# Oxygen Transport in the Human Brain — Analytical Solutions

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

## Abstract

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:
$$Capillary:\,v(1 + \frac{{Nkn{P^{n - 1}}}}{{c{{(1 + k{P^n})}^2}}})\frac{{\partial P}}{{\partial z}} = D\frac{1}{r}\frac{\partial}{{\partial r}}(r\frac{{\partial P}}{{\partial r}})$$
(1)
$$Tissue:\,{D_T}\frac{1}{r}\frac{\partial }{{\partial r}}(r - \frac{{\partial {P_T}}}{{\partial r}}) - \frac{A}{{{C_T}}}\, = 0$$
(2)
$${\text{Boundary Conditions}}:\,at\,z = 0,\,P = known\,P(r)$$
(3)
$$at\,r = 0,\,\frac{{\partial P}}{{\partial r}} = 0$$
(4)
$$at\,r = R,\,P = {P_T} = {P_i}(z)$$
(5)
$$at\,r = R,\,Dc\frac{{\partial P}}{{\partial r}} = {D_T}{c_T}\frac{{\partial {P_T}}}{{\partial r}}$$
(6)
$$at\,r = {R_T},\,\frac{{\partial P}}{{\partial r}} = 0$$
(7)

## Keywords

Oxygen Partial Pressure Tissue Oxygenation Oxygen Transport Radial Profile Approximate Analytical Solution
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## 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