Mathematical assessment of properties of precursor-dependent indirect pharmacodynamic response models.

Precursor-dependent indirect response (PDIDR) models may describe the tolerance and rebound phenomenon observed for many pharmacodynamic responses where these characteristics are manifested due to the depletion or accumulation of a physiological precursor or cofactor pool responsible for generating drug effects. The purpose of this report is to extend the concepts and applications of these models and to approximate responses and limiting conditions for very large doses of drugs. Asymptotic analysis was performed for qualitative determination of various parameters, such as maximum response (Rmax) and rebound (RBmax), time to maximum response and rebound (TRmax and TRBmax), and area under the effect and rebound curve (ABEC and ABRC) for large doses. Computer simulations were performed to assess the role of dose for both cases where drugs act either by depleting (Model V) or by blocking (Model VI) the endogenous precursor. Simulations showed that Rmax, RBmax, TRBmax, ABEC and ABRC increase with dose, eventually reaching a plateau when Dose/V is very large compared to the efficacy parameters (SC50 or IC50) of the drug. However, TRmax either increased or decreased with dose depending on various system and drug parameters. The limits for these parameters at large doses qualitatively determined by asymptotic analysis closely approximated the plateaus observed from the simulated curves. At large doses, the drug response could be approximated by a Bateman-like function for both Models V and VI. Qualitative analyses along with simulation studies provide a fundamental basis for understanding the temporal aspects of the PDIDR models especially at large doses to describe the tolerance and rebound phenomenon.