Oxygen Transport in the Human Brain — Analytical Solutions

  • Richard R. Stewart
  • Catherine A. Morrazzi
Part of the Advances in Experimental Medicine and Biology book series (AEMB, volume 191)


Reneau et al. (1966, 1967) performed a digital computer simulation of steady-state oxygen transport in the human brain. The Krough capillary-tissue cylinder geometry was used in which the blood flows axially in a cylindrical capillary of radius, R, surrounded by a layer of tissue of outer radius, RT. The mathematical model used is given by:
$$Capillary:\,v(1 + \frac{{Nkn{P^{n - 1}}}}{{c{{(1 + k{P^n})}^2}}})\frac{{\partial P}}{{\partial z}} = D\frac{1}{r}\frac{\partial}{{\partial r}}(r\frac{{\partial P}}{{\partial r}})$$
$$Tissue:\,{D_T}\frac{1}{r}\frac{\partial }{{\partial r}}(r - \frac{{\partial {P_T}}}{{\partial r}}) - \frac{A}{{{C_T}}}\, = 0$$
$${\text{Boundary Conditions}}:\,at\,z = 0,\,P = known\,P(r)$$
$$at\,r = 0,\,\frac{{\partial P}}{{\partial r}} = 0$$
$$at\,r = R,\,P = {P_T} = {P_i}(z)$$
$$at\,r = R,\,Dc\frac{{\partial P}}{{\partial r}} = {D_T}{c_T}\frac{{\partial {P_T}}}{{\partial r}}$$
$$at\,r = {R_T},\,\frac{{\partial P}}{{\partial r}} = 0$$


Oxygen Partial Pressure Tissue Oxygenation Oxygen Transport Radial Profile Approximate Analytical Solution 
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Literature Cited

  1. Reneau, D.D.: Ph.D. Thesis, Clemson University, Clemson, S.C., December, 1966.Google Scholar
  2. Reneau, D.D., D.F. Bruley, and M.H. Knisely: Chemical Engineering in Biology and Medicine, pp. 135–241, Plenum Press, 1967.Google Scholar

Copyright information

© Plenum Press, New York 1973

Authors and Affiliations

  • Richard R. Stewart
    • 1
  • Catherine A. Morrazzi
    • 1
  1. 1.Department of Chemical EngineeringNortheastern UniversityBostonUSA

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