We have successfully incorporated probabilistic modeling into a PBPK model and applied it to the prediction of meat withdrawal times. We were able to predict the upper limit of a 95% confidence interval for the 99th percentile of the population. Thus, the method applied allows simulations that meet the recommendations for establishment of meat withdrawal times established by FDA to be conducted.
The validity of this model is directly dependent upon the distributions used to define key parameters. It was noted during the creation of these distributions that several probability density functions that would describe the same mean and range that were found in the published literature could be used. The different distributions did produce different population spreads (data not shown). Ranges of values for the parameter distributions were taken to be the broadest in terms of both variability in the population and uncertainty in the distributions reported. This would increase the overall spread of the Monte Carlo output and allow a more conservative estimate of the withdrawal time. The distributions could also overestimate population variance since we were unable to correct for the variance inherent within the literature studies. The wider distributions contribute to a more conservative estimate for a meat withdrawal time and should be considered when the model is applied to practical situations. However, from a public health standpoint, it is better to err by creating a more conservative meat withdrawal time than risk the consequences of the possible presence of residues in tissue. Furthermore, the model can easily be updated as more data on the true distributions of the parameters are generated.
Lognormal distributions were assumed for the parameters on the basis of their acceptance within regulatory agencies such as FDA and the U.S. Environmental Protection Agency with regards to population estimation. Again, further research into the exact distributions for both parameters and populations would continue to increase the accuracy of this model. It should be noted that a strength of this type of probabilistic modeling is the transparency with which the assumptions and results are reported. The 99th percentile of the population was established without assuming any specific distribution.
The accuracy of the model was determined by its ability to predict concentrations similar to those found in the external data set. Each point of the external data set represented a mean concentration from the pigs (the number of which ranged from three to six) used in the published studies. In order to get a more robust estimate of individual pig variations, a small in vivo pilot study was performed. While this study produced only a single datum point for each tissue at each slaughter time, the results of the pilot study coincided well with previously published results for all tissues and allowed us to accept the means with a greater degree of confidence. In fact, although the data from the pilot study had minor idiosyncrasies, the data were graphically indistinguishable from data from other studies when they were plotted together (Fig. ). This again helped to establish robustness within the external data set. The external data set also incorporated a wide range of dosing regimens. Dose independence, due to the mechanistic nature of PBPK models, is a strength of PBPK models. This allows validation against a wide range of doses and application to a dose not found within the external data set. The model tended to overpredict the concentrations at early time points. This may reflect a difference between absorption in vivo and how absorption was calculated in the model. However, the time points in the elimination portion of the curves were well covered by the Monte Carlo analysis. Since we are applying the model to the prediction of meat withdrawal times, the accuracy of prediction for later time points is more important. Further refining of the model could be made to increase the accuracy of the predictions for early time points.
Another source of variability within the population could be due to breed differences in metabolism and protein binding. To our knowledge, there are no reports of this for sulfamethazine or for any other drug in swine. Breed differences are most likely incorporated into the parameter distributions that were taken from the literature since these studies were carried out with various breeds and cross-breeds of swine. Variability within pigs can also be increased if the drug was given in feed to a pen of animals. Differences in social hierarchy and interpig personalities will mean a difference in overall drug intake and, thus, in the dose administered. For all studies used for validation that used oral dosing, dosing was done in such a manner (i.e., gavage) as to be able to accurately determine the true dose given to each pig. Variability in the pharmacokinetics of the N4-acetyl metabolite could also affect the population pharmacokinetics of sulfamethazine. The metabolite was included in the model due to the unique deaceytlation pathway that increases the concentration of the parent compound at later time points. In fact, the plasma protein binding of this metabolite was determined to be a sensitive parameter and was included in the Monte Carlo analysis. Ultimately, there are an infinite number of sources of variability between pigs in terms of drug disposition. Sensitivity analyses help to narrow the scope by identifying those parameters which affect pharmacokinetic predictions. Therefore, the Monte Carlo analysis did not include parameter distributions where parameters were determined to be insensitive.
The oral withdrawal time predicted by the Monte Carlo method is 6 days longer than the labeled withdrawal time. Sulfamethazine had an original withdrawal time of 5 days, which was established in 1968. In 1980, the current withdrawal time of 15 days was established by using an algorithm based on the sensitivity of the analytical tests available at that time. While this was the standard practice at that time, the sensitivities and specificities of analytical techniques have substantially increased over the last 25 years. Thus, the labeled withdrawal time may not cover the population according to the current rigorous standards now required by FDA. Our model shows that the withdrawal time for approximately 20% of the population is often greater than the 15 days currently used. This could account for the significant amount of tissue residue violations found with this drug (2
). Beyond the differences between the methods used, other reasons for the differences in withdrawal times could be related to the distributions used. As was discussed above, multiple distributions can provide curves with the same shapes and ranges. Also, we are comparing the means of several studies rather than data for individuals. One would expect an even greater spread between datum points if more individuals were included. Data for individual pigs could contribute to an even longer withdrawal time if even larger variability was shown. The addition of data from the in vivo study for i.v. route did create a more robust external data set and allowed us to evaluate the model in terms of individual variability. Thus, the model provided excellent coverage of individuals as well as means.
The tolerance limit method predicted a withdrawal time 3 days less than the label withdrawal time. This is most likely because the data set did not include points beyond 5 days posttreatment, a limitation not present in a PBPK model. A major assumption in this method is that there are enough time points on the depletion portion of the concentration-time curve to accurately assess the terminal slope. Given the kinetics of sulfamethazine in swine, the data set most likely does not have enough data to accurately determine that terminal slope. Thus, the tolerance limit method is descriptive in nature and dependent upon the sample used. The PBPK model, on the other hand, is not dependent upon the data set for the exact determination of terminal slopes. Thus, its predictive nature, rather than the descriptive nature of the tolerance limit method, provides a strength for the prediction of withdrawal times.
In conclusion, we were able to incorporate probabilistic modeling into a PBPK model using Monte Carlo sampling and then successfully use the model to predict the tissue kinetics of sulfamethazine in swine. As a result of this, we believe that the current withdrawal time of 15 days may be inadequate to cover the upper limit of the 95% confidence interval for the 99th percentile of the swine population and should be reevaluated in light of public health concerns over the presence of sulfamethazine residue.