A Mathematical Model on the Immune System Role in Achieving Better Outcomes of Cancer Chemotherapy
DOI:
https://doi.org/10.5540/tema.2019.020.02.343Palavras-chave:
Cancer, Immune System, Chemotherapy, Mathematical Modeling, Ordinary Differential Equations.Resumo
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.Referências
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}
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