Automated Shape and Process Parameter Optimization for Scaling Up Geometrically Non-Similar Bioreactors

Abstract

Scaling bioprocesses remains a major challenge. Since it is physically impossible to increase all process parameters equally, a suitable scale-up strategy must be selected for a successful bioprocess. One of the most widely used criteria when scaling up bioprocesses is the specific power input. However, this represents only an average value. This study aims to determine the Kolmogorov length scale distribution by means of computational fluid dynamics (CFD) and to use it as an alternative scale-up criterion for geometrically non-similar bioreactors for the first time. In order to obtain a comparable Kolmogorov length scale distribution, an automated geometry and process parameter optimization was carried out using the open-source tools OpenFOAM and DAKOTA. The Kolmogorov–Smirnov test statistic was used for optimization. A HEK293-F cell expansion (batch mode) from benchtop (Infors Minifors 2 with 4 L working volume) to pilot scale (D-DCU from Sartorius with 30 L working volume) was carried out. As a reference cultivation, the classical scale-up approach with constant specific power input (233 W m⁻³) was used, where a maximum viable cell density (VCD${}_{\max }$) of $5.02·{10}^6$ cells mL⁻¹ was achieved (VCD${}_{\max }$ at laboratory scale $5.77·{10}^6$ cells mL⁻¹). Through the automated optimization of the stirrer geometry (three parameters), position and speed, comparable cultivation results were achieved as in the small scale with a maximum VCD of $5.60·{10}^6$ cells mL⁻¹. In addition, even on the pilot scale, cell aggregate size distribution was seen to strictly follow a geometric distribution and can be predicted with the help of CFD with the previously published correlation.

1. Introduction

Biopharmaceuticals are a multi-billion-dollar business with a continuously increasing market value [1]. The production of such biopharmaceuticals traditionally takes place in stirred bioreactors on a cubic meter scale [2]. Even though process intensification is an important market trend, the challenge of transferring the optimized bioprocess from laboratory scale to production scale remains. Scaling up bioreactors is considered one of the biggest challenges [3], and there are various ways in which scale transfer can take place. Typically, scale-independent variables such as pH, dissolved oxygen concentration (DO), temperature and inoculation density are kept constant during scale-up. Ideally, geometrically similar systems are used for scaling up, as they allow for similar conditions, which in practice is not always possible. As Kaiser et al. [4] and Böhm et al. [5] demonstrated in their studies, it is physically impossible to scale up all process parameters equally. Therefore, typically one or more scale-up criteria are defined. Classical scale-up criteria are specific power input P/V, volumetric oxygen mass transfer coefficient $k_{\mathrm{L}}a$, mixing time ${\Theta }_{\mathrm{M}}$, tip speed $v_{\mathrm{tip}}$, superficial gas velocity $v_g$ and the Reynolds number Re [6,7,8,9,10]. Interested readers will find a detailed overview of scale-up criteria used in Neubauer and Junne [8] and Löffelholz et al. [10]. Regardless of the frequent and often successful use of these simple scale transfer approaches, some limitations exist [7]. The choice of bioreactors is enormous and exact geometric similarity is rarely given [11]. Different geometries lead to varying flow patterns and mixing regimes in the vessels for the same power inputs, which can be accounted for by the different distribution profiles of the local turbulent energy dissipation rates [12]. Furthermore, the higher heterogeneity at the production scale leads to higher cell-to-cell variability, which cannot always be represented with sufficient accuracy in scale-down models [13].

However, there are also more sophisticated alternatives, mostly used in conjunction with computational fluid dynamics (CFD). For example, Haringa [14] used CFD to model the view of the cell with his lifeline analysis, which can also be used for scale-up. Villiger et al. [15] used maximum hydrodynamic stress as a scale-up criterion in addition to the mixing time and volumetric oxygen mass transfer coefficient. Li et al. [16] defined a three-dimensional shear space consisting of shear strain rate in the impeller zone, shear strain rate in the tank bulk zone and the overall average shear strain rate at which the two systems should be located. Böhm et al. [5] recommend the consideration of the impeller swept volume $V_l$ respectively the resulting energy dissipation circulation function (EDCF). The EDCF, introduced by Jüsten et al. [17], is a mixing parameter originally defined according to Equation (1) where $t_c$ corresponds to the circulation time. However, there are different ways of determining and calculating the EDCF. A detailed overview is provided by Böhm et al. [5] and Esperança et al. [18].

$$\mathrm{EDCF}=\frac{P}{V_l·t_c}$$

(1)

Nevertheless, the specific power input is the most frequently used scale-up criterion [8,9]. This may be due to the fact that homogenization, dispersion of gas bubbles and suspension of the cells depend on the specific power input and that the specific power input is comparatively easy to determine. The specific power input can be determined experimentally in various ways, whereby the torque measurement (Equation (2)) is recommended by the DECHEMA expert group for single-use technology and is used most frequently [19]. M corresponds to the torque, N to the stirrer speed and V to the liquid volume. This method is both operatively and skill-wise less demanding than the method used by Villiger et al. [15] to determine the maximum hydrodynamic stress.

