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Volumn 4, Issue 2, 2011, Pages 185-206

A study of differential equation modeling malignant tumor cells in competition with immune system

Author keywords

Hopf bifurcation; Immunology; modeling; ordinary and delay differential equations; stability; tumor

Indexed keywords


EID: 84857540899     PISSN: 17935245     EISSN: 17937159     Source Type: Journal    
DOI: 10.1142/S1793524511001404     Document Type: Article
Times cited : (16)

References (55)
  • 1
    • 0005448153 scopus 로고    scopus 로고
    • Effects of vascularization on lymphocyte/tumor cell dynamics: Qualitative features
    • J. A. Adam, Effects of vascularization on lymphocyte/tumor cell dynamics: qualitative features, Math. Comput. Model. 23(6) (1996) 1-10. (Pubitemid 126686205)
    • (1996) Mathematical and Computer Modelling , vol.23 , Issue.6 , pp. 1-10
    • Adam, J.A.1
  • 3
    • 0036000662 scopus 로고    scopus 로고
    • Generalized kinetic (Boltzman) models: Mathematical structures and application
    • L. Arlotti, N. Bellomo and E. De Angelis, Generalized kinetic (Boltzman) models: Mathematical structures and application, Math. Models Meth. Appl. Sci. 12 (2002) 567-592.
    • (2002) Math. Models Meth. Appl. Sci. , vol.12 , pp. 567-592
    • Arlotti, L.1    Bellomo, N.2    De Angelis, E.3
  • 4
    • 37549035821 scopus 로고    scopus 로고
    • Delay induced model for tumor-immune interaction and control of malignant tumor growth
    • S. Banerjee and R. R. Sarkar, Delay induced model for tumor-immune interaction and control of malignant tumor growth, Biosystems 91 (2007) 268-288.
    • (2007) Biosystems , vol.91 , pp. 268-288
    • Banerjee, S.1    Sarkar, R.R.2
  • 5
    • 0034254401 scopus 로고    scopus 로고
    • Modelling and mathematical problems related to tumor evolution and its interaction with the immune system
    • DOI 10.1016/S0895-7177(00)00143-6, PII S0895717700001436
    • N. Bellomo and L. Preziosi, Modeling and mathematical problems related to tumor immune system interactions, Math. Comput. Model. 31 (2000) 413-452. (Pubitemid 30610360)
    • (2000) Mathematical and Computer Modelling , vol.32 , Issue.3-4 , pp. 413-452
    • Bellomo, N.1    Preziosi, L.2
  • 7
    • 0001020135 scopus 로고    scopus 로고
    • Special issue on the modeling in applied sciences by methods of transport and kinetic theory
    • N. Bellomo and M. Pulvirenti (eds.), Special issue on the modeling in applied sciences by methods of transport and kinetic theory, Math. Comput. Model. 12 (2002) 909-990.
    • (2002) Math. Comput. Model. , vol.12 , pp. 909-990
    • Bellomo, N.1    Pulvirenti, M.2
  • 8
    • 0007894647 scopus 로고    scopus 로고
    • The nonnegativity of solutions of delay differential equations
    • M. Bodnad, The nonnegativity of solutions of delay differential equations, Appl. Math. Lett. 13 (2000) 91-95.
    • (2000) Appl. Math. Lett. , vol.13 , pp. 91-95
    • Bodnad, M.1
  • 9
    • 0000756289 scopus 로고    scopus 로고
    • Behavior of solutions to Marchuk's model depending on a time delay
    • M. Bodnar and U. Foryś, Behavior of solutions to Marchuk's model depending on a time delay, Int. Math. Comput. Sci. 10(1) (2000) 97-112.
    • (2000) Int. Math. Comput. Sci. , vol.10 , Issue.1 , pp. 97-112
    • Bodnar, M.1    Foryś, U.2
  • 10
    • 0000600543 scopus 로고    scopus 로고
    • Periodic dynamics in the model of immune system
    • M. Bodnar and U. Foryś, Periodic dynamics in the model of immune system, Appl. Math. 27(1) (2000) 113-126.
    • (2000) Appl. Math. , vol.27 , Issue.1 , pp. 113-126
    • Bodnar, M.1    Foryś, U.2
  • 12
    • 0030822132 scopus 로고    scopus 로고
    • The effect of time delays on the dynamics of avascular tumor growth
    • DOI 10.1016/S0025-5564(97)00023-0, PII S0025556497000230
    • H. M. Byrne, The effect of time delay on the dynamics of avascular tumor growth, Math. Biosci. 144(2) (1997) 83-117. (Pubitemid 27413208)
