# Multivariate Parameter Determination of Multi-Component Isotherms for Chromatography Digital Twins

^{1}

^{2}

^{3}

^{*}

*Processes*in 2023)

## Abstract

**:**

## 1. Introduction

## 2. Modelling Chromatography: General Rate Model

_{p,i}and surface diffusion D

_{S,i}[30,33]:

_{eff}[10,33].

_{i}. All notations can be transferred into the other with:

## 3. Materials and Methods

#### 3.1. Feed, Buffer Components and Columns

#### 3.2. Preparative Chromatography

#### 3.3. Software

## 4. Design of Experiments

- Counter ion: 25–100 mM;
- Buffer: 50–150 mM;
- pH: 3.35–3.7.

- A high counter ion (CI) value; and
- A high pH value at a low buffer concentration; or
- A low pH value at a high buffer concentration.

## 5. Parameter Extraction

_{1,i}, a

_{2,1}, b

_{1,i}and b

_{2,i}. These factors influence the Langmuir parameters Henry coefficient H

_{i}and maximum loading capacity q

_{max,I}depending on the modifier concentration:

_{i}is the component dependent linear factor. The latter should be determinable directly with JMP. To do so, ${a}_{1,i}$, ${a}_{2,i}$, ${b}_{1,i}$ and ${b}_{2,i}$ were added as new target values to the DoE. The factors themselves were obtained by fitting the simulation results to the experimental chromatograms. Again, the linear correlations looked very good. The p-values were between 0.003 and 0.0009, R² was above 0.98 and the Root Mean Square Errors were also very low. All the values can be found in Table 2. The observed vs. predicted plots are given in Figure 3, exemplified with b1 and b2. Again, the confidence intervals were narrow. An interesting observation can be undertaken for data point (---). Despite being on the line with the others, it is relatively far outside.

