(327e) Uncertainty Analysis of Hybrid Modeling: A Case Study of Partial Least Squares and Population Balance Model in Continuous Twin-Screw Wet Granulation

A hybrid modeling approach combining a mechanistic model and a data-driven model has been recognized as a powerful tool along with the interest in Industry 4.0 [1]. Hybrid models retain the merits of both mechanistic and data-driven models by compensating for the demerits of each technique. Different types of hybrid models have been applied to different processes, e.g., parallel hybrid modeling for water systems [2] and serial hybrid modeling for pharmaceutical manufacturing [3]. Whereas the hybrid modeling approach is an appropriate tool for industrial applications, uncertainty analysis should be performed simultaneously to consider variability and different sources of uncertainty. A large number of researchers have worked on analyses of different types of uncertainties for different applications, as summarized in the review paper [4]. However, uncertainty analysis of hybrid models has yet to be standardized because both mechanistic and data-driven models can possess uncertainty in addition to original measurement data uncertainty.

This study aims to develop a methodology of uncertainty analysis for hybrid models. We have established a hybrid model of partial least square (PLS) regression and a population balance model (PBM) for simulating granule size distribution in continuous twin-screw wet granulation (TSWG) [5]. This hybrid model is a serial model, where the outputs of PLS are used as the inputs of PBM and was used as the case study of uncertainty analysis. A two-step Markov chain Monte Carlo (MCMC) simulation was performed to obtain probabilistic density functions (PDFs) of model parameters. First, MCMC was applied to one-dimensional compartmental PBM [6] to quantify model uncertainty from experimental data. The Gaussian log-likelihood function was defined, where standard deviations of experimental data were calculated from granule size distributions of multiple samples characterized. Subsequently, mean values and standard deviations of PBM parameters were calculated from PDFs generated by the first MCMC. MCMC of the PBM was performed for all the compartments considered in the TSWG with ten different formulations and liquid-to-solid (L/S) ratios. Second, MCMC was used for PLS models to simulate model uncertainty of the PBM parameters. PLS models can calculate PBM parameters based on the values of material properties as well as the L/S ratio as inputs [5]. The second MCMC considers uncertainty quantified by the first MCMC in the Gaussian log-likelihood function. Finally, PDFs were generated for PLS coefficients of material properties and L/S ratio.

The first MCMC results revealed the dependencies of PBM parameters. In the first compartment, which is named as the wetting zone, two PBM parameters related to aggregations were inversely proportional. An aggregation kernel and a positive breakage constant were slightly correlated in the kneading zones. In some cases, PDFs of PBM parameters have bimodality because of heterogeneous liquid distribution or missing phenomena captured in the PBM. In the current PBM, PBM kernels of aggregation and breakage were described only by size and defined based on experimental investigation with identifiability analysis [6]. 1D-PBM has a limitation in that it cannot capture certain phenomena or consider other properties than size. Hence, it is necessary to develop multi-dimensional PBMs, e.g., 2D-PBM computing size and porosity [7], for understanding heterogeneity and capturing missing phenomena. After the execution of the second MCMC, the developed uncertainty-conscious hybrid model was validated by applying it to new formulations with different L/S ratios. The expected ranges of granule size distributions were calculated by Monte Carlo simulation (i.e. uncertainty propagation) of PLS coefficients following the defined PDFs and checked if experimental data were within the expected ranges. The proposed method can reflect measurement errors and uncertainty of different types of models simultaneously. Different case studies are expected to prove the value and the applicability of the proposed method.

[1] Joel Sansana, Mark N. Joswiak, Ivan Castillo, Zhenyu Wang, Ricardo Rendall, Leo H. Chiang, Marco S. Reis. Recent trends on hybrid modeling for Industry 4.0, Computers & Chemical Engineering, 151, 107365 (2021).

[2] Ward Quaghebeur, Elena Torfs, Bernard De Baets, Ingmar Nopens. Hybrid differential equations: Integrating mechanistic and data-driven techniques for modelling of water systems, Water Research, 213, 118166 (2022).

[3] Yingjie Chen, Marianthi Ierapetritou. A framework of hybrid model development with identification of plant-model mismatch, AIChE Journal, 66(10), e16996 (2020).

[4] Can Li, Ignacio E. Grossmann. A review of stochastic programming methods for optimization of process systems under uncertainty, Frontiers in Chemical Engineering, 2, 622241 (2021).

[5] Kensaku Matsunami, Ana Alejandra Barrera Jiménez, Michiel Peeters, Daan Van Hauwermeiren, Thomas De Beer, Ingmar Nopens. In-depth analysis of the impacts of material properties on size distributions in continuous twin-screw wet granulation to construct a generic 1D population balance model, 2022 AIChE Annual Meeting, 641b (2022).

[6] Ana Alejandra Barrera Jiménez, Daan Van Hauwermeiren, Michiel Peeters, Thomas De Beer, Ingmar Nopens. Improvement of a 1D population balance model for twin-screw wet granulation by using identifiability analysis, Pharmaceutics, 13(5), 692 (2021).

[7] Ana Alejandra Barrera Jiménez, Kensaku Matsunami, Michael Ghijs, Daan Van Hauwermeiren, Michiel Peeters, Fanny Stauffer, Thomas De Beer, Ingmar Nopens. Model development and calibration of two-dimensional population balance model for twin-screw wet granulation based on particle size distribution and porosity, Powder Technology, 419, 118334 (2023).