Analytical and Bioanalytical Chemistry

, Volume 410, Issue 16, pp 3905–3915 | Cite as

Analyzing chromatographic data using multilevel modeling

  • Paweł Wiczling
Research Paper


It is relatively easy to collect chromatographic measurements for a large number of analytes, especially with gradient chromatographic methods coupled with mass spectrometry detection. Such data often have a hierarchical or clustered structure. For example, analytes with similar hydrophobicity and dissociation constant tend to be more alike in their retention than a randomly chosen set of analytes. Multilevel models recognize the existence of such data structures by assigning a model for each parameter, with its parameters also estimated from data. In this work, a multilevel model is proposed to describe retention time data obtained from a series of wide linear organic modifier gradients of different gradient duration and different mobile phase pH for a large set of acids and bases. The multilevel model consists of (1) the same deterministic equation describing the relationship between retention time and analyte-specific and instrument-specific parameters, (2) covariance relationships relating various physicochemical properties of the analyte to chromatographically specific parameters through quantitative structure–retention relationship based equations, and (3) stochastic components of intra-analyte and interanalyte variability. The model was implemented in Stan, which provides full Bayesian inference for continuous-variable models through Markov chain Monte Carlo methods.

Graphical abstract

Relationships between log k and MeOH content for acidic, basic, and neutral compounds with different log P. CI credible interval, PSA polar surface area


Liquid chromatography pH Multilevel modeling Bayesian analysis 



This project was supported by the National Science Centre, Poland (grant 2015/18/E/ST4/00449).

Compliance with ethical standards

Conflict of interest

The author declares that he has no competing interests.

Supplementary material

216_2018_1061_MOESM1_ESM.pdf (1.7 mb)
ESM 1 (PDF 1706 kb)
216_2018_1061_MOESM2_ESM.csv (109 kb)
ESM 2 (CSV 108 kb)


  1. 1.
    Wiczling P, Struck-Lewicka W, Kubik L, Siluk D, Markuszewski MJ, Kaliszan R. The simultaneous determination of hydrophobicity and dissociation constant by liquid chromatography-mass spectrometry. J Pharm Biomed Anal. 2014;94:180–7.CrossRefPubMedGoogle Scholar
  2. 2.
    Kubik Ł, Struck-Lewicka W, Kaliszan R, Wiczling P. Simultaneous determination of hydrophobicity and dissociation constant for a large set of compounds by gradient reverse phase high performance liquid chromatography-mass spectrometry technique. J Chromatogr A. 2015;1416:31–7.CrossRefPubMedGoogle Scholar
  3. 3.
    Pappa-Louisi A, Nikitas P, Balkatzopoulou P, Malliakas C. Two- and three-parameter equations for representation of retention data in reversed-phase liquid chromatography. J Chromatogr A. 2004;1033(1):29–41.CrossRefPubMedGoogle Scholar
  4. 4.
    Rosés M, Subirats X, Bosch E. Retention models for ionizable compounds in reversed-phase liquid chromatography: effect of variation of mobile phase composition and temperature. J Chromatogr A. 2009;1216(10):1756–75.CrossRefPubMedGoogle Scholar
  5. 5.
    Andrés A, Téllez A, Rosés M, Bosch E. Chromatographic models to predict the elution of ionizable analytes by organic modifier gradient in reversed phase liquid chromatography. J Chromatogr A. 2012;1247:71–80.CrossRefPubMedGoogle Scholar
  6. 6.
    Téllez A, Rosés M, Bosch E. Modeling the retention of neutral compounds in gradient elution RP-HPLC by means of polarity parameter models. Anal Chem. 2009;81(21):9135–45.CrossRefPubMedGoogle Scholar
  7. 7.
    Kaliszan R. QSRR: quantitative structure-(chromatographic) retention relationships. Chem Rev. 2007;107(7):3212–46.CrossRefPubMedGoogle Scholar
  8. 8.
    Park SH, Haddad PR, Amos RIJ, Talebi M, Szucs R, Pohl CA, et al. Towards a chromatographic similarity index to establish localised quantitative structure-retention relationships for retention prediction. III combination of Tanimoto similarity index, logP, and retention factor ratio to identify optimal analyte training sets for ion chromatography. J Chromatogr A. 2017;1520:107–16.CrossRefPubMedGoogle Scholar
  9. 9.
    Tyteca E, Talebi M, Amos R, Park SH, Taraji M, Wen Y, et al. Towards a chromatographic similarity index to establish localized quantitative structure-retention models for retention prediction: Use of retention factor ratio. J Chromatogr A. 2017;1486:50–8.CrossRefPubMedGoogle Scholar
  10. 10.
    Daghir-Wojtkowiak E, Wiczling P, Waszczuk-Jankowska M, Kaliszan R, Markuszewski MJ. Multilevel pharmacokinetics-driven modeling of metabolomics data. Metabolomics. 2017;13(3):31.CrossRefPubMedPubMedCentralGoogle Scholar
  11. 11.
    Wiczling P, Bartkowska-Śniatkowska A, Szerkus O, Siluk D, Rosada-Kurasińska J, Warzybok J, et al. The pharmacokinetics of dexmedetomidine during long-term infusion in critically ill pediatric patients. A Bayesian approach with informative priors. J Pharmacokinet Pharmacodyn. 2016;43(3):315–24.CrossRefPubMedPubMedCentralGoogle Scholar
  12. 12.
    Lin C, Gelman A, Price PN, Krantz DH. Analysis of local decisions using hierarchical modeling, applied to home radon measurement and remediation. Stat Sci. 1999;14(3):333–7.Google Scholar
  13. 13.
    Wiczling P, Kaliszan R. How much can we learn from a single chromatographic experiment? A Bayesian perspective. Anal Chem. 2016;88(1):997–1002.CrossRefPubMedGoogle Scholar
  14. 14.
    Wiczling P, Kubik Ł, Kaliszan R. Maximum a posteriori Bayesian estimation of chromatographic parameters by limited number of experiments. Anal Chem. 2015;87(14):7241–9.CrossRefPubMedGoogle Scholar
  15. 15.
    Wiczling P. Evaluation of sequential Bayesian-based method development procedures for chromatographic problems involving one, two, and three analytes. Sep Sci Plus. 2018;1(2):63–75. Scholar
  16. 16.
    Wiczling P, Kawczak P, Nasal A, Kaliszan R. Simultaneous determination of pKa and lipophilicity by gradient RP HPLC. Anal Chem. 2006;78(1):239–49.CrossRefPubMedGoogle Scholar
  17. 17.
    Canals I, Portal J, Bosch E, Roses M. Retention of ionizable compounds on HPLC. 4. Mobile-phase pH measurement in methanol/water. Anal Chem. 2000;72(8):1802–9.CrossRefPubMedGoogle Scholar
  18. 18.
    Wiczling P, Markuszewski MJ, Kaliszan M, Kaliszan R. pH/organic solvent double-gradient reversed-phase HPLC. Anal Chem. 2005;77(2):449–58.CrossRefPubMedGoogle Scholar
  19. 19.
    Snyder LR, Dolan JW. High-performance gradient elution: the practical application of the linear-solvent-strength model. Hoboken: Wiley; 2006.CrossRefGoogle Scholar
  20. 20.
    Nikitas P, Pappa-Louisi A. Expressions of the fundamental equation of gradient elution and a numerical solution of these equations under any gradient profile. Anal Chem. 2005;77(17):5670–7.CrossRefPubMedGoogle Scholar
  21. 21.
    Wiczling P, Kaliszan R. Retention time and peak width in the combined pH/organic modifier gradient high performance liquid chromatography. J Chromatogr A. 2010;1217(20):3375–81.CrossRefPubMedGoogle Scholar
  22. 22.
    Wiczling P, Kaliszan R. Influence of pH on retention in linear organic modifier gradient RP HPLC. Anal Chem. 2008;80(20):7855–61.CrossRefPubMedGoogle Scholar
  23. 23.
    Snyder LR, Kirkland JJ, Dolan JW. Introduction to modern liquid chromatography. 3rd ed. Oxford: Wiley-Blackwell; 2010.Google Scholar
  24. 24.
    Gelman A. Bayesian data analysis. 2nd ed. Boca Raton: CRC; 2004.Google Scholar
  25. 25.
    Carpenter B, Gelman A, Hoffman M, Lee D, Goodrich B, Betancourt M, et al. Stan: a probabilistic programming language. J Stat Softw. 2017;76(1):1–29.CrossRefGoogle Scholar
  26. 26.
    Gelman A, Hill J. Data analysis using regression and multilevel/hierarchical models. Cambridge: Cambridge University Press; 2007.Google Scholar
  27. 27.
    McElreath R. Statistical rethinking: a Bayesian course with examples in R and Stan. Boca Raton: CRC; 2016.Google Scholar
  28. 28.
    Neue U, Phoebe C, Tran K, Cheng Y, Lu Z. Dependence of reversed-phase retention of ionizable analytes on pH, concentration of organic solvent and silanol activity. J Chromatogr A. 2001;925(1-2):49–67.CrossRefPubMedGoogle Scholar
  29. 29.
    Hansch C, Leo A, Taft R. A Survey of Hammett substituent constants and resonance and field parameters. Chem Rev. 1991;91(2):165–95.CrossRefGoogle Scholar
  30. 30.
    Gritti F, Guiochon G. Critical contribution of nonlinear chromatography to the understanding of retention mechanism in reversed-phase liquid chromatography. J Chromatogr A. 2005;1099(1-2):1–42.CrossRefPubMedGoogle Scholar
  31. 31.
    Gritti F, Guiochon G. Overloaded elution band profiles of ionizable compounds in reversed-phase liquid chromatography: influence of the competition between the neutral and the ionic species. J Sep Sci. 2008;31(21):3657–82.CrossRefPubMedGoogle Scholar
  32. 32.
    Haddad PR. Seeking the holy grail—prediction of chromatographic retention based only on chemical structures. LCGC. 2017;35(8):499–502.Google Scholar
  33. 33.
    Kubik Ł, Wiczling P. Quantitative structure-(chromatographic) retention relationship models for dissociating compounds. J Pharm Biomed Anal. 2016;127:176–83.CrossRefPubMedGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Biopharmaceutics and PharmacodynamicsMedical University of GdańskGdańskPoland

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