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

${c}_{i}$ | (g/L) | Concentration of component i |

${c}_{p,i}$ | (g/L) | Concentration of component i inside the pores |

CTCC | Continuous Twin Column Chromatography | |

CV | Column Volume | |

${D}_{ax}$ | (cm²/s) | Axial dispersion coefficient |

D_{eff} | (cm²/s) | Effective diffusion coefficient |

${D}_{m,i}$ | (cm²/s) | Molecular diffusion coefficient |

${d}_{p}$ | (cm) | Particle diameter |

D_{p,i} | (cm²/s) | Pore diffusion coefficient |

D_{S,i} | (cm²/s) | Surface diffusion coefficient |

DoE | Design of Experiments | |

${\epsilon}_{p,i}$ | (-) | Porosity |

${\epsilon}_{s}$ | (-) | Voidage |

H_{i} | (-) | Henry coefficient of component i |

K_{i} | (l/g) | Langmuir coefficient of component i |

${k}_{eff}$ | (cm/s) | Effective mass transport coefficient |

k_{f} | (cm/s) | Mass transport coefficient |

l | (cm) | Length |

MCSGP | Multicolumn Countercurrent Solvent Gradient Purification | |

PAT | Process Analytical Technology | |

$P{e}_{i}$ | (-) | Peclet-Number |

q_{i} | (g/L) | Loading of component i |

q_{max,i} | (g/L) | Maximum loading capacity of component i |

r | (cm) | Radius |

Re | (-) | Reynolds-Number |

RMSE | Root Mean Square Error | |

${R}_{p}$ | (cm) | Particle Radius |

$S{h}_{i}$ | (-) | Sherwood-Number |

t | (s); (min) | Time |

$\overline{{t}_{i}}$ | (s); (min) | Mean residence time |

${u}_{int}$ | (cm/s) | Interstitial velocity |

v | (cm/s) | Velocity |

$\dot{V}$ | (mL/min) | Volumetric flow |

${V}_{column}$ | (mL) | Volume of column |

$\eta $ | (mg/cm∗s) | Dynamic viscosity |

$\rho $ | (g/L) | Density |

${\sigma}^{2}$ | (s²) | Variance |

## References

- Guiochon, G.; Felinger, A.; Shirazi, D.G.; Katti, A.M. Fundamentals of Preparative and Nonlinear Chromatography, 2nd ed.; Elsevier Academic Press: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Strube, J. Technische Chromatographie: Auslegung, Optimierung, Betrieb und Wirtschaftlichkeit; Univ., Habil.-Schr.–Dortmund, Als Ms. gedr; Shaker: Aachen, Germany, 1999; ISBN 3826568974. [Google Scholar]
- Altenhöner, U.; Meurer, M.; Strube, J.; Schmidt-Traub, H. Parameter estimation for the simulation of liquid chromatography. J. Chromatogr. A
**1997**, 769, 59–69. [Google Scholar] [CrossRef] - Felinger, A. (Ed.) 3 Models of chromatography. In Data Analysis and Signal Processing in Chromatography; Elsevier: Amsterdam, The Netherlands, 1998; pp. 43–78. ISBN 9780444820662. [Google Scholar]
- Zobel-Roos, S. Entwicklung, Modellierung und Validierung von Integrierten Kontinuierlichen Gegenstrom-Chromatographie-Prozessen; Shaker: Herzogenrath, Germany, 2018; ISBN 3844061878. [Google Scholar]
- Seidel-Morgenstern, A. Modeling of Chromatographic Processes. In Preparative Chromatography, 3rd ed.; Schmidt-Traub, H., Schulte, M., Seidel-Morgenstern, A., Eds.; WILEY-VCH: Weinheim, Germany, 2020; pp. 311–354. ISBN 9783527344864. [Google Scholar]
- Levenspiel, O. Chemical Reaction Engineering, 3rd ed.; Wiley: New York, NY, USA, 1999; ISBN 9780471254249. [Google Scholar]
- Hejtmánek, V.; Schneider, P. Axial dispersion under liquid-chromatography conditions. Chem. Eng. Sci.
**1993**, 48, 1163–1168. [Google Scholar] [CrossRef] - Tallarek, U.; Albert, K.; Bayer, E. Measurement of transverse and axial apparent dispersion coefficients in packed beds. AIChE J.
**1996**, 42, 3041–3054. [Google Scholar] [CrossRef] - Carta, G.; Jungbauer, A. Protein Chromatography: Process Development and Scale-Up; WILEY-VCH: Weinheim, Germany, 2010; ISBN 978-3-527-31819-3. [Google Scholar]
- Seidel-Morgenstern, A. Experimental determination of single solute and competitive adsorption isotherms. J. Chromatogr. A
**2004**, 1037, 255–272. [Google Scholar] [CrossRef] [PubMed] - Hahn, T.; Huuk, T.; Heuveline, V.; Hubbuch, J. Simulating and Optimizing Preparative Protein Chromatography with ChromX. J. Chem. Educ.
**2015**, 92, 1497–1502. [Google Scholar] [CrossRef] - Osberghaus, A.; Hepbildikler, S.; Nath, S.; Haindl, M.; von Lieres, E.; Hubbuch, J. Determination of parameters for the steric mass action model—A comparison between two approaches. J. Chromatogr. A
**2012**, 1233, 54–65. [Google Scholar] [CrossRef] - Huuk, T.C.; Hahn, T.; Osberghaus, A.; Hubbuch, J. Model-based integrated optimization and evaluation of a multi-step ion exchange chromatography. Sep. Purif. Technol.
**2014**, 136, 207–222. [Google Scholar] [CrossRef] - Mouellef, M.; Vetter, F.L.; Zobel-Roos, S.; Strube, J. Fast and Versatile Chromatography Process Design and Operation Optimization with the Aid of Artificial Intelligence. Processes
**2021**, 9, 2121. [Google Scholar] [CrossRef] - Wang, G.; Briskot, T.; Hahn, T.; Baumann, P.; Hubbuch, J. Estimation of adsorption isotherm and mass transfer parameters in protein chromatography using artificial neural networks. J. Chromatogr. A
**2017**, 1487, 211–217. [Google Scholar] [CrossRef] - Gao, W.; Engell, S. Neural Network-Based Identification of Nonlinear Adsorption Isotherms. IFAC Proc. Vol.
**2004**, 37, 721–726. [Google Scholar] [CrossRef] - Kreusser, J.; Jirasek, F.; Hasse, H. Influence of pH value and salts on the adsorption of lysozyme in mixed-mode chromatography. Eng. Life Sci.
**2021**, 21, 753–768. [Google Scholar] [CrossRef] - Zhu, M.; Carta, G. Protein adsorption equilibrium and kinetics in multimodal cation exchange resins. Adsorption
**2016**, 22, 165–179. [Google Scholar] [CrossRef] - Schmidt, A.; Zobel-Roos, S.; Helgers, H.; Lohmann, L.; Vetter, F.; Jensch, C.; Juckers, A.; Strube, J. Digital Twins for Continuous Biologics Manufacturing. In Process Control, Intensification, and Digitalisation in Continuous Biomanufacturing; Subramanian, G., Ed.; Wiley: Hoboken, NJ, USA, 2022; pp. 265–350. ISBN 9783527347698. [Google Scholar]
- Vetter, F.L.; Zobel-Roos, S.; Mota, J.P.B.; Nilsson, B.; Schmidt, A.; Strube, J. Toward Autonomous Production of mRNA-Therapeutics in the Light of Advanced Process Control and Traditional Control Strategies for Chromatography. Processes
**2022**, 10, 1868. [Google Scholar] [CrossRef] - Baumann, P.; Huuk, T.; Hahn, T.; Osberghaus, A.; Hubbuch, J. Deconvolution of high-throughput multicomponent isotherms using multivariate data analysis of protein spectra. Eng. Life Sci.
**2016**, 16, 194–201. [Google Scholar] [CrossRef] - Field, N.; Konstantinidis, S.; Velayudhan, A. High-throughput investigation of single and binary protein adsorption isotherms in anion exchange chromatography employing multivariate analysis. J. Chromatogr. A
**2017**, 1510, 13–24. [Google Scholar] [CrossRef] - Zobel-Roos, S.; Mouellef, M.; Ditz, R.; Strube, J. Distinct and Quantitative Validation Method for Predictive Process Modelling in Preparative Chromatography of Synthetic and Bio-Based Feed Mixtures Following a Quality-by-Design (QbD) Approach. Processes
**2019**, 7, 580. [Google Scholar] [CrossRef] - Zobel-Roos, S.; Vetter, F.L.; Scheps, D.; Pfeiffer, M.; Gunne, M.; Boscheinen, O.; Strube, J. Digital Twin Based Design and Experimental Validation of a Continuous Peptide Polishing Step. Processes
**2023**, 11, 1401. [Google Scholar] [CrossRef] - Zobel-Roos, S.; Mouellef, M.; Siemers, C.; Strube, J. Process Analytical Approach towards Quality Controlled Process Automation for the Downstream of Protein Mixtures by Inline Concentration Measurements Based on Ultraviolet/Visible Light (UV/VIS) Spectral Analysis. Antibodies
**2017**, 6, 24. [Google Scholar] [CrossRef] - Vetter, F.L.; Zobel-Roos, S.; Strube, J. PAT for Continuous Chromatography Integrated into Continuous Manufacturing of Biologics towards Autonomous Operation. Processes
**2021**, 9, 472. [Google Scholar] [CrossRef] - Uhl, A.; Schmidt, A.; Hlawitschka, M.W.; Strube, J. Autonomous Liquid–Liquid Extraction Operation in Biologics Manufacturing with Aid of a Digital Twin including Process Analytical Technology. Processes
**2023**, 11, 553. [Google Scholar] [CrossRef] - Mouellef, M.; Vetter, F.L.; Strube, J. Benefits and Limitations of Artificial Neural Networks in Process Chromatography Design and Operation. Processes
**2023**, 11, 1115. [Google Scholar] [CrossRef] - Kaczmarski, K.; Cavazzini, A.; Szabelski, P.; Zhou, D.; Liu, X.; Guiochon, G. Application of the general rate model and the generalized Maxwell–Stefan equation to the study of the mass transfer kinetics of a pair of enantiomers. J. Chromatogr. A
**2002**, 962, 57–67. [Google Scholar] [CrossRef] [PubMed] - Kaczmarski, K.; Gubernak, M.; Zhou, D.; Guiochon, G. Application of the general rate model with the Maxwell–Stefan equations for the prediction of the band profiles of the 1-indanol enantiomers. Chem. Eng. Sci.
**2003**, 58, 2325–2338. [Google Scholar] [CrossRef] - Felinger, A.; Guiochon, G. Comparison of the Kinetic Models of Linear Chromatography. Chromatographia
**2004**, 60, S175–S180. [Google Scholar] [CrossRef] - Piątkowski, W.; Antos, D.; Kaczmarski, K. Modeling of preparative chromatography processes with slow intraparticle mass transport kinetics. J. Chromatogr. A
**2003**, 988, 219–231. [Google Scholar] [CrossRef] - Asnin, L. Adsorption models in chiral chromatography. J. Chromatogr. A
**2012**, 1269, 3–25. [Google Scholar] [CrossRef] - Blümel, C.; Kniep, H.; Seidel-Morgenstern, A. Measuring adsorption isotherms using a closed-loop perturbation method to minimize sample consumption. In Proceedings of the 6th International Conference of Fundamentals of Adsorption—FOA 6, Presqu’ile de Giens, France, 23–27 May 1998; Elsevier: Amsterdam, The Netherlands, 1998; pp. 449–454. [Google Scholar]
- Cavazzini, A.; Felinger, A.; Guiochon, G. Comparison between adsorption isotherm determination techniques and overloaded band profiles on four batches of monolithic columns. J. Chromatogr. A
**2003**, 1012, 139–149. [Google Scholar] [CrossRef] - Ching, C.B.; Chu, K.H.; Ruthven, D.M. A study of multicomponent adsorption equilibria by liquid chromatography. AIChE J.
**1990**, 36, 275–281. [Google Scholar] [CrossRef] - Gamba, G.; Rota, R.; Storti, G.; Carra, S.; Morbidelli, M. Absorbed solution theory models for multicomponent adsorption equilibria. AIChE J.
**1989**, 35, 959–966. [Google Scholar] [CrossRef] - Hu, X.; Do, D.D. Comparing various multicomponent adsorption equilibrium models. AIChE J.
**1995**, 41, 1585–1592. [Google Scholar] [CrossRef] - Heinonen, J.; Rubiera Landa, H.O.; Sainio, T.; Seidel-Morgenstern, A. Use of Adsorbed Solution theory to model competitive and co-operative sorption on elastic ion exchange resins. Sep. Purif. Technol.
**2012**, 95, 235–247. [Google Scholar] [CrossRef] - Emerton, D.A. Profitability in the Biosimilars Market: Can You Translate Scientific Excellence into a Healthy Commercial Return? BioProcess Int.
**2013**, 11, 6–23. [Google Scholar] - Erto, A.; Lancia, A.; Musmarra, D. A modelling analysis of PCE/TCE mixture adsorption based on Ideal Adsorbed Solution Theory. Sep. Purif. Technol.
**2011**, 80, 140–147. [Google Scholar] [CrossRef] - Myers, A.L.; Prausnitz, J.M. Thermodynamics of mixed-gas adsorption. AIChE J.
**1965**, 11, 121–127. [Google Scholar] [CrossRef] - Costa, E.; Calleja, G.; Marron, C.; Jimenez, A.; Pau, J. Equilibrium adsorption of methane, ethane, ethylene, and propylene and their mixtures on activated carbon. J. Chem. Eng. Data
**1989**, 34, 156–160. [Google Scholar] [CrossRef] - Brooks, C.A.; Cramer, S.M. Steric mass-action ion exchange: Displacement profiles and induced salt gradients. AIChE J.
**1992**, 38, 1969–1978. [Google Scholar] [CrossRef] - Langmuir, I. The adsorption of gases on plane surfaces of glass, mica and platinum. J. Am. Chem. Soc.
**1918**, 40, 1361–1403. [Google Scholar] [CrossRef]

**Figure 5.**Contour plots showing the influence of two factors on target values, normalized yield on the left and normalized productivity on the right. Top row: Buffer over counter ion, middle row: pH over counter ion, bottom row: pH over buffer. Due to the stepwise change in color instead of a steady color gradient, there were rounding errors leading to the display of more inflection points.

Factors | Target Values | ||||
---|---|---|---|---|---|

CI | Buffer | pH | Normalized Yield | Normalized Productivity | |

Pattern | [mM] | [mM] | [-] | [-] | [-] |

(000) | 62.5 | 100 | 3.53 | 1.01 | 1.00 |

(000) | 62.5 | 100 | 3.53 | 0.99 | 0.99 |

(000) | 62.5 | 100 | 3.53 | 1.00 | 1.00 |

(---) | 25 | 50 | 3.35 | 0.73 | 0.61 |

(--+) | 25 | 50 | 3.7 | 1.06 | 1.08 |

(-+-) | 25 | 150 | 3.35 | 1.09 | 1.08 |

(-++) | 25 | 150 | 3.7 | 0.93 | 1.02 |

(+--) | 100 | 50 | 3.35 | 0.93 | 0.91 |

(+-+) | 100 | 50 | 3.7 | 1.20 | 1.20 |

(++-) | 100 | 150 | 3.35 | 1.23 | 1.22 |

(+++) | 100 | 150 | 3.7 | 0.97 | 0.99 |

p | R² | RMSE | |
---|---|---|---|

a1 | 0.003 | 0.98 | 2.8 × 10^{−7} |

a2 | 0.0009 | 0.99 | 0.0494 |

b1 | 0.0023 | 0.98 | 0.007 |

b2 | 0.0006 | 0.99 | 0.0115 |

Simulation | Experiment | Deviation | ||||
---|---|---|---|---|---|---|

Pattern | Normalized Yield | Normalized Productivity | Normalized Yield | Normalized Productivity | Yield | Productivity |

[-] | [-] | [-] | [-] | [%] | [%] | |

(000)_1 | 1.00 | 1.00 | 1.11 | 1.08 | 9.88 | 7.32 |

(000)_2 | 1.00 | 1.00 | 1.09 | 1.07 | 8.65 | 6.70 |

(000)_3 | 1.00 | 1.00 | 1.10 | 1.08 | 9.12 | 7.59 |

(--+) | 1.24 | 1.23 | 1.17 | 1.16 | −5.51 | −5.60 |

(-+-) | 1.22 | 1.21 | 1.20 | 1.16 | −1.74 | −4.52 |

(-++) | 1.01 | 1.04 | 1.03 | 1.10 | 1.66 | 5.03 |

(+--) | 1.00 | 0.98 | 1.03 | 0.98 | 2.59 | −0.19 |

(+-+) | 1.29 | 1.27 | 1.32 | 1.29 | 1.91 | 1.81 |

(++-) | 1.28 | 1.26 | 1.35 | 1.32 | 5.11 | 4.23 |

(+++) | 1.02 | 1.03 | 1.06 | 1.07 | 3.89 | 3.26 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zobel-Roos, S.; Vetter, F.; Scheps, D.; Pfeiffer, M.; Gunne, M.; Boscheinen, O.; Strube, J.
Multivariate Parameter Determination of Multi-Component Isotherms for Chromatography Digital Twins. *Processes* **2023**, *11*, 1480.
https://doi.org/10.3390/pr11051480

**AMA Style**

Zobel-Roos S, Vetter F, Scheps D, Pfeiffer M, Gunne M, Boscheinen O, Strube J.
Multivariate Parameter Determination of Multi-Component Isotherms for Chromatography Digital Twins. *Processes*. 2023; 11(5):1480.
https://doi.org/10.3390/pr11051480

**Chicago/Turabian Style**

Zobel-Roos, Steffen, Florian Vetter, Daniel Scheps, Marcus Pfeiffer, Matthias Gunne, Oliver Boscheinen, and Jochen Strube.
2023. "Multivariate Parameter Determination of Multi-Component Isotherms for Chromatography Digital Twins" *Processes* 11, no. 5: 1480.
https://doi.org/10.3390/pr11051480