EXCHANGE OF SUBSTANCES THROUGH CAPILLARY WALLS 



IOO3 



velocity of flow at the center of the pore is twice the 

 average velocity and the effective target area pre- 

 sented by the pore to the incoming molecules is 

 slightly increased (95). Under these conditions the 

 restricted diffusion equation becomes 



(\) : [2('- a rf-(l- a T)]h'0(f) 

 ^2.09(ff -0.95(ff] 



P f/Ou ' 



(7. 10) 



Figure 7.2 shows that equation 7.9 describes 

 observed diffusion through artificial porous 

 membranes with considerable accuracy. The data 

 were obtained using seven molecular species and 

 three membranes having porosities in the range of 

 interest for capillary physiology. Analogous develop- 

 ment of theory for restriction to diffusion through 

 rectangular slits, rather than cylindrical pores, leads 

 to a theoretical curve closely approximating that 

 shown in figure 7.2 (281). However, electron micro- 

 graphs indicate that true pore geometry of artificial 

 membranes is closer to the cylindrical than to the 

 rectangular model (27). 



Study of figure 7.2 reveals that pores of sufficient 

 size to allow the slow penetration of plasma proteins 

 (i.e., 30-40 A) will nevertheless impose differential 

 restriction to diffusion of much smaller molecules. 

 Thus diffusion of glucose (a = 3.7 A) through pores 

 of radius 40 A will be slowed by 34 per cent, whereas 

 diffusion of water through the same pores will be 

 slowed by only 14 per cent. This differential restric- 

 tion to diffusion of small solutes and water is the 



% 



, tf / D -o-f) 8 ['-«-«>(f)+"»e-f-ft*(?f] 



fic. 7.2. Restricted diffusion through artificial porous 

 membranes of various pore sizes. The smooth curve is drawn 

 from the theory of restricted diffusion, equation 7.9. [Adapted 

 from Renkin (298).] 



essential factor underlying transcapillary fluid shifts 

 caused by transient changes in the concentration of 

 small molecules in either plasma or tissue fluids. 



The theory of restricted diffusion provides a 

 method for estimation of effective pore radius, both 

 in artificial membranes and in living capillaries. Thus 

 equation 7.9 contains only two unknowns, D' and r, 

 and it is therefore possible to solve for r from observed 

 diffusion rates of two molecular species of known free 

 diffusion coefficients and molecular radii. Greater 

 accuracy can be obtained from the best fit of equation 

 7.9 to results obtained from several molecular species 

 as shown in figure 7.1. Pore dimensions calculated 

 from the theory of restricted diffusion agree well with 

 values obtained by independent methods (298). 



C. Diffusion and Hydrodynamic Flow, 

 Relation to Pore Dimensions 



Hydrostatic or osmotic forces, acting across a 

 porous membrane, cause net fluid movement in 

 proportion to the difference between hydrostatic 

 and effective osmotic pressure (equation 1.1). Two 

 different mechanisms are involved, diffusion and 

 hydrodynamic flow. 



diffusion. The effective concentration (thermo- 

 dynamic activity) of water depends upon pressure, 

 temperature, and solute concentration. An increase 

 of pressure or temperature increases the kinetic 

 energy of the water molecules and therefore increases 

 the statistical probability of net movement toward a 

 region of lower pressure or temperature. Conversely, 

 the addition of solute molecules to water decreases 

 the probability of net diffusion of water to a region of 

 lower solute concentration. For an ideal semiperme- 

 able membrane Fick's law may be restated as follows 

 to take account of these variables 



dn 



dt 



V ho Ax 



h 2 o * 



/AP-A/7 ) 

 1 RT ' 



(7. II) 



where q is rate of net water flow (ml/sec) and F H2 o 

 is the partial molal volume of water (18 cm 3 /mole). 

 The term (A j> — Ml)/R T replaces the concentration 

 term in Fick's law and represents the difference in 

 activity of water molecules on the two sides of the 

 membrane. Formal derivations of equation 7.1 1 may 

 be found in references (38) or (170). 



hydrodynamic flow. The minimum dissipation of 

 energy for net water flow through a membrane 



