(61e) Reliable Design and Optimization of Crystallization Systems Under Uncertainty

Model-based design approaches are very popular for the design and control of various crystallization processes. To implement these approaches for the design of a new crystallization process, systematic digital design frameworks have been developed, which include steps such as thermodynamic and kinetic operating space investigation, population-balance model (PBM) development, and model-based process optimization and control. A common feature of most of the proposed frameworks is open-loop process optimization with nominal kinetic parameters as the last step.¹ However, as these kinetic parameters are estimated from experiments, it is very difficult to generate highly accurate models because of the limited quality and quantity of the experimental data and due to the experimental errors, that often affect the accuracy and precision of the data. Thus, during the practical implementation of the optimized model-based results, deviations between experimental and model-predicted trends are often observed, thereby defeating the entire purpose of model-based design. To address this issue, it is essential to incorporate parameter uncertainties in the design process.²

In this study, we aim to extend existing digital design frameworks by integrating different concepts associated with uncertainty quantification, reliable design, and robust optimization to design crystallization systems that are robust to different parameter uncertainties. The system considered is the cooling crystallization of the pharmaceutical ingredient diphenhydramine hydrochloride. Population-balance models for both batch and continuous multistage stirred tank crystallizers were developed and solved using a high-resolution finite volume method. The existing digital design framework discussed above was implemented for the compound to estimate the unknown kinetic parameters and the associated uncertainty in the estimates by solving a nonlinear optimization problem that minimized the deviation between the model-predicted and experimentally observed trajectories.

In the first step of the extended framework, the effects of kinetic parameter uncertainties and control trajectory uncertainties on the final properties of the product crystal size distribution were studied through uncertainty propagation and global sensitivity techniques. To circumvent the high computational expense associated with multiple model evaluations for uncertainty analysis, the sparse polynomial chaos expansion method was used to generate the surrogate model of the high-fidelity PBM model.³ For the batch crystallization process design, different temperature trajectories were identified to optimize different process objectives, such as maximizing the mean crystal size or minimizing the span of the crystal size distribution, with nominal values of the kinetic parameters, and the influence of variations in these optimized trajectories on the optimal objectives was studied. These studies provided insights into which sections of the optimal trajectories need more robust control implementations to avoid deviations from the nominal behavior under parameter uncertainty. The above analysis was also extended to a multi-stage continuous crystallization process.

Furthermore, novel optimization problem reformulations for reliable crystallization process design are presented and solved using the reliability-based design optimization framework (RBDO) to produce crystals satisfying the target crystal attributes with a given probability under parameter uncertainty. For both batch and continuous studies, it was found that during process optimization with nominal values of kinetic parameters, the optimized operating point falls within the boundaries of the limit-state constraints considered in the problem. However, by using RDBO, it is possible to push this operating point in the feasible/safe region in the design space at the expense of an increase in the objective function value. However, operation at this point ensures the reliable and robust production of crystals satisfying the target crystal attributes under kinetic parameter uncertainties.

Acknowledgements

This work is supported by the US Food and Drug Administration (FDA) under contract number 75F40121C00106.

References:

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3. Makrygiorgos, G., Maggioni, G. M. & Mesbah, A. Surrogate modeling for fast uncertainty quantification: Application to 2D population balance models. Comput Chem Eng 138, 106814 (2020).