Explicit solutions for a class of indirect pharmacodynamic response models

Explicit solutions for four, ordinary differential equation (ODE)-based, types of indirect response models are presented. These response models were introduced by Dayneka et al. in 1993 [J. Pharmacokinet. Biopharm. 21 (1993) 457] to describe pharmacodynamic responses utilizing inhibitory or stimulatory Emax type functions. The explicit solutions are expressed in terms of hypergeometric 2F1 functions and their analytical continuations. A practical application is demonstrated for modeling the kinetics of drug action for ibandronate, a potent bisphosphonate that suppresses bone turnover resulting in a reduction in the markers of bone turnover. Ten times shorter model evaluation times, with the explicit solution compared with the differential equation implementation, may enhance situations where a large number of model evaluations are needed, such as clinical trial simulations and parameter estimation.