Am J Physiol Heart Circ Physiol 290: H844-H852, 2006.
First published September 23, 2005; doi:10.1152/ajpheart.00695.2005
0363-6135/06 $8.00
Mechanism of osmotic flow in a periodic fiber array
Xiaobing Zhang,1
Fitz-Roy Curry,2 and
Sheldon Weinbaum1
1Departments of Biomedical and Mechanical Engineering, The City College of The City University of New York, New York, New York; and 2Department of Physiology and Membrane Biology, University of California, Davis, California
Submitted 26 June 2005
; accepted in final form 19 September 2005
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ABSTRACT
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The classic analysis by Anderson and Malone (Biophys J 14: 957–982, 1974) of the osmotic flow across membranes with long circular cylindrical pores is extended to a fiber matrix layer wherein the confining boundaries are the fibers themselves. The equivalent of the well-known result for the reflection coefficient 
0 = (1 – 
)2, where is the partition coefficient, is derived for a periodic fiber array of hexagonally ordered core proteins. The boundary value problem for the potential energy function describing the solute distribution surrounding each fiber is solved by defining an equivalent fluid annulus in which the pressures and osmotic forces are determined. This model is of special interest in the osmotic flow of water across a capillary wall, where recent experimental studies suggest that the endothelial glycocalyx is a quasiperiodic fiber array that serves as the primary molecular sieve for plasma proteins. Results for the reflection coefficient are presented in terms of two dimensionless numbers, 
= a/R and 
= b/R, where a and b are the solute and fiber radii, respectively, and R is the outer radius of the fluid annulus. In general, the results differ substantially from the classic expression for a circular pore because of the large difference in the shape of the boundary along which the osmotic force is generated. However, as in circular pore theory, one finds that the reflection coefficients for osmosis and filtration are the same.
endothelial glycocalyx layer; reflection coefficient
THE THERMODYNAMIC THEORY for the osmotic flow through porous membranes is largely based on the movement of water and solute through long, circular, cylindrical pores as described by Anderson and Malone (2). This theory is applicable to cell membrane proteins with hydrophilic interiors that either allow only the passage of water, such as aquaporins, or have larger channels that permit the passage of water and the restricted convection and diffusion of small solutes. In contrast to this classic theory, Michel (13) and Weinbaum (21) have independently proposed that the osmotic forces that are exerted across the capillary wall are due to the sieving of plasma proteins by a surface fiber matrix layer of proteoglycans and glycoproteins that is called the endothelial glycocalyx layer (EGL). The Michel-Weinbaum model further suggests that the pressure and osmotic forces in the Starling equation for fluid exchange should be applied just across this layer, rather than globally between plasma and tissue as previously believed. Because of the presence of a junction strand in the interendothelial cleft behind the EGL, there is a strong asymmetry in the effectiveness of solutes located on the plasma and tissue fronts, resulting from the washout of solutes from the protected region on the lumen side of the tight junction strand. Strong evidence in support of this hypothesis is provided by recent experiments in frog (9) and rat mesentery capillaries (1), where the tissue was back loaded with albumin at the same concentration as the lumen and nearly the full plasma osmotic force was observed.
Although a rigorous theory based on classic transport and thermodynamic relations was developed by Anderson and Malone (2) to determine the reflection coefficient for long, circular, cylindrical pores, no equivalent analysis has been developed for a sieving fiber matrix layer. The central tenet in the Anderson-Malone model is that there is an exclusion region near the walls of the pores producing a radial discontinuity in hydrostatic pressure and concentration that provides the driving force for the osmotic flow. For the theory derived in the present study, we examined the equivalent effect that develops along the walls of the fibers in a simplified model for a periodic fiber array. This model is based on the recent experiments of Squire et al. (18), which showed that the EGL is a three-dimensional quasiperiodic structure in which the core protein fibers form bushlike clusters associated with the underlying actin cortical cytoskeleton (ACC). This conceptual picture was then converted into an idealized mathematical model with the use of the hexagonal symmetry in Weinbaum et al. (22). In this study, we used this idealized fiber geometry to develop a theory for determining the reflection coefficient in an ordered fiber matrix layer.
Osmotic or oncotic flow is generated between solutions of different concentrations separated by a discriminating barrier. If the solutes are larger than the pores, the barrier is semipermeable: solvent can pass through it freely, whereas solutes will be impermeable. For semipermeable membranes, Mauro (12) and Ray (15) advanced independently a model to describe the equivalence between the pressure and osmotic driving forces at the membrane-solution interface (fluid boundary between bulk and pore fluid). Whereas the solutes exert their full osmotic pressures across a semipermeable barrier, Staverman (19) proposed that for a barrier that is partially permeable to solutes, one must introduce a reflection coefficient, 0, to account for the fraction of the full osmotic pressure that would be exerted across a porous (leaky) membrane. For a leaky membrane, the total flow is determined by the difference in osmotic pressures on each side of the barrier, as well as the difference in hydrostatic pressures. Therefore,
 | (1) |
where JV/A is the fluid filtration flux across the barrier, Lp is the hydraulic permeability, 
denotes the bulk solution conditions, and 
P and 
are the differences in the hydrostatic and osmotic pressures across the barrier, respectively.
Because of the striking similarity between the sieving of macromolecules by artificial porous membranes and the ultrafiltration of plasma proteins at the capillary wall, many investigators have used pore or slit theory to describe the osmotic force exerted at the capillary wall (5). The basic mechanism for osmotic flow in pores is shown in Fig. 1, adapted from Anderson and Malone (2). The steric exclusion of solute from the pore wall, r = rw, establishes an equilibrium pressure drop, Pe(z) – Pc(z) = –c(z), across the imaginary boundary r = rw – a, where Pe is the pressure in the solute-excluded layer and Pc and c are the fluid and osmotic pressures in the core region, respectively. This pressure drop produces the driving force for osmotic flow. The theoretical model by Anderson and Malone assumes that the pore length is much greater than the pore radius so that end effects are negligible. This theory is extended by Yan et al. (23) for short pores, where pore entrance and exit behavior are described. Using a rigorous theory based on classic transport and thermodynamic relations, Anderson and Malone (2) showed that for a circular pore with spherical solutes that experience only steric exclusion,
 | (2) |
where is the partition coefficient, defined as the ratio between the area available to solute and that available to water, i.e.,
 | (3) |
In the absence of an equivalent theory for a fiber matrix layer, Curry and Michel (6) suggested an expression identical to Eq. 2 to describe the reflection coefficient for the osmotic forces developed by the surface matrix in capillary endothelium, because the flow between fibers can be approximated as pseudo-Poiseuillean. The partition coefficient for a periodic fiber matrix is defined by the ratio between the area available to the solute and that of a periodic unit of the fiber matrix. This is an intuitive extrapolation without detailed derivation. One of the objectives of this study was to examine the validity of this simple intuitive extrapolation for a periodic fiber array.
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Fig. 1. Sketch for the pore theory [adapted from Anderson and Malone(2)]. The velocity in the excluded zone, rw – a < r < rw, is a function of r, whereas the velocity in the core region, r < rw – a, is relatively flat when the hydrostatic pressures at both ends are the same. See text for definitions.
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MODEL DESCRIPTION
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In the past, the molecular sieve associated with the surface glycocalyx was assumed to be a fiber matrix composed of glycoproteins with extended glycan side chains of typically 0.6 nm in radius and 7–8 nm of gap spacing associated with the disaccharide repeat along the core protein. This repeat distance was assumed by Fu et al. (7) and in subsequent studies (9, 10) to provide the dimensions of the molecular sieve for albumin. It is typical of the two-dimensional appearance of glycan side chains along chondroitin sulfate proteoglycans when observed on carbon filters (3).
The observations by Squire et al. (18) provide an alternative picture for the sieving structure of the EGL, shown in Fig. 2A. Using computed autocorrelation functions and Fourier transforms of electron microscopic images obtained from both new (18) and previous studies (4) of frog mesenteric capillaries, Squire et al. were able to identify for the first time the quasiperiodic substructure of the glycocalyx and the anchoring foci that appear to emanate from the underlying ACC. The computer-enhanced images showed that the glycocalyx is a three-dimensional fibrous meshwork with a characteristic spacing of 20 nm in all directions and that the effective diameter of the periodic scattering centers was 10–12 nm. These periodic scattering centers could be either aggregated glycan side chains that take on a spherical appearance or plasma proteins that are attracted by negative charge repeats. Using a freeze-fracture replica from a rare section where the fracture plane passed parallel and close to the endothelial surface, they also showed that anchoring foci formed a hexagonal array with an intercluster spacing of typically 100 nm in frog lung capillary.
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Fig. 2. A: sketch of endothelial layer [adapted from Weinbaum et al. (22) with permission]. B: en face view of the endothelial glycocalyx layer (EGL) with the underlying actin cortical web.
