Periodic and chaotic dynamics in a map-based model of tumor-immune interaction

Journal of Theoretical Biology

Volumn 334, Issue , 2013, Pages 130-140

  Moghtadaei, Motahareh ^a   Hashemi Golpayegani, Mohammad Reza ^a   Malekzadeh, Reza ^b  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

^b TEHRAN UNIVERSITY OF MEDICAL SCIENCES   (Iran)

Author keywords

Chaos; Delay difference equation system; Discrete map; Limit cycle; Tumor growth model

Indexed keywords

CANCER; DISEASE TREATMENT; IMMUNE RESPONSE; IMMUNE SYSTEM; NUMERICAL MODEL; TUMOR;

ANIMAL CELL; ANIMAL EXPERIMENT; ANIMAL MODEL; ARTICLE; CALCULATION; CANCER INHIBITION; COMPARATIVE STUDY; CONTROLLED STUDY; DELAY DIFFERENCE EQUATION MODEL; IMMUNE RESPONSE; IMMUNOCOMPETENT CELL; KUZNETSOV MODEL; LYMPHOMA; MATHEMATICAL MODEL; MOUSE; NONHUMAN; PRIORITY JOURNAL; SIMULATION; TUMOR CELL; TUMOR GROWTH; TUMOR IMMUNITY; VALIDITY;

CHAOS; DELAY DIFFERENCE EQUATION SYSTEM; DISCRETE MAP; LIMIT CYCLE; TUMOR GROWTH MODEL;

ALGORITHMS; COMPUTER SIMULATION; HUMANS; IMMUNE SYSTEM; MODELS, BIOLOGICAL; NEOPLASMS; NONLINEAR DYNAMICS; TUMOR BURDEN;

EID: 84880374241     PISSN: 00225193     EISSN: 10958541     Source Type: Journal    
DOI: 10.1016/j.jtbi.2013.05.031     Document Type: Article

References (49)

