A Three-dimensional Junction-pore-matrix Model for Capillary Permeability - S

MICROVASCULAR RESEARCH 4‘%, 85-111 (1992)

A Three-Dimensional Junction-Pore-Matrix Model for Capillary Permeability

S. WEINBAUM,*” R. TsAY,t AND F. E. CURRYZ~?

Departments of *Mechanical Engineering and tChemica1 Engineering, The City College of the City University of New York, New York, New York 10031; and $Department of Human

Physiology, School of Medicine, University of California at Davis, Davis, California 95616

Received February 14, 1991

A three-dimensional model is presented for the hydraulic conductivity and diffusive permeability of capillary endothelial clefts with a junctional strand with discrete pores and a fiber matrix in its wide parts. The model attempts to provide new insight into long-standing issues concerning the relative importance of open junction discontinuities, restricted slit regions, and matrix components in determining the permeability and selectivity of the capillary wall. The predictions drawn from the model are used to formulate new experiments to test two hypotheses concerning the molecular organization of the junction strand and the location of matrix structures in the wide part of the cleft. Using the three-dimensional theoretical approach recently developed by Tsay, Weinbaum, and Pfeffer (Chem. Eng. Comm. %2,67- 102, 1989), the model first explores the behavior of three different molecular models for the junctional strand discontinuities: (i) a more frequent circular pore of 5.5-nm radius formed by isolated missing junction proteins; (ii) a restricted rectangular slit of four to eight missing proteins and 8-nm gap height; and (iii) larger more infrequent breaks of four to eight missing proteins with a gap height of 22 nm, equal to the width of the wide part of the cleft. For the circular and 8-nm gap height pores the primary molecular sieve can be located at the level of the junction strand, whereas for the 22-nm gap height pores, matrix components must be present in at least some portion of the cleft to provide the molecular filter. The water flow through the cross-bridging fibers in the wide part of the cleft is described either by a new exact three-dimensional theory (Tsay and Weinbaum, J. Fluid Mech. 226, 125-148, 1991) for an ordered periodic array or by a new approximate theory for a random array of perpendicular fibers. Both this theory and the new approximate theory for diffusion presented herein take into account for the first time the interaction between the fibers and plasmalemma boundaries. The principal predictions of the model are that (i) infrequent larger breaks are most likely required to account for small solute permeability; (ii) these larger breaks must be accompanied by a sieving matrix, but this matrix probably occupies only a small portion of the depth of the cleft and/or its entrance at the luminal surface; (iii) neither junctional pore, restricted slit, or fiber matrix models can by themselves satisfy permeability and selectivity data; and (iv) one-dimensional models are a poor description of a cleft with infrequent larger breaks since the solute will be confined to small wakelike regions on the downstream side of the junction strand discontinuities and thus not fill the wide part of the cleft. 0 1~2 Academic PKSS, I~C.

’ To whom correspondence and reprint requests should be addressed. * Present address: Institute of Biomedical Engineering, National Yang-Ming Medical College,

Taipei, Taiwan 11221.

85

w26-286292 $5.00 Copyright 0 1992 by Academic Press, Inc.

All rights of reproduction in any form reserved. Printed in U.S.A.

86 WEINBAUM, TSAY, AND CURRY

6-611 width not speoiffed

Ilf

w

17-22nm

1, J

1 ,

FIG. 1. Two one-dimensional models of intercellular pathway. (a) Constricted channel model. This model proposed that the junctional cleft is 17-22 nm wide except for one or more constrictions. The constriction is assumed to be the main molecular filter and its width, 6-8 nm, is chosen to fit reflection coefficient data. (b) Fiber matrix model. The fiber matrix is assumed to be contained in the wide part of the channel and acts as the molecular filter. Both models assume that the permeability is proportional to the fractional length of the tight junction which is effectively open.

INTRODUCTION

The reasons for developing the theoretical model described in this paper were, first, to more rigorously evaluate two alternative hypotheses describing the size- selective structures that determine capillary permeability and, second, to use the theory to conceptualize new experiments to test these two hypotheses. One hypothesis proposes that the molecular filter lies within the small discontinuities in the strand of molecules forming the junctional complex between adjacent endothelial cells. These discontinuities also determine the length of the junction effectively open for the diffusion of small hydrophilic solutes. The second hypothesis also proposes that the breaks in the junctional strand determine small- solute permeability, but assumes that the primary molecular filter is a fibrous network associated with the cell surface and the wide part of the junctional cleft. To quantitatively test the feasibility of these two hypotheses we have developed specific molecular models for the structure of the junctional pore and the fiber matrix. The prediction of the models are evaluated using data on the hydraulic conductivity, permeability, and selectivity of microvessels with continuous en- dothelium and the ultrastructure of microvessels of known permeability properties.

The starting point for our analysis is the model in Fig. la, which is based on published studies of the ultrastructure of the cleft between adjacent endothelial cells, most of which were completed prior to 1980 using electron microscopy on conventional thin sections of 40-50 nm thickness, and protein tracer molecules, which could be used as size-specific molecular probes of the pathways through the cleft (Perl, 1971; Casley-Smith et al., 1975; Wissig, 1979; Perry, 1980; Crone and Levitt, 1984). The molecular filter was assumed to be associated with slitlike openings in the junctional complex, where the adjacent membranes appear to be very close, but not fused. This size-selective pore could also be formed by a tortuous pathway across a multistranded array in which these slitlike openings could appear in different planes of section. These views lead several authors to

A MODEL FOR CAPILLARY PERMEABILITY 87

propose a simple one-dimensional constricted channel model, as shown in Fig. la, in which the permeability was proportional to the fractional length of the junction strand, which was effectively open for the diffusion of small solutes, and the size-selective structure was a restrictive slit of 6- to 8-nm gap height, connecting the wider regions of the cleft whose width was estimated to be 17-22 nm by various investigators. In muscle capillaries the measured hydraulic conductivity and permeability coefficient for small solutes could be accounted for if 10% of the junctional strand was open, while in frog mesenteric capillaries close to 90% of the junctional strand needed to be open to account for these permeability properties. The restricted cleft height was the equivalent slit dimension that would satisfy the measured reflection coefficients for solutes of 1.5 to 3.5-nm radius (Curry, 1986; Crone and Levitt, 1984).

Over the past decade there have been two significant challenges to important details of this model. First, three-dimensional reconstructions of serial sections of capillaries in rat heart muscle indicated that the discontinuous breaks in the junctional strand were only of the order of a single conventional thin-section thickness (40 nm) and thus was much shorter than the breaks proposed by earlier investigators, who had based their estimates on random thin sections. Further- more, the 40-nm breaks observed by Bundgaard (1984) were very infrequent (approximately one per micrometer of junction length). These discontinuities thus represented a fractional length of open junction that was considerably shorter than the length of the discontinuities predicted by the one-dimensional models. The 3-D reconstructions also showed that the pattern of the junctional complexes representing the points of contact between adjacent cells was very similar to the pattern of ridges and groves seen in freeze-fracture electron micrographs of the junctional complex. This was the result expected if the junctional complex seen in conventional EM corresponds to the arrays of intramembranous particles seen in freeze-fracture as is also suggested by Wissig (1979). Bundgaard pointed out that discontinuities in the junctional strand shorter than 40 nm could not be identified in conventional thin sections. After reexamining a small sample of ultrathin sections (12-15 nm) from a previous study, he suggested that discontinuities as small as 12 nm in the strand might represent a population of smaller pores. We also evaluate this hypothesis in more detail in this paper.

The second challenge to the model in Fig. la is the observation that the permeability and selectivity of a variety of continuous capillaries can be accounted for if a matrix of fibrous molecules covered the endothelial cell surface and filled all or part of the cleft as shown in Fig. lb. The fiber matrix hypothesis placed the primary molecular filter in the matrix and not at the junctional complex. Reviews evaluating this hypothesis have been published elsewhere (Michel, 1984, 1988; Curry, 1986; Crone and Levitt, 1984; Tsay et al., 1989). Further experimental support for this concept is the recent demonstration that enzymatic removal of the endothelial glycocalyx in frog mesenteric capillaries decreases the hydraulic resistance by roughly 50% (Adamson, 1990). Since the matrix does not constitute a significant barrier to small solutes, the predicted permeability to small solutes, assuming the cleft is open, is much too large. To account for this shortcoming, it is necessary to postulate that small solute permeability is regulated by pores or breaks in the junctional strand. One fundamental difference between the proposed junctional strand-pore-fiber matrix model considered herein and the straight junctional strand-pore model considered previously in Fig. la is that the structure of


b)

‘missing junction partihes

f3nm

%ght regibn ?ong rectangular

44423nm

........................... ......................... ........................ FIG. 2. Three schematic diagrams of junction protein arrangements. These protein arrangements

correspond to (a) small gaps suggested by Bungaard’s (1984) ultrathin sections (the zigzag protein pattern was first proposed in Tsay et al. (1989)); (b) 1 on rectangular junction pores, similar to those g in the one-dimensional constricted channel model; and (c) large pores formed by discontinuities of 44-88 nm in the junctional strands.

the junctional strand discontinuities does not have to be restricted to dimensions required of a molecular sieve if a matrix is present.

The development of the present models was also driven by the need to obtain more precise molecular models of the junctional complex. Either the discontinuities are so small that they are not observed in 40-nm sections or they are larger, but infrequent, so that most of the junctional strand is a tight barrier impenetrable to water and solutes. Figure 2 summarizes several arrangements of the junction proteins considered in this manuscript. The interlocking structure in Fig. 2a was suggested by Tsay et al. (1989) (TWP) as a way in which size-selective pores could be formed by individual missing proteins. This interlocking arrangement was suggested by the observation in Firth et al. (1983) that the average spacing between junctional proteins in each membrane of the guinea pig placental capillaries was 22 nm, just twice the diameter of the individual proteins (11 nm). An important consequence of the model in TWP was that the predicted spacing between the pores, required to account for measured value of L, in frog mesenteric capillaries, (-100 nm) was too small to neglect the three-dimensional interaction of the velocity fields between pores and a three-dimensional model for the junctional strand was developed.

The alternate possibility that the discontinuities in the junctional strand are larger than a single missing protein, and widely spaced, is usually represented as shown in Fig. 2b. The rectangular slit length is shown as 44-88 nm, representing four to eight missing proteins, and the slit height is shown as 6-8 nm in order that the slit can function as the primary molecular filter. However, it is difficult to explain how this slit height is maintained. One expects that, if junction proteins were absent in these longer discontinuities, the membrane forces that determine the equilibrium spacing in the wide part of the cleft would also be operative in


Junction

FIG. 3. Sketch of the three-dimensional junction-pore-matrix model of the intercellular cleft. Junctional strand with periodic pores lies parallel to luminal front. Lz is the depth of pores in the junctional strand and 15, and 15, are the depths between the junctional strand and the luminal and abluminal fronts. Within the wide portion of cleft, cross-bridging fibers are represented by either a periodic square array or a random array of cylindrical posts. The radius of these posts is a and the gap spacing between posts for periodical array is A. The distance between two adjacent pores in the junctional strand is 20.

the region where these breaks appear. Thus, if the membrane-bending resistance over these longer breaks was small, it is anticipated that their gap height would be nearly the same as the background 17- to 22-nm height of the wide part of the channel, as shown in Fig. 2c. The larger breaks observed in Bundgaard (1984), whose lengths are typically single conventional FM sections, support this conjec- ture concerning the gap height of the larger breaks. If such pores are present they cannot be the primary size-selective structure and a fiber matrix in the wide part of the cleft is also required to provide the molecular sieve.

The theoretical model described in this paper is an extension of the 3-D model in TWP. The simplest form of the model is shown in Fig. 3, where the interlocking junction strands were treated as an impermeable barrier with periodically distributed discrete circular pores representing the individual missing proteins as shown in Fig. 2a. In order to represent the fiber matrix components in the wide part of the cleft, a two-dimensional square array of slender cross-bridging fibers of radius a and height 2B was introduced with a periodic spacing 2w and gap spacing A as sketched in the left hand diagram in Fig. 3. The fibers are shown in an ordered array to account for the fact that part of the interaction of albumin with the matrix appears to involve ordering of the fibers as described by Michel (1988).

In the present paper we shall explore three important extensions of the model in TWP. First, the pore in TWP was assumed to be formed by individual missing proteins in the junction protein strand whose cross-sectional shape was a circular hole of radius rp = 5.5 nm. This value of pore radius was selected to fit the osmotic reflection coefficient data for intermediate-size solutes smaller than al-


burnin (effective hydraulic diam 7 nm). In this study the various rectangular junction pore geometries shown in Fig. 2 will also be examined and the permeability properties of small clusters of up to eight missing proteins will be analyzed. Second, a much more rigorous hydrodynamic theory has been developed for the fiber matrix, which is valid for an ordered array of cross-bridging fibers of arbitrary aspect ratio and spacing (Tsay and Weinbaum, 1991). This new theory will be used to (i) describe the additional hydrodynamic resistance due to an ordered matrix in the wide part of the cleft and (ii) to develop a simple model for randomly distributed cross-bridging fibers. Third, the model in TWP considered only filtration, In the present study an equivalent theory for the junctional strand-pore- matrix model has also been developed for diffusion. Thus, it will be possible to examine the effect of pore geometry and fiber matrix components on the solute permeability coefficient P.

METHODS

1. Hydrodynamic Interaction between Junction Pores

The model for filtration through the idealized intercellular cleft geometry sketched in Fig. 3 combines two different types of channel flows; the three- dimensional hydrodynamic interaction between the pores in the junction strand and the flow through the fiber array. The expression for L, for the entire cleft can be written as the sum of three resistances in series,

L, = (R, + R2 + R&l, (1)

where RI and R3 describe the mixing interaction of the Stokes jets within the wide parts of the channel and Rz the resistance within the junction pore itself, The expressions for these resistances are given for a circular pore of radius ri, in TWP as

3~ & + I z tanh (X,L,) sin* (h,d)

Ri = B2Lj, 2 & -[ A3, 1 , i = 1,3 (2)

A, = F, d=g

R 2

= 16pL2D 7rri Lj, ’

(3)

(4)

where p is the fluid viscosity, all the geometric lengths are defined in Fig. 3, and Lj, is the total cleft length (total cell perimeter) per unit endothelial surface area. For a pore of rectangular cross-section of height 26 and width 2d, Eq. (4) for R2 is replaced by

R 2

(5)

Gr, y = b/d.

The foregoing results are derived by matching the volumetric flow and average


pressure entering and leaving the junction pore with the entrance and exit conditions in regions 3 and 1. The nonuniformity of the pressure at the pore entrance and exit obtained from the Hele-Shaw solutions in regions 1 and 3 is small and gives results that are nearly indistinguishable from those obtained by requiring the pressure to be uniform across the entrance and exit of the pore.

2. Hydrodynamic Resistance of Fiber Matrix

The effect of the hydrodynamic interaction with the cross-bridging fibers in the wide portion of channel can be treated by replacing the actual fluid viscosity ,U in Eq. (2) by an effective viscosity peff. peff is defined by V (P) = - heff (w/B’, where ( ) denotes an average value over a region that is small compared to the depth of the cleft L. For the periodic fiber arrangement shown in Fig. 3 the average is taken over a single periodic unit containing one fiber in the array. The effective viscosity peff can be written as pf, where f is an hydrodynamic interaction function that depends on the fiber configuration, the fiber volume fraction S, and the aspect ratio B/a of the fibers.

The boundary value problem for determining peff requires that the no-slip conditions be satisfied on the surface of each fiber or on the channel walls. In Tsay and Weinbaum (1991) an exact infinite series solution to the full boundary value problem for the doubly periodic fiber array is constructed using the fundamental singularities for a single circular post developed in Lee and Fung (1969) and a quasi-periodic function that is a linear combination of the Weierstrass 4 function and its derivatives to describe periodicity conditions for each fiber unit cell. This highly accurate solution is far too complicated to be used with the same ease as a Carmen-Kozeny equation for an infinite medium. However, this solution provides important new insight into the velocity field surrounding the fibers and is used to justify the validity of the Brinkman approximation introduced in the next section. The important observation is that for long slender fibers the viscous boundary layers surrounding the fibers can be much larger than their radius and when the fiber boundary layers overlap there will be a large increase in drag or peff (decrease in Lp). In contrast, if these boundary layers remain distinctly separated there will be very little increase in drag beyond what is caused by the walls of the channel. One thus anticipates that large increases in drag will start to occur when A/B < 1.

3. Brinkman Approximation for a Bounded Fiber Matrix

Based on the foregoing discussion, the fiber matrix will be an important consideration in determining the filtration resistance if A is less than 10 nm since B is approximately 10 nm. In this regime the two-term approximate solution in Lee (1969) is not accurate and we seek a simpler approximation to the exact infinite series solution in Tsay and Weinbaum (1991) for a bounded porous media using a Brinkman-type approximation. In the Brinkman approach the resistance due to the fibers is taken into account by adding a Darcy-type resistance term in the governing equation for the channel flow:

VP=j+LV%. P


Here p is the fluid viscosity and I& is a Darcy permeability coefficient, which describes the flow through an infinite matrix that has the same fiber geometry as the interior of the bounded flow under consideration. In the present case the unbounded flow geometry corresponds to flow perpendicular to a periodic array of infinitely long fibers. The exact solution to this two-dimensional unbounded flow problem is given in Sangani and Acrivos (1982). Tsay and Weinbaum (1991) showed that the following simple approximation is accurate to within 10% for s < 0.7:

Kp = 0.0572 a2 (A/a)2.377 (7)

The solution to the Brinkman equation that satisfies the no-slip conditions on the channel walls is

is _ - Kp dP _ cash (2/I&)

P dx (

1

1 cash (Pimp) *

By averaging this solution for the velocity across the channel height we can define an effective Darcy law permeability by

K p,eff = -CL - \ I dx

which describes the effect of the channel The above expression for Kp,eff can be

coefficient f using the expression Kp,eff =

(9)

walls in increasing Kp. directly related to the channel friction B2/3f for Poiseuille flow in a channel.

After some algebra f can be written using Eq. (9) in the compact form

f= p3 3@ - tanh /3)’

where p = B/fip. The effective hydraulic permeability of a uniform channel with a fiber matrix is related to K,,eff by

where L, is the depth of the channel with matrix components and Lj, is the total cleft length.

The accuracy of the Brinkman approximation and the various approximate solutions described in this and the previous subsection are shown in Fig. 4, where the solutions for the increase in the channel drag correction factor f as a function of the solidity fraction S are plotted for three different aspect ratio fibers B/a = 0.5, 5, and 20. B/a = 20 is representative of a glycoprotein side chain spanning the wide part of the cleft, B/a = 5 describes the protein core of the proteoglycan, and B/a = 0.5 is typical of the septal posts in lung alveoli for which the theory in Lee and Fung (1969) and Lee (1969) was developed. Four sets of curves are shown. The solid set is the exact infinite series solution in Tsay and Weinbaum (Ml), the dashed set is the two-dimensional solution in Sangani and Acrivos (1982) for unbounded infinitely long fibers, the dash-dot set is the two-term


:E-3 lE-2 lE-1 1

S

FIG 4. A comparison between the drag coefficients, f , obtained by the three-dimensional exact solution (-, Tsay and Weinbaum, 1991) the Brinkman approximation (. . . , eq.(lO), the two-term asymptotic solution (-..-, Lee, 1%9), and the 20 solution (- - -, Sangani and Acrivos, 1982). The long dashed curve with symbols corresponds to the fiber array when its gap spacing equals B/2.

asymptotic solution previously used in TWP (Lee, 1969), and the dotted set in the new Brinkman approximation described by Eq. (10). The difficulty with the two-term asymptotic approximation used in TWP is clearly evident in Fig. 5. This asymptotic solution accurately describes the additional hydraulic resistance when the viscous boundary layers surrounding the fibers do not overlap and breaks down entirely for high aspect ratio fibers when A/B < 1. From the geometry of the fiber array in Fig. 3, A and S are related by

(12)

The value of S at which A/B = 1 is readily obtained by setting A = B in (12) and solving for S. For B/a = 0.5, which is the case treated by (Lee, 1969) in his

1.00

l.OE-2

FIG. 5. The ratio of D,,,&D,, for square array of cross-bridging fibers vs the effective fiber solid fraction S,.


model for the septal posts in the lung, A/B = 1 when S = 0.5. Thus, for B/a = 0.5 the two-term approximation is valid for nearly all values of S except for the dense septal post limit. However, for B/a = 20 the two-term approximation breaks down for S > 0.0065 or A < 12.0 nm if 2B is 22 nm. The corresponding values for B/a = 5 are S > 0.063 or A < 15.5 nm. This approximation, therefore, cannot be used for fiber spacings where the fiber matrix is the primary molecular filter for molecules smaller than albumin, effective diameter 7.1 nm. In contrast, the results clearly show that the Brinkman approximate solution is a remarkably accurate approximation for all values of S when B/a > 5, the aspect ratios of interest in the present application.

The excellent approximation provided by the Brinkman approximation when B/u > 5 suggests a very simple solution for a random matrix of cross-bridging perpendicular fibers in a channel. Instead of using the Sangani and Acrivos solution for K,, for a square periodic array, we introduce instead in the Brinkman equation a random cell Carman-Kozeny approximation for the two-dimensional perpendicular array. The Carman-Kozeny approximation is given by

K P

= (1 - 9”. g ) s* ( 1

03)

where C is a fiber density correction factor. When the fibers are circular cylinders perpendicular to the flow, Happel (1959) used a periodic unit cell model with vanishing shear at the edge of the periodic unit to obtain the following approximate expression for C:

c = 2 (1 - S13 s [In($) - i: J z1il-l. (14)

The randomness in the distribution of fibers can now be taken into account using a stochastic model developed by Yu and Soong (1975). In this model N fibers are randomly inserted into M subregions and a nonuniform fiber density distribution is generated. Each subregion has a different value of S given as

S(k) = #) !?! s 1 1 0 N ’

when it i(k) is the number of fibers in subregion i. By assuming that there is at least one fiber in each subregion, a different value of Kp is obtained for each region. The average value of K,,, which we denote by KPr is calculated as:

KPr = i i . . . i e) (N - M)!

n,=l n*=l mu=’ fi (,#d _ 1)!M(-‘)’

i=l

n1 + 122 + ..- + nM = N. (16)

The mean KPr is used to replace Kp in Eq. (6). In the results presented in Fig. 8a and 8b we have chosen N and M as 25 and 10, respectively.


4. Diffusive Permeability In the same way that a separate theory was required to describe the Hele-

Shaw flow through the junction strand with its pores and the effective hydraulic resistance of the cross-bridging fibers in the wide part of the cleft, separate theoretical models are required to describe (i) the basic diffusive resistance of the cleft with its junction strand and (ii) the added resistance due to fiber matrix components in the wide part of the cleft for solutes of finite size. The theoretical model for the lateral spreading of solute and the diffusive interaction between pores closely parallels the analysis developed in TWP for the derivation of Eqs. (l)-(4) in the present study, since the governing equation and boundary conditions for the pressure field in the Hele-Shaw flow through the junction strand are closely related to the boundary value problem for pure diffusion of a solute through this same geometry. The details of this solution are therefore omitted and only the final results are given below. The solution of problem (ii) for the diffusion of a finite-size solute through a bounded periodic fiber array is much more difficult since the effective diffusion coefficient D. rw,eff in the wide part of the cleft in regions 1 and 3 in Fig. 3 varies spatially and depends on the solution to a complicated hydrodynamic boundary value problem in which the no-slip conditions are satisfied on both the solute and channel-fiber surfaces. However, a simplified approximate expression for Diw,eff in regions 1 and 3 based on the Brinkman approximation will be proposed, which should provide reasonable accuracy provided the solute diameter 2r, does not approach the fiber gap spacing A. Current statistical theories for diffusion in a fiber matrix take account of the steric exclusion of the particles and introduce a collision-mean-free path to take account of the interaction of the solute with the matrix fibers using a statistical model for the fiber density distribution, see Ogston et al. (1973). However, this statistical theory does not consider the hydrodynamic interaction of the solute molecule with the disturbance velocity field of the matrix. Therefore, the statistical model would be anticipated to un- derestimate the diffusive resistance of the periodic fiber array considered in Fig. 3.

We consider first the diffusion of the solute through a cleft with a junction strand with periodic circular or rectangular pores, but no fiber matrix components in the wide part. For this simplified junction-pore model the diffusion coefficient in the wide part Di, can be treated as a constant. The finite size of the solute can be taken into account by considering the additional hydraulic resistance due to the channel walls using the solutions in Ganatos et al. (1981) for the motion of a finite sphere for this geometry as described in Curry (1984). In place of Eqs. (4) and (5) one has for circular or rectangular pores, in that order.

R=2L, D-1 2 ()O -

nri Ljt 02 (17)

R=2L, Dl 2 ()O 2bd Ljt z’ (18)

where D2, the restricted diffusion coefficient, depends on the cross-sectional geometry of the pore. Expressions for Dz are given in Curry (1984) for both a circular pore and a pore that is a slit with parallel walls. The latter is a good approximation for the rectangular pores considered herein since b < d. In contrast to the hydraulic resistance, which varies as the fourth power of the pore radius


or the third power of the slit height, the diffusional resistance of the junction strand is proportional to the open pore area and thus the junction strand constitutes a smaller fraction of the total resistance of the cleft.

The solution to the boundary value problem for the concentration field in regions 1 and 3 in Fig. 3 leads to the following expression for the diffusional resistance in these regions:

m tanh (h,LJsin*(h,d) 1 An3 ’ i = 1,3 (19

where

A, = F, d= ‘TT(rp - rs)2

4 (B - rs)

Here Di, is the restricted diffusion coefficient for a finite sphere in a channel obtained from Ganatos et al. (1981). Note that (19) is of the same form as (2) except for multiplicative parameters. The permeability P is related to the diffusional resistances in (17)-(19) by

P = [R, + R2 + R3]-‘. (20)

We consider next problem (ii), the added resistance due to matrix components in the wide part of the cleft. Expression (19) could be easily modified to take account of cross-bridging fiber components in regions 1 and 3 if one could find a suitable expression for Diw,eff to replace Di, in these regions. A rigorous treatment of the additional attenuation of the velocity disturbance from the sphere due to the presence of the fibers would require a solution that satisfies zero-slip conditions on the surface of the sphere and all the surrounding fibers. A reasonable approximation to this shielding behavior for the high aspect ratio fibers of current interest is suggested by the excellent approximation provided by the Brinkman equation (6) shown in Fig. 5, when B/a > 5. An exact solution of (6) that satisfies the no-slip conditions of the channel walls and the solute (sphere) surface is beyond the scope of the present investigation, but a very good approximation to this much more difficult problem can be obtained by combining the solution in Ganatos et al. (1980) for a sphere in a channel and the known solution for a sphere in an unbounded Brinkman medium.

The solution to Eq. (6) for an unbounded sphere is given in the original theory of Brinkman (1947). This solution leads to the expression for the drag on the sphere,

F,=6rpVr, l+L+&, (

1 r*

l&7 p ) (21)

where KP for our periodic fiber array is given by equation (7), r, is the sphere radius, and the term in parentheses is the Brinkman correction to the Stokes drag on a sphere. Result (21) neglects the walls of the channel, but Fig. 4 shows that for A < B (see Eq. (12)) the walls of the channel provide only a modest correction to the unbounded solution of Sangani and Acrivos (1982), whereas for A % B one should approach the exact limiting solution of Ganatos et al. (1981) in the same way that the Brinkman approximation (10) approached the solution for


Poiseuille flow in a channel when the sphere was not present in the dilute fiber limit. Therefore, using the Stokes-Einstein relation, we propose the simple approximate relation for the solute diffusion coefficient &, in the available space between fibers,

1r2 -’ 1 +L+3+

I& p I i = 1,3, (22)

where Di, is the average diffusion coefficient integrated across the channel height in the absence of fibers obtained from the solutions in Ganatos et al. (1981). Equation (22) approaches the correct limiting behavior for both dilute fiber arrays, where the walls of the channel provide the primary interaction, and more dense fiber arrays (A < B), where wall effects are not important.

The above expression for Dim still does not include the steric exclusion of the solute by the fibers and the channel walls and the spatial variation of the solute concentration in the available space of the matrix. The latter effects can be treated by solving the solute diffusion equation for the periodic fiber array in Fig. 3 in which a zero-flux boundary condition is applied at the exclusion radius, r, = a +

r, of the fibers. This last boundary value problem is analogous to the boundary value problem for the potential pressure field for the Hele-Shaw flow past a periodic fiber array. The solution to the latter problem in Tsay and Weinbaum (1991) can be readily applied to derive the expression for Diw,eff,

(23)

where b1 is the coefficient of the leading term of the doubly periodic Wierstrasse expansion series that is used to describe the disturbance produced by each fiber in the array. Although (23) contains only the coefficient of the leading term, it is an exact expression since the value of b1 is determined by solving an infinite matrix equation that contains the coefficients b, of all the higher order terms. Once can show that this solution for bl depends only on the effective fiber solid fraction S, = S(l + r-,/a)‘. The coefficient of Dim is, therefore, only a function of S, and is plotted in Fig. 5. For a pure fiber matrix model in which there is no junction strand, P is related to Diw,eff by

P = 2BLj, Di, eff, L ’ (24)

whereas if a junction strand is present, Di, in Eq. (17) is replaced by the expression for Diw,eff in Eq. (23). A comparison of the hydrodynamic theory and the stochastic theory is given in the appendix.

RESULTS AND ANALYSIS

The results for L, and P are presented in three sections: (1) the cleft with only a junction strand with pores and no fiber matrix components in its wide parts, (2) a cleft with fiber matrix components distributed uniformly through the entire depth and no junction strand, and (3) a combined model with both junction strand and fiber matrix components. In addition, three types of junction pores are ex-

98 WEINBAUM, TSAY, AND CUBBY

2b=22 2d=4 = pb_‘585 - 2d=44

b4OOkl

.

2B=22nm L2=l lnm

10 100

2D 1000 lE4

h-4 FIG. 6. Solutions for the hydraulic conductivity for clefts with pores in the junctional strand whose

cross-sectional shape is one of three basic types are shown. Results are shown for cylindrical pores of 5.5nm radius (-..-), rectangular pores 8 nm high and 44 nm long (-), and pores 22 nm high and either 44 nm long (...) or 88 nm long (--).

amined: (i) a small circular pore representing individual missing proteins in an otherwise impermeable junction strand, (ii) a longer rectangular pore with four missing proteins and an 8-nm slit height (for these rectangular openings the junction pores would serve as the primary molecular filter since this slit height provides an optimum fit for the reflection coefficient data for all the intermediate size solutes between 1.5 and 3.5nm radius, see Fig. 4 in Curry (1986)), and (iii) a wider rectangular pore with four missing proteins and a 22-nm slit height. In the latter case the junction strand would contribute to L, and P but a fiber matrix would need to be present, at least in some portion of the cleft or at its entrance, since this large pore could not possibly function as the primary molecular sieve.

A exp. results

. 2b=22 2d=44 2D=1220. -- 2b=22 2d=88 2D=1650’

FIG. 7. Solutions for the diffusive permeability for clefts with pores in the junctional strand having geometry as in Fig. 6. The spacing between pores (20) is determined from the hydraulic conductivity data for frog mesentery 5.9 x lo-’ cm/set/cm-HzO. The solid triangles are directly measured values for the solute permeability.


1. Cleft with Junction Strand Only

In Figs. 6 and 7 we have plotted, respectively, the solutions given by Eqs. (1) to (5) for the hydraulic conductivity L, and Eqs. (17) to (24) for the diffusive permeability P for a cleft with a junction strand with pores whose cross-sectional shape is one of the three basic types summarized above. The values for the cleft depth, L = 400 nm, channel height, 22 nm, and the total cleft length per unit area, Lj, = 2000 cm/cm2, are based on the measurements for single perfused vessels in frog mesentery by Clough and Michel, (1988). These data were selected because morphometric electron microscopic studies were performed on the same vessels for which the measurements of hydraulic permeability and osmotic reflection coefficient for serum albumin were obtained. The measured average value of L, was 5.9 x 10m7 cm/set/cm H20.

The lowest curve in Fig. 6 is the solution for a circular pore of 5.5 nm radius. This value for rP is estimated from the measured osmotic reflection coefficient for albumin u = 0.83 using centerline pore theory (Curry, 1984). The second curve in Fig. 6 is the solution for a rectangular slit of 8 nm height and 44 nm width. One notes that to achieve a value of L, of 5.9 x 10m7 cm/set/cm HzO, the spacing between pores, 20, will be 90 nm for the 5.5-nm radius pore and 370 nm for the rectangular slit. The fractional length of open junction for the circular pore and the rectangular slits is roughly 12%. Results for the 22-nm-high rectangular pore are shown by the upper two curves in Fig. 6. For a pore 22 x 44 nm a spacing of 1.22 pm would account for the measured L,. An important aspect of this model is that L,‘s of the order of 20 x 1O-7 cm/set/cm H20, measured after albumin is removed from the perfusate, can be accounted for if the pore spacing is reduced and if the resistance of the fiber matrix is also lowered by the removal of albumin. For example, with a 44-nm-wide pore a value of L, close to 20 x 10e7 cm/set/cm H20 can be achieved with a pore spacing of 480 nm. In contrast, with the 5.5~nm-radius pore these large values of L, are not possible for any pore spacing. The horizontal line in the left-hand margin of Fig. 6 is the value of L, for a cleft without a junctional strand.

In Fig. 7 the corresponding solutions for the solute permeability P have been plotted as a function of solute radius for the four different pores shown in Fig. 6. In obtaining the predicted curves, the averge periodic spacing 20 between pores is first determined from Fig. 6 by requiring L, to be equal to the average measured value for frog mesentery 5.9 x 10e7 cm/set/cm H20 cited above. The solid triangular symbols in Fig. 7 are directly measured values for the solute permeability that have been obtained for a spectrum of solutes up to the size of albumin, 3.5 nm radius, in studies using single perfused frog mesentery capillaries. The continuous theoretical curves for the solute permeability P use the Stokes- Einstein relation to determine the solute diffusion coefficient in an unbounded free solution.

In Fig. 7 the predicted values of P for 5.5-nm-radius pores closely fit the measured data for very small solutes and also provide a reasonable approximation for albumin, but deviate significantly from the measured values (by a factor of 2- to lo-fold) for all intermediate size solutes. The combined results for the 5.5- nm pore thus closely satisfy the measured values for u and L, and small solute permeability. The deviation of the predicted curves in Fig. 7 from the measured values of P for a 5.5-nm pore is an indication of the inability of the junction-


circular pore model to describe the full range of permeability data on the permeability and selectivity of the capillary wall in frog mesentery. The reason is that although the pores in the junctional strand are the major resistance to water, the pores being quite short (11 nm), offer relatively little resistance to the diffusion of solutes until molecules the size of albumin are approached. This is not the pattern displayed by the experimental data in which permeability decreases quite significantly with successive increases in molecular size over the entire range of solute sizes.

Figure 7 also shows the same computations for S-nm-high rectangular junction pores that are 44 nm wide. These values correspond to periodic junction discontinuities, which would be created by four missing proteins assuming each protein is 11 nm wide. The 44-nm-wide break is typical of a single standard section thickness. The predicted curves for P in Fig. 7, which satisfy a value of L, of 5.9 x 10e7 cm/set/cm HzO, show that the 44-nm-wide rectangular slit model with an 8-nm gap height that satisfies the requirements for the osmotic sieve, fits the measured data for small solutes up to the size of sucrose (0.48 nm radius), but substantially overestimates the measured data for P for all the larger solutes, including albumin, which could be better matched by the 5.5nm circular pore. The values of the small-solute permeability coefficients calculated in Fig. 7 for the 22-nm-high rectangular pores, which account for the measured L, are about one half the experimental values. This reflects the fact that, per unit of open junctional strand, the hydraulic conductivity of these junctional pores is larger than for the more restrictive pores. For the larger solutes the predicted values of P are too large because the gap height of the 22-nm-high rectangular pore is too large to offer significant resistance to solutes between 0.8 and 3.5 nm radius. It is unlikely that this junction-pore model could exist by itself and fiber matrix components would have to play an important role to account for the osmotic sieving properties of the cleft. The importance of this model is that it is flexible enough to enable an additional selective barrier to be inserted. In the next section the properties of a cleft filled with fiber matrix alone is considered first.

2. Cleft with Fiber Matrix Only

All the results in the preceding section are for a cleft in which there is no fiber matrix present in its wide portions. In Fig. 8a we have plotted the various solutions for L, for a cross-bridging proteoglycan matrix in a uniform cleft without junctional strands with cross-bridging fibers of aspect ratio B/a = 18.3. The cleft height and depth again correspond to the measurements of frog mesentery interendothelial clefts in (Clough and Michel, 1988). Starting from the left the curves in this figure represent the two-term asymptotic approximation in TWP, the square periodic array solution given by either the infinite series solution in Tsay and Weinbaum (1991) or the Brinkman approximation for L, obtained from equations (9)-(ll), the Brinkman approximation for the statistically random perpendicular array based on the value of Z$ described by Eqs. (15) and (16), and the original random array solution for an infinite medium used by Curry and Michel, (1980) where the Carman-Kozeny parameter is C = 5.

One notes in Fig. 8a that there is a relatively small difference between the solutions for the square and random arrays at the fiber densities of interest, S < 0.02 or A > 6.3 nm, and that the two-term solution used in TWP, as suggested by the results in Fig. 4, deteriorates rapidly for S > 0.005. One also observes


a

x

4

I

1 .bE-3

I

1 .OE-2 0.1

Solidity Ratio S

b 100.0

In 0

; 10.0

IIT ma,

3-r: 1.0 ZE 6.5 CL P 0.1

3 without junction strand

isi 1 .OE-2

0.0 1.0 2.0 3.0 4.0 5.0

SOLUTE RADIUS (nm)

FIG. 8. (a) Solutions for hydraulic conductivity for ciefts with fiber matrix present only. The figure shows various solutions for the hydraulic conductivity I!,, for a cleft with cross-bridging fibers of B/a

= 18.3. From the left, the curves represent two-term asymptotic approximation (TWP, 1989) (.a*), square array solution (Tsay and Weinbaum, 1991) (-), the Brinkman approximation for two-dimensional random array (-..-), and the Carman-Kozney solution with C = 5 (Curry and Michel, 1980) (- - -). (b) Solutions for the solute permeability P as a function of the solute radius r,. The solid curve is for a = 0.6 nm, dashed curve for a = 2 nm, and dotted curve for a = 5. The top three curves represent solutions for clefts with random cross-bridging fibers and the bottom ones are for periodic fiber arrays.

that the measured value of L, of 5.9 x lo-’ cm/set/cm Hz0 can be achieved by a fiber matrix in a cleft without junctional strands with a solid volume fraction that lies between 0.017 and 0.025 for the square periodic and random arrays, respectively. A fiber matrix with these values of S, which fills the entire cleft, reduces the hydraulic conductivity of the cleft by more than sevenfold if the resistance of the junction strand is neglected. When arranged in square array, a matrix of 0.6-nm-radius fibers has an open spacing between fibers A of 7 nm when S = 0.017. This is close to the spacing required if the fiber matrix is to serve as the primary molecular sieve. It follows that if there is any additional resistance

102 WEINBAUM, TSAY, AND CUBBY

to water movement through the cleft from the junctional strand, there must be reduced resistance to water flow from the matrix to satisfy the measured value of L,. Therefore, if one requires that the matrix continue to be the primary filter, the matrix must occupy less than the total cleft depth.

