Ultrastructural model for size selectivity in glomerular filtration

Aurélie Edwards1, Barbara S. Daniels2, and William M. Deen3

1 Department of Chemical Engineering, Tufts University, Medford, Massachusetts 02155; 2 Department of Medicine, University of Minnesota, Minneapolis, Minnesota 55455; and 3 Department of Chemical Engineering and Division of Bioengineering and Environmental Health, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139


    ABSTRACT
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ABSTRACT
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MATHEMATICAL MODEL
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REFERENCES

A theoretical model was developed to relate the size selectivity of the glomerular barrier to the structural characteristics of the individual layers of the capillary wall. Thicknesses and other linear dimensions were evaluated, where possible, from previous electron microscopic studies. The glomerular basement membrane (GBM) was represented as a homogeneous material characterized by a Darcy permeability and by size-dependent hindrance coefficients for diffusion and convection, respectively; those coefficients were estimated from recent data obtained with isolated rat GBM. The filtration slit diaphragm was modeled as a single row of cylindrical fibers of equal radius but nonuniform spacing. The resistances of the remainder of the slit channel, and of the endothelial fenestrae, to macromolecule movement were calculated to be negligible. The slit diaphragm was found to be the most restrictive part of the barrier. Because of that, macromolecule concentrations in the GBM increased, rather than decreased, in the direction of flow. Thus the overall sieving coefficient (ratio of Bowman's space concentration to that in plasma) was predicted to be larger for the intact capillary wall than for a hypothetical structure with no GBM. In other words, because the slit diaphragm and GBM do not act independently, the overall sieving coefficient is not simply the product of those for GBM alone and the slit diaphragm alone. Whereas the calculated sieving coefficients were sensitive to the structural features of the slit diaphragm and to the GBM hindrance coefficients, variations in GBM thickness or filtration slit frequency were predicted to have little effect. The ability of the ultrastructural model to represent fractional clearance data in vivo was at least equal to that of conventional pore models with the same number of adjustable parameters. The main strength of the present approach, however, is that it provides a framework for relating structural findings to the size selectivity of the glomerular barrier.

sieving coefficient; Ficoll; glomerular basement membrane


    INTRODUCTION
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ABSTRACT
INTRODUCTION
MATHEMATICAL MODEL
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MATHEMATICAL MODELS for glomerular filtration are used most often to calculate the values of membrane parameters from micropuncture and/or clearance data. By taking into account the expected effects of plasma flow rate and other hemodynamic variables on the rates of filtration of water or macromolecules, models allow one to evaluate the ultrafiltration coefficient, apparent pore radius, or other quantities that characterize the overall barrier properties of the capillary wall. A closely related use of models is in predicting the effects of hemodynamic changes on filtration rates, for a given set of barrier properties. Many examples of these types of applications are reviewed in Maddox et al. (21).

A different and particularly challenging use of theoretical models is in predicting the properties of the glomerular barrier from basic structural information. Advances in computational fluid dynamics have made it practical to calculate the resistance to flow through three-dimensional assemblies of fibers or small channels of complex shape (11, 24). Drumond and Deen (11) determined pressure-flow relations for various representations of the filtration slit diaphragm, using dimensions taken from electron microscopy studies. The model for the slit diaphragm was combined with ones for the fenestrated endothelium and glomerular basement membrane (GBM) to predict the overall hydraulic permeability of the capillary wall (12). The predictions agreed well with values derived from micropuncture data in normal rats. It was inferred that roughly half of the resistance to water flow is due to the GBM and half to the slit diaphragm, with the endothelial resistance normally being negligible. The model for hydraulic permeability has offered insight into the structural basis for changes in glomerular filtration rate (GFR) in various human glomerulopathies (14, 19).

The objective of the present study was to extend the structural-hydrodynamic approach to describe the filtration of uncharged macromolecules of varying size. A model for sieving across the slit diaphragm was already available (13), and recent studies using isolated rat glomeruli (5, 15, 16) provided key data. What was needed was to synthesize descriptions of the sieving behavior of the three-layer capillary wall and of filtration in a whole glomerulus in vivo.


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Geometric assumptions. The model was based on the idealized structural unit shown in Fig. 1, which was assumed to be repeated along the length of a capillary. This unit of width W consisted of a single filtration slit (with slit diaphragm), bounded by epithelial foot processes, and representative areas of basement membrane and fenestrated endothelium. The fenestrae have been reported to be channels of circular cross section with varying radius, much like an hourglass (18); the minimum radius of a fenestra is denoted as rf. The GBM, which is a complex network, consisting of collagen, laminin, glycosaminoglycans, and other polymers, was treated as a homogenous material of thickness L. The width of a filtration slit is denoted as w. Two geometric quantities derived from those shown in Fig. 1 are epsilon f and epsilon s, the fractions of the capillary surface occupied by fenestrae and filtration slits, respectively. The slit diaphragm was modeled as a single row of cylindrical fibers spanning the filtration slit, as described below.


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Fig. 1.   Idealized structural unit of glomerular capillary wall, corresponding to 1 filtration slit (not to scale). GBM, glomerular basement membrane.

Transport across the GBM. The endothelial fenestrae were assumed either to offer negligible resistance to the passage of macromolecules or were modeled as a functional extension of the GBM, so that the calculations for the GBM are outlined first. With the assumption that the GBM is an isotropic fibrous material with uniform properties, the flux N of a particular macromolecule is given by
<B>N</B> = −<IT>K</IT><SUB>d</SUB><IT>D</IT><SUB>∞</SUB>∇C + <IT>K</IT><SUB>c</SUB><B>v</B>C (1)
where C and v are the solute concentration and fluid velocity, respectively. Both of these quantities are based on total volume (fluid plus solid) and are assumed to be averaged over a length scale that is large compared with the interfiber spacing of the GBM but small compared with L or W. The diffusivity of the macromolecule in free solution is Dinfinity , which is related to the Stokes-Einstein radius (rs) by Dinfinity  = kBT/(6pi µrs), where kB is Boltzmann's constant, T is the absolute temperature, and µ is the viscosity of water. The coefficients Kd and Kc represent size-based hindrances to diffusion and convection, respectively. They were evaluated from data we obtained previously with isolated rat GBM (15, 16), as described below.

