Isobole Methods

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

Toxicological Risk Assessments of Chemicals A Practical Guide 

The theoretical line of additivity is the straight line that connects the individual doses of each of the single agents that produce a predetermined, hxed effect alone, for example an ED50 (50% response) of a given toxic or biochemical effect. 

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. 

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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... 

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). 

The ADM is valid for studying the interactions of herbicides with other herbicides or nonherbicides that exhibit similar action. Under the assumptions of the ADM, the growth responses of a test plant to the joint application of two components of a herbicide mixture are equated to the sum of the responses to each chemical tested separately.Thus, the predicted dose-response relationships for a herbicide mixture using the ADM are such that if one component of the mixture is replaced at a constant proportion by the other, the predicted response remains unchanged. Methods developed for the prediction of plant responses to herbicide mixtures using the ASM include the isobole method, the ANOVA method, and the two-parameter method. " ... 

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). 

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

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. 

In the absence of empirically intensive studies of cross adaptation (Kohbara and Ca-prio 1996) there currently are no models that accurately predict the nature of the interaction of mixtures, regardless of the sensory system studied or bioassay used. In the absence of any theoretical models that describe the interactions of agents, investigators have used descriptive models based on data. Isobole approaches use averaging models to define the null condition of no interaction, then go on to provide a descriptive index of enhancement or suppression using concentrations, responses, and empirical concentration-response relationships (Suhnel, 1993). This approach is useful because it can be applied to systems for which mechanistic information driving the concentration-response is not available (see methods). 

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