$$P/V=\frac{2·\pi ·N·M}{V}=\overline{\epsilon }·\rho$$

(2)

Furthermore, the specific power input can be estimated using literature values and empirical formulae, or calculated using CFD [20]. However, the specific power input is only an average value without any indication of variability in the system. In contrast, hydrodynamic heterogeneity Φ is defined as the ratio of maximum ${\epsilon }_{\max }$ to mean $\overline{\epsilon }$ energy dissipation rate (Equation (3)) [21]. An overview of stirred systems can be found in Zhou and Kresta [22], and for different types of bioreactors in Seidel et al. [23]. If not only the mean value P/V but also the hydrodynamic heterogeneity is considered as a type of variance measure, the system can be characterized more precisely. However, it must be taken into account that the maximum energy dissipation rate ${\epsilon }_{\max }$ is difficult to measure, since it is the maximum directly at the stirrer [5].

$$\Phi =\frac{{\epsilon }_{\max }}{\overline{\epsilon }}$$

(3)

Freiberger et al. [24] went one step further and looked not only at the maximum and average values or their ratio Φ, but at the entire energy dissipation rate distribution comparing two bioreactors on the same scale. Johnson et al. [25] studied the Kolmogorov length scale distribution ${\lambda }_k$ for five different bioreactors from 200 L to 15,000 L and found that only with the same stirrers and number of baffles can a similar Kolmogorov length scale distribution be obtained for different working volumes. The Kolmogorov length scale ${\lambda }_k$ describes the smallest vortices that are formed before they dissipate into heat and is a function of the energy dissipation rate ε and the kinematic viscosity ν (Equation (4)). Various authors describe that cell damage is likely if the Kolmogorov length scale is equal to or smaller than the cells to be cultivated [15,26,27,28].

$${\lambda }_k={\left(\frac{{\nu }^3}{\epsilon }\right)}^{\frac{1}{4}}$$

(4)

Our hypothesis is that similar Kolmogorov length scale distributions lead to comparable growth of mammalian cell cultures such as human embryonic kidney cells (HEK293) and can thus be used as a scale-up criterion. The aim of this study is to develop a method that, based on CFD, automatically optimizes stirrer geometry, position and speed in order to achieve a similar energy dissipation rate distribution, and therefore Kolmogorov length scale distribution, in a geometrically dissimilar system that allows for a successful scale-up and comparable cell growth. The method will be implemented using open-source resources from optimization to CFD simulations and evaluation.

Shape optimization with the help of CFD is a widely used strategy, especially in optimizing aerospace foils [29,30]. Furthermore, this method is also used to optimize wind turbines [31,32], static mixers [33], pumps [34,35] and other equipment [36]. Both Hoseini et al. [37] and Wu et al. [38] have studied stirrer optimization in stirred tanks, although their focus was neither an open-source nor a biotechnological application. Wu et al. [38] carried out a multi-objective optimization in which mixing time and specific power input were to be minimized. Jossen et al. [39] dealt with stirrer optimization for bioreactors, whereby only a primitive optimization approach was used. Nine CFD simulations were carried out and the optimum was selected from these simulations.

Because CFD simulations are both time and computationally intensive, a surrogate-based optimization (SBO) is typically preferred [34,40]. Here, the design space is investigated using design and analysis of computer experiments (DACEs) and a surrogate model (also known as response surface model (RSM)) is created. Latin hypercube sampling (LHS) is commonly used for these types of experiments [36,41,42,43]. Alternatively, a random or regular grid sampling can be applied [44]. Based on the surrogate model, the actual optimization is then carried out, whereby the surrogate model and optimization algorithm differ depending on the problem under investigation and the optimization objective.

The optimization objective depends on the bioprocess and thus on the needs of the used cells. After Chinese hamster ovary (CHO), HEK293 cells are among the most widely used mammalian cell cultures for the production of biopharmaceuticals [26,45,46]. HEK293 cells are used to produce recombinant proteins, viral vectors and vaccines [47,48,49,50]. It should be noted that HEK293 cells grow adherently, such as HEK293-E and HEK293-T, or in suspension, such as HEK293-H and HEK293-F, with only the latter being considered here [51,52]. HEK293 cells growing in suspension have a maximum specific growth rate between 0.020 h⁻¹ and 0.036 h⁻¹ [53,54,55,56,57] and typical cell diameters from 14 μm to 16 μm [58,59,60]. In Seidel et al. [26], cell growth of HEK FreeStyle^TM 293-F suspension cells was improved by the authors at a laboratory scale by adjusting the hydrodynamic stress. This optimized process at 4 L scale serves as a baseline for the scale-up method proposed in this article. To verify the results, the data were compared with cultivations in which the specific power input was used as a scale-up criterion. The reason for choosing this criterion is that several authors have already shown (or at least proposed) that the use of this criterion has worked for geometrically similar bioreactors and HEK293 cells [61,62,63,64].

