Pharmacokinetic stochastic model with Weibull-distributed residence times of drug molecules in the body

The use of a function to fit blood concentration-time data points is equivalent, under certain assumptions, to specifying a model of the distribution of residence times of the drug molecules in the body (stochastic pharmacokinetic model). An empirical density function of the Weibull type is offered to describe this distribution. The model gives the following disposition function describing the time course of the drug concentrations in blood after an intravenous bolus input: $$C_\delta (t) = \frac{\operatorname{D} }{{\operatorname{CL} }}s\lambda t^{s - 1} \exp ( - \lambda t^s )$$ It contains only three parameters: λ is like an ‘elimination rate constant’ in the single-exponential model into which the Weibull function reduces when the shape parameters becomes equal to unity; CL is the conventional systemic drug clearance, and, D is the dose injected. The Weibull function gives an analytical solution of the convolution integral for zero-order input, thereby permitting use of the model for intravenous infusion data and for extravascular administration, when the absorption may be considered to be zero-order. Using examples from the literature it is shown that in some cases the Weibull function gives a better fit than may be obtained with two- and three-exponential or gamma functions.