Research article

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

  • Received: 02 May 2019 Accepted: 30 July 2019 Published: 14 August 2019
  • 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.

    Citation: Shihe Xu, Junde Wu. Qualitative analysis of a time-delayed free boundary problem for tumor growth with angiogenesis and Gibbs-Thomson relation[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 7433-7446. doi: 10.3934/mbe.2019372

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  • 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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    [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.
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