# American Institute of Mathematical Sciences

January  2011, 15(1): 293-308. doi: 10.3934/dcdsb.2011.15.293

## Analysis of a delayed free boundary problem for tumor growth

 1 Department of Mathematics, Zhaoqing University, Zhaoqing, 526061, China

Received  December 2009 Revised  February 2010 Published  October 2010

In this paper we study a delayed free boundary problem for the growth of 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 process of proliferation is delayed compared to apoptosis. By $L^p$ theory of parabolic equations and the Banach fixed point theorem, we prove the existence and uniqueness of a local solutions and apply the continuation method to get the existence and uniqueness of a global solution. We also study the asymptotic 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 dormant state as $t\rightarrow\infty.$
Citation: Shihe Xu. Analysis of a delayed free boundary problem for tumor growth. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 293-308. doi: 10.3934/dcdsb.2011.15.293
##### 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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##### 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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