Analysis of volumetric mass transfer coefficient (k_La) in small- (250 mL) to large-scale (2500 L) orbitally shaken bioreactors

Abstract

In this study, the combination of dimensional analysis (DA) and analysis of variance (ANOVA) was used to predict the volumetric mass transfer coefficient (k_La) values under different operating conditions for orbitally shaken bioreactors (OSRs) with different filling volumes. It was found that Reynolds number and the interaction between Froude number and geometric number have the largest impact on k_La with impact indexes of 7.41 and 7.50, respectively. Moreover, the volume number has the largest negative impact on k_La, with an impact index of − 5.34. Thus, an effective way to increase the oxygen supply is by increasing the shaking speed and shaking diameter or decreasing the vessel diameter. However, cell cultivation with a higher filling volume will have an increased risk of oxygen scarcity. Therefore, with the help of the k_La prediction model, a suitable operating condition can be determined effectively and easily.

Introduction

Orbitally shaken bioreactors (OSRs) are used for the cultivation of animal and plant cells in suspension at small- and large-scales (Liu and Hong 2001; De Jesus et al. 2004; Muller et al. 2005; Klockner et al. 2014a, b; Raven et al. 2015). OSRs are a promising alternative to stirred-tank bioreactors (STR) or wave-agitated systems, particularly for applications in large volumetric scales with suspension-adapted animal cells (Liu et al. 2001; Muller et al. 2005; Stettler et al. 2007; Hacker et al. 2009; Zhang et al. 2009a, b; Tissot et al. 2011; Monteil et al. 2013; Klockner et al. 2014a, b; Raven et al. 2015). Furthermore, a prototype 2500-L OSR has recently been used for the suspension cultivation of Chinese hamster ovary cells (Zhang et al. 2009a, b). OSRs have inherent advantages over other bioreactor systems because they use the same agitation principle at any scale and the liquid flow is not obstructed because they lack impellers and baffles. This makes it likely that the liquid distribution pattern and key engineering parameters are easier to measure or calculate in OSRs than in other bioreactor systems. Consequently, the scale-up process in OSRs is expected to be more predictable than that in other bioreactor systems (Liu and Hong 2001; Zhang et al. 2009a, b; Tissot et al. 2011; Klockner et al. 2014a, b).

Sufficient oxygen transfer is a fundamental requirement for cultures in a bioreactor (Klockner and Büchs 2012). The volumetric mass transfer coefficient of oxygen (k_La) is commonly used as a critical scale-up parameter for bioprocesses, as it represents the general gas transfer efficiency of a bioreactor (Xing et al. 2009). A considerable database is available for k_La for various STRs and airlift and bubble column bioreactors, and several empirical equations predicting the k_La for these systems have been established (Garcia-Ochoa and Gomez 2009), but such equations have not been established for OSRs (Weheliye et al. 2013). We have previously measured the k_La values for OSRs at a variety of volumetric scales and attempted to use k_La as the main parameter for scaling-up OSR-based bioprocesses (Zhang et al. 2008; Zhang et al. 2009a, b; Xing et al. 2009). Here, we describe the development of a quantitative correlation between k_La and operating conditions in 250 mL–2500 L OSRs.

Materials and methods

Description of OSRs

We used 250 mL, 500 mL, 1 L, 2 L, and 5 L cylindrical glass bottles each with a slightly opened GL45 screw cap (1/4 turn), which allowed gas exchange in a diffusive manner while maintaining a sufficiently efficient sterility barrier (Corning; New York, USA). Additionally, we used the commercially available disposable OSR models SB50-X, SB200-X, and the prototype SB2500-X (Kühner AG; Birsfelden, Switzerland). The disposable OSRs required single-use 50, 200, and 2500 L cultivation bags (CultiBag^® ORB; Sartorius AG; Goettingen, Germany). The OSR dimensions and their respective displacement diameters during orbital shaking are shown in Table 1. All experiments were performed at 37 °C.

Table 1.

Vessel dimensions and shaking diameters

Working volume	Height	Diameter	Shaking diameter
250 mL	138 mm	70 mm	20–50 mm
500 mL	176 mm	86 mm	20–50 mm
1 L	225 mm	101 mm	20–50 mm
2 L	262 mm	136 mm	20–50 mm
5 L	335 mm	186 mm	20–50 mm
50 L	550 mm	460 mm	50 mm
200 L	0.821 m	0.754 m	50 mm
2000 L	1.85 m	1.5 m	100 mm

Different scales of orbitally shaken bioreactors were analyzed. The corresponding filling volumes ranged from 250 mL to 2000 L and shaking diameters for different scales changed from 20 to 100 mm

Static gassing-out method for measuring k_La

The standard static gassing-out method was applied to determine the k_La of OSRs filled with water, as previously described (Monteil et al. 2013). Nitrogen was pumped into the vessel until the air saturation in the liquid reached a constant value close to zero. Then, the OSR was supplied with air to replace the nitrogen. All measurements were performed while the shaker motor with a defined shaking diameter (d₀) was set at a specific speed. OSRs with a minimum volume of 50 L were actively aerated by pumping 10 mL air/L/minute into the head-space. Using a noninvasive optical sensor spot (PreSens GmbH; Regensburg, Germany), the dissolved oxygen concentration was recorded online. The k_La was calculated using the following mass balance equation (see Fig. 1):

$$ln\frac{(C_{\text{L}}^{\ast }-C_1)}{(C_{\text{L}}^{\ast }-C_2)}=k_{\text{L}}a\times (t_2-t_1),$$ 1

Fig. 1.

Containers used as orbitally shaken bioreactors (OSRs) in this study. Experiments determining the volumetric mass transfer coefficient (k_La) were carried out at different scales of OSRs with different filling volumes and shaking speeds. For the 250 mL, 500 mL, 1 L, 2 L, and 5 L glass vessels used as OSRs, the filling volumes varied from 50–150 mL, 100–300 mL, 200–600 mL, 250–700 mL, and 625 mL–3 L, respectively, and the shaking speeds varied from 70–110 rpm, 110–150 rpm, 70–220 rpm, 100–160 rpm, and 70–220 rpm, respectively. For the 50 L, 200 L, and 2500 L OSRs, the filling volumes varied from 10–20 L, 50–150 L, and 1000–2000 L, respectively, and the shaking speeds varied from 70–120 rpm, 60–75 rpm, and 35–55 rpm, respectively

where $C_{\text{L}}^{\ast }$ (mol/L) is the oxygen saturation concentration in the liquid and C₁ (mol/L) and C₂ (mol/L) are the measured dissolved oxygen concentrations at times t₁ (s) and t₂ (s), respectively. A schematic diagram of the experimental system is shown in Fig. 2.

Fig. 2.

Experimental setup of the static gassing-out method for determining the volumetric mass transfer coefficient (k_La). The orbitally shaken bioreactors (OSRs) were shaken on a platform connected to a shaker motor. The distance between the attachment point of the platform and the center axis of the shaker motor represents half the shaking diameter (= displacement radius). The OSRs contained deionized water maintained at 37 °C. A noninvasive submerged optical sensor was pasted to the OSR inner wall to measure the oxygen concentration in the liquid. To measure the k_La, nitrogen was pumped into the liquid to decrease the oxygen concentration in the liquid, then oxygen was supplied instead of nitrogen. The changing curve of oxygen saturation in the liquid was recorded by the optical sensor to calculate k_La value

Dimensional analysis (DA)

DA (Jensen 2013) is an effective approach for determining the correlation among variables. Here, Buckingham’s π-Theorem (Curtis et al. 1982) was used to analyze the factors influencing k_La ($\frac{1}{s}$), including filling volume (V_L; m³), vessel diameter (d; m), shaking diameter (d₀; m), shaking speed (n; $\frac{1}{s}$), gravitational acceleration (g; $\frac{m}{s^2}$), and kinematic viscosity ($\nu ;\frac{m^2}{s}$). Considering that the diffusion coefficient for oxygen ($D_{{\text{O}}_2};\frac{m^2}{s}$) is constant at a fixed temperature (37 °C) (Hanand Bartels 1996), the effect of $D_{{\text{O}}_2}$ on k_La was not taken into account in this analysis. According to Buckingham’s π-Theorem, five dimensionless numbers were obtained as previously suggested (Klockner et al. 2013):

$$\text{Mass transfer number given by}{k}_{\text{L}}a\times {\left(\frac{\upsilon }{g^2}\right)}^{\frac{1}{3}}$$

$$\text{Froude number}\left(\text{Fr}\right)\text{given by}\frac{n^2\times d_0}{g}$$

$$\text{Volume number}\left(V_{\text{L}}\right)\text{given by}\frac{V_{\text{L}}}{d^3}$$

$$\text{Geometric number}\left(\text{Ge}\right)\text{given by}\frac{d_0}{d}$$

$$\text{Reynolds number}\left(\text{Re}\right)\text{given by}\frac{n{d}^2}{\upsilon }.$$

According to the DA theory, a function that could correlate these dimensionless numbers exists (Eq. 2):

$$F\left(k_{\text{L}}a\times {\left(\frac{\upsilon }{g^2}\right)}^{\frac{1}{3}},\frac{n^2·d_0}{g},\frac{V_{\text{L}}}{d^3},\frac{d_0}{d},\frac{n{d}^2}{\upsilon }\right)=0\iff k_{\text{L}}a\times {\left(\frac{\upsilon }{g^2}\right)}^{\frac{1}{3}}=f\left(\frac{n^2\times d_0}{g},\frac{V_{\text{L}}}{d^3},\frac{d_0}{d},\frac{n{d}^2}{\upsilon }\right).$$ 2

A power law function was proposed to analyze the effects of these dimensionless numbers on k_La (Eq. 3):

$$k_{\text{L}}a\times {\left(\frac{\upsilon }{g^2}\right)}^{\frac{1}{3}}={\left(\frac{n^2\times d_0}{g}\right)}^{\alpha }\times {\left(\frac{V_{\text{L}}}{d^3}\right)}^{\beta }\times {\left(\frac{d_0}{d}\right)}^{\gamma }\times {\left(\frac{n{d}^2}{\upsilon }\right)}^{\delta }.$$ 3

A critical minimal shaking speed (N_C) is needed to efficiently circulate the liquid in the vessel (Klöckner et al. 2012; Reclari et al. 2014). The N_C was calculated using Eq. 4:

$$N_{\text{C}}=\frac{1}{d^2}\times \sqrt{0.28\times V_{\text{L}}\times g}.$$ 4

Regression analysis

Based on the DA results, MLR analysis was used to investigate the effects of the dimensionless numbers on k_La. Considering that the effect of V_L was independent of those of other factors, the interactions involving V_L were removed (Eq. (5)):

$$\begin{array}{cc}\hfill Y& =a_0+a_1{x}_1+a_2{x}_2+a_3{x}_3+a_4{x}_4\hfill \\ \hfill & +a_{13}{x}_1{x}_3+a_{14}{x}_1{x}_4+a_{34}{x}_3{x}_4+\epsilon ,\hfill \\ \hfill \end{array}$$ 5

where Y, x₁, x₂, x₃, x₄, represent the response (k_La), Fr, V_L, Ge, and Re, respectively. Moreover, a₀, a₁, a₂, a₃, a₄, a₁₃, a₁₄, a₃₄, and ε represent a constant term, coefficient of Fr, coefficient of V_L, coefficient of Ge, coefficient of Re, coefficient of interaction between Fr and Ge, coefficient of interaction between Fr and Re, coefficient of interaction between Ge and Re, and experimental error, respectively.

The matrix of X is represented as follows:

$$X=\left(\begin{array}{cccccccc}1& x_{11}& x_{12}& x_{13}& x_{14}& x_{11}{x}_{13}& x_{11}{x}_{14}& x_{13}{x}_{14}\\ 1& x_{21}& x_{22}& x_{23}& x_{24}& x_{21}{x}_{23}& x_{21}{x}_{24}& x_{23}{x}_{24}\\ ·& ·& ·& ·& ·& ·& ·& ·\\ ·& ·& ·& ·& ·& ·& ·& ·\\ 1& x_{N1}& x_{N2}& x_{N3}& x_{N4}& x_{N1}{x}_{N3}& x_{N1}{x}_{N4}& x_{N3}{x}_{N4}\end{array}\right),$$ 6

where N represents the number of experiments. x₁₁ is the value of x₁ in the first-round experiment and x₁₁x₁₃ denotes the product of x₁ and x₃ in the same experiment. In this study, N = 84, indicating 84 random experimental runs were executed.