2. Materials and Methods

In Seidel et al. [26], the authors showed that HEK FreeStyle^TM 293-F cells can be cultivated with a higher specific power input (233 W m⁻³) than typically stated in the literature (≈60 W m⁻³) and to yield 24% higher maximum viable cell densities (VCD${}_{\max }$). The cultivation optimized in Seidel et al. [26] in the Minifors 2 bioreactor from Infors AG (Bottmingen, Switzerland) with 4 L working volume serves as the basis for the automated scale-up concept presented here (Figure 1A). These CFD simulations were validated using particle image velocimetry (PIV) and literature data. Figure 1B shows the new scale-up method proposed here, in which a similar Kolmogorov length scale distribution is achieved by stirrer optimization. The Kolmogorov–Smirnov (KS) test, which compares the cumulative distribution functions (CDF) of the Kolmogorov scale length distribution, was used as the optimization criterion. The scale-up bioreactor was intentionally changed from a cell culture bioreactor to a system that is rather atypical for mammalian cell cultures. However, there are now some companies that successfully cultivate mammalian cells in bioreactors that they also use for processes with microorganisms. The 30 L D-DCU bioreactor (Sartorius AG, Göttingen, Germany) is a classic bioreactor for microbial fermentations with four baffles, three Rushton stirrers and a bioreactor height to diameter ratio H/D of 3:1 [65]. As a reference cultivation, duplicate cultivations were carried out with the same specific power input (233 W m⁻³) as in the Minifors 2 bioreactor (Figure 1C). A duplicate was also carried out using the system with optimized stirrer geometry, position and speed. The process is described in detail in Figure 1 and the following sections.

2.1. Computational Fluid Dynamics

The CFD model validated in Seidel et al. [26] using PIV and data from literature serves as the basis for the simulations carried out here. All CFD simulations were carried out on the 30 L D-DCU from Sartorius AG, using different stirrers, stirrer positions and speeds (see Section 2.2). All geometries excluding the stirrers, which were required for the simulations, were drawn with Inventor Professional 2023 (Autodesk Inc., San Rafael, CA, USA). To automate the optimization of the stirrer geometry, the stirrers were drawn using Onshape (PTC Inc., Needham, MA, USA) and adapted using the Python application programming interface (API) for each simulation (this option is not available as standard in Autodesk Inventor). For the CFD simulations, OpenFOAM version 10 was used and meshes were generated with the integrated meshing tool SnappyHexMesh. In order to estimate the discretization error, a mesh study with four different computational meshes was performed. Due to the low aeration rates and the low stirrer speeds (in combination with baffles, no vortex formation could be observed), a single-phase, steady-state simulation was carried out. However, since Reynolds numbers higher than 10,000 were achieved in the simulations, the Reynolds-Averaged Navier–Stokes (RANS) approach was chosen. As in Seidel et al. [26], the k-ω shear stress transport (SST) model of Menter [66] was employed as turbulence model, which is described in detail in Seidel et al. [26]. This turbulence model is suitable for low Reynolds numbers as they occur under the selected process conditions (Section 3.1) and in Seidel et al. [26]. The resulting momentum equation corresponds to Equation (5) and the continuity equation to Equation (6).

$$\frac{\partial \overrightarrow{v}}{\partial t}+\nabla ·(\overrightarrow{v}\overrightarrow{v})-\nabla ·{\nu }_{\mathrm{eff}}\nabla \overrightarrow{v}=-\frac{1}{\rho }\nabla p_p+\nabla ·S_{ij}$$

(5)

$$\nabla ·\overrightarrow{v}=0$$

(6)

For the rotation of the stirrers, the multiple reference frame (MRF) approach was utilized since these are steady-state simulations. A no-slip boundary condition was applied for all walls and a symmetry plane for the fluid surface [67,68]. For the pressure–velocity coupling, the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm was employed, which can be utilized in OpenFOAM using simpleFOAM. Also, as in Seidel et al. [26], an undershoot of the residuals of $1·{10}^{-5}$ was chosen as a convergence criterion. All CFD simulations were performed with water at a temperature of $T=310.15 \mathrm{K}$, which corresponds to a density of $\rho =993.37 \mathrm{kg} {\mathrm{m}}^{-3}$ and a kinematic viscosity of $\nu =0.6959·{10}^{-6} {\mathrm{m}}^2 {\mathrm{s}}^{-1}$ [69]. The calculations were decomposed into 32 parts using the Scotch algorithm and computed in parallel on the high-performance computing (HPC) system described in Seidel and Eibl [70]. The visualization of the simulations was carried out using Paraview 5.10.0 [71].

2.2. Optimization Process

As described in Johnson et al. [25], a similar Kolmogorov length scale distribution cannot be expected if different geometries such as the number of baffles and different stirrers are used. To obtain comparable Kolmogorov length scale distributions that serve as a scale-up criterion, five parameters were defined, varied and optimized. These five parameters are the stirrer speed N, stirrer diameter $d_{\mathrm{s}}$, the blade angle ${\alpha }_1$, the pitch angle of the stirrer blades ${\alpha }_2$ and the stirrer height H (Figure 2).

Table 1 shows the design space investigated with the limits set by the system. Based on this design space, a DACE was created using LHS for design space exploration. For the five parameters, 100 CFD simulations were performed automatically. Daymo et al. [43] also used 100 simulations for five variable parameters for the optimization of a methane catalytic partial oxidation monolith reactor. Chen et al. [41] described ten times the number of parameters as standard for LHS and CFD-based optimization. Afzal et al. [42] used 15 times the number of parameters and Benchikh Le Hocine et al. [36] performed 148 simulations for seven parameters.

Table 1. Overview of the input variables and the resulting design space.

Parameter	Minimum	Maximum
Stirrer speed N	50 rpm	500 rpm
Stirrer height H	0.10 m	0.45 m
Stirrer diameter $d_{\mathrm{s}}$	50 mm	170 mm
Blade angle ${\alpha }_1$	20${}^{\circ }$	120${}^{\circ }$
Pitch angle ${\alpha }_2$	$-45$${}^{\circ }$	90${}^{\circ }$

Table 1. Overview of the input variables and the resulting design space.

Parameter	Minimum	Maximum
Stirrer speed N	50 rpm	500 rpm
Stirrer height H	0.10 m	0.45 m
Stirrer diameter $d_{\mathrm{s}}$	50 mm	170 mm
Blade angle ${\alpha }_1$	20${}^{\circ }$	120${}^{\circ }$
Pitch angle ${\alpha }_2$	$-45$${}^{\circ }$	90${}^{\circ }$

The Design Analysis Kit for Optimisation and Terascale Applications (DAKOTA) version 16.5 developed by Sandia National Laboratories was used for the entire optimization process, which allowed the LHS, DACE, evaluation and optimization to be carried out with one command. How DAKOTA interacts with OpenFOAM is depicted in Figure 1 and described in detail in Guerrero et al. [33] and Daymo et al. [43]. As described in Section 2.1, the stirrer geometry was drawn and parameterized using the web-based CAD service Onshape [72]. Using the Python API, the geometry and thus the computational mesh could be automatically adapted for the CFD simulations. In order to obtain a single value per simulation for the optimization process, the Kolmogorov length scale distribution was analyzed using the two-sample KS test with the null hypothesis $H_0$ of equal cumulative distributions F (Equation (7)) [73]. The test statistic D served as the optimization criterion (Equation (8)). In the KS test, the cumulative Kolmogorov length scale distribution $F_{\mathrm{Minifors}}$ from the optimized Minifors 2 cultivation by Seidel et al. [26] was used.

$$H_0:F_{\mathrm{Minifors}}\left(\lambda \right)=F_{\mathrm{D}-\mathrm{DCU}}\left(\lambda \right)\forall \lambda \in {\mathbb{R}}^{\ge 0}$$

(7)

$$D=max|F_{\mathrm{Minifors}}\left(\lambda \right)=F_{\mathrm{D}-\mathrm{DCU}}\left(\lambda \right)|$$

(8)

The automated analysis was carried out with Python 3.10, using PyVista 0.38.5 for the analysis in addition to the usual modules, which has an integrated OpenFOAM case reader [74]. A quadratic polynomial model was used as a surrogate model. DAKOTA allows asynchronous evaluation, which means that not only a single CFD simulation was parallelized (Section 2.1) but also several simulations and evaluations could be carried out in parallel for the design space exploration. Here, four simulations were run in parallel on the HPC system. Once all 100 simulations were completed, the surrogate model was created, and the optimization was carried out.

2.3. Cultivations for Biological Evaluation

In order to evaluate the optimized stirrer geometry, position and speed predicted by CFD, four cultivations were carried out in the 30 L D-DCU system from Sartorius AG. Two cultivations were carried out with the standard configuration of three Rushton stirrers ($d_{\mathrm{s}}=105 \mathrm{mm}$), four baffles and ring sparger and two cultivations with the same configuration but with optimized stirrer geometry. Both the inoculum production and the cultivations were carried out analogously to the experiments described in Seidel et al. [26] and serve as the basis for the experiments conducted here. Batch cultivations were carried out with the HEK FreeStyle^TM 293-F cell line (Thermo Fisher Scientific, Waltham, MA, USA [75]) and the chemically defined FreeStyle^TM 293 medium (Thermo Fisher Scientific) which contains L-alanyl-L-glutamine (GlutaMAX^TM) as a stabilized form of L-glutamine. The inoculum for cultivations in the 30 L D-DCU system and reference shake flasks was prepared from a cryovial of a working cell bank with $1·{10}^7$ cells mL⁻¹. The thawed cells were transferred into 30 mL of pre-warmed FreeStyle^TM 293 medium (125 mL unbaffled shake flask) and then passaged into 500 mL unbaffled shake flasks at a VCD of about $0.3·{10}^6$ cells mL⁻¹. The inoculum production lasts 7 days and was performed in a Multitron shaker from Infors AG at a temperature of $T=310.15 \mathrm{K}$, a shaking speed of $N=100 \mathrm{rpm}$, a shaking amplitude of $d_0=50$ mm, a CO${}_2$ concentration of $c_{{\mathrm{CO}}_2}=8\%$ and a relative humidity of $RH=80\%$ [26]. The cultivations were carried with a working volume of 30 L and an inoculation cell density of $0.3·{10}^6$ cells mL⁻¹. The inoculum was transferred to the bioreactor using a transfer flask (5 L plain bottom Erlenmeyer flask with a disposable 100 mm aseptic transfer cap, from Corning Inc. (Corning, NY, USA)). The cultivation temperature, which was controlled through the double jacket, was 310.15 K in each case. The oxygen concentration was kept above 40% with 0.1 vvm air being added via the headspace and, when necessary, O${}_2$ via the sparger (Appendix A, Figure A2A). The pH was controlled by adding CO${}_2$ through the sparger to maintain a pH of 7.10 ± 0.05 (Appendix A, Figure A2B). The stirrer speeds were determined with CFD resulting in 213 rpm ($v_{\mathrm{tip}}=1.17 \mathrm{m} {\mathrm{s}}^{-1}$) for the Rushton stirrer configuration and 67 rpm ($v_{\mathrm{tip}}=0.56 \mathrm{m} {\mathrm{s}}^{-1}$) for the optimized stirrer.

