1004 



HANDBOOK OF PHYSIOLOGY- 



CIRCULATION II 



containing A' cylindrical pores of radius r will occur 

 if the velocity profile assumes the parabolic distribu- 

 tion of Poiseuille's law. 



^(ar-ah) 



i.o 



8yA> 



" A-x-8-J Ap - AJI ) 



(7 12) 



Equation 7.12 implies that hydrodynamic stream- 

 ing occurs even when the hydrostatic pressure differ- 

 ence across the membrane is zero, i.e., during flow 

 caused by purely osmotic forces. The question is often 

 raised as to how Poiseuille flow could occur in the 

 apparent absence of a difference in hydrostatic 

 pressure. An explanation of this apparent paradox 

 was first offered in an important paper by Schlogl 

 (317) who pointed out that a hydrostatic pressure 

 drop accounting for hydrodynamic flow does indeed 

 exist along most of the length of the membrane 

 pores, even though the hydrostatic pressures on the 

 two sides of the membrane are equal. The intra- 

 membrane hydrostatic pressure gradient reverses 

 sharply near the edge of the pore where it becomes 

 equal and opposite to the steep gradient of diffusion 

 potential. A more detailed treatment of this hypoth- 

 esis will be found in the recent paper by Ray (292). 



Comparison of equation 7.1 1 with equation 7.12 

 reveals that for a given total pore area, A p , net flow 

 by diffusion is independent of pore radius, whereas 

 hydrodynamic flow varies with the second power of 

 the pore radius. It follows that for a given difference 

 in hydrostatic pressure the hydrodynamic component 

 of flow will increase rapidly as a function of pore 

 size. Figure 7.3 shows the relative importance of 

 diffusion and hydrodynamic flow as a function of 

 pore radius. For porosities in the range of interest for 

 capillary permeability (e.g., 20-50 A) the hydro- 

 dynamic component of net flow is overwhelmingly 

 greater than the diffusion component. The capillary 

 filtration coefficient discussed in section 6 is therefore 

 a measure of hydrodynamic conductivity rather than 

 diffusion permeability. Detailed discussions of the 

 relations between diffusion permeability and hydro- 

 dynamic conductivity will be found in papers by 

 Koefoed- Johnson & Ussing (177), Pappenheimer 

 (276, 277), Garby (1 13), Durbin et al. (83) Kedem & 

 Katchalsky (170) Katchalsky (169), Mauro (230), 

 and Ray (292). A recent experimental evaluation of 

 diffusion and hydrodynamic flow through artificial 

 membranes has been published by Robbins & 

 Mauro (303). 



Combination of diffusion data with hydrodynamic 



o 



< .6 



01 ? 



20 



-N- 



30 



40 



50, A 



> 



REGION of uncertainty re- 



STRICTIOM TOHrORODrNANIC 

 flOW AND DIFFUSION UNKNOWN 

 1MIS RANGE OF INTEREST FOR 

 CELLULAR PERMEABILITY 



RECiil IN WHICH EVIDENCE 

 SUPPORTS VALIDITY OF 

 POISEUILLE FLOW THIS 

 NANCE OF INTEREST FOR 

 CAPILLARY PERMEABILITY 



fig. 7.3. Net diffusion and hydrodynamic flow of water as 

 a function of pore size during flow induced by hydrostatic or 

 osmotic forces. For membranes with effective pore radii greater 

 than about 20 A the net flow of water by diffusion is negligible 

 compared to hydrodynamic flow. [From Pappenheimer (276).] 



data leads to a solution for pore dimensions. Equation 

 7.11 describing hydrodynamic or osmotic flow through 

 cylindrical pores can be rearranged to give 



A /Ax 



But Q/(Ap — All) is the filtration coefficient defined 

 by equation 1.1 and {A u ,/Ax) can be determined from 

 diffusion of labeled water as shown in figure 7.1. 

 Therefore, the effective pore radius is defined by 

 measurable quantities 



8yK f 



A. /Ax 



(7 13) 



Similar equations, based on diffusion and hydro- 

 dynamic flow, can be derived to estimate the dimen- 

 sions of slit pores (18, 281) or any other pore geom- 

 etry for which the laws of hydrodynamic flow are 

 known. 



Equation 7.13 has been used to estimate pore size 

 in artificial membranes (82, 298) and in living 

 capillaries (281). In general there is good agreement 

 between pore size estimated from flow and diffusion 

 and pore size estimated from restricted diffusion 

 (fig. 7.1). 



D. Simultaneous Flow and Restricted Diffusion: 

 Theory of Molecular Sieving 



In the capillary circulation both filtration and 

 restricted diffusion usually occur simultaneously and 



