Skip to main content
SpringerLink
Log in

Chemotactic effects in reaction-diffusion equations for inflammation

  • Original Paper
  • Published:
Journal of Biological Physics Aims and scope Submit manuscript

Abstract

Predator–prey systems are used to model time-dependent virus and lymphocyte population during a liver infection and to discuss the influence of chemotactic behavior on the chronification tendency of such infections. Therefore, a model family of reaction-diffusion equations is presented, and the long-term behavior of the solutions is estimated by a critical value containing the reaction strength, the diffusion rate, and the extension of the liver domain. Fourier techniques are applied to evaluate the influence of chemotactic behavior of the immune response to the long-term behavior of locally linearized models. It turns out that the chemotaxis is a subordinated influence with respect to the chronification of liver infections.

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

Similar content being viewed by others

References

  1. Rehermann, B., Nascimbeni, M.: Immunology of hepatits B virus and hepatitis C virus infection. Nat. Rev. Immunol. 5, 218–229 (2005)

    Article  Google Scholar 

  2. Thomas, E., Liang, T.J.: Experimental models of hepatitis B and C — new insights and progress. Nat. Rev. Gastroenterol. Hepat. 13, 362–374 (2016)

    Article  Google Scholar 

  3. Kerl, H.-J., Langemann, D., Vollrath, A.: Reaction-diffusion equations and the chronification of liver infections. Math. Comput. Simulat. 82, 2145–2156 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Campbell, J.J., Butcher, E.C.: Chemokines in tissue specific and microenvironment-specific lymphocyte homing. Curr. Opin. Immunol. 12, 336–341 (2000)

    Article  Google Scholar 

  5. Carambia, A., Herkel, J.: CD4 T cells in hepatic immune tolerance. J. Autoimmun. 34, 23–28 (2010)

    Article  Google Scholar 

  6. Carragher, D.M., Rangel-Moreno, J., Randall, T.D.: Ectopic lymphoid tissues and local immunity. Semin. Immunol. 20, 26–42 (2008)

    Article  Google Scholar 

  7. Frebel, H., Richter, K., Oxenius, A.: How chronic viral infections impact on antigen-specific T cell responses. Eur. J. Immunol. 40, 654–663 (2010)

    Article  Google Scholar 

  8. Wiegard, C., Wolint, P., Frenzel, C., Cheruti, U., Schmitt, E., Oxenius, A., Lohse, A.W., Herkel, J.: Defective T helper 1 response by hepatocyte-stimulated CD4 T cells impairs anti-viral CD8 response and viral clearance. Gastroenterology 133, 2010–2018 (2007)

    Article  Google Scholar 

  9. Kumar, R., Clermont, G., Vodovotz, Y., Chow, C.C.: The dynamics of acute inflammation. J. Theor. Biol. 240, 145–155 (2004)

    Article  MathSciNet  Google Scholar 

  10. Wodarz, D.: Hepatitis C virus dynamics and pathology: the role of CTL and antibody response. J. Gen. Virology 84, 1743–1750 (2003)

    Article  Google Scholar 

  11. Ibragimov, A.I., McNeal, C.J., Ritter, L.R., Walton, J.R.: A mathematical model of atherogenesis as an inflammatory response. Math. Med. Biol. 22, 305–333 (2005)

    Article  MATH  Google Scholar 

  12. Ciupe, S.M., Catllá, A. J., Forde, J., Schaeffer, D.G.: Dynamics of hepatitis B virus infection: what causes viral clearance?. Math. Popul. Stud. 18, 87–105 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Long, C., Qi, H., Huang, S.H.: Modeling the mechanism of HBV infection. In: Proc. Int. Conf. Biomedical Engineering and Computer Science, Wuhan (2010)

  14. Perelson, A.S., Mirmirani, M., Oster, G.F.: Optimal strategies in immunology, 1. B-cell differentiation and proliferation. J. Math. Biol. 3, 325–367 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bevan, M.J.: Helping the CD8 + T cell response. Nat. Rev. Immunol. 4, 595–602 (2004)

    Article  Google Scholar 

  16. Bowen, D.G., Warren, A., Davis, T., Hoffmann, M.W., McCaughan, G.W., De St Groth, D.F., Bertolino, P.: Cytokine-dependent bystander hepatitis due to intrahepatic murine CD8 T cell activation by bone marrow-derived cells. Gastroenterology 123, 1252–1264 (2002)

    Article  Google Scholar 

  17. Lang, P.A., Contaldo, C., Georgiev, P., El-Badry, A.M., Recher, M., Kurrer, M., Cervantes-Barragan, L., Ludewig, B., Calzascia, T., Bolinger, B., Merkler, D., Odermatt, B., Bader, M., Graf, R., Clavien, P.A., Hegazy, A.N., Löhning, M., Harris, N.L., Ohashi, P.S., Hengartner, H., Zinkernagel, R.M., Lang, K.S.: Aggravation of viral hepatitis by platelet-derived serotonin. Nat. Med. 14, 756–761 (2008)

    Article  Google Scholar 

  18. Scheuer, P.J., Lefkowitch, J.H.: Liver Biopsy Interpretation. W.B. Saunders, London (2000)

    Google Scholar 

  19. Evans, L.C.: Partial differential equations. American Mathematical Society (2010)

  20. Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Springer, New York (1983)

    Book  MATH  Google Scholar 

  21. Perthame, B.: Parabolic Equations in Biology. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, Cham (2015)

    Google Scholar 

  22. Murray, J.D.: Mathematical Biology I/II. Springer, New York (2002)

    Book  Google Scholar 

  23. Allee, W.C.: Principles of Animal Ecology. W.B. Saunders, Philadelphia (1949)

    Google Scholar 

  24. Hartman, P.: Ordinary Differential Equations. John Wiley & Sons, New York (1964)

    MATH  Google Scholar 

  25. Chalub, F.A.C.C., Markowich, P.A., Perthame, B., Schmeiser, C.: Kinetic models for chemotaxis and their drift-diffusion limits. In: Jüngel, A., Manasebvich, R., Markowich, P.A., Shahgholian, H. (eds.) Nonlinear Differential Equation models, pp 123–141. Springer, Vienna (2004)

  26. Hinch, E.J.: Perturbation Methods. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. TU Braunschweig, Institute of Computational Mathematics, AG PDE, Universitätsplatz 2, 38106, Braunschweig, Germany

    Cordula Reisch & Dirk Langemann

Authors
  1. Cordula Reisch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dirk Langemann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cordula Reisch.

Ethics declarations

Conflict of interests

The authors declare that they have no conflict of interest.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Reisch, C., Langemann, D. Chemotactic effects in reaction-diffusion equations for inflammation. J Biol Phys 45, 253–273 (2019). https://doi.org/10.1007/s10867-019-09527-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10867-019-09527-3

Keywords

Search

Navigation