(346p) Chemometric Techniques for the Combined Calibration/Parameter Estimation of Pharmaceutical Drug Substance Manufacture

Numerous initiatives have been pursued over the past decades by the FDA, industry and academia to promote advances in pharmaceutical manufacturing. These efforts have resulted in guiding concepts such as Quality by Design -QbD- and Quality by Control -QbC- [1], [2], which aim to systematically lead to improvements in product quality and process understanding. An essential tool in achieving such improvements are process models based, to the extent possible, on fundamental understanding of the modeled phenomena. However, since even the most fundamental, first principles-based models involve system specific parameters, the estimation of these parameters is a key step that must be successfully executed. The resulting models calibrated to the specific system under investigation enable determination of which variables most significantly impact product quality, processing costs, and process operation within safe and stable limits.

For effective parameter estimation, several aspects associated with the model formulation must be considered. First, from a structural point of view, a numerically robust model formulation is essential for reliable execution of the optimization runs required for parameter estimation. One relevant example arises in the choice of kinetic expressions, where various reformulations have been proposed [3]–[5] to facilitate optimization execution. Second, scaling of the state variables employed in the model also plays a role in the efficiency of model solution, since the solution methods invariably make use of numerical approximations. Lastly, for derivative-based optimization, reliable gradient information from the modeled system has direct impact on the convergence speed and accuracy of the of the parameter estimates.

The other important element of the parameter estimation problem is the experimental data. The experimental data obtained for operations such as reaction or crystallization typically come from offline/online spectroscopic measurements, resulting in datasets that relate variables of interest to multivariate measurements. This permits the use of rich datasets that contribute to the robustness of the measurements, in particular for multicomponent systems, but also poses some challenges related to handling the increased dimensionality of the mathematical problem which must be solved, and the feasibility of executing the parameter estimation within reasonable computation time.

Multivariate data can be incorporated in various ways into a parameter estimation framework. One of the most widespread approaches is to build chemometric models using samples prepared with known concentration of the analyzed components. The measured spectra of the samples are related to their corresponding concentration through these calibration models, either using data at a single wavelength or using multivariate, reduced-dimension linear models such as principal component regression (PCR) or partial least squares (PLS) [6]. On the other hand, multiple curve resolution coupled with alternating least squares (MCR-ALS) [7], [8] approaches the problem by directly analyzing the spectra resulting from the experiment. Here, the spectra changes as a result of the change in chemical composition of the multicomponent system under study are directly considered. Both the concentration of the species of interest and their individual absorptivities are unknown in this method, hence its two-layer structure. Depending whether a first-principles or a data-driven model representing the system is used, the combined methodology is termed as hard-modeling or soft-modeling [9]. The MCR assumes that the spectroscopic response of the system is caused by a linear combination of the absorbing components present in the system, a multicomponent version of the Beer-Lambert law with a bilinear mathematical structure [10].

At the top level of the hard-modeling variant of the MCR-ALS methodology, a first-principles model that contains the parameters to be estimated is used to calculate the concentration of interest, given an initial parameter estimate. Then, determination of the single-component absorptivity is dealt within the MCR component, which calculates a modeled spectra by using the concentrations returned by the model and the experimental spectra. Then, the ALS component performs the parameter optimization by comparing the experimental and calculated spectra and changing the values of the parameters until sufficient agreement is reached. In this manner, both the calibration and the parameter estimation are achieved in a simultaneous fashion. Finally, the mathematical structure of the MCR problem allows it to be posed as the calculation of the modeled spectra as a matrix orthogonal projection calculation, for which expressions for derivative calculations are available [11], [12]. Calculation of the derivative of the orthogonal projectors can be further coupled to the information gained by sensitivity of the original modeled states (e.g. concentration) to enhance the quality of the derivative information passed to the optimization algorithm.

In this work, we report a case study involving the synthesis and crystallization of lomustine, [13], [14]. We demonstrate using a dynamic simulation-optimization approach to determine the parameters of the first-principles models for the reaction and crystallization steps drawn from the PharmaPy library [17]. Liquid-phase reaction experiments are carried out in batch and semi-batch reaction setups with automated sampling, and several kinetic reaction mechanisms are evaluated and compared. Also, cooling batch crystallization experiments are used to determine nucleation, growth and dissolution parameters, using the method of moments [15] to model the crystallization process and determine the crystal size distribution. UV-Vis and infrared spectroscopic data are generated for both the reaction and crystallization systems, and analyzed under the MCR-ALS framework. Focused beam reflectance measurement (FBRM) and particle vision and measurement (PVM) are used for the crystallization experiments to determine nucleation and obtain in-situ particle images. Dynamic parametric sensitivities are calculated by a forward sensitivity method, using analytical jacobians for both reaction and crystallization [16], [17]. Lastly, parametric uncertainty in the models is quantified using sampling techniques (i.e. bootstrapping).

References

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[17] D. Casas-Orozco et al., “PharmaPy: an object-oriented tool for the development of hybrid pharmaceutical flowsheets,” Comput. Chem. Eng., 2021 (submitted).