A Simulation of Three Dimensional Oxygen Transport in Brain Tissue with a Single Neuron-Single Capillary System by the Williford-Bruley Technique

  • K. A. Kang
  • D. F. Bruley
Part of the Advances in Experimental Medicine and Biology book series (AEMB, volume 180)


Despite the importance of biological mass transfer, its mathematical models have been difficult to develop because of the complexity of biological systems and the lack of sufficient and efficient computational techniques to solve the resulting equations. The Williford-Bruley (W-B) Technique is a probabilistic numerical method which is found to be a better method to solve this kind of problems because of following advantages. In computation of the solution, this technique takes only mean distances of the probability distribution function instead of performing actual random walks which require enormous computation time. This property makes it simple to apply and reduces the computation time. The W-B technique can solve three dimensional time dependent diffusion-convection-reaction problems, which are very difficult to solve by other methods. The first objective of this simulation is the application of the W-B technique for solving a three dimensional time dependent bio-mass transport problem in a heterogeneous system (2, 21, 22).


Brain Tissue Oxygen Partial Pressure Oxygen Transport Single Neuron Oxygen Consumption Rate 
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  1. 1.
    Bharucha-Reid, A., Elements of the theory of Markov Processes and their application, McGraw-Hill, New York, 1970.Google Scholar
  2. 2.
    Bruley, D., Probabalistic Solutions and Models: Oxygen Transport in the Brain Microcirculation, Mathematics of Microcirculation Phenomena, 1980, pp. 134–157.Google Scholar
  3. 3.
    Carslaw, H. and J. Jaeger, Conduction of Heat in Solids, Oxford University Press, London, 1959.Google Scholar
  4. 4.
    Chung, K., Elementary Probabiliby Theory with Stochastic Process, Springer-Verlag, New York Inc., 1974.CrossRefGoogle Scholar
  5. 5.
    Chung, K., Markov Chains with Stationary Transition Probabilities, Springer-Verlag, Berlin, 1960.CrossRefGoogle Scholar
  6. 6.
    Crank, J., The Mathematics of Diffusion, Oxford University Press, London, 1956.Google Scholar
  7. 7.
    Gnedenko, B., The Theory of Probability, Chelsea Publishing Co., New York, 1968.Google Scholar
  8. 8.
    Hertz, L. and A. Schousboe, Ion and Energy Metabolism of the Brain at the Cellular Level, International Review of Neurology, 18, 1975, pp. 141–211.CrossRefGoogle Scholar
  9. 9.
    Hyden, H., and P. Lange, The Steady State and Endogenous Respiration in Neuron and Glia, ACTA Physiol. Scand., 64, 1965, pp. 6–14.PubMedCrossRefGoogle Scholar
  10. 10.
    Ivanov, K., Y. Kislyakov, and M. Samoilov, Microcirculation and Transport of Oxygen to Neurons of the Brain, Pavlov Institute of Physiology, USSR, Sechenov Institute of Evolutionalry Physiology and Biochemistry, USSR.Google Scholar
  11. 11.
    Jost, W., Diffusion, Academic Press, New York, 1960.Google Scholar
  12. 12.
    Kaplan, W., Advanced Mathematics for Engineers, Addison Wesley publishing Company Inc., Reading, Massachusetts, 1981.Google Scholar
  13. 13.
    Keilson, J., Green’s Function Methods in Probability Theory, Hafner Publishing Co., New York, 1965.Google Scholar
  14. 14.
    Korey, S. and M. Orchen, Relative Respiration of Neuronal and Glial Cells, Journal of Neurochenmistry, Vol. 3, 1959, pp. 277–285.CrossRefGoogle Scholar
  15. 15.
    Ozisik, N., Boundary Value Problems of Heat Conduction, International Textbook Company, Scranton, Pennsylvania, 1968.Google Scholar
  16. 16.
    Ozisic, N., Heat Conduction, John Wiley and Sons, New York, 1980.Google Scholar
  17. 17.
    Reneau, Jr., D., D. Bruley, and M. Knisely, A Mathematical Simulation of Oxygen Release, Diffusion and Consumption in the Capillaries and Tissue of the Human Brain, Chemical Engineering in Medicine and Biology, Plenum Press, New York, 1967, pp. 135–241.Google Scholar
  18. 18.
    Reneau, Jr., D., D. Bruley, and M. Knisely, A Digital Simulation of Transient Oxygen Transport in Capillary-Tissue Systems Cerebral Gray Matter, AIChE Journal, Nov. 1969, pp. 916–925.Google Scholar
  19. 19.
    Reneau, Jr., D. and M. Knisely, A Mathematical Simulation of Oxygen Transport in the Human Brain under Conditions of Countercurrent Capillary Blood Flow, Chemical Engineering Progress, 67, No. 114, 1971, pp. 18–27.Google Scholar
  20. 20.
    Syski, R., Potential Theory for Markov Chains, In: Probabilistic Methods in Applied Mathematics, Ed: Bharucha-Reid, Academic Press, New York, 1973, pp. 214–276.Google Scholar
  21. 21.
    Williford, Jr., C., The probabailistic modeling of Oxygen Transport in Brain Tissue, Ph D Dissertation, Dept. of Chemical Engineering, Tulane Univ., 1978.Google Scholar
  22. 22.
    Williford, Jr., C., D. Bruley, and R. Artigue, Probabilistic Modeling of Oxygen Transport in Brain Tissue, Neurological Research, 2, No. 2, 1980, pp. 153–170.PubMedGoogle Scholar

Copyright information

© Plenum Press, New York 1984

Authors and Affiliations

  • K. A. Kang
    • 1
  • D. F. Bruley
    • 1
  1. 1.Department of Biomedical EngineeringLouisiana Tech UniversityRustonUSA

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