This project is intended to increase our understanding of the use and application of mathematical and statistical models in toxicology and biochemistry and to implement new models to aid in explaining current research findings. The research effort explores a diverse range of biological areas including carcinogenesis, pharmacology, developmental biology, neurology and immunology. In carcinogenesis; (1) it was found that tumor incidence rates cannot be used to provide accurate estimates of carcinogenic mechanism; (2) an estimate of carcinogenic potency which adjusts for chemically related changes in survival was developed and is being evaluated; (3) a mathematical model of carcinogenesis which explicitly incorporates DNA repair into the multistage process was developed and is being applied to initiation-promotion experiments in the skin and liver models; (4) two chemicals, 2,3,7,8-TCDD and 1,3-butadiene, are being explored as examples for the application of mechanistically-based mathematical models to the estimation of carcinogenic risks; (5) in examining potential life-shortening effects of carcinogens in breast milk it was found that only extreme levels of contaminants in breast milk represent more of a hazard than failure to breast feed; (6) the shape of the carcinogenesis dose-response curve using information on biological activity of the chemical has been studied and suggests that chemical structure correlates well with dose-response shape. In teratology, several models (e.g., Rai and Van Ryzin model) for estimating teratological risks were evaluated for agreement with existing data and no model was found to be adequate for these purposes. In immunotoxicology, it was found that there is little or no relationship between immunotoxicity and mutagenicity but that carcinogenicity and immunotoxicity are related with immunotoxic compounds having a high probability of being carcinogenic. In neurology, a research effort is underway with NINDS to apply multivariate smoothing techniques to characterize the relationship between areas of the brain and different parts of the body.