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The schematic organization of the EGL proposed by Squire et al. (18) is shown in Fig. 2, adapted from Weinbaum et al. (22). Figure 2B is an en face view of the idealized mathematical model of the EGL proposed by Weinbaum et al., which assumes both a hexagonal arrangement of the core proteins in each cluster and a hexagonal arrangement of the actin filaments in the underlying ACC. To simplify the description of the EGL's function as a molecular sieve, the core proteins with their 10- to 12-nm scattering centers are replaced by circular cylindrical fibers whose diameter is the same as the scattering centers. Figure 3A is a periodic unit in this idealized fiber array for the EGL; b is the fiber radius, and 
is the open spacing between fibers. On the basis of the measurements by Squire et al. (18), Weinbaum et al. (22) assumed b = 6 nm and = 8 nm. The partition coefficient for the hexagonally ordered fiber array in Fig. 3A is given as
 | (4) |
in which Vf is the solid fraction, defined as
 | (5) |
Equation 4 takes account of the steric exclusion of a solute of radius a by a fiber of radius b.
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Fig. 3. A: dashed lines AB, CB, and DB are lines of flow symmetry along which  U/n = 0, where U is the local velocity along the fiber axis and n is normal to AB, CB, and DB. The lines ABC, ABD, and CBD can be replaced by circular arcs with an effective radius R. R is chosen such that one-half of the resulting fluid annulus in B has the same area as the open-flow cross section in A. b, fiber radius; , is the open spacing between fibers. B: fluid annulus with radius R surrounding the fiber of radius b. C: sketch of the current fiber matrix model. The axial velocity in the excluded zone is a function of r, whereas the axial velocity in the core region is relatively flat when the hydrostatic pressures at both sides of the fiber matrix layer are the same.
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METHODS
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The flow geometry in the periodic unit in Fig. 3A is difficult to treat theoretically. Thus we wanted to construct an equivalent flow geometry around each fiber that has nearly the same flow characteristics as the periodic unit in Fig. 3A. This geometric construction, which is shown in Fig. 3B, is similar to the construction first proposed by Happel (8) for the flow through periodic fiber arrays.
Simplified fiber flow geometry.
For the flow parallel to the fiber axis in the periodic unit shown in Fig. 3A, the dashed lines AB, CB, and DB are lines of flow symmetry along which U/n = 0, where U is the local velocity along the fiber axis and n is normal to AB, CB, or DB. The lines ABC, ABD, and CBD can be replaced by circular arcs with an effective radius R, as shown in Fig. 3B. R is chosen such that one-half of the resulting fluid annulus has the same area as the open-flow cross section in Fig. 3A, i.e.,
 | (6) |
Solving for R, one obtains
 | (7) |
For b = 6 nm and = 8 nm, R = 10.5 nm. Therefore, the periodic unit in Fig. 3A can be approximated by a fiber surrounded by a fluid annulus whose outer radius is R, as shown in Fig. 3B. R is much smaller than the EGL thickness, L, which has been estimated to vary between 150 and 400 nm (18, 20). This assumption in Fig. 3B enables us to take advantage of a cylindrical coordinate system with symmetry about the longitudinal axis of the fiber and, also, to neglect end effects.
Applying Eq. 7 to Eqs. 5 and 6, one finds that
 | (8) |
Reflection coefficient 0.
One can use the Gibbs-Duhem equation to relate pressure and potential energy gradients in the radial direction in the fluid annulus b < r 
R in Fig. 3B. Because the chemical potential is constant in the radial direction,
 | (9) |
where P is the hydrostatic pressure, C is the solute concentration, and 
(r) is the solute potential energy field that controls the radial distribution of the solute in the annulus. (r) could include London forces and electric effects, as well as steric exclusion. The Boltzmann equation is valid if one assumes a constant activity coefficient of solute and a low volume fraction. Therefore, the solute concentration is given by
 | (10) |
where CR is the value of C at r = R in Fig. 3B and z is the axial direction of the fibers, 
is the universal gas constant, and T is the absolute temperature.
Our model differs from that of Anderson and Malone (2) in that our model describes the fluid annulus surrounding individual fibers, as shown in Fig. 3B, rather than circular cylindrical pores. Substituting Eq. 10 into Eq. 9 and integrating the resulting equation, one finds,
 | (11) |
where R(z) is the axial variation of the osmotic pressure, TCR(z), along the fiber axis. The subscript R denotes that the functions are evaluated at the symmetry line, where r = R. Equation 11 describes the coupling of osmotic and hydrostatic pressures in generating the driving force for the bulk flow through the EGL.
Neglecting the end effects, one can write the momentum balance for the fluid flow as
 | (12) |
where U = U(r,z) is the axial velocity. The boundary conditions for Eq. 12 are
 | (13A) |
 | (13B) |
where = r/R. Equation 13A is the no-slip condition at the fiber surface, whereas Eq. 13B is the symmetry condition at the outer boundary of the fluid annulus. Substituting Eq. 11 into Eq. 12, integrating, and applying the boundary conditions 13A and 13B, one finds
 | (14) |
Therefore, the average velocity over an annular cross section is
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By continuity, 
is independent of z, and Eq. 15 can be directly integrated along the fiber axis to obtain
 | (16) |
where
 | (17) |
and
 | (18) |
where LFM is the hydraulic permeability of the fiber matrix, and 
is the void volume fraction of the fiber matrix. Therefore,
 | (19) |
The Boltzmann and Gibbs-Duhem relations are applied at the cylinder ends.
 | (20A) |
 | (20B) |
 | (20C) |
 | (20D) |
where 0, L, P0, and PL denote the osmotic and hydrostatic pressures in the bulk fluid phase at the EGL entrance and exit planes. Substituting Eqs. 20A–20D into Eq. 16, one can rewrite the resulting equation in the form of Eq. 1, since = JV/A. The final result is
 | (21) |
where
 | (22A) |
 | (22B) |
and
 | (23A) |
By using Eq. 18, Eq. 23A also can be written as
 | (23B) |
Steric exclusion with spherical solute molecules.
One cannot calculate 0 using Eq. 23 unless (r) is a known function. The simplestmodel is that for pure steric exclusion of the solute molecules treated as rigid spheres of radius a. For this case
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 | (24B) |
Accordingly, the potential energy is described by the Heaviside or step function
 | (25) |
Substituting Eq. 25 into Eq. 23, one finds
 | (26) |
Evaluating the integrals in Eq. 26, one obtains
 | (27) |
Equation 27 can be written in more compact form using the definition Eq. 13 for and introducing a second dimensionless length parameter, = a/R,
 | (28) |
In terms of and , the expression for in Eq. 8 can be simplified to
 | (29) |
Pressure and velocity fields for spherical solute molecules.
We next wanted to examine the pressure distribution and velocity profiles in the excluded zone, b r a + b, and in the fluid annulus, a + b r R. In the excluded zone of thickness a adjacent to the fiber wall, there is no solute. In the fluid annulus, the solute equilibrium distribution is only a function of z. This radial partitioning establishes an equilibrium pressure drop across the imaginary boundary r = a + b:
 | (30) |
because Pa(z) = PR(z) and a(z) = R(z). The subscript "e" denotes the excluded region, subscript "a" denotes fluid annulus region outside the excluded region, and a(z) is the thermodynamic osmotic pressure one would measure at the concentration C(z) in the fluid annulus region. The mechanism for osmotic flow across the fiber matrix is clear from Eq. 30. The steric exclusion of solute near the fiber wall creates an abrupt pressure change proportional to the solute concentration; thus an axial concentration gradient in the fluid annulus region generates an axial pressure gradient in the excluded region near the fiber boundary that is the driving force for the fluid in the annulus.
The equation for the velocity profile is derived from Eq. 14. In the excluded zone, b r a + b,
 | (31A) |
in which
 | (31B) |
and
 | (31C) |
In the fluid annulus region, a + b r R,
 | (32A) |
in which
 | (32B) |
and
 | (32C) |
One notices that the hydrostatic components of the velocity in both the excluded region and the fluid annulus region are the same as in Happel's periodic fiber model for axial flow (8). However, the osmotic components of the velocity differ from the hydrostatic components. Note that Eq. 32C, which describes the osmotic contribution to the velocity profile in the fluid annulus, is independent of r and thus non-Poiseuillean in nature. However, the osmotic contribution in the excluded zone is a function of r containing both parabolic and logarithmic terms.
The average velocity derived from Eq. 15 is
 | (33) |
As mentioned in the model by Anderson and Malone (2), the exact determination of Pa/z and a/z requires the solution of the mass flux equation for solute, which is coupled to the velocity profile. When the bulk hydrostatic pressures at both sides are the same, Pa/z is close but not equal to zero, and the velocity in the fluid annulus is relatively flat because Va,os(z) is not a function of r. On the other hand, Ue(r,z) is a function of r. A sketch of a typical axial velocity profile when P(0) = P(L) is shown in Fig. 3C.
