Kidney International, Vol. 16 (1979), pp.
353 —365
Macromolecule transport across glomerular capillaries:
Application of pore theory
WILLIAM M. DEEN, MICHAEL P. BOHRER, and BARRY M. BRENNER
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, and Laboratory of Kidney
and Electrolyte Physiology, Departments of Medicine, Peter Bent Brigham Hospital and Harvard Medical School,
Boston, Massachusetts
refinements of their approaches, have been re-
Measurements of transport rates of various highmolecular-weight solutes across capillaries have
viewed by a number of authors [12-15]. The treatment of the theory by Anderson and Quinn [14], although not as rigorous as that of Brenner and Gay-
contributed significantly to our understanding of the
functional properties of both renal and extrarenal
microcirculatory systems [1—7]. In addition to
serum albumin and other plasma proteins, non-
dos [15], has been particularly useful in that it
provides several improvements over earlier ap-
protein polymers such as dextran and poly-
proaches yet remains relatively simple to apply to
the analysis of transcapillary movement of macromolecules. Accordingly, the treatment which follows closely parallels that of Anderson and Quinn,
with some additional simplifications made in an effort to minimize the mathematical background required of the reader, while still making clear the
physical concepts involved.
The pore model, in its simplest form, considers
transport to take place through numerous identical
right cylindrical pores of radius r0 and length f, as
shown in Fig. I. Solute molecules are regarded as
hard spheres of radius a; the solvent is treated as a
continuum. An assumption fundamental to this the-
vinylpyrrolidone (PVP) also have been used successfully as transport probes. Complementing these
quantitative measurements of macromolecular
transport are the more qualitative insights into permeation pathways gained from studies with macromolecules as ultrastructural tracers [6, 8, 9]. As a
result of these investigations, it has become apparent that several factors influence the transcapillary
movement of macromolecules, including molecular
size, molecular charge, and perhaps even 'shape.
In addition, transport rates are known also to be in-
fluenced by plasma flow rate and other hemodynamic variables.
In an effort to better understand how these various factors combine to govern the transport of mac-
ory is that the solute molecule exhibits not only
Brownian motion (by which it diffuses), but also
romolecules, several theoretical descriptions of
transcapillary exchange have been developed.
possesses hydrodynamic characteristics. This concept is embodied in Einstein's equation [16]:
Much of this work has been based on hydrodynamic
D,,. =
models of solute transport through pores, first applied to capillaries by Pappenheimer [4] and Pappenheimer, Renkin, and Borrero [10]. This review
kT
(1)
where D,. is the Fick's Law diffusivity of the solute,
will summarize the current status of the pore model,
k is Boltzmann's constant, T is absolute temper-
with emphasis on its application to the study of
macromolecule transport across glomerular capil-
ature, and f,,. is a molecular friction coefficient. For
a sphere, f,,. = 6Ha, where is the viscosity of the
solvent. Substituting this expression for f,. into Eq.
laries.
Theory of restricted transport through pores
Principles underlying pore model. The pioneering
efforts of Pappenheimer et al [4, 10] and Renkin [11]
Received for publication June 22, 1978
in the use of hydrodynamic descriptions of transport through porous membranes, and subsequent
© 1979 by the International Society of Nephrology
0085-2538/79/0016—0353 $02.60
353
354
Deen et a!
1 yields the Stokes-Einstein equation (Eq. 2), which
relates "effective" molecular radius to diffusivity.
kT
D= 6ITa
cose [18].
The flux of an uncharged solute within a pore can
be derived by examining the forces acting on a representative solute molecule. For flux in the z direc-
tion (along the pore axis), the force responsible for
solute diffusion may be shown by thermodynamic
arguments to be minus the gradient of the chemical
potential. For an uncharged molecule, this is given
by
d ln C
—kT ______
dz
where C is solute concentration. Solute movement
is retarded by a hydrodynamic force, — F, which
arises from friction between the solute and solvent.
At steady state, these forces sum to zero, giving
(4)
V is the local solvent velocity in the axial (z) direction1, and K and G are coefficients which express
the retarding effect of the pore wall on the solute.
For an unbounded solution, K = G = 1, and the
hydrodynamic drag reduces to the product of f and
the relative velocity of solute and solvent, U — V.
The coefficients K and G, which depend principally
on the ratio air0, will be discussed further below.
The local solute flux within the pore, N5, is given
by the solute concentration times the solute velocity
N5 = CU
N = -K-1 Dr,.
(5)
Except in the immediate neighborhood of a solute molecule or
the ends of the pore, the variation of solvent velocity, V. with
radial position in the pore exhibits the familiar parabolic profile
characteristic of laminar flow. This velocity profile is given by V
2V [1 — (rIr0)2], where V is the mean velocity and r is the radial
position coordinate.
+ GCV
(6)
The first term on the right hand side of Eq. 6 repre-
sents the component of the solute flux due to diffusion, and the second term represents that due to
bulk flow or "solvent drag." Hence, Eq. 6 is a special form of Fick's law, one which includes effects
of convection and of hindrances due to the presence
of the pore wall.
