EXCHANGE OF SUBSTANCES THROUGH CAPILLARY WALLS 



membrane can be maintained constant. This con- 

 dition is frequently encountered in the capillary 

 circulation where blood on the luminal side of the 

 capillary membrane is maintained at constant 

 composition by virtue of an adequate blood flow and 

 fluid on the tissue side of the capillary membrane is 

 maintained at a different constant composition as a 

 result of tissue metabolism. A specific example will 

 serve to illustrate the use of ecjuation 7.2 and at the 

 same time indicate the magnitude of diffusion in 

 systems of capillary dimensions. Consider the diffusion 

 of glucose across an aqueous boundary .5 /i (0.5 X 

 io -4 cm) thick and with a surface area of 10 cm-. The 

 diffusion coeflicient of glucose is 0-9 X io~~ 5 cm 2 per 

 sec (see table 9.1). Let the concentration of glucose 

 on one side of the boundary be maintained constant 

 at 100 mg per cent and the concentration on the 

 other side of the boundary at 99 mg per cent, thus 

 producing a constant concentration difference of 0.01 

 mg per cm 3 . Substituting these values in equation 

 7.2, we have 



cm 2 .01 mg/cm 3 



n = .9 X 1 o -5 X 10 cm 2 X 



0.5X1 cr 4 cm 

 = 0.018 mg/sec or 1.08 mg/min 



This rate of transfer is greater than the normal 

 metabolic consumption of glucose in 100 g of skeletal 

 muscle containing more than 5,000 cm- of total 

 capillary surface and it is thus obvious that even a 

 small concentration difference operating over a 

 relatively small aqueous area will provide a physio- 

 logically sufficient diffusion flow of glucose through 

 distances comparable in thickness with the capillary 

 wall. 



Fick realized that the driving force for diffusion 

 results from random kinetic motions of the diffusing 

 molecules, but he did not perceive the physical sig- 

 nificance of the diffusion coefficient. It remained for 

 Nernst (1888) to relate diffusion coefficient to osmotic 

 and frictional forces in solution (260). Nernst showed 

 that 



D ' RT/fN 



(7.3) 



where / is the frictional force opposing unit linear 

 velocity of each molecule and N is the number of 

 molecules per mole (Avogadro's number). For the 

 case of large spherical molecules the frictional force 

 opposing diffusion is given by Stokes' law describing 

 the motion of a sphere falling at unit velocity in a 

 viscous medium 



f = 6ir V a (7. 4) 



where r\ is the viscosity of the medium and a is the 

 molecular radius. In 1905 Sutherland (355) and 

 Einstein (91) independently noted the possibility 

 of combining equations 7.3 and 7.4 to obtain the 

 relationship between free diffusion coefficient and 

 molecular radius 



D = RT/6vya N 



(7.5) 



Equation 7.5 indicates that diffusion coefficient is 

 inversely related to molecular radius and to the 

 viscosity of the diffusion medium; conversely the 

 equation allows calculation of molecular radius from 

 measurements of free diffusion coefficient. It should 

 perhaps be emphasized that molecules are rarely 

 spherical and the molecular radius calculated from 

 the Einstein-Stokes relation (equation 7.5) is a 

 virtual quantity represented by a sphere of equivalent 

 diffusion coefficient. Moreover, the equation is 

 derived on the assumption that the diffusing molecules 

 are large compared to the solvent molecules; for 

 molecules smaller than glucose it is necessary to 

 apply corrections such as those given by Gierer & 

 W'irz (116). Additional methods for estimating 

 molecular dimensions include calculations from 

 density, intrinsic viscosity, and X-ray diffraction data. 

 Table 9.1, based on more detailed tables published in 

 references 82, 281, and 2g8, shows free diffusion 

 coefficients and approximate molecular radii of a 

 variety of molecular species which have been used 

 in studies of capillary permeability. 



B. Diffusion Through Porous Membranes, 

 Restricted Diffusion 



The diffusion of small molecules through thin, 

 large-pored membranes takes place according to 

 Fick's law; the only effect of the membrane is to 

 reduce the total area available for free diffusion. 

 Indeed, the most accurate method of estimating the 

 pore area, A p , in a membrane with large water-filled 

 pores is to measure the diffusion rate, ii, through the 

 membrane of small, uncharged molecules of known 

 free diffusion coeflicient. From rearrangement of 

 Fick's law 



n x 



Ax 

 DAc 



In most practical applications the path length, A\, 

 through the membrane is also unknown and it is more 

 useful to solve for the pore area per unit path length, 



A p /Ax 