The solutions for the solute permeability P as a function of the solute radius r, for a cleft with either periodic or random cross-bridging fibers are shown in Fig. 8b. These calculations are based on Eqs. (22) and (23) for the ordered square array. For the random array the statistical theory in Ogston et al. (1973) has been used to calculate the steric exclusion of the fibers, while the hydrodynamic predictions to determine the relation between L, and S are based on Eqs. (11) and (16). Curves for three fiber radii are presented and for each curve the fiber solid fraction S has been selected to satisfy the measured value of L,, 5.9 x lop7 cm/set/cm H20, for frog mesentery. The results show that the fiber matrix offers little resistance to small solutes as indicated by the fact that the predicted permeability coefficient of the fiber-filled cleft for solutes <.48 nm radius is close to that calculated for an open cleft. The present higher estimates for small solute permeability compared with previous fiber matrix calculations for frog mesenteric capillaries (Curry and Michel, 1980; Curry, 1986; Michel, 1988) reflect the larger value for cleft width and reduced cleft depth used in the present model. To account for small-solute permeability it is, therefore, necessary to assume that junctional strands act to reduce the effective length of open junction. Other combinations of fiber radius and fractional fiber density describe a relation between permeability and solute size for molecules up to 2 nm in radius that is similar to that for the matrix with 0.6-nm fiber radius, but these combinations fail to describe the large solute permeabilities due to the wider spacing between fibers. The calculations in Figs. 6-8 indicate that neither the junction pore or fiber matrix model by itself can account for all the data.

3. Combined Model for Cleft with Junction Strands and Fiber Matrix

In Fig. 9a the solutions for the junction pore interaction given by Eqs. (l)-(5) are combined with the solution for pen for the fibers in the wide part of the channel. Three sets of curves are shown, a junction strand with a circular cylindrical pore of 5.5 nm radius and two rectangular slit pores of 44 nm width, one with a gap height of 8 nm and the other an unconstricted height of 22 nm. In each case there are fibers of 0.6 nm radius and a spacing A = 7 nm (S = 0.017) filling the wide portion of the cleft. As expected from our results presented above, there is a limited range of parameters that are consistent with the experimental data when the matrix fills the whole cleft. The predictions in Fig. 9a indicate that the measured value of L, = 5.9 x 10m7 cm/set/cm Hz0 cannot be achieved by a junction strand with 5.5-nm-radius circular pores even if every other protein is missing (20 = 22 nm). If we consider the latter junction strand as the most permeable possible for the circular pore geometry, the model predicts that the maximum value of S for which Clough and Michel’s measured value of L, can be achieved is S = 0.01 (A = 10 nm) if the matrix fills the entire wide part of the cleft. For the rectangular slit geometry with 8 nm gap height and 44 nm width, with A = 7 nm, L, approaches a limiting value close to 5.5 x 10M7 cm/set/cm HZ0 when there is only a single protein separating junctional pores (20 = 55 nm). For the 22 x 44-nm rectangular pore this limiting value of L, is 6 x 1O-7 cm/set/cm H20. Based on these predictions, the fiber matrix could not serve as


a

I

‘.. 2b=22 2d=44 nm -- 2b=6 2d=44 nm

a=0.6nm. S=.O17

5 10 100 1000 lE4

2D (nm)

A experimental results

i?i A

l.OE-2 ....‘...“,““..‘.‘.‘., 0.0 1.0 2.0 3.0 4.0 5.0

SOLUTE RADIUS (nm) FIG. 9. Solutions for the hydraulic conductivity and the diffusive permeability for clefts with

junction strands and fiber matrix are presented in (a) and (b). The set of curves shows junction- pore-matrix model with fiber density, S = 0.017. The three curves shown in (a) are for junction strands with (i) circular pores for 5.5nm radius; (ii) rectangular slit pores of 8 x 44 nm, and (iii) rectangular pores of 22 x 44 nm. In (b), results are presented only for the cases in (a) in which the measured value of L, can be achieved and in which the fiber spacing A is of the dimensions for an adequate molecular sieve for albumin.

the primary molecular sieve if the matrix is to occupy all or nearly all of the wide part of the cleft with either the 5.5nm radius or S-nm gap height rectangular pores. However, matrix components with primary sieving properties filling the entire cleft are just on the borderline of possibility for the larger 22-nm gap height pore.

An important property of the combined junction-pore-matrix model is that for all pore spacings 20 > 200 nm, the presence of a fiber matrix reduces the L, relative to that of a cleft without matrix by approximately the same extent for all spacings between the junctional pores for each matrix geometry. This reflects the fact that all the L, curves in Figs. 6 and 9a are parallel in this region. For


example, the addition of a matrix formed by 0.6-nm fibers (S = 0.017) decreases the L, of a cleft with junctional pores (2d = 44 nm, 2b = 22 nm) spaced 200 nm apart from 24 x 10d7 cm/set/cm Hz0 without matrix (see Fig. 6) to 4.0 x 10e7 cm/set/cm HZ0 (a factor of 6) and decreases the L, of a cleft with the same pores spaced a distance of 600 nm apart from 10 x 10e7 cm/set/cm HZ0 (see Fig. 6) to 1.7 x 10m7 cm/set/cm HZ0 (also a factor of 6).

Solutions for P are shown in Fig. 9b only for the single case in which the measured value of L, can be achieved. This corresponds to the 22-nm gap height rectangular pore, where the fiber spacing is just adequate for the matrix to be the primary molecular sieve when the pore spacing 20 = 55 nm.

DISCUSSION

The calculations described in this paper provide new detailed theoretical predictions to evaluate some of the principal questions in current studies of the structure-function relationship for water and solute flow through the interendothelial clefts. While none of the present models provide an entirely satisfactory description of the available experimental data, two fundamental new observations can be drawn from the calculations. One is that widely separated small breaks in the junctional strand represented by the 22 x 44-nm rectangular pore are consistent with the measured value of L, in Clough and Michel, 5.9 x low7 cm/set/cm HzO. The solutions in Fig. 6 for this rectangular pore show that the spacing can be as large as 20 = 1220 nm if no matrix is present. However, to account for the measured small-solute permeabilities and the selectivity of the microvessel wall to large molecules, these gaps are too far apart and a fiber matrix is required in some portion of the cleft to further reduce the hydraulic conductivity and provide the molecular sieve. There are a variety of combinations of pore spacing, matrix density, and depth of matrix penetration into the cleft that need to be investigated in future extensions of the present model. The second observation is that the small sieving pores in the junctional strand (the 5.5-nm-radius circular pores spaced 90 nm apart or the 8 x 44-nm rectangular pores spaced 370 nm apart), with no additional resistance due to a matrix, can account for the measured L,, the small solute permeability, and also the measured selectivity properties. These small-pore models, however, do not properly describe the data for intermediate size solute permeability.

The above calculations also indicate two experimental approaches for future studies. One is that it should be possible to determine the site of the principal molecular filter in the cleft based on the frequency of discontinuities in the junctional cleft. Specifically, if small (5.5-nm radius) junctional pores are the principal molecular filter, then they must be present at a very high frequency in the junctional strand (approximately 1 protein in 8 missing 20 = 90 nm). On the other hand, if the fiber matrix is the principal molecular filter, larger discontinuities spaced approximately several hundred to lo3 nm apart are required to account for the small solute permeability. Preliminary experimental data that address these considerations are described at the end of the discussion.

Before we elaborate on the experiments mentioned above, we first discuss the approximations and limitations of the model and the analysis. The approximations fall under two headings: approximations related to the hydrodynamic modeling


and approximations related to the assumed geometry of the cleft and its junctional structures. These are discussed separately.

