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Purpose: To examine and quantify bias in the Wagner-Nelson estimate of the fraction of drug absorbed resulting from the estimation error of the elimination rate constant (k), measurement error of the drug concentration, and the truncation error in the area under the curve., Methods: Bias in the Wagner-Nelson estimate was derived as a function of post-dosing time (t), k, ratio of absorption rate constant to k (r), and the coefficient of variation for estimates of k (CVk), or CV% for the observed concentration, by assuming a one-compartment model and using an independent estimate of k. The derived functions were used for evaluating the bias with r = 0.5, 3, or 6; k = 0.1 or 0.2; CV, = 0.2 or 0.4; and CV, =0.2 or 0.4; for t = 0 to 30 or 60., Results: Estimation error of k resulted in an upward bias in the Wagner-Nelson estimate that could lead to the estimate of the fraction absorbed being greater than unity. The bias resulting from the estimation error of k inflates the fraction of absorption vs. time profiles mainly in the early post-dosing period. The magnitude of the bias in the Wagner-Nelson estimate resulting from estimation error of k was mainly determined by CV,. The bias in the Wagner-Nelson estimate resulting from to estimation error in k can be dramatically reduced by use of the mean of several independent estimates of k, as in studies for development of an in vivo-in vitro correlation. The truncation error in the area under the curve can introduce a negative bias in the Wagner-Nelson estimate. This can partially offset the bias resulting from estimation error of k in the early post-dosing period. Measurement error of concentration does not introduce bias in the Wagner-Nelson estimate., Conclusions: Estimation error of k results in an upward bias in the Wagner-Nelson estimate, mainly in the early drug absorption phase. The truncation error in AUC can result in a downward bias, which may partially offset the upward bias due to estimation error of k in the early absorption phase. Measurement error of concentration does not introduce bias. The joint effect of estimation error of k and truncation error in AUC can result in a non-monotonic fraction-of-drug-absorbed-vs-time profile. However, only estimation error of k can lead to the Wagner-Nelson estimate of fraction of drug absorbed greater than unity.

Purpose: Sparse sampling in rodent toxicokinetics usually involves the collection of a single blood sample on a given study day from each animal in a treatment group. The samples are allocated to different time points, often allowing some replicates, and statistical inferences are then made about the concentration-time behavior of the test compound. The present study compared the results of one such analysis with those obtained from serial sampling as might be applied using satellite animals., Methods: Ten rats each received a single oral dose of tritium-labeled compound X. Blood concentrations in each rat at 10 time points post-dose were determined by liquid scintillation counting. Individual peak concentrations of blood radioactivity (Cmax) and peak times (tmax) were recorded, and area-under-curve (AUC) values were calculated by trapezoidal rule. The mean AUC and Cmax of all 10 animals over all 10 time points, referred to as AUCtrue and Cmax,true, were used as points of reference. These values were then estimated using subsets of the data that simulated satellite-animal or sparse-sampling designs. First, several different sampling schedules of 5 bleeding times were stimulated by taking subsets of the full data set. For analysis using satellite animals, serial blood concentrations from subsets of 3 or 4 rats were used to calculate point and confidence-interval estimates of AUCtrue and Cmax,true by standard methods; all possible subsets of 3 or 4 of the 10 rats were considered. For sparse data analysis, a single concentration from each of the 10 rats was used to calculate both point and confidence-interval estimates of AUCtrue by the Bailer-Satterthwaite method, and point estimates of Cmax,true, according to several different designs of replication. Animals were randomly assigned to time points, and 1000 of over 50000 possible combinations were evaluated for each bleeding schedule. The average percent absolute errors of the point estimates were computed and, for the 95% confidence intervals, average widths were determined., Results: For point estimates of AUCtrue, sparse sampling yielded average percent absolute errors of 7-13%. Percent errors for 3 and 4 satellite animals were 6-12% and 5-10%, respectively. For 95% confidence intervals, sparse sampling yielded widths of 24-90% of AUCtrue, whereas for 3 and 4 satellite animals widths were 37-50% and 24-34%, respectively. For Cmax,true point estimates from sparse-sampling and satellite-animal approaches had average percent absolute errors of 5-12% and 3-8%, respectively. The confidence-interval widths for Cmax,true from the satellite-animal approach were 15-24% of Cmax,true, but coverage did not achieve the nominal 95% for some choices of sampling times., Conclusions: By using proper study designs, one can limit the number of samples and the amount of blood drawn so as not to affect the animals' health status, yet still achieve the customary pharmacokinetic objectives in a toxicity study.

A nonlinear mixed-effects model simultaneously modeled two pharmacokinetic (PK) variables in patients administered cyclosporine twice daily: (i) concentration of drug in blood at the end of the 12-hr dosing interval (C12) and (ii) area under the concentration-time curve within the dosing interval (AUC). For two formulations (Neoral and Sandimmune), the model assessed the following: nonlinearity with respect to dose, interoccasion (intraindividual) variability, interindividual variability, and within- and across-individual correlation between C12 and AUC. Data were pooled from six clinical studies in stable renal transplant patients administered each formulation. PK samples on two occasions were taken usually for each formulation. Each individual's random effect was eight-dimensional consisting of two PK variables for each formulation on two occasions. An ANOVA-like partitioning worked well and reduced the variance matrix for the random effect to a known function of 13 parameters to be estimated, thereby making a numerically intensive computation feasible. Simulations were used to check the model fit, to compute standard errors, and to account for peculiarities in the residual analysis. Outcomes of tests comparing formulations, most of which were statistically significant, favored Neoral (dose proportional, lower interoccasion variability, lower interindividual variability, and higher correlation between C12 and AUC).

Since the pioneering work of Haggard and Teorell in the first half of the 20th century, and of Bischoff and Dedrick in the late 1960s, physiologically based pharmacokinetic (PBPK) modeling has gone through cycles of general acceptance, and of healthy skepticism. Recently, however, the trend in the pharmaceuticals industry has been away from PBPK models. This is understandable when one considers the time and effort necessary to develop, test, and implement a typical PBPK model, and the fact that in the present-day environment for drug development, efficacy and safety must be demonstrated and drugs brought to market more rapidly. Although there are many modeling tools available to the pharmacokineticist today, many of which are preferable to PBPK modeling in most circumstances, there are several situations in which PBPK modeling provides distinct benefits that outweigh the drawbacks of increased time and effort for implementation. In this Commentary, we draw on our experience with this modeling technique in an industry setting to provide guidelines on when PBPK modeling techniques could be applied in an industrial setting to satisfy the needs of regulatory customers. We hope these guidelines will assist researchers in deciding when to apply PBPK modeling techniques. It is our contention that PBPK modeling should be viewed as one of many modeling tools for drug development.

Bailer developed a method for constructing confidence intervals for areas under the concentration-vs-time curve (AUC's) with only one sample per subject but with multiple subjects sampled at each of several time points post dose. We have modified this method to account for estimation of the variances. How the need to estimate variances affects study design is discussed. An extension of Bailer's method is proposed where variances are modeled as a function of the means, in order to get more precise estimates of variances. The modified and extended methods are applied to a rat toxicokinetic study with only two rats per time point per treatment group.

Five major objectives for pharmacokinetic investigations in support of toxicity studies are identified as follows: Assess whether animals exhibited measurable blood concentrations in a dose-dependent manner; estimate average area under the concentration-time curve (AUC) and maximal concentration (Cmax) for each treatment group; elucidate general patterns in the concentration-time (CxT) profile, and summarize relationships between CxT and treatment group; determine CxT dependence on day into study; and judge interanimal variability and identify any animals with unusual concentration response. Such objectives are generally addressed in rodent toxicity studies by including "satellite" animals in the study. Satellite animals are extra animals dosed as per protocol but not subjected to toxicological and pathological observations and tests. Instead, they are used exclusively for the evaluation of pharmacokinetic characteristics of the test compound. In this paper, methods are described for achieving the five listed pharmacokinetic objectives in rodent toxicity studies without the use of satellite animals. A rat toxicity study is presented as an example.