Introduction

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Two drugs that produce overtly similar effects may exhibit nonadditive action when they are administered together. Considerable interest exists in superadditive (synergistic) interactions. When two drugs are synergistic in producing a desired effect level, the required dosages of each are lower than expected from their individual potencies, a result with possible clinical significance. Synergism may also provide information on mechanism. Experiments designed to distinguish between synergistic and simply additive interactions often present challenges for the investigator, who must also utilize the most efficient study design. These challenges arise from the inherent variability of the experimental data, a phenomenon further complicated by the variety of new and existing compounds and the many different experimental methods used to study them. Thus, new and efficient experimental designs are continually needed. Toward this end, we recently developed a new experimental and statistical algorithm and applied it to testing synergism in a combination of morphine and clonidine (Tallarida et al., 1997), a pair previously shown to be synergistic (Ossipov et al., 1990a,b). In our previous study the two agents were administered in various doses of a single fixed-ratio combination, using an efficient experimental design. As expected, the combination demonstrated clear synergism, but that finding and the algorithm employed prompted us to ask whether other fixed-ratio combinations of these compounds would also be synergistic, and to what degree? In the current study two additional fixed-ratio combinations of these compounds were tested and the responses over this more extensive set of doses were determined and analyzed in a way that measured the strength of the synergism. Viewed graphically this design produces a surface in a three-dimensional plot in which the magnitude of the effect is the surface height over the planar domain of doses. An additional aim of the current study was to determine whether the value of the model(s) parameter that measures the synergism over the response surface could also be estimated by the method of isoboles.

	Theory and Methods

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Additive Combinations. Considered here is a situation in which each of the two drugs (denoted A and B) produces a dose-dependent effect, i.e., each yields a dose-effect relation for the common effect being studied. This study is concerned with cases in which the effect is measured on a continuous scale, and each dose-effect relation is well fitted to some appropriate smooth, nondecreasing curve, E = f(A) for compound A, and E = g(B) for compound B. The two agonist agents will produce either additive or nonadditive actions when they are given together as the dose combination (a, b). An additive action occurs when the constituents contribute to the effect in accordance with their individual potencies. For example, if drug A has twice the potency of drug B, then in combination B may be substituted for A in an amount that is double the amount that would be required of A. Using the relative potency of the agents, the combination may be referred to either compound. For example, if A is the reference compound, then (a, b) may be expressed as an equivalent amount of A when it acts alone. For a simply additive interaction with relative potency R (= doseA/doseB) the same at every level of effect, the calculation of the dose of A that is equivalent to (a, b) is given by

(1)

Because the effect E for any dose of A is known (from its dose-effect curve), the dose combination (a, b) that yields A_eq from the above equation can be paired with this effect by use of the curve described by the relation, E = f(A). A graphical view of this pairing is provided in a plot of E against (a, b), a procedure that yields a surface above the domain of (a, b) points; i.e., the pair (a, b) is a point in the plane and the effect is plotted as a vertical distance above it. The response surface obtained this way applies to simply additive combinations and is therefore an additive response surface. (An example of an additive response surface will be given subsequently.)

Superadditive Combinations. When the combination is superadditive (synergistic) and R is constant, the dose pair (a, b) acts like the greater dose (a + Rb)/, where , the interaction index, is less than unity. This relation follows from the simultaneous solution of R = A/B and a/A + b/B = . With reference to compound A's dose-effect relation (curve), this greater dose produces a different (greater) effect than the additive effect. The three-dimensional plot of effect versus (a, b) would therefore be a surface positioned above the additive response surface. In tests with the combination, the effect of (a, b) is determined and this value is related to the corresponding dose of A given by

(2)

a relation that allows a determination of .

