Biomechanics and Modeling in Mechanobiology

, Volume 13, Issue 4, pp 713–734 | Cite as

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

  • F. J. Vermolen
  • M. M. Mul
  • A. Gefen
Original Paper


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.


Immune system Stochastic modeling Antibiotics 


  1. 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 369–387Google Scholar
  2. 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):946–950Google Scholar
  3. Badolato R (2013) Defects of leukocyte migration in primary immunodeficiencies. Eur J Immunol 43(6):1436–1440CrossRefGoogle Scholar
  4. Bergman RA, Afifi AK, Heidger PM (2011) Blood (Section 4), Atlas of microscopic anatomy. Accessed 23 July 2013
  5. Britton NF, Chaplain MAJ (1993) A qualitative analysis of some models of tissue growth. Math Biosci 113:77–89CrossRefzbMATHGoogle Scholar
  6. Byrne H, Drasdo D (2009) Individual-based and continuum models of growing cell populations: a comparison. J Math Biol 58:657–687CrossRefMathSciNetGoogle Scholar
  7. 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):1122–1130CrossRefGoogle Scholar
  8. Evans LC (1998) Partial differential equations. Americal Mathematical Society, Providence, Rhode IslandzbMATHGoogle Scholar
  9. Filion J, Popel AP (2004) A reaction diffusion model of basic fibroblast growth factor interactions with cell surface receptors. Ann Biomed Eng 32(5):645–663CrossRefGoogle Scholar
  10. Freund JB, Orescanin MM (2011) Cellular Flow in a small blood vessel. J Fluid Dyn 671:466–490zbMATHGoogle Scholar
  11. 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 43:369–378CrossRefGoogle Scholar
  12. Friesel RE, Maciang T (1995) Molecular mechanisms of angiogenesis: fibroblast growth factor signal transduction. J Fed Am Soc Exp Biol 9:919–925Google Scholar
  13. Gaffney EA, Pugh K, Maini PK (2002) Investigating a simple model for cutaneous wound healing angiogenesis. J Math Biol 45(4): 337–374CrossRefzbMATHMathSciNetGoogle Scholar
  14. 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):4501–4505Google Scholar
  15. Graner F, Glazier J (1992) Simulation of biological cell sorting using a two-dimensional extended Potts model. Phys Rev Lett 69:2013–2016CrossRefGoogle Scholar
  16. Grimmett GR, Stirzaker DR (1992) Probability and random processes. Clarendon Press, OxfordGoogle Scholar
  17. Groh A, Louis AK (2010) Stochastic modeling of biased cell migration and collagen matrix modification. J Math Biol 61:617–647CrossRefzbMATHMathSciNetGoogle Scholar
  18. Hill HR, Augustine NH, Rallison ML, Santos JI (1983) Defective monocyte chemotactic responses in diabetes mellitus. J Clin Immunol 3(1):70–78CrossRefGoogle Scholar
  19. 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 294–312Google Scholar
  20. Ignarro LJ (2000) Nitric oxide: biology and pathobiology. Academic Press, San DiegoGoogle Scholar
  21. 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):3597–3606CrossRefzbMATHGoogle Scholar
  22. Jones GW, Chapman SJ (2012) Modeling growth in biological materials. SIAM Rev 54(1):52–118CrossRefzbMATHMathSciNetGoogle Scholar
  23. 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 CrossRefGoogle Scholar
  24. Kim Y-C (1997) Diffusivity of bacteria. Korean J Chem Eng 13(3):282–287CrossRefGoogle Scholar
  25. Maggelakis SA (2003) A mathematical model for tissue replacement during epidermal wound healing. Appl Math Model 27(3):189–196Google Scholar
  26. Mantzaris NV, Webb S, Othmer HG (2004) Mathematical modeling of tumor-induced angiogenesis. J Math Biol 49:111–187CrossRefzbMATHMathSciNetGoogle Scholar
  27. Merks MH, Koolwijk P (2009) Modeling morphogenesis in silico and in vitro: towards quantitative, predictive, cell-based modeling. Math Mod Natur Phenom 4(4):149–171CrossRefzbMATHMathSciNetGoogle Scholar
  28. Middleton J, Patterson AM, Gardner L, Schmutz C, Ashton BA (2002) Leukocyte extravasation: chemokine transport and presentation by the endothelium. Blood 100(2):3853–3860CrossRefGoogle Scholar
  29. Neilson MP, MacKenzie JA, Webb SD, Insall RH (2011) Modeling cell movement and chemotaxis using pseudopod-based feedback. SIAM J Sci Comput 33(3):1035–1057CrossRefzbMATHMathSciNetGoogle Scholar
  30. Oda T, Katori M, Hatanaka K, Yamashina S (1992) Five steps in leukocyte extravasation in the microcirculation by chemoattractants. Mediat Inflamm 1(6):403–409CrossRefGoogle Scholar
  31. 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 177:113–128CrossRefGoogle Scholar
  32. Reinhart-King CA, Dembo M, Hammer DA (2008) Cell-cell mechanical communication through compliant substrates. Biophys J 95:6044–6051CrossRefGoogle Scholar
  33. Sherratt JA, Murray JD (1991) Mathematical analysis of a basic model for epidermal wound healing. J Math Biol 29:389–404CrossRefzbMATHGoogle Scholar
  34. Steele JM (2001) Stochastic calculus and financial applications. Springer, New YorkCrossRefzbMATHGoogle Scholar
  35. Todar K (2013) Todar’s online textbook of bacteriology. Textbook of bacteriology. Accessed 23 July 2013
  36. Tranquillo RT, Murray JD (1992) Continuum model of fibroblast-driven wound contraction inflammation-mediation. J Theor Biol 158(2):135–172 Google Scholar
  37. Vermolen FJ, Javierre E (2012) A finite-element model for healing of cutaneous wounds combining contraction, angiogenesis and closure. J Math Biol 65(5):967–996CrossRefzbMATHMathSciNetGoogle Scholar
  38. 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):183–95CrossRefGoogle Scholar
  39. 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):301–323CrossRefGoogle Scholar
  40. Vermolen FJ, Gefen A (2013b) A semi-stochastic cell-based model for in-vitro infected ’wound’ healing through motility reduction. J Theor Biol 318:68–80CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Delft Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands
  2. 2.Department of Biomedical EngineeringTel Aviv UniversityTel AvivIsrael

Personalised recommendations