doi: 10.3934/dcdsb.2021150

The model of nutrients influence on the tumor growth

1. 

Department of Applied Mathematics, Faculty of Mechanical Engineering, University of Žilina, Žilina, 010 26, Slovak Republic

2. 

Commenius University in Bratislava, Biomedical Center Martin JFM CU, Malá Hora 4C, Martin, 036 01, Slovak Republic

* Corresponding author: Božena Dorociaková

Received  November 2020 Revised  March 2021 Early access  May 2021

In this article a model of tumor growth is considered. The model is based on the reaction-diffusion equation that describes the distribution of nutrients within the tissue. Our aim was to predict the influence of nutrients on the tumor development. In the tissue the nutrients are transformed into energy, which supports the transfer of chemical and electrical signals and also transfer and copy the information in the tumor cells. We investigate, from a mathematical point of view, under which conditions this process takes place and how it affects the evolution of the tumor.

Citation: Rudolf Olach, Vincent Lučanský, Božena Dorociaková. The model of nutrients influence on the tumor growth. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021150
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U. Forys and M. Bodnar, Time delays in proliferation process for solid avascular tumor, Math. Comput. Modelling, 37 (2003), 1201-1209.   Google Scholar

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U. Forys and M. Bodnar, Time delays in regulatory apoptosis for solid avascular tumor, Math. Comput. Modelling, 37 (2003), 1211-1220.   Google Scholar

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B. M. Levitan, Some questions of the theory of almost periodic functions (in Russian), Uspechi Matem. Nauk (N.S.), 2 (1947), 133-192.   Google Scholar

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A. MarusykD. P. TabassumP. M. AltrockV. AlmendroF. Michor and K. Polyak, Non-cell autonomous tumor-growth driving supports sub-clonal heterogeneity, Nature, 514 (2014), 54-58.   Google Scholar

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N. Navin, Tumor evolution inferred by single cell sequencing, Nature, 472 (2011), 90-94.   Google Scholar

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S. Y. ParkM. GönenH. J. KimF. Michor and K. Polyak, Cellular and genetic diversity in the progression of in situ human breast carcinomas to an invasive phenotype, JCI, 120 (2010), 636-644.  doi: 10.1172/JCI40724.  Google Scholar

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V. PoltavetsM. KochetkovaS. M. Pitson and M. S. Samuel, The role of the extracellular matrix and its molecular and cellular regulators in cancer cell plasticity, Front. Oncol., 8 (2018), 1-19.  doi: 10.3389/fonc.2018.00431.  Google Scholar

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J. Schauder, Der fixpunktsatz in functionalraümen, Studia Math., 2 (1930), 171-180.   Google Scholar

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S. Xu, Global stability of solutions to a free boundary problem of ductal carcinoma in situ, Nonlinear Analysis: RWA, 27 (2016), 238-245.  doi: 10.1016/j.nonrwa.2015.08.003.  Google Scholar

[18]

S. XuM. Bai and X. Q. Zhao, Analysis of a solid avascular tumor growth model with time delays in proliferation process, J. Math. Anal. Appl., 391 (2012), 38-47.  doi: 10.1016/j.jmaa.2012.02.034.  Google Scholar

[19]

Y. Xu, A free boundary problem model of ductal carcinoma in situ, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 337-348.  doi: 10.3934/dcdsb.2004.4.337.  Google Scholar

[20]

P. ZarogoulidisV. PapadopoulosE. MaragouliG. PapatsibasI. KarapantzosC. Bai and H. Huang, Tumor heterogenicity: Multiple needle biopsies from different lesion sites–key to successful targeted therapy and immunotherapy, Transl. Lung Cancer Res., 7 (2018), 46-48.  doi: 10.21037/tlcr.2018.01.07.  Google Scholar

show all references

References:
[1]

N. M. BadrF. Berditchevski and A. M. Shaaban, The immune microenvironment in breast carcinoma: Predictive and prognostic role in the neoadjuvant setting, Pathobiology, 86 (2019), 1-14.  doi: 10.1159/000504055.  Google Scholar

[2]

