American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Periodic orbits for the perturbed planar circular restricted 3–body problem
  • DCDS-B Home
  • This Issue
  • Next Article
    Preface to the special issue "Dynamics and control in distributed systems: Dedicated to the memory of Valery S. Melnik (1952-2007)"
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  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Comparison of normal and cancer cells of chemotherapy with/without delays
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Chemotherapy with delays approaching a preferred steady state
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Comparison of normal and cancer cells of chemotherapy with/without delays
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Chemotherapy with delays approaching a failure steady state
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Comparison of normal and cancer cells of chemotherapy with/without delays
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

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
[1]

Nasser Sweilam, Fathalla Rihan, Seham AL-Mekhlafi. A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020120
[2]

Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445
[3]

Urszula Foryś, Jan Poleszczuk. A delay-differential equation model of HIV related cancer--immune system dynamics. Mathematical Biosciences & Engineering, 2011, 8 (2) : 627-641. doi: 10.3934/mbe.2011.8.627
[4]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689
[5]

Avner Friedman, Harsh Vardhan Jain. A partial differential equation model of metastasized prostatic cancer. Mathematical Biosciences & Engineering, 2013, 10 (3) : 591-608. doi: 10.3934/mbe.2013.10.591
[6]

Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations & Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493
[7]

Christoph Sadée, Eugene Kashdan. A model of thermotherapy treatment for bladder cancer. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1169-1183. doi: 10.3934/mbe.2016037
[8]

Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 129-150. doi: 10.3934/dcdsb.2006.6.129
[9]

Urszula Ledzewicz, Heinz Schättler. Controlling a model for bone marrow dynamics in cancer chemotherapy. Mathematical Biosciences & Engineering, 2004, 1 (1) : 95-110. doi: 10.3934/mbe.2004.1.95
[10]

Urszula Ledzewicz, Heinz Schättler, Shuo Wang. On the role of tumor heterogeneity for optimal cancer chemotherapy. Networks & Heterogeneous Media, 2019, 14 (1) : 131-147. doi: 10.3934/nhm.2019007
[11]

Eugen Stumpf. On a delay differential equation arising from a car-following model: Wavefront solutions with constant-speed and their stability. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3317-3340. doi: 10.3934/dcdsb.2017139
[12]

Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529
[13]

Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361
[14]

Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577
[15]

Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993
[16]

Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099
[17]

P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220
[18]

Urszula Ledzewicz, Heinz Schättler, Mostafa Reisi Gahrooi, Siamak Mahmoudian Dehkordi. On the MTD paradigm and optimal control for multi-drug cancer chemotherapy. Mathematical Biosciences & Engineering, 2013, 10 (3) : 803-819. doi: 10.3934/mbe.2013.10.803
[19]

Wei Feng, Shuhua Hu, Xin Lu. Optimal controls for a 3-compartment model for cancer chemotherapy with quadratic objective. Conference Publications, 2003, 2003 (Special) : 544-553. doi: 10.3934/proc.2003.2003.544
[20]

Urszula Ledzewicz, Heinz Schättler. The Influence of PK/PD on the Structure of Optimal Controls in Cancer Chemotherapy Models. Mathematical Biosciences & Engineering, 2005, 2 (3) : 561-578. doi: 10.3934/mbe.2005.2.561

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (219)
  • HTML views (146)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences