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.
 |
INTRODUCTION |
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
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
f and
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.
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|
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
|
(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
(rs) by
D
= kBT/(6
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
|
(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
|
(3A)
|
|
(3B)
|
|
(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
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
|
(3B`)
|
where
CB is the local concentration in
Bowman's space,
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
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
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
bm is a function only of
Kc,
f,
s,
L/W,
sd, and the Péclét number for the basement membrane, which is defined as
|
(4)
|
The
relevant variables were incorporated into expressions of the form
|
(5)
|
|
(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
f and
s
1, then
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,
bm
1 if
Pebm
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
Kc
1, 0.01
s
1, 0.05
L/W
1, and 10
4
sd
1, with
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
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
Kd and
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),
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
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
Kd and
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
Kd and
Kc for a
point-size molecule should be 1
(5/3)
and 1, respectively,
where
is the volume fraction of fibers in the GBM; the result for
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
|
(7A)
|
|
(7B)
|
where
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
= 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
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
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
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
|
(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
from the downstream surface of the GBM; the cylinder
radius is rc; the
center-to-center spacing of the cylinders is
2
; 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,
sd. For a
diaphragm with uniform cylinder size and spacing, where the sieving
coefficient is denoted as
, the
theoretical results are summarized as
|
(9)
|
|
(10)
|
|
(11)
|
|
(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.
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|
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
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
|
(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
|
(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
|
(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 (
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
|
(16)
|
and
the observable sieving coefficient for the tracer (
) was computed as
|
(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 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.
An additional key parameter is the Darcy permeability of the GBM (
),
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
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
|
(18)
|
where
is in units of square nanometers and
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 (
) 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
per se were not needed to compute
CB/CS
or
. 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
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
for any molecule that is not
completely excluded by the membrane; in the limit as
Jv
0,
1. At the other extreme, large values of
Jv lead to solute
fluxes that are almost entirely convective; as
Jv
,
declines to a minimum value equal to 1
, where
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
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
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 (
bm) and slit diaphragm
(
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
sd varies inversely
with Jv, as
expected for a "simple" membrane, concentration polarization
causes
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
bm = 1 and
CB/CS =
sd. However, in contrast to
what would occur for a simple membrane, the constant values of
CB/CS and
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
( bm) and slit diaphragm
( 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.
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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
will decrease. This behavior is seen with the pore model, but the prediction from the structural model is that
will be almost constant. That constancy reflects a balance between the tendency of high volume fluxes
to increase
(due to concentration polarization) and the effect of
the reduced SNFF to lower
.

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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
0 = 1.67 × 10 4 (10).
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The dependence of the sieving coefficient on the mean transmembrane
pressure difference (
P) is illustrated in Fig.
7. Selective increases in
P increase
both SNGFR and SNFF, suggesting that a cancellation of effects might
leave
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,
is predicted to increase significantly as
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
P
is somewhat less than with the pore model, because of the dependence of
on
P described by Eq. 18.

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Fig. 7.
Effects of selective variations in mean transcapillary pressure ( 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.
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The dependence of
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
decreased with increasing QA under
those conditions. The plasma-flow dependence of
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
P.
Effects of ultrastructural parameters.
The sensitivity of the predicted values of
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
, 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
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.
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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
, 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
Kc or
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
will increase or decrease in parallel with changes in
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
Kd is
increased. What underlies this behavior is that increases in
Kd reduce the
Péclét number for the basement membrane
(Pebm), which in turn lessens
the extent of concentration polarization within the GBM. Accordingly,
bm and
are both lowered as
Pebm is reduced. Because
Pebm contains the ratio of
Kc to
Kd
(Eq. 4), it is that ratio which is
most critical. Thus, as shown in Fig. 10, the effects of a twofold
increase in
Kc
are much the same as a twofold decrease in
Kd, and vice
versa. When both hindrance coefficients are multiplied or divided by
the same factor, Pebm is
unaffected and the predicted changes in
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
( Kc) and
diffusion
( Kd). Curves
correspond to the following: curve a,
baseline values given by Eq. 7;
curve b,
Kc reduced by
half or Kd
doubled (results indistinguishable); curve
c,
Kc doubled;
and curve d,
Kd reduced by
half. When Kc
and 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.
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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
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(19)
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where
n is the number of data points in the
sieving curve,
i(meas) and
i(calc) are the measured and
calculated sieving coefficient of solute
i, respectively, and
i is the standard error of
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
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
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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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.
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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 (
,
Kd,
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
, 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
(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.
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.
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).