January  2011, 15(1): 293-308. doi: 10.3934/dcdsb.2011.15.293

Analysis of a delayed free boundary problem for tumor growth

1. 

Department of Mathematics, Zhaoqing University, Zhaoqing, 526061, China

Received  December 2009 Revised  February 2010 Published  October 2010

In this paper we study a delayed free boundary problem for the growth of tumors. The establishment of the model is based on the diffusion of nutrient and mass conservation for the two process proliferation and apoptosis(cell death due to aging). It is assumed the process of proliferation is delayed compared to apoptosis. By $L^p$ theory of parabolic equations and the Banach fixed point theorem, we prove the existence and uniqueness of a local solutions and apply the continuation method to get the existence and uniqueness of a global solution. We also study the asymptotic behavior of the solution, and prove that in the case $c$ is sufficiently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to a dormant state as $t\rightarrow\infty.$
Citation: Shihe Xu. Analysis of a delayed free boundary problem for tumor growth. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 293-308. doi: 10.3934/dcdsb.2011.15.293
References:
[1]

M. Bodnar, U. Forys, Time delay in necrotic core formation,, Math. Biosci. Eng., 2 (2005), 461.   Google Scholar

[2]

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

[3]

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

[4]

H. Byrne and M. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors,, Math. Biosci., 135 (1996), 187.  doi: 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.  doi: doi:10.1007/s002850100130.  Google Scholar

[6]

S. Cui and A. Friedman, Analysis of a mathematical model of the effact of inhibitors on the growth of tumors,, Math. Biosci., 164 (2000), 103.  doi: doi:10.1016/S0025-5564(99)00063-2.  Google Scholar

[7]

S. Cui, Analysis of a free boundary problem modeling tumor growth,, Acta. Math. Sinica., 21 (2005), 1071.  doi: doi:10.1007/s10114-004-0483-3.  Google Scholar

[8]

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.  doi: doi:10.1016/j.jmaa.2007.02.047.  Google Scholar

[9]

M. Dorie, R. Kallman, D. Rapacchietta and et al, Migration and internalization of cells and polystrene microspheres in tumor cell sphereoids,, Exp. Cell Res., 141 (1982), 201.  doi: doi:10.1016/0014-4827(82)90082-9.  Google Scholar

[10]

U. Forys and M. Bodnar, Time delays in proliferation process for solid avascular tumour,, Math. Comput. Modelling, 37 (2003), 1201.  doi: doi:10.1016/S0895-7177(03)80019-5.  Google Scholar

[11]

U. Forys and M. Kolev, Time delays in proliferation and apoptosis for solid avascular tumour,, Mathematical Modelling of Population Dynamics, 63 (2004), 187.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

J. Hale, "Theory of Functional Differential Equations,", Springer-Verlag, (1977).   Google Scholar

[16]

M. J. Piotrowska, Hopf bifurcation in a solid asascular tumor growth model with two discrete delays,, Math. and Compu. Modeling, 47 (2008), 597.  doi: doi:10.1016/j.mcm.2007.02.030.  Google Scholar

[17]

R. R. Sarkar and S. Banerjee, A time delay model for control of malignant tumor growth,, National Conference on Nonlinear Systems and Dynamics, (2006), 1.   Google Scholar

[18]

K. Thompson and H. Byrne, Modelling the internalisation of labelled cells in tumor spheroids,, Bull. Math. Biol., 61 (1999), 601.  doi: doi:10.1006/bulm.1999.0089.  Google Scholar

[19]

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

[20]

X. Wei and S. Cui, Existence and uniqueniss of global solutions of a free boundary problem modeling tumor growth (in chinese)., Math. Acta. Scientia., 26A (2006), 1.   Google Scholar

[21]

S. Xu, Hopf bifurcation of a free boundary problem modeling tumor growth with two time delays,, Chaos, 41 (2009), 2491.  doi: doi:10.1016/j.chaos.2008.09.029.  Google Scholar

[22]

S. Xu, Hopf bifurcation of tumor growth under direct effect of inhibitors with two time delays,, Inter. J. Appl. Math. Comput., 1 (2009), 97.   Google Scholar

[23]

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

show all references

References:
[1]

M. Bodnar, U. Forys, Time delay in necrotic core formation,, Math. Biosci. Eng., 2 (2005), 461.   Google Scholar

[2]

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

[3]

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

[4]

H. Byrne and M. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors,, Math. Biosci., 135 (1996), 187.  doi: 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.  doi: doi:10.1007/s002850100130.  Google Scholar

[6]

S. Cui and A. Friedman, Analysis of a mathematical model of the effact of inhibitors on the growth of tumors,, Math. Biosci., 164 (2000), 103.  doi: doi:10.1016/S0025-5564(99)00063-2.  Google Scholar

[7]

S. Cui, Analysis of a free boundary problem modeling tumor growth,, Acta. Math. Sinica., 21 (2005), 1071.  doi: doi:10.1007/s10114-004-0483-3.  Google Scholar

[8]

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.  doi: doi:10.1016/j.jmaa.2007.02.047.  Google Scholar

[9]

M. Dorie, R. Kallman, D. Rapacchietta and et al, Migration and internalization of cells and polystrene microspheres in tumor cell sphereoids,, Exp. Cell Res., 141 (1982), 201.  doi: doi:10.1016/0014-4827(82)90082-9.  Google Scholar

[10]

U. Forys and M. Bodnar, Time delays in proliferation process for solid avascular tumour,, Math. Comput. Modelling, 37 (2003), 1201.  doi: doi:10.1016/S0895-7177(03)80019-5.  Google Scholar

[11]

U. Forys and M. Kolev, Time delays in proliferation and apoptosis for solid avascular tumour,, Mathematical Modelling of Population Dynamics, 63 (2004), 187.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

J. Hale, "Theory of Functional Differential Equations,", Springer-Verlag, (1977).   Google Scholar

[16]

M. J. Piotrowska, Hopf bifurcation in a solid asascular tumor growth model with two discrete delays,, Math. and Compu. Modeling, 47 (2008), 597.  doi: doi:10.1016/j.mcm.2007.02.030.  Google Scholar

[17]

R. R. Sarkar and S. Banerjee, A time delay model for control of malignant tumor growth,, National Conference on Nonlinear Systems and Dynamics, (2006), 1.   Google Scholar

[18]

K. Thompson and H. Byrne, Modelling the internalisation of labelled cells in tumor spheroids,, Bull. Math. Biol., 61 (1999), 601.  doi: doi:10.1006/bulm.1999.0089.  Google Scholar

[19]

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

[20]

X. Wei and S. Cui, Existence and uniqueniss of global solutions of a free boundary problem modeling tumor growth (in chinese)., Math. Acta. Scientia., 26A (2006), 1.   Google Scholar

[21]

S. Xu, Hopf bifurcation of a free boundary problem modeling tumor growth with two time delays,, Chaos, 41 (2009), 2491.  doi: doi:10.1016/j.chaos.2008.09.029.  Google Scholar

[22]

S. Xu, Hopf bifurcation of tumor growth under direct effect of inhibitors with two time delays,, Inter. J. Appl. Math. Comput., 1 (2009), 97.   Google Scholar

[23]

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

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