On the correction for radioactive decay in pharmacokinetic modeling.

The question of how to include radioactive decay during biological modeling with first-order differential equations was considered. Modeling may involve either experimental data y(t) or decay-corrected data z(t) [identical to exp(lambda t)y(t) where lambda is the decay constant] for each compartment. It is sometimes assumed that the latter are solutions to corresponding purely pharmacokinetic models (no decay). We primarily compared the two analyses in the case where the model did not require simultaneous consideration of both labeled and unlabeled material. A general theorem was found which limits the use of decay-corrected data to pharmacokinetic models containing linear, homogeneous differential equations. By way of verification, an example of this model type was analyzed for a chimeric monoclonal antibody biodistribution in man. Even in this case, statistically significant differences between the two solutions showed that one may find different model parameters depending upon which data set (y or z) was analyzed. For other mathematical forms, the analyst must include the physical decay in all relevant compartments. By analyzing an open, quadratic model, effects of not including decay were seen to be maximized if the biological rate constant was > or = lambda, the physical decay constant. Finally, using monoclonal antibody-antigen reactions, similar discrepancies between the z functions and the pharmacokinetic variables were demonstrated. This result was found to persist even if competitive molecules were included. We conclude that decay-corrected data may be shown, but should not be entered into the modeling equations unless the latter are of the linear, homogeneous form.