## Abstract

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

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.

## MATHEMATICAL MODEL

*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*r*
_{f}. 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 ε_{f} and ε_{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.

*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
Equation 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*D*
_{∞}, which is related to the Stokes-Einstein radius (*r*
_{s}) by*D*
_{∞} =*k*
_{B}
*T*/(6πμ*r*
_{s}), where *k*
_{B} is Boltzmann’s constant, *T* is the absolute temperature, and μ is the viscosity of water. The coefficients *K*
_{d}and *K*
_{c} 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
Equation 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
Equation 3A
Equation 3B
Equation 3C *Equation3A
* relates the component of the flux in the*z* direction (*N _{z}
*) to the flux averaged over the entire width of the structural unit (

*J*

_{s}).

*Equation 3B*relates the average concentration at the upstream end of the filtration slit, C

_{0}, to that at the adjacent surface of the GBM; and Φ 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 Equation 3B` where C

_{B}is the local concentration in Bowman’s space, Θ

_{sd}= C

_{B}/C

_{0}is the local sieving coefficient for the filtration slit only (determined mainly by the slit diaphragm), and

*J*

_{v}is the mean volume flux (fluid velocity) in the structural unit. (“Local” quantities such as C

_{B}and Θ

_{sd}vary from one structural unit to another, due to variations in solute concentration and

*J*

_{v}along a capillary.) It is worth noting that

*J*

_{s}and

*J*

_{v}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 Θ_{bm} = C_{0}/C_{1}, where C_{1} 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 Θ_{bm} is a function only of Φ*K*
_{c}, ε_{f}, ε_{s},*L*/*W*, Θ_{sd}, and the Péclét number for the basement membrane, which is defined as
Equation 4 The relevant variables were incorporated into expressions of the form
Equation 5
Equation 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*→ ∞, or if both ε_{f} and ε_{s} → 1, then Θ_{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, Θ_{bm} → 1 if Pe_{bm} → 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} ≤ Pe_{bm} ≤ 10, 10^{−4} ≤ Φ*K*
_{c} ≤ 1, 0.01 ≤ ε_{s} ≤ 1, 0.05 ≤*L*/*W*≤ 1, and 10^{−4} ≤ Θ_{sd} ≤ 1, with ε_{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 Θ_{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 Φ*K*
_{d} and Φ*K*
_{c} 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), Φ for Ficoll appears to be increased markedly by the repulsive interactions of Ficoll with BSA. For Ficoll with*r*
_{s} = 3.6 nm, it was estimated that Φ 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 Φ*K*
_{d} and Φ*K*
_{c} (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 Φ*K*
_{d} and Φ*K*
_{c} for a point-size molecule should be 1 − (5/3)φ and 1, respectively, where φ is the volume fraction of fibers in the GBM; the result for Φ*K*
_{d} 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
Equation 7A
Equation 7B where ΔP_{bm} 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 φ = 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 ΔP_{bm}and *r*
_{s} 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,*r*
_{f} = 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 Θ_{fbm} = C_{0}/C_{S}, where C_{S} 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 ε_{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
Equation 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 δ from the downstream surface of the GBM; the cylinder radius is *r*
_{c}; the center-to-center spacing of the cylinders is 2ℓ; and the surface-to-surface spacing of the cylinders is 2*u*. Of importance,*u* is of the same order of magnitude as the radius of a macromolecular solute,*r*
_{s}. 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, Θ_{sd}. For a diaphragm with uniform cylinder size and spacing, where the sieving coefficient is denoted as
, the theoretical results are summarized as
Equation 9
Equation 10
Equation 11
Equation 12 where Pe_{sd} is the Péclét number for the slit diaphragm.

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 Θ vs.*r*
_{s}) is predicted. This is because macromolecules with*r*
_{s} >*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
Equation 13 where G(*u*)d*u*is the fraction of filtrate volume passing through gaps of half-width between *u* and*u* + d*u*. The hydraulic permeability for the epithelial filtration slit (*k*
_{s}) was expressed as
Equation 14 where g(*u*)d*u*is the probability that the gap half-width is between*u* and*u* + d*u*. 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
Equation 15 where*u*
_{m} 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,*u*
_{m} and*s*. The number of degrees of freedom was reduced by fixing the value of the slit hydraulic permeability (*k*
_{s}) at that estimated previously (11). For any given value of*r*
_{c}, this implied a certain relationship between*u*
_{m} and*s*. We chose to regard*s* as the independent parameter and used *Eq. 14
* to determine*u*
_{m}.

*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, C_{B}/C_{S}, must vary with position along a capillary because of the decreases in*J*
_{v} 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 *J*
_{v}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 (*K*
_{f}), the mean transcapillary pressure (ΔP), and the afferent values of the plasma flow rate (Q_{A}), total protein concentration (C_{PA}), and tracer concentration (C_{SA}), 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
Equation 16 and the observable sieving coefficient for the tracer (Θ) was computed as
Equation 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

*Parameter values*. Input quantities representative of normal euvolemic rats are summarized in Table1. Except for the slit diaphragm parameters (*r*
_{c},*s*, and*u*
_{m}), 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.

An additional key parameter is the Darcy permeability of the GBM (κ), 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 κ 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
Equation 18 where κ is in units of square nanometers and ΔP_{bm} 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. Figure3 shows the local sieving coefficient (C_{B}/C_{S}) 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*J*
_{v} 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 cm^{2}/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, C_{B}/C_{S}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*).

