A Statistical Approach to Estimating Adsorption-Isotherm Parameters in Gradient-Elution Preparative Liquid Chromatography

Determining the adsorption isotherms is an issue of significant importance in preparative chromatography. A modern technique for estimating adsorption isotherms is to solve an inverse problem so that the simulated batch separation coincides with actual experimental results. However, due to the ill-posedness, the high non-linearity, and the uncertainty quantification of the corresponding physical model, the existing deterministic inversion methods are usually inefficient in real-world applications. To overcome these difficulties and study the uncertainties of the adsorption-isotherm parameters, in this work, based on the Bayesian sampling framework, we propose a statistical approach for estimating the adsorption isotherms in various chromatography systems. Two modified Markov chain Monte Carlo algorithms are developed for a numerical realization of our statistical approach. Numerical experiments with both synthetic and real data are conducted and described to show the efficiency of the proposed new method.

[1]  A. Seidel-Morgenstern,et al.  Frontal analysis method to determine competitive adsorption isotherms. , 2001, Journal of chromatography. A.

[2]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[3]  Jiguo Cao,et al.  Parameter Estimation of Partial Differential Equation Models , 2013, Journal of the American Statistical Association.

[4]  Georges Guiochon,et al.  Determination of the single component and competitive adsorption isotherms of the 1-indanol enantiomers by the inverse method. , 2003, Journal of chromatography. A.

[5]  A. Belloni,et al.  On the Computational Complexity of MCMC-Based Estimators in Large Samples , 2007, 0704.2167.

[6]  M. Gulliksson,et al.  An adjoint method in inverse problems of chromatography , 2017 .

[7]  M. Girolami,et al.  Bayesian Solution Uncertainty Quantification for Differential Equations , 2013 .

[8]  Mark A. Girolami,et al.  Bayesian Probabilistic Numerical Methods , 2017, SIAM Rev..

[9]  Douglas M. Ruthven,et al.  Principles of Adsorption and Adsorption Processes , 1984 .

[10]  R. Tweedie,et al.  Exponential convergence of Langevin distributions and their discrete approximations , 1996 .

[11]  Torgny Fornstedt,et al.  An improved algorithm for solving inverse problems in liquid chromatography , 2006, Comput. Chem. Eng..

[12]  Shamsul Qamar,et al.  Efficient and accurate numerical simulation of nonlinear chromatographic processes , 2011, Comput. Chem. Eng..

[13]  M. Gulliksson,et al.  A regularizing Kohn–Vogelius formulation for the model-free adsorption isotherm estimation problem in chromatography , 2018 .

[14]  Xiaoliang Cheng,et al.  A modified coupled complex boundary method for an inverse chromatography problem , 2018 .

[15]  Stephen C. Jacobson,et al.  Determination of isotherms from chromatographic peak shapes , 1991 .

[16]  Xiaoliang Cheng,et al.  A regularization method for the reconstruction of adsorption isotherms in liquid chromatography , 2016 .

[17]  Ajay Jasra,et al.  Markov Chain Monte Carlo Methods and the Label Switching Problem in Bayesian Mixture Modeling , 2005 .

[18]  J. Rosenthal,et al.  Optimal scaling of discrete approximations to Langevin diffusions , 1998 .

[19]  Aad van der Vaart,et al.  Fundamentals of Nonparametric Bayesian Inference , 2017 .