Variable Relative Potency. When the relative potency varies with the effect level, the individual dose-effect curves, E_A = f(A) and E_B = g(B), provide the values of R and, thus, A_eq. To get the additive equivalent of A in this situation, the effects are equated, yielding f(A) = g(B), and this equation is coupled to the additive relation

(3)

the latter equation being the same as eq. 1, except that R is now a variable whose value is obtained from the simultaneous solution. The value of A determined from these simultaneous equations is A_eq, and it can be used to calculate the effect level from drug A's dose-response equation (curve) as previously described. This calculated effect level is the expected effect for pair (a, b) under additivity, and a plot of these gives the additive response surface. Whether the relative potency is constant or variable, it is seen that the additive surface is completely determined from the individual dose-effect curves. Synergism in this situation of varying R, described by, a/A + b/B = , is not characterized by a simple relation such as eq. 2. Instead, a new relation is derived in the following way. The effect of (a, b) is determined by testing and is referred to drug A's dose-effect curve to get the corresponding dose, A_corr. Simultaneous solution of the equations, a/A+b/B =  and f(A)= g(B), eliminates B to give the relation between  and A (=A_corr), thereby providing the value of  for the dose pair.

Dose-Effect Relations. The previous analysis shows that whether the relative potency of the two compounds is constant or variable over the range of effects it is still possible to determine the value of  from the combination data and thereby distinguish additivity from superadditivity. This analysis requires suitable equations for modeling each compound's dose-effect relation. The (graded) dose-effect relation of a drug has been modeled in a number of ways; a common model is the hyperbolic relation given by E = E_max (D)/ [(D) + C], where the constant C is equivalent to the dose (D) that gives the half-maximal effect. This dose is denoted "D₅₀". If each drug also produces the same maximum effect then R, the relative potency, determined from the hyperbolic relation is a constant equal to the ratio of the C of each agent: R = C_A/C_B. Another common model is the linear log (dose)-effect relation. When the two linear relations give parallel lines the relative potency is constant, whereas nonparallel lines mean a varying R. Whether R is a constant or a variable the parameter  may be calculated as previously described. It is additionally noted that the dose-effect curves of the individual drugs allow one to calculate the additive total dose, Z_add, for a specified effect level, and this is given by

(4)

where A denotes the dose of drug A acting alone that gives this effect, R is the relative potency (A/B) at that effect level, and p is the proportion of the total that is drug A. A synergistic interaction gives a total dose Z_mix that is equal to  Z_add.

Experimental Methods. The mouse tail immersion test was used with hot water (55°C) as previously described (Raffa and Stone, 1996). Intrathecal administration via injection into the subvertebral space between L5 an L6 (Hylden and Wilcox, 1980) was employed in all tests. Antinociception was measured as an increase in tail-withdrawal latency and was converted to percent of maximum percent effect (MPE) according to the formula: %MPE = 100 × (test latency  control latency)/(15  control latency). The 15-s cutoff was used to avoid injury to the tail. For construction of the dose-effect curves, the effect was expressed as mean %MPE, from 10 mice per dose, and was assessed at the time of peak effect (10 min after drug administration).

	Results

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Tests of antinociception used the tail immersion test in mice given intrathecal doses of morphine sulfate, clonidine hydrochloride, and various fixed-ratio dose combinations of these. This choice of test produced data that are continuous on the effect scale and are shown in Table 1. Graded dose-effect data for each drug, used alone, are also given in Table 1. These data allow a calculation of the additive total dose for each fixed-ratio combination, which may then be compared statistically to the actual total dose for the same fixed-ratio combination in order to distinguish synergism from simple additivity. That kind of analysis (use of total dose), previously made for one combination of these agents, is described by Tallarida et al., (1997). We here present the data for two additional combinations and, in so doing, utilize the response surface approach that uses the dose pairs as previously described.