Reflection coefficient for filtration f
The reflection coefficient for filtration, f, is defined by Curry (5) as
 | (34) |
in which JS is the solute convective flux, JV is the solvent flux, C1 is the solute concentration in the bulk solution at the entrance to the fiber layer, and U is the pressure-driven velocity through a periodic unit of the fiber matrix. The lower limit of integration r = a + b for the solute convective flux arises from the steric exclusion due to the finite solute size. Equation 34 assumes that the solute is convected at the local velocity of the solvent and thus neglects the hydrodynamic interaction with the fiber boundary. As shown by Happel (8),
 | (35) |
Therefore,
 | (36) |
which is exactly the same as the osmotic reflection coefficient given in Eq. 28.
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RESULTS
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In Fig. 4 we have plotted the dimensionless solvent velocity profile resulting from the hydrostatic pressure gradient, Vhy (dashed curve), and compared this profile with the velocity profiles resulting from the osmotic pressure gradient given by Eq. 31C in the excluded zone, Ve,os (dotted curves), and Eq. 32C in the fluid annulus, Va,os (solid curves), for three different size solutes: sucrose (a = 0.46 nm), -lactalbumin (a = 2.01 nm), and albumin (a = 3.5 nm). We applied the fiber matrix structure proposed in Refs. 18 and 22 for the EGL, where b = 6 nm and = 8 nm. Equation 7 predicts that R = 10.5 nm. The vertical reference lines show the interface between the excluded region and the fluid annulus, and the vertical boundary on the right is the edge of the periodic unit. The hydrostatic velocity component is independent of solute size, whereas the osmotic velocity components are strongly size dependent. For large molecules like albumin, the velocity component resulting from the osmotic pressure gradient (or concentration gradient) is substantial. On the other hand, for small molecules like sucrose, the osmotic contribution to the velocity profile is trivial.
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Fig. 4. Comparison of the dimensionless velocity profiles attributed to the hydrostatic pressure gradient (Vhy) and the osmotic pressure gradient in the excluded region (Ve,os) and in the fluid annulus (Va,os) for albumin, -lactalbumin, and sucrose.
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In Fig. 5, we have plotted 0 given by Eq. 28 as a function of one of the two dimensionless lengths, and , with the other treated as a fixed parameter. 2 is used instead of , because this is the widely used fiber fraction, Vf. In Fig. 5A, we have shown 0 as a function of for three values of 2. As shown by Squire et al. (18), the radii of the scattering centers (fiber radii b) varied between 5 and 6 nm. 2 = 0.33 corresponds to b = 6 nm, = 8 nm, and a periodicity between fibers of 20 nm. 2 = 0.23 corresponds to b = 5 nm, = 10 nm, and a periodicity between fibers of, again, 20 nm. 2 = 0.10 corresponds to b = 2 nm, a typical radius of the center core protein monomers in proteoglycans, and = 8 nm so that the matrix is still suitable to function as a molecular sieve for albumin. For each value of 2, 0 
0 as 0, and 0 1 as 1 – . In Fig. 5B, we have shown 0 as a function of 2 for several values of . For each value of , 0 2 as 2 0, and 0 1 as 2 (1 – )2.
Equations 3 and 28 for 0 are quite different in form. We wanted to see how they compared with one another for the structural model proposed in Refs. 18 and 22 for the EGL. Figure 6 compares the predictions for 0 as a function of the partition coefficient, . In Fig. 6A, is described, and in Fig. 6B, 2 is described. If the core proteins are spaced at 20-nm intervals and are of 10–12 nm in diameter, Eq. 4 predicts that falls in the range 0.182–0.345 for b = 5–6 nm. This range of is indicated by the vertical dashed lines in Fig. 6. For b = 6 nm, R = 10.5 nm, and a = 3.5 nm, = 0.33, and 2 
0.33.One observes that for these values of and 2, and for this range of , the predictions for 0 from the pore theory are close to those for Eq. 28. For other values of or 2, the discrepancy is much larger.
