Tablet coating is an important processing step in the pharmaceutical industry in which tablets are coated with an aqueous or organic film coating for both aesthetic and functional reasons.
Mathematical models of tablet film coating are important in pharmaceutical development as they aid in design of experiments, scale-up and determining optimal process conditions as formulations change. A universal steady-state film coating model has previously been developed (am Ende, Berchielli, 2005) with the aim of providing process engineers with a means of predicting target operating conditions for optimization, scale-up and robustness studies. In that work, the model was validated with experimental conditions and the predictions obtained were found to be in good agreement with data not used for model validation.
In this study, the steady-state model was first validated against steady state plant data using gSOLIDS (Process Systems Enterprise, UK). It was found that starting with one experiment, as experiments were added, the 95% confidence interval decreased rapidly reducing to less than 10% with four or more experiments.
Figure 1: Change in estimated value of heat loss factor as experimental data is added
In order to quantify whether sufficient data was used for model validation, the 95% T-value was compared to the reference T-value. (If the 95% T-value is greater than the reference T-value, this indicates that sufficient data was used to for model validation). The results obtained indicated that at least three experiments are needed to accurately estimate the unknown model parameter. This was consistent with the results obtained for the 95% confidence interval which was more than 50% when only two experiments were used.
Figure 2: Changes in reference and 95% T-values as experimental data is added
As indicated by the 95% confidence intervals, the estimated value of the model parameter is not a perfect estimate. Therefore in order to determine a reasonable estimate of the model prediction a Monte Carlo simulation was carried out. In this simulation several instances of the model were run with each instance having the unknown model parameter sampled from a normal distribution centred at the parameter estimate and with a standard deviation obtained from the 95% confidence interval. The results from these numerous simulations were then averaged to obtain an estimate of the model prediction. In addition, the results from these simulations were used to obtain a 95% confidence interval on the model predictions. The results obtained indicated that the model predictions were in good agreement with the experimental data used for parameter estimation. Further, given the confidence interval of the estimated value of the model parameter, the standard deviation of the model prediction was much smaller than that of the data indicating that the predictions of the model vary very little with respect to the uncertainty in the parameter estimate.
Figure 3: Comparison of model predictions with experimental data
A similar analysis was conducted using dynamic data. In this case however, although the data used for parameter estimation was deemed sufficient, the quality of the fit was quantified as being unsatisfactory even if several experiments were used. This indicates insufficiencies in the model which is expected as a steady state model was used to study dynamic plant data.
In summary, the results presented above indicate that with relatively few steady state experiments, an accurate estimate can be obtained of the unknown model parameter in the steady state film coating model. The model predictions were found to be in good agreement with the experimental data. The uncertainty in the model parameter was quantified and it was found that the model predictions were relatively unaffected given the uncertainty in the parameter estimate.
The analysis presented above is generic and can be used to quantity uncertainty in any mathematical model. This information can be especially useful as it can be used to quantify the risk associated with decisions made using these mathematical models. Further, the rigour of the analysis helps determine whether more experimentation is needed or whether further model development is needed (or possibly a combination of both).
M. am Ende, A Berchielli (2005) A thermodynamic model for organic and aqueous tablet film coating. Pharmaceutical Development and Technology, 1:47-58
See more of this Group/Topical: Topical I: Comprehensive Quality by Design in Pharmaceutical Development and Manufacture