At steady state, conservation of mass for the macromolecular solute requires that
∇ ⋅ <B>N</B> = 0 (2)
at all points within the GBM. This differential equation was solved to determine C(x,z) (see coordinates in Fig. 1). The velocity field in the GBM, v(x,z), was calculated using Darcy's law (12), and the hindrance coefficients were evaluated as described below. In reality, the concentration and velocity fields in the GBM are three-dimensional. To simplify the problem to one involving only x and z, the circular fenestrae were replaced by functionally equivalent slits, as justified previously (12). The boundary conditions imposed on the concentration field were
<IT>N</IT><SUB><IT>z</IT></SUB> = <FR><NU><IT>J</IT><SUB>s</SUB></NU><DE>&egr;<SUB>f</SUB></DE></FR> (fenestra-GBM boundary) (3A)
C = &PHgr;C<SUB>0</SUB> (GBM-slit boundary) (3B)
<B>n</B> ⋅ <B>N</B> = 0 (cell surfaces and symmetry planes) (3C)
Equation 3A relates the component of the flux in the z direction (Nz) to the flux averaged over the entire width of the structural unit (Js). Equation 3B relates the average concentration at the upstream end of the filtration slit, C0, to that at the adjacent surface of the GBM; and Phi  is the equilibrium partition coefficient that describes the steric exclusion of macromolecules from the GBM. Equation 3C embodies the assumption that there is no flux across any cell membrane; n is a unit vector normal to a given surface. In the actual calculations Eq. 3B was replaced by the equivalent condition
C = <FR><NU>&PHgr;C<SUB>B</SUB></NU><DE>&THgr;<SUB>sd</SUB></DE></FR> = <FR><NU>&PHgr;<IT>J</IT><SUB>s</SUB></NU><DE>&THgr;<SUB>sd</SUB><IT>J</IT><SUB>v</SUB></DE></FR> (GBM-slit boundary) (3B`)
where CB is the local concentration in Bowman's space, Theta sd = CB/C0 is the local sieving coefficient for the filtration slit only (determined mainly by the slit diaphragm), and Jv is the mean volume flux (fluid velocity) in the structural unit. ("Local" quantities such as CB and Theta sd vary from one structural unit to another, due to variations in solute concentration and Jv along a capillary.) It is worth noting that Js and Jv correspond to the fluxes used in traditional models, which do not involve the structural details of the glomerular capillary wall (21).

Equation 2 was solved using Galerkin finite element methods. Using a mesh with 1,600 quadrilateral elements and bilinear basis functions, the CPU time needed to solve this problem was ~20 s on a DEC station 5000/133. Once the concentration field was determined, the local sieving coefficient for the GBM was calculated as Theta bm = C0/C1, where C1 is the average concentration at the downstream end of a water-filled fenestral opening, next to the GBM.

For calculations involving a wide range of molecular sizes at many axial locations along a capillary, the time required to run the finite element code was judged to be impractical. Accordingly, explicit formulas were sought which would adequately approximate the finite element results. Dimensional analysis shows that Theta bm is a function only of Phi Kc, epsilon f, epsilon s, L/W, Theta sd, and the Péclét number for the basement membrane, which is defined as
Pe<SUB>bm</SUB> = <FR><NU>(&PHgr;<IT>K</IT><SUB>c</SUB>)<IT>J</IT><SUB>v</SUB><IT>L</IT></NU><DE>(&PHgr;<IT>K</IT><SUB>d</SUB>)<IT>D</IT><SUB>∞</SUB></DE></FR> (4)
The relevant variables were incorporated into expressions of the form
&THgr;<SUB>bm</SUB> = <FR><NU>&PHgr;<IT>K</IT><SUB>c</SUB></NU><DE>&THgr;<SUB>sd</SUB> − (&THgr;<SUB>sd</SUB> − &PHgr;<IT>K</IT><SUB>c</SUB>) exp[−Pe<SUB>bm</SUB>(1 + <IT>f</IT> )]</DE></FR> (5)
<IT>f</IT> = <IT>a</IT>(1 − &egr;<SUB>f</SUB>  
 
&egr;<SUB>s</SUB>)<SUP><IT>b</IT></SUP>(<IT>L</IT>/<IT>W</IT>)<SUP>−<IT>c</IT></SUP> (6)
where a, b, and c in Eq. 6 are positive constants. The functional forms of Eqs. 5 and 6, which are to some extent arbitrary, were selected to ensure the correct behavior in certain limits. Namely, if L/W right-arrow infinity , or if both epsilon f and epsilon s right-arrow 1, then Theta bm approaches the exact solution of the one-dimensional problem corresponding to bare GBM (i.e., with none of the surface blocked by cells); also, Theta bm right-arrow 1 if Pebm right-arrow 0. In generating a set of finite element results that could be fitted to determine the unknown constants, the input parameters were varied over the ranges 10-5 <=  Pebm <=  10, 10-4 <=  Phi Kc <=  1, 0.01 <=  epsilon s <=  1, 0.05 <=  L/W <=  1, and 10-4 <=  Theta sd <=  1, with epsilon f = 0.20. Using Powell's method (25) to determine the best-fit values, we obtained a = 0.7366, b = 11.9864, and c = 1.2697. For the 10,600 data points used, the root-mean-square error in Theta bm was only 5%, confirming that Eqs. 5 and 6 were satisfactory approximations.

Ficoll, a copolymer of sucrose and epichlorohydrin, has been favored in recent years as a test macromolecule for clearance studies, because it is neither secreted nor reabsorbed by the tubules and because it has been shown to behave like an ideal, neutral sphere (4, 23). The hindrance coefficients Phi Kd and Phi Kc for Ficoll were evaluated from two studies of isolated rat GBM. In the first, isolated glomeruli were denuded of cells to consist mostly of bare GBM (>95%), and their diffusional permeability to four narrow fractions of Ficoll of varying sizes was assessed by confocal microscopy (16). Diffusion and convection were both present in the second study, where Ficoll was filtered across packed acellular glomeruli (i.e., multiple layers of GBM) (15). Results from both sets of experiments were used previously to determine the GBM hindrance coefficients as a function of both Ficoll size and applied pressure (15). However, in pooling the results, we did not account for the effect of BSA, present in the buffer for the filtration studies but not in that for the confocal microscopy experiments. As discussed by Bolton et al. (5), Phi  for Ficoll appears to be increased markedly by the repulsive interactions of Ficoll with BSA. For Ficoll with rs = 3.6 nm, it was estimated that Phi  was increased by a factor of 1.7 in the presence of 4 g/dl BSA, the concentration used in the filtration experiments. At present there is no theory to estimate the magnitude of this effect for other molecular sizes. Lacking more complete information, we assumed that in the diffusion studies of Edwards et al. (16), where BSA was absent, the hindrance coefficients for all sizes of Ficoll were 1.7 times less than what they would have been in the presence of BSA.