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DISCUSSION
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Heretofore, the molecular sieving structure of the EGL had been widely assumed to be associated with extended GAG side chains periodically arranged along the core proteins of proteoglycans (6, 7, 10). These extended fibers were thought to be sialic acid side chains with typically 0.6-nm radius. Because the structure of the proteoglycans was largely deduced from their two-dimensional appearance when splayed out on carbon filters (3), it was never clear how the fibers formed a three-dimensional lattice. It was assumed that in their extended state, the protein monomers had a bottle brush appearance. The structural model depicted in Fig. 2, adapted from Weinbaum et al. (22), is an appealing alternative in that it provides a reasonable three-dimensional organization of the matrix that is consistent with the latest ultrastructural studies (18). For a = 3.5 nm (albumin), if b = 0.6 nm and = 8 nm, values typical of extended glycosaminoglycan side chains, Eq. 3 predicts that 0 = 0.52. In contrast, if b = 6 nm and = 8 nm, Eq. 28 predicts that 0 = 0.64. The measured value of 0 is typically >0.9 (14). This can be achieved simply by decreasing to 6.5 nm while keeping b = 6 nm. For a = 2.0 nm (-lactalbumin), b = 6 nm, and = 6.5 nm, Eq. 28 predicts that 0 = 0.32, which is close to the lower limit of the measured 0 for -lactalbumin in frog mesenteric capillaries of 0.34–0.69 (11). The current model may underestimate the value of 0 because other factors, such as electric charge, cross-links between core protein fibers, and clustering of core proteins at the base of each cluster have not been taken into consideration. Therefore, the structural model proposed in Refs. 18 and 22 provides more realistic predictions for 0 than a model based on extended GAG side chains.
Our own unpublished electron micrographs and those of Rostgaard and Qvortrup (16) suggest that the filamentous strands depicted in Fig. 2 are primarily perpendicular to the lipid bilayer. In other models of the glycocalyx, these fibers are aligned in a periodic array parallel to the bilayer. We have not attempted to develop a theory for the reflection coefficient for the latter fiber geometry because its analysis is much more difficult. Fluid mechanicians have questioned the validity of the zero vorticity (zero shear stress) boundary condition applied by Happel (8) at r = R for flow transverse to the fiber axis. This flow geometry was later analyzed by Sangani and Acrivos (17) using an infinite series solution combined with boundary collocation methods. These numerical solutions could be employed for the parallel fiber array, but one would not be able to obtain easily usable closed form solutions for the reflection coefficient corresponding to Eqs. 28 and 36.
The basic principle behind the osmotic flow in a fiber matrix is that impermeable fiber surfaces within the fiber matrix create radial gradients in solution properties normal to these surfaces that are necessary to maintain thermodynamic equilibrium. A solute potential energy is generated, because the impermeable fiber surfaces interact differently with the solute than with the solvent. The Gibbs-Duhem equation can be applied to relate the hydrostatic pressure and the solute concentration in the vicinity of the fiber surface as proposed by Anderson and Malone (2) for the walls of a cylindrical pore. However, as in the cylindrical pore model (2), factors other than steric exclusion enter into the evaluation of Eq. 23 for 0. These include, as mentioned earlier, solute shape, electric charge, core protein cross-links, and other fiber geometries.
There are two components to the velocity profile in the excluded region, b r a + b. One is attributed to the hydrostatic pressure gradient, Ve,hy, which is proportional to the dashed curve in Fig. 4, and the other is attributed to the osmotic pressure or concentration gradient, Ve,os, which is proportional to the dotted curves in Fig. 4. Similarly, there are two components to the velocity profile in the fluid annulus, a + b r R, the hydrostatic component, Va,hy, which is proportional to the dashed curve in Fig. 4, and the osmotic component, Va,os, which is proportional to the solid portion of the velocity profile curves in Fig. 4. The coefficients of all four dimensionless velocities in Eqs. 31 and 32 have the same scale factor, 4R2/µ, multiplying the hydrostatic or osmotic pressure gradient. As shown in Fig. 4, Vhy (which includes Ve,hy and Va,hy) and Ve,os are functions of r. However, Va,os is independent of r and describes the continuity in the osmotic component of the velocity at the interface, r = a + b. Therefore, the velocity profiles in an osmotic-driven flow, where C(0) \= C(L), differ from those in a pressure-driven flow, where C(0) = C(L). It is only in the limit where the solute is too large to pass through the fiber array that the profiles are the same.
The difference in pressure and osmotic velocity profiles raises a basic question: if T[C(0) – C(L)] = P(0) – P(L), will there be no flow? If the fiber matrix is impermeable to the solute, the answer is yes. The fluids on both sides of the matrix layer are exposed to the same net pressure, P(z) – (z), where z = 0 or L. The osmotic flow is completely balanced by the pressure-driven flow. There is no net flow in the fiber matrix. In contrast, if the fiber matrix is leaky to the solute, the flow across the fiber matrix layer is caused by the difference in the velocity profiles driven by the hydrostatic pressure gradient and the osmotic pressure gradient, and these two profiles differ. The reflection coefficient actually describes this difference (see Eq. 18 or 28). In Eq. 18, the numerator is proportional to the average velocity for the osmotic flow and the denominator is proportional to that for pressure-driven flow when the pressure gradient is the same as the osmotic pressure gradient.
