Previous |  Up |  Next

Article

Keywords:
tumor growth modeling; mean field theory; parabolic-ODE system; global-in-time existence; chemotaxis
Summary:
We study a class of parabolic-ODE systems modeling tumor growth, its mathematical modeling and the global in time existence of the solution obtained by the method of Lyapunov functions.
References:
[1] Anderson, A. R. A., Chaplain, M. A. J.: Continuous and discrete mathematical models of tumor-induced angeogenesis. Bull. Math. Biol. 60 (1998), 857-899. DOI 10.1006/bulm.1998.0042
[2] Anderson, A. R. A., Pitcairn, A. W.: Application of the hybrid discrete-continuum technique. Polymer and Cell Dynamics-Multiscale Modeling and Numerical Simulations, Birkhäuser, Basel, 2003, pp. 261-279.
[3] Anderson, A. R. A., Quaranta, V.: Integrative mathematical oncology. Cancer 8 (2008), 227-234.
[4] Anderson, A. R. A., Weaver, A. M., Cummings, R. T., Quaranta, V.: Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell 127 (2006), 905-915. DOI 10.1016/j.cell.2006.09.042
[5] Biler, P., Stańczy, R.: Mean field models for self-gravitating particles. Folia Matematica 13 (2006), 3-19. MR 2675439 | Zbl 1181.35290
[6] Childress, S., Percus, J. K.: Nonlinear aspects of chemotaxis. Math. Biosci. 56 (1981), 217-237. DOI 10.1016/0025-5564(81)90055-9 | MR 0632161 | Zbl 0481.92010
[7] Corrias, L., Perthame, B., Zaag, H.: A chemotaxis model motivated by angeogenesis. C.R. Acad. Sci. Paris, Ser. I 336 (2003), 141-146. DOI 10.1016/S1631-073X(02)00008-0 | MR 1969568
[8] Corrias, L., Perthame, B., Zaag, H.: Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milan J. Math. 72 (2004), 1-28. DOI 10.1007/s00032-003-0026-x | MR 2099126
[9] Fontelos, M. A., Friedman, A., Hu, B.: Mathematical analysis of a model for initiation of angiogeneis. SIAM J. Math. Anal. 33 (2002), 1330-1355. DOI 10.1137/S0036141001385046 | MR 1920634
[10] Friedman, A., Tello, J. I.: Stability of solutions of chemotaxis equations in reinforced random walks. J. Math. Anal. Appl. 272 (2002), 138-163. DOI 10.1016/S0022-247X(02)00147-6 | MR 1930708 | Zbl 1025.35005
[11] Ichikawa, K., Rouzimaimaiti, M., Suzuki, T.: Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete and Continuous Dynamical Systems S 5 (2012), 115-126 doi:10.3934/dcdss/2011.5. DOI 10.3934/dcdss/2011.5 | MR 2836554
[12] Jäger, W., Luckhaus, S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. 329 (1992), 819-824. DOI 10.1090/S0002-9947-1992-1046835-6 | MR 1046835 | Zbl 0746.35002
[13] Keller, E. F., Segel, L. A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970), 399-415. DOI 10.1016/0022-5193(70)90092-5 | Zbl 1170.92306
[14] Kubo, A., Hoshino, H., Suzuki, T.: Asymptotic behavior of soltuions to a parabolic ODE system. Proceedings of the 5th East Asia PDE Conference H. J. Choe, C.-S. Lin, T. Suzuki, J. Wei Gakkotosho, Tokyo (2005), 121-136.
[15] Kubo, A., Saito, N., Suzuki, T., Hoshino, H.: Mathematical modelds of tumor angiogenesis and simulations. Kokyuroku RIMS 1499 (2006), 135-146. MR 2320335
[16] Kubo, A., Suzuki, T.: Asymptotic behavior of the solution to a parabolic ODE system modeling tumor growth. Differ. Integral Equ. 17 (2004), 721-736. MR 2074683
[17] Kubo, A., Suzuki, T.: Mathematical models of tumour angiogenesisi. J. Comp. Appl. Math. 204 (2007), 48-55. DOI 10.1016/j.cam.2006.04.027 | MR 2320335
[18] Levine, H. A., Sleeman, B. D.: A system of reaction and diffusion equations arising in the theory of reinforced random walks. SIAM J. Appl. Math. 57 (1997), 683-730. DOI 10.1137/S0036139995291106 | MR 1450846
[19] Lions, J. L.: Quelques Méthodes de Résolution de Problèmes aux Limites Non Linéaires. Dunod-Gauthier-Villars, Paris (1969), French. MR 0259693
[20] Murray, J. D.: Mathematical Biology, I: An Introduction, third edition. Springer, New York (2001). MR 1908418
[21] Nanjundiah, V.: Chemotaxis, signal relaying, and aggregation morphology. J. Theor. Biol. 42 (1973), 63-105. DOI 10.1016/0022-5193(73)90149-5
[22] Okubo, A.: Diffusion and Ecological Problems---Modern Perspectives, second edition. Springer, New York (2001). MR 1895041
[23] Othmer, H. G., Dumber, S. R., Alt, W.: Models of dispersal in biological systems. J. Math. Biol. 6 (1988), 263-298. DOI 10.1007/BF00277392 | MR 0949094
[24] Othmer, H. G., Stevens, A.: Aggregation, blowup, and collapse: The ABC's of taxis in reinforced random walks. SIAM J. Appl. Math. 57 (1997), 1044-1081. DOI 10.1137/S0036139995288976 | MR 1462051 | Zbl 0990.35128
[25] Rascle, M.: Sur une équation intégro-différentielle non linéaire issue de la biologie. J. Differ. Equations 32 (1979), 420-453. DOI 10.1016/0022-0396(79)90043-3 | MR 0535172 | Zbl 0389.45013
[26] Ribba, B., Saut, O., Colin, T., Bresch, D., Grenier, E., Boissel, J. P.: A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents. J. Theor. Biol. 243 (2006), 532-541. DOI 10.1016/j.jtbi.2006.07.013 | MR 2306343
[27] Senba, T.: Blow-up of radially symmetric solutions to some systems of partial differential equations modelling chemotaxis. Adv. Math. Sci. Appl. 7 (1997), 79-92. MR 1454659 | Zbl 0877.35022
[28] Sleeman, B. D., Levine, H. A.: Partial differential equations of chemotaxis and angiogenesis. Math. Meth. Appl. Sci. 24 (2001), 405-426. DOI 10.1002/mma.212 | MR 1821934 | Zbl 0990.35014
[29] Suzuki, T.: Free Energy and Self-Interacting Particles. Birkhäuser, Boston (2005). MR 2135150 | Zbl 1082.35006
[30] Suzuki, T.: Mean Field Theories and Dual Variation. Atlantis Press, Amsterdam (2008). MR 2510744 | Zbl 1247.35001
[31] Suzuki, T., Takahashi, R.: Global in time solution to a class of tumour growth systems. Adv. Math. Sci. Appl. 19 (2009), 503-524. MR 2605731
[32] Yang, Y., Chen, H., Liu, W.: On existence and non-existence of global solutions to a system of reaction-diffusion equations modeling chemotaxis. SIAM J. Math. Anal. 33 (1997), 763-785. DOI 10.1137/S0036141000337796 | MR 1884721
Partner of
EuDML logo