doi: 10.3934/dcdsb.2021219
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The impact of time delay and angiogenesis in a tumor model

School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China

*Corresponding author: Huijuan Song

Received  March 2021 Revised  June 2021 Early access September 2021

We consider a free boundary tumor model under the presence of angiogenesis and time delays in the process of proliferation, in which the cell location is incorporated. It is assumed that the tumor attracts blood vessels at a rate proportional to $ \alpha $, and a parameter $ \mu $ is proportional to the 'aggressiveness' of the tumor. In this paper, we first prove that there exists a unique radially symmetric stationary solution $ \left(\sigma_{*}, p_{*}, R_{*}\right) $ for all positive $ \alpha $, $ \mu $. Then a threshold value $ \mu_\ast $ is found such that the radially symmetric stationary solution is linearly stable if $ \mu<\mu_\ast $ and linearly unstable if $ \mu>\mu_\ast $. Our results indicate that the increase of the angiogenesis parameter $ \alpha $ would result in the reduction of the threshold value $ \mu_\ast $, adding the time delay would not alter the threshold value $ \mu_\ast $, but would result in a larger stationary tumor, and the larger the tumor aggressiveness parameter $ \mu $ is, the greater impact of time delay would have on the size of the stationary tumor.

Citation: Zejia Wang, Haihua Zhou, Huijuan Song. The impact of time delay and angiogenesis in a tumor model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021219
References:
[1]

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[23]

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F. ZhouJ. Escher and S. Cui, Bifurcation for a free boundary problem with surface tension modeling the growth of multi-layer tumor, J. Math. Anal. Appl., 337 (2008), 443-457.  doi: 10.1016/j.jmaa.2007.03.107.  Google Scholar

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Y. Zhuang and S. Cui, Analysis of a free boundary problem modeling the growth of spherically symmetric tumors with angiogenesis, Acta Appl. Math., 161 (2019), 153-169.  doi: 10.1007/s10440-018-0208-8.  Google Scholar

show all references

References:
[1]

R. P. Araujo and D. L. S. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling, Bull. Math. Biol., 66 (2004), 1039-1091.  doi: 10.1016/j.bulm.2003.11.002.  Google Scholar

[2]

H. M. Byrne, The effect of time delays on the dynamics of avascular tumor growth, Math. Biosci., 144 (1997), 83-117.  doi: 10.1016/S0025-5564(97)00023-0.  Google Scholar

[3]

H. M. Byrne and M. A. J. 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]

H. M. Byrne and M. A. J. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135 (1996), 187-216.  doi: 10.1016/0025-5564(96)00023-5.  Google Scholar

[5]

S. Cui, Analysis of a mathematical model for the growth of tumors under the action of external inhibitors, J. Math. Biol., 44 (2002), 395-426.  doi: 10.1007/s002850100130.  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]

S. Cui and Y. Zhuang, Bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesis, J. Math. Anal. Appl., 468 (2018), 391-405.  doi: 10.1016/j.jmaa.2018.08.022.  Google Scholar

[8]

M. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal., 35 (2003), 187-206.   Google Scholar

[9]

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

[10]

A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262-284.  doi: 10.1007/s002850050149.  Google Scholar

[11]

A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equations, 227 (2006), 598-639.  doi: 10.1016/j.jde.2005.09.008.  Google Scholar

[12]

A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem arising in a tumor model, Arch. Ration. Mech. Anal., 180 (2006), 293-330.  doi: 10.1007/s00205-005-0408-z.  Google Scholar

[13]

A. Friedman and B. Hu, Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model, Trans. Amer. Math. Soc., 360 (2008), 5291-5342.  doi: 10.1090/S0002-9947-08-04468-1.  Google Scholar

[14]

A. Friedman and F. Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth, Trans. Amer. Math. Soc., 353 (2001), 1587-1634.  doi: 10.1090/S0002-9947-00-02715-X.  Google Scholar

[15]

A. Friedman, Mathematical analysis and challenges arising from models of tumor growth, Math. Models Methods Appl. Sci., 17 (2007), 1751-1772.  doi: 10.1142/S0218202507002467.  Google Scholar

[16]

A. Friedman and K-Y. Lam, Analysis of a free-boundary tumor model with angiogenesis, J. Differential Equations, 259 (2015), 7636-7661.  doi: 10.1016/j.jde.2015.08.032.  Google Scholar