Another observation is that if one applies equal hydrostatic pressures but unequal concentration or osmotic pressures at both sides of a leaky fiber matrix layer, the velocity in the fluid annulus, Ua(r,z), is not zero, primarily because of the osmotic pressure difference, and there must be convective flux for the solute, which leads to a nonuniform concentration or osmotic pressure gradient, a/z, along the z direction. On the other hand, the average velocity, , defined in Eq. 33, is constant because of the continuity in fluid flux. One concludes that the deviation of a/z from a constant gradient must be compensated for by a nonvanishing gradient for Pa/z. Therefore, the nonlinear effect of convection on the axial concentration profile requires that Pa/z 
0, although P(0) = P(L).
The reflection coefficient for filtration (f) is exactly the same as the osmotic reflection coefficient in the fiber matrix model when only steric exclusion is considered (see Eqs. 28 and 36). The same result also is obtained for pore theory, where it has been shown (2) that 0 = f. For the fiber matrix model examined in this study, 0 is actually the ratio of the average velocity (or the dimensionless flow rate) for the dimensionless velocity profile ADiEi, i = 1, 2, 3, in Fig. 4 to the average velocity for the velocity profile ABiC, i = 1, 2, 3. f is the ratio of the average velocity (or the dimensionless flow rate) for the dimensionless velocity profile ABi to the average velocity for the profile ABiC, i = 1, 2, 3. The integral of Eq. 31C from b/R to (a + b)/R plus the integral of Eq. 32C from (a + b)/R to 1 yields the same result as integrating Eq. 31B from b/R to (a + b)/R. These integrals involve multiplying the local velocities by 2r. Therefore, 0 = f. This is not unexpected, because both reflection coefficients describe the steric exclusion of solutes from the confining boundaries in a leaky barrier.
The large differences in the expressions for 0, Eqs. 3 and 28, arise from two sources. One is that the velocity profiles generated by the osmotic gradient (see Eqs. 31C and 32C) are in fact not Poiseuillean and do not have a parabolic profile. The other is that the confining boundary geometries differ greatly, one representing flow interior to a circular cylinder and the other representing flow exterior to the cylinder surface. As shown in Fig. 6B, there is only good agreement between Eqs. 3 and 28 for 0 for all solute radii when 2 
0.16.
The curves in Fig. 6A for Eq. 28 are truncated at values of <1. = 1 requires that all the area within each periodic unit in the fiber matrix be available to solute, i.e., the solid fraction for the fiber matrix is zero. For each value of or 2, there is a maximum value of that satisfies the constraint = 1 – ( + )2. In the limit 2 0, the fiber radius becomes vanishingly small and the minimum value of 0 is determined by the exclusion volume of the solute surrounding the fiber axis, given by 0 2, whereas in the limit 0, 0 0, because the solute radius vanishes and no osmotic force can be generated.
The dimensionless parametric curves in Figs. 5 and 6 are convenient for quickly estimating 0 for a given periodic fiber array in which either or 2 are specified. For a given 2 (see Fig. 5A or 6B), the fiber radius and fiber matrix periodicity are specified and one varies or solute radius. For a given (see Fig. 5B or 6A), the solute radius and fiber matrix periodicity are specified and one varies 2 or fiber radius.
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GRANTS
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This research is supported by National Heart, Lung, and Blood Institute Grant HL-44485. This research was performed in partial fulfillment of the Ph.D. dissertation of Xiaobing Zhang from the City University of New York.
