Skip to main content

Advertisement

SpringerLink
Log in

Semi-stochastic cell-level computational modeling of the immune system response to bacterial infections and the effects of antibiotics

  • Original Paper
  • Published:
Biomechanics and Modeling in Mechanobiology Aims and scope Submit manuscript

Abstract

A mathematical model for the immune system response to bacterial infections is proposed. The formalism is based on modeling the chemokine-determined transmigration of leukocytes from a venule through the venule walls and the subsequent in-tissue migration and engulfment of the pathogens that are responsible for the infection. The model is based on basic principles, such as Poiseuille blood flow through the venule, fundamental solutions of the diffusion–reaction equation for the concentration field of pathogen-released chemokines, linear chemotaxis of the leukocytes, random walk of pathogens, and stochastic processes for the death and division of pathogens. Thereby, a computationally tractable and, as far as we know, original framework has been obtained, which is used to incorporate the interaction of a substantial number of leukocytes and thereby to unravel the significance of biological processes and parameters regarding the immune system response. The developed model provides a neat way for visualization of the biophysical mechanism of the immune system response. The simulations indicate a weak correlation between the immune system response in terms of bacterial clearing time and the leukocyte stiffness, and a significant decrease in the clearing time with increasing in-blood leukocyte density, decreasing pathogen motility, and increasing venule wall transmissivity. Finally, the increase in the pathogen death rate and decrease in pathogen motility induce a decrease in the clearing time of the infection. The adjustment of the latter two quantities mimic the administration of antibiotics.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

References

  • Alarcon T, Byrne H, Maini P, Panovska J (2006) Mathematical modeling of angiogenesis and vascular adaptation. In: Paton R, McNamara L (ed) Multidisciplinary approaches to the theory of medicine, vol 3, pp 369–387

  • Arazynski SE, Tucker BJ, Aratow M, Crenshaw A, Hargens AR (1993) Direct measurement of capillary blood pressure in the human lip. J Appl Physiol 74(2):946–950

    Google Scholar 

  • Badolato R (2013) Defects of leukocyte migration in primary immunodeficiencies. Eur J Immunol 43(6):1436–1440

    Article  Google Scholar 

  • Bergman RA, Afifi AK, Heidger PM (2011) Blood (Section 4), Atlas of microscopic anatomy. http://www.anatomyatlases.org/MicroscopicAnatomy/Section04/Section04.shtml. Accessed 23 July 2013

  • Britton NF, Chaplain MAJ (1993) A qualitative analysis of some models of tissue growth. Math Biosci 113:77–89

    Article  MATH  Google Scholar 

  • Byrne H, Drasdo D (2009) Individual-based and continuum models of growing cell populations: a comparison. J Math Biol 58:657–687

    Article  MathSciNet  Google Scholar 

  • Damiano ER, Westheider J, Tözeren A, Ley K (1996) Variation in the velocity, deformation, and adhesion energy density of leukocytes rolling within venules. Circ Res 79(6):1122–1130

    Article  Google Scholar 

  • Evans LC (1998) Partial differential equations. Americal Mathematical Society, Providence, Rhode Island

    MATH  Google Scholar 

  • Filion J, Popel AP (2004) A reaction diffusion model of basic fibroblast growth factor interactions with cell surface receptors. Ann Biomed Eng 32(5):645–663

    Article  Google Scholar 

  • Freund JB, Orescanin MM (2011) Cellular Flow in a small blood vessel. J Fluid Dyn 671:466–490

    MATH  Google Scholar 

  • Friedl P, Zänker KS, Bröcker E-B (1998) Cell migration strategies in 3-D extracellular matrix: differences in morphology, cell matrix interactions, and integrin function. Microsc Res Tech 43:369–378

    Article  Google Scholar 

  • Friesel RE, Maciang T (1995) Molecular mechanisms of angiogenesis: fibroblast growth factor signal transduction. J Fed Am Soc Exp Biol 9:919–925

    Google Scholar 

  • Gaffney EA, Pugh K, Maini PK (2002) Investigating a simple model for cutaneous wound healing angiogenesis. J Math Biol 45(4): 337–374

    Article  MATH  MathSciNet  Google Scholar 

  • Gefen A (2010) Effects of virus size and cell stiffness on forces, work and pressures driving membrane invagination in a receptor-mediated endocytosis. J Biomech Eng (ASME) 132(8):4501–4505

    Google Scholar 

  • Graner F, Glazier J (1992) Simulation of biological cell sorting using a two-dimensional extended Potts model. Phys Rev Lett 69:2013–2016

    Article  Google Scholar 

  • Grimmett GR, Stirzaker DR (1992) Probability and random processes. Clarendon Press, Oxford

    Google Scholar 

  • Groh A, Louis AK (2010) Stochastic modeling of biased cell migration and collagen matrix modification. J Math Biol 61:617–647

    Article  MATH  MathSciNet  Google Scholar 

  • Hill HR, Augustine NH, Rallison ML, Santos JI (1983) Defective monocyte chemotactic responses in diabetes mellitus. J Clin Immunol 3(1):70–78