In addition, the functional forms proposed previously for Phi Kd and Phi Kc (16) were not ideal, in that they did not exhibit the proper behavior for small molecules. Namely, for a given applied pressure, the values of Phi Kd and Phi Kc for a point-size molecule should be 1 - (5/3)phi and 1, respectively, where phi  is the volume fraction of fibers in the GBM; the result for Phi Kd is based on the theory for diffusion or heat conduction through an array of randomly oriented cylinders (9). The expressions used here for the hindrance coefficients were
&PHgr;<IT>K</IT><SUB>d</SUB> = <FENCE>1 − <FR><NU>5</NU><DE>3</DE></FR> &phgr;</FENCE> exp[−<IT>A</IT>(1 + <IT>C</IT>&Dgr;P<SUB>bm</SUB>)<IT>r</IT><SUB>s</SUB>] (7A)
&PHgr;<IT>K</IT><SUB>c</SUB> = exp[−<IT>B</IT>(1 + <IT>C</IT>&Dgr;P<SUB>bm</SUB>)<IT>r</IT><SUB>s</SUB>] (7B)
where Delta Pbm is the pressure drop across the GBM. The constants A, B, and C were determined by fitting Eqs. 7A and 7B to the measured diffusional permeabilities of GBM (16), corrected for the effect of BSA as described above, and GBM sieving coefficients (15). Assuming that phi  = 0.10 (8, 15, 27), data for four Ficoll radii ranging from 3.0 to 6.2 nm yielded A = 1.064 nm-1, B = 0.472 nm-1, and C = 0.00295 mmHg-1. Note that in these correlations the units of Delta Pbm and rs are in millimeters Hg and nanometers, respectively.

Transport across the endothelium. The extent to which the endothelial fenestrae hinder the passage of macromolecules is unclear. The minimum radius of the fenestrae, rf = 30 nm (20), greatly exceeds the Stokes-Einstein radii of macromolecules of physiological interest, which range from ~2 to 6 nm. Thus, if the fenestrae are filled only with water, then they will offer little hindrance based on molecular size. In contrast, it has been suggested that the fenestrae are filled with a sparse glycocalyx (1, 2). To determine whether the endothelial barrier to the transport of uncharged solutes can be neglected, we calculated an upper bound on its contribution by assuming that the fenestrae are filled with the same dense matrix as the GBM. We computed solute sieving coefficients across the two layers (i.e., endothelium plus GBM) using that assumption and compared the results with those obtained by neglecting the endothelial contribution.

The concentration field in the composite region composed of the fenestrae and GBM was obtained by again solving Eqs. 2 and 3 using finite elements, but with Eq. 3A applied at the lumen-fenestra boundary. The local sieving coefficient for the fenestrae plus GBM was then computed as Theta fbm = C0/CS, where CS is the concentration of the test solute in the capillary lumen. To simplify the finite element calculations, we assumed that the fenestrae were straight channels 60 nm in length and of a width such that epsilon f = 0.20 and the number density of fenestrae was 1/120 nm-1 (12). The sieving coefficients calculated for water-filled or matrix-filled fenestrae never differed by more than 20%. Because this is an upper bound, it seems reasonable to neglect the resistance to macromolecule transport offered by the endothelial fenestrae. Accordingly, the overall sieving coefficient for one structural unit is assumed to be given by
<FR><NU>C<SUB>B</SUB></NU><DE>C<SUB>S</SUB></DE></FR> = &THgr;<SUB>bm</SUB>&THgr;<SUB>sd</SUB> (8)
Transport across the epithelium. The filtration slit between epithelial foot processes was modeled as a water-filled channel interrupted by a thin barrier perpendicular to the channel walls. The barrier (representing the slit diaphragm) was assumed to consist of a single row of parallel, cylindrical fibers, like the rungs of a ladder. The key geometric parameters for the filtration slit and slit diaphragm are defined in Fig. 2. The slit diaphragm is located at a distance delta  from the downstream surface of the GBM; the cylinder radius is rc; the center-to-center spacing of the cylinders is 2ell ; and the surface-to-surface spacing of the cylinders is 2u. Of importance, u is of the same order of magnitude as the radius of a macromolecular solute, rs. A hydrodynamic analysis of the transport of spherical macromolecules through such a channel, using dimensions derived from various electron microscopic studies of the filtration slit and slit diaphragm, led to the conclusion that the slit diaphragm provides the dominant resistance to the movement of macromolecules through the slit (13). Thus, to good approximation, the sieving coefficient for the slit equals that for the slit diaphragm, Theta sd. For a diaphragm with uniform cylinder size and spacing, where the sieving coefficient is denoted as &THgr;*<SUB>sd</SUB>, the theoretical results are summarized as
&THgr;*<SUB>sd</SUB>
= <FR><NU>1 − &lgr;</NU><DE>1 − &lgr;{1 − exp(−Pe<SUB>sd</SUB>&dgr;/<IT>r</IT><SUB>c</SUB>)[1 − exp(−<IT>F</IT>Pe<SUB>sd</SUB>)]}</DE></FR> (9)
&lgr; = <FR><NU><IT>r</IT><SUB>s</SUB></NU><DE>ℓ − <IT>r</IT><SUB>c</SUB></DE></FR> (10)
Pe<SUB>sd</SUB> = <FR><NU><IT>J</IT><SUB>v</SUB><IT>r</IT><SUB>c</SUB></NU><DE>&egr;<SUB>s</SUB><IT>D</IT><SUB>∞</SUB></DE></FR> (11)
<IT>F</IT> = <FR><NU>3.65</NU><DE>(<IT>r</IT><SUB>c</SUB>/ℓ  )</DE></FR> + <FR><NU>0.573</NU><DE>1 − (<IT>r</IT><SUB>c</SUB>/ℓ  )</DE></FR> (12)
where Pesd is the Péclét number for the slit diaphragm.


