# American Institute of Mathematical Sciences

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

 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.
Citation: Shihe Xu, Yinhui Chen, Meng Bai. Analysis of a free boundary problem for avascular tumor growth with a periodic supply of nutrients. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [20] H. Greenspan, Models for the growth of solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340. doi: 10.1002/sapm1972514317. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [20] H. Greenspan, Models for the growth of solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340. doi: 10.1002/sapm1972514317. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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