Estimation of Cmax and Tmax in populations after single and multiple drug administrations

Following the oral administration of drugs, the plasma concentration generally reaches, in principle, a single, well-defined peak (Cmax) at the time of Tmax. A complication for the direct estimation of Cmax and Tmax is that measurements of concentrations are recorded only at discrete time points. Theoretical equations characterizing the population distribution of Cmax and Tmax are derived in relationship to the pharmacokinetic model, its parameters, their variabilities, and experimental errors. These equations can be solved by numerical integration. The resulting means, variances and other summary statistics of Cmax and Tmax are evaluated under various conditions involving single and multiple drug administrations. Results gained by the proposed numerical method agree closely with results gained by Monte-Carlo simulations. It is argued that the numerical method could be useful to study the statistical properties of the investigated measures and could, in some cases, provide a viable alternative to simulations. It is demonstrated that Cmax is estimated directly with positive bias, especially following repeated drug administrations. As a consequence, the recorded peak-trough fluctuation (PTF), measured in the steady state, can be excessively large (even by orders of magnitude) particularly when drug accumulation is high. These results have practical implications for the development of drugs and drug formulations.