Guide to Essential Biostatistics XXII: Demonstrating synergy from efficacy data - the Additive Dose Model (Wadley/Isobole method)

In the other articles in this series, we explore Designing and Implementing Experiments (the?Scientific Method,?Proposing Hypotheses?and?Type-I and Type-II?errors, Significance, Power, Effect,?Variance,?Replication,?Experimental Degrees of Freedom and Randomization), Descriptive statistics (critically evaluating experimental data: Q-test;?SD, SE, and 95%CI and Chi-square test), Inferential statistics (accepting or rejecting hypotheses by the parametric and non-parametric methods:?F-test, One- and Two-Sample Means Comparisons (the t-test), ANOVA and post-ANOVA tests and dose-response evaluation (linear and non-linear regression) as well as determining synergy from efficacy data.

?In the previous article, we considered the Multiplicative Dose Model (MSM) for differing modes of action. In this article, we will consider Additive Dose Model (ADM)for similar modes of action.

1.???ADM for similar modes of action

A more accurate (but less common, as it requires more experimental work) determination of the interaction between active ingredients in a mixture can be made by the Additive Dose Model (ADM).

The ADM method assumes that effects originate from similar modes of action, and that one component can substitute at a constant ratio for the other (the "equally effective rate"). The expected effectiveness of a mixture A+B(e) can thus be predicted from the effectiveness of the individual active ingredients when their relative ratio is known.

Example, equally effective rates:

In this example, the EC50 of active ingredient A is experimentally determined as 250g/Ha, the EC50 of active ingredient B is 0,9g/Ha. In a mix, A and B substitute for each other at a constant ratio to give an "equally effective rate" – for example in mix 2 (below): a mix of A+B where 50% of A (125g ai/Ha) is replaced by 50% of B (0,45g ai/Ha). The “weight ratio” is defined as the ratio of the individual active ingredient (A or B) to the mix (A+B). Other ratios are given in the following, and will be the basis of the following method examples:

1.1.?Wadley's method for binary mixes

Wadley’s method is based on EC50 values (effective concentration required to obtain a 50% effect) typically derived by linear or non-linear regression from dose-response experiments, and thus avoids errors arising from the nonlinear nature of the dose-response curve.

In the Wadley method, for a given combination of two active components, E (expected EC50) can be expressed as:

where a and b are the weight ratios (EC50 values in g/Ha) of compound A and B in the mixture and EC50(Ao) and EC50(Bo) are the experimentally observed (o) EC50 values obtained using the dose response curves for the individual compounds.?The synergism ratio (R) is calculated as the ratio between the expected values and observed values, EC50 A+B(e)/EC50 A+B(o).

Example, Wadley's method for binary mixes:

The EC50 of active ingredient A is experimentally determined as 250g/Ha, the EC50 of active ingredient B is 0,9g/Ha. In a mix, A and B substitute for each other at a constant ratio where 50% of A (125g ai/Ha) is replaced by 50% of B (0,45g ai/Ha). The EC50 is 70 g ai/Ha for the mix of A+B. If A+B (observed) is > A+B (expected) then synergy (R>1) is exhibited. If R=1 then the effect is additive and if R<1 then antagonism is exhibited:

In this example, the observed EC50 is 70 g ai/Ha for the mix of A+B and the synergism factor (R) between observed and expected is 1,79. The mix of A+B is synergistic.

One objective of studying synergy is to maximize the cost/benefit ratio of active ingredients in a mix, to attain maximal efficacy with minimal active ingredient. Wadley’s method can be used to evaluate multiple “equally effective rate” ratios (see above):

In this example, the synergy ratio (R=1,79) is greatest for the 50:50 mi, with A and B substituting for each other at a constant ratio where 50% of A (125g ai/Ha) is replaced by 50% of B (0,45g ai/Ha).

?1.2.?Isobole method for binary mixes and multiple dose ratios

?The Isobole method (also called Tammes method) is an extension of the Wadley method and uses a graphic representation to determine the effect of multiple dose ratios on the level of synergistic effect, to determine the mix ratio at which the greatest synergy is observed.?

Like the Wadley method, the Tammes method is based on EC50 values and avoids errors arising from the nonlinear nature of the dose-response curve.

Example, Isobole method for binary mixes with multiple dose ratios:

In this example, the EC50 of active ingredient A is experimentally determined as 250g/Ha, the EC50 of active ingredient B is 0,9g/Ha. In a mix, A and B substitute for each other at a constant ratio to give an "equally effective rate" – for example in mix 3 (below): a mix of A+B where 50% of A (125g ai/Ha) is replaced by 50% of B (0,45g ai/Ha).

The EC50s are used to plan a factorial trial with 25 treatments / 50 rates (g ai/Ha) comprising five rates (0,25X to 4X) and five mix ratios (100:0; 75:25; 50:50; 25:75; 0:100):

The factorial trial with 25 treatments is carried out and EC50 values are determined by e.g. nonlinear regression for each mix ratio, and the EC50s for each AI are calculated as a factor of their weight ratios (e.g. for mix 3 (above), the rates 250 : 0,9g ai/Ha corresponds to a ratio of 99,6 : 0,4%):?

The outcome of mix 3 corresponds to the example given earlier for Wadley's method - the observed EC50 is 70 g ai/Ha for the "equally effective dose" of A+B (where 50% of A (125g ai/Ha) is replaced by 50% of B (0,45g ai/Ha).

Finally, the EC50s for each AI are plotted as an isobole (a plot in rectangular coordinates where the axes represent the doses of active ingredient A and B):

The dotted straight line represents the EC50 isobole for an expected additive effect (additive isobole) of differing “equally effective” ratios of A and B, while the curved full line represents the actual experimental EC50s for the specified mixture ratios. If mixture EC50s are positioned below the additive isobole (as they are here) they are considered synergistic (synergistic isobole), and if the mixtures are positioned above, they are considered antagonistic (antagonistic isobole). The mix ratio with the greatest distance from the isobole (in this example mix 3) has the greatest synergy. This corresponds to the Wadley analysis, where the synergy ratio (R=1,79) was greatest for the 50:50 mi, with A and B substituting for each other at a constant ratio where 50% of A (125g ai/Ha) is replaced by 50% of B (0,45g ai/Ha).

1.3. Co-toxicity model) for binary mixes

A further method occasionally encountered in the literature is that of co-toxicity model. The outcome on the same dataset is the same as for the Wadley model (see below). In contrast to the Wadley model, this model is based on converting EC50 values for the individual active ingredients and the mix to a toxicity index (TI):

The cotoxicity coefficient (CC) of the mix can then be calculated as:

When the cotoxicity coefficient (CC) of the mix is greater than 100, there is a synergistic effect. If the CC equals 100 then the effect is additive and if greater than 100, antagonism is exhibited).

Example, co-toxicity method for binary mixes:

The EC50 of active ingredient A is experimentally determined as 250g/Ha, the EC50 of active ingredient B is 0,9g/Ha and the EC50 is 70 g ai/Ha for the mix of A+B. The theoretical toxicity index for the mix is 0,72 and the observed toxicity index for the mix is 1,3, giving a cotoxicity coefficient (CC) of the mix of 179:

This outcome again corresponds to the example given earlier for Wadley's method - for both methods (Wadley and co-toxicity) the synergism factor (R) between observed and expected is 1,79 and the mix of A+B is synergistic.

Thanks for reading - please feel free to?read and share my other articles?in this series!

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