Abstract
Delivery of oxygen to tissue is an essential function of the circulatory system. The distance that oxygen can diffuse into oxygen-consuming tissue is small, and so tissue oxygenation is critically dependent on the spatial arrangement of microvessels in tissue. Theoretical methods have been developed to simulate the spatial distribution of oxygen levels in tissue surrounding a network of microvessels. Here, numerical methods based on a Green's function approach are presented, for realistic three-dimensional network geometries derived from observations of skeletal muscle, brain, and tumor tissues. Relative to finite-difference methods, the Green's function approach reduces the number of unknowns in the numerical formulation and allows rapid computations even for complex vascular geometries. Generally, the boundary conditions on the exterior of the computational domain are not known. Imposition of a no-flux boundary condition can lead to exaggerated estimates of the extent of hypoxia in the tissue region. A new version of the method is described that avoids this problem and can be applied to arbitrarily shaped tissue domains.
Similar content being viewed by others
REFERENCES
Beard, D. A. Computational framework for generating transport models from databases of microvascular anatomy. Ann. Biomed. Eng 29: 837–843, 2001.
Beard, D. A., and J. B. Bassingthwaighte. Modeling advection and diffusion of oxygen in complex vascular networks. Ann. Biomed. Eng 29: 298–310, 2001.
Beard, D. A., K. A. Schenkman, and E. O. Feigl. Myocardial oxygenation in isolated hearts predicted by an anatomically real-istic microvascular transport model. Am.J.Physiol.Heart.Circ. Physiol. 285: H1826–H1836, 2003.
Bentley, T. B., H. Meng, and R. N. Pittman. Temperature de-pendence of oxygen diffusion and consumption in mammalian striated muscle. Am. J. Physiol. 264: H1825–H1830, 1993.
Brizel, D. M., B. Klitzman, J. M. Cook, J. Edwards, G. Rosner, and M. W. Dewhirst. A comparison of tumor and normal tissue microvascular hematocrits and red cell fluxes in a rat window chamber model. Int. J. Radiat. Oncol. Biol. Phys. 25: 269–276, 1993.
Chi, O. Z., H. M. Wei, J. Tse, S. L. Klein, and H. R. Weiss. Cere-bral microregional oxygen balance during chronic versus acute hypertension in middle cerebral artery occluded rats. Anesth. Analg. 82: 587–592, 1996.
Dewhirst, M. W., E. T. Ong, B. Klitzman, T. W. Secomb, R. Z. Vinuya, R. Dodge, D. Brizel, and J. F. Gross. Perivascular oxygen tensions in a transplantable mammary tumor growing in a dorsal flap window chamber. Radiat. Res. 130: 171–182, 1992.
Duling, B. R., and R. M. Berne. Longitudinal gradients in peri-arteriolar oxygen tension. A possible mechanism for the partic-ipation of oxygen in local regulation of blood flow. Circ. Res. 27: 669–678, 1970.
Fletcher, J. E. On facilitated oxygen diffusion in muscle tissues. Biophys. J. 29: 437–458, 1980.
Goldman, D., and A. S. Popel. A computational study of the ef-fect of capillary network anastomoses and tortuosity on oxygen transport. J. Theor. Biol. 206: 181–194, 2000.
Goldman, D., and A. S. Popel. A computational study of the ef-fect of vasomotion on oxygen transport from capillary networks. J. Theor. Biol. 209: 189–199, 2001.
Gray, L. H., and J. M. Steadman. Determination of the oxy-haemoglobin dissociation curves for mouse and rat blood. J. Physiol 175: 161–171, 1964.
Groebe, K. A versatile model of steady state O2 supply to tis-sue. Application to skeletal muscle. Biophys. J. 57: 485–498, 1990.
Hellums, J. D. The resistance to oxygen transport in the capil-laries relative to that in the surrounding tissue. Microvasc. Res. 13: 131–136, 1977.
Hellums, J. D., P. K. Nair, N. S. Huang, and N. Ohshima. Simu-lation of intraluminal gas transport processes in the microcircu-lation. Ann. Biomed. Eng. 24: 1–24, 1996.
Hoofd, L. Updating the Krogh model—assumptions and exten-sions. In: Oxygen Transport in Biological Systems: Modelling of Pathways from Environment to Cell, edited by S., Egginton and H. F. Ross. Cambridge University Press, 1992, pp. 197–229.
