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Theory in Biosciences

, Volume 137, Issue 1, pp 67–78 | Cite as

Mathematical modeling of cancer–immune system, considering the role of antibodies

  • Sumana Ghosh
  • Sandip Banerjee
Original Article

Abstract

A mathematical model for the quantitative analysis of cancer–immune interaction, considering the role of antibodies has been proposed in this paper. The model is based on the clinical evidence, which states that antibodies can directly kill cancerous cells (Ivano et al. in J Clin Investig 119(8):2143–2159, 2009). The existence of transcritical bifurcation, which has been proved using Sotomayor theorem, provides strong biological implications. Through numerical simulations, it has been illustrated that under certain therapy (like monoclonal antibody therapy), which is capable of altering the parameters of the system, cancer-free state can be obtained.

Keywords

Cancer cells B cells Plasma cells Antibodies Global stability Transcritical bifurcation 

Notes

Acknowledgements

We are grateful to the anonymous reviewers for their comments and useful suggestions to improve the quality of the paper. This study was supported by the Indo-French Centre for Applied Mathematics (IFCAM) (Grant No. MA/IFCAM/13/120) and the Ministry of Human Resource Development (MHRD) (Grant No. MHR02-41-113-429).

References

  1. Adam J, Bellomo N (1997) A survey of models for tumor immune dynamics. Birkhauser, Boston, MACrossRefGoogle Scholar
  2. Adams GP, Weiner LM (2005) Monoclonal antibodies therapy of cancer. Nat Biotechnol 23(9):1147–1157CrossRefPubMedGoogle Scholar
  3. Banerjee S, Sarkar RR (2008) Delay Induced-model for tumor–immune interaction and control of malignant tumor growth. Bio Syst 91:268–288Google Scholar
  4. Banerjee S, Khajanchi S, Chowdhuri S (2015) Mathematical model to elucidate brain tumor abrogation by immunotherapy with T11 target structure. PLoS ONE 10(5):1–21CrossRefGoogle Scholar
  5. Banerjee S, Tsygvintsev A (2015) Stability and bifurcations of equilibria in a delayed Kirschner–Panetta model. Appl Math Lett 40:65–71CrossRefGoogle Scholar
  6. Bodnar M, Foryś U (2000) Periodic dynamics in the model of immune system. Int J Appl Math Comput Sci 10(1):1201–1209Google Scholar
  7. Chaplain MAJ, Matzavinos A (2006) Mathematical modeling of spatio-temporal phenomena in tumor immunology. Springer, BerlinGoogle Scholar
  8. de Pillis LG, Gu W, Radunskaya AE (2006) Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretation. J Theor Biol 238:841–862CrossRefPubMedGoogle Scholar
  9. Foryś U (2002) Marchuk’s Model of Immune System Dynamics with Application to Tumor Growth. J Theor Med 4(1):85–93CrossRefGoogle Scholar
  10. Fink M (2006) myAD: fast automatic differentiation code in Matlab. http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=15235. Accessed 28 Oct 2016
  11. Fink M, Batzel JJ, Tran H (2008) A respiratory system model: parameter estimation and sensitivity analysis. Cardiovasc Eng 8:120–134CrossRefPubMedGoogle Scholar
  12. Gabriel JA (2007) The biology of cancer, 2nd edn. Wiley, Boca RatonCrossRefGoogle Scholar
  13. Ivanov A, Beers SA, Walshe CA, Honeychurch J, Alduaij W, Cox KL, Potter KN, Murray S, Chan CHT, Klymenko T, Erenpreisa J, Martin GJ, Illidge Tim M, Cragg MS (2009) Monoclonal antibodies directed to CD20 and HLA-DR can elicit homotypic adhesion followed by lysosome-mediated cell death in human lymphoma and leukemia cells. J Clin Investig 119(8):2143–2159PubMedPubMedCentralGoogle Scholar
  14. Khajanchi S, Banerjee S (2014) Stability and bifurcation analysis of delay induced tumor immune interaction model. Appl Math Comput 248:652–671Google Scholar
  15. Kirschner D, Panetta JC (1998) Modeling the immunotherapy of tumor–immune interaction. J Math Biol 37(3):235–252CrossRefPubMedGoogle Scholar
  16. Kirschner D, Tsygvintsev A (2009) On global dynamics of a model for tumor immunotherapy. Math Biosci Eng 6(3):579–583CrossRefGoogle Scholar
  17. Kolev M (2003a) Mathematical modeling of the competition between acquired immunity and cancer. Appl Math Comput Sci 13(3):289–296Google Scholar
  18. Kolev M (2003b) Mathematical modeling of the competition between tumors and the immune systems considering the role of antibodies. J Math Comput Model 37:1143–1152CrossRefGoogle Scholar
  19. Kuznetsov VA, Makalkin IA, Taylor MA, Perelson AS (1994) Non-linear Dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull Math Biol 56(2):295–321CrossRefPubMedGoogle Scholar
  20. Mallet DG, de Pillis LG (2006) A cellular automata model of tumor–immune system interactions. J Theor Biol 239:334–350CrossRefPubMedGoogle Scholar
  21. Nagumo N (1942) Uber die Lage der Integralkurven gewonlicher Differantialgleichungen. Proc Phys Math Soc Jpn 24:551–567Google Scholar
  22. Nossal GJV, Makela O (1962) Elaboration of antibodies by single cell. Ann Rev Microbiol 16:53–74CrossRefGoogle Scholar
  23. Perelson AS, Mimirani M, Oster GF (1976) Optimal strategies in immunology, B-cell differentiation and proliferation. J Math Biol 3:325–367CrossRefPubMedGoogle Scholar
  24. Perko L (1991) Differential equations and dynamical systems, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
  25. Preziosi L (1996) From population dynamics to modeling the competition between tumor and immune system. Math Model 23(6):135–152CrossRefGoogle Scholar
  26. Rosenberg S, Yang J, Restifo N (2004) Cancer immunotherapy: moving beyond current vaccines. Nat Med 10:909915Google Scholar
  27. Sarkar RR, Banerjee S (2005) Cancer self remission and tumor stability—a stochastic approach. J Math Biosci 196:65–81CrossRefGoogle Scholar
  28. Schulz WA (2007) Molecular biology of human cancer. Springer, BerlinGoogle Scholar
  29. Szymanska Z (2003) Analysis of immunotherapy models in the context of cancer dynamics. Int J Appl Math Comput Sci 13(3):407–418Google Scholar
  30. Tsygvintsev A, Banerjee S (2014) Bounded immune response in immunotherapy described by delay Kirschner–Panetta model. Appl Math Lett 35:90–94CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia

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