Results

Results of DA

All k_La values were measured at shaking speeds higher than the critical minimal value. Applying logarithm on both sides of (Eq. (3)), an equivalent form was obtained (Eq. 7):

$$\begin{array}{cc}\hfill ln\left[k_{\text{L}}a\times {\left(\frac{\upsilon }{g^2}\right)}^{\frac{1}{3}}\right]& =\alpha ln\left(\frac{n^2\times d_0}{g}\right)+\beta ln\left(\frac{V_{\text{L}}}{d^3}\right)\hfill \\ \hfill & +\gamma ln\left(\frac{d_0}{d}\right)+\delta ln\left(\frac{n{d}^2}{\upsilon }\right).\hfill \\ \hfill \end{array}$$ 7

For each OSR, the effect of V_L was determined by changing the filling volume, while keeping all other parameters constant. The average exponent ($\overline{\beta }$) was found to be about − 1.28 (Fig. 3), which matched the previously published results (Klockner et al. 2013). The valid V_L range was between 0.091 and 1.14. However, it was difficult to determine the exponents of other dimensionless numbers similarly because it was impractical to change them independent of other dimensionless numbers. To analyze the effect of other dimensionless numbers, we attempted to solve two effect exponents together. Unfortunately, the algebraic equations were found to be ill-conditioned, which indicated highly unreasonable solutions for these equations (data not shown).

Fig. 3.

The relationship between volumetric mass transfer coefficient (k_La) and volume number for four orbitally shaken bioreactors (OSRs) with different working volumes at different shaking speeds. The k_La values were determined using the static gassing-out method using three or four different filling volumes for each OSR (250 mL, 1 L, 5 L, and 2500 L). For each OSR, the filling volumes were changed, keeping all other operating conditions constant and the temperature was maintained at 37 °C. The average slope for the obtained data in the four OSRs with different volumes was about − 1.28

Results of the multi-linear regression analysis

Evaluation of the experimental design

Dispersion matrix

The dispersion matrix (Eq. 8) represents the transfer of variance between the factors and coefficients (Ma and Genton 2001). All the main diagonal elements of this matrix should be small to avoid variance amplification. As Table 2 shows, no diagonal elements > 1 were found, indicating that the variance was not amplified between the factors and coefficients.

$${\upsilon }_{\text{ij}}={(X^{\text{T}}\times X)}^{-1},$$ 8

Table 2.

The matrix of dispersion

	a0	a1	a2	a3	a4	a1a3	a1a4	a3a4
a0	0.02	0.21	0.08	0.01	− 0.03	0.04	0.03	− 0.02
a0	0.21	0.27	− 0.01	0.17	− 0.04	0.16	0.04	0.03
a0	0.08	− 0.01	0.13	0.01	0.02	0.01	0.06	− 0.01
a0	0.01	0.17	0.01	0.11	0.10	0.02	0.16	− 0.03
a0	− 0.03	− 0.04	0.02	0.10	0.08	0.14	0.02	− 0.04
a0	0.04	0.16	0.01	0.02	0.14	0.03	0.35	− 0.04
a0	0.03	0.04	0.06	0.16	0.02	0.35	0.17	0.08
a0	− 0.02	0.03	− 0.01	− 0.03	− 0.04	− 0.04	0.08	0.20

The dispersion matrix for all the influencing coefficients was determined. The data located on a matrix diagonal represent the dispersion coefficient values

Bold values indicate the amplification level of the variance from the coefficients to the factors

where ${\upsilon }_{\text{ij}}$ is the element of the dispersion matrix, and i, j = 1, 2, 3, 4

Variance inflation factors (VIFs)

The VIF predicts the accuracy of coefficient estimation in a model (O’brien 2007; Miles 2014). For example, a VIF close to 1 indicates good coefficient estimation, while a VIF larger than 10 indicates that the estimated coefficient was disturbed by experimental errors. As Table 3 shows, all VIF values were less than 10, indicating that coefficient estimation seemed to be good.

Table 3.

Diagonal elements of the variance inflation factors

Coefficient	a0	a1	a2	a3	a4	a1a3	a1a4	a3a4
VIF	8.2	9.3	2.4	8.6	3.4	9.1	2.6	2.0

The VIFs were calculated using Excel tool

Analysis of variance (ANOVA)

ANOVA (Cohen and Cohen 2008; Henson 2015) of the k_La prediction model was applied to evaluate the significance of factors affecting k_La. The P value is a crucial value estimated using ANOVA, and values larger than 0.05 imply that the corresponding effects of factors are too insignificant to be considered (Gelman 2013). Given the high P value for interaction between Fr and Re (0.42; Table 4), the interaction was probably not reasonable. Thus, it was acceptable to remove x₁₄ from the k_La prediction model. Consequently, an improved model without x₁₄ was formed, and a series of new P values (P* values) were obtained (Table 4). All the P* values were smaller than 0.05 in the improved k_La model and the factor coefficients were determined using the following equation (Eq. 9):

$$\overrightarrow{a}={(X^{\text{T}}\times X)}^{-1}{X}^{\text{T}}\times Y$$ 9

Table 4.

Analysis of varianceANOVA table of the k_La predicting equation

Factors	Coefficients	P values	P* values
Constant term	7.07	< 0.01%	< 0.01%
Froude number (Fr)	4.21	< 0.01%	< 0.01%
Volume number (VL)	− 5.34	< 0.01%	< 0.01%
Geometric number (Ge)	2.14	0.05%	0.05%
Reynolds number (Re)	7.41	< 0.01%	< 0.01%
Interaction of Fr and Ge	7.5	4.04%	4.05%
Interaction of Fr and Re	–	42.26%	–
Interaction of Ge and Re	− 3.46	1.07%	0.83%

The values of the influencing coefficients were predicted using ANOVA. The P and P*-values were also calculated

The final result is shown in Eq. (10) and the valid ranges for Fr, V_L, Ge, and Re are $0.38<\frac{n^2\times d_0}{g}<9.85$, $0.091<\frac{V_{\text{L}}}{d^3}<1.138$, $1.4<\frac{d_0}{d}<15$, and $7222<\frac{n{d}^2}{\upsilon }<82752$, respectively.

Consequently, the calculated k_La values were obtained using Eq. (10)

$$\begin{array}{cc}\hfill ln\left[k_{\text{L}}a\times {\left(\frac{\upsilon }{g^2}\right)}^{\frac{1}{3}}\right]& =7.\text{07 + 4.21ln}\left(\frac{n^2\times d_0}{g}\right)+5.\text{34ln}\left(\frac{V_{\text{L}}}{d^3}\right)\hfill \\ \hfill & +2.\text{14ln}\left(\frac{d_0}{d}\right)+7.\text{41}ln\left(\frac{n{d}^2}{\upsilon }\right)+7.\text{50ln}\left(\frac{n^2{d}_0^2}{g\times d}\right)-3.\text{46ln}\left(\frac{n{d}_{0}d}{\upsilon }\right)\hfill \\ \hfill \end{array}$$ 10

where the unit of k_La was 1/h.

The calculated k_La values were compared to those measured experimentally. As shown in Fig. 4, the various OSR k_La values were located based on calculated and measured values, that is, those symbols that are on or near the black line indicate nearly identical measured and calculated k_La values. Most predicted k_La values, up to a value of 15/h, appeared to match the experimental results with only a small deviation. The experimental results highly deviated from the predicted k_La values for the 500 and 250 mL OSRs, when operated under conditions where the k_La values were higher than 15/h. We assume that the results under these conditions are due to experimental errors caused by technical difficulties in the experimental setup.

Fig. 4.

Comparison between measured and calculated volumetric mass transfer coefficient (k_La) values at 37 °C for various orbitally shaken bioreactors (OSRs). The measured and calculated k_La values were determined using the static gassing-out method and equation predicting k_La, respectively. The coordinates of the points represent the calculated and measured k_La values, respectively. The points on the solid line indicate that the calculated k_La value was equal to the measured value. The experiment was repeated 100 times, which was different from those used in the model

Discussion

DA and ANOVA were used to find a model for predicting the k_La values of OSRs with different volumetric scales. All potential factors were sorted into groups (dimensionless numbers) using DA. V_L was found to influence k_La with an effect exponent of − 1.28 in a power law function. This result was reasonable because resistance (gravity) against orbital shaking increases with filling volume increase. Moreover, the driving force (centrifugal force) for moving the liquid decreases with vessel diameter decrease. Thus, large filling volume or small vessel diameter will produce an unfavorable oxygen transfer in OSRs. Considering the drawbacks of DA, MLR was applied to support the correction between dimensionless numbers and k_La. By analyzing the dispersion matrix and VIFs, the chosen factors were of good quality. Fr and Re affect the k_La value independently, which might be the reason for the insignificant effect of interaction between Fr and Re on k_La. The interaction between Fr and Ge had the largest positive influence on k_La. Re also generated a strong positive influence on k_La with a coefficient of 7.41. As Re is a key indicator for flowing state, it played an important role in predicting the k_La value. Thus, an efficient way to increase the OTR is to increase both Fr and Ge. However, increasing the volume number or interaction between Re and Ge will negatively influence the k_La value. The k_La prediction model is a promising application for OSR scale-up and saves time in the determination of k_La. However, further work is needed to improve the model, especially for OSRs of volumetric scales other than those described here.

Compliance with ethical standards

Conflict of interest

The authors confirm that there are no conflicts of interest.