Latent variable regression model (LVRM) inversion is a useful tool to support the development of new products and their manufacturing conditions. The objective of the model inversion exercise is that of finding the best combination of regressors (e.g., raw material properties, process parameters) that are needed to obtain a desired response (e.g., product quality) from the model. Each of the published applications where model inversion has been applied utilizes a tailored approach to achieve the inversion, given the specific objectives and needs. These approaches range from the direct inversion of the LVRM to the formulation of an objective function that is optimized using nonlinear programming. In this paper we present a framework that aims to give a holistic view of the optimization formulations that can arise from the need to invert an LVRM. The different sets of equations that become relevant (either as a term within the objective function or as a constraint) are discussed, and an example of these scenarios is also provided. Additional to the formulation of the different scenarios and their objective functions, this work proposes a new metric (the P 2 statistic) to cross-validate the ability of the model to reconstruct the regressor vector (analogous to the Q 2 statistic aimed at the predictability of the response). This new metric comes from the need to not only predict the response from the regressor, but to also reconstruct the regressors from the scores values. In this context, a discussion is provided on the effect of uncertainty in the reconstruction of the regressor (the actual design) as these values are normally given upstream as targets to the supplier of materials, or as set points to the process.

In this article we present a new and more accurate model for the prediction of the solubility of proteins overexpressed in the bacterium Escherichia coli. The model uses the statistical technique of logistic regression. To build this model, 32 parameters that could potentially correlate well with solubility were used. In addition, the protein database was expanded compared to those used previously. We tested several different implementations of logistic regression with varied results. The best implementation, which is the one we report, exhibits excellent overall prediction accuracies: 94% for the model and 87% by cross-validation. For comparison, we also tested discriminant analysis using the same parameters, and we obtained a less accurate prediction (69% cross-validation accuracy for the stepwise forward plus interactions model).

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Primary drying is the most time-consuming and energy-intensive step in pharmaceutical freeze-drying. Minimizing the duration of this stage is of paramount importance to speed up process development and product manufacturing. In this study, we propose a stochastic modeling framework that can help to reach this target. The framework is composed of five sequential steps: model development, sensitivity analysis, model calibration, model validation, and dynamic optimization. Three critical issues are addressed and accounted for in the model structure, namely, (i) the effect of time-varying operating conditions on the process key performance indicators (KPIs); (ii) the dynamic evolution of the water vapor partial pressure inside the drying chamber; and (iii) the impact of drying heterogeneity on the primary drying duration. We cope with the first two issues by introducing macroscopic energy and mass balances within the model formulation. The third issue is addressed by allocating intralot variability as a parametric uncertainty in the model parameter with the strongest sensitivity toward the process KPIs. The proposed stochastic model is calibrated and validated with data generated from industrial experiments. Nonlinear dynamic optimization is then exploited to minimize the duration of primary drying while simultaneously guaranteeing the fulfillment of tight constraints on the product temperature and sublimation rate. Experimental results show a reduction of ∼20% of the primary drying duration with the optimized protocol when compared to standard (i.e., at constant shelf temperature and chamber pressure) protocols, while ensuring the same product quality.

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