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 mathematical 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 cancer modeling, major accomplishments include (1) two new models of carcinogenesis (a model with true stem cells for skin carcinogenesis and a multipathway, multistage model for studying colon and liver cancer); (2) development of new methods for utilizing data on malignant and premalignant states when building a mechanistic model; (3) applying mechanistic models to data on premalignant lesions and tumors simultaneously; and (4) the development of a general algorithm for estimating tumor incidence from arbitrarily complicated multistage models. For non-cancer endpoints, this group has focused on (1) evaluating existing models in developmental toxicology with the intention of finding better models; and (2) developing new methods for analyzing teratology screens and modeling the relationship between immune function and host resistance following exposure to immunotoxicants. In physiologically-based pharmacokinetic modeling, this group has (l) developed a PBPK model for TCDD; (2) developed a PBPK model for 1-3, butadiene; (3) theoretically studied models for receptor binding and gene expression and (4) explored the effects of constitutive expression on risk estimates when using biomarkers. In risk assessment, this group has (l) collaborated with EPA on the reassessment of TCDD; (2) evaluated the shapes of dose-response curves and their relationship to chemical structure and activity; and (3) developed/applied a measure of carcinogenic potency which estimates and corrects for dose-response shape. 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. A project was completed on the application of smoothing spline methods to the estimation of rate functions in pharmacology and biochemistry.