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 (*r*
_{s} = 3.6 nm) and the average value of*J*
_{v} 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 (Φ) 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 Φ per se were not needed to compute C_{B}/C_{S}or Θ. 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.

The predicted increase in concentration across the GBM, as shown in Fig. 4, implies that Θ_{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*r*
_{s} = 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*J*
_{v} increase the time available for diffusional equilibration between filtrate and retentate and thereby tend to increase Θ for any molecule that is not completely excluded by the membrane; in the limit as*J*
_{v} → 0, Θ → 1. At the other extreme, large values of*J*
_{v} lead to solute fluxes that are almost entirely convective; as*J*
_{v} → ∞, Θ declines to a minimum value equal to 1 − ς, where ς 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 Θ 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 Θ 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 (C_{B}/C_{S}) and the individual contributions of the GBM (Θ_{bm}) and slit diaphragm (Θ_{sd}). It is seen that C_{B}/C_{S}is constant at low*J*
_{v}, decreases with increasing flux at intermediate*J*
_{v}, and then increases with increasing flux at high*J*
_{v}. Focusing first on the intermediate and high volume fluxes, it is seen that although Θ_{sd} varies inversely with *J*
_{v}, as expected for a “simple” membrane, concentration polarization causes Θ_{bm} to increase with*J*
_{v}. This competition between the GBM and the slit is what underlies the biphasic response of C_{B}/C_{S}. 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 Θ_{bm} = 1 and C_{B}/C_{S}= Θ_{sd}. However, in contrast to what would occur for a simple membrane, the constant values of C_{B}/C_{S}and Θ_{sd} reached in Fig. 5 for low *J*
_{v} 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.,*u*
_{m} = 1.0 nm), so that the sieving coefficient remains zero for most of the filtrate, even at small *J*
_{v}. 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.

Moving now to the level of a whole glomerulus, the dependence of the sieving coefficient on glomerular plasma flow rate (Q_{A}) 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 Q_{A} = 150 nl/min. Selective increases in Q_{A} increase SNGFR and decrease SNFF, indicating that for a homogeneous barrier Θ will decrease. This behavior is seen with the pore model, but the prediction from the structural model is that Θ will be almost constant. That constancy reflects a balance between the tendency of high volume fluxes to increase Θ (due to concentration polarization) and the effect of the reduced SNFF to lower Θ.

The dependence of the sieving coefficient on the mean transmembrane pressure difference (ΔP) is illustrated in Fig.7. Selective increases in ΔP increase both SNGFR and SNFF, suggesting that a cancellation of effects might leave Θ 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, Θ is predicted to increase significantly as ΔP increases, especially for the larger molecules. It should be mentioned that with the present model the sensitivity of SNGFR and SNFF to changes in ΔP is somewhat less than with the pore model, because of the dependence of κ on ΔP described by *Eq. 18
*.

The dependence of Θ on Q_{A} was studied in rats by measuring fractional clearances for dextrans of moderate size (2 ≤*r*
_{s} ≤ 4 nm), with plasma volume expansion used to increase Q_{A} (7). It was found that Θ decreased with increasing Q_{A} under those conditions. The plasma-flow dependence of Θ for a more ideal tracer, such as Ficoll, has not yet been investigated. As with Q_{A}, there are not yet suitable data with which to test the predicted effects of ΔP.

*Effects of ultrastructural parameters*. The sensitivity of the predicted values of Θ 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*r*
_{c} decreases, so does Θ, because with *s* and*k*
_{s} constant, a smaller value of*r*
_{c} 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 Θ 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.

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 Θ, 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 Φ*K*
_{c} or Φ*K*
_{d}, 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 Θ will increase or decrease in parallel with changes in Φ*K*
_{c}, 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 Φ*K*
_{d} is increased. What underlies this behavior is that increases in Φ*K*
_{d} reduce the Péclét number for the basement membrane (Pe_{bm}), which in turn lessens the extent of concentration polarization within the GBM. Accordingly, Θ_{bm} and Θ are both lowered as Pe_{bm} is reduced. Because Pe_{bm} contains the ratio of Φ*K*
_{c} to Φ*K*
_{d}(*Eq. 4
*), it is that ratio which is most critical. Thus, as shown in Fig. 10, the effects of a twofold increase in Φ*K*
_{c}are much the same as a twofold decrease in Φ*K*
_{d}, and vice versa. When both hindrance coefficients are multiplied or divided by the same factor, Pe_{bm} is unaffected and the predicted changes in Θ 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.

*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,*r*
_{c} 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*r*
_{c} and*s* that minimized the least-square error, defined as
Equation 19 where*n* is the number of data points in the sieving curve, Θ_{i}(meas) and Θ_{i}(calc) are the measured and calculated sieving coefficient of solute*i*, respectively, and ς_{i} is the standard error of Θ_{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 ≤*r*
_{s} ≤ 6.2 nm, the fractional clearance data used were restricted to that range of molecular sizes.

The results for*r*
_{c} and*s* are shown in Table2, along with the corresponding values of χ^{2}. The range for*r*
_{c} 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 *r*
_{c}. In each case, the fit to the experimental sieving curve was excellent, as evidenced by the low values of χ^{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.

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*r*
_{c} 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 *r*
_{c} 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*r*
_{c} and*s*. In contrast, the fractional clearances in vivo are very sensitive to*r*
_{c} 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*r*
_{s} > 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 (κ, Φ*K*
_{d}, Φ*K*
_{c}) 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 κ, 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 κ (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.

## Acknowledgments

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

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).

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- Copyright © 1999 the American Physiological Society