Average solute flux for cylindrical pores. In gen-
eral, N will vary with radial position in the pore,
taking on different values near the pore center than
near the wall. It is desired to know the average so-
lute flux in the z direction, R5, since this is the
quantity that can be related to experimental data.
To obtain this information, we must integrate the
above equation for local solute flux over the crosssectional area of the pore. This averaging yields
= -HD,-+ WCS'
(7)
H (1 — x)2K1(X)
(8)
W = (I
(3)
If U is defined as the net solute velocity along the
axis of the pore, the hydrodynamic force takes the
form [13, 14, 19]
F5 = fK(U5 — GV)
equation
(2)
Although Eq. 2 is most appropriate for very large
solutes [17], it has also been shown to adequately
describe the behavior of molecules as small as glu-
_kTdC=FS
Combining Eqs. 1 to 5 gives the basic solute flux
—
x)2(2
—
[1
—
.]2) G(X)
(9)
AaJr0
In Eq. 7,
(10)
is the mean fluid velocity. The in-
tegration was carried out assuming no radial variation in C, except that C = 0 in a ring of fluid around
the pore wall of thickness a. This condition is required because the center of the solute sphere cannot approach the pore wall more closely than the
solute radius a. The term (1 — X)2 in Eqs. 8 and 9
arises from this "steric partitioning" effect. In addition, radial variations in K and G were neglected.
Although not strictly correct, this is made neces-
C0
z=O
-
I
Solute molecule
z=1
Fig. 1. Schematic representation of a solute molecule traversing
a membrane through a cylindrical pore. Symbols are defined in
the text. Reproduced with permission of Biophys J [221.
355
Glomerular capillary permeability to macromolecules
sary by the fact that tabulated values of these hy-
C(z) and using the latter to evaluate the derivative,
drodynamic functions are limited largely to the case
where the solute molecule moves along the centerline of the pore, as depicted in Fig. 1. There is some
dC!dz. This allows I to be expressed in terms of
the measurable solute concentrations in the solu-
evidence that this limitation may not be serious
[14], although further work is needed. As pointed
out by Bean [13], much of the literature on pore the-
tions on either side of the membrane. The following
is obtained
IT1 = wc0'cT [1
ory is based on the incorrect assumption that G =
K1, leading to errors in the hindrance factor for
a=
convection (in our notation, W). Anderson and
Quinn [14] also discuss this point in some detail.
The retarding effects of the pore wall are now expressed completely by H and W, as defined by Eqs.
8 and 9. Note that these quantities are functions only of X, the ratio of solute radius to pore radius. Fig.
2 shows a plot of H and W derived from the numerical results of Paine and Scherr [20]. It can be seen
that H and W — 1 for X — 0, indicating that solute
movement is not hindered for very small solutes or
large pores. In the other extreme, H and W —* 0 for
X — 1, indicating that transport diminishes toward
zero for large solutes or small pores, as expected. It
is significant that H declines very sharply with increasing A, falling to 0.64 for A of only 0.1. Hence, it
is rarely possible to completely neglect the hinder-
ing effects of the pore wall on diffusion. The effects on convective solute transport (expressed by
W) are not as pronounced as those on diffusion, especially at small values of A.
A more convenient form of Eq. 7 for IT can be
obtained by solving for the concentration profile
—
(Cc!Co)e]
—
1
WV1
(11)
(12)
HD0.
where C0 and C1 are the solute concentrations on
either side of the membrane, at z = 0 and z =
respectively.
The parameter a is a measure of the relative importance of convection and diffusion to solute transport through the pore. This can be seen by examin-
ing the two limiting cases, a — and a — 0. For
relatively high fluid flow rates through the pore and!
or large solutes for which D. is small, a —
Eq. 11 reduces to
lim I = WCOV
a —p
and
(13)
In this limit, solute movement is dominated by convection.
For the opposite case, that of relatively low fluid
flow rates through the pore and!or small solutes,
a — 0 and Eq. 11 reduces to
J
N1
=
(C0 — C1)
(14)
As expected, solute movement is governed solely
by diffusion for this limiting case. Thus, a can be
seen to be the ratio of quantities which give the
characteristic rates of solute transport by convection (WV) and diffusion (HDa/€).
Comparison of pore theory with the theory of irreversible thermodynamics. An alternative, widely
a
U
a
0
U
used theory for transport of water and uncharged
I
a,
U
0
solutes across membranes is that derived by Kedem
and Katchalsky [21], who based their analysis on
the principles of irreversible thermodynamics. The
equation for the solute flux, J, which they derived,
can be written as
W(X}
0
U
E
a
0
>V
0
V
= PS(CO
I>
—
CL = (C0
C1)
—
+ (1 —
a-) CLJV
C1)/ln (C0/C,)
(15)
(16)
As can be seen in Eq. 15, J is given by the sum of a
0.2
0.4
0.6
0.8
1.0
Fig. 2. Restriction factors for convection (W)and diffusion (H) as
a function of the ratio of solute radius to pore radius (X'.
diffusive term which contains the solute permeability, P, and a convective term which contains
the solute reflection coefficient, a-, and the volume
flux, J. In the convective term of Eq. 15, the con-
356
Deen et a!
centration that appears (CL) is the logarithmic mean
of C0 and Cf.
A correspondence between Eq. 15 and the solute
flux predicted by pore theory can be obtained by a
ries based on irreversible thermodynamics are given elsewhere [13, 14, 22].
Application of pore theory to the study of the
permeability properties of the glomerulus
rearrangement of Eq. 11. First, it must be recognized that N, represents the rate of solute transport
per unit cross-sectional area of pore, whereas J.
gives solute transport per unit surface area of the
entire membrane. For equivalent amounts of solute
transported, I, therefore exceeds J by a factor SI
S', where S is the total membrane area and S' is the
total cross-sectional area of all pores. A similar relationship exists between J,,. and '. Accordingly:
=
Js
Is
(17)
(18)
Using Eqs. 17 and 18, and performing some algebraic manipulations, we may recast Eq. 11 in a form
identical to that of Eq. 15:
(C —
Js =
?'
'—a
C'
-0ea
'—'1 —
l_ea
C1)
+ WC0J
(19)
C' — C'
—
a
(20)
—
5' HD,.
tration as it does in plasma water. If neither test nor
reference solutes are secreted or reabsorbed, clear-
ance determined as a fraction of that of inulin is
equivalent to the ratio of the concentration of the
test macromolecule in Bowman's space to that in
plasma water. An example of data obtained using
this approach is given in Fig. 3, which shows results
of measurements of the transport of uncharged, tn-
tiated dextrans of various sizes across glomerular
capillaries in the normal hydropenic Munich Wistar
rat [23, 25]. In this figure, fractional dextran clearance is plotted as a function of effective dextran
radius, a value of one on the ordinate corresponding
By comparing Eq. 19 with Eq. 15, it is apparent that
the following relations exist between the membrane
parameters of the pore model and those of Kedem
and Katachaisky [21]
Ps—S £
o1—W
Fractional clearance tneasurements. The pore
model of solute transport has frequently been applied to the study of macromolecule movement
across mammalian glomerular capillaries [1, 4, 7,
10, 22-33]. The fundamental quantity measured in
these studies is the ratio of the concentration of a
test macromolecule in Bowman's space to that in
plasma. Typically, the clearance of the test macromolecule (usually dextran or PVP) is compared to
that of a reference solute such as inulin, the latter
appearing in Bowman's space in the same concen-
to a dextran clearance equal to that of inulin. As
illustrated, measurable restriction to dextran transport (that is, fractional clearance ratios less than 1)
(21)
1.0
0.9
(22)
The correspondence between a- and 1 — W is not
exact because the mean concentrations defined by
Eqs. 16 and 20 are not identical.
The flux expressions obtained with these two approaches therefore differ, not in their overall form,
but rather in the significance of the coefficients
(membrane parameters) which they contain. Pore
theory is based on specific models for both the
membrane (regular porous structure) and the solute
molecule (hard sphere), which are expressed in the
quantities H and W. The coefficients in the Kedern-
Katachaisky expression, on the other hand, are
purely phenomenologic. When the pore model is
applicable, it is capable of yielding more detailed
information about the membrane being studied. Additional comparisons between pore theory and theo-
0.8
a,
0.7
Neutral dextran
C
cc
cc
0.6
0.5
C
0
C,
U-
0.4
0.3
0.2
0.1
0
18 20 22 24 26 28 30 32 34 36 38 40 42
Effective molecular radius, A
Fig. 3. Fractional clearance of tritiated, neutral dextran as a
function of effective molecular radius. Values are expressed as
1 SEM for 14 normal hydropenic rats. Data are from
means
Refs. 23 and 25.
357
Glomerular capillary permeability to macromolecules
does not occur until the effective radius exceeds ap-
proximately 18 A. For larger dextrans, fractional
clearance decreases progressively with increasing
size, approaching zero at dextran radii greater than
about 42 A. Fractional solute clearance profiles
Table 1. Membrane parameters derived from fractional clearance
values for neutral dextran in the normal Munich Wistar rata
Effective
dextran radius
20
similar to that shown in Fig. 3 for the normal hydro-
24
28
penic rat have also been reported in other experimental animals and man [7, 28-30, 32-34].
Membrane parameters derived from pore theory
and fractional clearance data. Chang et at [22]
have made a detailed theoretical analysis of the
size-selectivity of the glomerular capillary wall, but
Pore radius (r0)
A
A
32
36
40
Pore area/length (S'Ie)
cm
67
48
13
7
48
13
49
49
48
13
12
13
a Based on fractional clearance data taken from Refs. 23 and
25.
also taking into account the important and pre-
viously neglected variations in solute and volume
flux along a glomerular capillary. Application of this
model to the fractional clearance data for the Munich Wistar rat (Fig. 3) yields the results given in
Table 1. Shown are the calculated values for pore
radius, r0, for several effective dextran radii. For
values of effective dextran radius greater than —24
A, values for r0 prove to be relatively independent
of dextran size, averaging approximately 48 A. One
measure of the success of pore theory in describing
transport across membranes (isoporous) is the extent to which pore radius is found to be independent
of molecular size. Presumably all molecules "see"
the same size pores. Except for molecular radii <24
A, where pore theory appears to overestimate hindrances to dextran transport, pore theory can be
seen to correlate the data very well. Values for r0
similar to those shown for the rat in Table 1 have
been found for glomerular capillaries in dog [29, 30]
and man [28].
