# 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

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**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 |
---|---|---|

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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