Each drug's dose-effect data (Table1; Fig. 1) were fitted to the hyperbolic relation, E = E_max (D)/[(D) + C] over the range of effects, 0 to 100 = E_max. Correlation coefficients, 0.996 for the morphine relation and 0.985 for clonidine's relation, confirm the good fits shown in Fig. 2. The constant C (=D₅₀) for each is given in Table 1. As previously noted, this kind of fit means that the potency ratio R is also a constant: C_A/C_B = 1.546 in this case. It is thus possible to construct the additive response surface for these drugs in a three-dimensional plot (Fig.3A). Before examining the results of the combination experiments it is instructive to view the response surface for a synergistic (superadditive) interaction of these drugs. To accomplish this illustration we have constructed such a surface using  = 0.1 (Fig. 3B). This is an illustrative example in which a value of  indicative of strong (but realistic) synergism is used (as will be shown subsequently); moreover, this illustration includes the assumption of a single value of  that characterizes the drug combination. This assumption was examined in the current study by calculating the values of  for all dose pairs tested, as we describe next; but the graph of Fig. 3B, based on a single value of , is nevertheless revealing, as it shows a uniformly smooth response surface, convex and clearly positioned above the simply additive surface. The extent to which a single value of  with this magnitude (0.1) applies to the current data is revealed in an analysis of the actual combination data obtained.

View larger version (8K): [in this window] [in a new window]   Fig. 1.   An additive combination pair (a, b) is equivalent to a dose Aeq = (a + Rb) of drug A and yields effect level shown as E1. If tests with combination give a greater effect E2, then the combination is synergistic and acts like the greater dose Acorr. When relative potency is constant, Aeq and Acorr are related by eq. 2, which shows that  is the factor that relates these quantities.

View larger version (8K): [in this window] [in a new window]   Fig. 2.   Dose-effect curves in mice for morphine SO4 (A) and clonidine HCl (B) using the common hyperbolic model, E = Emax (D)/[ (D) + C ], where D is intrathecal dose of the compound and antinociception is obtained from the 55°C tail immersion test. In this model equation the constant C is the D50 dose.

View larger version (60K): [in this window] [in a new window]   Fig. 3.   A, additive response surface for the combination of morphine and clonidine as calculated from the curves of Fig. 2. Contour of constant effect on the surface (PQ) projects onto the plane to give the line of additivity (AB) that is also shown in the inset. B, illustration of a superadditive response surface based on a constant value (0.1) of the interaction index for the drug combination. This surface is positioned above the surface of additivity; it is shown in part to enhance the visual. Amount of surface elevation for a synergistic dose (a, b) can be determined from its equation,

The data shown in Table 2 provides the results of the combination experiments along with the calculated additive equivalent of drug A (morphine) as well as the amount of A (A_corr) that corresponds to the actual combination effect observed. Three different sets of fixed-ratio combinations were used. In the first set the proportion of morphine SO₄ was 0.605, while in sets 2 and 3, the proportions were 0.338 and 0.821, respectively. For each drug combination the parameter  was calculated by relating the observed effect to get A_corr, calculating A_eq and applying eq. 2, as previously described. These are given in the table. It is noteworthy that the magnitudes of the  values are indicative of marked synergism and indicate that the  value used above to illustrate the synergistic response surface (Fig. 3B) has the correct order of magnitude. But the actual  values suggest a difference for each combination set tested. To test whether the mean value of this interaction index differs among the three dose sets, we examined the groups in an ANOVA followed by the Neuman Keuls test (see Tallarida and Murray, 1987). The result, shown in Table 3, indicates significance, P < .05. These statistical tests indicate that the mean value of  for set 1 is greater than the values for the other two sets, which do not differ significantly. In other words, there is synergism for each of the three dose proportions tested, but it is more pronounced in sets 2 and 3 than in set 1. 

View this table: [in this window] [in a new window]   TABLE 3 ANOVA of interaction index () for different fixed-ratio combinations

