American Institute of Mathematical Sciences

Advanced
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), 1039-1091. 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), 217-223.  Google Scholar

[3]

M. Bodnar and U. Foryś, Time delay in necrotic core formation, Math. Biosci. Eng., 2 (2005), 461-472. 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), 83-117. 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), 151-181. 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), 187-216. 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), 1563-1578. 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), 395-426. 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), 781-796. 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), 103-137. 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), 1071-1082. 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), 523-541. 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), 201-209. 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), 2-32. 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), 745-753. 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), 1201-1209. 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), 593-600. 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), 187-196.  Google Scholar

[19]

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

[20]

H. Greenspan, Models for the growth of solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340. 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), 229-242. 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), 381-418. 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), 597-603. 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), 606-623. 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), 171-211. 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), 4181-4207. 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), 1-8.  Google Scholar

[28]

S. Xu, Analysis of tumor growth under direct effect ofinhibitors with time delays in proliferation, Nonlinear Anal. RWA, 11 (2010), 401-406. 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), 293-308. 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), 3295-3304. 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), 38-47. 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), 1317-1336. 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), 1039-1091. 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), 217-223.  Google Scholar

[3]

M. Bodnar and U. Foryś, Time delay in necrotic core formation, Math. Biosci. Eng., 2 (2005), 461-472. 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), 83-117. 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), 151-181. 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), 187-216. 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), 1563-1578. 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), 395-426. 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), 781-796. 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), 103-137. 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), 1071-1082. 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), 523-541. 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), 201-209. 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), 2-32. 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), 745-753. 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), 1201-1209. 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), 593-600. 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), 187-196.  Google Scholar

[19]

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

[20]

H. Greenspan, Models for the growth of solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340. 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), 229-242. 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), 381-418. 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), 597-603. 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), 606-623. 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), 171-211. 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), 4181-4207. 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), 1-8.  Google Scholar

[28]

S. Xu, Analysis of tumor growth under direct effect ofinhibitors with time delays in proliferation, Nonlinear Anal. RWA, 11 (2010), 401-406. 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), 293-308. 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), 3295-3304. 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), 38-47. 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), 1317-1336. doi: 10.1017/S0308210510001423.  Google Scholar

[1]

Junde Wu, Shangbin Cui. Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 625-651. doi: 10.3934/dcds.2009.24.625
[2]

Chengxia Lei, Yihong Du. Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 895-911. doi: 10.3934/dcdsb.2017045
[3]

Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic & Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75
[4]

Fujun Zhou, Junde Wu, Shangbin Cui. Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1669-1688. doi: 10.3934/cpaa.2009.8.1669
[5]

Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019
[6]

Ling Mi. Asymptotic behavior for the unique positive solution to a singular elliptic problem. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1053-1072. doi: 10.3934/cpaa.2015.14.1053
[7]

Belkacem Said-Houari, Radouane Rahali. Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III. Evolution Equations & Control Theory, 2013, 2 (2) : 423-440. doi: 10.3934/eect.2013.2.423
[8]

Xiaofeng Ren. Shell structure as solution to a free boundary problem from block copolymer morphology. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 979-1003. doi: 10.3934/dcds.2009.24.979
[9]

Junde Wu. Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3399-3411. doi: 10.3934/dcds.2019140
[10]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319
[11]

Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017
[12]

Bernard Brighi, S. Guesmia. Asymptotic behavior of solution of hyperbolic problems on a cylindrical domain. Conference Publications, 2007, 2007 (Special) : 160-169. doi: 10.3934/proc.2007.2007.160
[13]

Zhijun Zhang. Optimal global asymptotic behavior of the solution to a singular monge-ampère equation. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1129-1145. doi: 10.3934/cpaa.2020053
[14]

Juan Dávila, Louis Dupaigne, Marcelo Montenegro. The extremal solution of a boundary reaction problem. Communications on Pure & Applied Analysis, 2008, 7 (4) : 795-817. doi: 10.3934/cpaa.2008.7.795
[15]

Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001
[16]

Toyohiko Aiki. On the existence of a weak solution to a free boundary problem for a model of a shape memory alloy spring. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 1-13. doi: 10.3934/dcdss.2012.5.1
[17]

Jian Yang. Asymptotic behavior of solutions for competitive models with a free boundary. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3253-3276. doi: 10.3934/dcds.2015.35.3253
[18]

Jing Li, Boling Guo, Lan Zeng, Yitong Pei. Global weak solution and smooth solution of the periodic initial value problem for the generalized Landau-Lifshitz-Bloch equation in high dimensions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1345-1360. doi: 10.3934/dcdsb.2019230
[19]

Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737
[20]

Yuan Wu, Jin Liang, Bei Hu. A free boundary problem for defaultable corporate bond with credit rating migration risk and its asymptotic behavior. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1043-1058. doi: 10.3934/dcdsb.2019207

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (97)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences