A Mathematical Simulation of Oxygen Release, Diffusion, and Consumption in the Capillaries and Tissue of the Human Brain

  • Daniel D. ReneauJr.
  • Duane F. Bruley
  • Melvin H. Knisely


The purpose of this paper is to describe an attempt to obtain a better understanding of the oxygen supply to the cells of the brain by examining all phases of the release, diffusion, and consumption processes in a mathematical model. Since the original investigations of August Krogh in 1919, the mechanisms by which molecular oxygen is transported from red blood cells while they are carried in blood flowing longitudinally through capillaries, into plasma, thence radially out to and through the capillary wall into the surrounding tissues for tissue cell respiration, have been a topic of major interest. During maximal good health, and during several states of special physiology and disease states or pathologic conditions, what specific factors, or summation of factors, limit in the mathematical sense of the term the rates of oxygen supply to individual tissue cells located in individual geometric positions in the living organs? As examples, what are the effects of inhaling air containing low concentrations of oxygen, such as at high altitudes, on the individual nerve cells surrounding and all along the lengths of brain capillaries? What are the specific effects of forcibly reducing the linear rates of flow of blood through capillaries on nerve cells along the lengths of, and at various positions between, capillaries? What are the specific effects on nerve cells so located produced by the inhalation of air containing excessively high concentrations of oxygen, or of pure oxygen, perhaps under pressure such as during hyperbaric conditions?


Oxygen Partial Pressure Oxygen Activity Mathematical Simulation Axial Diffusion Axial Gradient 
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  1. 1.
    Ananthakrishnan, V., W. N. Gill, and Allen J. Barduhn: Laminar Dispersion in Capillaries, A. I. Ch. E. Journal, 11, No. 6, 1063–1072 (1965).CrossRefGoogle Scholar
  2. 2.
    Ananthakrishnan, V., W. N. Gill, and A. J. Barduhn: Laminar Dispersion in Capillaries, Document 8544, Am. Doc. Inst., Washington, D. C.Google Scholar
  3. 3.
    Anrep, G. V., A. Blalock, and Adli Samaan: The Effect of Muscular Contraction Upon the Blood Flow in the Skeletal Muscle, Proc. Roy. Soc., B 114, 223–245 (1933).CrossRefGoogle Scholar
  4. 4.
    Anrep, G. V., S. Cerqua, and Adli Samaan: The Effect of Muscular Contraction Upon the Blood Flow in the Skeletal Muscle, in the Diaphragm and in the Small Intestine, Proc. Roy. Soc., B 114, 245–257 (1933).CrossRefGoogle Scholar
  5. 5.
    Anrep, G. V., and Saalfeld: The Blood Flow Through the Skeletal Muscle in. Relation to its Contraction, Journal of Physiology, 85, 375–399 (1935).Google Scholar
  6. 6.
    Bailey, H. R., and W. B. Gogarty: “Numerical and Experimental Results on the Dispersion of a Solute in a Fluid in Laminar Flow Through a Tube, ” Unpublished Paper, The Ohio Oil Co., Littleton, Colo.Google Scholar
  7. 7.
    Barcroft, Joseph, and F. Robert: The Dissociation Curve of Haemoglobin, J. Physiol., 39, 143–148 (1909–1910).Google Scholar
  8. 8.
    Barcroft, J., et al.: On the Hydrogen-Ion Concentration and Some Related Properties of Normal Human Blood, J. Physiol., 56, 157–178 (1922).Google Scholar
  9. 9.
    Bard, Phillip: “Medical Physiology, ” C. V. Mosby Co., St. Louis, Mo., 1956, 10 ed.Google Scholar
  10. 10.
    Becker, Ernest L., Rosemarie G. Cooper, and George D. Hataway: Capillary Vascularization in Puppies Born at a Simulated Altitude of 20, 000 Feet, J. Appl. Physiol., 8, 166–168 (1955–56).Google Scholar
  11. 11.
    Bennett, H. Stanley, John H. Luff, and James C. Hampton: Morphological Classifications of Vertebrate Blood Capillaries, Am. J. Physiol., 196, 381–390 (1959).Google Scholar
  12. 12.
    Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot: “Transport Phenomena, ” John Wiley and Sons, New York, N. Y., 1960, 1 ed.Google Scholar
  13. 13.
    Bloch, E. H.: A Quantitative Study of the Hemodynamics in the Living Microvascular System, Am. J. Anat., 110, 125–153 (1962).CrossRefGoogle Scholar
  14. 14.