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Fig. 2.   Assumed structure of slit diaphragm in relation to GBM (not to scale). View is parallel to axis of the fibers in the slit diaphragm and perpendicular to the podocyte membranes that bound the filtration slit.

According to this model, the filtrate must pass through the spaces between the cylinders, as through the bars of a cage. If u is uniform, as assumed in Eq. 9, a sharp cutoff in the sieving curve (the plot of Theta  vs. rs) is predicted. This is because macromolecules with rs > u cannot pass through the slit diaphragm. However, there is abundant evidence that there is no such sharp cutoff, even in healthy animals or humans (3, 21, 23, 26). As discussed previously (13, 16), this finding can be explained by assuming that u is not uniform, but rather follows a continuous probability distribution. Adopting this approach, the average sieving coefficient for the slit diaphragm of one structural unit is given by
&THgr;<SUB>sd</SUB> = <LIM><OP>∫</OP><LL>0</LL><UL>∞</UL></LIM> &THgr;*<SUB>sd</SUB>(<IT>u</IT>)G(<IT>u</IT>)d<IT>u</IT> (13)
where G(u)du is the fraction of filtrate volume passing through gaps of half-width between u and u + du. The hydraulic permeability for the epithelial filtration slit (ks) was expressed as
<IT>k</IT><SUB>s</SUB> = <FR><NU><IT>J</IT><SUB>v</SUB></NU><DE>&egr;<SUB>s</SUB>&Dgr;P<SUB>s</SUB></DE></FR> = <FR><NU><LIM><OP>∫</OP><LL>0</LL><UL>∞</UL></LIM> (<IT>r</IT><SUB>c</SUB> + <IT>u</IT>)g(<IT>u</IT>)V(<IT>u</IT>)d<IT>u</IT></NU><DE>&Dgr;P<SUB>s</SUB><LIM><OP>∫</OP><LL>0</LL><UL>∞</UL></LIM> (<IT>r</IT><SUB>c</SUB> + <IT>u</IT>)g(<IT>u</IT>)d<IT>u</IT></DE></FR> (14)
where g(u)du is the probability that the gap half-width is between u and u + du. In the absence of quantitative data regarding structural heterogeneities in slit diaphragms, we chose either a gamma or a lognormal distribution for the cylinder spacings, assuming that g(u) is centered around a single value and vanishes as u goes to zero or to infinity. Using a gamma distribution, the model tended to predict unrealistically small sieving coefficients for large solute radii. Accordingly, the lognormal distribution of cylinder spacings was used for all results reported here. This distribution is given by
g(<IT>u</IT>) = <FR><NU>1</NU><DE><RAD><RCD>2&pgr;</RCD></RAD><IT> u</IT> ln <IT>s</IT></DE></FR> exp<FENCE>−<FR><NU>1</NU><DE>2</DE></FR> <FENCE><FR><NU>ln (<IT>u</IT>/<IT>u</IT><SUB>m</SUB>)</NU><DE>ln <IT>s</IT></DE></FR></FENCE><SUP>2</SUP></FENCE> (15)
where um is the mean gap half-width and ln s is the standard deviation of the distribution. The probability density G(u) was calculated from g(u) and theoretical results for low-speed flow through a row of cylinders (see equation 24 in Ref. 13).

The lognormal distribution contains two parameters, um and s. The number of degrees of freedom was reduced by fixing the value of the slit hydraulic permeability (ks) at that estimated previously (11). For any given value of rc, this implied a certain relationship between um and s. We chose to regard s as the independent parameter and used Eq. 14 to determine um.

Observable sieving coefficient. It has not been technically feasible to measure sieving coefficients at the level of a single filtration slit. Micropuncture techniques have occasionally been employed to determine sieving coefficients for single glomeruli in experimental animals, but more often the approach has been to use fractional clearance measurements to assess sieving at the whole kidney level. Accordingly, to relate the model predictions to measured sieving coefficients, it is necessary to average the local values along the length of a representative capillary. We made the usual assumption that glomerular filtration at the whole kidney level amounts to many such capillaries functioning in parallel.

The local sieving coefficient, CB/CS, must vary with position along a capillary because of the decreases in Jv that take place from the afferent to the efferent end. The decline in the volume flux results mainly from the progressive increase in oncotic pressure associated with production of a nearly protein-free ultrafiltrate. Decreases in Jv affect the local sieving coefficient by causing the Péclét numbers for the GBM and filtration slit (Eqs. 4 and 11) to decline. Axial variations in concentrations and fluxes along a capillary were described using steady-state mass balance equations applied to total blood plasma, total plasma protein, and a test macromolecule (e.g., Ficoll) assumed to be present at tracer levels. The differential equations were identical to those used in many previous studies (21), and so will not be repeated here. Using as inputs the glomerular ultrafiltration coefficient (Kf), the mean transcapillary pressure (Delta P), and the afferent values of the plasma flow rate (QA), total protein concentration (CPA), and tracer concentration (CSA), these equations were solved numerically to determine plasma flow rate, protein concentration, and tracer concentration as functions of axial position (x). In these calculations, the tracer flux was evaluated as
<IT>J</IT><SUB>s</SUB>(<IT>x</IT>) = &THgr;<SUB>bm</SUB>(<IT>x</IT>)&THgr;<SUB>sd</SUB>(<IT>x</IT>)C<SUB>S</SUB>(<IT>x</IT>) <IT>J</IT><SUB>v</SUB>(<IT>x</IT>) (16)
and the observable sieving coefficient for the tracer (Theta ) was computed as
&THgr; = <FR><NU><LIM><OP>∫</OP><LL>0</LL><UL>1</UL></LIM> <IT>J</IT><SUB>s</SUB>(<IT>x</IT>)d<IT>x</IT></NU><DE>C<SUB>SA</SUB> <LIM><OP>∫</OP><LL>0</LL><UL>1</UL></LIM> <IT>J</IT><SUB>v</SUB>(<IT>x</IT>)d<IT>x</IT></DE></FR> (17)
In this expression the axial coordinate has been normalized, such that x = 0 and x = 1 correspond to the afferent and efferent ends of the capillary, respectively. The numerator is the actual transmembrane solute flux averaged over the length of the capillary. The denominator is the average solute flux that would exist if there were no hindrances based on molecular size (i.e., if the solute behaved like water).

The differential equations were solved using a fourth-order Runga-Kutta scheme (25), and all integrals were evaluated using Romberg's method. Using an IBM RS 6000 (model 370) workstation, the CPU time required to compute the sieving coefficients of 26 solutes ranging from 2 to 7 nm in radius was ~10 s.


    RESULTS AND DISCUSSION
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Parameter values. Input quantities representative of normal euvolemic rats are summarized in Table 1. Except for the slit diaphragm parameters (rc, s, and um), all ultrastructural and microhydrodynamic quantities shown are those estimated by Drumond and Deen (11-13) from electron microscopy studies in the literature or calculated from their model for glomerular hydraulic permeability. The slit diaphragm parameters are representative of those obtained by fitting Ficoll sieving data in normal rats and humans, as will be discussed. The hemodynamic inputs were obtained from the review of Maddox et al. (21). To the extent that ultrastructural information is available for healthy humans, the main differences are in the GBM thickness and the width of a structural unit (L = 518 vs. 200 nm in rats and W = 465 vs. 360 nm in rats); there is not a significant difference in slit width, w (14). Simulations using structural and hemodynamic quantities representative of humans yielded results very similar to those for rats, so that most results to be presented were based on input parameters for rats.

                              
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Table 1.   Parameter values for normal euvolemic rats

An additional key parameter is the Darcy permeability of the GBM (kappa ), which is its intrinsic conductance to water. The Darcy permeability influences the fluxes of macromolecules by affecting the fluid velocity. Studies using isolated GBM have shown that kappa  decreases as the applied pressure is increased, thereby compressing the membrane. Based on the data of Edwards et al. (15) for rat GBM, this relationship was expressed as
&kgr; = 2.40 − 0.0264&Dgr;P<SUB>bm</SUB> (18)
where kappa  is in units of square nanometers and Delta Pbm is in units of millimeters Hg.

Contributions of individual layers. A unique feature of the current model is its ability to differentiate the effects of the individual layers (endothelium, GBM, epithelium) on glomerular size selectivity. (Although, as already mentioned, the concentration drop within the fenestrae was assumed to be negligible, the endothelial cells are still able to influence macromolecule transport by blocking much of the upstream surface of the GBM.) To illustrate the effects of each layer, we used a synthetic approach, adding one structural feature at a time. Figure 3 shows the local sieving coefficient (CB/CS) calculated for four hypothetical barriers: bare GBM (curve a); GBM with endothelial cells (curve b); GBM with both endothelial and epithelial cells, but without a slit diaphragm (curve c); and the complete structure (curve d). The results are based on the average value of Jv for a rat glomerulus and therefore represent the situation at an intermediate position along a capillary. With single-nephron GFR (SNGFR) = 45 nl/min and a filtration surface of 0.002 cm2/glomerulus (21), the average volume flux is 3.8 × 10-6 m/s. As might be expected, Fig. 3 shows that the sieving coefficient computed for any given size of macromolecule decreased as each structural element was added. The "surface blockage" effect of the endothelial cells was minimal. The blockage effect of the epithelial foot processes was more significant; for large solute radii, CB/CS was reduced by more than a factor of two in going from curve b in Fig. 3 to curve c. The most dramatic effect was that of the epithelial slit diaphragm, the addition of which reduced the sieving coefficient of large macromolecules by some two orders of magnitude (curve d vs. curve c).


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Fig. 3.   Local sieving coefficient (CB/CS) as a function of Stokes-Einstein radius (rs) for 4 hypothetical structures: curve a, bare GBM; curve b, GBM with endothelial cells; curve c, GBM with endothelial and epithelial cells, but no slit diaphragm; and curve d, complete capillary wall. Calculations were based on ultrastructural and microhydrodynamic inputs in Table 1 and a volume flux of Jv = 3.8 × 10-6 m/s. As each structure was added, the sieving coefficient for any given molecular size was reduced.

The individual layers of the capillary wall do not act independently, so that it is misleading to view the glomerular barrier to macromolecules simply as a set of resistances in series. In particular, the foot processes and slit diaphragm influence the velocity and concentration fields within the GBM, and thereby affect the sieving coefficient for that layer. The interactions between the GBM and epithelium are illustrated in Fig. 4, which shows a representative solute concentration profile within the capillary wall. The results are for a macromolecule of the size of albumin (rs = 3.6 nm) and the average value of Jv given above. The concentration at the center of a structural unit is shown for positions ranging from the capillary lumen to Bowman's space. The concentrations are normalized by that in the lumen and plotted on a log scale. Due to steric exclusion, the concentration in the GBM immediately adjacent to the lumen was smaller than that in the lumen. For this plot only, the partition coefficient (Phi ) was determined using the model of Ogston (22) for a random fiber matrix, assuming a GBM fiber radius of 3 nm and a solid volume fraction of 0.10 (8, 15, 27); values of Phi  per se were not needed to compute CB/CS or Theta . Continuing across the GBM, the solute concentration increased as a result of the more selective slit diaphragm downstream. Thus the model predicts a form of concentration polarization within the GBM. At the interface between the GBM and the slit channel, another concentration jump occurred, because the partition coefficient in the channel (relative to free solution) was different than that in the GBM. The concentration continued to rise slightly until the slit diaphragm, at which point there was a large drop. After that step-like change, the concentration remained almost constant in the remainder of the slit. (For that reason, the total length of the slit was unimportant.) A final, slight, increase in concentration occurred where the slit joined Bowman's space, as partitioning effects were canceled.


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Fig. 4.   Concentration variations within capillary wall for a solute with rs = 3.6 nm. Concentration was evaluated along the center of the structural unit (x = 0 in Fig. 1) and was normalized by that in the capillary lumen. Positions are distances from the upstream surface of GBM. Calculations were based on volume flux and other conditions of Fig. 3.

The predicted increase in concentration across the GBM, as shown in Fig. 4, implies that Theta bm > 1 for the intact capillary wall. Thus the overall sieving coefficient for the capillary wall was predicted to be larger with the GBM present than with it absent! This emphasizes that, for macromolecule transport, it is inaccurate to think in terms of series resistances.

