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May  2016, 21(3): 997-1008. doi: 10.3934/dcdsb.2016.21.997

Analysis of a free boundary problem for avascular tumor growth with a periodic supply of nutrients

Shihe Xu 1, , Yinhui Chen 2, and  Meng Bai 3,

1. 

Department of Mathematics, Zhaoqing University, Zhaoqing, 526061
2. 

College of Mathematics and Informatics, South China Agricultural University, Guangzhou, Guangdong 510642, China
3. 

School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, Guangdong 526061, China

Received  July 2015 Revised  September 2015 Published  January 2016

In this paper we study a free boundary problem for the growth of avascular 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 supply of external nutrients is periodic. We mainly study the long time 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 positive periodic state.
Keywords: global solution, periodic solution, free boundary problem, asymptotic behavior., Tumors.
Mathematics Subject Classification: Primary: 35K57, 35K20; Secondary: 35B1.
Citation: Shihe Xu, Yinhui Chen, Meng Bai. Analysis of a free boundary problem for avascular tumor growth with a periodic supply of nutrients. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 997-1008. doi: 10.3934/dcdsb.2016.21.997
References:
[1]

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

[2]

M. Bai and S. Xu, Qualitative analysis of a mathematical model for tumor growth with a periodic supply of external nutrients,, Pacific J. Appl. Math., 5 (2013), 217.   Google Scholar

[3]

M. Bodnar and U. Foryś, Time delay in necrotic core formation,, Math. Biosci. Eng., 2 (2005), 461.  doi: 10.3934/mbe.2005.2.461.  Google Scholar

[4]

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

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

[6]

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

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H. M. Byrne et al., Modelling aspects of cancer dynamics: A review,, Trans. Royal Soc. A, 364 (2006), 1563.  doi: 10.1098/rsta.2006.1786.  Google Scholar

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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: 10.1007/s002850100130.  Google Scholar

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S. Cui, Fromation of necrotic cores in the growth of tumors: Analytic results,, Aata. Math. Scientia., 26 (2006), 781.  doi: 10.1016/S0252-9602(06)60104-5.  Google Scholar

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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: 10.1016/S0025-5564(99)00063-2.  Google Scholar

[11]

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

[12]

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: 10.1016/j.jmaa.2007.02.047.  Google Scholar

[13]

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: 10.1016/0014-4827(82)90082-9.  Google Scholar

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R. Eftimie, J. L. Bramson and D. J. D. Earn, Interactions between the immune system and cancer: A brief review of non-spatial mathematical models,, Bull Math. Biol., 73 (2011), 2.  doi: 10.1007/s11538-010-9526-3.  Google Scholar

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J. Folkman and M. Hochberg, Self-Regulation of growth in three dimensions,, J. Exp. Med., 138 (1973), 745.  doi: 10.1084/jem.138.4.745.  Google Scholar

[16]

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

[17]

U. Foryś and A. Mokwa-Borkowska, Solid tumour growth analysis of necrotic core formation,, Math. Comput. Modelling, 42 (2005), 593.  doi: 10.1016/j.mcm.2004.06.022.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

J. Nagy, The ecology and evolutionary biology of cancer: A review of mathematical models of necrosis and tumor cell diversity,, Math.Biosci. Eng., 2 (2005), 381.  doi: 10.3934/mbe.2005.2.381.  Google Scholar

[23]

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

[24]

F. A. Rihan and D. H. Abdel Rahman,et al., A time delay model of tumour-immune system interactions: Global dynamics, parameter estimation, sensitivity analysis,, Appl. Math. Comput., 232 (2014), 606.  doi: 10.1016/j.amc.2014.01.111.  Google Scholar

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J. Ward and J. King, Mathematical modelling of avascular-tumor growth II: Modelling growth saturation,, IMA J. Math. Appl. Med. Biol., 16 (1999), 171.  doi: 10.1093/imammb16.2.171.  Google Scholar

[26]

J. Wu and F. Zhou, Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with fluid-like tissue under the action of inhibitors,, Trans. Amer. Math. Soc., 365 (2013), 4181.  doi: 10.1090/S0002-9947-2013-05779-0.  Google Scholar

[27]

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

[28]

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

[29]

S. Xu, Analysis of a delayed free boundary problem for tumor growth,, Discrete & Contin. Dyn. Syst. B., 15 (2011), 293.  doi: 10.3934/dcdsb.2011.15.293.  Google Scholar

[30]