The values of the interaction index  shown in Table 2 were determined from each combination's observed effect, the values of A_corr, and the additive equivalent dose, A_eq, of reference drug A (morphine) calculated from eq. 1. The data for these three sets, however, also permit estimations of  based on the 50% effect level. Toward this end we utilize the calculated additive total dose for %MPE=50, denoted Z_add, and the total dose, Z_mix, which gives %MPE=50. The value Z_mix was obtained by curve fitting the total dose-effect data to the hyperbolic model, while Z_add is calculated from eq. 4. The values Z_add and Z_mix for the 50% effect level are shown in Table 4. The ratio, Z_mix/Z_add, provides an estimation of  for the 50% level even though %MPE=50 was not an effect actually attained by any dose combination. The values of  determined this way are given in Table 4 for each of the three fixed-ratio combination sets. This method of determining  used equieffective total doses, Z_add, and Z_mix, and is therefore the isobole method we have previously used to test for synergism (Tallarida, 1992; Tallarida et al., 1997). It is seen that these estimates of the interaction index for the three sets have the same order relation as the mean values of the index that were obtained from actual effects over the surface (Table 2).

	Discussion

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In each of the fixed-ratio combinations tested, morphine and clonidine were synergistic in the tail immersion test. The value of  however, was not the same for each drug ratio. This finding illustrates that the degree of superadditivity of a drug combination depends not only on the drugs but also on the dose ratio of the combination. Efforts were also made to examine the graphical aspects of these findings. When dose pairs are plotted the effect values define a surface in this three-dimensional plot. An additive response surface is completely defined from the dose-effect relation of each drug as these provide the relative potency (R) used in calculating the equivalent dose of either drug. In our calculation each dose pair was expressed as an equivalent dose of morphine sulfate (here called drug A). This calculation follows directly from the definition of additivity, meaning that one drug may be substituted for the other in an amount determined by the relative potency of the pair. For morphine and clonidine in the tail immersion test the relative potency was constant at every effect level, a finding that is a consequence of the hyperbolic fit that characterized the dose-effect relation of each drug. The constancy of R simplifies the calculation of the additive equivalent (A_eq) and, thus, the construction of the additive response surface. But, even when R is not constant, the calculation of A_eq can still be made from the dose-effect data of each drug and the simultaneous solution of the equations described in the Theory and Methods section Variable Relative Potency.

It is seen from the graph (Fig. 3) that the additive response surface is a smooth convex surface. On this surface a contour of equal effects is the projection in the dose plane that is the familiar additive isobole, a line with intercepts A and B on the dose axes. In contrast, a superadditive surface will have a projection that is off the additive line and contained within the region formed by the line and the axes. The amount by which this trace is off the line is a visual indicator of the degree of synergism but is not obviously represented on the isobologram in a way that shows its precise measure. That measure is more precisely accomplished by the parameter , which directly relates the additive equivalent to the dose corresponding to the observed effect of the combination. This parameter or interaction index is a mathematical factor (multiplier) that indicates the degree of dosage reduction (of reference drug A) obtained with the combination. In the illustrative plot (Fig. 3B), we used  = 0.1, a value having the correct order of magnitude as shown in our tests. In words, this says that the interaction of the drugs is such that the dose pairs acted like 10 times the expected dose of the reference drug or, equivalently, only 0.1 of the expected amount of the reference drug is needed. We saw, however, that this factor varied with the combinations employed. The actual synergistic response surface is therefore more complicated than that illustrated in Fig. 3.

Isoboles are dose combinations that produce a constant effect; hence, these are planar projections of constant height from the response surface. The use of isoboles was succinctly reviewed by Loewe (1957), who introduced the concept without specifically addressing the statistical aspects. These became a major focus of studies by our group (Tallarida et al., 1989, 1992, 1997). The main statistical method uses a specified effect level (such as 1/2 E_max) and compares dose pairs that give this effect with the calculated additive dose pairs. This method also uses a graph, or isobologram, a plot in Cartesian coordinates of the doses (or concentrations) that give the specified effect level. Thus, the dose of each, when it acts alone, is a point on the axis. The line that connects these intercepts is the line of additivity. Synergism is indicated by a pair (a, b) off the additive line and contained in the region bounded by the line and the coordinate axes. This method has been employed in a number of studies with analgesic combinations (Yeung and Rudy, 1980; Roerig and Fujimoto, 1988; Ossipov et al.,1990a,b, Porreca et al., 1990) and in many other studies. It has also been used, along with statistical considerations, by our group (Tallarida et al., 1997) in an earlier study with one fixed-ratio combination (set 1) of morphine and clonidine. In contrast to that earlier study the current work employed a method in which each effect value produced by the combination was used and referred to one of the drugs (morphine) to yield the corresponding dose of A for comparison with the equivalent additive dose (A_eq). This allowed a calculation of , a measure of the synergism, and the finding that this quantity varied significantly with the drug ratio. Therefore this study showed that the actual three-dimensional response surface could not be well described by a single value of this parameter. Although planar isoboles are common in the study of drug combinations a plot of the actual response surface, whose contours of constant effect define the isobole, is far less common. We have graphed the surface for an additive drug combination of these drugs and illustrated how it would look if the synergistic interaction index was a constant over the domain of doses. Although our choice of index = 0.1 has the proper order of magnitude, actual analysis of the data revealed a more complicated superadditive surface.

