A dynamic model is proposed for photopolymerization of 1,6-hexane-diol diacrylate (HDDA) with bifunctional initiator bis-acylphosphine oxide (BAPO) in the presence of oxygen. This partial-differential-equation (PDE) model predicts time- and spatially-varying vinyl-group conversion as well as concentrations of monomer, initiator, oxygen, and seven types of radicals. Experiments to obtain diffusivities of oxygen, BAPO and HDDA are reported. Oxygen-related parameters are estimated using real-time Fourier-transform infrared (FTIR) conversion data. FTIR experiments were conducted using a range of film thicknesses (8-17 μm), BAPO levels (1-4 wt%) and light intensities (200-6000 W/m^2). The model predicts qualitative trends. Conversion predictions for runs with high intensities (≥5000 W/m^2) and high BAPO (4 wt%) are accurate with a root-mean-squared error (RMSE) of 0.04. Larger RMSE (0.13) for runs with lower intensities and BAPO indicates that improved parameter estimates are required. Parameter estimates will be updated using in future using a model that accounts for shrinkage during polymerization.

A methodology is proposed to aid parameter estimation in fundamental models of pharmaceutical processes. This methodology addresses situations with insufficient data to reliably estimate all parameters, when the estimation is complicated by uncertain independent variables. The proposed method uses an augmented sensitivity matrix to rank the combined set of parameters and uncertain inputs from most estimable to least estimable. An updated mean-squared-error criterion is then used to determine the appropriate parameters and inputs that should be estimated, based on the ranked list. A model for one step in a batch pharmaceutical production process with an uncertain initial reactant concentration is used to illustrate the method, revealing that the initial reactant concentration in each batch should be estimated along with three out of six model parameters. Non-estimable parameters are fixed at their initial values to prevent overfitting. The method will aid error-in-variables parameter estimation in many situations involving limited data.

Error-in-variables model (EVM) methods are used for parameter estimation when independent variables are uncertain. During EVM parameter estimation, output measurement variances are required as weighting factors in the objective function. These variances can be estimated based on data from replicate experiments. However, conducting replicates is complicated when independent variables are uncertain. Instead, pseudo-replicate runs may be performed where the target values of inputs for repeated runs are the same, but the true input values may be different. Here, we propose a method to estimate output-measurement variances for use in multivariate EVM estimation problems, based on pseudo-replicate data. We also propose a bootstrap technique for quantifying uncertainties in resulting parameter estimates and model predictions. The methods are illustrated using a case study involving n-hexane hydroisomerization in a well-mixed reactor. Case-study results reveal that assumptions about input uncertainties can have important influences on parameter estimates, model predictions and their confidence intervals.

Error-in-variables model (EVM) methods require information about input and output measurement variances when estimating model parameters. In EVM, using replicate experiments for estimating output measurement variances is complicated, because true values of inputs may be different when multiple attempts are made to repeat an experiment. To address this issue, we categorize attempted replicate experiments as: i) true replicates (TRs) when uncertain inputs are the same in replicated runs and ii) pseudo-replicates (PRs) when measured inputs are the same, but unknown true values of inputs are different. We propose methodologies to obtain output measurement variance estimates and associated parameter estimates for both situations. We also propose bootstrap methods for obtaining joint-confidence information for the resulting parameter estimates. A copolymerization case study is used to illustrate the proposed techniques. We show that different assumptions noticeably affect the uncertainties in the resulting reactivity-ratio estimates.