EXCHANGE OF SUBSTANCES THROUGH CAPILLARY WALLS 



IOO5 



these two processes, operating together, are largely 

 responsible for observed concentrations of large 

 molecules in lymph (126) and renal glomerular 

 nitrate (194, 278, 366). From the theory of 

 molecular sieving described below it is possible to 

 make deductions concerning capillary permeability 

 for comparison with results obtained by independent 

 methods. The degree of molecular sieving of a given 

 solute may be defined as the ratio of its concentration 

 in the filtrate (r 2 ) to its concentration in the filtrand 

 (ci). It is often supposed that during ultrafiltra- 

 tion of a monodisperse solute through an isoporous 

 membrane, the value of ( 2 /< 1 will be zero when the 

 pores are smaller than the solute molecules and unity 

 when the pores are larger than the solute molecules. 

 If intermediate values are actually observed they are 

 said to be evidence for heteroporosity. If, for example, 

 the concentration of a given solute in a capillary 

 ultrafiltrate is 50 per cent of that in plasma it is 

 supposed that half the capillary pores were smaller 

 than the solute molecules and half were larger (21, 

 208, 232, 254). This reasoning fails to explain the 

 dependence of molecular sieving on filtration rate. 

 The following considerations show that molecular 

 sieving of a monodisperse solute through an isoporous 

 membrane is determined by the ratio of restricted 

 diffusion to rate of filtration. 



If the passage of solute through a porous membrane 

 is restricted relative to passage of solvent, then the 



filtration, c 2 approaches c-y (dialysis) and at high rates 

 of filtration c, c, approaches the ratio of restricted 

 pore areas A a /A w . Equation 7.14 is derived on the as- 

 sumption that the concentration gradient through the 

 membrane is linear. Grotte (126), Garby (113), and 

 Kuhn (192) have pointed out that the concentration 

 gradient in the membrane will in general be an ex- 

 ponential function of flow velocity, but this correction 

 was shown by Grotte to be a small one and will be 

 neglected here. 



The restricted pore areas, A w and A s , have been 

 expressed by equation 7.10 as a function of molecular 

 radius and pore radius. Substitution of this function 

 in equation 7.14 yields a cumbersome but explicit 

 expression for molecular sieving as a function of 

 filtration rate when molecular and membrane pore 

 dimensions are known; conversely, it provides an 

 independent method for calculation of pore size from 

 experimental measurements of molecular sieving and 

 filtration rate. 



Figure 7.4 shows experimentally determined values 

 of molecular sieving as a function of filtration rate 

 through Visking dialysis membrane. The theoretical 

 curves were drawn according to equation 7.15, using 

 the value of A w /Ax determined from diffusion of 

 tritiated water and choosing pore radii to provide the 

 best fits to the experimental data for each molecular 

 species. Satisfactory fits were obtained with pore radii 

 15 to 17 A. Pore radius for the same membrane 

 estimated irom the theory of restricted diffusion 



C 2 



I + 



Ax 



/*{<-&'-{'-?/ % ][<-2'0(^) + 2.09(±-f-0.95(^) 5 ] + °< 



(7. 15) 



filtrate will be diluted during filtration, thus giving 

 rise to a concentration difference for diffusion at a 

 rate determined by the restricted diffusion coefficient, 

 D' , through the membrane. The ultimate steady- 

 state composition of the filtrate relative to filtrand 

 (ci/ci) is therefore determined by a race between 

 hydrodynamic flow (Qj) tending to dilute the filtrate 

 and restricted diffusion tending to restore the concen- 

 tration difference. A quantitative expression for 

 molecular sieving through isoporous membranes 

 was derived by Pappenheimer (276) 



C ? 



I + 



J-= Of Ax 



*.+$. 



(7. 14) 



Ax 



Inspection of equation 7.14 shows that at low rates of 



Ax 



(fig. 7.1) was 16 A, and pore radius estimated from 

 combination with Poiseuille's law was ig A (equation 

 7.13). The internal consistency of these various 

 estimates of pore radius in artificial membranes 

 constitutes the chief evidence justifying the application 

 of similar techniques to biological membranes. 



E. Distribution of Pore Sizes 



Observed values for diffusion and molecular sieving 

 through artificial porous membranes are in reasonable 

 accord with theoretical predictions for isoporous 

 membranes. Equal or slightly better agreement 

 between experiment and theory can be obtained by 

 assuming certain limited distributions of pore sizes 

 (298). An upper limit to pore size may be determined 



