Isoboles

The location of the points is then compared to the location expected for mixtures with additive interaction, which is the straight diagonal line between points for A alone or B alone (e.g., LD50A and LD50B). If the points fall outside the triangle, we have antagonism, whereas when inside, we have potentiation. 

If one substance is nontoxic but modifies the toxicity of another substance, we get isoboles, as shown underneath. In this case, (B) is nontoxic but functions as a synergist or antagonist to (A). 

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An isobole is a contour line representing equi-effective quantities of two agents or their mixtures (Loewe and Muischnek 1926). 

The isobole method is widely used to evaluate the effects of binary mixmres. However, a large number of different mixtures of the two compounds have to be tested in order to identify combinations that produce the fixed effect. 

In practice, the interpretation of test results strongly depends on the accuracy of the estimated intercepts of the theoretical isobole with the axis, which represents the doses of the single compounds that induce the desired effect. In fact, large standard deviations of these intercepts prevent a rehable conclusion as to the deviation from additivity. 

For binary mixtures, a straight line (isobole) is produced joining Di and D2 and passing through di, d-2). The interaction index (Cl) is 1, <1, or >1 when the combinations show zero interaction, synergism, or antagonism using dose addition, respectively. 

One difficulty in using this approach is to determine when a specific Cl actually deviates from 1 (additivity), as the method of isoboles as developed does not include measures to decide whether deviations from the line of additivity are systematic or simply due to chance or experimental error (Cassee et al. 1998). One way of dealing with this problem is to calculate confidence intervals for the iso-effective doses of the single compounds and to add a confidence belt to the line of additivity (Kortenkamp and Altenburger 1998). This envelope of additivity is an area in which those combinations of two compounds are lying that has a specific effect and may reasonably be considered as showing no interaction (for details see Cassee et al. 1998). 

The isobole method can also be applied to mixtures where only one of the two agents produces the effect under consideration. In case agent A produces an effect, whereas agent B does not, the equation is reduced to... 

In this case the iso-effective dose, D2, of the agent lacking the effect of interest can be regarded as infinitely large, so that the resulting additivity isobole runs parallel to the respective dose axis. 

One of the strengths of the isobole method is that it can be used to analyze combined effects of compounds irrespective of the shape of their dose-response curves. It is possible to assess mixtures of agents with dissimilar dose-response relationships, even when the maximal effects are not identical (Kortenkamp and Altenburger 1998). 

Kortenkamp, A. and R. Altenburger. 1998. Synergisms with mixtures of xenoestrogens A reevalution using the method of isoboles. Sci. Total Environ. 221 59-73. 

These interactive effects may be visualized graphically as isoboles (Fig. 2.3) or alternatively, there are simple formulae, which may be used for detecting them ... 

Siihnel J. 1992. Assessment of interaction of biologically active agents by means of the isobole approach fundamental assumptions and recent developments. ACES 4 35 14. 

More recently Brochot et al. [89] reported an extension of the isobolographic approach to interaction studies for convulsant interaction among pelloxacin, norfloxacin, and theophylline in rats. Their contribution is unique in that they started out by explaining pharmacodynamic interactions for two drugs, but then extended the approach to derive an isobol for three drug interaction. In addition they included Bayesian analysis and developed a population model with Markov chain Monte Carlo methods. 

Response Surface Model A dose-response surface is an extension of dose-response lines (isobols) to three dimensions. In this representation there can be a dose-response surface representing additivity and surfaces above and below suggesting deviation from additivity. Tam et al. [90] studied the combined pharmacodynamic interactions of two antimicrobial agents, meropenem and tobramycin. Total bacterial density data, expressed as CFU (colony forming units), were modeled using a three-dimensional surface. Effect summation was used as the definition of additivity (null interaction hypothesis) and the pharmacodynamic model was assumedi to take the functional form... 

Figure 11-1- Curves of equitoxic effect (isoboles), calculated using simple additive action, and observed data of synergism at LD s and LD.j.s achieved by topical application In house dies using pyrethrins/PBO.

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