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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 |
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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