(372k) Development of a Black-Box Parameter Estimation Methodology of a Batch Anti-Solvent Protein Crystallisation Process with Sparse Measurements | AIChE

(372k) Development of a Black-Box Parameter Estimation Methodology of a Batch Anti-Solvent Protein Crystallisation Process with Sparse Measurements

Authors 

Papathanasiou, M. M., Imperial College London
Heng, J., Imperial College London
Numerical models of a crystallisation process can support in-vitro experiments and be used to predict the crystal size distribution (CSD), optimise the process and aid scale-up experiments to reduce costs (Mitchell et al., 2023). Developing and parametrising a high-fidelity model of a biomacromolecule batch crystallisation process is challenging due to limited measurement data available, complex crystallisation behaviour of proteins and other biomacromolecules and limited parameter identifiability from the model’s mathematical structure (Vekilov, 2010; Quilló et al., 2021; Casas-Orozco et al., 2023; Orosz et al., 2024). Although process analytical technology (PAT) developments has enabled in-situ high-frequency measurements of IR, UV, Raman spectra, particle counts and size, such probes do not operate perfectly with every solute (Tian, Li & Yang, 2023). To tackle such challenges, in this work we present a modular, black-box parameter estimation strategy for a stiff and highly non-linear crystallisation process with few and sparse off-line measurements. Specifically, a black-box, gradient-free methodology is presented for the parametrisation of high-fidelity crystallisation models of a batch anti-solvent lysozyme crystallisation system at 40ml with sparse measurements. The models leverage a semi-discrete Monotonic Upstream-centred Scheme for Conservation Laws (MUSCL) finite-volume scheme to solve the 1-D population balance equation and accurately reconstruct the CSD with reasonable computational expenditure (LeVeque, 2002). All formulated models and optimisation schemes were solved in Julia (Rackauckas et al., 2017).

Two models were developed with Classical Nucleation Theory (CNT) kinetics, and empirical and birth-and-spread (BpS) growth kinetics respectively to assess which scheme models lysozyme crystallisation more accurately (referred to as CNT-Emp and CNT-BpS respectively). Global Sensitivity Analysis (GSA) was performed to calculate Sobol’ sensitivity indices of the developed models across batch operation and investigate which parameters dominate output variation. The critical parameters were estimated using data generated from nine in-vitro lysozyme crystallisation experiments. For the parameter estimation, a gradient-free, algorithm was developed, based on a maximum likelihood estimation (MLE) loss function with nonconstant measurement variance, using a differential evolution algorithm. The parameterised model was validated using experiments at intermediate and out-of-bounds initial concentrations. The CNT-Emp model developed had lower likelihood estimation compared to CNT-BpS, however none of the tested combinations can predict error-free crystallisation behaviour. Approximate Bayesian Computation was used to investigate parameter uncertainty and approximate the parameter’s posterior distributions, avoiding model linearisation or the need of an analytical function of the likelihood.

The presented methodology allows for a-priori calculation of time-indexed sensitivity indices supports decision-making regarding in-vitro measurement sampling, while the calculated likelihood and approximated parameter posterior distributions are used to assess and ultimately select the parametrised models.

Acknowledgments

This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) for the Imperial College London Doctoral Training Partnership (DTP) and by AstraZeneca UK Ltd through a CASE studentship award.

References

Casas-Orozco, D., Laky, D., Mackey, J., Reklaitis, G. & Nagy, Z. (2023) Reaction kinetics determination and uncertainty analysis for the synthesis of the cancer drug lomustine. Chemical Engineering Science. 275, 118591. doi:10.1016/j.ces.2023.118591.

LeVeque, R.J. (2002) Finite volume methods for hyperbolic problems. Cambridge texts in applied mathematics. Cambridge ; New York, Cambridge University Press.

Mitchell, H.M., Jovannus, D., Rosbottom, I., Link, F.J., Mitchell, N.A. & Heng, J.Y.Y. (2023) Process modelling of protein crystallisation: A case study of lysozyme. Chemical Engineering Research and Design. 192, 268–279. doi:10.1016/j.cherd.2023.02.016.

Orosz, Á., Szilágyi, E., Spaits, A., Borsos, Á., Farkas, F., Markovits, I., Százdi, L., Volk, B., Kátainé Fadgyas, K. & Szilágyi, B. (2024) Dynamic Modeling and Optimal Design Space Determination of Pharmaceutical Crystallization Processes: Realizing the Synergy between Off-the-Shelf Laboratory and Industrial Scale Data. Industrial & Engineering Chemistry Research. 63 (9), 4068–4082. doi:10.1021/acs.iecr.3c03954.

Quilló, G.L., Bhonsale, S., Gielen, B., Van Impe, J.F., Collas, A. & Xiouras, C. (2021) Crystal Growth Kinetics of an Industrial Active Pharmaceutical Ingredient: Implications of Different Representations of Supersaturation and Simultaneous Growth Mechanisms. Crystal Growth & Design. 21 (9), 5403–5420. doi:10.1021/acs.cgd.1c00677.

Rackauckas, C. & Nie, Q. (2017) DifferentialEquations.jl – A Performant and Feature-Rich Ecosystem for Solving Differential Equations in Julia. Journal of Open Research Software. 5 (1), 15. doi:10.5334/jors.151.

Tian, W., Li, W. & Yang, H. (2023) Protein Nucleation and Crystallization Process with Process Analytical Technologies in a Batch Crystallizer. Crystal Growth & Design. doi:10.1021/acs.cgd.3c00411.

Vekilov, P.G. (2010) Nucleation. Crystal Growth & Design. 10 (12), 5007–5019. doi:10.1021/cg1011633.