An additional parameter that can be derived from
fractional clearance values is the ratio of effective
pore area to pore length, S'/€ (Table 1). If pore
length is assumed to be constant, this parameter becomes a useful measure of the number of pores. S'I
t can be calculated directly from r0 and the gbmerular ultrafiltration coefficient2. Taking an aver-
age value of S'/t =
cm, we can obtain a very
rough estimate of the fraction of the capillary sur13
face area occupied by pores (S'/S). Assuming that
corresponds to the thickness of the glomerular
basement membrane, we éan estimate that
1.5 x 10 cm [35] and S' 2 x l0 cm2 (total pore
area). The total capillary surface area may be estimated as 2 x 10 cm2 [36], so that S'/S is about 0.1.
This suggests that some 10% of the capillary surface
area
is occupied by pores, an extraordinarily large
value compared to capillaries of mammalian skeletal muscle [3, 4, 10]. This difference can be rationalized, at least in part, by noting that glomerular capillaries are highly fenestrated, whereas capillaries of
skeletal muscle are of the continuous type [37].
Relationship between fractional clearance values
and the filtration rate of water. In the interpretation
of fractional clearance profiles (Fig. 3), it is important to recognize that the convective terms in Eqs. 7
and 11 cause solute and water transport across the
glomerular capillary to be coupled closely. Several
studies have shown that the Bowman's space-toplasma water concentration ratio of a macromolecule should, in general, depend inversely on the
single nephron gbomerular filtration rate (SNGFR)
[4, 7, 22, 29]. Accordingly, the filtration of macromolecules will be influenced not only by the permeability properties of the glomerular capillary wall
(such as pore radius and pore area), but also by the
other determinants of SNGFR.
We consider now in more detail the relationship
between the fractional clearance of a test macromolecule and SNGFR. For ideal test and reference
solutes (as discussed above), fractional clearance of
the test solute is equal to CB/CA, where C8 and CA
are the concentrations of the test solute in Bowman's space and systemic plasma, respectively.
The concentration in Bowman's space is determined by the ratio of the total amount of test solute
filtered per unit time to the filtration rate of water.
Accordingly,
CB
CA
—
I Jsdx
1 Jo
CA f1
0
2
According to pore theory, the glomerular ultrafiltration coef-
ficient, Kf, is related to r0 and S'/ as follows [22]: Kf = (S/I')
(r02/8) where is the viscosity of water at 370
= S
CA
ps
Jdx
fi (C —
(1
CB)dx
+WI
Jo
Jo
SNGFR
Jdx
(23)
Deen et al
358
where x is position along a glomerular capillary,
1. Factors that alter
normalized so that 0 x
SNGFR will affect primarily the convective component of solute transport, the term containing J in
the numerator of Eq. 23. If the diffusive component
of solute transport (the term involving P8) is not neg-
ligible, the denominator in Eq. 23 will tend to
change proportionately more than the numerator.
Accordingly, fractional clearance (CB/CA) will tend
to vary inversely with SNGFR.
It is instructive to consider the extreme values of
fractional clearance reached at very high or very
low values of SNGFR, since these bracket the actual physiologic situation. Because glomerular
plasma flow rate has been shown to be the principal
determinant of SNGFR [38], these variations in
SNGFR will be considered to result from changes in
plasma flow rate. For the limiting case of very high
filtration rates (high plasma flow rates), the solute
flux will be governed almost entirely by convection.
The solute flux equation is then equivalent to that
given above for large values of the parameter a (Eq.
13). In this case, the expression for fractional clearance (Eq. 23) reduces to
CA
J
==
(24)
JJvdx
The right hand equality in Eq. 24 holds approximately because at high plasma flow rate, C will vary
little along the capillary, so that C CA everywhere. Thus, for the case of high plasma flow and
filtration rates, the fractional clearance depends only on the ratio of solute radius to pore radius, X (see
curve for W in Fig. 2).
In the other extreme, when filtration rate (and
plasma flow) is small, the appropriate solute flux ex-
pression is the same as that for small values of a
(Eq. 14). The fractional clearance becomes (with
some rearrangement of Eq. 23)
CB
CA
I (C/C)dx
Jo
f1c—dxl
CA
(25)
+ (SNGFRIPSS) Jo
remaining in the capillary (C). Accordingly, CB/CA
can be at most unity, whereas the integral in Eq. 25
is no less than unity. Hence, CB/CA 1 is needed to
satisfy both requirements. This result is expected
intuitively, because equilibrium between plasma
and Bowman's space should be approached as the
ifitration rate declines toward zero. Taken together,
Eqs. 24 and 25 suggest that fractional clearance of a
given solute will take some value between W and 1,
depending on the glomerular filtration rate.
Effects of individual determinants of SNGFR on
fractional clearances. Thus far we have discussed
situations where either convection or diffusion is
dominant, leading to simple expressions for fractional clearance (Eqs. 24 and 25). In general, when
both convection and diffusion contribute appreciably to solute transport, fractional clearance of
the solute will depend in a complex manner on
SNGFR and its determinants. Recently, two analyses of the relationship between fractional clearance
and SNGFR have been reported [22, 31]. These
analyses have considered the effects of variations in
each of the individual determinants of SNGFR on
fractional clearances of macromolecules. Results
from one of these analyses, that from our laboratory
[22], are summarized in Fig. 4. Theoretical fractional clearance values for three sizes of some neutral
polymer, M, are plotted as a function of the four
determinants of SNGFR. These determinants are:
the glomerular capillary ultrafiltration coefficient,
Kf; initial glomerular plasma flow rate, QA; mean
transcapillary hydraulic pressure difference, P;
and systemic (afferent arteriolar) plasma protein
concentration, CPA. Input quantities representative
of the normal Munich-Wistar rat have been used for
these calculations [38]. M is assumed to be neither
secreted nor reabsorbed by the tubule, so that its
fractional clearance can be equated with the concentration ratio of M in Bowman's space to plasma
water (CB/CA). Panel A illustrates the effects of selective variations in QA, the approximate mean value for normal hydropenia being 75 nL'min. Because
whole kidney and single nephron glomerular filtra-
tion rates (GFR and SNGFR, respectively) have
been shown to be highly plasma-flow dependent in
the rat [38], it follows from the discussion above
The second equality in Eq. 25 follows from the fact
that SNGFR << PS in this limit. The last equality
is a consequence of the fact that for a neutral mac-
that CB/CA would be expected to vary inversely
romolecule, any "sieving" effect of the capillary
wall will reduce the solute concentration in the fil-
greater increase in the average transcapillary volume flux than solute flux (Eq. 23). Moreover, these
trate (CB) below that of arterial plasma (CA), thereby increasing the solute concentration in the plasma
decreases in CB/CA are predicted to be less marked
for very small and very large macromolecules than
with QA. As predicted, increases in QA tend to decrease CBICA, an effect due to a disproportionately
359
Glomerular capillary permeability to macromolecules
A
B
1.0
1.0
22A
0.8
C-)
0
0.6
0.8
28A
0.6
28A
0.4
0.2
0
22A
36A
0.4
0.2
36A
20 25 30 35 40 45 50
50 100 150 200 250 300
LW, mm Hg
ni/mm
28A
D
C
1.0
0'C
0
E
22A 1.0
36A
0.2
0
4.0
5.0
6.0
7.0
22A
28A
26A
0.2
.02 .04 .06 .08 .10 .12
Kf,nI/secmm Hg
C,g/lOOmi
Fig. 4. Theoretical relationships between fractional clearance of an uncharged macromolecule, GB/C4, and the determinants of
SNGFR. Each panel shows the effects of selective variations in a single quantity (QA P, CPA, or Kf). Curves are shown for three
molecular radii, with pore radius assumed to be 50 A. Calculations were performed as described by Chang et a! [22], and unless otherwise
indicated, are based on input values representative of the normal hydropenic Munich Wistar rat [38]: QA = 75 nllmin, LW = 35 mm Hg,
CPA = 5.7 g/l00 ml, and Kf = 0.08 nlI(secmm Hg). Reprinted with permission from Fed Proc 36:2614-2618, 1977
they are for a molecule of intermediate size. In-
same as those of inverse changes in iW, because
creasing QA without limit, however, is predicted to
eliminate the flow dependency of CB/CA, due to the
diminishing contribution to total solute transport of
diffusion, relative to convection [22]. For this case,
CPA and AP exert opposing influences on SNGFR.
Decreases in CPA below the normal value of —5.7 gI
100 ml can be seen to have essentially no effect on
CB/CA, whereas increases in CA above the normal
value are predicted to raise CB/CA, again the effect
being greater for the larger molecules.
Shown in Fig. 4D are the effects on CB/CA of se-
the limiting value of CRC/A is given by Eq. 24 above.
Despite the fact that CB!CA declines with increases
in QA, it is important to recognize that the absolute
solute clearance per glomerulus (SNGFR x CR/CA)
would be expected to increase, because the magnitude of the rise in SNGFR will exceed the decline in
CB/CA when QA is increased to high levels [22]. A
decline in QA will tend to have the opposite effect on
CB/CA, as shown in Fig. 4A.
As shown in Fig. 4B, selective decreases in iP
below the normal value of 35 mm Hg are expected
to raise the value of CR/CA, the effect being greater
for the larger molecules. Increases in AP above 35
mm Hg, however, are predicted to have essentially
no effect for any molecular size. This predicted insensitivity of CB/CA to increases in P occurs because, as P is raised, the convective and diffusive
components of solute transport are enhanced to an
equivalent extent, and both in proportion to the increases in the volume filtered (Eq. 23).
The effects of CPA (or systemic oncotic pressure)
on CB/CA, shown in Fig. 4C, are essentially the
lective variations in K1. As a reference state, we
have chosen a value of K1 = 0.08 nIJ(sec mm Hg),
representative of the normal Munich-Wistar rat
studied in our laboratory. As indicated, changes in
K1 are associated with directionally similar changes
in CB/CA. According to pore theory, Kf can be
shown to be a function of pore radius, r0, as well as
the ratio of pore cross-sectional area to pore length,
S'/e (see Footnote 2 and Ref. 22). Thus, selective
changes in K1 can be obtained by altering either r0,
or S'/€, or both. The variations in K1 shown in Fig.
4D are the result of variations in S 'IC alone. Larger
but directionally similar changes are obtained when
Kf is altered by changing r0. The approach taken in
Fig. 4D was chosen because in the experimental
models of glomerular injury discussed below,
changes in K1 were found to be the result of changes
in S'/€ whereas r0 remained essentially constant.
Several of the predictions contained in Fig. 4
Deen et a!
360
have been verified in experimental studies that used
in vivo clearance and micropuncture techniques.
Chang et al [23] measured fractional clearances of
tritiated neutral dextrans in Munich Wistar rats at
than those for neutral dextran for any given molecular radius. The fractional clearance of dextran sulfate with an effective radius of —--36 A averaged
0.01, a value considerably less than that for neutral
normal and elevated values of QA. By using isoncot-
dextran of the same size, and more closely ap-
ic plasma infusions to induce volume expansion,
they were able to increase QA, on average from 70
proaching that obtained for albumin. Because neu-
to 220 ni/mm. This increase in QA resulted in a fall in
tral dextrans and dextran sulfate are neither secreted nor reabsorbed by the renal tubules to any
the fractional dextran clearance profile similar to
that predicted in Fig. 4A. The opposite effect, that
of a reduction in QA, was examined recently by
measurable extent [23, 41], differences in transport
across tubule epithelia cannot account for these differences in their fractional clearances. In addition,
Bohrer et a! [25]. In these experiments, QA was reduced from a mean of 83 to 60 ni/mm by means of an
intravenous infusion of angiotensin II (All). This
fall in QA was accompanied by a concomitant increase in P, on average from 34 to 44 mm Hg. As a
result of the All infusion, fractional dextran clearances were found to increase significantly for dextrans of radii from 22 to 42 A. These increases in
because the measured glomerular pressures and
fractional dextran clearances could be accounted
for solely by the measured average fall inQA, because, as shown in Fig. 4B, increases in P above
charged component(s) of the glomerular capillary
flows in the rats given neutral dextran and those given dextran sulfate were virtually identical [23, 41],
these dissimilar clearance profiles cannot be ascribed to hemodynamic influences. This observed
restriction to the transport of polyanions, relative to
neutral macromolecules, is believed to be the result
of electrostatic repulsion by fixed, negatively
wall. This selective restriction to the filtration of cir-
culating polyanions, including dextran sulfate and
—35 mm Hg are predicted to have no effect on frac-
albumin, serves to minimize urinary losses of
tional clearance. Verification of the predictions
plasma proteins while allowing the filtration of wa-
shown in Fig. 4C is not yet possible because experiments examining the effects of variations in CPA on
ter to proceed at very high rates. This same conclusion was reached independently and simultane-
fractional clearances of macromolecules have not
been reported.
ously by Rennke et al [42].
The existence of fixed negative charges has been
demonstrated morphologically in all layers of the
normal glomerular capillary wall [43-50]. This finding is based on the observation that a variety of cationic "stains," including colloidal iron, alcian blue,
ruthenium red, and lysozyme, bind to the various
structures comprising the glomerular capillary wall,
including the surfaces of endothelial and epithelial
cells, and the basement membrane interposed between these cells.
In addition to retarding the filtration of circulating
polyanions, the highly anionic glomerular capillary
wall mighht be expected to enhance the filtration of
circulating polycations. Support for this hypothesis
was initially furnished by Rennke et al [42, 51, 52]
who found that cationic forms of ferritin and horseradish peroxidase were transported across the gbmerular capillary wall of the normal mouse and rat
Influence of molecular charge on glomnerular
permeability properties
Studies using normal animals. In addition to molecular size and the various hemodynamic determinants of SNGFR, experimental studies reveal that
molecular charge also influences the transglomerular passage of circulating macromolecules.
The chief clue that led to experimental proof for this
conclusion was the finding that the Bowman's
space-to-plasma water concentration ratio for albumin, a polyanion in physiologic solution, is —0.001
or less [39, 40]. As indicated in Fig. 3, the fractional
clearance of neutral dextran of the same effective
molecular radius, 36 A, averages '—0. 14, some two
orders of magnitude greater than that for albumin.
Thus, the filtration of the negatively charged albu-
min is restricted to a much greater extent than is
that of neutral macromolecules of the same size. To
investigate the effects of molecular charge on the
glomerular filtration of macromolecules, Chang et al
[41] examined the transglomerular transport of dextran sulfate, an anionic polymer structurally similar
to neutral dextran. These workers found that the
fractional dextran sulfate clearances were lower
to an extent considerably greater than were their
neutral counterparts. Similarly, Bohrer et al [53]
found that the fractional clearances of a highly cationic form of dextran, diethylaminoethyl dextran,
were greatly enhanced, relative to neutral dextran,
over a wide range of molecular radii studied. The
average value for fractional clearance of the cationic dextran with an effective radius equivalent to that
Glomerular capillary permeability to tnacromolecuies
361
of albumin, -—-36 A, was found to be —0.4, exceeding that of albumin and comparably sized dex-
were remarkably similar to those in NSN rats. Frac-
tran sulfate molecules by some two orders of magni-
creased, in accord with the findings of Buerkert et a!
[59] and Robson et al [60], who reported reductions
tude, and that of neutral dextran by a factor of 3.
Thus, the finding of enhanced filtration of polycations adds further support for the existence of a
functionally significant electrostatic interaction between circulating, charged macromolecules and the
negatively charged components of the glomerular
capillary wall.
Studies using animals with experimental proteinuria. The importance of electrostatic factors in regulating the filtration of charged macromolecules is
further emphasized by the results of studies on the
pathogenesis of proteinuria. In rats with a mild form
of nephrotoxic serum nephritis (NSN), Chang et al
[24] found fractional dextran clearances (molecular
radii 18 to 42 A) to be reduced, relative to values for
normal controls. Values for SNGFR and QA were
found to be unchanged from normal in the NSN
rats, the former being the result of offsetting effects,
namely increases in zP and decreases in Kf. As indicated in Fig. 4, the increase in AP should have had
little effect on the fractional clearances of dextran,
whereas the fall in Kf would be expected to reduce
neutral dextran transport, as observed experimentally. The observed decrease in the fractional clearance of dextran, however, cannot explain the proteinuria present in these NSN rats. If albumin behaved like neutral dextran, albumin filtration and
excretion would be expected to be decreased rather
than increased.
To test the explanation that molecular charge was
responsible for the proteinuria in NSN rats, Bennett
et a! [54] measured fractional clearances of negatively charged dextran sulfate. At any given molecular size, dextran sulfate fractional clearances in
NSN rats were substantially greater than those in
normal rats. These findings suggest that albuminuria is a specific consequence of the reduction in
fixed negative charges on the diseased glomerular
capillary wall, leading to less restriction of circulating negatively charged macromolecules such as albumin and dextran sulfate. In support of this possi-
bility, glomerular polyanion content has been
shown to be diminished in a variety of glomerulopathies associated with proteinuria [43, 46, 50, 55—58],
a finding also confirmed in NSN rats in the study of
Bennett et al [54].
A second experimental model of proteinuria, examined in detail by Bohrer et al [26], was induced
by administration of puromycin aminonucleoside
(PAN) to Munich Wistar rats. The results obtained
tional clearances of neutral dextran were dein the transglomerular passage of neutral PVP of
broad size distribution in PAN-treated rats, and in
children with minimal change nephrotic syndrome,
respectively. Fractional clearances of dextran sulfate, however, were shown by Bohrer et al [26] to
be increased in PAN-treated rats, relative to normal
controls. These enhanced fractional clearances of
anionic dextran sulfate, in the absence of similar
changes with neutral dextran, suggest that PAN reduces the electrostatic barrier to circulating poly-
anions. In accord with this explanation, several
studies have shown that binding of cationic stains to
anionic sites on the glomerular capillary wall is reduced appreciably in aminonucleoside treated rats,
when compared to controls [43, 46, 50, 56].
Extensions and modifications of the theory
The foregoing illustrates the success of the hydrodynamic theory of solute transport through porous
membranes in describing the size-selectivity of the
glomerular capillary wall. In combination with suitable mass balance relations for the glomerular capilanes [22], this theory is also able to accurately pre-
dict the influence of various hemodynamic per-
turbations on fractional solute clearances.
Alternative models exist, however, and other phenomena, such as the influence of molecular charge,
are not dealt with at all by the theory discussed. The
purpose of this section is to comment on some ex-
tensions and modifications of this theory which
have been or could be developed, at least some of
which may improve our quantitative understanding
of glomerular capillary permeability to macromolecules.
Possible modifications of the basic pore model
may be divided into two categories: (a) those representing the membrane pores as other than uniform,
right circular cylinders, and (b) those representing
the solute molecules as other than electrically neu-
tral, rigid spheres. These two categories are
adopted primarily for convenience of discussion
and are, of course, not mutually exclusive.
Pore configurations. The simplest variation of the
theory is obtained by postulating a distribution of
pore diameters (heteroporous membrane), an approach taken by numerous authors [3, 4, 7, 10, 11,
61—64]. This distribution, in principle, can take any
form, but in practice it must contain few enough pa-
rameters so as to retain the simplicity and predic-
362
Deen et al
tive power of the model. Two populations of pores,
with "large" and "small" radii, have been chosen
to model capillaries in several organs [61, 62]. Such
a model is particularly useful in describing capillaries where the molecular size cutoff is not as sharp as
it is in the glomerulus; a few large pores can account
for the transport of small amounts of tracer molecules which have a very large size, whereas the se-
lectivity exhibited toward smaller tracers is explained by postulating that most of the pores are of
a smaller dimension. A log-normal distribution of
pore sizes has also been used [63, 64]. Lambert et a!
[64] have used this approach in the analysis of fractional PVP clearances in the dog. Variations in pore
radius with position along a pore have also been discussed [65] but seem to have found little application
to date in capillary permeability studies. Several authors have used a model where the pore takes the
form of a "slit," a narrow channel with flat, parallel
walls [4, 7, 10, 66-69]. Renkin and Gilmore [7]
found that filtration data for proteins across mammalian glomerular capillaries could be fitted equally
well with a model using uniform cylindrical pores or
a model using slits with a distributed population of
half-widths. Uniform slits were found to be less
adequate. This conclusion must be regarded as only
tentative in that these authors assumed a correspon-
dence between the restriction factors for convection and diffusion which is apparently incorrect
(analogous to the assumption for cylindrical pores
that K-1 = G). Unfortunately, available hydrodynamic results [70] allow estimation of the restriction factor for diffusion only. Using data on macromolecule transport in extrarenal capillaries, several
authors have endeavored to establish a correspon-
dence between the width of gaps between endothelial cells and the slit dimensions inferred from
permeation rates of macromolecules, with some degree of success [6, 67—69].
Much of the morphologic evidence now available
points to the glomerular basement membrane as the
primary barrier to neutral macromolecules [8]. A
model which may provide a fairly realistic represen-
tation of the basement membrane is one based on
the theory of molecular sieving in polymeric gels.
As discussed by Renkin and Gilmore [7], this approach considers the barrier to consist of a complex
network of fibers through which solutes and water
move in tortuous paths. Ogston [71] and Ogston,
Preston, and Wells [72] have developed theoretical
expressions for steric partitioning and restricted diffusion in such a system. Diffusion was modeled as a
stochastic process in which the solute molecule has
a certain probability of moving from one "hole" in
the polymer network to another [72]. Their results
indicate that the transport rate of a globular molecule should decrease exponentially with the molecular diameter [72]. As was the case for the slit model, satisfactory application of the fiber network ap-
proach to macromolecule movement across the
glomerular capillary wall awaits the development of
suitable expressions for the convective component
of solute transport.
Molecular properties. The only characteristic of
the permeating macromolecule used in pore theory,
as discussed above, is its effective molecular radius.
This effective radius is defined most often by using
diffusivity data and the Stokes-Einstein equation
(Eq. 2). As discussed above, there is now abundant
experimental evidence [1, 9, 25, 26,41,42, 5 1—54] to
indicate that the molecular charge is also a significant determinant of the transglomerular movement
of macromolecules. In modifying the pore model to
account for interactions between a charged macromolecule and a membrane containing fixed charges,
it must be recognized that at least two additional
phenomena may play a role. First, there is an electrostatic partitioning effect which may either contribute to or oppose that due purely to molecular
size (the latter "steric" effect is represented by the
term (1 — A)2 in Eqs. 8 and 9). If, for example, the
pore walls contain fixed negative charges, polyanions will tend to be selectively excluded from,
and polycations attracted to, the pore interior, relative to uncharged molecules of similar size. Second,
if there is a significant gradient in electrical potential
along the pore, migration of the charged molecule in
this field will contribute to its flux, in addition to
diffusion and convection. Theoretical analyses are
available to describe the transport of "small" ions
(A << 1) through a charged, porus membrane [73,
74]. For spherical molecules of appreciable size,
Brenner and Gaydos [15] included in their hydrodynamic analysis a potential energy term that could
be taken to represent the electrostatic interaction
between a macromolecular ion and a charged pore
wall, but numerical results for this case were not
given.
in all of the treatments discussed above, the
macromolecular solute has been regarded as a hard
sphere. Although this molecular model has been
used with some success, it is likely that most macromolecules could be better represented as ellipsoids or as relatively open, random coils [17]. Giddings et al [75] have derived expressions for the
steric partitioning of rigid particles of various
Glomerular capillary permeability to macromolecules
shapes, and Casassa [76] has treated this problem
for random coil macromolecules. DeGennes [77]
has developed a stochastic model for the diffusion
of linear, flexible macromolecules through strongly
cross-linked polymeric gels. It is concluded that the
diffusion coefficient in the gel, D, should vary as
M2 [77], where M is the molecular weight of the
solute, suggesting that DID,, should vary approximately as a-3 [17]. Detailed comparison with the
hard sphere model, and application to capillary permeability data, will require further development of
the theory.
It is our view that for describing transport of mac-
romolecules across the glomerular capillary wall,
the most significant future developments of the theory will involve molecular properties, the foremost
of these being molecular charge. Whether molecular shape or flexibility is a significant factor in gb-
merular filtration, as it has been found to be for
sedimentation and diffusion through polymer networks in vitro [78-80], is not yet clear. Additional
experimental studies are needed to ascertain the importance of including these latter effects in the theory for glomerular capillaries.
Summary
Developments in the hydrodynamic theory of sol-
ute transport through porous membranes are reviewed, with emphasis on their application to mac-
romolecule movement across capillary walls. A
model that treats the capillary wall as a barrier containing uniform cylindrical pores, and permeating
solutes as hard spheres, is shown to be successful in
describing the size-selectivity of the glomerulus. Influences of various hemodynamic perturbations on
solute transport are also accounted for by this approach. Possible extensions and modification of the
theory, to account for the influence of molecular
charge and other factors on glomerular permeability
properties, are discussed.
Acknowledgments
This work was supported largely by grants from
the National Institutes of Health (AM 20368 and
AM 19467) and the Whitaker Health Sciences Fund.
Reprint requests to Dr. William M. Deen, Department of
Chemical Engineering, Room 66-544, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA
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