Skip to main content
Log in

Simulation of intraluminal gas transport processes in the microcirculation

  • Review Article
  • Published:
Annals of Biomedical Engineering Aims and scope Submit manuscript

Abstract

Intraluminal resistance to gas transport between the microcirculation and tissue was neglected for a half-century following the early work of Krogh. In recent years it has come to be understood that this neglect is seriously in error. This paper reviews the background for the long period of misdirection, and progress in placing the simulation of gas transport processes on a more accurate, quantitative basis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adair, G. S. The hemoglobin system. VI. The oxygen dissociation curve of hemoglobin.J. Biol. Chem. 63:529–545, 1925.

    CAS  Google Scholar 

  2. Aroesty, J., and J. F. Gross. Convection and diffusion in the microcirculation.Microvasc. Res. 2:247–267, 1970.

    Article  PubMed  CAS  Google Scholar 

  3. Artigue, R. S., and E. F. Bruley. The transport of oxygen, glucose, carbon dioxide and lactic acid in the human brain: Mathematical models. In:Oxygen Transport to Tissue IV: Advances in Experimental Medicine and Biology, edited by D. F. Bruley and H. E. Bischer. New York: Plenum Press, 1983, pp. 227–234.

    Google Scholar 

  4. Bassingthwaighte, J. B., and C. A. Goresky. Modeling in the analysis of solute and water exchange in the microvasculature. In:Handbook of Physiology, (Sect. 2, The Cardiovascular System; vol. IV, The Microcirculation, edited by E. M. Renkin and C. C. Michel. Bethesda, MD: American Physiological Society: City, 1984, pp. 549–626.

    Google Scholar 

  5. Baxley, P. T., and J. D. Hellums. A simple model for simulation of oxygen transport in the microcirculation.Ann. Biomed. Eng. 11:401–416, 1983.

    Article  PubMed  CAS  Google Scholar 

  6. Benn, J. A., K. A. Smith, P. A. Drinker, and B. B. Mikic. The effects of carbonic anhydrase and oxygen uptake on carbon dioxide transfer in an oxygenator.Proc. 28th Annu. Conf. Eng. Med. Biol. 17:241, 1975.

    CAS  Google Scholar 

  7. Bird, R. B., E. W. Stewart, and E. N. Lightfoot.Transport Phenomena. New York: John Wiley and Sons, 1960, pp. 42–47.

    Google Scholar 

  8. Boland, E. J., P. K. Nair, D. D. Lemon, J. S. Olson, and J. D. Hellums. An in vitro capillary system for studies on microcirculatory O2 transport.J. Appl. Physiol. 62:791–797, 1987.

    PubMed  CAS  Google Scholar 

  9. Boland, E. J., J. S. Olson, and J. D. Hellums. Development of an in vitro method for simulation of oxygen transport in the microcirculation. In:Oxygen Transport to Tissue VII: Advances in Experimental Medicine and Biology, edited by F. Kreuzer, S. M. Cain, Z. Turek, and T. K. Goldstick. New York: Plenum Press, 1984, pp. 923–936.

    Google Scholar 

  10. Boland, E. J., H. Unno, J. S. Olson, and J. D. Hellums. An in vitro method for simulation of oxygen transport in the microcirculation.Adv. Exp. Med. Biol. 180:251–259, 1984.

    PubMed  CAS  Google Scholar 

  11. Buckles, R. G., E. W. Merrill, and E. R. Gilliland. An analysis of oxygen absorption in a tubular membrane oxygenator.Am. Inst. Chem. Eng. J. 14:703–708, 1968.

    CAS  Google Scholar 

  12. Bugliarello, G., and G. C. Hsau. A mathematical model of the flow in the axial plasmatic gaps of the smaller vessels.Biorheology 7:5–15, 1970.

    PubMed  CAS  Google Scholar 

  13. Cabantchik, Z. I., P. A. Knauf, and A. Rothstein. The anion transport system of the red blood cell: the role of membrane protein evaluated by the use of probes.Biochim. Biophys. Acta 515:239–302, 1978.

    PubMed  CAS  Google Scholar 

  14. Clark, A., W. J. Federspiel, P. A. A. Clark, and G. R. Cokelet. Oxygen delivery from red cells.Biophys. J. 47: 171–181, 1985.

    PubMed  Google Scholar 

  15. Deussen, A., and J. B. Bassingthwaighte. Modeling15O-oxygen tracer data for estimating oxygen consumption.Am. J. Physiol. 1995, in press.

  16. Diller, T. E., and B. B. Mikic. Oxygen diffusion in blood: a translational model of shear-induced augmentation.J. Biomech. Eng. 105:346–352, 1983.

    Article  PubMed  CAS  Google Scholar 

  17. Diller, T. E., B. B. Mikic, and P. A. Drinker. Shear-induced augmentation of oxygen transfer in blood.J. Biomech. Eng. 102:67–72, 1980.

    PubMed  CAS  Google Scholar 

  18. Diller, T. E., I. A. Pattantyus, and W. C. Gritts. Augmentation and facilitation of oxygen transfer in flowing hemoglobin solutions.Adv. Exp. Med. Biol. 180:545–550, 1984.

    PubMed  CAS  Google Scholar 

  19. Dorson, W. J. Mass transfer modeling for membrane oxygenation. In:Artificial Organs, edited by W. J. Kolff. London: Macmillian & Co., 1977, pp. 39–49.

    Google Scholar 

  20. Dorson, W. J., K. G. Larsen, R. J. Elgas, and M. E. Voorhees. Oxygen transfer to blood: Data and theory.Trans. Am. Soc. Artif. Int. Organs XVII:309–316, 1971.

    Google Scholar 

  21. Dorson, W. J. and M. Voorhees. Limiting models for the transfer of CO2 and O2 in membrane oxygenators.Trans. Am. Soc. Artif. Int. Organs XX:219–226, 1974.

    Google Scholar 

  22. Duda, J. L., and J. S. Vrentas. Steady flow in the region of closed streamlines in a cylindrical cavity.J. Fluid Mech. 45:247–261, 1971.

    Article  Google Scholar 

  23. Duling, B. R., and R. M. Berne. Longitudinal gradients in periarteriolar oxygen tension.Circ. Res. 27:669–676, 1975.

    Google Scholar 

  24. Duling, B. R., and R. N. Pittman. Oxygen tension: dependent or independent variable in local control of blood flow.Fed. Proc. 34:2012–2019, 1975.

    PubMed  CAS  Google Scholar 

  25. Fair, J. C., and M. H. Weissman. Oxygen transfer to blood flowing in round tubes including Bohr and Haldane effects.Chem. Eng. Sci. 26:963–967, 1971.

    Article  CAS  Google Scholar 

  26. Falke, J. J., and S. I. Chan. Evidence that anion transport by band 3 proceeds via a ping-pong mechanism involving a single transport site.J. Biol. Chem. 260:9537–9544, 1985.

    PubMed  CAS  Google Scholar 

  27. Federspiel, W. J., and A. S. Popel. A theoretical analysis of the effect of the particulate nature of blood on oxygen release in capillaries.Microvasc. Res. 32:164–189, 1986.

    Article  PubMed  CAS  Google Scholar 

  28. Federspiel, W. J., and I. H. Sarelius. An examination of the contribution of red cells spacing to the uniformity of oxygen flux at the capillary wall.Microvasc. Res. 27:273–285, 1984.

    Article  PubMed  CAS  Google Scholar 

  29. Fletcher, J. E., and R. W. Schubert. Axial diffusion and wall permeability effects in perfused capillary-tissue structure.BioSystems 20:153–174, 1987.

    Article  PubMed  CAS  Google Scholar 

  30. Frohlich, O., and R. B. Gunn. Erythrocyte anion transport: the kinetics of a single-site obligatory exchange systems.Biochim. Biophys. Acta 864:169–194, 1986.

