs, using agonist B as a reference is: and analogously:
When the obtained isoboles (eqs. 9 and 11) are compared, they look very similar, but they are not the same. Figure 1 represents some of the curved isoboles (using drug B as reference) taken from Grabovsky and Tallarida (2004) along with the corresponding inversely transformed isoboles (using drug A as reference). Both isoboles can only be the same when p = q, where: Given that A_{50}, B_{50}, and E are fixed for a given isobole, eq. 12 represents a straight line. Hence, the assumption that E_{a + b} = E_{a + a′} = E_{b + b′} is not valid for full agonists with different relative potencies.
The other case described by Grabovsky and Tallarida (2004) uses the concept of equivalent doses to prove the existence of curved isoboles in the presence of full and partial agonists. In this case, the transformation of b into a′ is even more questionable. The transformation is based on comparing the concentration of different agonists having the same effect. But a 50% effect is different in a full agonist, being on its halfmaximal dose, than in a partial agonist with a maximum effect of 50%. In this last case, 50% represents the saturation of that drug. It can be also proven that two different curved isoboles are obtained depending on which drug is transformed into an equivalent dose of the other one, but the procedure is much more tedious. Hence, just as an example, the same case mentioned in Grabovsky and Tallarida (2004) will be studied here: B_{50} = 15, A_{30} = 80, E_{B max} = 100, and E_{A max} = 60. For a given combination of both drugs, such as a = 80 and b = 15, the following transformed values would be obtained: a′ = 400 and b′ = 6.429. Therefore, the calculated effects using A or B as reference drug would be E_{a + a′} = 51.43% or E_{b + b′} = 58.82%.
The main assumption that E_{a + b} = E_{a + a′} = E_{b + b′} is not valid either for full agonist with different toxicity curve slopes or for agonists with different efficacy (a full and a partial agonist), and the formulas of the expected curved isoboles presented in Grabovsky and Tallarida (2004) should not be used to test drug additivity.
Thus, in addition to the stress Grabovsky and Tallarida (2004) placed on the effect of different relative potencies on the transformation to get drug equivalents to invalidate the general concept of linear isoboles and the definition of additivity (eq. 1), we would like to stress here the problem of using the concept of equivalent doses of toxicants in those same cases. In fact, equivalent doses for one level of effect do not behave as equivalents when added to other doses, due to the different slopes of the doseresponse curves of each agonist. Straight isoboles are expected from drugs or toxicants that have the same slope as showed by Tallarida (2001). But linear isoboles lack proof for other cases, and the shape of the expected isoboles remains uncertain. More studies like the one from Grabovsky and Tallarida (2004) should be performed to resolve this crucial issue, paying attention to all the implicit and explicit assumptions that were made for obtaining the definition of additivity (eq. 1) and attempting to attain a new and truly generally valid model.
Acknowledgments
We thank Drs. Ricardo Beiras and Hans de Wolf for useful comments.
Footnotes

This work was supported by a postdoctoral fellowship from Fundación Alfonso Martín Escudero.

Article, publication date, and citation information can be found at http://jpet.aspetjournals.org.

doi:10.1124/jpet.105.095091.

ABBREVIATIONS: None.
 Received September 2, 2005.
 Accepted September 29, 2005.
 The American Society for Pharmacology and Experimental Therapeutics