Approximations in the Hydrodynamic and Diffusion Models

The principal motivation for the development of the three-dimensional junctional model is to account for the lateral spreading of water and solute proximal and distal to the junction strand pore in the wide portions of the cleft. This spreading has been ignored in all previous one-dimensional models (Fig. 1). The main approximation in the junction-pore model is that the junction strand array can be replaced by an idealized single equivalent strand with periodically spaced pores whose average spacing is the same as in the junction strand array. As expected the largest deviations between the 1-D and 3-D models for the wider rectangular pores arise whenever the wide portions of the cleft are the major contribution to the total resistance. This appears to be nearly always the case for solute diffusion. The agreement between the 1-D and 3-D models is particularly poor when the spacing between pores is large. For example, for the 22 nm x 44 nm rectangular pore that satisfies the measured L, in Fig. 6 (20 = 1220 nm), the junctional region would account for only 11% of the resistance to a small solute, whereas the junctional region would account for 44% of the diffusive resistance in the 1-D model. If one were to plot solute flux lines (these lines are everywhere perpendicular to isoconcentration lines), one would observe that when the pores are far apart these lines would be confined to small wakelike regions surrounding each pore exit and thus not fill the wide part of the cleft. When this occurs 1-D theory grossly underpredicts the junctional resistance of the cleft since it neglects the finite distance for the lateral spreading of the solute to fill the wide portion of the cleft on the downstream side of the junction strand barrier.

The behavior, just described, is also observed for filtration, where the fluid streamlines leaving the pore exit replace the solute flux lines. The principal difference is that in the case of filtration the junctional strand offers a more significant fraction of the total cleft resistance when the pore dimensions are sufficiently small for the junction strand to function as the primary size selective structure (5.5nm pores and S-nm-high rectangular pores). In this case the model for the pores is more important than the lateral spreading of the water downstream of the junction strand. In contrast, for the larger 22 x 44-nm rectangular pore one finds that the junction strand provides 15% of the total hydraulic resistance when 20 = 1226 nm. This is less than one-third the value predicted by the 1-D model.

The principal approximation introduced in the treatment of the ordered fiber array is that the fibers can be treated as rigid, periodic structures arranged at right angles to the direction of flow. The solution for L, for this geometry is close to exact. The important insight gained from the hydrodynamic analysis of water flow through this mathematically idealized geometry is that the increased viscous drag is due primarily to the additional viscous dissipation that occurs when the viscous layers surrounding each fiber overlap. This behavior is confirmed by the detailed theoretical analysis of the three-dimensional velocity profiles for water flow around each fiber in Tsay and Weinbaum (1991), which show that the thickness of the viscous layers is of order B. Thus, if B/A > 1 the region of viscous overlay will fill the entire region between adjacent fibers. This increase in resistance can occur even if the fiber density is low (B/a + 1). As seen in Fig. ga, for values


of fiber density ~0.1, the new model predicts a much larger resistance than the Carman-Kozeny equation. For example, the fiber volume density S = 0.017 predicted by the present model to achieve an L, of 5.9 x 1O-7 cm/set/cm Hz0 is significantly less than the value of S = 0.05 obtained in the original calculations of Curry and Michel (1980). The latter value was based on a Carman-Kozeny equation in which the density correction factor C in Eq. (13) was assigned a constant value of 5, representative of the dense fiber limit. As observed in Fig. 8a, the new model approaches the predicted curve for C = 5 in Curry and Michel (1980) for S > 0.1, but the result deviates significantly from the predictions of the Carman-Kozeny relation at lower value of S. As shown in Levick (1987) the Kozeny factor C needs to be increased from 5 to more than 20 when S is decreased from 0.1 to 0.01 if it is to fit the experimental data.

The principal approximation used in the development of the new model for diffusion in the ordered array is that the decrease in effective diffusion coefficient in the matrix can be accounted for by the increased drag on the solute, (Eqs. (21) and (22)) and the increased exclusion of the solute imposed by the hydrodynamic flow between the fibers, (Eq. (23)). Equation (23) includes not only the excluded volume effect due to the presence of the fibers but also takes into account the nonuniform concentration distribution of the solute that arises from the interaction between the fibers. The approach is quite different from that used in the stochastic description of the reduction in diffusion within a matrix introduced by Ogston et al. (1973). The latter approach describes the reduction in the diffusion coefficient in terms of the probability of collision between solute and fibers and the classical excluded volume effect of the fibers. The predictions of the Ogston approach are compared with the new hydrodynamic formulation in Appendix 1 for the case S = 0.017 or A = 7 nm. It is shown for small and large solutes that the reductions in diffusion coefficient using the new hydrodynamic formulation are quite close to those of the Ogston model for the case in which the fibers are in an ordered array, even though the relative importance of the exclusion and drag/collision terms differ significantly for the two approaches.

While the present theory for solute diffusion includes, for the first time, the far-field hydrodynamic interaction of the solute molecule with the surrounding fibers, it does not accurately describe the near-field interaction with the closest neighboring fibers. This near field interaction has not previously been studied, since the undisturbed velocity field in a bounded matrix in the absence of the solute molecule was not known. The new solutions in Tsay and Weinbaum (1991) for the velocity field in the fiber array, when combined with the Faxen reciprocal theorems for Stokes flow, should enable us to examine problems of this type in the future.

Approximations Related to Assumed Cleft and Pore Geometry

The data for cleft depth, cleft height, and effective cleft length per unit area in Clough and Michel (1988) are mean values based on measurements in single perfused microvessels in which vessel hydraulic conductivity was also measured. While this is arguably the best data available to correlate structure and function, the shortcomings of the data should be noted. The length of junction per unit area 2000 cm/cm’ is similar to that estimated by Bundgaard and Frokjaer-Jensen (1984) in frog mesentery and Bundgaard (1984) in the rat heart. The cleft height (22 nm) is larger than that reported by Mason et al. (1979) and Bundgaard and


Frokjaer-Jensen (1984) (17 nm) and the cleft depth (0.395 pm) is shorter than the value of 0.7 pm described in these earlier publications. There is no simple explanation for these discrepancies. In addition, there was considerable variation in the measured value for cleft depth both within a given vessel, and between vessels. For example, the value of 0.395 pm for the cleft depth is calculated from a population that varied from 0.104 pm to 1.7 pm.

Measured values of L, in Clough and Michel (1988) also varied over a sevenfold range from 1.84 to 12.5 x lo-’ cm/set/cm H20. Clough and Michel found a reasonably good correlation between L, and the inversion of the harmonic mean cleft depth (Z/L). This correlation would not be expected if the pores in the junction strand are the primary resistance to water flow. The Z/L correlation suggests that the resistance resides in the wide part of the cleft for frog mesentery. Thus, although it was noted above that the calculations for L, based on mean values for cleft geometry might be compatible with a junctional pore model with 5.5-nm pores in the strand, this model is not compatible with observed relation between L, and Z/L because the predictions of our 3-D model for the junction pores with rP = 5.5 nm and 20 = 90 nm show that only 18% of the total resistance to flow resides in the wide part of the cleft. Thus the variation in the depth of the wide portion of the cleft can not be explained by this small circular pore model for the junction pores. In contrast, our predictions for the 22 x 44- nm rectangular pore with 20 = 1220 nm that satisfy the measured value of L, require that only 15% of the total hydraulic resistance resides in the junction strand. This model is, therefore, more consistent with the Z/L correlation.

Another area of uncertainty concerns the geometry of breaks in the junctional strand. There is no definitive data on the geometry of the junctional strands. The size of individual missing proteins is at best an educated guess, constrained as much by the measured selectivity of the vessel wall as any precise knowledge of the channel geometry. There is also no definitive data on the effective depth of the strand. In one-dimensional models (see Fig. 1) the depth of the channel in the region of the junction strand has been assumed to be as much as 10% of the total cleft depth, while in the present model this depth has been related to the diameter of the missing proteins, 11 nm or <3% of cleft depth. This assumption reduces the relative contribution of the junctional strand to the total resistance.