    PubMed  CAS  Google Scholar 

  31. Gaehtgens, P., K. H. Albrecht, and F. Kreutz. Fahraeus effect and cell screening during tube flow of human blood. I. Effect of variation of flow rate.Biorheology 15:147–154, 1978.

    PubMed  CAS  Google Scholar 

  32. Gaehtgens, P., C. Duhrssen, and K. H. Albrecht. Motion, deformation, and interaction of blood cells and plasma during flow through narrow capillary tubes.Blood Cells 6:799–812, 1980.

    PubMed  CAS  Google Scholar 

  33. Gayeski, T. E. J., and C. R. Honig. O2 gradients from sarcolemma to cell interior in red muscle at maximal VO 2.Am. J. Physiol. 251:H789-H799, 1986.

    PubMed  CAS  Google Scholar 

  34. Gayeski, T. E. J., and C. R. Honig. IntracellularP O 2 in long axis of individual fibers in working dog gracilis muscle.Am. J. Physiol. 254:H1179-H1186, 1988.

    PubMed  CAS  Google Scholar 

  35. Groebe, K. A versatile model of steady state O2 supply to tissue: application to skeletal muscle.Biophys. J. 57:485–498, 1990.

    PubMed  CAS  Google Scholar 

  36. Groebe, K. An easy-to-use model for O2 supply to red muscle: validity of assumptions, sensitivity to errors in data.Biophys. J. 68:1246–1269, 1995.

    PubMed  CAS  Google Scholar 

  37. Groebe, K., and Thews, G. Theoretical analysis of oxygen supply to contracted skeletal muscle.Adv. Exp. Med. Biol. 200:495–514, 1986.

    PubMed  CAS  Google Scholar 

  38. Groebe, K., and G. Thews. Effects of cell spacing and red cell movement upon oxygen release under conditions of maximally working skeletal muscle.Adv. Exp. Med. Biol. 248:175–184, 1989.

    PubMed  CAS  Google Scholar 

  39. Hellums, J. D. The resistance to oxygen transport in the capillaries relative to that in the surrounding tissue.Microvasc. Res. 13:131–136, 1977.

    Article  PubMed  CAS  Google Scholar 

  40. Hellums, J. D. Deformation of blood cells in capillaries: a commentary.Blood Cells 6:815–817, 1980.

    Google Scholar 

  41. Hill, A. V. The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curve.J. Physiol. (Lond.) 41:iv, 1910.

    Google Scholar 

  42. Hochmuth, R. M., R. N. Marple, and S. P. Sutera. Capillary blood flow. I. Erythrocyte deformation in glass capillaries.Microvasc. Res. 2:409–417, 1970.

    Article  PubMed  CAS  Google Scholar 

  43. Homer, L. D., P. K. Weathersby, and L. A. Kiesow. Oxygen gradients between red blood cells in the microcirculation.Microvasc. Res. 22:308–323, 1981.

    Article  PubMed  CAS  Google Scholar 

  44. Honig, E. R., T. E. J. Gayeski, W. Federspiel, A. Clark, and P. Clark. Muscle oxygen gradients from hemoglobin to cytochrome: new concepts, new complexities.Adv. Exp. Med. Biol. 169:23–28, 1984.

    PubMed  CAS  Google Scholar 

  45. Hsu, R., and T. W. Secomb. Analysis of oxygen exchange between arterioles and surrounding capillary-perfused tissue.J. Biomech. Eng. 114:227–231, 1992.

    PubMed  CAS  Google Scholar 

  46. Huang, N. S., and J. D. Hellums. A theoretical model for gas transport and acid/base regulation by blood flowing in microvessels.Microvasc. Res. 48:364–388, 1994.

    Article  PubMed  CAS  Google Scholar 

  47. Huang, N. S., J. D. Hellums, and J. S. Olson. Mathematical simulation of gas transport and acid/base regulation by blood flowing in microvessels: the Cl/HCO 3 exchange across the red cell membrane. In:Oxygen Transport to Tissue XV: Advances in Experimental Medicine and Biology, edited by P. Vaupel. New York: Plenum Press, 1994, pp. 167–174.

    Google Scholar 

  48. Hyman, W. A. A simplified model of the oxygen supply function of capillary blood flow. In:Advances in Experimental Medicine and Biology, edited by D. F. Bruley and H. E. Bicher. New York: Plenum Press, 1973, pp. 835–871.

    Google Scholar 

  49. Jennings, M. L. Structure and function of the red blood cell anion transport protein.Ann. Rev. Biophys. Chem. 18:397–430, 1989.

    Article  CAS  Google Scholar 

  50. Keller, K. H. Effect of fluid shear on mass transport in flowing blood.Fed. Proc. 30:5, 1971.

    Google Scholar 

  51. Keller, K. H., and E. K. Friedlander. The steady-state transport of oxygen through hemoglobin solutions.J. Gen. Physiol. 49:663–679, 1966.

    Article  PubMed  CAS  Google Scholar 

  52. Klocke, R. A. Carbon dioxide transport. In:Handbook of Physiology, edited by L. E. Farhi and S. M. Ternney. Bethesda, MD: American Physiology Society, 1987, pp. 173–198.

    Google Scholar 

  53. Knisley, M. H., D. D. Reneau, and D. F. Bruley. The development and use of equations for predicting the limits on the rates of oxygen supply to the cells of living tissue and organs.Angiology 20:1–56, 1969.

    Article  Google Scholar 

  54. Krogh, A. The number and distribution of capillaries in muscles with calculations of the oxygen pressure head.J. Physiol. 52:409–415, 1919.

    PubMed  CAS  Google Scholar 

  55. Krogh, A.The Anatomy and Physiology of Capillaries. New Haven, CT: Yale University Press, 1922, 000 pp.

    Google Scholar 

  56. Kutchai, H., and N. C. Staub. Steady-state, hemoglobinfacilitated O2 transport in human erythrocytes.J. Gen. Physiol. 53:576–589, 1969.

    Article  PubMed  CAS  Google Scholar 

  57. Lemon, D. D., P. K. Nair, E. J. Boland, J. S. Olson, and J. D. Hellums. Physiological factors affecting O2 transport by hemoglobin in an in vitro capillary system.J. Appl. Physiol. 62:798–806, 1987.

    PubMed  CAS  Google Scholar 

  58. Lightfoot, E. N., Jr. In:Transport Phenomena in the Cardiovascular System. New York: Wiley, 1972, pp. 334–343.

    Google Scholar 

  59. Lih, M. M. A mathematical model for the axial migration of suspended particles in tube flow.Bull. Math. Biophys. 31: 143–152, 1969.

    Article  PubMed  CAS  Google Scholar 

  60. Lowe, A. G., and A. Lamber. Chloride-bicarbonate exchange and related transport processes.Biochim. Biophys. Acta 694:353–374, 1983.

    Google Scholar 

  61. Moll, W. The influence of hemoglobin diffusion on oxygen uptake and release by red cells.Respir. Physiol. 6:1–15, 1969.

    Article  Google Scholar 

  62. Nair, P. K. Simulation of Oxygen Transport in Capillaries. Houston: Rice University, Ph.D. Thesis, 1988.

    Google Scholar 

  63. Nair, P. K., J. D. Hellums, and J. S. Olson. Prediction of oxygen transport rates in blood flowing in large capillaries.Microvasc. Res. 38:269–285, 1989.

    Article  PubMed  CAS  Google Scholar 

  64. Nair, P. K., N. S. Huang, J. D. Hellums, and J. S. Olson. A simple model for prediction of oxygen transport rates by flowing blood in large capillaries.Microvasc. Res. 39:203–211, 1990.

    Article  PubMed  CAS  Google Scholar 

  65. Nunn, J. F.Applied Respiratory Physiology. New York: Butterworths, 1987, 000 pp.

    Google Scholar 

  66. O'Riordan, J. F., T. K. Goldstick, J. Ditzel and J. T. Ernest. Characterization of oxygen-hemoglobin equilibrium curves using nonlinear regression of the Hill equation: parameter values for normal adults.Adv. Exp. Med. Biol. 159:435–444, 1983.

    PubMed  Google Scholar 

  67. Pittman, R. N. Influence of microvascular architecture on oxygen exchange in skeletal muscle.Microcirculation 2:1–18, 1995.

    PubMed  CAS  Google Scholar 

  68. Pittman, R. N., and M. L. Ellsworth. Estimation of red cell flow in microvessels: Consequences of the Baker-Wayland spatial averaging model.Microvasc. Res. 32:371–388, 1986.

    Article  PubMed  CAS  Google Scholar 

  69. Popel, A. S. Theory of oxygen transport to tissue.Crit. Rev. Biomed. Eng. 17:257–321, 1989.

    PubMed  CAS  Google Scholar 

  70. Popel, A. S., and J. F. Gross. Analysis of oxygen diffusion from arteriolar networks.Am. J. Physiol. 237(6):H681-H689, 1979.

    PubMed  CAS  Google Scholar 

  71. Popel, A. S., R. N. Pittman, and M. L. Ellsworth. Rate of oxygen release from arterioles is an order of magnitude higher than expected.Am. J. Physiol. 256:H921-H924, 1989.

    PubMed  CAS  Google Scholar 

  72. Reneau, D. D., D. F. Bruley, and M. H. Knisley. A mathematical simulation of oxygen release, diffusion and consumption in the capillaries and tissue of the human brain. In:Chemical Engineering in Medicine and Biology, edited by D. Hershey. New York: Plenum Press, 1967, pp. 135–241.

    Google Scholar 

  73. Reneau, D. D., D. F. Bruley, and M. H. Knisley. A digital simulation of transient oxygen transport in capillary tissue systems (cerebral grey matter).Am. Inst. Chem. Eng. J. 15:916–925, 1969.

    CAS  Google Scholar 

  74. Salathe, E. P. Mathematical modeling of oxygen transport in skeletal muscle.Math. Biosci. 58:171–177, 1982.

    Article  Google Scholar 

  75. Salathe, E. P., T. C. Wang, and J. C. Gross. Mathematical analysis of oxygen transport to tissue.Math. Biosci. 51:89–96, 1980.

    Article  Google Scholar 

  76. Schmukler, R., and S. Chien. Rapid deoxygenation of red cells and hemoglobin solution using hollow capillary fibers.Biorheology 22:21–29, 1985.

    PubMed  CAS  Google Scholar 

  77. Secomb, T. W., and R. Hsu. Simulation of O2 transport in skeletal muscle: diffusive exchange between arterioles and capillaries.Am. J. Physiol. 267:H1214-H121, 1994.

    PubMed  CAS  Google Scholar 

  78. Secomb, T. W., M. Intaglietta, and J. F. Gross. Effects of vasomotion on microcirculatory mass transport. In:Vasomotion and Flow Modulation in the Microcirculation, Prog. Appl. Microcirc, edited by M. Intaglietta. Basel, Switzerland: Karger, 1989, pp. 49–61.

    Google Scholar 

  79. Seshadri, V., R. M. Hochmuth, P. A. Groce, and S. P. Sutera. Capillary blood flow. II. Deformable model cells compared to erythrocytesin vitro.Microvasc. Res. 2:434–442, 1970.

    Article  PubMed  CAS  Google Scholar 

  80. Sheth, B. V., and J. D. Hellums. Transient oxygen transport in hemoglobin layers under conditions of the microcirculation.Ann. Biomed. Eng. 8:183–196, 1980.