    • (1997) Mathematical Biosciences , vol.144 , Issue.2 , pp. 83-117
    • Byrne, H.M.1
  • 13
    • 0008314607 scopus 로고    scopus 로고
    • Special issue on mathematical models for the growth, development and treatment of tumors
    • M. A. J. Chaplain (eds.), Special issue on mathematical models for the growth, development and treatment of tumors, Math. Models Meth. Appl. Sci. 9 (1999).
    • (1999) Math. Models Meth. Appl. Sci. , vol.9
    • Chaplain, M.A.J.1
  • 14
    • 0000209631 scopus 로고
    • Discrete delay, distributed delay and stability switches
    • K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl. 86 (1982) 592-627.
    • (1982) J. Math. Anal. Appl. , vol.86 , pp. 592-627
    • Cooke, K.L.1    Grossman, Z.2
  • 15
    • 0017332977 scopus 로고
    • Immune surveillance and neoplasia. I. A minimal mathematical model
    • C. DeLisi and A. Rescigno, Immune surveillance and neoplasia - I: A minimal mathematical model, Bull. Math. Biol. 39 (1977) 201-221. (Pubitemid 8067322)
    • (1977) Bulletin of Mathematical Biology , vol.39 , Issue.2 , pp. 201-221
    • DeLisi, C.1    Rescigno, A.2
  • 16
    • 0021894588 scopus 로고
    • Macrophage T lymphocye interactions in the anti-tumor immune response: A mathematical model
    • R. J. DeBoer, P. Hogeweg, H. F. J. Dullens, R. A. DeWeger and W. DenOtter, Macrophage T lymphocyte interactions in the anti-tumor immune response: A mathematical model, J. Immunol. 134(4) (1985) 2748-2758. (Pubitemid 15146857)
    • (1985) Journal of Immunology , vol.134 , Issue.4 , pp. 2748-2758
    • De Boer, R.J.1    Hogeweg, P.2    Dullens, H.F.J.3
  • 21
    • 0036244827 scopus 로고    scopus 로고
    • Marchuk's model of immune system dynamics with application to tumour growth
    • DOI 10.1080/10273660290052151
    • U. Foryś, Marchuk's model of immune system dynamics with application to tumor growth, J. Theoret. Med. 4(1) (2002) 85-93. (Pubitemid 34506364)
    • (2002) Journal of Theoretical Medicine , vol.4 , Issue.1 , pp. 85-93
    • Forys, U.1
  • 24
    • 85196200749 scopus 로고    scopus 로고
    • A delay model for viral infection in toxin producing phytoplankton and zooplankton system
    • in press
    • S. Gakkhar and A. Singh, A delay model for viral infection in toxin producing phytoplankton and zooplankton system, Commun. Nonlinear Sci. Numer. Simul., in press.
    • Commun. Nonlinear Sci. Numer. Simul.
    • Gakkhar, S.1    Singh, A.2
  • 25
    • 23144456670 scopus 로고    scopus 로고
    • Dynamics of the tumor-immune system competition the effect of time delay
    • M. Galach, Dynamics of the tumor-immune system competition the effect of time delay, Int. J. Appl. Comput. Sci. 13(3) (2003) 395-406.
    • (2003) Int. J. Appl. Comput. Sci. , vol.13 , Issue.3 , pp. 395-406
    • Galach, M.1
  • 26
    • 0036561472 scopus 로고    scopus 로고
    • Complex discrete dynamics from simple continuous population models
    • DOI 10.1006/bulm.2002.0286, PII S0092824002902867
    • J. G. P. Gamarra and R. V. Solé, Complex discrete dynamics from simple continuous population models, Bull. Math. Biol. 64 (2002) 611-620. (Pubitemid 38370209)
    • (2002) Bulletin of Mathematical Biology , vol.64 , Issue.3 , pp. 611-620
    • Gamarra, J.G.P.1    Sole, R.V.2
  • 27
    • 0030417280 scopus 로고    scopus 로고
    • Tumor heterogeneity and progression: Conceptual foundations for modeling
    • L. Greller, F. Tobin and G. Poste, Tumor hetereogenity and progression: Conceptual foundation for modeling, Invasion Metastasis 16 (1996) 177-208. (Pubitemid 27399786)
    • (1996) Invasion and Metastasis , vol.16 , Issue.4-5 , pp. 177-208
    • Greller, L.D.1    Tobin, F.L.2    Poste, G.3
  • 28
    • 0024435736 scopus 로고
    • Quiescence as an explanation of Gompertzian tumor growth
    • M. Gyllenberg and G. Webb, Quiescence as an explanation of gompertzian tumor growth, Growth, Development and Aging, Report-Mat-A264, (1989), pp. 25-33. (Pubitemid 19237680)
