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April  2020, 19(4): 1875-1890. doi: 10.3934/cpaa.2020082

A mathematical model of chemotherapy with variable infusion

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

*Corresponding author

Dedicated to Prof. Dr. Tomás Caraballo's 60th birthday

Received  June 2019 Revised  September 2019 Published  January 2020

Fund Project: This work was partially supported by Simons Foundation, USA (Collaboration Grants for Mathematicians No. 429717).

A nonautonomous mathematical model of chemotherapy cancer treatment with time-dependent infusion concentration of the chemotherapy agent is developed and studied. In particular, a mutual inhibition type model is adopted to describe the interactions between the chemotherapy agent and cells, in which the chemotherapy agent is modeled as the prey being consumed by both cancer and normal cells, thereby reducing the population of both. Properties of solutions and detailed dynamics of the nonautonomous system are investigated, and conditions under which the treatment is successful or unsuccessful are established. It can be shown both theoretically and numerically that with the same amount of chemotherapy agent infused during the same period of time, a treatment with variable infusion may over perform a treatment with constant infusion.

Citation: Ismail Abdulrashid, Xiaoying Han. A mathematical model of chemotherapy with variable infusion. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1875-1890. doi: 10.3934/cpaa.2020082
References:
[1]

I. Abdulrashid, A. M. A. Abdallah and X. Han, Stability analysis of a chemotherapy model with delays, 2019.

[2]

I. Abdulrashid, T. Caraballo and X. Han, Effects of delays in mathematical models of cancer chemotherapy, preprint.

[3]

P. Boyle and B. Levin, The World Cancer Report, World Health Oganization, 2008.

[4]

T. Caraballo and X. Han, Applied Nonautonomous and Random Dynamical Systems, SpringerBriefs in Mathematics, Springer, Cham, 2016. doi: 10.1007/978-3-319-49247-6.

[5]

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. 

[6]

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

[7]

H. Cui, P. E. Kloeden and M. Yang, Forward omega limit sets of nonautonomous dynamical systems, Disc. Cont, Dyn, Sys.- S, to appear. doi: 10.3934/dcdss.2020065.

[8]

L. de PillisW. Gu and A. 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.

[9]

L. de PillisA. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Res., 65 (2005), 7950-7958. 

[10]

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

[11]

M. Eisen, Mathematical models in cell biology and cancer chemotherapy, in Lect. Notes Biomath., vol. 30

[12]

J. K. Hale, Ordinary Differential Equations, Dover Publication, Mineola, NY, 1980.

[13]

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

[14]

P. E. Kloeden and C. Potzsche, Nonautonomous Dynamical Systems in the Life Sciences, Lecture Note in Math., Springer, New York, 2013. doi: 10.1007/978-3-319-03080-7_1.

[15]

P. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, in Mathematical Surveys and Monographs, Vol. 176, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.

[16]

P. Krishnapriya and M. Pitchaimani, Optimal control of mixed immunotherapy and chemotherapy of tumors with discrete delay, Int. J. Dynam. Control, 5 (2017), 872-892.  doi: 10.1007/s40435-015-0221-y.

[17]

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.

[18]

J. Mackay and G. Mensah, The Atlas of Disease and Stroke, Published by the World Health Organization in Collaboration with the Centers for Disease Control and Prevention, 2004.

[19]

J. Murray, Optimal drug regimens in cancer chemotherapy for single drug that block progression through the cell cycle, Math. Biosci., 123 (1994), 183-213. 

[20]

F. 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.

[21]

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

[22]

S. 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.

[23]

M. S. Rajput and P. Agrawal, Microspheres in cancer therapy, Indian J. Cancer, 47 (2010), 458-468. 

[24]

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

[25]

E. D. Sontag, Lecture Notes on Mathematical Systems Biology, Northeastern University, Boston, 2018.

[26]

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

show all references

References:
[1]

I. Abdulrashid, A. M. A. Abdallah and X. Han, Stability analysis of a chemotherapy model with delays, 2019.

[2]

I. Abdulrashid, T. Caraballo and X. Han, Effects of delays in mathematical models of cancer chemotherapy, preprint.

[3]

P. Boyle and B. Levin, The World Cancer Report, World Health Oganization, 2008.

[4]

T. Caraballo and X. Han, Applied Nonautonomous and Random Dynamical Systems, SpringerBriefs in Mathematics, Springer, Cham, 2016. doi: 10.1007/978-3-319-49247-6.

