Mathematical Biosciences and Engineering, 2009, 6(3): 573-583. doi: 10.3934/mbe.2009.6.573.

Primary: 37N25, 62P10; Secondary: 35B40.

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

On the global dynamics of a model for tumor immunotherapy

1. Department of Microbiology and Immunology, The University of Michigan Medical School, Ann Arbor, MI 48109-0620
2. Unité de mathématiques pures et appliquées, Ecole Normale Supérieure de Lyon, Lyon, F-69364 LYON Cedex 07

Understanding the dynamics of human hosts and tumors is of critical importance. A mathematical model was developed that explored the immune response to tumors that was used to study a special type of treatment [3]. This treatment approach uses elements of the host to boost its immune response in the hopes that the host can clear the tumor. This model was extensively studied using numerical simulation, however no global analytical results were originally presented. In this work we explore the global dynamics to show under what conditions tumor clearance can be achieved.
  Figure/Table
  Supplementary
  Article Metrics

Keywords cancer; Lyapunov functions; immunotherapy; dynamical systems.

Citation: Denise E. Kirschner, Alexei Tsygvintsev. On the global dynamics of a model for tumor immunotherapy. Mathematical Biosciences and Engineering, 2009, 6(3): 573-583. doi: 10.3934/mbe.2009.6.573

 

This article has been cited by

  • 1. Andrea Minelli, Francesco Topputo, Franco Bernelli-Zazzera, Controlled Drug Delivery in Cancer Immunotherapy: Stability, Optimization, and Monte Carlo Analysis, SIAM Journal on Applied Mathematics, 2011, 71, 6, 2229, 10.1137/100815190
  • 2. Konstantin E. Starkov, Luis N. Coria, Global dynamics of the Kirschner–Panetta model for the tumor immunotherapy, Nonlinear Analysis: Real World Applications, 2013, 14, 3, 1425, 10.1016/j.nonrwa.2012.10.006
  • 3. N. S. Ravindran, M. Mohamed Sheriff, P. Krishnapriya, Analysis of tumour-immune evasion with chemo-immuno therapeutic treatment with quadratic optimal control, Journal of Biological Dynamics, 2017, 11, 1, 480, 10.1080/17513758.2017.1381280
  • 4. Deming Zhu, Shigui Ruan, Dan Liu, Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions, Mathematical Biosciences and Engineering, 2012, 9, 2, 347, 10.3934/mbe.2012.9.347
  • 5. Shiferaw Feyissa, Sandip Banerjee, Delay-induced oscillatory dynamics in humoral mediated immune response with two time delays, Nonlinear Analysis: Real World Applications, 2013, 14, 1, 35, 10.1016/j.nonrwa.2012.05.001
  • 6. J. Leonel Rocha, Sandra M. Aleixo, An extension of Gompertzian growth dynamics: Weibull and Fréchet models, Mathematical Biosciences and Engineering, 2013, 10, 2, 379, 10.3934/mbe.2013.10.379
  • 7. Mitra Shojania Feizabadi, Tarynn M Witten, Modeling drug resistance in a conjoint normal-tumor setting, Theoretical Biology and Medical Modelling, 2015, 12, 1, 10.1186/1742-4682-12-3
  • 8. Konstantin E. Starkov, Alexander P. Krishchenko, Ultimate dynamics of the Kirschner–Panetta model: Tumor eradication and related problems, Physics Letters A, 2017, 381, 39, 3409, 10.1016/j.physleta.2017.08.048
  • 9. Amine Hamdache, Smahane Saadi, A stochastic nominal control optimizing the adoptive immunotherapy for cancer using tumor-infiltrating lymphocytes, International Journal of Dynamics and Control, 2017, 5, 3, 783, 10.1007/s40435-016-0228-z
  • 10. F.A. Rihan, D.H. Abdel Rahman, S. Lakshmanan, A.S. Alkhajeh, A time delay model of tumour–immune system interactions: Global dynamics, parameter estimation, sensitivity analysis, Applied Mathematics and Computation, 2014, 232, 606, 10.1016/j.amc.2014.01.111
  • 11. Marc E. Songolo, Issa Ramadhani, Analysis of a mathematical model for cancer treatment by the nonstandard finite difference methods, Journal of Difference Equations and Applications, 2017, 23, 7, 1222, 10.1080/10236198.2017.1318863
  • 12. Alberto d’Onofrio, Francesca Gatti, Paola Cerrai, Luca Freschi, Delay-induced oscillatory dynamics of tumour–immune system interaction, Mathematical and Computer Modelling, 2010, 51, 5-6, 572, 10.1016/j.mcm.2009.11.005
  • 13. Giulio Caravagna, Alberto d’Onofrio, Paolo Milazzo, Roberto Barbuti, Tumour suppression by immune system through stochastic oscillations, Journal of Theoretical Biology, 2010, 265, 3, 336, 10.1016/j.jtbi.2010.05.013
  • 14. Murad Shibli, In Vivo Dynamic Image Characterization of Brain Tumor Growth Using Singular Value Decomposition and Eigenvalues, Journal of Biomedical Science and Engineering, 2011, 04, 03, 187, 10.4236/jbise.2011.43026
  • 15. KONSTANTIN E. STARKOV, ALEXANDER YU. POGROMSKY, ON THE GLOBAL DYNAMICS OF THE OWEN–SHERRATT MODEL DESCRIBING THE TUMOR–MACROPHAGE INTERACTIONS, International Journal of Bifurcation and Chaos, 2013, 23, 02, 1350020, 10.1142/S021812741350020X
  • 16. Alexei Tsygvintsev, Sandip Banerjee, Bounded immune response in immunotherapy described by the deterministic delay Kirschner–Panetta model, Applied Mathematics Letters, 2014, 35, 90, 10.1016/j.aml.2013.11.006
  • 17. F. A. Rihan, D. H. Abdelrahman, F. Al-Maskari, F. Ibrahim, M. A. Abdeen, Delay Differential Model for Tumour-Immune Response with Chemoimmunotherapy and Optimal Control, Computational and Mathematical Methods in Medicine, 2014, 2014, 1, 10.1155/2014/982978
  • 18. Sumana Ghosh, Sandip Banerjee, Mathematical modeling of cancer–immune system, considering the role of antibodies, Theory in Biosciences, 2018, 137, 1, 67, 10.1007/s12064-018-0261-x
  • 19. Konstantin E. Starkov, Alexander P. Krishchenko, On the global dynamics of one cancer tumour growth model, Communications in Nonlinear Science and Numerical Simulation, 2014, 19, 5, 1486, 10.1016/j.cnsns.2013.09.023
  • 20. Amine Hamdache, Smahane Saadi, Ilias Elmouki, Nominal and neighboring-optimal control approaches to the adoptive immunotherapy for cancer, International Journal of Dynamics and Control, 2016, 4, 3, 346, 10.1007/s40435-015-0205-y
  • 21. Murad Al-Shibli, Generalized electro-biothermo-fluidic and dynamicalmodeling of cancer growth: state-feedback controlled cesium therapy approach, Journal of Biomedical Science and Engineering, 2011, 04, 09, 569, 10.4236/jbise.2011.49073
  • 22. Konstantin E. Starkov, On dynamic tumor eradication conditions under combined chemical/anti-angiogenic therapies, Physics Letters A, 2018, 382, 6, 387, 10.1016/j.physleta.2017.12.025
  • 23. Yueping Dong, Rinko Miyazaki, Yasuhiro Takeuchi, Mathematical modeling on helper T cells in a tumor immune system, Discrete and Continuous Dynamical Systems - Series B, 2013, 19, 1, 55, 10.3934/dcdsb.2014.19.55
  • 24. P. Krishnapriya, M. Pitchaimani, Optimal control of mixed immunotherapy and chemotherapy of tumours with discrete delay, International Journal of Dynamics and Control, 2017, 5, 3, 872, 10.1007/s40435-015-0221-y
  • 25. Alberto d’Onofrio, , Systems Biology of Tumor Dormancy, 2013, Chapter 7, 111, 10.1007/978-1-4614-1445-2_7
  • 26. Alexei Tsygvintsev, Simeone Marino, Denise E. Kirschner, , Mathematical Methods and Models in Biomedicine, 2013, Chapter 13, 367, 10.1007/978-1-4614-4178-6_13
  • 27. Renee Brady, Heiko Enderling, Mathematical Models of Cancer: When to Predict Novel Therapies, and When Not to, Bulletin of Mathematical Biology, 2019, 10.1007/s11538-019-00640-x
  • 28. Liuyong Pang, Sanhong Liu, Xinan Zhang, Tianhai Tian, Mathematical modeling and dynamic analysis of anti-tumor immune response, Journal of Applied Mathematics and Computing, 2019, 10.1007/s12190-019-01292-9
  • 29. Azadeh Aghaeeyan, Mohammad Javad Yazdanpanah, Jamshid Hadjati, A New Tumor-Immunotherapy Regimen based on Impulsive Control Strategy, Biomedical Signal Processing and Control, 2020, 57, 101763, 10.1016/j.bspc.2019.101763

Reader Comments

your name: *   your email: *  

Copyright Info: 2009, , licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved