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.

中文翻译：

---

单次和多次给药后人群中Cmax和Tmax的估算。

口服药物后，原则上血浆浓度通常在Tmax时达到一个明确的峰值（Cmax）。直接估算Cmax和Tmax的复杂之处在于，仅在离散时间点记录浓度测量值。与药代动力学模型，其参数，其变异性和实验误差有关，推导了表征Cmax和Tmax总体分布的理论方程式。这些方程可以通过数值积分求解。在涉及单次和多次给药的各种条件下，评估了Cmax和Tmax的均值，方差和其他汇总统计数据。所提出的数值方法获得的结果与蒙特卡洛模拟获得的结果非常吻合。有人认为，数值方法可能有助于研究所研究措施的统计特性，并且在某些情况下可以为模拟提供可行的替代方法。结果表明，Cmax可以直接估计为正偏差，尤其是在重复给药后。结果，在稳态下测得的记录的峰谷波动（PTF）可能会过大（甚至是数量级），尤其是在药物堆积较高的情况下。这些结果对药物和药物制剂的开发具有实际意义。特别是在重复给药之后。结果，在稳态下测得的记录的峰谷波动（PTF）可能会过大（甚至是数量级），尤其是在药物堆积较高的情况下。这些结果对药物和药物制剂的开发具有实际意义。特别是在重复给药之后。结果，在稳态下测得的记录的峰谷波动（PTF）可能会过大（甚至是数量级），尤其是在药物堆积较高的情况下。这些结果对药物和药物制剂的开发具有实际意义。