Skip to main content
SpringerLink
Log in

Application of the Global Optimization Methods for Solving the Parameter Estimation Problem in Mathematical Immunology

  • Conference paper
  • First Online:
Large-Scale Scientific Computing (LSSC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11958))

Included in the following conference series:

  • 1115 Accesses

Abstract

Mathematical modeling is widely used in modern immunology. The availability of biologically meaningful and detailed mathematical models permits studying the complex interactions between the components of a biological system and predicting the outcome of the therapeutic interventions. However, the incomplete theoretical understanding of the immune mechanism leads to the uncertainty of model structure and the need of model identification. This process is iterative and each step requires data-based model calibration. When the model is highly detailed, the considerable part of model parameters can not be measured experimentally or found in literature, so one has to solve the parameter estimation problem. Using the maximum likelihood framework, the parameter estimation leads to minimization problem for least square functional, when the observational errors are normally distributed. In this work we presented different computational approaches to the treatment of global optimization problem, arising in parameter estimation. We consider two high-dimensional mathematical models of HIV (human immunodeficiency virus)-infection dynamics as examples. The ODE (ordinary differential equations) and DDE (delay differential equations) versions of models were studied. For these models we solved the parameter estimation problem using a number of numerical global optimization techniques, including the optimization method, based on the tensor-train decomposition (TT). The comparative analysis of obtained results showed that the TT-based optimization technique is in the leading group of the methods ranked according to their performance in the parameter estimation for ODE and DDE versions of both models.

Supported by Russian Foundation for Basic Research (RFBR), research projects 18-31-00356 and research project 17-01-00636.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Germain, R., Meier-Schellersheim, M.: Systems biology in immunology: a computational modeling perspective. Annu. Rev. Immunol. 29, 527–85 (2011)

    Article  Google Scholar 

  2. Zheltkov, D., Oferkin, I., Katkova, E., Sulimov, A., Sulimov, V., Tyrtyshnikov, E.: TTDock: a docking method based on tensor train decompositions. Vychislitel’nye Metody i Programmirovanie 4(3), 279–291 (2013)

    Google Scholar 

  3. Bocharov, G., et al.: Mathematical Immunology of Virus Infections. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72317-4

    Book  MATH  Google Scholar 

  4. Ashyraliyev, M., Fomekong-Nanfack, Y., Kaandorp, J., Blom, J.: Systems biology: parameter estimation for biochemical models. FEBS J. 276, 886–902 (2009)

    Article  Google Scholar 

  5. Lillacci, G., Khammash, M.: Parameter estimation and model selection in computational biology. PLOS Comput. Biol. 6(3), e1000696 (2010)

    Article  MathSciNet  Google Scholar 

  6. Zheltkova, V., Zheltkov, D., Grossman, Z., Bocharov, G., Tyrtyshnikov, E.: Tensor based approach to the numerical treatment of the parameter estimation problems in mathematical immunology. J. Inverse Ill-posed Probl. 26(1), 51–66 (2018)

    Article  MathSciNet  Google Scholar 

  7. Oseledets, I., Tyrtyshnikov, E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432(1), 70–88 (2010)

    Article  MathSciNet  Google Scholar 

  8. Oseledets, I.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)

    Article  MathSciNet  Google Scholar 

  9. Bocharov, G., Chereshnev, V., et al.: Human immunodeficiency virus infection: from biological observations to mechanistic mathematical modelling. Math. Model. Nat. Phenom. 7(5), 78–104 (2012)

    Article  MathSciNet  Google Scholar 

  10. Grossman, Z., et al.: CD4+ T-cell depletion in HIV infection: are we closer to understanding the cause? Nat. Med. 8(4), 319 (2002)

    Article  Google Scholar 

  11. Perelson, A.: Modelling viral and immune system dynamics. Nat. Rev. Immunol. 2(1), 28 (2002)

    Article  Google Scholar 

  12. Adams, B., et al.: HIV dynamics: modeling, data analysis, and optimal treatment protocols. J. Comput. Appl. Math. 184(1), 10–49 (2005)

    Article  MathSciNet  Google Scholar 

  13. Marchuk, G.: Mathematical Modelling of Immune Response in Infectious Diseases, vol. 395. Springer, Heidelberg (2013). https://doi.org/10.1007/978-94-015-8798-3

    Book  Google Scholar 

  14. Simonov, M.: Modeling adaptive regulatory T-cell dynamics during early HIV infection. PLoS ONE 7(4), e33924 (2012)

    Article  Google Scholar 

  15. Baker, C., Bocharov, G., Rihan, F.: A report on the use of delay differential equations in numerical modelling in the biosciences. Manchester Centre for Computational Mathematics, Manchester, UK (1999)

    Google Scholar 

  16. Zheltkova, V., Zheltkov, D., Bocharov, G. : Modelling HIV infection: model identification and global sensitivity analysis. Math. Biol. Bioinform. 14(1), 19–33 (2019). (in Russian)

    Google Scholar 

  17. The NLopt nonlinear-optimization package. http://ab-initio.mit.edu/wiki/index.php/NLopt. Accessed 27 Feb 2019

  18. Kaelo, P., Ali, M.: Some variants of the controlled random search algorithm for global optimization. J. Optim. Theory Appl. 130(2), 253–264 (2006)

    Article  MathSciNet  Google Scholar 

  19. Kan, R.: Stochastic global optimization methods. Math. Program. 39(1) (1987)

    Google Scholar 

  20. Rowan, T.: Functional stability analysis of numerical algorithms (1990)

    Google Scholar 

  21. Runarsson, T., Yao X.: Search biases in constrained evolutionary optimization. IEEE Trans. Syst. Man Cybern. Part C (Appl. Rev.) 35(2), 233–243 (2005)

    Google Scholar 

  22. Santos, C., Goncalves, M., Hernandez-Figueroa, H.: Designing novel photonic devices by bio-inspired computing. IEEE Photonics Technol. Lett. 22(15), 1177–1179 (2010)

    Article  Google Scholar 

  23. Munier, M., Kelleher, A.: Acutely dysregulated, chronically disabled by the enemy within: T-cell responses to HIV-1 infection. Immunol. Cell Biol. 85(1), 6–15 (2007)

    Article  Google Scholar 

  24. Hindmarsh, A., et al.: SUNDIALS: suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw. (TOMS) 31(3), 363–396 (2005)

    Article  MathSciNet  Google Scholar 

  25. Bocharov, G., Romanyukha, A.: Numerical solution of delay-differential equations by linear multistep methods: algorithm and programme. Preprint No. 117. Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow (1986). (in Russian)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences, Moscow, Russia

    V. V. Zheltkova, Dmitry A. Zheltkov, G. A. Bocharov & Eugene Tyrtyshnikov

  2. Lomonosov Moscow State University, Moscow, Russia

    V. V. Zheltkova

Authors
  1. V. V. Zheltkova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitry A. Zheltkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. A. Bocharov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Eugene Tyrtyshnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dmitry A. Zheltkov .

Editor information

Editors and Affiliations

  1. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Lirkov

  2. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Svetozar Margenov

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Zheltkova, V.V., Zheltkov, D.A., Bocharov, G.A., Tyrtyshnikov, E. (2020). Application of the Global Optimization Methods for Solving the Parameter Estimation Problem in Mathematical Immunology. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2019. Lecture Notes in Computer Science(), vol 11958. Springer, Cham. https://doi.org/10.1007/978-3-030-41032-2_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-41032-2_23

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-41031-5

  • Online ISBN: 978-3-030-41032-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation