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Applicable Analysis
An International Journal
Volume 95, 2016 - Issue 5
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Articles

Homogenization of oxygen transport in biological tissues

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Pages 1013-1049 | Received 28 Feb 2015, Accepted 05 May 2015, Published online: 07 Jul 2015

References

  • Galdi GP, Rannacher R, Robertson AM, Turek S. Hemodynamical flows: modeling, analysis and simulation. Vol. 37, Oberwolfach seminars. Basel: Birkhäuser; 2008.
  • Bowles RI, Dennis SC, Purvis R, Smith FT. Multi-branching flows from one mother tube to many daughters or to a network. Philos. Trans. A Math. Phys. Eng. Sci. 2005;363:1045–1055.
  • Smith FT, Jones MA. AVM modelling by multi-branching tube flow: large flow rates and dual solutions. Math. Med. Biol. 2003;20:183–204.
  • Smith FT, Purvis R, Dennis SCR, Jones MA, Ovenden NC, Tadjfar M. Fluid flow through various branching tubes. J. Eng. Math. 2003;47:277–298.
  • Goldman D, Popel AS. A computational study of the effect of capillary network anastomoses and tortuosity on oxygen transport. J. Theor. Biol. 2000;206:181–194.
  • McDougall SR, Anderson AR, Chaplain MA, Sherratt JA. Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies. Bull. Math. Biol. 2002;64:673–702.
  • Pries AR, Secomb TW, Gaehtgens P. Biophysical aspects of blood flow in the microvasculature. Cardiovasc. Res. 1996;32:654–667.
  • Stéphanou A, McDougall SR, Anderson ARA, Chaplain MAJ. Mathematical modelling of the influence of blood rheological properties upon adaptative tumour-induced angiogenesis. Math. Comput. Modell. 2006;44:96–123.
  • Li X, Popel AS, Karniadakis GE. Blood-plasma separation in Y-shaped bifurcating microfluidic channels: a dissipative particle dynamics simulation study. Phys. Biol. 2012;9:026010.
  • Pozrikidis C, Farrow DA. A model of fluid flow in solid tumors. Ann. Biomed. Eng. 2003;31:181–194.
  • Chapman SJ, Shipley RJ, Jawad R. Multiscale modeling of fluid transport in tumors. Bull. Math. Biol. 2008;70:2334–2357.
  • Breward CJ, Byrne HM, Lewis CE. A multiphase model describing vascular tumour growth. Bull. Math. Biol. 2003;65:609–640.
  • O’Dea RD, Waters SL, Byrne HM. A two-fluid model for tissue growth within a dynamic flow environment. Eur. J. Appl. Math. 2008;19:607–634.
  • Matzavinos A, Kao CY, Green JE, Sutradhar A, Miller M, Friedman A. Modeling oxygen transport in surgical tissue transfer. PNAS. 2009;106:12091–12096.
  • Granzow JW, Levine JL, Chiu ES, Allen RJ. Breast reconstruction with the deep inferior epigastric perforator flap: history and an update on current technique. J. Plast. Reconstr. Aesthet. Surg. 2006;59:571–579.
  • Gill PS, Hunt JP, Guerra AB, Dellacroce FJ, Sullivan SK, Boraski J, Metzinger SE, Dupin CL, Allen RJ. A 10-year retrospective review of 758 DIEP flaps for breast reconstruction. Plast. Reconstr. Surg. 2004;113:1153–1160.
  • Allaire G. Homogenization and two-scale convergence. SIAM J. Math. Anal. 1992;23:1482–1518.
  • Nguetseng G. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 1989;20:608–623.
  • Cioranescu D, Damlamian A, Donato P, Griso G, Zaki R. The periodic unfolding method in domains with holes. SIAM J. Math. Anal. 2012;44:718–760.
  • Cioranescu D, Damlamian A, Griso G. The periodic unfolding method in homogenization. SIAM J. Math. Anal. 2008;40:1585–1620.
  • Cioranescu D, Damlamian A, Griso G, Onofrei D. The periodic unfolding method for perforated domains and Neumann sieve models. J. Math. Pures Appl. 2008;89:248–277.
  • Allaire G. Homogenization of the stokes flow in a connected porous medium. Asymp. Anal. 1989;2:203–222.
  • Arbogast T, Lehr HL. Homogenization of a Darcy--Stokes system modeling vuggy porous media. Comput. Geosci. 2006;10:291–302.
  • Hornung U. Homogenization and porous media. Vol. 6, Interdisciplinary applied mathematics. New York (NY): Springer-Verlag; 1997.
  • Mikelić A. Homogenization of nonstationary Navier--Stokes equations in a domain with a grained boundary. Ann. Mat. Pura Appl. 1991;CLVIII:167–179.
  • Tartar L. Incompressible fluid flow in a porous medium -- convergence of the homogenisation processes. Appendix In: Sanchez Palencia E, editor. Non-homogeneous media and vibration theory. Vol. 127, Lecture notes in physics. Berlin: Springer-Verlag; 1980. p. 368–377.
  • Jäger W, Mikelić A. On the boundary conditions at the contact interface between two porous media. In: Jäger W, Necas J, John O, Najzar K, Stará J, editors. Partial differential equations: theory and numerical solution. Florida (FL): Chapman & Hall; 1999. p. 175–186.
  • Maru\u{s}i\’{c} S, Maru\u{s}i\’{c}-Paloka E. Two-scale convergence for thin domains and its applications to some lower-dimensional models in fluid mechanics. Asymp. Anal. 2000;23:23–57.
  • Chavarría-Krauser A, Ptashnyk M. Homogenization approach to water transport in plant tissues with periodic microstructures. Math. Modell. Nat. Phenom. 2013;8:80–111.
  • Jäger W, Hornung U. Diffusion, convection, adsorption, and reaction of chemicals in porous media. J. Differ. Equ. 1991;92:199–225.
  • Jäger W, Hornung U, Mikelić A. Reactive transport through an array of cells with semi-permeable membranes. RAIRO Modél. Math. Anal. Numér. 1994;28:59–94.
  • Neuss-Radu M, Jäger W. Effective transmission conditions for reaction-diffusion processes in domains separated by an interface. SIAM J. Math. Anal. 2006;39:687–720.
  • Guyton AC, Hall JE. Textbook of medical physiology. 12th ed. Philadelphia (PA): Saunders; 2010.
  • Marciniak-Czochra A, Ptashnyk M. Derivation of a macroscopic receptor-based model using homogenization techniques. SIAM J. Math Anal. 2008;40:215–237.

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