Comments on Tallarida and Miaskowski et al

shift produced by a fixed dose of one of the drugs can also be additive. In fact, most analyses over the years have used parallel line assays. It is the amount of the shift that is important. i.e.. is it significantly to the left (superadditivity) or to the right (subadditivity) at each level of effect. Hence. not only do these authors fail to show the validity of applying this well-known statistical procedure. but also they negate their claim through their own graphical illustration which is built around parallelism. They appear to be confusing the meaning of a significant lnter~c~iu?I term (a possible outcome of the ANOVA) with deviations from additivity of drug combinations. Additivity is very well defined (see this writer’s review. cited above. for further references) and the definition is quite clear. and sensible. as we now show. t;uomnpie. Say that each of two drugs is capable of producing the same level of effect (for example 50% of the m~imum effect, though any other level common to each can be used) but they require different doses to achieve this (50%,) effect. Let these doses be denoted DA* and Dt. More specifically, let DA* = 400 mg and D5 = IO0 mg, which means that drug B has 4 times the potency of drug A. Suppose that an experimenter needs to achieve this effect but finds that he has only 300 mg of drug A and wants to use up his stock of this weaker drug. Thus. he will use all 300 mg, but he is still short by 100 mg which must be made up from drug B. How much of drug B should he add’? Clearly, only 25 mg since that amount is equivalent to the 100 mg of drug A. Thus, D, = 300 and D, = 25. and these numbers also satisfy the equation

This simple example illustrates the concept of additivity and the equation above is the definirrg equation. At any level of effect, where the respective doses used alone are Dz and Df. the combination of doses DA and D, are additive if. and only if, they obey this equation. Returning now to the claim of Miaskowski et al. one can see what would be needed to validate their claim. They use a fixed dose of drug B and examine the dose-effect relation of drug A. From the above equation it is seen that additivity requires that, for any level of effect, the doses of drugs A and B must obey D>,= Dz -(R)D, 9:: R = -. the potency ratio. DB* Their claim would require a mathematicul

where

rrbo1.e equalion,

or departures from

proof that connects the

it, with the significance

of the

i~zterffcf~o~ ierm that comes frum ANOVA. They have not established

this connection. Moreover, their illustration with parallelism (hence, non-significant interaction term). in light of easily constructed and actual examples of parallelism that show either sub- or superadditivity, suggests that they can provide no proof for their claim. For the tests they describe, in which one drug is present in a fixed amount, it is easy to show that additivity may lead to either parallel dose-effect curves or to non-parallel dose-effect curves. This will depend on wether R. the potency ratio, is constant at all levels of effect. It is equally easy to show that parallelism of the curves can mean either subadditivity, additivity or superadditivity. Thus. these authors are applying the results of ANOVA to this problem without having established a rigorous basis for such an application. Nowhere in this letter have I addressed the conclusions of Miaskowski et al. in regard to their combinations of intrathecal opioid agonists. My letter addresses only their claims related to the use of statistical methodology for demonstrating synergism.

PAIN 07217

Comments

on Tallarida

and Miaskowski

et al.

The April issue of PAIN (Vol. 49) contains two articles addressing evaluation of drug interactions, In his Basic Review, R.J. Tallarida describes a “Statistical analysis of drug combinations for synergism” that mathematically derives a method of analysis for determining the interactions between two drugs that have the same action. This is a brief review of his previous publication (Tatlarida et al.. Life Sci.. 45 (1989) 947-961) which describes the mathematics in detail. In his discussion, the author emphasizes the importance of fixed-ratio combinations in the statistical analysis. Individual dose-response curves for two agents given in combination are often not parallel and the drugs may have different efficacies. Thus, the potency ratio changes with the level of effect. Tallarida presents an isobolographic analysis method in which the median effect values from experimentally obtained dose-response curves for the drugs in combination are compared with the curve of additivity in the same proportional mix. The additivity curve is derived from median effect levels from dose-response curves obtained when the two drugs are administered separately. Confidence intervals for the curves are derived and significance is assigned to any shifts in the curves. The calculations allow comparisons at a common effect level and take into account the slopes of the individual drug dose-response curves and efficacies. A superadditive (synergistic) effect depends not only on the drugs and the test, but also on the specific mixture ratio of the two drugs. The other report, by Miaskowski et al., evaluates combinati[Jns of opioid receptor selective agents which produce antinociception. These investigators used variable, not fixed, dose ratio combinations and analyzed the data using ANOVA. Dose-response curves for the drugs administered separately in the paw pressure test are not parallel (see figure, estimated from manuscript), and it appears that U5~488H and DPDPE are partial agonists. Neither of these conditions is taken into account for the statistical analyses. Also. the dose of DAMGO that was given in combination with U50488H and DPDPE (0.5 ny) was more effective than the doses of U50488H (5 ng) or DPDPE (0.5 ng) that were given in the other combinations, which may account for the greater ‘enhancement’ effect of DAMGO as concluded by the authors. These two experimental methods are completely different. fixed vs. variable dose ratios, as are the statistical methods, isobolographic vs. ANOVA. The Tallarida method cannot be used to analyze the Miaskowski et al. data. I am concerned that readers of PAIN may conclude that the Miaskowski et al. paper is an example of applica-