The concentration profile in Fig. 4 is very different from that inferred for albumin in the ultrastructural visualization study of Ryan and Karnovsky (29). When superficial rat glomeruli were fixed in situ during normal blood flow, immunostaining of endogenous albumin was confined to the capillary lumen and endothelial fenestrae. This discrepancy may be due to molecular charge. Whereas the simulation in Fig. 4 was for an uncharged solute with rs = 3.6 nm, albumin is highly anionic. Although the GBM seems not to exhibit significant charge selectivity (5), charge-based restriction might occur at the level of the endothelial glycocalyx. Consistent with the findings of Ryan and Karnovsky (29), an assumption of the present model is that the albumin concentration within the GBM or slit is negligible. This is implicit in our description of water filtration, in which the net hydraulic-oncotic (Starling) pressure difference is assumed to act across the entire capillary wall (12).

Effects of hemodynamic factors. The existence of concentration polarization within the GBM, as well as other features of the present model, will influence the manner in which SNGFR and its determinants affect the sieving coefficients for molecules of varying size. In the discussion which follows, it is assumed that changes in SNGFR and the single-nephron filtration fraction for water (SNFF) are due to variations in glomerular plasma flow rate and/or pressure, without changes in the intrinsic properties of the barrier. As discussed previously (21), what underlies these "hemodynamic effects" are two physical relationships that occur generally in ultrafiltration processes. First, any change in filtrate velocity (volume flux) will tend to alter the relative contributions of convection and diffusion to the flux of a test molecule. Reductions in Jv increase the time available for diffusional equilibration between filtrate and retentate and thereby tend to increase Theta  for any molecule that is not completely excluded by the membrane; in the limit as Jv right-arrow 0, Theta  right-arrow 1. At the other extreme, large values of Jv lead to solute fluxes that are almost entirely convective; as Jv right-arrow infinity , Theta  declines to a minimum value equal to 1 - sigma , where sigma  is the traditional "reflection coefficient." To the extent that the glomerular capillary wall behaves as a homogeneous ultrafiltration membrane (for which these statements apply), there will be an inverse relationship between Theta  and SNGFR. The second physical effect arises from the increase in the concentration of any selectively retained solute as plasma moves from the afferent to the efferent end of a capillary. This increase in the luminal concentration above that in afferent plasma will increase the local solute flux, an effect which is magnified when SNFF is large. Thus there is a tendency for Theta  to change in the same direction as SNFF. In a given physiological setting, the effects of SNGFR and SNFF may either reinforce or cancel one another.

Figure 5 shows the predicted effects of the local volume flux on the sieving coefficient for one structural unit, for three sizes of test molecule. Included are the overall sieving coefficient for one unit (CB/CS) and the individual contributions of the GBM (Theta bm) and slit diaphragm (Theta sd). It is seen that CB/CS is constant at low Jv, decreases with increasing flux at intermediate Jv, and then increases with increasing flux at high Jv. Focusing first on the intermediate and high volume fluxes, it is seen that although Theta sd varies inversely with Jv, as expected for a "simple" membrane, concentration polarization causes Theta bm to increase with Jv. This competition between the GBM and the slit is what underlies the biphasic response of CB/CS. A second departure from the behavior of homogeneous membranes is in the asymptotic values of the sieving coefficients at small volume fluxes. Diffusion within the GBM is rapid enough then to make concentration polarization negligible, so that Theta bm = 1 and CB/CS = Theta sd. However, in contrast to what would occur for a simple membrane, the constant values of CB/CS and Theta sd reached in Fig. 5 for low Jv are all much less than unity. The underlying factor here is the nonuniform spacing between the cylinders used to represent the slit diaphragm. Diffusional equilibration across the slit diaphragm can occur only when these spaces are large enough to permit passage of the test solute. With the particular lognormal distribution used for these calculations, most of the spaces allow only filtration of water (i.e., um = 1.0 nm), so that the sieving coefficient remains zero for most of the filtrate, even at small Jv. In summary, the unexpectedly complex dependence of the sieving coefficient on the volume flux in Fig. 5 is the result of the capillary wall having elements of differing size selectivity arranged both in series (GBM and slit diaphragm) and in parallel (individual spaces in the slit diaphragm). Equivalent-pore models that postulate pores of nonuniform size (e.g., a lognormal distribution of pore radii) have the parallel but not the series feature.


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Fig. 5.   Local sieving coefficients as a function of volume flux. Overall sieving coefficient for capillary wall (CB/CS) is compared with values for GBM (Theta bm) and slit diaphragm (Theta sd), for each of 3 molecular radii. Calculations were based on ultrastructural and microhydrodynamic inputs in Table 1. For reference, volume flux in a rat glomerulus typically averages 4 × 10-6 m/s.

Moving now to the level of a whole glomerulus, the dependence of the sieving coefficient on glomerular plasma flow rate (QA) is shown in Fig. 6. Results are given both for the current model and for an equivalent-pore model (10), with membrane parameter values chosen to yield similar results at the baseline value of QA = 150 nl/min. Selective increases in QA increase SNGFR and decrease SNFF, indicating that for a homogeneous barrier Theta  will decrease. This behavior is seen with the pore model, but the prediction from the structural model is that Theta  will be almost constant. That constancy reflects a balance between the tendency of high volume fluxes to increase Theta  (due to concentration polarization) and the effect of the reduced SNFF to lower Theta .


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Fig. 6.   Effects of selective variations in glomerular plasma flow rate (QA) on sieving coefficients for 3 molecular sizes. Results are shown for the present (ultrastructural) model and for an equivalent-pore model. Inputs for the ultrastructural model were as given in Table 1. The pore model assumed a lognormal distribution of pore radii plus a nonselective shunt, with a mean radius of 6.54 nm, a variance parameter of 1.69 (analogous to s in Eq. 15), and a shunt parameter of omega 0 = 1.67 × 10-4 (10).

The dependence of the sieving coefficient on the mean transmembrane pressure difference (Delta P) is illustrated in Fig. 7. Selective increases in Delta P increase both SNGFR and SNFF, suggesting that a cancellation of effects might leave Theta  relatively constant. This expected behavior is seen again for the pore model but not the structural model. In the latter, concentration polarization in the GBM, which is aggravated by the increased SNGFR, reinforces the effect of changing SNFF. Consequently, Theta  is predicted to increase significantly as Delta P increases, especially for the larger molecules. It should be mentioned that with the present model the sensitivity of SNGFR and SNFF to changes in Delta P is somewhat less than with the pore model, because of the dependence of kappa  on Delta P described by Eq. 18.


