428 Diffusion, Permeability, and Active Transport /23 : 3 



Spherical-Cell Approximations 



A mathematically less cumbersome technique than that of the average- 

 value method is to replace the cell by a model of simple geometry and 

 solve the problem exactly. As indicated above, there will, in general, 

 exist a steady-state solution. This is usually the only one which can 

 be measured. 



In this subsection, the diffusion of oxygen into a living cell is approxi- 

 mated by a steady-state spherical model. The rate of use of oxygen 

 per unit volume q is assumed constant if the concentration c of oxygen 

 inside the cell is greater than zero. Depending upon the cell radius 

 r , the diffusion constants inside and outside the cell D and D' , and the 

 permeability k, there may exist a region in the center of the cell in 

 which the oxygen concentration is zero. The symbol r a will be used to 

 designate the radius of this last region. 



For this case, Equations 9 and 10 may be written as 



^ -j D'd 2 {r 2 c') _ 



Outside r > r n —^ — . Q = 



r A dr A 



t -a Dd 2 {r 2 c) 



Inside r > r > r a J^—^r = ~9 ( 9 ) 



r a < r c = 



dc' dr 



Membrane r = r Q D' ' =- = k(c - c') = D^- (10') 



dr dr 



In addition, one may require 



Far outside r = oo c' = c' 



This set of equations appears complex but, actually, it is one of the 

 simplest cases. The mathematically trained can easily show that a 

 solution is 



C ' = c '° + ¥^ir\ r>r ° w 



fa Q ( 2 2\ Q r o¥l(^ 1 1 1 \ 



c ~ c o + <xl~ anvo ~ r ) + 57v o~ 7v~ tt~ + TZz + 



3k 6D K0 3D' 3 \D'r Q Dr kr 2 3Dr) 



r Q > r > r a (14) 







r a > r 



If no oxygen-free region exists, slightly different equations and solutions 

 can be written. However, their general character is not altered. 



Equations 13 and 14 can be tested experimentally. There can be 

 little doubt that the experimental values for oxygen uptake by single cells 



