# A Framework for the Development of Integrated and Computationally Feasible Models of Large-Scale Mammalian Cell Bioreactors

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. The Importance of Reliable Unit Operation Models

#### 1.2. Bioreactor

#### 1.3. Existing Culture Models and Their Need for Improvement

## 2. Development of a Dynamic, Integrated, and Computationally Feasible Bioreactor Model

#### 2.1. Development of CFD Simulations

^{®}Fluent

^{®}15.0.7 for the prediction of spatial variations of environmental parameters. Conservation laws of mass, momentum, and energy are usually used to describe a single phase flow, gas or liquid. If the thermodynamic, transport, and chemical properties of a component need to be specified, the field equations may be accompanied by the constitutive equations of state, stress, chemical reactions, etc. The presence of interfacial surface in a multi-phase flow complicates the mathematical formulation of the problem. To derive the field and constitutive equations of a multi-phase flow, such as inside a bioreactor, local characteristics have to be considered. This is not straightforward due to unknown motions of multiple deformable interfaces, variable fluctuations due to turbulence and moving interfaces, and discontinuity of properties at the interface. Obtaining local mean values of flow properties has been shown to be an efficient way to eliminate instantaneous fluctuations. Three averaging methodologies have been developed: Eulerian, Lagrangian, and Boltzmann statistical averaging. In the Eulerian approach, time and space coordinates are independent and other variables are expressed with respect to them. In the Lagrangian averaging methodology, particle coordinates replace spatial coordinates. If the purpose of modeling is studying the group behavior of particles, the Eulerian approach is preferred. However, if the behavior of individual particles is of interest, the Lagrangian description has a clear advantage [45]. Tracking individual bubbles increases the computation. Additionally, it would only improve model predictive power if the extent of the interactions between individual bubbles and the liquid phase could be quantified. These interactions involve growth, breakage, and agglomeration of bubbles and energy dissipation due to bubble rupture. Therefore, in this study gas and liquid phases are treated as continua and Eulerian averaging is used. The Eulerian multiphase model creates sets of momentum and continuity equations for each phase and couples them through exchanging pressure and interphase coefficients [46]. Turbulence of flow is calculated using the k-ε viscosity model, which has been widely used for stirred tanks [47]. It is a robust model that gives reasonably accurate results for a wide range of turbulent flows [48]. A k-ε model consists of two transport equations, one each for the turbulent kinetic energy (k) and the energy dissipation rate (ε). The motion of the impeller is captured using a multiple reference frame (MRF). To implement the MRF model, the geometry is broken up into stationary and moving zones. The MRF model approximates the flow in the moving zone around the impeller by freezing the motion of the moving part in a specific position and observing the instantaneous flow field. To use flow variables of one zone for calculation of fluxes at the boundary of the adjacent zone, a local reference frame transformation is performed at the interface between cell zones. In the absence of large-scale transient effects due to weak impeller–wall interactions, the MRF approach provides a reasonable approximation of the flow [48].

#### 2.2. Development of the Integrated Model

^{2}/4 (mm/h) and r is cell radius in μm [49]. Cell radius is estimated assuming spherical shape, density equal to that of water, and average mass of 1.1165 × 10

^{−6}mg [20].

^{−3}[50,51]. Power input is calculated by multiplying the density of the liquid phase (kg/m

^{3}) by the turbulent energy dissipation rate (m

^{2}/s

^{3}). The turbulent energy dissipation rate is the rate of absorption of kinetic energy that breaks up large eddies. This is then converted to heat by viscous forces [52]. A rate of cell damage of 3.4% min

^{−1}has been reported for cells in high shear regions [51]. The volume fraction of high shear region in compartments is calculated using the data obtained from CFD simulations. Since cells are assumed to be homogenously distributed inside compartments, the volume fraction of the high shear region is equal to the fraction of cells exposed to shear beyond their tolerable threshold. Therefore, the rate of loss of viable cells under operating condition $op$ and in compartment $c$ is calculated using Equation (4):

^{−1}·h

^{−1}, which has been reported for carbon dioxide production [57]. The overall volumetric mass transfer coefficient, k

_{L}a, is calculated using Equation (7), in which ${U}_{G}{}_{op,q}$ is superficial gas velocity (m/s) for computational cell $q$ under operating condition $op$ [33]. Volumetric mass transfer coefficient is the product of liquid phase mass transfer coefficient; k

_{L}(m/s) and specific interfacial area; a (m

^{2}·m

^{−3}). Mass transfer stops after reaching the saturation concentration at 37 °C. The reported saturation mass fraction of oxygen is 3.43 × 10

^{−5}for [58]. The unstructured model is assumed to predict metabolite uptake and production rates when the culture is oxygen-saturated. Experimental data show the dependence of these rates on the concentration of dissolved oxygen [59]. The reported data are used to calculate correction factors for metabolites’ uptake and production rates at different concentrations of DO (Figure 2). Uptake and production rates of metabolites predicted by the unstructured model are multiplied by correction factors to take into account the effects of mass transfer mechanism on metabolism of cells.

#### 2.3. Coupling the Model with Nonlinear Solvers

^{−9}h. Instead, integration is carried out using appropriate solvers for stiff, nonlinear ODEs and the problem is formulated for the application of the interior point method [69]. Equations (8)–(11) represent the solution to the mathematical optimization problem. ${T}_{i}$ and ${C}_{i}$ are the time and composition of the ith feeding. In addition to initial nutrient concentrations, schedule, and composition of feeding, it also finds optimal criteria for setting aeration and agitation rates. Aeration is stopped or started based on the DO level. For the adjustment of impeller rotation speed, a measure of homogeneity is defined based on the relative standard deviation (RSD) of distribution of cells over compartments.

