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J Biopharm Stat. 2018;28(4):763-777. doi: 10.1080/10543406.2017.1397005. Epub 2017 Nov 27.

Parametric modeling and optimal experimental designs for estimating isobolograms for drug interactions in toxicology.

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a Division of Biostatistics , German Cancer Research Center , Heidelberg , Germany.
b Division of Cancer Drug Development , German Cancer Research Center , Heidelberg , Germany.


In toxicology and related areas, interaction effects between two substances are commonly expressed through a combination index [Formula: see text] evaluated separately at different effect levels and mixture ratios. Often, these indices are combined into a graphical representation, the isobologram. Instead of estimating the combination indices at the experimental mixture ratios only, we propose a simple parametric model for estimating the underlying interaction function. We integrate this approach into a joint model where both the parameters of the dose-response functions of the singular substances and the interaction parameters can be estimated simultaneously. As an additional benefit, this concept allows to determine optimal statistical designs for combination studies optimizing the estimation of the interaction function as a whole. From an optimal design perspective, finding the interaction parameters generally corresponds to a [Formula: see text]-optimality resp. [Formula: see text]-optimality design problem, while estimation of all underlying dose response parameters corresponds to a [Formula: see text]-optimality design problem. We show how optimal designs can be obtained in either case as well as how combination designs providing reasonable performance in regard to both criteria can be determined by putting a constraint on the efficiency in regard to one of the criteria and optimizing for the other. As all designs require prior information about model parameter values, which may be unreliable in practice, the effect of misspecifications is investigated as well.


Combination index; c-optimal design; dose-response studies; isobologram; nonlinear regression; toxicology

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