1. Ahmed E. Fractals and chaos in cancer models. Int. J. Theor. Phys. 1992, 32(2):353-355.
2. Banerjee S., Sarkar R.R. Delay-induced model for tumor-immune interaction and control of malignant tumor growth. Biosystems 2008, 91(1):268-288.
3. Bellomo N., Preziosi L. Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math. Comput. Modelling 2000, 32:413-452.
4. Bonate P.L. Modeling tumor growth in oncology. Pharmacokinetics in Drug Development 2011, 1-19. Springer, US. P.L.L. Bonate, D.R.R. Howard (Eds.).
5. Cristini V., Lowengrub J. Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach 2010, Cambridge University Press.
6. Denis F., Letellier C. Chaos theory: a fascinating concept for oncologists. Cancer/Radiothérapie 2012, 16(3):230-236.
7. Denis F., Letellier C. Radiotherapy and chaos theory: the tit and the butterfly. Cancer/Radiothérapie 2012, 16(5-6):404-409.
8. de Pillis L.G., Radunskaya A. A mathematical tumor model with immune resistance and drug therapy: an optimal control approach. J. Theor. Med. 2001, 3(2):79-100.
9. d'Onofrio A., Gatti F., Cerrai P., Freschi L. Delay-induced oscillatory dynamics of tumour-immune system interaction. Math. Comput. Modelling 2010, 51.
10. El-Gohary A. Chaos and optimal control of cancer self-remission and tumor system steady states. Chaos Solitons Fractals 2008, 37(5):1305-1316.
11. El-Gohary A. Chaos and optimal control of equilibrium states of tumor system with drug. Chaos Solitons Fractals 2009, 41(1):425-435.
12. Elaydi S.N. Discrete Chaos: With Applications in Science and Engineering 2007, Chapman and Hall/CRC, Boca Raton, FL. 2ed.
13. Galach M. Dynamics of the tumor-immune system competition-the effect of time delay. Int. J. Appl. Math. Comput. Sci. 2003, 13(3):395-406.
14. Hilborn R.C. Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers 2000, Oxford University Press, USA, New York. 2ed.
15. Hwang M., et al. Rule-based simulation of multi-cellular biological systems-a review of modeling techniques. Cell. Mol. Bioeng. 2009, 2(3):285-294.
16. Ideta A., et al. A mathematical model of intermittent androgen suppression for prostate cancer. J. Nonlinear Sci. 2008, 18(6):593-614.
17. Ivancevic T.T., Bottema M.J., Jain L.C. A Mathematical Model of Chaotic Attractor in Tumor Growth and Decay 2008, http://arxiv:math-ph/0601028.
18. Ivancevic T.T., Bottema M.J., Jain L.C. A Theoretical Model of Chaotic Attractor in Tumor Growth and Metastasis 2008, http://arxiv:math-ph/0601028.
19. Kirschner D., Panetta J.C. Modeling immunotherapy of the tumor-immune interaction. J. Math. Biol. 1998, 37(3):235-252.
20. Kiyoshi M., Walker Shirley M., Motowo N. Influence of organ environment on the growth, selection, and metastasis of human colon carcinoma cells in nude mice. Cancer Res. 1988, 48(1):6863-6871.
21. Kocarev L., et al. Discrete chaos-I: theory. IEEE Trans. Circuits Syst.-I: Regular Pap. 2006, 53(6):1300-1309.
22. Kuznetsov V.A., Knott G.D. Modeling tumor regrowth and immunotherapy. Math. Comput. Modelling 2001, 33(12-13):1275-1287.
23. Kuznetsov V.A., et al. Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull. Math. Biol. 1994, 56(2):295-321.
24. Ledzewicz U., Naghnaeian M., Schättler H. Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics. J. Math. Biol. 2012, 64(3):557-577.
25. Letellier C., et al. Difference equations versus differential equations, a possible equivalence for the Rössler system?. Phys. D: Nonlinear Phenom. 2004, 195(1-2):29-49.
26. Letellier C., Denis F., Aguirre L.A. What can be learned from a chaotic cancer model?. J. Theor. Biol. 2013, 322(0):7-16.
27. Liu D., Ruan S., Zhu D. Bifurcation analysis in models of tumor and immune system interactions. Discrete Continuous Dyn. Sys. B 2009, 12(1):151-168.
28. Lorenz N.E. deterministic nonperiodic flow. J. Atmos. Sci. 1963, 20(2):130-148.
29. Masuda R., et al. Two cases of a solitary fibrous tumor with different growth patterns. Nihon Kyobu Geka Gakkai Zasshi 1996, 44(12):2177-2182.
30. Mickens R.E. Nonstandard finite difference schemes for differential equations. J. Difference Equations Appl. 2002, 8(9):823-847.
31. Moghaddasi F.L., Bezak E., Marcu L. In silico modelling of tumour margin diffusion and infiltration: review of current status. Comput. Math. Methods Med. 2012, 2012:16.
32. Moghtadaei M., Golpayegani M.R.Hashemi Complex dynamic behaviors of the complex Lorenz system. Sci. Iran. 2012, 19(3):733-738.
33. Moghtadaei M., Golpayegani M.R.H., Malekzadeh R. A variable structure fuzzy neural network model of squamous dysplasia and esophageal squamous cell carcinoma based on a global chaotic optimization algorithm. J. Theor. Biol. 2012, (0).
34. Moreira J., Deutsch A. Cellular automaton models of tumor development: a critical review. Adv. Complex Syst. 2002, 5(2 & 3):247-267.
35. Obcemea C. Chaotic dynamics of tumor growth and regeneration. Unifying Themes in Complex Systems 2006, 349-354. Springer, Berlin Heidelberg. A.A. Minai, Y. Bar-Yam (Eds.).
36. Posadas E.M., Criley S.R., Coffey D.S. Chaotic oscillations in cultured cells: rat prostate cancer. Cancer Res. 1996, 56(1):3682-3688.
37. Preziosi L. Cancer Modelling and Simulation. Mathematical Biology and Medicine Series 2003, vol. 3. Chapman & Hall/CRC.
38. Rejniak K.A., Anderson A.R.A. Hybrid models of tumor growth. Wiley Interdiscip. Rev.: Syst. Biol. Med. 2011, 3(1):115-125.
39. Rew D.A. Tumour biology, chaos and nonlinear dynamics. Eur. J. Surg. Oncol. 1999, 25:86-89.
40. Robertson-Tessi M., El-Kareh A., Goriely A. A mathematical model of tumor-immune interactions. J. Theor. Biol. 2012, 294(0):56-73.
41. Ruanxiaogang, X., Huricha, R., 2003. Differential equation and cellular automata model. In: Proceedings of the 2003 IEEE International Conference on Robotics, Intelligent Systems and Signal Processing. IEEE, Changsha, China.
42. Sachs R.K., Hlatky L.R., Hahnfeldt P. Simple ODE models of tumor growth and anti-angiogenic or radiation treatment. Math. Comput. Modelling 2001, 33(12-13):1297-1305.
43. Saleem M., Agrawal T. Chaos in a tumor growth model with delayed responses of the immune system. J. App. Math. 2012, 2012:16.
44. Siu H., et al. Tumor dormancy. I. Regression of BCL1 tumor and induction of a dormant tumor state in mice chimeric at the major histocompatibility complex. J. Immunol. 1986, 137(4):1376-1382.
45. Swanson K.R., et al. Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion. J. Neurol. Sci. 2003, 216(1):1-10.
46. Tan W.-Y., Hanin L. Handbook of Cancer Models with Applications (Series in Mathematical Biology and Medicine) 2008, World Scientific Publishing Company.
47. Thomlinson R.H., Gray L.J. The histological structure of some human lung cancers and the possible implications for radiotherapy. Br. J. Cancer 1955, 9:539-549.
48. Voitikova M. Strange Attractor in Immunology of Tumor Growth 1997, http://arxiv:math-ph/0601028.
49. von Bremen H.F., Udwadia F.E., Proskurowski W. An efficient QR based method for the computation of Lyapunov exponents. Phys. D: Nonlinear Phenom. 1997, 101(1-2):1-16.