# Mathematical Models in Experimental Design and Scale-up

The implementation of a mammalian cell-based biopharmaceutical manufacturing process demands robust methods for knowledge handling, from early-stage development and technology transfer to production scale. Mathematical process modeling can summarize this knowledge as the relationships of critical quality attributes to critical process parameters using mathematical equations and sound statistics.1
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In this article, the term “mathematical model” refers to a system of ordinary differential equations describing the timely progression of process state variables, such as cell, glucose, or product concentration. Please refer to3 and4 for more insights into this model class.

Based on our experience in the field of bioprocess development and optimization, mathematical modeling has multiple advantages because it can:

- Be used in novel computational tools to deepen process understanding during pharmaceutical process development, which can be applied during the process and product life cycles5
- Evaluate manufacturing data of already established processes to identify unknown dependencies (i.e., data mining)1
^{,}2 - Be a decision-making tool during routine manufacturing, e.g., to plan operator capacity or to evaluate batch-to-batch variability6
- Capture knowledge during the life cycle of the process, including the prediction of defects and transferability during the product life cycle6
- Virtually evaluate new configurations and feeding regimes prior to experimental testing5
^{,}6 - Show the validity of the process during technology transfer7
- Decrease the development costs for experimental design and determine fast and efficient cell expansion, resulting in accelerated time to the clinic8

Physical laws and a metabolic understanding of the biotechnological system are the basis for using mathematical modeling to represent the bioprocess. Several mathematical models of varying complexity for bioprocess control and optimization have been previously described in the literature.1
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However, to our knowledge, mathematical process models have been seldom used in industrial practice.

Disadvantages of mathematical modeling are ensuring that the necessary internal resources are available to work and handle them in a regulated environment. Well-trained staff with backgrounds in GMP, computer systems, and statistics with a clear technological focus are needed. Even then, the setup of a mathematical model can take several months and requires deep immersion into the process. Moreover, the qualification of model-based systems is challenging and not standard because it must verify the performance of the model with respect to product quality and process robustness. The operators and technical staff have extensive experience within their respective processes, but the translation of this experience into a computational tool may face personal reservations and not be well received.

Nevertheless, novel efforts for the application of mathematical models have been described for upstream and downstream processing.9
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The general difference between mathematical models is the structure of the underlying algorithms, which are specific to their intended use. Off-the-shelf software tools for the mathematical modeling of biomanufacturing processes are not commercially available because the complexity of the models used relies on a different number of measurements, available data, and computational power.

Our view on the use of mathematical models in the design of experiments and the evaluation of process transfer and scale-up is described as follows.

## Mathematical Modeling and Design of Experiments

The early-stage development of novel bioprocesses (upstream) at laboratory scale is mainly based on the experience of the research team involved and the performance of the cell line screened for production. For fed-batch processes, the first steps in process development partly comprise media adjustments and the investigation of the most appropriate feeding regime for the platform technologies used. During this phase, mathematical process models can summarize the metabolic dependencies, e.g., glucose consumption for cell growth and viability or the effects of supplementation on the glycoprofile.11

- 3 a b Moser, A., C. Appl, S. Brüning, and V. Hass. “Mechanistic Mathematical Models as a Basis for Digital Twins.” In Digital Twins. Advances in Biochemical Engineering/Biotechnology, vol 176. Edited by C. Herwig, R. Pörtner, and J. Möller. Cham: Springer, 2020. doi:10.1007/10_2020_152
- 4Kuchemüller, K. B., R. Pörtner, and J. Möller. “Digital Twins and Their Role in Model-Assisted Design of Experiments.” In Digital Twins. Advances in Biochemical Engineering/Biotechnology, vol. 177. Edited by C. Herwig, R. Pörtner, and J. Möller. Cham: Springer, 2020. doi:10.1007/10_2020_136
- 5 a b Arndt, L., V. Wiegmann, K. B. Kuchemüller, F. Baganz, R. Pörtner, and J. Möller. “Model-Based Workflow for Scale-Up of Process Strategies Developed in Miniaturized Bioreactor Systems.” Biotechnology Progress 37, no. 3 (May 2021): e3122. doi:10.1002/btpr.3122
- 1 a b
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- 6 a b c Hernández Rodríguez, T., C. Posch, R. Pörtner, and B. Frahm. “Dynamic Parameter Estimation and Prediction over Consecutive Scales, Based on Moving Horizon Estimation: Applied to an Industrial Cell Culture Seed Train.” Bioprocess and Biosystems Engineering 44, no. 4 (April 2021): 793–808. doi:10.1007/s00449-020-02488-1
- 7Möller, J., T. Hernández Rodríguez, J. Müller, L. Arndt, K. B. Kuchemüller, B. Frahm, R. Eibl, D. Eibl, and R. Pörtner. “Model Uncertainty-Based Evaluation of Process Strategies During Scale-Up of Biopharmaceutical Processes.” Computers & Chemical Engineering 134 (2020): 106693. doi:10.1016/j.compchemeng.2019.106693
- 8Möller, J., and R. Pörtner. “Digital Twins for Tissue Culture Techniques—Concepts, Expectations, and State of the Art.” Processes (Basel, Switzerland) 9, no. 3 (2021): 447. doi:10.3390/pr9030447
- 9Hutter, C., M. von Stosch, M. N. Cruz Bournazou, and A. Butté. “Knowledge Transfer Across Cell Lines Using Hybrid Gaussian Process Models with Entity Embedding Vectors.” Biotechnology and Bioengineering (2021): 1–13. doi:10.1002/bit.27907
- 10Rolinger, L., M. Rüdt, and J. Hubbuch. “A Critical Review of Recent Trends, and a Future Perspective of Optical Spectroscopy as PAT in Biopharmaceutical Downstream Processing.” Analytical and Bioanalytical Chemistry 412, no. 9 (April 2020): 2047–64. doi:10.1007/s00216-020-02407-z
- 11Erklavec Zajec, V., U. Novak, M. Kastelic, B. Japelj, L. Lah, A. Pohar, and B. Likozar. “Dynamic Multiscale Metabolic Network Modeling of Chinese Hamster Ovary Cell Metabolism Integrating N-Linked Glycosylation in Industrial Biopharmaceutical Manufacturing.” Biotechnology and Bioengineering 118, no. 1 (January 2021): 397–411. doi:10.1002/bit.27578

By using standard cell lines and media, the expected growth characteristic is efficiently transferred into a mathematical process model.12 This mathematical model can be combined with design of experiment (DoE) methods, which show great potential for the development of process strategies and media supplementation.13

In an intensified DoE method (iDoE), the factors in the planned experiments are changed within each individual experiment, and the model is then used to analyze the results. Due to the complexity of such staged experimental results, the process analysis is enhanced by the model.14

In model-based DoE (mbDoE), experiments are planned to properly identify the mathematical model and its parameters.15
In model-assisted DoE (mDoE), a process-related target (i.e., maximum product concentration) is efficiently optimized using a low number of experiments, and the model assists in the evaluation and recommendation of DoE designs.12
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The use of these mDoE results in typical savings of 40%–80% in the number of experiments, depending on the specific study.12
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For all methods, the available data and the known cellular effects obtained from screening studies or media test experiments can be used as the basis for setting up cause and effect relationships for cell growth, metabolism, and productivity.

An exemplary workflow for mDoE involves multiple steps (Figure 1).5
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First, the objective of the study (i.e., maximization of product concentration) should be well defined. Then, the biotechnological system is mathematically modeled based on the identified cause and effect relationships, and an mDoE is planned. Typically, two to four influencing variables are chosen with consideration of the technological constraints present in the production scale. The space for equipment operation is usually well known due to extensive qualification/validation activities, and optimizations are within narrow technology-related borders.

After planning the experiments, they are performed at laboratory/pilot scale, preferably using a scaled-down model of the manufacturing process. If the experimental data are available and the aim of the study is fulfilled, the data are included in the mathematical model, and it is transferred to production, together with the process settings. If the aim is not fulfilled, the data are used to adapt the process understanding in the form of the mathematical model, and new iterative experiments are planned.

By using universally understood principles (i.e., mathematical model), the experimental strategy can be enhanced, which could lead to reduced experimentation (mDoE/iDoE) and/or better factor understanding (mbDoE).

- 12 a b c Möller, J., K. B. Kuchemüller, T. Steinmetz, K. S. Koopmann, and R. Pörtner. “Model-Assisted Design of Experiments as a Concept for Knowledge-Based Bioprocess Development.” Bioprocess and Biosystems Engineering 42, no. 5 (May 2019): 867–82. doi:10.1007/s00449-019-02089-7
- 13Abt, V., T. Barz, M. N. Cruz-Bournazou, C. Herwig, P. Kroll, J. Möller, R. Pörtner, and R. Schenkendorf. “Model-Based Tools for Optimal Experiments in Bioprocess Engineering.” Current Opinion in Chemical Engineering 22 December (2018): 244–52. doi:10.1016/j.coche.2018.11.007
- 14von Stosch, M., and M. J. Willis. “Intensified Design of Experiments for Upstream Bioreactors.” Engineering in Life Sciences 17, no. 11 (September 2016): 1173–84. doi:10.1002/elsc.201600037
- 15 a b Hertweck, D., V. N. Emenike, A. C. Spiess, and R. Schenkendorf. “Rigorous Model-Based Design and Experimental Verification of Enzyme-Catalyzed Carboligation Under Enzyme Inactivation.” Catalysts 10, no. 1 (2020): 96. doi:10.3390/catal10010096
- 16 a b Moser, A., K. B. Kuchemüller, S. Deppe, T. Hernández Rodríguez, B. Frahm, R. Pörtner, V. C. Hass, and J. Möller. “Model-Assisted DoE Software: Optimization of Growth and Biocatalysis in Saccharomyces Cerevisiae Bioprocesses.” Bioprocess and Biosystems Engineering 44, no. 4 (April 2021): 683–700. doi:10.1007/s00449-020-02478-3
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## Evaluation of Process Transfer and Scale-up

After process development, the bioprocess, including its process strategy, is transferred to pilot or production scale. Currently, scale-up and scale-down of the derived process knowledge between different departments within a company are challenging because of varying process performance and cellular changes.5
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For scale-up criteria, hydrodynamic states such as power input per volume are often defined as constant between the different scales, even if a hydrodynamic characterization is not available at each scale.18

Additionally, a hydrodynamic scale-up procedure does not consider the dynamics of the bioprocess itself. It is not ensured that the previously developed process strategy is scaled up sufficiently and that the process dynamics stay constant during scale-up. In other words, how could the growth behavior and productivity be ensured from the smaller to larger scale?

Mathematical process models are key to compare and evaluate the process dynamics between different scales (Figure 2).

The development of process strategies requires fast methods to plan experiments and ensure efficient process transfer and scale-up.

Starting with the developed process and the mathematical model, scale-up is performed using known hydrodynamic criteria and experience of the bioreactors. Then, cultivations are performed at the different scales, and the same mathematical model is used to describe these cultivations considering experimental variations and analytical deviations. The model parameter distributions are derived to predict batch-to-batch variability and potential out-of-trends. Furthermore, the average and expected specification limits of the in-process controls are simulated. At the end, the individual model parameter distributions are statistically evaluated to identify if the process dynamics are the same between the tested scales. The same process dynamics are ensured if no changes in the parameter distributions are identified. Otherwise, if the parameters differ significantly, a validation of the process strategy is recommended using advanced DoE methods, which were introduced previously.7 This approach provides a novel, knowledge-driven decision-making tool for bioprocess scale-up and scale-down to guarantee the same process performance from a few milliliters to production scale.

## Conclusion

The development of process strategies requires fast methods to plan experiments and ensure efficient process transfer and scale-up. This article described the use of mDoE methods to consider well-known biological effects in the planning of experiments. This approach results in a reduced amount of laboratory work. Furthermore, a workflow was highlighted to evaluate process transfer and scale-up/scale-down using mathematical process models. The introduced approaches provide novel knowledge-driven decision-making tools for bioprocess development and implementation.

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- 17Kamravamanesh, D., D. Kiesenhofer, S. Fluch, M. Lackner, and C. Herwig. “Scale-Up Challenges and Requirement of Technology-Transfer for Cyanobacterial poly (3-hydroxybutyrate) Production in Industrial Scale.” International Journal of Biobased Plastics 1, no. 1 (2019): 60–71.
- 18Seidel, S., and D. Eibl. “Influence of Interfacial Force Models and Population Balance Models on the kLa Value in Stirred Bioreactors.” Processes (Basel, Switzerland) 9, no. 7 (2021): 1185. doi:10.3390/pr9071185
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