Characterization of Pharmacodynamic Recession Slopes for Direct and Indirect Response Models

Abstract

Direct pharmacologic effects are known to recede over time with largely linear slopes (Levy's k · m product, J. Pharm. Sci. 53: 342, 1964) and indirect responses have similar behavior. Pharmacodynamic slope properties were examined mathematically for the Hill function with monoexponential drug disposition and simulations were carried out for other pharmacokinetic functions. Both types of pharmacodynamic profiles exhibit a single terminal inflection point (fp) when drug concentrations exceed the EC₅₀ (that concentration causing one-half maximum effect, E_max ). For direct effects it was found that C_fp (the drug concentration at fp) =EC₅₀ , the determinants of inflection time were identified, and Slope_fp = −λ_zγE_max /4 where λ_z is the terminal disposition slope and γ is the Hill coefficient. These characteristics were explored for the four basic indirect response models which also exhibit recession profiles with slight sigmoidity and a single terminal inflection point at higher doses. The drug concentration at inflection C_fp is ≤IC₅₀ or SC₅₀ (drug concentrations causing half-maximal inhibition or stimulation), while the inflection response (R_fp ) attains constant values at larger doses. Indirect Response Models I, III, and IV have nearly linear return slopes for a wide range of doses which are governed by the disposition slope λ_z of the drug, loss constant k_out of the response, maximum inhibition (I_max ) or stimulation (S_max ) factors, and a unique fractional constant (0<G≤1). Model II exhibits more complex behavior with recession slopes which are less likely to be parallel for various doses. Most indirect responses are expected to show nearly linear recession slopes which are parallel for moderate to large doses and mainly governed by an identical combination of pharmacokinetic (λ_z ), system (k_out ), and dynamic capacity factors (I_max or S_max ).