Relationship between pharmacokinetic half-life and pharmacodynamic half-life in effect-time modeling.

A pharmacodynamic parameter relating time-dependent changes of the effect with time-dependent changes of concentrations has yet to be developed. In pharmacokinetics, half-lives (T1/2kin) are used to describe the relation between concentration (C) and time (t). In pharmacodynamics, often the sigmoid Emax model and the Hill equation are used (E = Emax CH/(EC50H + CH)) to describe the relation between effect (E) and concentration (C). To describe the correlation between effect (E) and time (t), a pharmacodynamic half-life (T1/2dyn) could be estimated if the use of the term half-life is not restricted only to log-linear first order processes. To bisect the drug effect a variable time (t1-2 = t2-t1) will be required for this nonlinear process. The bisection of the effect (E2 = 1/2 E1) is associated with a decrease in concentrations (C2 = C1 exp(-0.693 t1-2/T1/2kin)). A mathematical relationship can be derived between pharmacodynamic half-life (T1/2dyn = t1-2) and pharmacokinetic half-life (T1/2dyn = T1/2kin (ln (1 + ln(a)/ln(2))/H ) with (a = (EC50H + C1H)/(EC50H + C2H)). For concentrations in the range of the EC50 value with the Hill coefficient (H = 1), the pharmacodynamic half-life will be 1.6-2.0 times the kinetic half-life (T1/2dyn < or = 2.0 T1/2kin). For high concentrations (C1 > EC50), the dynamic half-life will grow much longer than the kinetic half-life, consequently the effect of a drug will not increase but it will last longer. The pharmacodynamic half-life turns out to be a specific estimate for the effect time relation, being a concentration-dependent function of the kinetic half-life.