×

On the derivation of angiogenesis tissue models: from the micro-scale to the macro-scale. (English) Zbl 1327.76175

Summary: This paper deals with the derivation of mathematical models at the macroscale of biological tissues corresponding to angiogenesis phenomena. The derivation is obtained by mathematical description delivered at the microscale of cells using a kinetic theory approach. A classical Chapman-Enskog expansion properly truncated is used to obtain the desired result. It is shown that the approach is general enough to describe a broad variety of different angiogenesis models corresponding to well-defined assumptions on interactions at the cellular scale.

MSC:

76Z05 Physiological flows
92C35 Physiological flow
92C10 Biomechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Weinberg RA. The Biology of Cancer. New York: Garland Sciences/Taylor and Francis, 2007.
[2] Zheng Z, Disc Cont Dyn Syst B 18 pp 1109– (2012)
[3] Bauer A, J Theoret Biol 264 pp 838– (2010) · doi:10.1016/j.jtbi.2010.03.025
[4] Anderson ARA, Bull Math Biol 60 pp 857– (1998) · Zbl 0923.92011 · doi:10.1006/bulm.1998.0042
[5] Levine HA, Bull Math Biol 63 pp 801– (2001) · Zbl 1323.92029 · doi:10.1006/bulm.2001.0240
[6] Manussaky D, Model Numer Anal 37 pp 581– (2003) · Zbl 1080.92012 · doi:10.1051/m2an:2003046
[7] Bellouquid A, Math Mod Meth Appl Sci 23 pp 949– (2013) · Zbl 1303.92040 · doi:10.1142/S0218202512500650
[8] Bellomo N, Math Mod Meth Appl Sci 22 pp 1130001– (2012) · Zbl 1328.92023 · doi:10.1142/S0218202512005885
[9] Perthame B, Bull Am Math Soc 41 pp 205– (2004) · Zbl 1151.82351 · doi:10.1090/S0273-0979-04-01004-3
[10] Bellomo N, Math Mod Meth Appl Sci 23 pp 1861– (2013) · Zbl 1315.35137 · doi:10.1142/S021820251350053X
[11] Othmer H, J Math Biol 26 pp 263– (1988) · Zbl 0713.92018 · doi:10.1007/BF00277392
[12] Bellomo N, Disc Cont Dyn Syst B 4 pp 59– (2004) · Zbl 1044.92021 · doi:10.3934/dcdsb.2004.4.59
[13] Alber M, Phys Rev Lett 99 pp 168102– (2007) · doi:10.1103/PhysRevLett.99.168102
[14] Bellouquid A, Nonlin Anal Real World Appl 12 pp 1111– (2011) · Zbl 1203.92020 · doi:10.1016/j.nonrwa.2010.09.005
[15] Hillen T, SIAM J Appl Math 61 pp 751– (2000) · Zbl 1002.35120 · doi:10.1137/S0036139999358167
[16] Lachowicz M, Lecture Notes in Mathematics 1940, in: Multiscale Problems in the Life Sciences pp 201– (2008) · Zbl 1264.92050 · doi:10.1007/978-3-540-78362-6_4
[17] Othmer H, SIAM J Appl Math 62 pp 1222– (2002) · Zbl 1103.35098 · doi:10.1137/S0036139900382772
[18] Bellomo N, Math Mod Meth Appl Sci 20 pp 1179– (2010) · Zbl 1402.92065 · doi:10.1142/S0218202510004568
[19] Keller EF, J Theoret Biol 30 pp 235U– (1971) · Zbl 1170.92308 · doi:10.1016/0022-5193(71)90051-8
[20] Patlak CS, Bull Math Biol Biophys 15 pp 311– (1953) · Zbl 1296.82044 · doi:10.1007/BF02476407
[21] Carrillo JA, Multiscale Model Sim 11 pp 336– (2013) · Zbl 1274.92007 · doi:10.1137/110851687
[22] Hillen T, J Math Biol 58 pp 183– (2009) · Zbl 1161.92003 · doi:10.1007/s00285-008-0201-3
[23] Marciniak-Czochra A, Math Mod Meth Appl Sci 20 pp 449– (2010) · Zbl 1194.35043 · doi:10.1142/S0218202510004301
[24] Perthame B, Trans Am Math Soc 361 pp 2319– (2009) · Zbl 1180.35343 · doi:10.1090/S0002-9947-08-04656-4
[25] DOI: 10.1016/S0092-8674(00)81683-9 · doi:10.1016/S0092-8674(00)81683-9
[26] Nowak MA, Evolutionary Dynamics. Exploring the Equations of Life (2006)
[27] Anderson ARA, J Theoret Med 2 pp 129– (2000) · Zbl 0947.92012 · doi:10.1080/10273660008833042
[28] Chaplain MAJ, J Neuro-Oncol 50 pp 37– (2000) · doi:10.1023/A:1006446020377
[29] Folkman J, Nat Rev Cancer 2 pp 727– (2002) · doi:10.1038/nrc905
[30] Folkman J, Sem Oncol 29 pp 15– (2002) · doi:10.1053/sonc.2002.37263
[31] Prusa V, Math Mod Meth Appl Sci 23 pp 1761– (2013) · Zbl 1452.76015 · doi:10.1142/S0218202513500516
[32] Bellomo N, Math Mod Meth Appl Sci 21 pp 1– (2011) · Zbl 1213.80002 · doi:10.1142/S0218202511004952
[33] Andreu A., Arch Rat Mech Anal 182 pp 269– (2006) · Zbl 1142.35455 · doi:10.1007/s00205-006-0428-3
[34] Calvo J, Math Mod Meth Appl Sci 21 pp 893– (2011) · Zbl 1223.35057 · doi:10.1142/S0218202511005416
[35] Bellomo N, Front Sci Eng (2014)
[36] Bellouquid A, Phys Life Rev 7 pp 477– (2013)
[37] Verbeni M, Phys Life Rev 10 pp 457– (2013) · doi:10.1016/j.plrev.2013.06.004
[38] Brenier Y, Lecture Notes in Mathematics 1813, in: Optimal Transportation and Applications, Lectures given at the C.I.M.E. Summer School help in Martina Franca pp 91– (2003) · doi:10.1007/978-3-540-44857-0_4
[39] Brenier Y, J Nonlin Sci 19 pp 547– (2009) · Zbl 1177.49064 · doi:10.1007/s00332-009-9044-3
[40] Burger M, SIAM J Math Anal 38 pp 1288– (2006) · Zbl 1114.92008 · doi:10.1137/050637923
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.