The CFD-optimized stirrer was manufactured using a selective laser sinter (SLS) 3D printing process. The material used is biocompatible polyamide 2200 (PA2200), which is United States Pharmacopeia (USP) class VI certified, Food and Drug Administration (FDA) approved and complies with EN ISO 10993-1 [76]. The stirrers were printed with the EOS p396 system (EOS GmbH, Krailling, Germany). In order to exclude the influence of the stirrer material despite biocompatibility, the standard stainless steel Rushton stirrers were replaced by SLS 3D-printed stirrers for maximum comparability.

To monitor the process, cell-specific parameters such as VCD, total cell density (TCD), viability, and cell and aggregate sizes were measured using a CedexHiRes analyzer (Roche Diagnostics GmbH, Basel, Switzerland) and NucleoCounter NC-200 (Chemometec, Allerod, Denmark). Furthermore, an EXcell 231 near-infrared (NIR) absorption sensor (Exner Process Equipment GmbH, Ettlingen, Germany) was used to determine the optical density (OD${}_{850\mathrm{n}\mathrm{m}}$) online at a wavelength of 850 nm. Morphology was regularly checked using differential interference microscopy (IX83 inverted microscope with UPlanSApo 100x/1.4 oil ∞/0.17/FN26.5 objective, both from Olympus Life Science (Waltham, MA, USA)). In addition to the cell-specific parameters, the substrates glucose and L-alanyl-L-glutamine, as well as metabolites lactate and ammonium, were measured. These measurements were carried out with a CedexBio analyzer (Roche Diagnostics GmbH) every 24 h. The pH value was likewise externally checked every 24 h with a FiveEasy pH-meter F20 (Mettler-Toledo GmbH, Greifensee, Switzerland). A detailed functional description of the measuring instruments used can be found in Seidel et al. [26]. The reference cultivations in 500 mL unbaffled shake flasks were each prepared from the same inoculum and were cultivated at a shaking rate of 130 rpm (the remaining conditions as for the production of the inoculum). Samples were also taken daily, and the same analyses were carried out as in the stirred bioreactor.

3. Results and Discussion

3.1. Optimization with CFD

The CFD model used has already been validated using PIV and literature data as described in Seidel et al. [26]. In order to quantify the discretization error for the new geometry and to ensure an economic implementation with regard to the 100 optimization simulations, a mesh study was carried out. For this study, the grid convergence index (GCI) method was used as described in Baker et al. [77], Ramírez et al. [78] and Pappalardo et al. [79]. The standard value of $1.25$ was chosen as the safety factor $F_S$ [23,26]. The mesh study was conducted with the standard 30 L D-DCU configuration with three Rushton impellers at a stirrer speed of 364 rpm ($v_{\mathrm{tip}}=2 \mathrm{m} {\mathrm{s}}^{-1}$) and 30 L working volume. The specific power input was used as an evaluation criterion, which was also calculated by Schirmer et al. [65] with the same configuration and speed. The refinement factor ranged from 1.08 to 1.17, whereas a value of 1.1 to 1.3 is typically recommended [80]. The quotient $\frac{{\mathrm{GCI}}_{i+2,i+1}}{r^{p_a}{\mathrm{GCI}}_{i+1,i}}$ was close to 1 in each case, suggesting an asymptotic behavior of the specific power input (

Table 2. Overview of GCI analysis for the 30 L D-DCU bioreactor with three Rushton impellers and a stirrer speed of 364 rpm ($v_{\mathrm{tip}}=2 \mathrm{m} {\mathrm{s}}^{-1}$). A detailed overview of the mesh properties can be found in Table A1.

