A Nonlinear Isobologram Model with Box-Cox Transformation to Both Sides for Chemical Mixtures

• Published in: Environmental health perspectives Vol. 106; no. suppl 6; pp. 1367 - 1371
• Main Authors: Ding G. Chen, Pounds, Joel G.
• Format: Journal Article
• Language: English
• Published: United States National Institute of Environmental Health Sciences. National Institutes of Health. Department of Health, Education and Welfare 01-12-1998
• Subjects: Additivity; Algorithms; Animals; Biomedical and Methodological Advances on Chemical Mixtures; Cadmium; Cadmium - toxicity; Chemical hazards; Chemical mixtures; Dosage; Dose response relationship; Drug Interactions; Humans; Mercury - toxicity; Modeling; Models, Biological; Models, Chemical; Nonlinear Dynamics; Parametric models; Synergism; Toxicity Tests - statistics & numerical data; Toxicology
• ISSN: 0091-6765; 1552-9924
• DOI: 10.1289/ehp.98106s61367

The linear logistical isobologram is a commonly used and powerful graphical and statistical tool for analyzing the combined effects of simple chemical mixtures. In this paper a nonlinear isobologram model is proposed to analyze the joint action of chemical mixtures for quantitative dose-response relationships. This nonlinear isobologram model incorporates two additional new parameters, yminand$y_{max}$, to facilitate analysis of response data that are not constrained between 0 and 1, where parameters yminand$y_{max}$represent the minimal and the maximal observed toxic response. This nonlinear isobologram model for binary mixtures can be expressed as:$y=y_{min}+y_{max}-y_{min}/1+e^{-(\beta_{0}+\beta_{1}d_{1}+\beta_{2} d_{2}+\beta_{12}d_{1}d_{2})}$. In addition, a Box-Cox transformation to both sides is introduced to improve the goodness of fit and to provide a more robust model for achieving homogeneity and normality of the residuals. Finally, a confidence band is proposed for selected isobols, e.g., the median effective dose, to facilitate graphical and statistical analysis of the isobologram. The versatility of this approach is demonstrated using published data describing the toxicity of the binary mixtures of citrinin and ochratoxin as well as a new experimental data from our laboratory for mixtures of mercury and cadmium.