A diffusion-consumption problem for oxygen in a living tissue perfused by capillaries

Nonlinear Differential Equations and Applications

Volumn 13, Issue 3, 2006, Pages 349-367

  Mikelić, Andro ^a   Primicerio, Mario ^b  

^a UNIVERSITÉ LYON 1   (France)

^b UNIVERSITY OF FLORENCE   (Italy)

Author keywords

Microcirculation; Parabolic hyperbolic problems

Indexed keywords

EID: 33750688010     PISSN: 10219722     EISSN: 14209004     Source Type: Journal    
DOI: 10.1007/s00030-006-4008-x     Document Type: Article

References (22)

1. C. BOS, L. HOOF and T. OOSTENDORP, Mathematical model of erytrocytes as point-like sources, Math. Biosci. 125 (1995), 165-189.
2. 2AN Mathematical Modelling and Numerical Analysis, Vol. 26 (1992), 867-892.
3. J. CARRILLO, Entropy Solutions for Nonlinear Degenerate Problems, Arch. Rational Mech. Anal., 147 (1999), 263-361.
4. A. DUTTA and A. S. POPEL, A Theoretical analysis of intracellular oxygen diffusion, J. Theoret. Biol. 176 (1995), 165-174.
5. C. D. EGGLETON, A. VARDAPALLI, T. K. ROY and A. S. POPEL, Calculation of intracapillary oxygen tension distribution in muscle, Math.Biosci., 167 (2000), 123-143.
6. J. E. FLETCHER, Mathematical modeling of the microcirculation, Math. Biosci. 38 (1978), 155-202.
7. J. E. FLETCHER and R. W. SHUBERT, On the computation of substrate levels in perfused tissues, Math. Biosci. 62 (1982), 75-106.
8. A. FRIEDMAN and A. TZAVARAS, A Quasilinear Parabolic System Arising in Modelling of Catalytic Reactors, J. of Differential Equations, 70 (1987), 167-196.
9. J. GONZALEZ - FERNANDEZ and S. ATTA, Transport and consumption of oxygen in capillary-tissue structure, Math. Biosci. 2 (1968), 225-261.
10. L. HOOFD, Calculation of oxygen pressures in tissue with anisotropic capillary orientation, Math. Biosci. 129 (1995), 1-23.
11. R. HSU, T. W. SECOMB, A Green's function method for analysis of oxygen delivery to tissue by microvascular networks, Math. Biosci. 96 (1989), 61-78.
12. J. KEENER and J. SNEYD, Mathematical Physiology, Interdisciplinar Appl. Math. 8 Springer Verlag, 1998.
13. A. KROGH, The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue, J. Physiol. (London) 52 (1919), 409-415.
14. A. KROGH, The Anatomy and Physiology of Capillaries, Yale Univ. Press, New Haven 1929.
15. O. A. LADYŽENSKAYA, V. A. SOLONNIKOV and N. N. URAL'CEVA, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs Vol. 23, American Mathematical Society, Providence, 1968.
16. A. MIKELIĆ and M. PRIMICERIO, Oxygen exchange between multiple capillaries and living tissues: An homogenization study, Rend. Mat. Acc. Lincei, Vol. 13 (2002), 149-164.
17. C. V. PAO, Stability analysis of a coupled diffusion-transport system for oxygen transport in blood and tissue, Nonl. Anal. Th. Met. App. 9 (1985), 1037-1059.
18. A. S. POPEL, Analysis of capillary-tissue diffusion multicapillary system, Math. Biosci. 35 (1978), 187-211.
19. O2, Math. Biosci. 55 (1981), 231-246.
20. A. S. POPEL, Theory of oxygen transport to tissue, Crit. Riv. Biomed. Engrs., 17 (1989), 257-321.
21. E. P. SALATHÉ, T. C. WANG and J. F. GROSS, Mathematical analysis of oxygen transport to tissue, Math. Biosci. 51 (1980), 89-115.
22. M. S. TITICOMBE, M. J. WARD, An asymptotic study of oxygen transport from multiple capillaries to skeletral muscle tissue, SIAM J. Appl. Math., 60 (2000), 1767-1788.