Hoofd, L., J. Olders, and Z. Turek. Oxygen pressures calculated in a tissue volume with parallel capillaries. Adv. Exp. Med. Biol. 277: 21–29, 1990.
Hoofd, L., Z. Turek, K. Kubat, B. E. Ringnalda, and S. Kazda. Variability of intercapillary distance estimated on histological sections of rat heart. Adv. Exp. Med. Biol. 191: 239–247, 1985.
Hsu, R., and T. W. Secomb. A Green's function method for analysis of oxygen delivery to tissue by microvascular networks. Math. Biosci. 96: 61–78, 1989.
Kellogg, O. D. Foundations of Potential Theory. New York: Dover, 1953.
Kimura, H., R. D. Braun, E. T. Ong, R. Hsu, T. W. Secomb, D. Papahadjopoulos, K. Hong, and M. W. Dewhirst. Fluctuations in red cell flux in tumor microvessels can lead to transient hypoxia and reoxygenation in tumor parenchyma. Cancer Res. 56: 5522–5528, 1996.
Klitzman, B., A. S. Popel, and B. R. Duling. Oxygen transport in resting and contracting hamster cremaster muscles: Experi-mental and theoretical microvascular studies. Microvasc. Res. 25: 108–131, 1983.
Krogh, A. The number and the distribution of capillaries in muscle with the calculation of the oxygen pressure necessary for supplying the tissue. J. Physiol. (Lond.) 52: 409–515, 1919.
Lo, A., A. J. Fuglevand, and T. W. Secomb. Oxygen delivery to skeletal muscle fibers: Effects of microvascular unit structure and control mechanisms. Am. J. Physiol. Heart Circ. Physiol. 285: H955–H963, 2003.
Middleman, S. Transport Phenomena in the Cardiovascular Sys-tem. New York: John Wiley, 1972.
Motti, E. D., H. G. Imhof, and M. G. Yasargil. The terminal vascular bed in the superficial cortex of the rat. An SEM study of corrosion casts. J. Neurosurg. 65: 834–846, 1986.
Popel, A. S. Theory of oxygen transport to tissue. Crit. Rev. Biomed. Eng. 17: 257–321, 1989.
Pozrikidis, C., and D. A. Farrow. A model of fluid flow in solid tumors. Ann. Biomed. Eng. 31: 181–194, 2003.
Pries, A. R., K. Ley, M. Claassen, and P. Gaehtgens. Red cell dis-tribution at microvascular bifurcations. Microvasc. Res. 38: 81–101, 1989.
Secomb, T. W., and R. Hsu. Analysis of oxygen delivery to tissue by microvascular networks. Adv. Exp. Med. Biol. 222: 95–103, 1988.
Secomb, T. W., and R. Hsu. Simulation of O2 transport in skeletal muscle: Diffusive exchange between arterioles and capillaries. Am. J. Physiol. 267: H1214–H1221, 1994.
Secomb, T. W., R. Hsu, N. B. Beamer, and B. M. Coull. The-oretical simulation of oxygen transport to brain by networks of microvessels: Effects of oxygen supply and demand on tissue hypoxia. Microcirculation 7: 237–247, 2000.
Secomb, T. W., R. Hsu, R. D. Braun, J. R. Ross, J. F. Gross, and M. W. Dewhirst. Theoretical simulation of oxygen transport to tumors by three-dimensional networks of microvessels. Adv. Exp. Med. Biol. 454: 629–634, 1998.
Secomb, T. W., R. Hsu, M. W. Dewhirst, B. Klitzman, and J. F. Gross. Analysis of oxygen transport to tumor tissue by microvas-cular networks. Int. J. Radiat. Oncol. Biol. Phys. 25: 481–489, 1993.
Unthank, J. L., J. M. Lash, J. C. Nixon, R. A. Sidner, and H. G. Bohlen. Evaluation of carbocyanine-labeled erythrocytes for microvascular measurements. Microvasc. Res. 45: 193–210, 1993.
Weiss, R. Parameter-Free Iterative Linear Solvers. Berlin: Akademie Verlag GmbH, 1996.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Secomb, T.W., Hsu, R., Park, E.Y.H. et al. Green's Function Methods for Analysis of Oxygen Delivery to Tissue by Microvascular Networks. Annals of Biomedical Engineering 32, 1519–1529 (2004). https://doi.org/10.1114/B:ABME.0000049036.08817.44
Issue Date:
DOI: https://doi.org/10.1114/B:ABME.0000049036.08817.44