    • (1989) Growth, Development and Aging , vol.53 , Issue.1-2 , pp. 25-33
    • Gyllenberg, M.1    Webb, G.F.2
  • 31
    • 0032160863 scopus 로고    scopus 로고
    • Modeling immunotherapy of the tumor-immune interaction
    • D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol. 37(3) (1998) 235-252.
    • (1998) J. Math. Biol. , vol.37 , Issue.3 , pp. 235-252
    • Kirschner, D.1    Panetta, J.C.2
  • 32
    • 0028388757 scopus 로고
    • Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis
    • V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol. 56(2) (1994) 295-321.
    • (1994) Bull. Math. Biol. , vol.56 , Issue.2 , pp. 295-321
    • Kuznetsov, V.A.1    Makalkin, I.A.2    Taylor, M.A.3    Perelson, A.S.4
  • 34
    • 0001682782 scopus 로고
    • A basic mathematical model of the immune response
    • H. Mayer, K. S. Zänker and U. der Heiden, A basic mathematical model of the immune response, Chaos 5(1) (1995) 155-161.
    • (1995) Chaos , vol.5 , Issue.1 , pp. 155-161
    • Mayer, H.1    Zänker, K.S.2    Der Heiden, U.3
  • 36
    • 77957176660 scopus 로고
    • Modelling and simulation of Rosenberg-type adoptive cellular immunotherapy
    • F. K. Nani and M. N. Oguztoreli, Modelling and simulation of Rosenberg-type adoptive cellular immunotherapy, IMA J. Math. Appl. Med. Biol. 11 (1994) 107-147. (Pubitemid 2112760)
    • (1994) IMA J. Math. Appl. Med. Biol. , vol.11 , Issue.2 , pp. 107-147
    • Nani, F.K.1    Oguztoreli, M.N.2
  • 37
    • 37549047236 scopus 로고    scopus 로고
    • Improving alloreactive CTL immunotherapy for malignant gliomas using a simulation model of their interactive dynamics
    • DOI 10.1007/s00262-007-0387-z
    • N. Kronik, Y. Kogan, V. Vainstein and Z. Agur, Improving alloreactive CTL immunotherapy for malignant gliomas using a simulation model of their interactive dynamics, Cancer Immunol. Immunother. 57(3) (2008) 425-439. (Pubitemid 50007307)
    • (2008) Cancer Immunology, Immunotherapy , vol.57 , Issue.3 , pp. 425-439
    • Kronik, N.1    Kogan, Y.2    Vainstein, V.3    Agur, Z.4
  • 38
    • 23144445934 scopus 로고    scopus 로고
    • A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences
    • DOI 10.1016/j.physd.2005.06.032, PII S0167278905002939
    • A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D 208 (2005) 220-235. (Pubitemid 41085168)
    • (2005) Physica D: Nonlinear Phenomena , vol.208 , Issue.3-4 , pp. 220-235
    • D'Onofrio, A.1
  • 39
    • 38849187163 scopus 로고    scopus 로고
    • Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy
    • A. d'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Math. Comput. Model. 47 (2008) 614-637.
    • (2008) Math. Comput. Model. , vol.47 , pp. 614-637
    • D'Onofrio, A.1
  • 40
    • 33746019012 scopus 로고    scopus 로고
    • Tumor evasion from immune system control: Strategies of a MISS to become a MASS
    • A. d'Onofrio, Tumor evasion from immune system control: Strategies of a MISS to become a MASS, Chaos Solitons Fractal 31 (2007) 261-268.
    • (2007) Chaos Solitons Fractal , vol.31 , pp. 261-268
    • D'Onofrio, A.1
  • 41
    • 33747088162 scopus 로고    scopus 로고
    • Tumor-immune system interaction: Modeling the tumor-stimulated proliferation of effectors and immunotherapy
    • DOI 10.1142/S0218202506001571, PII S0218202506001571
    • A. d'Onofrio, Tumor-immune system interaction: Modeling the tumor-stimulated proliferation of effectors and immunotherapy, Math. Models Meth. Appl. Sci. 16 (2006) 1375-1401. (Pubitemid 44221742)
    • (2006) Mathematical Models and Methods in Applied Sciences , vol.16 , Issue.8 , pp. 1375-1401
    • D'Onofrio, A.1
  • 43
    • 24744472075 scopus 로고    scopus 로고
    • A validated mathematical model of cell-mediated immune response to tumor growth
    • DOI 10.1158/0008-5472.CAN-05-0564
    • L. G. de Pillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Res. 65 (2005) 7950-7958. (Pubitemid 41297274)
    • (2005) Cancer Research , vol.65 , Issue.17 , pp. 7950-7958
    • De Pillis, L.G.1    Radunskaya, A.E.2    Wiseman, C.L.3
  • 44
    • 0022605668 scopus 로고
    • Cancer immunotherapy using interleukin-2 and interleukin-2-activated lymphocytes
    • S. A. Rosenberg and M. T. Lotze, Cancer immunotherapy using interleukin-2 and interleukin-2-activated lymphocytes, Ann. Rev. Immunol. 4 (1986) 681-709.
    • (1986) Ann. Rev. Immunol. , vol.4 , pp. 681-709
    • Rosenberg, S.A.1    Lotze, M.T.2
  • 46
    • 1442290947 scopus 로고    scopus 로고
    • A delay differential equation model for tumor growth
    • DOI 10.1007/s00285-003-0211-0
    • M. Villasana and A. Radunskaya, A delay differential equation model for tumour growth, J. Math. Biol. 47 (2003) 270-294. (Pubitemid 38272158)
    • (2003) Journal of Mathematical Biology , vol.47 , Issue.3 , pp. 270-294
    • Villasana, M.1    Radunskaya, A.2
  • 49
    • 67650302453 scopus 로고    scopus 로고
    • Dynamics analysis and limit cycle in a delayed model for tumor growth with quiescence
    • R. Yafia, Dynamics analysis and limit cycle in a delayed model for tumor growth with quiescence, Nonlinear Anal. Model. Contr. 11(1) (2006) 95-110.
    • (2006) Nonlinear Anal. Model. Contr. , vol.11 , Issue.1 , pp. 95-110
    • Yafia, R.1
  • 50
    • 57849145839 scopus 로고    scopus 로고
    • Periodic solutions for small and large in a tumor-immune system model
    • R. Yafia, Periodic solutions for small and large in a tumor-immune system model, Electron. J. Differential Equations 14 (2006) 241-248.
    • (2006) Electron. J. Differential Equations , vol.14 , pp. 241-248
    • Yafia, R.1
  • 51
    • 34547126471 scopus 로고    scopus 로고
    • Hopf bifurcation analysis and numerical simulations in an ODE model of the immune system with positive immune response
    • DOI 10.1016/j.nonrwa.2006.08.003, PII S146812180600099X
    • R. Yafia, Hopf bifurcation analysis and numerical simulations in an ODE model of the immune system with positive immune response, Nonlinear Anal. Real World Appl. 8 (2007) 1359-1369. (Pubitemid 47102232)
    • (2007) Nonlinear Analysis: Real World Applications , vol.8 , Issue.5 , pp. 1359-1369
    • Yafia, R.1
  • 52
    • 38049074515 scopus 로고    scopus 로고
    • Hopf bifurcation in differential equations with delay for tumor-immune system competition model
    • R. Yafia, Hopf bifurcation in differential equations with delay for tumor-immune system competition model, SIAM J. Appl. Math. 67(6) (2007) 1693-1703.
    • (2007) SIAM J. Appl. Math. , vol.67 , Issue.6 , pp. 1693-1703
    • Yafia, R.1
  • 53
    • 33749437893 scopus 로고    scopus 로고
    • Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response
    • Article ID 95296
    • R. Yafia, Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response, Discrete Dyn. Nat. Soc. 2006 (2006) 1-9 Article ID 95296.
    • (2006) Discrete Dyn. Nat. Soc. , vol.2006 , pp. 1-9
    • Yafia, R.1
  • 54
    • 33749437893 scopus 로고    scopus 로고
    • Stability of limit cycle in a delayed model for tumor-immune system competition with negative immune response
    • Article ID 58463
    • R. Yafia, Stability of limit cycle in a delayed model for tumor-immune system competition with negative immune response, Discrete Dyn. Nat. Soc. 2006 (2006) 1-13 Article ID 58463.
    • (2006) Discrete Dyn. Nat. Soc. , vol.2006 , pp. 1-13
    • Yafia, R.1
  • 55
    • 84857540275 scopus 로고    scopus 로고
    • The effect of time delay and Hopf bifurcation in a tumor-immune system competition model with negative immune response, to appear in
    • R. Yafia, The effect of time delay and Hopf bifurcation in a tumor-immune system competition model with negative immune response, to appear in Appl. Math.
    • Appl. Math
    • Yafia, R.1


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