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FOOTNOTES
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Address for reprint requests and other correspondence: S. Weinbaum, Depts. of Biomedical and Mechanical Engineering, The City College of New York, CUNY, 138th St. at Convent Ave., New York, NY 10031 (e-mail: weinbaum{at}ccny.cuny.edu)
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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REFERENCES
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- Adamson RH, Lenz JF, Zhang X, Adamson GN, Weinbaum S, and Curry FE. Oncotic pressures opposing filtration across non-fenestrated rat microvessels. J Physiol 557: 889–907, 2004.[Abstract/Free Full Text]
- Anderson JL and Malone DM. Mechanism of osmotic flow in porous membranes. Biophys J 14: 957–982, 1974.[Web of Science][Medline]
- Buckwalter JA and Rosenberg LC. Electron microscopic studies of cartilage proteoglycans. Direct evidence for the variable length of the chondroitin sulfate-rich region of proteoglycan subunit core protein. J Biol Chem 257: 9830–9839, 1982.[Free Full Text]
- Clough G, Michel CC, and Phillips ME. Inflammatory changes in permeability and ultrastructure of single vessels in the frog mesenteric microcirculation. J Physiol 395: 99–114, 1988.[Abstract/Free Full Text]
- Curry FE. Mechanics and thermodynamics of transcapillary exchange. In: Handbook of Physiology. The Cardiovascular System. Microcirculation. Bethesda, MD: Am Physiol Soc, 1984, sect. 2, vol. IV, p. 309–374.
- Curry FE and Michel CC. A fiber matrix model of capillary permeability. Microvasc Res 20: 96–99, 1980.[CrossRef][Web of Science][Medline]
- Fu BM, Weinbaum S, Tsay RY, and Curry FE. A junction-orifice-fiber entrance layer model for capillary permeability: application to frog mesenteric capillaries. J Biomech Eng 116: 502–513, 1994.[Web of Science][Medline]
- Happel J. Viscous flow relative to arrays of cylinders. AIChE J 5: 174–177, 1959.[CrossRef]
- Hu X, Adamson RH, Liu B, Curry FE, and Weinbaum S. Starling forces that oppose filtration after tissue oncotic pressure is increased. Am J Physiol Heart Circ Physiol 279: H1724–H1736, 2000.[Abstract/Free Full Text]
- Hu X and Weinbaum S. A new view of Starling's hypothesis at the microstructural level. Microvasc Res 58: 281–304, 1999.[CrossRef][Web of Science][Medline]
- Huxley VH, Curry FE, Powers MR, and Thipakorn B. Differential action of plasma and albumin on transcapillary exchange of anionic solute. Am J Physiol Heart Circ Physiol 264: H1428–H1437, 1993.[Abstract/Free Full Text]
- Mauro R. Some properties of ionic and nonionic semipermeable membranes. Circulation 21: 845–854, 1960.[Abstract/Free Full Text]
- Michel CC. Starling: the formulation of his hypothesis of microvascular fluid exchange and its significance after 100 years. Exp Physiol 82: 1–30, 1997.[Web of Science][Medline]
- Michel CC and Curry FE. Microvascular permeability. Physiol Rev 79: 703–761, 1999.[Abstract/Free Full Text]
- Ray PM. On the theory of osmotic water movement. Plant Physiol 35: 783–795, 1960.[Free Full Text]
- Rostgaard J and Qvortrup K. Sieve plugs in fenestrae of glomerular capillaries—site of the filtration barrier? Cells Tissues Organs 170: 132–138, 2002.[CrossRef][Web of Science][Medline]
- Sangani AS and Acrivos A. Slow flow past periodic arrays of cylinders with application to heat transfer. Int J Multiphase Flow 8: 193–206, 1982.
- Squire JM, Chew M, Nneji G, Neal C, Barry J, and Michel C. Quasi-periodic substructure in the microvessel endothelial glycocalyx: a possible explanation for molecular filtering? J Struct Biol 136: 239–255, 2001.[CrossRef][Web of Science][Medline]
- Staverman AJ. The theory of measurement of osmotic pressure. Rec Trav Chim Pays-Bas Belg 70: 344–352, 1951.
- Vink H and Duling BR. Identification of distinct luminal domains for macromolecules, erythrocytes, and leukocytes within mammalian capillaries. Circ Res 79: 581–589, 1996.[Abstract/Free Full Text]
- Weinbaum S. 1997 Whitaker Distinguished Lecture: models to solve mysteries in biomechanics at the cellular level; a new view of fiber matrix layers. Ann Biomed Eng 26: 627–643, 1998.[CrossRef][Web of Science][Medline]
- Weinbaum S, Zhang X, Han Y, Vink H, and Cowin SC. Mechanotransduction and flow across the endothelial glycocalyx. Proc Natl Acad Sci USA 100: 7988–7995, 2003.[Abstract/Free Full Text]
- Yan ZY, Weinbaum S, and Preffer R. On the fine structure of osmosis including three-dimensional pore entrance and exit behaviour. J Fluid Mech 162: 415–438, 1986.[CrossRef]
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