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Analysis of a delayed free boundary problem for tumor growth
1. | Department of Mathematics, Zhaoqing University, Zhaoqing, 526061, China |
References:
[1] |
M. Bodnar, U. Forys, Time delay in necrotic core formation, Math. Biosci. Eng., 2 (2005), 461-472. |
[2] |
H. Byrne, The effect of time delays on the dynamics of avascular tumor growth, Math. Biosci., 144 (1997), 83-117.
doi: doi:10.1016/S0025-5564(97)00023-0. |
[3] |
H. Byrne and M. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151-181.
doi: doi:10.1016/0025-5564(94)00117-3. |
[4] |
H. Byrne and M. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135 (1996), 187-216.
doi: doi:10.1016/0025-5564(96)00023-5. |
[5] |
S. Cui, Analysis of a mathematical model for the growth of tumors under the action of external inhibitors, J. Math. Biol., 44 (2002), 395-426.
doi: doi:10.1007/s002850100130. |
[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), 103-137.
doi: doi:10.1016/S0025-5564(99)00063-2. |
[7] |
S. Cui, Analysis of a free boundary problem modeling tumor growth, Acta. Math. Sinica., 21 (2005), 1071-1082.
doi: doi:10.1007/s10114-004-0483-3. |
[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), 523-541.
doi: doi:10.1016/j.jmaa.2007.02.047. |
[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), 201-209.
doi: doi:10.1016/0014-4827(82)90082-9. |
[10] |
U. Forys and M. Bodnar, Time delays in proliferation process for solid avascular tumour, Math. Comput. Modelling, 37 (2003), 1201-1209.
doi: doi:10.1016/S0895-7177(03)80019-5. |
[11] |
U. Forys and M. Kolev, Time delays in proliferation and apoptosis for solid avascular tumour, Mathematical Modelling of Population Dynamics, 63 (2004), 187-196. |
[12] |
A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262-284.
doi: doi:10.1007/s002850050149. |
[13] |
H. Greenspan, Models for the growth of solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340. |
[14] |
H. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theor. Biol., 56 (1976), 229-242.
doi: doi:10.1016/S0022-5193(76)80054-9. |
[15] |
J. Hale, "Theory of Functional Differential Equations," Springer-Verlag, New York, 1977. |
[16] |
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: doi:10.1016/j.mcm.2007.02.030. |
[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, 1-4. |
[18] |
K. Thompson and H. Byrne, Modelling the internalisation of labelled cells in tumor spheroids, Bull. Math. Biol., 61 (1999), 601-623.
doi: doi:10.1006/bulm.1999.0089. |
[19] |
J. Ward and J. King, Mathematical modelling of avascular-tumor growth II: Modelling growth saturation, IMA J. Math. Appl. Med. Biol., 15 (1998), 1-42. |
[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, 1-8. |
[21] |
S. Xu, Hopf bifurcation of a free boundary problem modeling tumor growth with two time delays, Chaos, Solitons & Fractals, 41 (2009), 2491-2494.
doi: doi:10.1016/j.chaos.2008.09.029. |
[22] |
S. Xu, Hopf bifurcation of tumor growth under direct effect of inhibitors with two time delays, Inter. J. Appl. Math. Comput., 1 (2009), 97-103. |
[23] |
S. Xu, Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation, Nonlinear Anal., 11 (2010), 401-406.
doi: doi:10.1016/j.nonrwa.2008.11.002. |
show all references
References:
[1] |
M. Bodnar, U. Forys, Time delay in necrotic core formation, Math. Biosci. Eng., 2 (2005), 461-472. |
[2] |
H. Byrne, The effect of time delays on the dynamics of avascular tumor growth, Math. Biosci., 144 (1997), 83-117.
doi: doi:10.1016/S0025-5564(97)00023-0. |
[3] |
H. Byrne and M. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151-181.
doi: doi:10.1016/0025-5564(94)00117-3. |
[4] |
H. Byrne and M. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135 (1996), 187-216.
doi: doi:10.1016/0025-5564(96)00023-5. |
[5] |
S. Cui, Analysis of a mathematical model for the growth of tumors under the action of external inhibitors, J. Math. Biol., 44 (2002), 395-426.
doi: doi:10.1007/s002850100130. |
[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), 103-137.
doi: doi:10.1016/S0025-5564(99)00063-2. |
[7] |
S. Cui, Analysis of a free boundary problem modeling tumor growth, Acta. Math. Sinica., 21 (2005), 1071-1082.
doi: doi:10.1007/s10114-004-0483-3. |
[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), 523-541.
doi: doi:10.1016/j.jmaa.2007.02.047. |
[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), 201-209.
doi: doi:10.1016/0014-4827(82)90082-9. |
[10] |
U. Forys and M. Bodnar, Time delays in proliferation process for solid avascular tumour, Math. Comput. Modelling, 37 (2003), 1201-1209.
doi: doi:10.1016/S0895-7177(03)80019-5. |
[11] |
U. Forys and M. Kolev, Time delays in proliferation and apoptosis for solid avascular tumour, Mathematical Modelling of Population Dynamics, 63 (2004), 187-196. |
[12] |
A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262-284.
doi: doi:10.1007/s002850050149. |
[13] |
H. Greenspan, Models for the growth of solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340. |
[14] |
H. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theor. Biol., 56 (1976), 229-242.
doi: doi:10.1016/S0022-5193(76)80054-9. |
[15] |
J. Hale, "Theory of Functional Differential Equations," Springer-Verlag, New York, 1977. |
[16] |
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: doi:10.1016/j.mcm.2007.02.030. |
[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, 1-4. |
[18] |
K. Thompson and H. Byrne, Modelling the internalisation of labelled cells in tumor spheroids, Bull. Math. Biol., 61 (1999), 601-623.
doi: doi:10.1006/bulm.1999.0089. |
[19] |
J. Ward and J. King, Mathematical modelling of avascular-tumor growth II: Modelling growth saturation, IMA J. Math. Appl. Med. Biol., 15 (1998), 1-42. |
[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, 1-8. |
[21] |
S. Xu, Hopf bifurcation of a free boundary problem modeling tumor growth with two time delays, Chaos, Solitons & Fractals, 41 (2009), 2491-2494.
doi: doi:10.1016/j.chaos.2008.09.029. |
[22] |
S. Xu, Hopf bifurcation of tumor growth under direct effect of inhibitors with two time delays, Inter. J. Appl. Math. Comput., 1 (2009), 97-103. |
[23] |
S. Xu, Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation, Nonlinear Anal., 11 (2010), 401-406.
doi: doi:10.1016/j.nonrwa.2008.11.002. |
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