An adjoint method in inverse problems of chromatography

Abstract

How to determine adsorption isotherms is an issue of significant importance in chromatography. A modern technique of obtaining adsorption isotherms is to solve an inverse problem so that the simulated batch separation coincides with actual experimental results. In this work, as well as the natural least-square approach, we consider a Kohn–Vogelius type formulation for the reconstruction of adsorption isotherms in chromatography, which converts the original boundary fitting problem into a domain fitting problem. Moreover, using the first momentum regularizing strategy, a new regularization algorithm for both the Equilibrium-Dispersive model and the Transport-Dispersive model is developed. The mass transfer resistance coefficients in the Transport-Dispersive model are also estimated by the proposed inverse method. The computation of the gradients of objective functions for both of the two models is derived by the adjoint method. Finally, numerical simulations for both a synthetic problem and a real-world problem are given to show the robustness of the proposed algorithm.

Keywords:

• Chromatography
• adsorption isotherm
• inverse problem
• regularization
• convection–diffusion equation
• adjoint method

AMS Subject Classifications:

• 35R25
• 35R30
• 65M32
• 65M08
• 65K15

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 2014 project ‘SOMI: Studies of Molecular Interactions for Quality Assurance, Bio-Specific Measurement & Reliable Supercritical Purification’ [grant number 20140179], by (ii) the ÅForsk Foundation in the project ‘Improved Purification Procedures to Satisfy Modern Drug Quality Assurance and Environ-mental Criteria’ [grant number 15/497] and by (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 NSCF [grant number 11571311].