=

400

E

300

-

-

.-~

DAMGO

.v,-h-

U50488 DPDPE

Ronald J. Tallarida 10

100 Dose

AgonisI

Fig. 1

lOO0 fngl

t oaoo

100000

383

tion of the Taliarida method. This is clearly not the case. The two methods and analyses are not the same. PAIN 02219 Sandra C. Roerig

Comments on the evaluation of drug interactions using isobol~aphi~ analysis and analysis of variance

“Different methods of analysis may give different results when applied to the same set of data, so that a combination may appear synergistic according to one method and antagonistic according to another. ” (Berenbaum I 989)

PAIN 02218

Comments on Miaskowski et al., Pain 49 (1992) 137144 In the above report, Dr. Miaskowski and co-workers have investigated the antinociceptive effects of delta/mu and kappa/mu drug combinations. The use of multiple agents has the potential to provide greater pain relief and/or decrease untoward effects, and the authors are to be complimented for contributing detailed information regarding opioid interactions. However, a careful examination of their data handling reveals certain inadequacies. Synergy is not only a function of the combination of drugs used but also of the ratios in which the agents are combined (for review with examples, see Berenbaum, 1989). Thus, studies in which the ratios of the individual agents are allowed to vary, and statistics are performed on data sets for which the ratios of combinations vary, are not appropriate (for a review of statistical methods appropriate to analyzing drug interaction studies see Tallarida et al., 1989, and Berenbaum, 1989). Since there is no a priori reason to expect that all combinations (ratios) of mu/delta or mu/kappa agents will produce the same interaction (additive, sub-additive, or supra-additive), the statistic used by the authors (test of parallelness of the delta/mu vs. mu and kappa/mu vs. mu dose-response curves) is not correct. Rather, the effect of each combination (ratio) should have been compared to the expected additive level of effect calculated from the individual dose-response relationships of the delta, kappa and mu agents. While the authors do present adequate data (some of it in the form of references to their previous work) to have generated expected additive levels of effect for each combination, they are to be taken to task for not manipulating their results in the correct manner and interpreting their data with the proper statistic.

References Berenbaum, M.C., What is synergy?, Pharmacol. Rev., 41 (1989) 93-141. Miaskowski, C., Sutters, K.A., Taiwo, Y.0 and Levine, J.D., Antinociceptive and motor effects of delta/mu and kappa/mu ~mbinations of intrathecal opioid agonists, Pain, 49 (1992) 137144. Tallarida, R.J., Porreca, F. and Cowan, A., Statistical analysis of drug-drug and site-site interactions with isobolograms, Life Sci., 45 (1989) 947-961. Gene Williams Neurobiology and Anesthesiology Branch National Instifute of Dental Research National Institutes of Health Bethesda, MD 20892, USA

This quote from Berenbaum provides a focal point for our response to the recent editorial by Gebhart (19921, the review on statistical analysis of synergistic interactions by Tallarida (1992) and the Letters to the Editor in this issue of PAIN. We would like to provide the readership of PAIN with some necessary background information on two methodological approaches that have been used to evaluate for drug interactions (i.e., isobolographic analysis and analysis of variance (ANOVA)) and then describe the strengths and the limitations of using either method to study drug interactions, as well as respond to the specific concerns raised in the letters to the editor.

Methodological approaches to the study of interactions In general, an interaction between biologically active agents “is defined as being present when the effect of a combination of agents differs from that expected from their individual dose-response curves” (Berenbaum 1989). Interactions can be categorized as more than additive (i.e., synergistic), or less than additive (i.e., antagonistic). Although the study of interactions between biologically active agents, including analgesic drugs, would appear to be relatively straightforward, a number of issues need to be addressed when designing experiments to evaluate for drug interactions (Berenbaum 1977, 1981. 1989: Wessinger 1986; Tallarida et al. 1989; Rideout and Chou 1991). Some of the methods of analysis that have been employed to study drug interactions include: isobolographic analysis (Litchfield and Wiicoxon 1949, Gessner 1974; Tallarida 1992); 2-factor, repeated-measures analysis of variance (ANOVA; Fisher and Mackenzie 1923; Lindman 1974; Cohen and Cohen 1983; Melton and Tsokos 1983) and measurement of the effect of a fixed dose of one agent on the dose-response curve of another agent (Berenbaum 1989). While the most widely published method of analysis used to study antinociceptive interactions is isobolographic analysis (Kissin et al. 1990; Ossipov et al. 1990; Porreca 19901, we have employed 2-factor, repeated-measures ANOVA, which has been used in a variety of other disciplines, to evaluate for interactions between selective opioid receptor agonists administered at spinal and supraspinal sites (Miaskowski et al. 1990. 1991, in press; Sutters et al. 19901. In order to choose the most appropriate method of analysis to study drug interactions, it is necessary to know the rationale for the use of each method.

Iso~lographic

analysis

To construct an isobologram for the evaluation of interactions between 2 analgesics, for example drugs A and B, single points representing doses of A and B that produce equal analgesic effects