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Fig. 7.   Effects of selective variations in mean transcapillary pressure (Delta P) on sieving coefficients for 3 molecular sizes. Results are shown for the present (ultrastructural) model and for an equivalent-pore model, using the same inputs as in Fig. 6.

The dependence of Theta  on QA was studied in rats by measuring fractional clearances for dextrans of moderate size (2 <=  rs <=  4 nm), with plasma volume expansion used to increase QA (7). It was found that Theta  decreased with increasing QA under those conditions. The plasma-flow dependence of Theta  for a more ideal tracer, such as Ficoll, has not yet been investigated. As with QA, there are not yet suitable data with which to test the predicted effects of Delta P.

Effects of ultrastructural parameters. The sensitivity of the predicted values of Theta  to changes in the ultrastructural parameters was examined by varying one parameter at a time. Figure 8 shows the effects of isolated changes in the radius of the cylindrical fibers used to represent the slit diaphragm. As rc decreases, so does Theta , because with s and ks constant, a smaller value of rc implies a smaller average spacing between the fibers. The effects of s, the parameter which describes the variance of the cylinder spacing, are illustrated in Fig. 9. The larger the value of s, the greater the number of large spaces, so that the barrier becomes less size selective. Thus Theta  for any given molecular size increases, and the slope of the sieving curve decreases, as s is increased. It is noteworthy that if the cylinder spacing is assumed to be uniform (i.e., s = 1), the gap half-width (u) is calculated to be 1.2 nm. Thus solutes with a radius larger than that could not enter Bowman's space, contrary to much experimental evidence. This emphasizes the need to postulate a nonuniform fiber spacing for the slit diaphragm. Overall, the results in Figs. 8 and 9 show that the sieving curve is very sensitive to the parameters used to describe the slit diaphragm.


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Fig. 8.   Effects on the sieving curve of selective changes in the radius of the slit diaphragm fibers (rc). Variance parameter for the fiber spacing was held constant at s = 1.50, and other inputs were as given in Table 1.



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Fig. 9.   Effects on the sieving curve of selective changes in variance parameter for the fiber spacing in the slit diaphragm (s). Fiber radius (rc) was held constant at 2.0 nm, and other inputs were as given in Table 1.

A twofold increase in the thickness of the GBM (L) and/or a threefold decrease in the filtration slit frequency (i.e., an increase in W) were found to have little effect on the predicted sieving curves. With these assumed changes in L and W, the maximum variations in Theta , seen for the largest solute radii, never exceeded a factor of 1.5. Such structural changes are representative of what has been observed in patients with membranous nephropathy or minimal change nephropathy, and they adequately explain the changes in the hydraulic permeability of the capillary wall (14). However, the present results suggest that the altered sieving characteristics in these proteinuric disorders must be due mainly to factors other than L and W. This emphasizes that the structural features that limit filtration of water are not necessarily the same as those that govern the size selectivity of the barrier.

The effects of uncertainties in the GBM hindrance coefficients were assessed by assuming twofold increases or decreases in Phi Kc or Phi Kd, yielding the results shown in Fig. 10. For given concentration and velocity fields in the GBM, the convective and diffusive fluxes of a macromolecule will vary in proportion to changes in the respective hindrance coefficients. This leads to the expectation that Theta  will increase or decrease in parallel with changes in Phi Kc, which is confirmed by the results in Fig. 10. The greatest percentage changes are seen for the largest solute sizes. Less intuitive is the fact that the calculated sieving curves are shifted downward as Phi Kd is increased. What underlies this behavior is that increases in Phi Kd reduce the Péclét number for the basement membrane (Pebm), which in turn lessens the extent of concentration polarization within the GBM. Accordingly, Theta bm and Theta  are both lowered as Pebm is reduced. Because Pebm contains the ratio of Phi Kc to Phi Kd (Eq. 4), it is that ratio which is most critical. Thus, as shown in Fig. 10, the effects of a twofold increase in Phi Kc are much the same as a twofold decrease in Phi Kd, and vice versa. When both hindrance coefficients are multiplied or divided by the same factor, Pebm is unaffected and the predicted changes in Theta  are minimal; those curves, very close to the baseline case, were omitted from Fig. 10 for clarity. Overall, it is seen that the hindrance coefficients for the GBM have a significant influence on the sieving curve, despite the fact that the slit diaphragm is calculated to be the more restrictive barrier.


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Fig. 10.   Effects on the sieving curve of selective changes in GBM hindrance factors for convection (Phi Kc) and diffusion (Phi Kd). Curves correspond to the following: curve a, baseline values given by Eq. 7; curve b, Phi Kc reduced by half or Phi Kd doubled (results indistinguishable); curve c, Phi Kc doubled; and curve d, Phi Kd reduced by half. When Phi Kc and Phi Kd were both doubled or both reduced by half, the results were indistinguishable from the baseline case. All other inputs were as given in Table 1.

Analysis of Ficoll sieving data in vivo. The ultrastructural parameters that could not be estimated reliably from electron microscopy were those related to the slit diaphragm, namely, rc and s. Their values were inferred by fitting the model to fractional clearance data for Ficoll in normal rats (23, 26) and humans (3). The hemodynamic inputs were obtained from data in the individual studies, and the ultrastructural and microhydrodynamic quantities were those listed in Table 1; the exception was L and W for humans, as noted above. The Ficoll sieving data were fitted by finding the values of rc and s that minimized the least-square error, defined as
&khgr;<SUP>2</SUP> = <LIM><OP>∑</OP><LL><IT>i</IT>=1</LL><UL><IT>n</IT></UL></LIM> <FENCE><FR><NU>&THgr;<SUB><IT>i</IT></SUB>(meas) − &THgr;<SUB><IT>i</IT></SUB>(calc)</NU><DE>&sfgr;<SUB><IT>i</IT></SUB></DE></FR></FENCE><SUP>2</SUP> (19)
where n is the number of data points in the sieving curve, Theta i(meas) and Theta i(calc) are the measured and calculated sieving coefficient of solute i, respectively, and sigma i is the standard error of Theta i(meas). Powell's method (25) was employed to find the best-fit parameter values. Because the expressions for the GBM hindrance coefficients in Eq. 7 were based on results only for 3.0 <=  rs <=  6.2 nm, the fractional clearance data used were restricted to that range of molecular sizes.

