A Mathematical Model on the Immune System Role in Achieving Better Outcomes of Cancer Chemotherapy

Wesley F. F. M. Gil, Tiago Carvalho, Paulo F. A. Mancera, Diego Samuel Rodrigues

Abstract


The incidence of neoplastic diseases shows that the search for more effective cancer treatments is still necessary. In addition to standard treatments such as chemo- and radiotherapy, new treatment modalities have been focused in recent advances in immunology. Since little has been discussed about the biological implications of chemotherapy with respect to its impact on the immune system and on the other normal, ``healthy'' cells, we devote the present mathematical modeling work to do so. First, we prove the invariance of the region where all state variables remain positive (i.e., the number of cancer cells, normal cells and immune cells, and the amount of chemotherapy); afterwards, we analyze the model in terms of the linear stability of the system, and we establish a necessary and sufficient condition for the local stability of the {\it{cure}} equilibrium point. Moreover, we simulate some scenarios involving both the immune system and chemotherapy, showing that a reasonable treatment strategy occurs when these are combined suitably.

Keywords


Cancer, Immune System, Chemotherapy, Mathematical Modeling, Ordinary Differential Equations.

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References


begin{thebibliography}{10}

bibitem{wwh}

{World Health Organization}, ``Cancer.''

newblock http://www.who.int/topics/cancer/en/. Accessed: August 2, 2014.

bibitem{pillis:2003b}

L.~G. de~Pillis and A.~Radunskaya, ``A mathematical model of immune response to

tumor invasion,'' in {em Computational {Fluid} and {Solid} {Mechanics} 2003}

(K.~J. Bathe, ed.), (Oxford), pp.~1661--1668, Elsevier Science Ltd, 2003.

bibitem{pillis:2005}

L.~G. de~Pillis, A.~E. Radunskaya, and C.~L. Wiseman, ``A validated

mathematical model of cell-mediated immune response to tumor growth,'' {em

Cancer Research}, vol.~65, pp.~7950--7958, Sept. 2005.

bibitem{pillis:2001}

G.~L. De~Pillis and A.~Radunskaya, ``A mathematical tumour model with immune

resistance and drug therapy: An optimal control approach,'' {em Journal of

Theoretical Medicine}, vol.~3, no.~2, pp.~79--100, 2001.

bibitem{pillis:2007}

L.~G. de~Pillis, W.~Gu, K.~R. Fister, T.~Head, K.~Maples, A.~Murugan, T.~Neal,

and K.~Yoshida, ``Chemotherapy for tumors: {An} analysis of the dynamics and

a study of quadratic and linear optimal controls,'' {em Mathematical

Biosciences}, vol.~209, pp.~292--315, Sept. 2007.

bibitem{kuznetsov:1994}

V.~A. Kuznetsov, I.~A. Makalkin, M.~A. Taylor, and A.~S. Perelson, ``Nonlinear

dynamics of immunogenic tumors: {Parameter} estimation and global bifurcation

analysis,'' {em Bulletin of Mathematical Biology}, vol.~56, pp.~295--321,

Mar. 1994.

bibitem{fassoni2016}

A.~C. Fassoni, {em Modelos matem'aticos em c^ancer abordando fase inicial e

tratamento de tumor avascular}.

newblock PhD thesis, Universidade Estadual de Campinas, Campinas, 2016.

bibitem{rodrigues:2018}

D.~S. Rodrigues, P.~F. de~Arruda~Mancera, T.~de~Carvalho, and L.~F.

Gonc{c}alves, ``A mathematical model for chemoimmunotherapy of chronic

lymphocytic leukemia,'' 2018.

newblock arXiv:1806.05790.

bibitem{vaidya:1982}

V.~G. Vaidya and F.~J. Alexandro~Jr, ``Evaluation of some mathematical models

for tumor growth,'' {em International Journal of Bio-Medical Computing},

vol.~13, pp.~19--35, Jan. 1982.

bibitem{rodrigues:2013}

D.~S. Rodrigues and P.~F.~A. Mancera, ``Mathematical analysis and simulations

involving chemotherapy and surgery on large human tumours under a suitable

cell-kill functional response,'' {em Mathematical Biosciences &

Engineering}, vol.~10, pp.~221--234, Dec. 2013.

bibitem{rodrigues:2016}

D.~S. Rodrigues, P.~F.~A. Mancera, and S.~T.~R. Pinho, ``Understanding the

antiangiogenic effect of metronomic chemotherapy through a simple

mathematical model,'' {em Physica A: Statistical Mechanics and its

Applications}, vol.~464, pp.~251--266, Dec. 2016.

bibitem{martin:1993}

R.~Martin and K.~L. Teo, {em Optimal {Control} of {Drug} {Administration} in

{Cancer} {Chemotherapy}}.

newblock Singapore: World Scientific Publishing Company, 1993.

bibitem{kay2007combination}

N.~E. Kay, S.~M. Geyer, T.~G. Call, T.~D. Shanafelt, C.~S. Zent, D.~F. Jelinek,

R.~Tschumper, N.~D. Bone, G.~W. Dewald, T.~S. Lin, N.~A. Heerema, L.~Smith,

M.~R. Grever, and J.~C. Byrd, ``Combination chemoimmunotherapy with

pentostatin, cyclophosphamide, and rituximab shows significant clinical

activity with low accompanying toxicity in previously untreated {B} chronic

lymphocytic leukemia,'' {em Blood}, vol.~109, pp.~405--411, Jan. 2007.

bibitem{awan:2016}

F.~T. Awan, ``Cure for {CLL}?,'' {em Blood}, vol.~127, pp.~274--274, Jan.

bibitem{mosteller}

R.~D. Mosteller, ``Simplified calculation of body-surface area,'' {em New

England Journal of Medicine}, vol.~317, p.~1098, Oct. 1987.

end{thebibliography}




DOI: https://doi.org/10.5540/tema.2019.020.02.343

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TEMA - Trends in Applied and Computational Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)
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