A regularizing Kohn–Vogelius formulation for the model-free adsorption isotherm estimation problem in chromatography

Abstract

Competitive adsorption isotherms must be estimated in order to simulate and optimize modern continuous modes of chromatography in situations where experimental trial-and-error approaches are too complex and expensive. The inverse method is a numeric approach for the fast estimation of adsorption isotherms directly from overloaded elution profiles. However, this identification process is usually ill-posed. Moreover, traditional model-based inverse methods are restricted by the need to choose an appropriate adsorption isotherm model prior to estimate, which might be very hard for complicated adsorption behavior. In this study, we develop a Kohn–Vogelius formulation for the model-free adsorption isotherm estimation problem. The solvability and convergence for the proposed inverse method are studied. In particular, using a problem-adapted adjoint, we obtain a convergence rate under substantially weaker and more realistic conditions than are required by the general theory. Based on the adjoint technique, a numerical algorithm for solving the proposed optimization problem is developed. Numerical tests for both synthetic and real-world problems are given to show the efficiency of the proposed regularization method.

Keywords:

• Chromatography
• adsorption isotherm
• inverse problem
• Kohn–Vogelius method
• convergence rate

AMS Subject Classifications:

• 65L09
• 35R30
• 76R50

Acknowledgements

We express our gratitude to the anonymous reviewers whose valuable comments and suggestions lead to an improvement of the manuscript.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by (i) the Swedish Knowledge Foundation for the KK HÖG 2015 project Predictive Separation of Biopharmaceuticals [grant number 20150233], by (ii) the ÅForsk Foundation in the project Improved Purification Procedures to Satisfy Modern Drug Quality Assurance and Environmental Criteria [grant number 15/497]; (iii) the Swedish Research Council (VR) in the project Fundamental Studies on Molecular Interactions aimed at Preparative Separations and Biospecific Measurements [grant number 2015-04627]. G. Lin and X. Cheng gratefully acknowledge the financial support from STINT [grant number IB2015-5989] and NSFC [grant number 11571311].