Mathematical models for hyperbolic type of timed dose-response relationship of drugs

Based on the time and space appearance of drug effect, the timed dose-response relationship (TDRR) was studied. According to the characteristics of the hyperbolic TDRR curve, the hyperbolic four-parameter model (HFPM) was designed: model I is y=cS+1/(|x-a|)s+b, where a is dosage parameter, b response parameter, c curvature parameter and s skewness parameter. By logarithm transformation of dosage, model I turns to be model II. In this study, 16 series of TDRR data with replication were obtained, in heparin-clotting times in vitro (14 series), thrombin-thrombin time (1 series) and atropine-lethal time of mice (1 series). These data were nonlinear least-square fit to 9 models, including 7 control models, weighting the square error inversely with the square of SE. Moreover, the metameter data of y and x according to 9 models were analysed by linear regression. From the results of nonlinear and linear fit, 9 models were compared in square error and goodness of fit. Combining Bartlett test of 16 series of original and metameter data, HFPM is proved the best model for hyperbolic TDRR. But, the use and biological explanation of HFPM, the design of models for other type TDRR remain unsolved.