Functions | Biological meaning |
Growth rate of the tumor | |
Functional response | |
External inflow of effector cells | |
Tumor-stimulated proliferation rate of effector cells | |
Tumor-induced loss of effector cells | |
Influx of effector cells | |
Immunotherapy |
In this paper, we consider a mathematical model of a tumor-immune system interaction when a periodic immunotherapy treatment is applied. We give sufficient conditions, using averaging theory, for the existence and stability of periodic solutions in such system as a function of the six parameters associated to this problem. Finally, we provide examples where our results are applied.
Citation: |
Table 1. Definition of the parameters in model (2)
Functions | Biological meaning |
Growth rate of the tumor | |
Functional response | |
External inflow of effector cells | |
Tumor-stimulated proliferation rate of effector cells | |
Tumor-induced loss of effector cells | |
Influx of effector cells | |
Immunotherapy |
Table 2. Definition of the parameters in model (3)
Parameter | Biological meaning |
Intrinsic growth rate of the tumor | |
Death malignant cells rate due to interaction with lymphocyte cells | |
Increased lymphocyte rate due to interaction with malignant cells | |
Death rate of the lymphocytes | |
Immunosuppression coefficient | |
Influx external of effector cells | |
Immunotherapy dosage frequency |
[1] |
P. Amster, L. Berezansky and L. Idels, Periodic solutions of angiogenesis models with time lags, Nonlinear Analysis: Real World Applications, 13 (2012), 299-311.
doi: 10.1016/j.nonrwa.2011.07.035.![]() ![]() ![]() |
[2] |
A. d'Onofrio, A general framework for modeling tumor-inmune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D: Nonlinear Phenomena, 208 (2005), 220-235.
doi: 10.1016/j.physd.2005.06.032.![]() ![]() ![]() |
[3] |
A. d'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Math. Comput. Model., 47 (2008), 614-637.
doi: 10.1016/j.mcm.2007.02.032.![]() ![]() ![]() |
[4] |
D. I. Gabrilovich, Combination of chemotherapy and immunotherapy for cancer: A paradigm revisited, Lancet Oncology, 8 (2007), 2-3.
doi: 10.1016/S1470-2045(06)70985-8.![]() ![]() |
[5] |
V. A. Kuznetsov, I. A. Makalkin, M. Taylor and A. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321.
doi: 10.1007/BF02460644.![]() ![]() |
[6] |
Z. Liu and C. Yang, A mathematical model of cancer treatment by radiotherapy, Comput. Math. Meth. Med., 124 (2014), 1-12.
doi: 10.1155/2014/172923.![]() ![]() ![]() |
[7] |
O. Sotolongo-Costa, L. Morales-Molina, D. Rodríguez-Pérez, J. C. Antonraz and M. Chacón-Reyes, Behaviour of tumors under nonstationary therapy, Physica D: Nonlinear Phenomena, 178 (2003), 242-253.
doi: 10.1016/S0167-2789(03)00005-8.![]() ![]() ![]() |
[8] |
F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, 2$^{nd}$ edition, Universitext, Springer-Verlag, Berlin Heidelberg, 1996.
doi: 10.1007/978-3-642-61453-8.![]() ![]() ![]() |
Intersection between the graph of the function
Intersection between the graph of the function
Intersection between the graph of the function
Intersection between the graph of the function
Intersection between the graph of the function
Intersection between the graph of the function
Malignant cells
Malignant cells
Malignant cells
Malignant cells
Malignant cells
Malignant cells
Malignant cells