    Blum, Jackob J.: Concentration Profiles in and Around Capillaries, Am. J. Physiol., 198, 991–998 (1960).Google Scholar
  15. 15.
    Brown, W. E. L., and A. V. Hill: The Oxygen-Dissociation Curve of Blood and its Thermodynamical Basis, Proc. Roy. Soc., B 94, 297–334 (1922).CrossRefGoogle Scholar
  16. 16.
    Bruce, G. H., D. W. Peaceman, H. H. Rackford, and J. D. Rice: Trans., A. I.M. E., 198, 79 (1953).Google Scholar
  17. 17.
    Bruley, Duane F., and J. W. Prados: The Frequency Response of a Wetted Wall Adiabatic Humidifier, A. I. Ch. E. Journal 10, 612–616 (1964).CrossRefGoogle Scholar
  18. 18.
    Carslaw, H. S., and J. C. Jaeger: “Heat Conduction in Solids, ” Oxford University Press, Oxford, Eng., 1959, 2 ed.Google Scholar
  19. 19.
    Chambers, Robert: “Blood Capillary Circulation Under Normal Conditions and in Traumatic Shock, Nature, 162, 835–837 (1948).CrossRefGoogle Scholar
  20. 20.
    Chance Britton: Cellular Oxygen Requirements, Fed. Proc., 16, 671–692 (1957).Google Scholar
  21. 21.
    Chrone, Christian: Does “Restricted Diffusion” Occur in Muscle Capillaries ? Proc. Soc. Exp. Biol. and Med., 112, 453–455 (1962).Google Scholar
  22. 22.
    Crank, J.: “The Mathematics of Diffusion, ” Oxford University Press, Oxford, Eng., 1956, 1 ed.Google Scholar
  23. 23.
    Crank, J., and P. Nicolson: A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat-Conduction Type, Proc. Cambridge Phil. Soc., 43, 50 (1947).Google Scholar
  24. 24.
    Davies, P. W., and D. W. Bronk: Oxygen Tension in Mammalian Brain, Fed. Proc., 16, 681–692 (1957).Google Scholar
  25. 25.
    Douglas, J.: The Application of Stability Analysis in the Numerical Solution of Quasi-Linear Parabolic Differential Equations, Trans. Am. Math. Soc., 89, 484–518 (1958).CrossRefGoogle Scholar
  26. 26.
    Drummond, Stanley P.: A Comparison of the Injection and Red Cell Staining Methods Used in Quantitative Studies of the Capillaries of the Central Nervous System, Anat. Rec., 89, 93–106 (1944).Google Scholar
  27. 27.
    Friede, R. L.: “A Histochemical Atlas of Tissue Oxidation in the Brain Stem of the Cat, ” Hafner Publishing Co., Inc., New York, N. Y., 1961, 1 ed.Google Scholar
  28. 28.
    Gonzalez, L. O., and E. H. Spencer: Studies on the Numerical Solution of a Model Simulating Fixed Bed Regeneration, Chem. Eng. Sci., 18, 753–766 (1963).CrossRefGoogle Scholar
  29. 29.
    Gurland, John: “Stochastic Models in Medicine and Biology, ” The University of Wisconsin Press, Madison, Wis., 1964, 1 ed.Google Scholar
  30. 30.
    Hershey, Daniel, R. E. Byrnes, R. L. Deddens, and A. M. Rao: Blood Rheology: Temperature Dependence of the Power Law Model, Presented at the Symposium on Chem. Engr. and Med., Fifty-Seventh Annual Meeting of A. I. Ch. E., Boston, Mass., 1964.Google Scholar
  31. 31.
    Hill, A. V.: The Diffusion of Oxygen and Lactic Acid Through Tissues, Proc. Roy. Soc., B 104, 39–96 (1928).CrossRefGoogle Scholar
  32. 32.
    Hillestad, Leif K.: The Peripheral Blood Flow in Intermittent Claudication, Acta Medica Scandinavica, 174, 671–685 (1963).CrossRefGoogle Scholar
  33. 33.
    Kety, Seymour S.: Determinants of Tissue Oxygen Tension, Fed. Proc., 16, 666–670 (1957).Google Scholar
  34. 34.
    King; C. Judson: Mass Transfer During Short Surface Exposures in Countercurrent Flow, I. EC Fundamentals, 4, No. 2, 125–129, (1965).CrossRefGoogle Scholar
  35. 35.