## 3. Case Study

## 4. Conclusions and Future Directions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Moorkens, E.; Meuwissen, N.; Huys, I.; Declerck, P.; Vulto, A.G.; Simoens, S. The Market of Biopharmaceutical Medicines: A Snapshot of a Diverse Industrial Landscape. Front. Pharmacol.
**2017**, 8, 314. [Google Scholar] [CrossRef] [PubMed] - Davidson, A.; Farid, S.S. Innovation in Biopharmaceutical Manufacture. BioProcess Int.
**2014**, 12, 12–19. [Google Scholar] - Rader, R.A.; Langer, E.S. 30 years of upstream productivity improvements. BioProcess Int.
**2015**, 13, 10–15. [Google Scholar] - Farid, S.S. Process economics of industrial monoclonal antibody manufacture. J. Chromatogr. B-Anal. Technol. Biomed. Life Sci.
**2007**, 848, 8–18. [Google Scholar] [CrossRef] [PubMed] - Terry, C.; Lesser, N. Balancing the R&D Equation; Deloitte Center for Health Solutions: London, UK, 2017. [Google Scholar]
- Gyurjyan, G.; Thaker, S.; Westhues, K.; Zwaanstra, C. Rethinking Pharma Productivity; McKinsey & Company: New York, NY, USA, 2017. [Google Scholar]
- Varma, V.A.; Reklaitis, G.V.; Blau, G.E.; Pekny, J.F. Enterprise-wide modeling & optimization—An overview of emerging research challenges and opportunities. Comput. Chem. Eng.
**2007**, 31, 692–711. [Google Scholar] - Grossmann, I.E. Advances in mathematical programming models for enterprise-wide optimization. Comput. Chem. Eng.
**2012**, 47, 2–18. [Google Scholar] [CrossRef] [Green Version] - Lara, A.R.; Galindo, E.; Ramírez, O.T.; Palomares, L.A. Living with heterogeneities in bioreactors. Mol. Biotechnol.
**2006**, 34, 355–381. [Google Scholar] [CrossRef] - Xie, L.; Zhou, W.; Robinson, D. Protein production by large-scale mammalian cell culture. New Compr. Biochem.
**2003**, 38, 605–623. [Google Scholar] - Spier, R.E. Encyclopedia of Cell Technology. In Wiley Biotechnology Encyclopedias; Wiley-Interscience: Hoboken, NJ, USA, 2000. [Google Scholar]
- Panda, T. Bioreactors: Analysis and Design; Tata McGraw-Hill Education Private Limited: New York, NY, USA, 2011. [Google Scholar]
- Ho, C.S.; Wang, D.I.C. Animal Cell Bioreactors; Biotechnology Series; Davies, J.E., Ed.; Butterworth-Heinemann: Oxford, UK, 1991. [Google Scholar]
- Mandenius, C.-F.; Titchener-Hooker, N.J. Measurement, Monitoring, Modelling and Control of Bioprocesses. In Advances in Biochemical Engineering/Biotechnology; Springer: Berlin, Germany, 2013. [Google Scholar] [Green Version]
- Meyer, H.-P.; Schmidhalter, D. Industrial Scale Suspension Culture of Living Cells; Wiley: Hoboken, NJ, USA, 2014. [Google Scholar]
- Shuler, M.L.; Kargi, F. Bioprocess Engineering, 2nd ed.; Prentice Hall: Upper Saddle River, NJ, USA, 2002. [Google Scholar]
- Prokop, A. Implications of Cell Biology in Animal Cell Biotechnology. In Animal Cell Bioreactors; Ho, C.S., Wang, D.I.C., Eds.; Butterworth-Heinemann: Oxford, UK, 1991. [Google Scholar]
- Pigou, M.; Morchain, J. Investigating the interactions between physical and biological heterogeneities in bioreactors using compartment, population balance and metabolic models. Chem. Eng. Sci.