S. F. Bakhoum and D. A. Landau, Chromosomal instability as a driver of tumor heterogeneity and evolution, Cold Spring Harb. Perspect. Med., 7 (2017), 1-13.  doi: 10.1101/cshperspect.a029611.  Google Scholar

[3]

H. Byrne and M. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151-181.  doi: 10.1016/0025-5564(94)00117-3.  Google Scholar

[4]

F. Castro-GinerP. Ratcliffe and I. Tomlinson, The mini-driver model of polygenic cancer evolution, Nat. Rev. Cancer, 15 (2015), 680-685.  doi: 10.1038/nrc3999.  Google Scholar

[5]

A. S. ClearyT. L. LeonardS. A. Gestl and E. J. Gunther, Tumor cell heterogeneity maintained by cooperating subclones in Wnt-driven mammary cancers, Nature, 508 (2014), 113-117.   Google Scholar

[6]

S. Cui and S. Xu, Analysis of mathematical models for the growth of tumors with time delays in cell proliferation, J. Math. Anal. Appl., 336 (2007), 523-541.  doi: 10.1016/j.jmaa.2007.02.047.  Google Scholar

[7]

A. C. Dudley, Tumor endothelial cells, Cold Spring Harb. Perspect. Med., 2 (2012), 1-18.  doi: 10.1101/cshperspect.a006536.  Google Scholar

[8]

L. H. Erbe, Q. K. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995.  Google Scholar

[9]

U. Forys and M. Bodnar, Time delays in proliferation process for solid avascular tumor, Math. Comput. Modelling, 37 (2003), 1201-1209.   Google Scholar

[10]

U. Forys and M. Bodnar, Time delays in regulatory apoptosis for solid avascular tumor, Math. Comput. Modelling, 37 (2003), 1211-1220.   Google Scholar

[11]

B. M. Levitan, Some questions of the theory of almost periodic functions (in Russian), Uspechi Matem. Nauk (N.S.), 2 (1947), 133-192.   Google Scholar

[12]

A. MarusykD. P. TabassumP. M. AltrockV. AlmendroF. Michor and K. Polyak, Non-cell autonomous tumor-growth driving supports sub-clonal heterogeneity, Nature, 514 (2014), 54-58.   Google Scholar

[13]

N. Navin, Tumor evolution inferred by single cell sequencing, Nature, 472 (2011), 90-94.   Google Scholar

[14]

S. Y. ParkM. GönenH. J. KimF. Michor and K. Polyak, Cellular and genetic diversity in the progression of in situ human breast carcinomas to an invasive phenotype, JCI, 120 (2010), 636-644.  doi: 10.1172/JCI40724.  Google Scholar

[15]

V. PoltavetsM. KochetkovaS. M. Pitson and M. S. Samuel, The role of the extracellular matrix and its molecular and cellular regulators in cancer cell plasticity, Front. Oncol., 8 (2018), 1-19.  doi: 10.3389/fonc.2018.00431.  Google Scholar

[16]

J. Schauder, Der fixpunktsatz in functionalraümen, Studia Math., 2 (1930), 171-180.   Google Scholar

[17]

S. Xu, Global stability of solutions to a free boundary problem of ductal carcinoma in situ, Nonlinear Analysis: RWA, 27 (2016), 238-245.  doi: 10.1016/j.nonrwa.2015.08.003.  Google Scholar

[18]

S. XuM. Bai and X. Q. Zhao, Analysis of a solid avascular tumor growth model with time delays in proliferation process, J. Math. Anal. Appl., 391 (2012), 38-47.  doi: 10.1016/j.jmaa.2012.02.034.  Google Scholar

[19]

Y. Xu, A free boundary problem model of ductal carcinoma in situ, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 337-348.  doi: 10.3934/dcdsb.2004.4.337.  Google Scholar

[20]

P. ZarogoulidisV. PapadopoulosE. MaragouliG. PapatsibasI. KarapantzosC. Bai and H. Huang, Tumor heterogenicity: Multiple needle biopsies from different lesion sites–key to successful targeted therapy and immunotherapy, Transl. Lung Cancer Res., 7 (2018), 46-48.  doi: 10.21037/tlcr.2018.01.07.  Google Scholar

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