    Article  PubMed  CAS  Google Scholar 

  81. Sinha, R. Untersuchungen uber die Viskositat von Suspensionen und Losungen.Kolloid. Z. 76:16–24, 1936.

    Article  Google Scholar 

  82. Skalak, R., and P.-I. Branemark. Deformation of red blood cells in capillaries.Science 164:717–722, 1969.

    Article  PubMed  CAS  Google Scholar 

  83. Stewart, R. R., and C. A. Morrazzi. Oxygen transport in the human brain—Analytical solution. In:Advances in Experimental Medicine and Biology, edited by D. F. Bruley and H. E. Bicher: New York: Plenum Press, 1979, pp. 843–848.

    Google Scholar 

  84. Teteishi N. N. Maeda, and T. Shiga. A method for measuring the rate of oxygen release from single microvessels.Circ. Res. 70:812–819, 1992.

    Google Scholar 

  85. Thews, G. Oxygen diffusion in the brain: a contribution to the question of the oxygen supply of the organs.Arch. Ges. Physiol. 271:197–205, 1960.

    Article  CAS  Google Scholar 

  86. Vandegriff, K. D., and J. S. Olson. Morphological and physiological factors affecting oxygen uptake and release by red blood cells.J. Biol. Chem. 259:12619–12627, 1984a.

    PubMed  CAS  Google Scholar 

  87. Vandegriff, K. D., and J. S. Olson. The kinetics of O release by human red blood cells in the presence of external sodium dithionite.J. Biol. Chem. 259:12609–12618, 1984b.

    PubMed  CAS  Google Scholar 

  88. Villarroel, F., C. E. Lanham, K. B. Bischoff, T. M. Regan, and J. M. Calkins. Gas transport to blood flowing in semipermeable tubes under steady and pulsatile flow conditions.Chem. Eng. Progr. Symp. Ser. 67:96–104, 1971.

    CAS  Google Scholar 

  89. Villarroel, F., and E. E. Lanham. A design calculation method for capillary-tube oxygenators.Med. Biol. Eng. 11: 732–742, 1973.

    Article  PubMed  CAS  Google Scholar 

  90. Voorhees, M. E. Mutual Transfer of Carbon Dioxide and Oxygen to and from Blood Flowing in Macrochannel Devices. Arizona State University, Tempe, AZ, Ph.D. Thesis, 1976.

    Google Scholar 

  91. Wang, C. H., and A. S. Popel. Effect of red blood cell shape on oxygen transport in capillaries.Math. Biosci. 116: 89–110, 1993.

    Article  PubMed  CAS  Google Scholar 

  92. Weerappuli, D. P. V., and A. S. Popel. A model of oxygen exchange between an arteriole or venule and the surrounding tissue.J. Biomech. Eng. 111:24–31, 1989.

    PubMed  CAS  Google Scholar 

  93. Weissman, M. H., and L. F. Mockros. Oxygen and carbon dioxide transfer in membrane oxygenators.Med. Biol. Eng. 7:169–184, 1969.

    Article  PubMed  CAS  Google Scholar 

  94. Yap, E. W., and J. D. Hellums. Use of Adair four-step kinetics in mathematical simulation of oxygen transport in the microcirculation.Adv. Exp. Med. Biol., 215:193–207, 1987.

    PubMed  CAS  Google Scholar 

  95. Zarda, P. R., S. Chien, and R. Skalak. Interaction of viscous incompressible fluid with an elastic body. Symp. Fluid-Structure Interaction. ASME, New York, 1977, pp. 65–82.

    Google Scholar 

  96. Zydney, A. L., and C. K. Colton. Augmented solute transport in the shear flow of a concentrated suspension.PhysicoChem. Hydrodynamics, 10:77–96, 1988.

    CAS  Google Scholar 

  97. Groebe, K., and G. Thews. Calculated intra- and extracellular PO 2 gradients in heavily working red muscle.Am. J. Physiol. 259 (Heart Circ. Physiol. 28):H84-H92, 1990.

    PubMed  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hellums, J.D., Nair, P.K., Huang, N.S. et al. Simulation of intraluminal gas transport processes in the microcirculation. Ann Biomed Eng 24 (Suppl 1), 1–24 (1995). https://doi.org/10.1007/BF02770991

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02770991

Keywords

Navigation