MATHEMATICAL SOLUTION OF A PHARMACOKINETIC MODEL WITH SIMULTANEOUS FIRST-ORDER AND HILL-TYPE ELIMINATION?

Mathematical studies on pharmacokinetic models are essentially important for drug development and optimal dose design. Considering the interaction between drug molecules and their receptors, the elimination of drug molecules can exhibit Hill-type kinetics. In this paper, motivated by the recombinant human granulocyte colony-stimulating factor (G-CSF) and the transcendent Lambert W function, we have studied the mathematical solu-tions of a one-compartment nonlinear pharmacokinetic model with simultane-ous first-order and Hill-type (n = 2) elimination for the case of intravenous bolus administration. By introducing three well-defined transcendental func-tions depending on three different scenarios, we have established the closed -form precise solutions of time course of drug concentration, which is a method to calculate drug concentrations at any time point. As a result, we also have derived the explicit expressions of some key pharmacokinetic surrogates such as the elimination half-life t1/2 and total drug exposure (i.e. area under the con-centration curve (AUC)), which are found as dose-dependent. Finally, a case study of a G-CSF drug is quantitatively illustrated to delineate our theoreti-cal results, including the elimination half-life and AUC for different dosages. Our findings can provide an effective guidance for drugs with simultaneous first-order and Hill-type (n = 2) elimination in clinical pharmacology.