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Formal asymptotic limit of a diffuse-interface tumor-growth model. (English) Zbl 1317.35123
The work considers the model of tumor growth supplied by the diffusive influx of nutrients. The original basic model is a generalization of the Cahn-Hillard equation but here it is reduced to the singular limit \(\epsilon\to 0\) (\(\epsilon^2\) is the diffusivity corresponding to the surface energy) and reverted to the phase-field representation. The most interesting corollary of the presented solution is the fact that reactive terms collapse to the interface layer instead of the bulk as considered in conventional approaches. This behaviour reflects mathematically the biophysical/physiological picture of the avascular tumor growth, the situation of a consumption of tumor-growth supporting nutrients on the boundary of the tumor and a tissue only.

MSC:
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
35R35 Free boundary problems for PDEs
92C50 Medical applications (general)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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References:
[1] DOI: 10.1016/j.jde.2008.01.014 · Zbl 1154.35006 · doi:10.1016/j.jde.2008.01.014
[2] DOI: 10.1007/BF00375025 · Zbl 0828.35105 · doi:10.1007/BF00375025
[3] DOI: 10.1016/0001-6160(79)90196-2 · doi:10.1016/0001-6160(79)90196-2
[4] DOI: 10.1142/S0218202502001878 · Zbl 1016.92016 · doi:10.1142/S0218202502001878
[5] DOI: 10.1146/annurev.fluid.30.1.139 · Zbl 1398.76051 · doi:10.1146/annurev.fluid.30.1.139
[6] S. Astanin and L. Preziosi, Selected Topics in Cancer Modeling: Genesis, Evolution, Immune Competition, and Therapy, Modeling and Simulation in Science, Engineering and Technology, eds. N. Bellomo, M. Chaplain and E. De Angelis (Birkhäuser, 2008) pp. 223–253.
[7] H. Byrne, Cancer Modelling and Simulation, Mathematical Biology and Medicine Series 3, ed. L. Preziosi (Chapman & Hall/CRC, 2003) pp. 75–119.
[8] DOI: 10.1016/0025-5564(94)00117-3 · Zbl 0836.92011 · doi:10.1016/0025-5564(94)00117-3
[9] DOI: 10.1093/imammb/20.4.341 · Zbl 1046.92023 · doi:10.1093/imammb/20.4.341
[10] DOI: 10.1016/S0893-9659(03)00038-7 · Zbl 1040.92015 · doi:10.1016/S0893-9659(03)00038-7
[11] DOI: 10.1007/BF00254827 · Zbl 0608.35080 · doi:10.1007/BF00254827
[12] DOI: 10.1103/PhysRevA.39.5887 · Zbl 1027.80505 · doi:10.1103/PhysRevA.39.5887
[13] DOI: 10.1017/S0956792598003520 · Zbl 0930.35024 · doi:10.1017/S0956792598003520
[14] DOI: 10.1137/070680965 · Zbl 1151.80006 · doi:10.1137/070680965
[15] DOI: 10.1137/0148029 · doi:10.1137/0148029
[16] V. Cristini, Selected Topics in Cancer Modeling: Genesis, Evolution, Immune Competition, and Therapy, Modeling and Simulation in Science, Engineering and Technology, eds. N. Bellomo, M. Chaplain and E. De Angelis (Birkhäuser, 2008) pp. 113–181.
[17] DOI: 10.1007/s00285-008-0215-x · Zbl 1311.92039 · doi:10.1007/s00285-008-0215-x
[18] DOI: 10.1017/CBO9780511781452 · doi:10.1017/CBO9780511781452
[19] DOI: 10.1007/s00285-002-0174-6 · Zbl 1023.92013 · doi:10.1007/s00285-002-0174-6
[20] Delfour M. C., SIAM Series on Advances in Design and Control 4, in: Shapes and Geometries: Analysis, Differential Calculus, and Optimization (2001)
[21] DOI: 10.1088/0951-7715/18/3/016 · Zbl 1125.35366 · doi:10.1088/0951-7715/18/3/016
[22] K. Eriksson, C. Johnson and A. Logg, Encyclopedia of Computational Mechanics, Fundamentals 1, eds. E. Stein, R. de Borst and T. J. R. Hughes (Wiley, 2004) pp. 675–702.
[23] DOI: 10.4171/IFB/111 · Zbl 1072.35150 · doi:10.4171/IFB/111
[24] DOI: 10.1137/1.9781611970180 · doi:10.1137/1.9781611970180
[25] DOI: 10.1016/S0022-5193(76)80054-9 · doi:10.1016/S0022-5193(76)80054-9
[26] DOI: 10.1007/s00285-012-0595-9 · Zbl 1280.35163 · doi:10.1007/s00285-012-0595-9
[27] DOI: 10.1002/cnm.1467 · Zbl 1242.92030 · doi:10.1002/cnm.1467
[28] DOI: 10.1142/S0218202514500304 · Zbl 1404.35459 · doi:10.1142/S0218202514500304
[29] DOI: 10.1098/rspa.1998.0273 · Zbl 0927.76007 · doi:10.1098/rspa.1998.0273
[30] M. Mimura, Mathematical Aspects of Evolving Interfaces, Lecture Notes in Mathematics 1812, eds. P. Colli and J. F. Rodrigues (Springer, 2003) pp. 89–121. · doi:10.1007/978-3-540-39189-0_3
[31] Nishiura Y., Translations of Mathematical Monographs, Iwanami Series in Modern Mathematics 209, in: Far-from-Equilibrium Dynamics (2002)
[32] DOI: 10.1142/S0218202510004313 · Zbl 1186.92024 · doi:10.1142/S0218202510004313
[33] DOI: 10.1098/rspa.1989.0027 · Zbl 0701.35159 · doi:10.1098/rspa.1989.0027
[34] DOI: 10.1137/S0036144504446291 · Zbl 1117.93011 · doi:10.1137/S0036144504446291
[35] Rudin W., International Series in Pure and Applied Mathematics, in: Principles of Mathematical Analysis (1976) · Zbl 0346.26002
[36] Sethian J. A., Cambridge Monographs on Applied and Computational Mathematics, in: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (1999) · Zbl 0973.76003
[37] DOI: 10.1002/num.20638 · Zbl 1428.35398 · doi:10.1002/num.20638
[38] DOI: 10.1016/j.jtbi.2008.03.027 · Zbl 1398.92135 · doi:10.1016/j.jtbi.2008.03.027
[39] DOI: 10.1002/cnm.2597 · doi:10.1002/cnm.2597
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