An important assumption in the fiber matrix model is that the fibers are distributed throughout the cleft depth. There is no clear evidence that fibers are present in the cleft. The fibers described here are conjectured to be either glycoprotein side chains of typically 0.6-nm radius (Curry and Michel, 1980; Curry, 1986) or core proteins of 2- to 2.5-nm radius, whose length is equal to the gap height of the wide part of the cleft. For bridging molecules within the cleft, the most likely fiber orientation is perpendicular to the wall. The evidence to support this suggestion is from studies of proteoglycan molecules similar to those described above, which have been found to perpendicularly span the space between neighboring collagen fibrils in subendothelial connective tissue matrix using rapid freeze- etching techniques (Frank and Fogelman, 1989). It is noted that conventional EM techniques do not show bridging fibers since these structures do not withstand chemical fixation. Rapid freeze-etching EM techniques are thus likely to be es- sential tools for investigating the structure of fiber matrices in the junction and on the cell surface in the future. It is noted that there is evidence that fibers associated with the endothelial cell surface glycocalyx account for 50% of the total


hydraulic resistance in frog mesentery , Adamson (1990). This surface glycocalyx is not included in the present model. At the end of the discussion we shall estimate the resistance of such a layer.

We also note that the model only indirectly accounts for the presence of ad- sorbed proteins that are assumed to be part of the matrix. Michel (1984) has hypothesized that one of the important actions of albumin may be to order the matrix. This ordering is suggested to arise from the electrostatic interaction between positively charged arginine groups on albumin and the negative charges on the proteoglycans. Thus the ordered array described in the model may be representative of the matrix structure in the presence of albumin. The additional resistance due to the albumin molecules, themselves, has been neglected in the model. The diffusion model also ignores the effect of charge on the channel walls and/or fibers. Charge effects reduce the permeability of negatively charged intermediate-size solutes to values that are l/3 to l/5 of the uncharged solutes of the same size (Adamson et al., 1987; Curry et al., 1989). These effects are roughly of the same order of magnitude as the deviation between theory and experiment for the 1-Znm-radius solutes shown in Fig. 7 for the 22 x 44-nm pore.

Structure-Function Correlation: Future Studies

In spite of the limitations and the uncertainties in the model there are important predictions to be investigated in new experiments. These are discussed below. If small breaks in the junction strand typical of a single EM thin section separated by distances of a few hundred to 1000 nm modulate small-solute permeability and the effective area for flow, it should be possible to determine their size and frequency from the lateral spreading of electron-dense tracers in the wide part of the cleft distal to the junctional strand. The spreading of the tracer can be described as a wake. Clough and Michel (1988) showed that, for the microvessels used to obtain the ultrastructure data used throughout this paper, the electron-dense tracer lanthanum was present in 50% of the random thin sections examined in EM. They interpreted these results as indicating that 50% of the junction strand was open to diffusion for small solutes and water flow. If, however, there is lateral spreading due to convection and diffusion on the distal side of the junction strand, the wake of a single break will occupy a number of section thicknesses, and far fewer breaks are required to explain the data. The number of breaks and their effective size could be determined from the three-dimensional information that would be obtained from a more detailed study of the wake structure. The present arguments provide a strong rationale for an extended study combining the approach of Clough and Michel and the theoretical framework outlined in this paper. Preliminary studies of this nature have been started (Adamson and Michel, 1991). The shape of the solute wake of the junction strand will also be influenced by the filtration flow if convection and diffusion are of comparable importance.

Adamson (1990) has recently shown that the treatment of the endothelial cell surface with proteolytic enzymes to remove all or a part of the endothelial cell glycocalyx reduces the hydraulic conductivity of the microvessel wall by roughly one half for frog mesentery. Gross ultrastructural observations suggests that the junction strand and cleft geometry are not altered. However, there is no quantitative data to show that the frequency of the discontinuities in the junction strand has not been changed. Assuming that a significant portion of the resistance to water flow resides in a fiber matrix extending into the microvessel lumen proximal


to the endothelial cleft, a rough estimate of the thickness of this matrix layer can be deduced from the results in Figs. 6 and 9a for the 22 x 44 nm pore. As seen in the Results section, L, for this rectangular pore is increased by approximately a factor of six (in the region where the L, curves are parallel) when a selective matrix (S = 0.017, fiber radius = 0.6 nm, A = 7 nm) is removed from the entire depth of the cleft, suggesting that the matrix components have roughly six times the resistance of the cleft with its junction strand. Therefore, a sieving matrix with A equal to 7 nm occupying about one-sixth the cleft depth, or roughly 60- 70 nm, would offer a resistance that is comparable to a cleft with a junctional strand only. From Fig. 9a the L, of a cleft with a junctional strand composed of pores 22 x 44 nm spaced 600 nm apart is 10 x lo-’ cm/set/cm H20. Since the resistance of this cleft would be doubled if there is a fiber matrix, 60-70 nm thick, near the cleft entrance, L, would be reduced to 5 x lo-’ cm/set/cm HZ0 if this thin fiber layer were present. A combined model that includes 22 x 44-nm rectangular pores and a fiber matrix that occupies only a small fraction of the cleft depth is thus compatible with Adamson’s measurements for L,. Note that the above result would not be predicted using the Carman-Kozeny equation because the predicted resistance of the thin matrix layer would be too small to reduce L, sufficiently.

The foregoing hypothesis could be more critically explored if similar experiments were performed to also assess the changes in solute permeability and selectivity that would result from the removal of the surface glycocalyx for different size solutes. To interpret these results the present theory would have to be modified by including a fourth region at the entrance to the cleft in Fig. 2 in which a thin region of ordered matrix elements would be present. This layer would be treated using the same Brinkman approach that has been described in the present study. In this manner it should be possible to evaluate the relative contributions of the junctional strand and a thin matrix layer to the diffusive resistance and molecular selectivity.

APPENDIX

The diffusive permeability coefficient of the cleft, neglecting channel wall effects, is determined by both the solute diffusion coefficient in the cleft (D/Do) and the partition coefficient for the solute in the matrix @. The reduction in permeability coefficient due to the presence of the matrix, P/PO, is proportional to the product (D/D0 . @ (Curry, 1984). The Ogston model for a random array predicts

P/PO = [exp - fi (1 + r,/a)] * [exp - S &,/a + rfla”)]. (Al) The first term on the right describes the reduction in diffusion coefficient due

to increased collisions of the solute with the fibers. The second term on the right describes exclusion of the solute from the matrix (Curry, 1984).

The relation for a random array overestimates P/P,, when the matrix is ordered as in the hydrodynamic model. For an ordered array the exponential terms on the right side of Eq. (Al) are replaced by linear relations. For an ordered array Curry (1985) used the term fi . (1 - 2r,/$a) to describe the reduction in diffusion coefficient. Michel(l984) used the term (1 - S,) to describe the partition coefficient.


In the hydrodynamic model, the corresponding equation for P/P,, with channel wall effects neglected, is given by the product of text Eqs. (21) and (23):

p/p, = [Cl + r,/l&) - ~~/3QJ)l-’ [(I - b%l(1 + h&)1. WI

Values calculated for the Ogston model and the hydrodynamic models using Eqs. (Al) and (A2), respectively, are compared below.

TABLE Al

r, (nm)

Ogston S = 0.017 Hydrodynamic S = 0.017

Diffusion Exclusion P/PO Diffusion Exclusion P/PO

0.2 0.951 0.970 0.922 0.928 0.94 0.870 0.5 0.878 0.943 0.828 0.832 0.89 0.740 1.0 0.755 0.880 0.664 0.701 0.78 0.547 2.0 0.510 0.682 0.341 0.513 0.52 0.207 3.0 0.264 0.390 0.102 0.070 0.22 0.085 3.5 0.129 0.190 0.031 - -

Note. In Equations Al and A2, call term one on the right “Diffusion” and term two on the right “Exclusion.”

ACKNOWLEDGMENTS

The first two authors acknowledge NIH Grant HL 19454 and NSF Grant CBT-8803116 for the support of those aspects of this research related to endothelial filtration in arteries. Research related to microvessels is supported by NIH Grant HL44485.

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A Three-dimensional Junction-pore-matrix Model for Capillary Permeability - S

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