S. Xu, Qualitative analysis of a delayed free boundary problem for tumor growth under the effect of inhibitors,, Nonlinear Anal.: TMA, 74 (2011), 3295.  doi: 10.1016/j.na.2011.02.006.  Google Scholar

[31]

S. Xu, M. 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.  doi: 10.1016/j.jmaa.2012.02.034.  Google Scholar

[32]

F. Zhou and J. Wu, Analyticity of solutions to a multidimensional moving boundary problem modelling tumour growth,, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1317.  doi: 10.1017/S0308210510001423.  Google Scholar

show all references

References:
[1]

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

[2]

M. Bai and S. Xu, Qualitative analysis of a mathematical model for tumor growth with a periodic supply of external nutrients,, Pacific J. Appl. Math., 5 (2013), 217.   Google Scholar

[3]

M. Bodnar and U. Foryś, Time delay in necrotic core formation,, Math. Biosci. Eng., 2 (2005), 461.  doi: 10.3934/mbe.2005.2.461.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

H. M. Byrne et al., Modelling aspects of cancer dynamics: A review,, Trans. Royal Soc. A, 364 (2006), 1563.  doi: 10.1098/rsta.2006.1786.  Google Scholar

[8]

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: 10.1007/s002850100130.  Google Scholar

[9]

S. Cui, Fromation of necrotic cores in the growth of tumors: Analytic results,, Aata. Math. Scientia., 26 (2006), 781.  doi: 10.1016/S0252-9602(06)60104-5.  Google Scholar

[10]

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: 10.1016/S0025-5564(99)00063-2.  Google Scholar

[11]

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

[12]

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: 10.1016/j.jmaa.2007.02.047.  Google Scholar

[13]

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: 10.1016/0014-4827(82)90082-9.  Google Scholar

[14]

R. Eftimie, J. L. Bramson and D. J. D. Earn, Interactions between the immune system and cancer: A brief review of non-spatial mathematical models,, Bull Math. Biol., 73 (2011), 2.  doi: 10.1007/s11538-010-9526-3.  Google Scholar

[15]

J. Folkman and M. Hochberg, Self-Regulation of growth in three dimensions,, J. Exp. Med., 138 (1973), 745.  doi: 10.1084/jem.138.4.745.  Google Scholar

[16]

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

[17]

U. Foryś and A. Mokwa-Borkowska, Solid tumour growth analysis of necrotic core formation,, Math. Comput. Modelling, 42 (2005), 593.  doi: 10.1016/j.mcm.2004.06.022.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

J. Nagy, The ecology and evolutionary biology of cancer: A review of mathematical models of necrosis and tumor cell diversity,, Math.Biosci. Eng., 2 (2005), 381.  doi: 10.3934/mbe.2005.2.381.  Google Scholar

[23]

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

[24]

F. A. Rihan and D. H. Abdel Rahman,et al., A time delay model of tumour-immune system interactions: Global dynamics, parameter estimation, sensitivity analysis,, Appl. Math. Comput., 232 (2014), 606.  doi: 10.1016/j.amc.2014.01.111.  Google Scholar

[25]

J. Ward and J. King, Mathematical modelling of avascular-tumor growth II: Modelling growth saturation,, IMA J. Math. Appl. Med. Biol., 16 (1999), 171.  doi: 10.1093/imammb16.2.171.  Google Scholar

[26]

J. Wu and F. Zhou, Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with fluid-like tissue under the action of inhibitors,, Trans. Amer. Math. Soc., 365 (2013), 4181.  doi: 10.1090/S0002-9947-2013-05779-0.  Google Scholar

[27]

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

[28]

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

[29]

S. Xu, Analysis of a delayed free boundary problem for tumor growth,, Discrete & Contin. Dyn. Syst. B., 15 (2011), 293.  doi: 10.3934/dcdsb.2011.15.293.  Google Scholar

[30]

S. Xu, Qualitative analysis of a delayed free boundary problem for tumor growth under the effect of inhibitors,, Nonlinear Anal.: TMA, 74 (2011), 3295.  doi: 10.1016/j.na.2011.02.006.  Google Scholar

[31]

S. Xu, M. 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.  doi: 10.1016/j.jmaa.2012.02.034.  Google Scholar

[32]

F. Zhou and J. Wu, Analyticity of solutions to a multidimensional moving boundary problem modelling tumour growth,, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1317.  doi: 10.1017/S0308210510001423.  Google Scholar

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