[5]

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. 

[6]

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

[7]

H. Cui, P. E. Kloeden and M. Yang, Forward omega limit sets of nonautonomous dynamical systems, Disc. Cont, Dyn, Sys.- S, to appear. doi: 10.3934/dcdss.2020065.

[8]

L. de PillisW. Gu and A. 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.

[9]

L. de PillisA. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Res., 65 (2005), 7950-7958. 

[10]

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

[11]

M. Eisen, Mathematical models in cell biology and cancer chemotherapy, in Lect. Notes Biomath., vol. 30

[12]

J. K. Hale, Ordinary Differential Equations, Dover Publication, Mineola, NY, 1980.

[13]

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

[14]

P. E. Kloeden and C. Potzsche, Nonautonomous Dynamical Systems in the Life Sciences, Lecture Note in Math., Springer, New York, 2013. doi: 10.1007/978-3-319-03080-7_1.

[15]

P. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, in Mathematical Surveys and Monographs, Vol. 176, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.

[16]

P. Krishnapriya and M. Pitchaimani, Optimal control of mixed immunotherapy and chemotherapy of tumors with discrete delay, Int. J. Dynam. Control, 5 (2017), 872-892.  doi: 10.1007/s40435-015-0221-y.

[17]

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.

[18]

J. Mackay and G. Mensah, The Atlas of Disease and Stroke, Published by the World Health Organization in Collaboration with the Centers for Disease Control and Prevention, 2004.

[19]

J. Murray, Optimal drug regimens in cancer chemotherapy for single drug that block progression through the cell cycle, Math. Biosci., 123 (1994), 183-213. 

[20]

F. 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.

[21]

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

[22]

S. 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.

[23]

M. S. Rajput and P. Agrawal, Microspheres in cancer therapy, Indian J. Cancer, 47 (2010), 458-468. 

[24]

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

[25]

E. D. Sontag, Lecture Notes on Mathematical Systems Biology, Northeastern University, Boston, 2018.

[26]

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

Figure 1.  Chemotherapy with time-dependent infusion $ \mu(t) = 4 + 2 \sin 0.04 t $, resulting a successful treatment where all cancer cells are removed and normal cells remain
Figure 2.  Chemotherapy with time-dependent infusion $ \hat{\mu} = 4 $, resulting a failed treatment where all normal cells are removed and cancer cells remain
Table 1.  Description of parameters in the chemotherapy model
Parameter Description
$ b_{1} $ (1/time) Per capita growth rate of cancer cells
$ b_{2} $ (1/time) Per capita growth rate of normal cells
$ \kappa_{1} $ (mass/vol) Environmental carrying capacity of cancer cells
$ \kappa_{2} $ (mass/vol) Environmental carrying capacity of normal cells
$ d_1 $ (vol/time$ \cdot $mass) Intraspecific competition coefficient of cancer on normal cells
$ d_2 $ (vol/time$ \cdot $mass) Intraspecific competition coefficient of normal on cancer cells
$ r_1 $ (1) Consumption effectiveness of cancer cells on the agent
$ r_2 $ (1) Consumption effectiveness of normal cells on the agent
Parameter Description
$ b_{1} $ (1/time) Per capita growth rate of cancer cells
$ b_{2} $ (1/time) Per capita growth rate of normal cells
$ \kappa_{1} $ (mass/vol) Environmental carrying capacity of cancer cells
$ \kappa_{2} $ (mass/vol) Environmental carrying capacity of normal cells
$ d_1 $ (vol/time$ \cdot $mass) Intraspecific competition coefficient of cancer on normal cells
$ d_2 $ (vol/time$ \cdot $mass) Intraspecific competition coefficient of normal on cancer cells
$ r_1 $ (1) Consumption effectiveness of cancer cells on the agent
$ r_2 $ (1) Consumption effectiveness of normal cells on the agent
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