Case	Mesh	r	${\widehat{\mathit{p}}}_{\mathit{a}}$	${\mathit{\epsilon }}_{\mathit{mn}}$	GCI [%]	$\frac{{\mathbf{GCI}}_{\mathit{i}+2,\mathit{i}+1}}{{\mathit{r}}^{{\mathit{p}}_{\mathit{a}}}{\mathbf{GCI}}_{\mathit{i}+1,\mathit{i}}}$
Case 1	M1-M2	1.08	2.46	$3.84·{10}^{-3}$	2.24	1.13
	M2-M3	1.13		$4.40·{10}^{-3}$	1.63
Case 2	M2-M3	1.13	2.04	$4.40·{10}^{-3}$	2.01	1.04
	M3-M4	1.17		$4.72·{10}^{-3}$	1.52

). For further simulations, mesh M4 with $4.88·{10}^6$ cells and its mesh settings was used, which in this case resulted in a simulation time of 3.5 h with 32 cores.

Schirmer et al. [65] also characterized this system in terms of biochemical engineering parameters using CFD, but with Ansys Fluent 16.2. They utilized the realizable k-ε model, whereas here the k-ω-SST model was used. The same boundary conditions were applied and Schirmer et al. [65] used slightly fewer grid cells ($4.20·{10}^6$ cells) than here ($4.88·{10}^6$ cells). As can be seen in Figure 3, the specific power inputs agree for the range from $v_{\mathrm{tip}}=2 \mathrm{m} {\mathrm{s}}^{-1}$ ($N=364 \mathrm{rpm}$ and Re = 66,592) to $v_{\mathrm{tip}}=5 \mathrm{m} {\mathrm{s}}^{-1}$ ($N=909 \mathrm{rpm}$ and Re = 166,481). As already mentioned, the 30 L D-DCU bioreactor is a bioreactor typically used for microorganisms [65,81,82]. Therefore Schirmer et al. [65] only investigated the range from $v_{\mathrm{tip}}$ = 2 m s⁻¹ to 5 m s⁻¹, which corresponds to a specific power input over 1 kW m⁻³. However, for the reference cultivation with the same specific power input as in Minifors 2, lower power inputs or speeds were required. With $N=213 \mathrm{rpm}$ ($v_{\mathrm{tip}}=1.17 \mathrm{m} {\mathrm{s}}^{-1}$ and Re = 38,956), the 233 W m⁻³ described in Seidel et al. [26] can be achieved. As can be seen in Figure 4B), the flow field typical for Rushton impellers is formed with the maximum velocities at the stirrer tips [38,83].

In order to obtain a Kolmogorov length scale distribution as similar as possible to that in Minfors 2, optimizations of the stirrer geometry, position and speed were then carried out. The 100 parameter combinations generated by the LHS and subsequently simulated using CFD are shown in Figure 5. For this purpose, the relative parameter values, which are described in Table 1, were mapped in the radar chart. Corresponding velocity plots are summarized in the appendix in Figure A1.

Based on the DACE, a different RSM can now be created, whereby the KS test variable was finally chosen for the scale-up. However, simpler analyses can also be carried out, such as modeling the specific power input. Equation (9) shows the results for the investigated design space ($n_s=100$, $R^2=0.96$). If only the specific power input should be kept constant for the scale-up, a Pareto front results, which is shown in Figure 6A, for the combination of stirrer speed and stirrer diameter. The same can be examined, for example, for the combination of pitch angle ${\alpha }_2$ and the stirrer speed N. This shows that a larger angle ${\alpha }_2$ or a larger stirrer diameter results in a larger projected cross-sectional area and thus a larger specific power input for the same stirrer speed N (Figure 6B).

$$P/V={10}^{a_1·H\left[\mathrm{m}\right]+a_2·d_{\mathrm{s}}\left[\mathrm{m}\mathrm{m}\right]-a_3·d_{\mathrm{s}}^2\left[{\mathrm{m}}^2\right]+a_4·{\alpha }_1{[}^{\circ }]+a_5·{\alpha }_1^2{[}^{{\circ }^2}]-a_6·{\alpha }_2{[}^{\circ }]+a_7·N\left[\mathrm{rpm}\right]-a_8·N^2\left[{\mathrm{rpm}}^2\right]-a_9}$$

(9)

$$\begin{array}{cccccccccc}\hfill a_1& =0.162\hfill & \hfill a_2& =0.049\hfill & \hfill a_3& =1.25·{10}^{-4}\hfill & \hfill a_4& =6.15·{10}^{-3}\hfill & \hfill a_5& =2.29·{10}^{-4}\hfill \\ \hfill a_6& =2.21·{10}^{-4}\hfill & \hfill a_7& =0.013\hfill & \hfill a_8& =1.39·{10}^{-5}\hfill & \hfill a_9& =3.67\hfill \end{array}$$

However, if the described test statistic D of the KS test is employed, the best agreement with the Minifors 2 bioreactor is obtained for $N=67 \mathrm{rpm}$, $H=0.124 \mathrm{m}$, $d_{\mathrm{s}}=160.23 \mathrm{mm}$, ${\alpha }_1={59.60}^{\circ }$ and ${\alpha }_2={41.56}^{\circ }$, which results in a test statistic of $D=0.117$. This corresponds to a Re of 32,080 and a tip speed of $0.56$ m s⁻¹. Visualizing the Kolmogorov length scale distributions, it can be seen that there are only minimal differences in the distribution between the optimized design of the 30 L D-DCU and the Minifors 2, whereas the differences with the standard 30 L D-DCU with three Rushton impellers are significantly larger (Figure 7). Since a similar Kolmogorov length scale distribution was achieved, a similar energy dissipation rate distribution was also achieved (Equation (4)) and therefore resulted in the same specific power input of 233 W m⁻³.

