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

Stability analysis of a chemotherapy model with delays

Ismail Abdulrashid , Abdallah A. M. Alsammani and  Xiaoying Han ,

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

* Corresponding author: Xiaoying Han

Received  January 2018 Revised  March 2018 Published  March 2019 Early access  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.

Keywords: Chemotherapy cancer treatment, delay differential equation, local stability.
Mathematics Subject Classification: Primary: 34D23, 34K20; Secondary: 92B05, 93D05.
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. Chambers, A. 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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

L. G. de Pillis, W. 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.  Google Scholar

[5]

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

[6]

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

[7]

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

[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.  Google Scholar

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

A. Lopez, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

S. T. R. Pinho, H. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

show all references

References:
[1]

A. F. Chambers, A. 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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

L. G. de Pillis, W. 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.  Google Scholar

[5]

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

[6]

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

[7]

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

[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.  Google Scholar

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

A. Lopez, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

S. T. R. Pinho, H. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

Figure 1.  Chemotherapy with delays approaching the axial steady state
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Figure 2.  Comparison of normal and cancer cells of chemotherapy with/without delays
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Figure 3.  Chemotherapy with delays approaching a preferred steady state
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Figure 4.  Comparison of normal and cancer cells of chemotherapy with/without delays
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Figure 5.  Chemotherapy with delays approaching a failure steady state
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Figure 6.  Comparison of normal and cancer cells of chemotherapy with/without delays
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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
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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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