**2015**, 126, 267–282. [Google Scholar] [CrossRef] [Green Version] - Meshram, M.; Naderi, S.; McConkey, B.; Ingalls, B.; Scharer, J.; Budman, H. Modeling the coupled extracellular and intracellular environments in mammalian cell culture. Metab. Eng.
**2013**, 19, 57–68. [Google Scholar] [CrossRef] [PubMed] - Sidoli, F.R.; Asprey, S.P.; Mantalaris, A. A Coupled Single Cell-Population-Balance Model for Mammalian Cell Cultures. Ind. Eng. Chem. Res.
**2006**, 45, 5801–5811. [Google Scholar] [CrossRef] - Mantzaris, N.V.; Daoutidis, P. Cell population balance modeling and control in continuous bioreactors. J. Process Control
**2004**, 14, 775–784. [Google Scholar] [CrossRef] - Mantzaris, N.V. Stochastic and deterministic simulations of heterogeneous cell population dynamics. J. Theor. Biol.
**2006**, 241, 690–706. [Google Scholar] [CrossRef] [PubMed] - Dorka, P.; Fischer, C.; Budman, H.; Scharer, J.M. Metabolic flux-based modeling of mAb production during batch and fed-batch operations. Bioprocess Biosyst. Eng.
**2009**, 32, 183–196. [Google Scholar] [CrossRef] [PubMed] - Fadda, S.; Cincotti, A.; Cao, G. A novel population balance model to investigate the kinetics of in vitro cell proliferation: Part I. model development. Biotechnol. Bioeng.
**2012**, 109, 772–781. [Google Scholar] [CrossRef] [PubMed] - Jandt, U.; Platas Barradas, O.; Pörtner, R.; Zeng, A.P. Synchronized Mammalian Cell Culture: Part II—Population Ensemble Modeling and Analysis for Development of Reproducible Processes. Biotechnol. Prog.
**2015**, 31, 175–185. [Google Scholar] [CrossRef] [PubMed] - Craven, S.; Whelan, J.; Glennon, B. Glucose concentration control of a fed-batch mammalian cell bioprocess using a nonlinear model predictive controller. J. Process Control
**2014**, 24, 344–357. [Google Scholar] [CrossRef] - Sbarciog, M.; Coutinho, D.; Wouwer, A.V. A simple output-feedback strategy for the control of perfused mammalian cell cultures. Control Eng. Pract.
**2014**, 32, 123–135. [Google Scholar] [CrossRef] - Amribt, Z.; Niu, H.X.; Bogaerts, P. Macroscopic modelling of overflow metabolism and model based optimization of hybridoma cell fed-batch cultures. Biochem. Eng. J.
**2013**, 70, 196–209. [Google Scholar] [CrossRef] - Mantzaris, N.V.; Liou, J.J.; Daoutidis, P.; Srienc, F. Numerical solution of a mass structured cell population balance model in an environment of changing substrate concentration. J. Biotechnol.
**1999**, 71, 157–174. [Google Scholar] [CrossRef] - Farzan, P.; Mistry, B.; Ierapetritou, M.G. Review of the Important Challenges and Opportunities related to Modeling of Mammalian Cell Bioreactors. AIChE J.
**2017**, 63, 398–408. [Google Scholar] [CrossRef] - Rocha, I. Model-Based Strategies for Computer-Aided Operation of Recombinant E. coli Fermentation; Universidade do Minho: Braga, Portugal, 2003. [Google Scholar]
- Lopez-Meza, J.; Araíz-Hernández, D.; Carrillo-Cocom, L.M.; López-Pacheco, F.; del Refugio Rocha-Pizaña, M.; Alvarez, M.M. Using simple models to describe the kinetics of growth, glucose consumption, and monoclonal antibody formation in naive and infliximab producer CHO cells. Cytotechnology
**2016**, 68, 1287–1300. [Google Scholar] [CrossRef] [PubMed] - Cachaza, E.M.; Díaz, M.E.; Montes, F.J.; Galán, M.A. Simultaneous Computational Fluid Dynamics (CFD) simulation of the hydrodynamics and mass transfer in a partially aerated bubble column. Ind. Eng. Chem. Res.
**2009**, 48, 8685–8696. [Google Scholar] [CrossRef] - Wang, H.N.; Jia, X.; Wang, X.; Zhou, Z.; Wen, J.; Zhang, J. CFD modeling of hydrodynamic characteristics of a gas-liquid two-phase stirred tank. Appl. Math. Model.
**2014**, 38, 63–92. [Google Scholar] [CrossRef] - Azargoshasb, H.; Mousavi, S.M.; Amani, T.; Jafari, A.; Nosrati, M. Three-phase CFD simulation coupled with population balance equations of anaerobic syntrophic acidogenesis and methanogenesis reactions in a continouos stirred bioreactor. J. Ind. Eng. Chem.
**2015**, 27, 207–217. [Google Scholar] [CrossRef] - Kerdouss, F.; Bannari, A.; Proulx, P.; Bannari, R.; Skrga, M.; Labrecque, Y. Two-phase mass transfer coefficient prediction in stirred vessel with a CFD model. Comput. Chem. Eng.
**2008**, 32, 1943–1955. [Google Scholar] [CrossRef] - Micale, G.; Montante, G.; Grisafi, F.; Brucato, A.; Godfrey, J. CFD simulation of particle distribution in stirred vessels. Chem. Eng. Res. Des.
**2000**, 78, 435–444. [Google Scholar] [CrossRef] - Farzan, P.; Ierapetritou, M.G. Integrated Modeling to Capture the Interaction of Physiology and Fluid Dynamics in Biopharmaceutical Bioreactors. Comput. Chem. Eng.
**2017**, 97, 271–282. [Google Scholar] [CrossRef] - Bezzo, F.; Macchietto, S.; Pantelides, C.C. A general methodology for hybrid multizonal/CFD models: Part I. Theoretical framework. Comput. Chem. Eng.
**2004**, 28, 501–511. [Google Scholar] [CrossRef] - Delafosse, A.; Collignon, M.L.; Calvo, S.; Delvigne, F.; Crine, M.; Thonart, P.; Toye, D. CFD-based compartment model for description of mixing in bioreactors. Chem. Eng. Sci.
**2014**, 106, 76–85. [Google Scholar] [CrossRef] - Kagoshima, M.; Mann, R. Development of a networks-of-zones fluid mixing model for an unbaffled stirred vessel used for precipitation. Chem. Eng. Sci.
**2006**, 61, 2852–2863. [Google Scholar] [CrossRef] - Vrabel, P.; Van der Lans, R.G.J.M.; Cui, Y.Q.; Luyben, K.C.A. Compartment model approach: Mixing in large scale aerated reactors with multiple impellers. Chem. Eng. Res. Des.
**1999**, 77, 291–302. [Google Scholar] [CrossRef] - Bashiri, H.; Heniche, M.; Bertrand, F.; Chaouki, J. Compartmental modelling of turbulent fluid flow for the scale-up of stirred tanks. Can. J. Chem. Eng.
**2014**, 92, 1070–1081. [Google Scholar] [CrossRef] - Vrabel, P.; van der Lans, R.G.; Luyben, K.C.A.; Boon, L.; Nienow, A.W. Mixing in large-scale vessels stirred with multiple radial or radial and axial up-pumping impellers: Modelling and measurements. Chem. Eng. Sci.
**2000**, 55, 5881–5896. [Google Scholar] [CrossRef] - Ishii, M.; Hibiki, T. Thermo-Fluid Dynamics of Two-Phase Flow; Springer: Berlin, Germany, 2011. [Google Scholar]
- ANSYS Inc. ANSYS Fluent Theory Guide, Release 15.0 ed.; ANSYS Inc.: Canonsburg, PA, USA, 2013. [Google Scholar]
- Schmalzriedt, S.; Jenne, M.; Mauch, K.; Reuss, M. Integration of physiology and fluid dynamics. In Process Integration in Biochemical Engineering; Springer: Berlin, Germany, 2003; pp. 19–68. [Google Scholar]
- ANSYS Inc. ANSYS Fluent User’s Guide, Release 15.0 ed.; ANSYS Inc.: Canonsburg, PA, USA, 2013. [Google Scholar]
- Adams, R.L.P. Cell Culture for Biochemists; Elsevier: New York, NY, USA, 1990. [Google Scholar]
- Kaiser, S.C.; Löffelholz, C.; Werner, S.; Eibl, D. CFD for Characterizing Standard and Single-use Stirred Cell Culture Bioreactors. In Computational Fluid Dynamics Technologies and Applications; Minin, I.V., Minin, O.V., Eds.; InTech: Vienna, Austria, 2011; pp. 97–122. [Google Scholar]
- Chalmers, J. Animal cell culture, effects of agitation and aeration on cell adaption. In Encyclopedia of Cell Technology; Spier, R.E., Ed.; Wiley-Interscience: Hoboken, NJ, USA, 2000. [Google Scholar]
- Sarkar, J.; Shekhawat, L.K.; Loomba, V.; Rathore, A.S. CFD of mixing of multi-phase flow in a bioreactor using population balance model. Biotechnol. Prog.