[17]

H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340.  doi: 10.1002/sapm1972514317.  Google Scholar

[18]

H. P. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theoret. Biol., 56 (1976), 229-242.  doi: 10.1016/S0022-5193(76)80054-9.  Google Scholar

[19]

Y. HuangZ. Zhang and B. Hu, Bifurcation for a free-boundary tumor model with angiogenesis, Nonlinear Anal. Real World Appl., 35 (2017), 483-502.  doi: 10.1016/j.nonrwa.2016.12.003.  Google Scholar

[20]

Y. HuangZ. Zhang and B. Hu, Bifurcation from stability to instability for a free boundary tumor model with angiogenesis, Discrete Contin. Dyn. Syst., 39 (2019), 2473-2510.  doi: 10.3934/dcds.2019105.  Google Scholar

[21]

Y. HuangZ. Zhang and B. Hu, Linear stability for a free-boundary tumor model with a periodic supply of external nutrients, Math. Methods Appl. Sci., 42 (2019), 1039-1054.  doi: 10.1002/mma.5412.  Google Scholar

[22]

Y. HuangZ. Zhang and B. Hu, Asymptotic stability for a free boundary tumor model with angiogenesis, J. Differential Equations, 270 (2021), 961-993.  doi: 10.1016/j.jde.2020.08.050.  Google Scholar

[23]

J. S. Lowengrub, H. B. Frieboes, F. Jin, Y-L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumours, Nonlinearity, 23 (2010), R1–R91. doi: 10.1088/0951-7715/23/1/R01.  Google Scholar

[24]

S. XuQ. Zhou and M. Bai, Qualitative analysis of a time-delayed free boundary problem for tumor growth under the action of external inhibitors, Math. Methods Appl. Sci., 38 (2015), 4187-4198.  doi: 10.1002/mma.3357.  Google Scholar

[25]

S. Xu and Z. Feng, Analysis of a mathematical model for tumor growth under indirect effect of inhibitors with time delay in proliferation, J. Math. Anal. Appl., 374 (2011), 178-186.  doi: 10.1016/j.jmaa.2010.08.043.  Google Scholar

[26]

S. Xu, Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation, Nonlinear Anal. Real World Appl., 11 (2010), 401-406.  doi: 10.1016/j.nonrwa.2008.11.002.  Google Scholar

[27]

S. Xu, Analysis of a free boundary problem for tumor growth with angiogenesis and time delays in proliferation, Nonlinear Anal. Real World Appl., 51 (2020), 103005. doi: 10.1016/j.nonrwa.2019.103005.  Google Scholar

[28]

X. Zhao and B. Hu, The impact of time delay in a tumor model, Nonlinear Anal. Real World Appl., 51 (2020), 103015. doi: 10.1016/j.nonrwa.2019.103015.  Google Scholar

[29]

X. Zhao and B. Hu, Symmetry-breaking bifurcation for a free-boundary tumor model with time delay, J. Differential Equations, 269 (2020), 1829-1862.  doi: 10.1016/j.jde.2020.01.022.  Google Scholar

[30]

F. ZhouJ. Escher and S. Cui, Bifurcation for a free boundary problem with surface tension modeling the growth of multi-layer tumor, J. Math. Anal. Appl., 337 (2008), 443-457.  doi: 10.1016/j.jmaa.2007.03.107.  Google Scholar

[31]

F. Zhou and S. Cui, Bifurcations for a multidimensional free boundary problem modeling the growth of tumor cord, Nonlinear Anal. Real World Appl., 10 (2009), 2990-3001.  doi: 10.1016/j.nonrwa.2008.10.004.  Google Scholar

[32]

Y. Zhuang and S. Cui, Analysis of a free boundary problem modeling the growth of multicell spheroids with angiogenesis, J. Differential Equations, 265 (2018), 620-644.  doi: 10.1016/j.jde.2018.03.005.  Google Scholar

[33]

Y. Zhuang and S. Cui, Analysis of a free boundary problem modeling the growth of spherically symmetric tumors with angiogenesis, Acta Appl. Math., 161 (2019), 153-169.  doi: 10.1007/s10440-018-0208-8.  Google Scholar

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