The results for rc and s are shown in Table 2, along with the corresponding values of chi 2. The range for rc of 1.2-8.6 nm corresponds fairly closely with the range of 2-10 nm inferred from slit diaphragm thicknesses in published electron micrographs (13). The values of s varied from 1.2 to 1.6, with the larger values of s being associated with the smaller values of rc. In each case, the fit to the experimental sieving curve was excellent, as evidenced by the low values of chi 2 in Table 2 and the comparisons of the measured and calculated sieving curves in Fig. 11. The sieving curve measured for healthy humans was quite different from that for either strain of rats, so that the wide range of values found for the slit-diaphragm parameters is not surprising.

                              
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Table 2.   Slit-diaphragm parameters calculated from fractional clearance data



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Fig. 11.   Comparison between measured and calculated Ficoll sieving coefficients in vivo. Experimental results, shown by the symbols as means ± SE, are those of Oliver et al. (23) and Remuzzi et al. (26) for normal rats and those of Blouch et al. (3) for healthy humans. Calculated values, shown by curves, were obtained by fitting the slit-diaphragm parameters to each set of data, yielding values of rc and s given in Table 2.

The ability of the present model to fit fractional clearance data was compared with that of equivalent-pore models (10). Using the data in Fig. 11 and that from a few other experimental conditions, it was found that the structural model tended to provide a better fit than one which assumes a lognormal distribution of pore sizes, although not as good a fit as one which postulates a lognormal distribution of pore sizes in parallel with a nonselective shunt pathway. The number of adjustable parameters (degrees of freedom) was two for the structural model, two for the lognormal pore model, and three for the lognormal-plus-shunt pore model. Thus the present model appears to be somewhat more accurate than a conventional pore model with the same number of adjustable parameters.

Predictions from Ficoll diffusion data in vitro. In a previous study (16), we determined the diffusional permeability of Ficoll in single capillaries of intact and cell-free glomeruli isolated from rats. The diffusional resistance of the cellular part of the barrier was assumed to be governed by the structure of the slit diaphragm. Interpreting the data using a lognormal distribution of spacings between cylindrical fibers, similar to the model presented here, it was found that the results could be explained most readily by assuming that a small fraction of the diaphragm area (~0.2%) was devoid of fibers, creating a "shunt." Our present estimates of epithelial slit parameters are largely consistent with those results. For example, when the values of rc and s derived from the fractional clearance data of Oliver et al. (23) were used to predict the diffusional permeabilities in vitro, there was good agreement with the in vitro results, albeit only if the lognormal distribution was augmented by a shunt.

The reverse approach, using the in vitro parameters to predict fractional clearances in vivo, was inconclusive. In general, the values of rc and s inferred from the diffusional data (16) yielded predicted sieving curves that did not agree with those measured in vivo (3, 23, 26). The main difficulty with this approach is that the diffusional permeabilities are predicted to be quite sensitive to the magnitude of the shunt pathway, which amplifies the uncertainties in the estimated values of rc and s. In contrast, the fractional clearances in vivo are very sensitive to rc and s (Figs. 8 and 9) and, in healthy subjects at least, seem not to be affected by a shunt in the slit diaphragm.

Conclusions. The present representation of the glomerular capillary wall is more realistic than in any previous model for glomerular filtration of macromolecules. By incorporating what is known about the individual structures, it allows one to predict the effects of specific alterations in any of the three layers of the barrier. The results suggest that glomerular size selectivity is most sensitive to the structural features of the slit diaphragm and to the hindrance coefficients of the GBM; variations in GBM thickness or filtration slit frequency are predicted to have little effect on fractional clearances. Although conventional models based on equivalent pores remain useful for comparative purposes (e.g., for showing that a disease or experimental maneuver caused a change in barrier properties), they do not provide a basis for structure-function correlations at the cellular or subcellular level. The ability of the ultrastructural model to represent sieving data in vivo is at least equal to that of pore models, although the structural approach requires more computational effort.

The most severe limitation of the present model stems from uncertainties in the fine structure of the epithelial slit diaphragm. The zipper-like configuration first described by Rodewald and Karnovsky (28), which involves a central filament connected to the podocyte membranes by alternating bridge fibers, is very appealing. However, the uniform dimensions of the openings (4 × 14 nm) are inconsistent with fractional clearance data in vivo; these dimensions imply that Ficoll molecules with rs > 2 nm will be excluded from urine, which is clearly not the case (Fig. 11). The concept of a central filament with regularly spaced bridges has been questioned by other electron microscopists (17, 18), but no quantitative alternative has emerged. As shown here, treating the slit diaphragm as a row of cylindrical fibers with variable spacing provides accurate functional predictions, but this representation must be viewed as provisional.

Another limitation of the present model is that the description of the GBM properties governing water and solute movement (kappa , Phi Kd, Phi Kc) is entirely empirical. Ultimately, we would like to relate those properties to the macromolecular composition of the GBM and to the spatial arrangement of those constituents. A reasonable starting point is to view the GBM as an array of uniformly sized fibers. Palassini and Remuzzi (24) assumed a regular polygonal arrangement of fibers in modeling kappa , and Booth and Lumsden (6) employed a randomly oriented fiber matrix in simulations designed to visualize GBM "pores." However, at least two populations of fibers may be needed to explain even the values of kappa  (15). Achieving the desired level of structural detail will probably require more quantitative information on the GBM composition, as well as advances in the theory for hindered transport of macromolecules through arrays of fibers.

Although our current understanding of several aspects of glomerular ultrastructure is severely limited, a major strength of the present approach (as opposed to equivalent-pore models) is that it provides a framework for relating future structural findings to the functional properties of the barrier.


    ACKNOWLEDGEMENTS

This work was supported by National Institute of Diabetes and Digestive and Kidney Diseases Grants DK-20368 and DK-45058. B. S. Daniels is the recipient of an American Heart Association Established Investigatorship.


    FOOTNOTES

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. §1734 solely to indicate this fact.

Address for reprint requests and other correspondence: W. M. Deen, Dept. of Chemical Engineering, Rm. 66-572, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139-4307 (E-mail: wmdeen{at}mit.edu).

Received 14 August 1998; accepted in final form 12 March 1999.


    REFERENCES
TOP
ABSTRACT
INTRODUCTION
MATHEMATICAL MODEL
RESULTS AND DISCUSSION
REFERENCES

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