(627a) Unveiling the Interplay between Mathematical Models and Data

Mathematical models are essential tools for understanding, designing, and optimising complex processes. However, a model is only as good as its data; poor-quality or insufficient data leads to non-identifiable models, where parameters cannot be uniquely determined from observations. This results in ill-posed models with high parameter uncertainty, limiting their predictability and practical applicability. Crystallisation, a key purification-separation step in pharmaceutical and agrochemical manufacturing, presents an ideal system to showcase the critical interplay between model and data. This is due to the presence of competing mechanisms (e.g., growth, nucleation, breakage) and challenges in accurately characterising both liquid and solid phases.[1]

In this work, we systematically quantify the impact of data on model identifiability and its consequences on the reliability of the model. In crystallisation, a process where the particle size and shape distribution (PSSD) affects product quality and process efficiency, we employ population balance equations (PBEs) to track its evolution with mass balance equations for the liquid phase, forming a system of integro-partial differential equation.[2] To study the influence of data, we simulated batch crystallisation using a PBE coupled with a digital twin that mimics process monitoring tools (e.g., Focused Beam Reflectance Measurement (FBRM), and single-projection imaging such as from microscope, particle vision measurement, or BlazeMetric probes). This approach reflects real-world data limitations while leveraging the digital twin to bypass resource and time constraints, ensuring a practical and representative study. To investigate key issues related to identifiability and their underlying causes, we conducted a series of carefully selected case studies. Through profile likelihood analysis of the parameters, we examine how different experimental strategies affect identifiability and model robustness. Examples include how FBRM struggles to capture the shape of crystal populations, leading to high uncertainty in estimation of growth kinetics, and how single-projection imaging introduces noise, which we quantify through parameter confidence intervals. Additionally, we demonstrate actions to improve identifiability, such as optimising sampling frequency and selecting key experimental variables (e.g., seed mass and PSSD variance). These findings provide practical guidelines for enhancing the predictability and generalisability of crystallisation models.[3]

This study highlights the critical role of data quality and experimental techniques in identifiability, offering guidelines to improve model performance. Using a digital twin, we assess the impact of real-world data limitations while avoiding resource constraints. These insights lay the foundation for designing efficient experiments that enhance the practicality and reliability of crystallisation models.[1]

References
(1) Wieland, F.-G.; Hauber, A. L.; Rosenblatt, M.; Tönsing, C.; Timmer, J. Current Opinion in Systems Biology 2021, 25, 60–69.
(2) Ramkrishna, D., Population balances: theory and applications to particulate systems in engineering; Academic Press: San Diego, CA, 2000.
(3) Kempkes, M.; Eggers, J.; Mazzotti, M. Chem. Eng. Sci 2008, 63, 4656–4675.