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Qualitative analysis of a time-delayed free boundary problem for tumor growth with angiogenesis and Gibbs-Thomson relation

1 School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, Guangdong 526061, China
2 Department of Mathematics, Soochow University, Suzhou, Jiangsu 215006, China

In this paper we consider a time-delayed mathematical model describing tumor growth with angiogenesis and Gibbs-Thomson relation. In the model there are two unknown functions: One is $\sigma(r,t)$ which is the nutrient concentration at time $t$ and radius $r$, and the other one is $R(t)$ which is the outer tumor radius at time $t$. Since $R(t)$ is unknown and varies with time, this problem has a free boundary. Assume $\alpha(t)$ is the rate at which the tumor attracts blood vessels and the Gibbs-Thomson relation is considered for the concentration of nutrient at outer boundary of the tumor, so that on the outer boundary, the condition $$\dfrac{\partial \sigma}{\partial r}+\alpha(t)\left(\sigma-N(t)\right)=0,~~r=R(t)$$ holds, where $N(t)=\bar{\sigma}\left(1-\dfrac{\gamma}{R(t)}\right)H(R(t))$ is derived from Gibbs-Thomson relation. $H(\cdot)$ is smooth on $(0,\infty)$ satisfying $H(x)=0$ if $x\leq \gamma$, $H(x)=1$ if $x\geq 2\gamma$ and $0\leq H'(x)\leq 2/\gamma$ for all $x\geq 0$. In the case where $\alpha$ is a constant, the existence of steady-state solutions is discussed and the stability of the steady-state solutions is proved. In another case where $\alpha$ depends on time, we show that $R(t)$ will be also bounded if $\alpha(t)$ is bounded and some sufficient conditions for the disappearance of tumors are given.
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References

1. H. Byrne, The effect of time delays on the dynamics of avascular tumor growth, Math. Biosci., 144 (1997), 83–117.

2. H. Byrne and M. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151–181.

3. H. Greenspan, Models for the growth of solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317–340.

4. M. J. Piotrowska, Hopf bifurcation in a solid asascular tumor growth model with two discrete delays, Math. Comput. Modeling, 47 (2008), 597–603.

5. K. Thompson and H. Byrne, Modelling the internalisation of labelled cells in tumor spheroids, Bull. Math. Biol., 61 (1999), 601–623.

6. J. Ward and J. King, Mathematical modelling of avascular-tumor growth II: Modelling growth saturation, IMA J. Math. Appl. Med. Biol., 15 (1998), 1–42.

7. S. Cui and A. Friedman, Analysis of a mathematical model of the effect of inhibitors on the growth of tumors, Math. Biosci., 164 (2000), 103–137.

8. S. Cui, Analysis of a free boundary problem modeling tumor growth, Acta. Math. Sinica., 21 (2005), 1071–1082.

9. 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.

10. A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262–284.

11. J. Wu and F. Zhou, Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs–Thomson relation, J. Differential Eqs., 262 (2017),4907–4930.

12. A.Friedman and K. Lam, Analysis of a free-boundary tumor model with angiogenesis, J. Differential Eqs., 259 (2015), 7636–7661.

13. U. Fory´ s and M. Bodnar, Time delays in proliferation process for solid avascular tumour, Math. Comput. Modelling, 37 (2003), 1201–1209.

14. S. Xu, M. Bai and X. Zhao, Analysis of a solid avascular tumor growth model with time delays in proliferation process, J. Math. Anal. Appl., 391 (2012), 38–47.

15. S. Xu, M. Bai and F. Zhang, Analysis of a time-delayed mathematical model for tumour growth with an almost periodic supply of external nutrients, J. Biol. Dynam., 11 (2017), 504–520.

16. 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.

17. H. Byrne and M. Chaplain, Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Math. Comput. Modelling, 24 (1996), 1–17.

18. J. Wu, Stationary solutions of a free boundary problem modeling the growth of tumors with Gibbs–Thomson relation, J. Differential Eqs., 260 (2016), 5875–5893.

19. J. Wu, Analysis of a mathematical model for tumor growth with Gibbs–Thomson relation, J. Math. Anal. Appl., 450 (2017), 532–543.

20. S. Xu, Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation, Nonlinear Anal., 11 (2010), 401–406.

21. S. Xu, M. Bai and F. Zhang, Analysis of a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays, Discrete Contin. Dyn. Syst. B., 23 (2018), 3535–3551.

22. J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.

23. M. Bodnar, The nonnegativity of solutions of delay differential equations, Appl. Math. Lett., 13 (2000), 91–95.

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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