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

  • V. V. Zheltkova
  • Dmitry A. ZheltkovEmail author
  • G. A. Bocharov
  • Eugene Tyrtyshnikov
Conference paper
  • 71 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11958)

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.

Keywords

Mathematical immunology Parameter estimation Global optimization Model identification 

References

  1. 1.
    Germain, R., Meier-Schellersheim, M.: Systems biology in immunology: a computational modeling perspective. Annu. Rev. Immunol. 29, 527–85 (2011)CrossRefGoogle Scholar
  2. 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. 3.
    Bocharov, G., et al.: Mathematical Immunology of Virus Infections. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-72317-4CrossRefzbMATHGoogle Scholar
  4. 4.
    Ashyraliyev, M., Fomekong-Nanfack, Y., Kaandorp, J., Blom, J.: Systems biology: parameter estimation for biochemical models. FEBS J. 276, 886–902 (2009)CrossRefGoogle Scholar
  5. 5.
    Lillacci, G., Khammash, M.: Parameter estimation and model selection in computational biology. PLOS Comput. Biol. 6(3), e1000696 (2010)MathSciNetCrossRefGoogle Scholar
  6. 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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Oseledets, I., Tyrtyshnikov, E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432(1), 70–88 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Oseledets, I.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)MathSciNetCrossRefGoogle Scholar
  9. 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)MathSciNetCrossRefGoogle Scholar
  10. 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)CrossRefGoogle Scholar
  11. 11.
    Perelson, A.: Modelling viral and immune system dynamics. Nat. Rev. Immunol. 2(1), 28 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Adams, B., et al.: HIV dynamics: modeling, data analysis, and optimal treatment protocols. J. Comput. Appl. Math. 184(1), 10–49 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Marchuk, G.: Mathematical Modelling of Immune Response in Infectious Diseases, vol. 395. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-94-015-8798-3CrossRefGoogle Scholar
  14. 14.
    Simonov, M.: Modeling adaptive regulatory T-cell dynamics during early HIV infection. PLoS ONE 7(4), e33924 (2012)CrossRefGoogle Scholar
  15. 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. 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. 17.
    The NLopt nonlinear-optimization package. http://ab-initio.mit.edu/wiki/index.php/NLopt. Accessed 27 Feb 2019
  18. 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)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kan, R.: Stochastic global optimization methods. Math. Program. 39(1) (1987)Google Scholar
  20. 20.
    Rowan, T.: Functional stability analysis of numerical algorithms (1990)Google Scholar
  21. 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. 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)CrossRefGoogle Scholar
  23. 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)CrossRefGoogle Scholar
  24. 24.
    Hindmarsh, A., et al.: SUNDIALS: suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw. (TOMS) 31(3), 363–396 (2005)MathSciNetCrossRefGoogle Scholar
  25. 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

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Marchuk Institute of Numerical Mathematics of the Russian Academy of SciencesMoscowRussia
  2. 2.Lomonosov Moscow State UniversityMoscowRussia

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