This study is motivated by optimal control applications that exhibit high complexity, high uncertainty, and very limited data [Wang et al., 2024, Zheng et al., 2023, Plotkin et al., 2017, Mirasol, 2017]. In particular, all of these challenges are present in the domain of biopharmaceutical manufacturing, used for production of essential life-saving treatments for severe and chronic diseases, including cancers, autoimmune disorders, metabolic diseases, genetic disorders, and infectious diseases such as COVID-19 [Zahavi and Weiner, 2020, Teo, 2022]. Using cells as factories, biomanufacturing involves hundreds of biological, physical, and chemical factors dynamically interacting with each other at molecular, cellular, and macroscopic levels and impacting production outcomes. Due to the complexity of these mechanisms, it is quite difficult to control production safely and effectively, especially in the presence of very limited data. Digital twins have proven very useful in guiding the control of complex physical systems [Tao et al., 2018].

To support interpretable predictions and optimal control of biomanfuacturing processes, in this paper, we develop a digital twin calibration approach for multi-scale bioprocess mechanistic model or Biological System-of-Systems (Bio-SoS) [Zheng et al., 2024] characterizing causal interdependence from molecular-to cellular-to macro-kinetics. Even though this study is motivated by cell culture process, it can be extended to calibrate general Bio-SoS with modular design. Basically, cell culture process dynamics and variations depend on the modules: (1) a single cell mechanistic model characterizing each living cell behaviors and their interactions with environment; (2) a metabolic shift model characterizing the change of cell metabolic phase and behaviors as a response to culture conditions and cell age; and (3) macro-kinetic model of a bioreactor system composed of many living cells under different metabolic phases. The benefits of considering the Bio-SoS mechanistic model with modular design include: a) support flexible manufacturing through assembling a system of modules to account for biomanufacturing processes under different conditions and inputs; and b) facilitate the integration of heterogeneous data from different production processes, such as 2D culture and 3D aggregate culture for Induced Pluripotent Stem Cells (iPSCs) [Wang et al., 2024, Zheng et al., 2024]. By incorporating the structure property of the Bio-SoS mechanistic model into the calibration method, we can quantify how the model uncertainties or approximation errors of different modules interact with each other and propagate through the reaction pathways to the prediction of outputs (e.g., yield and product quality attributes), which can guide interpretable and most informative Design of Experiments (DoEs) to efficiently improve model fidelity with less experiments. The model uncertainty quantification approaches for digital twin calibration can be divided into two main categories: Bayesian and frequentist approaches [Corlu et al., 2020]. Bayesian approaches treat unknown model parameters as random variables and quantify our belief by posterior distributions. It involves specifying prior distributions for model parameters and updating these distributions based on the information from observed data by applying Bayes' theorem.