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March 2019, 24(3): 989-1005. doi: 10.3934/dcdsb.2019002

Stability analysis of a chemotherapy model with delays

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

* Corresponding author: Xiaoying Han

Received  January 2018 Revised  March 2018 Published  January 2019

Fund Project: This work was partially supported by Simons Foundation, USA (Collaboration Grants for Mathematicians No. 429717) and MINECO/FEDER, EU (Project No. MTM2015-63723-P)

A chemotherapy model for cancer treatment is studied, where the chemotherapy agent and cells are assumed to follow a predator-prey type relation. The time delays from the instant that the chemotherapy agent is injected to the instant that the treatment is effective are taken into account and dynamics of systems with or without delays are compared. First basic properties of solutions including existence and uniqueness, boundedness and positiveness are discussed. Then conditions on model parameters are established for different outcomes of the treatment. Numerical simulations are provided to illustrate theoretical results.

Citation: Ismail Abdulrashid, Abdallah A. M. Alsammani, Xiaoying Han. Stability analysis of a chemotherapy model with delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 989-1005. doi: 10.3934/dcdsb.2019002
References:
[1]

A. F. ChambersA. C. Groom and I. C. MacDonald, Metastasis: Dissemination and growth of cancer cells in metastatic sites, Nature Rev. Cancer, 2 (2002), 563-572. doi: 10.1038/nrc865.

[2]

M. I. S. Costa and J. L. Boldrini, Chemotherapeutic treatments: A study of the interplay among drugs resistance, toxicity and recuperation from side effects, Bull. Math. Biol., 59 (1997), 205-232. doi: 10.1007/BF02462001.

[3]

L. de Pillis, A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Res., 65 (2005), 7950-7958.

[4]

L. G. de PillisW. Gu and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretaions, J. Theoret. Bio., 238 (2006), 841-862. doi: 10.1016/j.jtbi.2005.06.037.

[5]

R. T. Dorr and D. D. Von Hoff, Cancer Chemotherapy Handbook, Appleton and Lange, Connecticut, 1994.

[6]

M. Eisen, Mathematical Models in Cell Biology and Cancer Chemotherapy, Lect. Notes Biomath., Vol. 30, Springer-Verlag, Berlin, 1979.

[7]

J. K. Hale, Theory of Functional Differential Equations, Springer, New York, NY, Second Edition, 1977.

[8]

X. Han, Dynamical analysis of chemotherapy model with time-dependent infusion, Nonlinear Analysis: Real World Applications, 34 (2017), 459-480. doi: 10.1016/j.nonrwa.2016.09.001.

[9] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston MA, 1993.
[10]

A. LopezJ. Seoane and M. Sanjuan, A validated mathematical model of tumor growth including tumor host interaction, cell-mediated immune response and chemotherapy, Bull. Math. Biol., 76 (2014), 2884-2906. doi: 10.1007/s11538-014-0037-5.

[11]

J. M. Murray, Optimal drug regimens in cancer chemotherapy for single drug that block progression through the cell cycle, Math. Biosci., 123 (1994), 183-213. doi: 10.1016/0025-5564(94)90011-6.

[12]

F. K. Nani and H. I. Freedman, A mathematical model of cancer treatment by immunotherapy, Math. Biosci., 163 (2000), 159-199. doi: 10.1016/S0025-5564(99)00058-9.

[13]

J. C. Panetta and J. Adam, A mathematical model of cycle-specific chemotherapy, Math. Comput. Modeling, 22 (1995), 67-82.

[14]

S. T. R. PinhoH. I. Freedman and F. Nani, A chemotherapy model for the treatment of cancer with metastasis, Math. Comput. Modeling, 36 (2002), 773-803. doi: 10.1016/S0895-7177(02)00227-3.

[15]

V. Sree Hari Rao and P. Raja Sekhara Rao, Dynamic Models and Control of Biological Systems, Springer-Verlag, New York, NY, First Edition, 2009.

[16]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer-Verlag, 2011. doi: 10.1007/978-1-4419-7646-8.

[17]

E. D. Sontag, Lecture Notes in Mathematical Biology, 2006.

[18]

T. E. Wheldon, Mathematical Models in Cancer Research, Adam Hilger, Bristol, 1988.

show all references

References:
[1]

A. F. ChambersA. C. Groom and I. C. MacDonald, Metastasis: Dissemination and growth of cancer cells in metastatic sites, Nature Rev. Cancer, 2 (2002), 563-572. doi: 10.1038/nrc865.

[2]

M. I. S. Costa and J. L. Boldrini, Chemotherapeutic treatments: A study of the interplay among drugs resistance, toxicity and recuperation from side effects, Bull. Math. Biol., 59 (1997), 205-232. doi: 10.1007/BF02462001.

[3]

L. de Pillis, A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Res., 65 (2005), 7950-7958.

[4]

L. G. de PillisW. Gu and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretaions, J. Theoret. Bio., 238 (2006), 841-862. doi: 10.1016/j.jtbi.2005.06.037.

[5]

R. T. Dorr and D. D. Von Hoff, Cancer Chemotherapy Handbook, Appleton and Lange, Connecticut, 1994.

[6]

M. Eisen, Mathematical Models in Cell Biology and Cancer Chemotherapy, Lect. Notes Biomath., Vol. 30, Springer-Verlag, Berlin, 1979.

[7]

J. K. Hale, Theory of Functional Differential Equations, Springer, New York, NY, Second Edition, 1977.

[8]

X. Han, Dynamical analysis of chemotherapy model with time-dependent infusion, Nonlinear Analysis: Real World Applications, 34 (2017), 459-480. doi: 10.1016/j.nonrwa.2016.09.001.

[9] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston MA, 1993.
[10]

A. LopezJ. Seoane and M. Sanjuan, A validated mathematical model of tumor growth including tumor host interaction, cell-mediated immune response and chemotherapy, Bull. Math. Biol., 76 (2014), 2884-2906. doi: 10.1007/s11538-014-0037-5.

[11]

J. M. Murray, Optimal drug regimens in cancer chemotherapy for single drug that block progression through the cell cycle, Math. Biosci., 123 (1994), 183-213. doi: 10.1016/0025-5564(94)90011-6.

[12]

F. K. Nani and H. I. Freedman, A mathematical model of cancer treatment by immunotherapy, Math. Biosci., 163 (2000), 159-199. doi: 10.1016/S0025-5564(99)00058-9.

[13]

J. C. Panetta and J. Adam, A mathematical model of cycle-specific chemotherapy, Math. Comput. Modeling, 22 (1995), 67-82.

[14]

S. T. R. PinhoH. I. Freedman and F. Nani, A chemotherapy model for the treatment of cancer with metastasis, Math. Comput. Modeling, 36 (2002), 773-803. doi: 10.1016/S0895-7177(02)00227-3.

[15]

V. Sree Hari Rao and P. Raja Sekhara Rao, Dynamic Models and Control of Biological Systems, Springer-Verlag, New York, NY, First Edition, 2009.

[16]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer-Verlag, 2011. doi: 10.1007/978-1-4419-7646-8.

[17]

E. D. Sontag, Lecture Notes in Mathematical Biology, 2006.

[18]

T. E. Wheldon, Mathematical Models in Cancer Research, Adam Hilger, Bristol, 1988.

Figure 1.  Chemotherapy with delays approaching the axial steady state
Figure 2.  Comparison of normal and cancer cells of chemotherapy with/without delays
Figure 3.  Chemotherapy with delays approaching a preferred steady state
Figure 4.  Comparison of normal and cancer cells of chemotherapy with/without delays
Figure 5.  Chemotherapy with delays approaching a failure steady state
Figure 6.  Comparison of normal and cancer cells of chemotherapy with/without delays
Table 1.  Description of parameters in the chemotherapy model
Parameter Description
$ D $ Injection rate of the chemotherapy agent
$ I $ Injection concentration of the chemotherapy agent
$ {\alpha} $ Killing rate of the chemotherapy agent on cells
$ \delta $ Intraspecific competition coefficient between cancer and normal cells
$ \beta_{1} $ Intrinsic growth rate of cancer cells
$ \beta_{2} $ Intrinsic growth rate of normal cells
$ \kappa_{1} $ Environmental carrying capacity of cancer cells
$ \kappa_{2} $ Environmental carrying capacity of normal cells
$ \gamma_1 $ Effectiveness of chemotherapy agent on cancer cells
$ \gamma_2 $ Effectiveness of chemotherapy agent on normal cells
Parameter Description
$ D $ Injection rate of the chemotherapy agent
$ I $ Injection concentration of the chemotherapy agent
$ {\alpha} $ Killing rate of the chemotherapy agent on cells
$ \delta $ Intraspecific competition coefficient between cancer and normal cells
$ \beta_{1} $ Intrinsic growth rate of cancer cells
$ \beta_{2} $ Intrinsic growth rate of normal cells
$ \kappa_{1} $ Environmental carrying capacity of cancer cells
$ \kappa_{2} $ Environmental carrying capacity of normal cells
$ \gamma_1 $ Effectiveness of chemotherapy agent on cancer cells
$ \gamma_2 $ Effectiveness of chemotherapy agent on normal cells
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