    Knisely, Melvin H., Edward H. Bloch, and Louise Warner: “Selective Phagocytosis I, ” K. Danske Videnskabernes Selskab. Biologiske Skrifter, No. 7, Vol. 4, (1948).Google Scholar
  36. 35a.
    Knisely, Melvin H.: Intravascular Erythrocyte Aggregation (Blood Sludge), in “Handbook of Physiology, Circulation III, ”ed. by W. F. Hamilton and P. Dow. Chapter 63, pp. 2249–2292. Williams and Wilkins, Baltimore, 1965.Google Scholar
  37. 36.
    Krogh, August: “The Anatomy and Physiology of Capillaries,” Yale University Press, New Haven, Conn., 1922, 1 ed.Google Scholar
  38. 37.
    Krogh, August: The Number and Distribution of Capillaries in Muscles with Calculations of the Oxygen Pressure Head Necessary for Supplying the Tissue, J. Physiol., 52, 409415 (1918–1919).Google Scholar
  39. 38.
    Krogh, August: The Rate of Diffusion of Gases Through Animal Tissues with Some Remarks on the Coefficient of Invasion, J. Physiol., 52, 391–408 (1918–1919).Google Scholar
  40. 39.
    Krogh, August: The Supply of Oxygen to the Tissues and the Regulation of the Capillary Circulation, J. Physiol., 52, 457–474 (1918–1919).Google Scholar
  41. 40.
    Lapidus, Leon: “Digital Computation for Chemical Engineers, ” McGraw-Hill Book Co., Inc., New York, N. Y., 1962, 1 ed.Google Scholar
  42. 41.
    Lockard, I.: Capillary Patterns in the Central Nervous System and Their Possible Relation to Disease, Anat. Rec., 136, 236 (1960).Google Scholar
  43. 42.
    Lockard, I., J. R. Barham, Jr., N. G. Forlidas, Jr., and R. B. Myers: Simultaneous Histologic Demonstration of Blood Vessels, Nerve Cells and Nerve Fibers Within the Central Nervous System, J. Comp. Neur., 112, 169–183, 1959.Google Scholar
  44. 43.
    Merrill, E. W., E. R. Gilliland, G. Cokelet, H. Shin, A. Britten, and R. E. Wells, Jr.: Rheology of Blood and Flow in the Microcirculation, J. Appl. Physiol., 18, 255–260 (1963).Google Scholar
  45. 44.
    Miller, S. E.: “A Textbook of Clinical Pathology, ” Williams and Wilkins Co., Baltimore, Md., 1955, 5 ed.Google Scholar
  46. 45.
    Myers, Wayne W., and Carl R. Honig: Number and Distribution of Capillaries as Determinants of Myocardial Oxygen Tension, Am. J. Physiol., 207, 653–660 (1964).Google Scholar
  47. 46.
    Nicolson, Phyllis and F. J. W. Roughton: A Theoretical Study of the Influence of Diffusion and Chemical Reaction Velocity on the Rate of Exchange of Carbon Monoxide and Oxygen Between the Red Blood Corpuscle and the Surrounding Fluid, Proc. Roy. Soc., B 138, 241–264 (1951).CrossRefGoogle Scholar
  48. 47.
    Niessel, W., and G. Thews: Ein elektrisches Analogrechenverfahren zur Losung physiologischer Diffusionsprobleme, Pflugers Archiv, 269, 282–305 (1959).Google Scholar
  49. 48.
    Neisel, W., G. Thews, and D. Lubbers: Die Messung des zeitlichen Verlaufes der 02-Aufsattigung und-Entsattigung menschlicher Erythrocyten mit dem Kurzzeit-Spektralanalysator, Pflugers Archiv, 268, 296–307 (1959).CrossRefGoogle Scholar
  50. 49.
    Nunge, Richard J., and William N. Gill: Analysis of Heat or Mass Transfer in Some Countercurrent Flows, J. Heat Mass Transfer, 8, 873–886 (1965).CrossRefGoogle Scholar
  51. 50.
    Opitz, Erich: Increased Vascularization of the Tissue Due to Acclimatization to High Altitude and its Significance for the Oxygen Transport, Exp. Med. Surg., 9, 389–402 (1951).Google Scholar
  52. 51.