**2016**, 32, 613–628. [Google Scholar] [CrossRef] [PubMed] - Alves, S.S.; Maia, C.I.; Vasconcelos, J.M.T.; Serralheiro, A.J. Bubble size in aerated stirred tanks. Chem. Eng. J.
**2002**, 89, 109–117. [Google Scholar] [CrossRef] - Chatterjee, A. An introduction to the proper orthogonal decomposition. Curr. Sci.
**2000**, 78, 808–817. [Google Scholar] - Chen, H.; Reuss, D.L.; Sick, V. On the use and interpretation of proper orthogonal decomposition of in-cylinder engine flows. Meas. Sci. Technol.
**2012**, 23, 085302. [Google Scholar] [CrossRef] - Xiu, Z.-L.; Deckwer, W.-D.; Zeng, A.-P. Estimation of rates of oxygen uptake and carbon dioxide evolution of animal cell culture using material and energy balances. Cytotechnology
**1999**, 29, 159–166. [Google Scholar] [CrossRef] [PubMed] - Mostafa, S.S.; Gu, X.J. Strategies for improved dCO(2) removal in large-scale fed-batch cultures. Biotechnol. Prog.
**2003**, 19, 45–51. [Google Scholar] [CrossRef] [PubMed] - Kolev, N.I. Solubility of O
_{2}, N_{2}, H_{2}and CO_{2}in water. In Multiphase Flow Dynamics 4 Turbulence, Gas Adsorption and Release, Diesel Fuel Properties; Kolev, N.I., Ed.; Springer: Berlin, Germany, 2012; pp. 209–239. [Google Scholar] - Ozturk, S.S.; Palsson, B.O. Growth, metabolic, and antibody-production kinetics of hybridoma cell-culture: 2. Effects of serum concentration, dissolved-oxygen concentration, and medium PH in a batch reactor. Biotechnol. Prog.
**1991**, 7, 481–494. [Google Scholar] [CrossRef] [PubMed] - Xing, Z.Z.; Bishop, N.; Leister, K.; Li, Z.J. Modeling Kinetics of a Large-Scale Fed-Batch CHO Cell Culture by Markov Chain Monte Carlo Method. Biotechnol. Prog.
**2010**, 26, 208–219. [Google Scholar] [CrossRef] [PubMed] - Biegler, L.T.; Lang, Y.D.; Lin, W.J. Multi-scale optimization for process systems engineering. Comput. Chem. Eng.
**2014**, 60, 17–30. [Google Scholar] [CrossRef] - Bryson, J.A.E.; Ho, Y.-C. Applied Optimal Control: Optimization, Estimation and Control; CRC Press: Boca Raton, FL, USA, 1975. [Google Scholar]
- Flores-Tlacuahuac, A.; Moreno, S.T.; Biegler, L.T. Global optimization of highly nonlinear dynamic systems. Ind. Eng. Chem. Res.
**2008**, 47, 2643–2655. [Google Scholar] [CrossRef] - Mahadevan, R.; Doyle, F.J. On-line optimization of recombinant product in a fed-batch bioreactor. Biotechnol. Prog.
**2003**, 19, 639–646. [Google Scholar] [CrossRef] [PubMed] - Banga, J.R.; Balsa-Canto, E.; Moles, C.G.; Alonso, A.A. Dynamic Optimization of Bioreactors: A Review. Proc. Ind. Natl. Sci. Acad.
**2003**, 69, 257–265. [Google Scholar] - Cuthrell, J.E.; Biegler, L.T. Simultaneous-Optimization and Solution Methods for Batch Reactor Control Profiles. Comput. Chem. Eng.
**1989**, 13, 49–62. [Google Scholar] [CrossRef] - Hedengren, J.D.; Shishavan, R.A.; Powell, K.M.; Edgar, T.F. Nonlinear modeling, estimation and predictive control in APMonitor. Comput. Chem. Eng.
**2014**, 70, 133–148. [Google Scholar] [CrossRef] - Constantinides, A.; Mostoufi, N. Numerical Methods for Chemical Engineers with MATLAB Applications; Prentice Hall: Upper Saddle River, NJ, USA, 2000. [Google Scholar]
- Byrd, R.H.; Gilbert, J.C.; Nocedal, J. A trust region method based on interior point techniques for nonlinear programming. Math. Program.
**2000**, 89, 149–185. [Google Scholar] [CrossRef] [Green Version] - Farzan, P. A framework for development of integrated and computationally feasible models of large-scale mammalian cell bioreactors. In Chemical and Biochemical Engineering; Rutgers, The State University of New Jersey: New Brunswick, NJ, USA, 2018. [Google Scholar]
- Zhao, Y.; Amemiya, Y.; Hung, Y. Efficient Gaussian Process Modeling using Experimental Design-Based Subagging. In Proceedings of the Conference on Experimental Design and Analysis (CEDA), Taipei, Taiwan, 15–17 December 2016; Institute of Statistical Science, Academia Sinica: Taipei, Taiwan, 2016. [Google Scholar]

**Figure 5.**Comparison of two operational polices: near-optimal policy (solid lines); and alternative policy with uniform feeding schedule (dashed lines).

No Aeration | Aerated System | |||||
---|---|---|---|---|---|---|

Fill Level (mm) | Impeller Rotation Speed (RPM) | Power Input (W%·m^{−3}) | Fill Level (mm) | Impeller Rotation Speed (RPM) | Power Input (W%·m^{−3}) | Volumetric Mass Transfer (h^{−1}) |

130 | 150 | 3.8 | 130 | 150 | 3.5 | 12.8 |

225 | 11.0 | 225 | 11.1 | 16.6 | ||

300 | 24.5 | 300 | 26.5 | 14.9 | ||

155 | 150 | 3.2 | 155 | 150 | 3.1 | 10.9 |

225 | 9.5 | 225 | 9.7 | 11.3 | ||

300 | 21.6 | 300 | 23.6 | 16.1 | ||

180 | 150 | 2.7 | 180 | 150 | 2.4 | 14.4 |

225 | 8.4 | 225 | 7.9 | 14.8 | ||

300 | 19.2 | 300 | 19.6 | 20.4 | ||

205 | 150 | 2.3 | 205 | 150 | 2.7 | 11.9 |

225 | 7.0 | 225 | 6.5 | 9.6 | ||

300 | 16.1 | 300 | 15.5 | 13.7 |

Parameter | Value | Unit |
---|---|---|

${m}_{Glc}$ | $6.92\times {10}^{-11}$ | $\mathrm{mmol}\xb7{\mathrm{cell}}^{-1}\xb7{\mathrm{h}}^{-1}$ |

${a}_{1}$ | $3.2\times {10}^{-12}$ | $\mathrm{mmol}\xb7{\mathrm{cell}}^{-1}\xb7{\mathrm{h}}^{-1}$ |

${a}_{2}$ | $2.1$ | $\mathrm{mM}$ |

${\mu}_{max}$ | $0.029$ | ${\mathrm{h}}^{-1}$ |

${\mu}_{d}{}_{max}$ | $0.016$ | ${\mathrm{h}}^{-1}$ |

${K}_{Glc}$ | $0.084$ | $\mathrm{mM}$ |

${K}_{Gln}$ | $0.047$ | $\mathrm{mM}$ |

$K{I}_{Lac}$ | $43$ | $\mathrm{mM}$ |

$K{I}_{Amm}$ | $6.51$ | $\mathrm{mM}$ |

$K{D}_{Lac}$ | $45.8$ | $\mathrm{mM}$ |

$K{D}_{Amm}$ | $6.51$ | $\mathrm{mM}$ |

${d}_{Gln}$ | $7.2\times {10}^{-3}$ | ${\mathrm{h}}^{-1}$ |

${Y}_{X/Glc}$ | $1.69\times {10}^{8}$ | $\mathrm{cell}\xb7{\mathrm{mmol}}^{-1}$ |

${Y}_{X/Gln}$ | $9.74\times {10}^{8}$ | $\mathrm{cell}\xb7{\mathrm{mmol}}^{-1}$ |

${Y}_{Lac/Glc}$ | $1.23$ | $\mathrm{mmol}\xb7{\mathrm{mmol}}^{-1}$ |

${Y}_{Amm/Gln}$ | $0.67$ | $\mathrm{mmol}\xb7{\mathrm{mmol}}^{-1}$ |

$D{O}_{eq}$ | $1.0699$ | $\mathrm{mM}$ |

$OUR$ | $3.5\times {10}^{-10}$ | $\mathrm{mmol}\xb7{\mathrm{cell}}^{-1}\xb7{\mathrm{h}}^{-1}$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Farzan, P.; Ierapetritou, M.G.
A Framework for the Development of Integrated and Computationally Feasible Models of Large-Scale Mammalian Cell Bioreactors. *Processes* **2018**, *6*, 82.
https://doi.org/10.3390/pr6070082

**AMA Style**

Farzan P, Ierapetritou MG.
A Framework for the Development of Integrated and Computationally Feasible Models of Large-Scale Mammalian Cell Bioreactors. *Processes*. 2018; 6(7):82.
https://doi.org/10.3390/pr6070082

**Chicago/Turabian Style**

Farzan, Parham, and Marianthi G. Ierapetritou.
2018. "A Framework for the Development of Integrated and Computationally Feasible Models of Large-Scale Mammalian Cell Bioreactors" *Processes* 6, no. 7: 82.
https://doi.org/10.3390/pr6070082