The magnitude of the interaction, measured by the index , varied, not only with the ratio of morphine to clonidine in the combination, but also with the total dose within each fixed-ratio combination. In the combination of set 1, which contained the constituents in the greatest total dose, there is an order relation suggesting greater synergism (decreasing ) as the total dose increased. In contrast, sets 2 and 3, which have greater synergism than set 1, display no obvious order relation as the total dose changes. It is not clear whether the trend in set 1 or the lack of trend in sets 2 and 3 are mere chance phenomena, and the quality and quantity of these data suggest no easy way to answer this question. Yet the findings are revealing, as some investigators, working with other drugs and other end points, have detected dose-related synergism within a fixed-ratio combination (Meissler et al., 1998; R. W. Hurley, T. S. Grabow, R. J. Tallarida, and D. L. Hammond, submitted for publication). Also Ossipov et al. (1990a) showed that fixed-ratio combinations of morphine-clonidine may be either additive or synergistic depending on the route of administration, thereby suggesting the importance of concentrations. Another result of the current study is the near agreement in the values of the interaction index for each set as determined by the isobole method and the more detailed surface method. This suggests that the values of Z_add and Z_mix at some effect level, e.g., %MPE=50, and calculation of  as the ratio, Z_mix/Z_add, may be an acceptable indicator of the strength of synergism, a concept that was not mentioned explicitly by the authors in our previous work, in which these values were used only in testing for synergism. It therefore seems that this surface analysis has no advantage that would recommend its routine use in combination studies.

Interest in synergism, especially in tests with analgesics, has increased since Yeung and Rudy (1980) first demonstrated this kind of interaction for morphine administered at spinal and supraspinal sites. This site-site application of isobolar analysis is formally equivalent to the more usual drug-drug studies and points out the importance of synergism in steering us toward mechanism. Ossipov et al. (1990a), in tests with combinations of the same two drugs discussed in this report, provided evidence based on synergism that strongly implicate a spinal site of interaction between opiates and the -adrenoceptors stimulated by clonidine. Also relevant to mechanism are the studies conducted in Porreca's laboratory (Horan et al., 1992) in which different opioid -agonists and morphine, administered together, produced either subadditive or synergistic interactions. These findings suggested a possible regulatory role for the endogenous ligands of the opioid -receptor. Following the same line of investigation is a study (Adams et al., 1993), also with - and µ-opioids, in which synergism was detected in one test of antinociception (cold water test), while simple additivity was demonstrated in the hot water tail-flick test. That study, which also showed the importance of the dose ratio, concluded that the nociceptive stimulus was important and pointed out the different neuronal mechanisms that probably underlie each stimulus and the different modulatory role of these opioids.

The current study used combinations of morphine and clonidine mainly because this combination is well known to be synergistic and, thus, provided data that could be used to examine and compare values of the interaction index over the effect surface with values from the contour of the surface that represents one half the maximum effect (isobole method). This drug pair is also of interest clinically as clonidine has been used as a spinal analgesic and analgesic coadjuvant. Thus, both clinical and mechanistic considerations underscore the interest in this and other studies of synergism.