    Opitz, Erich, and Max Schneider: The Oxygen Supply of the Brain and the Mechanism of Deficiency Effects, Ergebnisse der Physiologie, biologischem Chemie, und experimentallen Pharmakologie, 46, 126–260 (1950).CrossRefGoogle Scholar
  53. 52.
    Pappenheimer, John R.: Passage of Molecules Through Capillary Walls, Physiol. Review, 33, 387–423 (1953).Google Scholar
  54. 53.
    Peaceman, D. W., and H. H. Rachford, Jr.: The Numerical Solution of Parabolic and Elliptic Differential Equations, J. Soc. Indust. Appl. Math., 3, No. 1, 28–41 (1955).CrossRefGoogle Scholar
  55. 54.
    Romanul, F. C.: Distribution of Capillaries in Relation to Oxidative Metabolism of Skeletal Muscle Fibres, Nature, 201, 307–308 (1964).CrossRefGoogle Scholar
  56. 55.
    Roughton, F. J. W.: Diffusion and Chemical Reaction Velocity as Joint Factors in Determining the Rate Uptake of Oxygen and Carbon Monoxide by the Red Corpuscle, Proc. Roy. Soc., B 111, 1–36 (1932).CrossRefGoogle Scholar
  57. 56.
    Roughton, F. J. W.: Diffusion and Chemical Reaction Velocity in Cylindrical and Spherical Systems of Physiological interest, Proc. Roy. Soc., B 140, 203–229 (1952).CrossRefGoogle Scholar
  58. 56a.
    Roughton, F. J. W.: The Oxygen Equilibrium of Mammalian Hemoglobin, in “Oxygen, ” Proceedings of a Symposium Sponsored by the New York Heart Association. Little, Brown and Company, Boston, Mass., 1965, pp. 105–126.Google Scholar
  59. 57.
    Scarborough, James B.: “Numerical Mathematical Analysis, ” The Johns Hopkins Univ. Press, Baltimore, Md., 1962, 5 ed.Google Scholar
  60. 58.
    Schmidt-Nielson, Knut, and James L. Lorimer: Oxygen Dissociation Curves of Mammalian Blood in Relation to Body Size, Am. J. Physiol., 195, 424–428 (1958).Google Scholar
  61. 59.
    Schmidt-Nielson, Knut, and Pamela Pennycuik: Capillary Density in Mammals in Relation to Body Size and Oxygen Consumption, Am. J. Physiol., 200, 746–750 (1961).Google Scholar
  62. 60.
    Snail, Lloyd L.: “Analytic Geometry and Calculus, ” AppletonCentury-Crofts, Inc., New York, 1953, 1 ed.Google Scholar
  63. 61.
    Thews, Gerhard: Oxygen Diffusion in the Brain. A Contribution to the Question of the Oxygen Supply of the Organs, Pflugers Archiv, 271, 197–226 (1960).CrossRefGoogle Scholar
  64. 62.
    Thews, G., and W. Niesel: Zur Theorie der Sauerstoffdiffusion im Erythrocyten, Pflugers Archiv, 268, 318–333 (1959).CrossRefGoogle Scholar
  65. 63.
    von Rosenberg, D. V., P. L. Durrill, and E. H. Spencer: “Numerical Solution of Partial Differential Equations Representing Fixed Bed Reactions, ” unpublished paper, ESSO Research Laboratories, Baton Rouge, La. (1960).Google Scholar
  66. 64.
    Wells, B. B.: “Clinical Pathology, ” Saunders, Philadelphia, Pa., 1962, 3 ed.Google Scholar
  67. 65.
    Wells, Roe E., Jr.: Rheology of Blood in the Microvasculature, New Eng. J. of Med., 270, 832–839 (1964).CrossRefGoogle Scholar
  68. 66.
    Wells, Roe E., Jr.: Rheology of Blood in the Microvasculature (concluded), New Eng. J. Med., 270, 889–893 (1964).CrossRefGoogle Scholar
  69. 67.
    Whetsell, William O., Jr., and Isabel Lockard: Responses to Intracarotid Hypaque in Rabbits With and Without Low Molecular Weight Dextran, J. Neur. -Path. and Exp. Neur., 25, No. 2, 283–295 (1966).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1967

Authors and Affiliations

  • Daniel D. ReneauJr.
    • 1
  • Duane F. Bruley
    • 1
  • Melvin H. Knisely
    • 2
  1. 1.Department of Chemical EngineeringClemson UniversityClemsonUSA
  2. 2.Department of AnatomyMedical College of South CarolinaCharlestonUSA

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