As can be seen in Figure 7, various simulations were performed that are far from the optimum. This is due to the fact that an LHS was performed for the optimization and thus the entire design space described in Table 1 was considered in an SBO. This approach, which has been used by various authors, also has the advantage that several simulations can be performed simultaneously [36,41,42,84,85]. Nevertheless, optimization might be performed with fewer simulations if direct optimization of the geometry were performed instead of using a surrogate model. For this purpose, optimization algorithms such as the genetic algorithm [86,87] or Bayesian optimization [88,89], which adapt the geometry up to a certain convergence criterion, would be suitable.

3.2. Biological Verification

The stirrer design optimized in Section 3.1 and the standard configuration were used for the cultivation of HEK FreeStyle^TM 293-F suspension cells (

Table A2. Comparison of the process parameters for the benchtop scale bioreactor (Minifors 2), the reference cultivation (D-DCU, same $P/V$) and the optimized scale-up approach (D-DCU, similar $F\left({\lambda }_k\right)$).

Parameter	Minifors 2	D-DCU Reference	D-DCU Optimized
V	4 L	30 L	30 L
Stirrer type	3-blade segment	3x Rushton	3-blade segment
$d_{\mathrm{s}}$	85 mm	105 mm	160 mm
N	275 rpm	213 rpm	67 rpm
$v_{\mathrm{tip}}$	$1.22 \mathrm{m} {\mathrm{s}}^{-1}$	$1.17 \mathrm{m} {\mathrm{s}}^{-1}$	$0.56 \mathrm{m} {\mathrm{s}}^{-1}$
$P/V$	233 W m−3	233 W m−3	233 W m−3
Same $F\left({\lambda }_k\right)$ as in the Minifors 2	–	No	Yes
DO set point	40%	40%	40%
pH set point	$7.1$	$7.1$	$7.1$

). The batch experiments, which were performed in duplicates, lasted 168 h each, where the maximum VCD was reached after 120 h. In the reference cultivation with Rushton impellers and the same specific power input as at the laboratory scale (Minifors 2), a maximum VCD of $5.02·{10}^6$ cells mL⁻¹ was achieved, which is lower compared to the laboratory scale with $5.77·{10}^6$ cells mL⁻¹. With the optimized stirrer geometry, however, a VCD${}_{\max }$ of $5.60·{10}^6$ cells mL⁻¹ was achieved, which is comparable to the laboratory scale (Figure 8). At this point, the viability of all cultivations was above 95%. The maximum VCD in all cultivations was slightly higher than the values described in the literature for cultivations in unbaffled shake flasks where values between $4.2·{10}^6$ cells mL⁻¹ and $4.6·{10}^6$ cells mL⁻¹ were achieved [26,57]. The maximum specific growth rates ${\mu }_{\max }$ were achieved for the laboratory scale and for all 30 L D-DCU experiments between the time points of 24 h and 72 h and were ${\mu }_{\max ,\mathrm{Minifors}}=0.0258 {\mathrm{h}}^{-1}$ for the laboratory scale, ${\mu }_{\max ,\mathrm{D}-\mathrm{DCU}}=0.0248 {\mathrm{h}}^{-1}$ for the reference cultivation with the same specific power input and ${\mu }_{\max ,\mathrm{D}-\mathrm{DCU},\mathrm{opt}.}=0.0262 {\mathrm{h}}^{-1}$ for the bioreactor with optimized stirrer design. All maximum specific growth rates were in the range of 0.020 ${\mathrm{h}}^{-1}$ to 0.036 ${\mathrm{h}}^{-1}$, which is described in the literature [53,54,55,56,57]. The glucose concentration dropped from 4.6 g L⁻¹ to 0.80 g L⁻¹ in all cultivations (Appendix A, Figure A3A). The same glucose concentration drop was described by Maschke et al. [57] for the same cell line and medium in the 250 mL shake flask. As in the laboratory scale, the lactate concentration rose up to the time point $t=96 \mathrm{h}$ and dropped to the end of cultivation (Appendix A, Figure A3B). Slightly higher maximum lactate concentrations (2.11 g L⁻¹) were measured in the reference cultivations with the same specific power input as in the laboratory scale than in the cultivations with optimized stirrer geometry (1.98 g L⁻¹). This increased lactate concentration could be due to the fact that the hydrodynamic stress was too high for the HEK293 cells [62].

In Seidel et al. [26], no statistically significant difference in the measured cell diameter could be observed with different specific power inputs ($63 \mathrm{W} {\mathrm{m}}^{-3}$ to $451 \mathrm{W} {\mathrm{m}}^{-3}$). The cell diameter in the 4 L system also remained constant over the cultivation period, although the variance in the cell diameter increased slightly over the course of the cultivation. For the reference cultivations in the 30 L system, the cell diameter ((14.49 ± 1.77) μm) was slightly smaller than at laboratory scale ((15.28 ± 2.03) μm) and that of the system with optimized stirrer geometry ((15.17 ± 2.33) μm). Nevertheless, cell diameter in all cultivations was within the typical range described in the literature (14 μm to 16 μm) [58,59,60]. As described in Seidel et al. [26], the aggregate size distribution at the time of maximum VCD strictly follows a geometric distribution with the free parameter p equal to the proportion of non-aggregated cells. The fraction of cells that occur as an aggregate of size n can thus be calculated according to Equation (10). Seidel et al. [26] could further show that Equation (11) applies for shake flasks both with and without baffles, as well as for different operating parameters in the Minfors 2 bioreactor. Using this equation, a value of $p=0.6876$ would result in the reference cultivation, which would correspond to 68.76% of non-aggregated cells at the time of maximum VCD ($\overline{{\lambda }_k}=5.064·{10}^{-5} \mathrm{m}$). As measurements of the two cultivations showed, 68.39% and 69.17%, respectively, were present as non-aggregated cells at this time. The same situation occurs for the cultivations with the optimized stirrer design. Whereas, the mean Kolmogorov length scale of $5.730·{10}^{-5} \mathrm{m}$ predicts a 65.70% proportion of non-aggregated cells, and 64.64% and 64.22% were measured. The aggregate size distribution also follows the correlation described by Seidel et al. [26] at the pilot scale.

$$f\left(n\right)={(1-p)}^{n-1}p,\left\{p\right|0\le p\le 1\}$$

(10)

$$p=-4589\left[{\mathrm{m}}^{-1}\right]·\overline{{\lambda }_k}\left[\mathrm{m}\right]+0.92$$

(11)

With the help of the OD${}_{850\mathrm{n}\mathrm{m}}$ sensor, the growth of cells was monitored online [90,91,92]. As only the turbidity can be measured, no distinction can be made between viable and dead cells. Since there is only a minimal difference between TCD and VCD (viability above 95%) before the maximum VCD is reached at 120 h, the OD signal can be used for process monitoring (Figure 9). As shown in Figure 10, the OD${}_{850\mathrm{n}\mathrm{m}}$ signal correlates with the offline measured TCDs for all performed cultivation cycles over the entire measuring range (0.30 · 10⁶ cells mL⁻¹ to 6.46 · 10⁶ cells mL⁻¹).

4. Conclusions and Outlook

In this study, two different scale-up strategies for HEK FreeStyle^TM 293-F suspension cells were quantitatively compared. On the one hand, the cultivation parameters proposed in Seidel et al. [26] were utilized to scale-up from a 4 L working volume to a 30 L working volume by keeping the specific power input constant at $233 \mathrm{W} {\mathrm{m}}^{-3}$ (in addition to all scale-independent parameters). Even though a system designed for microbial cultivation was used, decent growth was observed, although it was significantly lower than at the laboratory scale (VCD${}_{\max ,\mathrm{D}-\mathrm{DCU}}$ = $5.02·{10}^6\mathrm{cells} {\mathrm{mL}}^{-1}$ and VCD${}_{\max ,\mathrm{Minifors}}$ = $5.77·{10}^6\mathrm{cells} {\mathrm{mL}}^{-1}$). On the other hand, a CFD-optimized stirrer design was used for the same D-DCU bioreactor, which showed a similar Kolmogorov length scale distribution as in the small scale. Through the automated optimization of the stirrer, a comparable cultivation could be carried out with respect to the maximum VCD (VCD${}_{\max ,\mathrm{D}-\mathrm{DCU},\mathrm{opt}.}$ = $5.60·{10}^6\mathrm{cells} {\mathrm{mL}}^{-1}$ and VCD${}_{\max ,\mathrm{Minifors}}$ = $5.77·{10}^6\mathrm{cells} {\mathrm{mL}}^{-1}$). Thus, it could be shown that in this case, the simple and frequently used scale-up strategy of constant specific power input did not work as well as the use of the same Kolmogorov length scale distribution. Furthermore, these results support the findings of Sandadi et al. [93], Nienow et al. [28] and Nienow [27] that today’s mammalian cell cultures are much less fragile in the applied chemically defined culture media than many authors still assume.

With the method presented here, further, and more complex optimizations for bioreactors can be carried out on an open-source basis. For example, the $k_{\mathrm{L}}a$ value could also be modeled using CFD coupled with population balance modeling and utilized as a scale-up criterion in combination with the specific power input, or the EDCF described by Böhm et al. [5]. In addition, it could also be interesting to perform transient simulations to investigate the mixing time as Wu et al. [38] did as a scale-up criterion, since this can be problematic especially for large reactors. Furthermore, not only the stirrer geometry but also the addition of a tracer can be optimized, which is also a challenge when scaling up bioreactors. Another possible application of the approach demonstrated in this publication is to create scale-down bioreactors with similar Kolmogorov length scale distributions by adjusting the laboratory scale stirrer designs to given large-scale bioreactor geometries. This would allow the cost-efficient bioprocess optimization at laboratory scale without changing the large-scale bioreactor design, which is not easily possible with single-use bioreactors or validated systems, for example.