    Article  Google Scholar 

  • Hogg JC (1995) Leukocyte dynamics in regional circulations. In: Granger DN, Schmid-Schoenbein G (eds) Physiology pathophysiology of leukocyte adhesion. Oxford University, New York, pp 294–312

    Google Scholar 

  • Ignarro LJ (2000) Nitric oxide: biology and pathobiology. Academic Press, San Diego

    Google Scholar 

  • Javierre E, Moreo P, Doblare M, Garcia-Aznar JM (2009) Numerical modeling of a mechano-chemical theory for wound contraction analysis. Int J Solids Struct 46(20):3597–3606

    Article  MATH  Google Scholar 

  • Jones GW, Chapman SJ (2012) Modeling growth in biological materials. SIAM Rev 54(1):52–118

    Article  MATH  MathSciNet  Google Scholar 

  • Khandoga AG, Khandoga A, Reichel CA, Bihari P, Rehberg M, Krombach F (2009) In vivo imaging and quantitative analysis of leukocyte directional migration and polarization in inflamed tissue. PLoS-ONE 4(3):e4693. doi:10.1371/journal.pone.0004693

    Article  Google Scholar 

  • Kim Y-C (1997) Diffusivity of bacteria. Korean J Chem Eng 13(3):282–287

    Article  Google Scholar 

  • Maggelakis SA (2003) A mathematical model for tissue replacement during epidermal wound healing. Appl Math Model 27(3):189–196

    Google Scholar 

  • Mantzaris NV, Webb S, Othmer HG (2004) Mathematical modeling of tumor-induced angiogenesis. J Math Biol 49:111–187

    Article  MATH  MathSciNet  Google Scholar 

  • Merks MH, Koolwijk P (2009) Modeling morphogenesis in silico and in vitro: towards quantitative, predictive, cell-based modeling. Math Mod Natur Phenom 4(4):149–171

    Article  MATH  MathSciNet  Google Scholar 

  • Middleton J, Patterson AM, Gardner L, Schmutz C, Ashton BA (2002) Leukocyte extravasation: chemokine transport and presentation by the endothelium. Blood 100(2):3853–3860

    Article  Google Scholar 

  • Neilson MP, MacKenzie JA, Webb SD, Insall RH (2011) Modeling cell movement and chemotaxis using pseudopod-based feedback. SIAM J Sci Comput 33(3):1035–1057

    Article  MATH  MathSciNet  Google Scholar 

  • Oda T, Katori M, Hatanaka K, Yamashina S (1992) Five steps in leukocyte extravasation in the microcirculation by chemoattractants. Mediat Inflamm 1(6):403–409

    Article  Google Scholar 

  • Olsen L, Sherratt JA, Maini PK (1995) A mechanochemical model for adult dermal wound closure and the permanence of the contracted tissue displacement role. J Theor Biol 177:113–128

    Article  Google Scholar 

  • Reinhart-King CA, Dembo M, Hammer DA (2008) Cell-cell mechanical communication through compliant substrates. Biophys J 95:6044–6051

    Article  Google Scholar 

  • Sherratt JA, Murray JD (1991) Mathematical analysis of a basic model for epidermal wound healing. J Math Biol 29:389–404

    Article  MATH  Google Scholar 

  • Steele JM (2001) Stochastic calculus and financial applications. Springer, New York

    Book  MATH  Google Scholar 

  • Todar K (2013) Todar’s online textbook of bacteriology. Textbook of bacteriology. http://www.textbookofbacteriology.net/growth_3. Accessed 23 July 2013

  • Tranquillo RT, Murray JD (1992) Continuum model of fibroblast-driven wound contraction inflammation-mediation. J Theor Biol 158(2):135–172

    Google Scholar 

  • Vermolen FJ, Javierre E (2012) A finite-element model for healing of cutaneous wounds combining contraction, angiogenesis and closure. J Math Biol 65(5):967–996

    Article  MATH  MathSciNet  Google Scholar 

  • Vermolen FJ, Gefen A (2012) A semi-stochastic cell-based formalism to model the dynamics of migration of cells in colonies. Biomech Model Mechanobiol 11(1–2):183–95

    Article  Google Scholar 

  • Vermolen FJ, Gefen A (2013a) A phenomenological model for chemico-mechanically induced cell shape changes during migration and cell-cell contacts. Biomech Model Mechanobiol 12(2):301–323

    Article  Google Scholar 

  • Vermolen FJ, Gefen A (2013b) A semi-stochastic cell-based model for in-vitro infected ’wound’ healing through motility reduction. J Theor Biol 318:68–80

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

    F. J. Vermolen & M. M. Mul

  2. Department of Biomedical Engineering, Tel Aviv University, 69978 , Tel Aviv, Israel

    A. Gefen

Authors
  1. F. J. Vermolen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. M. Mul
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Gefen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. J. Vermolen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vermolen, F.J., Mul, M.M. & Gefen, A. Semi-stochastic cell-level computational modeling of the immune system response to bacterial infections and the effects of antibiotics. Biomech Model Mechanobiol 13, 713–734 (2014). https://doi.org/10.1007/s10237-013-0529-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10237-013-0529-5

Keywords

Search

Navigation