Department of Chemical and Biological Engineering, Tufts
University, Medford, Massachusetts 02155
In this study, we have extended a mathematical model of
microvascular exchange in the renal medulla to elucidate the mechanisms by which plasma proteins are transported between vasa recta and the
interstitium. In contrast with other work, a distinction was made
between the paracellular pathway and the transcellular route (i.e.,
water channels) in descending vasa recta (DVR). Our model first
indicates that concentration polarization on the interstitial side of
vasa recta has a negligible effect on medullary function. Our results
also suggest that, whereas proteins are cleared from the interstitium
by convection, both diffusion and convection play a role in carrying
proteins to the interstitium. In those regions where transcapillary
oncotic pressure gradients favor volume influx through the paracellular
pathway in DVR, diffusion is the only means by which proteins can
penetrate the interstitium. Whether the source of interstitial protein
is DVR or ascending vasa recta depends on medullary depth, vasa recta
permeability to proteins, and vasa recta reflection coefficients to
small solutes and proteins. Finally, our model predicts significant
axial protein gradients in the renal medullary interstitium.
microcirculation; medullary interstitium; urine concentration; mathematical model
 |
INTRODUCTION |
SEVERAL STUDIES HAVE
SHOWN that albumin is present in the medullary interstitium in
significant concentrations (11, 12, 15). The mechanisms by
which this extravascular pool of albumin is generated and maintained
have long remained uncertain. Radiolabeled albumin injected in the
medullary interstitium is rapidly cleared (12), but it is
unclear how. Lymphatics are absent in the inner medulla and sparse in
the outer medulla, and the possibility of clearance of albumin by
drainage via prelymphatic channels is not supported by experimental
evidence (12).
The most likely mechanism is that of protein clearance by the
microcirculation itself. Pallone (14) suggested descending vasa recta (DVR) as the source of interstitial proteins and postulated that accumulation of albumin in the interstitium results from convective transport processes. Because the reflection coefficient of
DVR to albumin is higher than that of ascending vasa recta (AVR), it is
possible in principle to maintain steady fluxes of albumin from DVR
through the interstitium to AVR (12). Wang and Michel
(25) recently developed a model of microvascular exchange
of fluid, plasma proteins, and small solutes among DVR, AVR, and the
medullary interstitial fluid (ISF) to examine this hypothesis. Their
results suggest that convection may indeed be the main mechanism by
which plasma proteins are transported from DVR to AVR via the interstitium.
Their model, however, does not distinguish between two parallel
transport pathways in DVR that differ significantly. The first one
consists of aquaporin-1 (AQP1) water channels, which are present in DVR
only and are impermeable to all solutes; transcellular volume fluxes
across water channels cannot, therefore, carry albumin by solvent drag
into the ISF. The second route, the paracellular pathway, appears to
have a reflection coefficient to small solutes that is close to zero
(16) and to favor water transport from the ISF toward the
lumen in most parts of the medulla (4). In those
regions where the paracellular flux is directed toward the lumen, the
convective transport of albumin from DVR to the interstitium is not
possible, even though there is overall volume efflux from DVR. The
objective of this work was to reexamine the mechanisms of albumin
exchange with a model that accounts for the presence of two separate
transport pathways in DVR as well as for concentration polarization.
We first evaluated albumin concentration differences between the bulk
interstitium and the interstitial side of vasa recta walls (i.e.,
immediately adjacent to the capillaries) to calculate accurately the
driving forces for transcapillary transport. We then used conservation
equations in the interstitium to determine both interstitial protein
concentrations and the processes by which proteins are transported
across vasa recta (i.e., diffusion and/or convection from AVR to DVR or
vice versa). Because the latter mechanisms appear to vary according to
the values of vasa recta permeability to proteins and reflection
coefficient to small solutes and macromolecules, corresponding
parameter sensitivity studies are conducted.
Glossary
Aim |
Cross-sectional area of inner medulla
|
Aint |
Cross-sectional area of inner medullary interstitium
|
AVR |
Ascending vasa recta
|
C , C |
Interstitial albumin concentration in bulk and at the vasa recta wall,
respectively
|
C , C ,
C |
Concentration of solute i in plasma, red blood cells, and
interstitium, respectively
|
Chb, C |
Molar and molal concentrations of hemoglobin in red blood cells,
respectively
|
D |
Vessel diameter
|
Di |
Diffusivity of solute i
|
DVR |
Descending vasa recta
|
f |
Fractional volume of distribution of urea in red blood cells
|
fp |
Fraction of capillary surface occcupied by pores
|
FVR |
Fraction of the inner medullary cross-sectional area occupied by vasa
recta
|
Hi |
Hydrodynamic hindrance factor for diffusive transport of solute
i
|
IM |
Inner medulla
|
INT |
Interstitium
|
ISF |
Interstitial fluid
|
Ji |
Paracellular molar flux of solute i across capillaries
|
Jv, J |
Volume fluxes across capillaries and red blood cell membranes,
respectively
|
Jvp, Jvt |
Paracellular and transcellular volume fluxes across capillaries,
respectively
|
Juc, J |
Carrier-mediated transcapillary molar flux of urea, and molar flux of
urea across red blood cell membranes
|
l |
Capillary pore length
|
L |
Length of renal medulla
|
Lim |
Length of inner medulla
|
Lp, Lt |
Hydraulic conductivities of paracellular and transcellular pathways,
respectively
|
LR |
Hydraulic conductivity of red blood cell membrane
|
N |
Number of vasa recta
|
Nv |
AVR-to-DVR number ratio
|
OM |
Outer medulla
|
P, PI |
Hydraulic pressure in plasma and interstitium, respectively
|
Pe |
Peclet number
|
Pi |
Permeability of capillary wall to solute i
|
Puc, Pur |
Permeability of urea transporter in capillary wall and red blood cell
membrane, respectively
|
QB |
Blood flow rate
|
QP |
Plasma flow rate
|
QR |
Red blood cell flow rate
|
r |
Radius
|
rp |
Capillary pore radius
|
ri |
Radius of solute i
|
RBC |
Red blood cell
|
v |
Fluid velocity in interstitium
|
Xim |
Dimensionless inner medullary coordinate, based on length of inner
medulla
|
W |
Half-width of slit pore
|
|
Solute distribution coefficient
|
|
Red blood cell-to-vessel surface area ratio
|
i |
Activity coefficient of solute i
|
i |
Oncotic pressure due to solute i
|
i |
Reflection coefficient of the paracellular pathway to solute
i
|
v, Na, u |
Generation rate of volume, sodium, and urea, respectively, per unit
area of interstitium
|
a |
Albumin
|
hb |
Hemoglobin
|
Na |
Sodium
|
pr |
Plasma protein
|
ss |
Small solute (sodium and urea)
|
u |
Urea
|
 |
METHODS |
Mathematical Model
The fundamental assumptions of our model of renal medullary
microvascular transport have been extensively described earlier (4-6). We consider only those vasa recta that are
destined for the inner medulla (IM), i.e., those that lie in the center
of the vascular bundles and do not perfuse the capillary plexus of the
outer medulla (OM). The deposition of NaCl, urea, and water into the IM
interstitium from the loops of Henle and the collecting ducts is
simulated with generation rates that undergo spatial variation within
the IM interstitium. In the vascular bundles, exchanges occur only
between vasa recta and the interstitium, so that generation rates are
taken to be zero. Plasma and red blood cells (RBC) are considered as
two separate compartments. Two transcellular pathways are present in
DVR only: AQP1 water channels and urea transporters.
Conservation and transport equations in vasa recta plasma.
If x is the axial coordinate along the corticomedullary
axis, changes in the plasma flow rate (QP) in DVR and AVR
at steady state are given by the following equation, based on mass
conservation
|
(1)
|
where Jv and
J
are the volume fluxes (per
unit membrane area) across the capillary wall and the RBC membrane,
respectively,
is the cell-to-vessel surface area ratio, N denotes the number of vessels and D their
diameter, and + and
apply to AVR and DVR, respectively.
Jv is the sum of two contributions, the
paracellular (Jvp) and transcellular
(Jvt) volume fluxes, which are given by
|
(2a)
|
|
(2b)
|
where Lp and Lt represent the hydraulic
conductivities of the paracellular and transcellular pathways,
respectively,
P is the transcapillary hydraulic pressure difference,

a and 
pr are the transcapillary
oncotic pressure differences due to albumin and all plasma proteins,
respectively, and
a is the reflection coefficient of the
paracellular pathway to albumin. The plasma and interstitial
concentrations of solute i are denoted by
C
and C
,
respectively;
i is the activity coefficient
of i, and
i is the reflection coefficient of the paracellular pathway to i. Note that
reflection coefficients are taken to be one for the solute-impermeable
transcellular pathway and that Jvt is zero
across AVR, where no AQP1 has been found. The oncotic pressures due to
albumin and all plasma protein are calculated as, respectively
|
(3)
|
where Ca and Cpr are the albumin and
total protein concentration, respectively, in grams per deciliter.
Conservation of solutes to which RBCs are impermeable, such as
sodium, albumin, and other proteins, can be written as
|
(4)
|
where Ji is the (paracellular)
molar flux of solute i (per unit membrane area) from plasma
to interstitium. With the assumption of negligible loss of protein
other than albumin to the interstitium, the flux of nonalbumin protein
is taken to be zero. Conservation of urea, which is exchanged across
the RBC membrane, yields
|
(5)
|
where Ju and Juc
are the paracellular and carrier-mediated transcapillary molar fluxes
of urea, respectively, and J
is the molar
flux of urea across RBCs. The paracellular flux of solute i
(i = sodium, albumin, urea) across capillary walls can be written as (2)
|
(6)
|
where Pi is the permeability of the
vessel to solute i, and the Peclet number, Pe, is a measure
of the importance of convection relative to diffusion. The
carrier-mediated trancapillary and transmembrane fluxes of urea,
respectively, are given by
|
(7)
|
where Puc and Pur
are the permeabilities of the urea transporter in the capillary wall
and in the RBC membrane, respectively, and C
is the
RBC concentration of urea.
Conservation and transport equations in RBCs.
Conservation of mass in RBCs can be expressed as
|
(8)
|
where QR is the RBC flow rate. If we assume that
there is no hydraulic pressure difference across the RBC membrane,
J
is given by
|
(9)
|
where LR is the RBC membrane hydraulic conductivity,
pr and
hb are the oncotic pressures due
to plasma proteins and to hemoglobin in the cells, respectively, and
C
denotes the RBC concentration of solute
i. As described in Edwards and Pallone (5), the
oncotic pressure due to hemoglobin in RBCs is calculated as
|
(10)
|
where Chb and C
are the molar and
molal RBC concentrations of hemoglobin, respectively, and
= 0.75 mg/l is the partial specific volume of hemoglobin.
Conservation of hemoglobin and other nonurea solutes (e.g., potassium,
magnesium, and associated intracellular anions) in RBCs yields
|
(11)
|
The RBC concentration of urea can be obtained on the basis of
the conservation equation
|
(12)
|
where f is the fractional volume of distribution of urea within
RBCs, taken to be 0.86.
Conservation equations in interstitium.
As described in Edwards et al. (4), the deposition of
NaCl, urea, and water into the medullary interstitium from the loops of
Henle and collecting ducts is simulated with generation rates that
undergo spatial variation within the IM interstitium. The interstitial
hydraulic pressure (PI) and small solute concentrations
(C
and C
) are determined by
considering that, at any location along the corticomedullary axis, the
sum of the fluxes from DVR and AVR, weighted according to their
respective surface area, must be equal and opposite to the rate of
generation in the interstitium
|
(13a)
|
|
(13b)
|
|
(13c)
|
where Aint is the cross-sectional area of
the medullary interstitium (in cm2), and
v,
Na, and
u are the local generation rates
of volume, sodium, and urea, respectively, per unit area of
interstitium. The latter three terms are taken to be zero in the OM,
where in the vascular bundles the exchange of water, sodium, and urea
can occur only between vasa recta and interstitium.
The cross-sectional area of the IM interstitium is calculated on the
basis of that of the inner medulla,
Aim(6)
|
(14)
|
where Xim is the dimensionless IM axial
coordinate based on the length of the IM.
Concentration polarization: annular space model.
As water is reabsorbed from the interstitium into AVR, the accumulation
of albumin near the AVR wall on the interstitial side, a phenomenon
known as concentration polarization, reduces the transcapillary oncotic
pressure difference and therefore decreases the driving force for water
reabsorption. Conversely, during volume efflux from DVR, the
concentration of albumin on the interstitial side of the DVR wall will
be lower than that averaged radially over the interstitium; i.e.,
reverse polarization will occur, leading to a decreased rate of fluid
filtration. Polarization (or its reverse) is not expected to be
significant within vasa recta, due to the presence of RBCs, which
create a circulatory flow that homogenizes plasma concentrations
(3).
Transport of fluid and solutes in the interstitium is predominantly in
the radial direction (i.e., normal to the corticomedullary axis); not
only do the orientation and density of lipid-laden IM interstitial
cells appear to hinder axial diffusion (10), but the
length of the renal medulla is about a thousand times the distance
between adjacent vasa recta. In our analysis, interstitial transport in
the axial direction is therefore deemed negligible compared with radial
diffusion and radial convection.
To assess the effects of concentration polarization at a given depth in
the medulla, we used a one-dimensional, cylindrical model of
polarization in the radial direction, following the approach of Lee
(9). The objective was to estimate the albumin
concentration difference between the bulk interstitium and in the
interstitium immediately adjacent to the capillary membrane. With the
assumption that each vas rectum can be represented as a cylinder
embedded in a coaxial, conic interstitium, as shown in Fig.
1, conservation of albumin in the
annular space (i.e., in the interstitium) is written as
|
(15)
|
where Ja, Ca, and
Da are the flux, concentration, and diffusivity
of albumin in the interstitium, respectively, and v is the
fluid velocity. Conservation of water implies that
|
(16)
|
where rA is the radius of the inner
cylinder (i.e., of DVR or AVR) and vA is the
(known) velocity at that boundary. Equations 15 and 16 can be combined to yield
|
(17)
|
Given the bulk interstitial albumin concentration,
C
, the following boundary conditions have to be satisfied
|
(18a)
|
|
(18b)
|
where rB is the outer radius, and
Ja, the (specified) transcapillary flux of
albumin, is constant in the r-direction in the annular space
(see Eq. 15). The differential equation Eq. 17,
coupled with the boundary conditions (Eq. 18, a and
b), has an explicit solution
|
(19)
|
By substituting r = rA
into Eq. 19, the albumin concentration at the vasa recta
wall, C
, can be determined.

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|
Fig. 1.
Idealized representation of vasa recta and interstitium,
used to evaluate the effects of concentration polarization at capillary
walls.
|
|
If diffusion is negligible across the capillary wall, the
transcapillary flux of albumin Ja is directly
proportional to the paracellular water flux (see Eq. 24
below in the limit when Pe
). In AVR where there are no
water channels, the paracellular water flux (per unit membrane area) is
equal to vA, so that
Ja = (1
a) · C
· vA
in that limit and the wall-to-bulk albumin concentration ratio is given
by
|
(20)
|
where
a is the reflection coefficient of vasa
recta to albumin. When water flows from the interstitium toward AVR,
Pea is negative, and the concentration ratio is >1. The
interstitial oncotic pressure at the AVR wall is, therefore, greater
than that which would be calculated based on the bulk interstitial
concentration. Polarization thus reduces the driving force for volume
flux toward AVR, thereby limiting fluid uptake.
In DVR, even when convection is highly dominant,
Ja is not directly proportional to
vA, because there is a transcellular component to the water flux and water channels are impermeable to albumin. Equation 19, with r = rA, cannot therefore be reduced to Eq. 20. It can be shown, however, that C
is less than C
when water flows out of DVR. The reduction in
the interstitial oncotic pressure then serves to limit volume efflux.
Interstitial space size calculations.
Because both the number of AVR and their diameters are greater than
those of DVR, the average distance between the bulk of the interstitium
and the capillary wall (i.e., rB
rA in the model described above) is different
for AVR and DVR. At every depth and for each type of vessel,
rB is approximated using the equation
|
(21)
|
Because we consider only those vasa recta that are destined for
the IM, the number of vasa recta in the OM remains constant. In the IM,
the number of DVR and AVR is given by (4)
|
(22)
|
where FVR, the fraction of the inner medullary
cross-sectional area occupied by vasa recta, and
Nv, the AVR-to-DVR number ratio, are taken to be
constant and equal to 0.3 and 2.25, respectively (26).
With those assumptions,
rB/rA varies between 1.3 at the OM-IM junction and 2.4 at the papillary tip for DVR and between 1.1 and 1.5 at the same boundaries for AVR. (Note that, even though rA is constant and Aint
decreases along the corticomedullary axis, the number N of
vessels decreases more rapidly, which is why the ratio
rB/rA increases.) In the
OM, we assume that rB/rA
remains constant and equal to its value at the OM-IM junction. In
addition, in those simulations of concentration polarization, the
diffusivity of albumin in the interstitium, Da, is
estimated to be ten times smaller than that in water (8).
Albumin interstitial concentration.
In our previous approaches (4, 6), the interstitial
concentration of albumin was taken to be fixed and constant along the
corticomedullary axis. The issue of albumin polarization having been
addressed, the transport of albumin may now be modeled more rigorously.
An interstitial mass conservation equation can be written to determine
the concentration of albumin in the interstitium. If there are no
interstitial sources or sinks of albumin (such as transcytosis,
proteolysis, or lymphatic drainage), the amount of albumin carried from
DVR and AVR should counterbalance
|
(23)
|
where Ja is the transcapillary flux of
albumin. Implicit in Eq. 23 is the assumption that axial
transport in the interstitium is negligible, as discussed earlier. The
albumin flux can be written as
|
(24)
|
where Pe, the Peclet number, is given by Eq. 6. Note
that Eq. 24 includes the interstitial concentration of
albumin immediately adjacent to the capillary wall,
C
, which is related to the bulk interstitial
concentration, C
, as described in Concentration
Polarization. At every depth along the corticomedullary axis, as
flow rates and concentrations in plasma are determined, Eq. 23, which relates albumin concentrations in vasa recta to
C
, is solved to determine interstitial albumin concentrations.
Parameter selection.
Parameter values for our model are given in Table
1. The hydraulic pressure P is assumed to
remain constant in AVR and IMDVR, with fixed values of 7.8 and 9.2 mmHg, respectively. In OMDVR, P is assumed to decrease linearly from 20 to 9.2 mmHg (5). The fraction of the filtered load
recovered by IM vasa recta for water, NaCl, and urea is taken as 1, 1, and 40%, respectively; the filtered load is calculated as described in
Edwards et al. (4), based on the values of
corticomedullary DVR concentrations and whole kidney glomerular
filtration rate (GFR) that are given in Table 1. In the baseline case,
the interstitial area-weighted generation rate of water decreases
linearly between the OM-IM junction and the papillary tip, whereas
those of sodium and urea increase linearly and exponentially,
respectively (4).
Permeability of vasa recta to albumin.
The permeability of AVR to albumin is <10
5 cm/s and is
therefore too low to be measured by present methods (14).
If we assume that the paracellular pathway consists of parallel pores
of uniform size, estimates of pore dimensions can be obtained using
pore theory and other available measurements, and the permeability of
vasa recta to albumin may be calculated using pore theory as well.
Calculations are made for both cylindrical and slit pores.
For cylindrical pores, the (osmotic) reflection coefficient in the
absence of electrical interactions between the solute and pore wall is
given by (2)
|
(25)
|
where rs and rp
are the radii of the solute and pore, respectively, and
is the
distribution coefficient, i.e., the ratio of the average intrapore
concentration to that in bulk solution at equilibrium. The reflection
coefficient of DVR to albumin (rs = 3.5 nm)
has been measured as 0.89 (23), yielding 4.6 nm as the
pore radius. In AVR, where the average value of
a is
~0.70 (12, 14), the pore radius is calculated to be 5.9 nm. Even if there are electrical interactions between the negatively
charged albumin and the endothelial glycocalyx, those values should
represent reasonable order-of-magnitude estimates of
rp.
For slit pores, the osmotic reflection coefficient can be written as
(2)
|
(26)
|
where W, the half-width of the slit, is calculated to
be 3.8 nm in DVR and 4.4 nm in AVR, following the procedure described immediately above.
The permeability Pi of the porous pathway to a
given solute i can be written as
|
(27)
|
where Di is the solute diffusivity in
dilute bulk solution, the coefficient Hi
expresses the hydrodynamic hindrance to diffusive solute transport,
fp is the fraction of capillary surface occupied by pores,
and l is the pore length. The permeability of the
paracellular pathway to urea (or sodium) being known, the permeability
to albumin can then be calculated as
|
(28)
|
where the subscripts a and u refer to albumin and urea
(rs = 0.28 nm), respectively. An expression
for Hi for uncharged solutes in cylindrical
pores is given by Bungay and Brenner (1) as a function of
= rs/rp
|
(29)
|
In slit pores, with
= rs/W, Hi can
be determined as (2)
|
(30)
|
The diffusivity of albumin in dilute bulk solution is calculated
using the Stokes-Einstein equation, yielding 9.3 × 10
7 cm2/s, and that of urea is estimated as
2.0 × 10
5 cm2/s on the basis of the
Wilke-Chang correlation for small solutes (20). In this
manner, the permeability to albumin of DVR and AVR is calculated to be
5.6 × 10
8 and 9.9 × 10
7 cm/s,
respectively, assuming that the pores are cylindrical and 1.3 × 10
6 and 5.4 × 10
6 cm/s, respectively,
in the case of slit pores. A range of parameter values for
Pa must therefore be explored.
Numerical Methods
In the microcirculation, nine variables must be determined along
both DVR and AVR: plasma flow rate, RBC flow rate, albumin plasma
concentration, other protein plasma concentration, sodium plasma
concentration, urea plasma concentration, urea RBC concentration, hemoglobin RBC concentration, and the RBC concentration of other nonurea solutes. Equations 1, 4, 5, 8, 11, and 12
form the corresponding set of ordinary differential equations
(ODEs) that need to be integrated to determine the profiles of these
variables. The initial values in DVR at the corticomedullary junction
are specified (see Table 1). At the papillary tip, i.e., at the
entrance to AVR, DVR and AVR values have to match.
The set of ODEs expressing mass conservation in DVR and AVR is highly
coupled. At each point along the corticomedullary axis, evaluating
fluxes across DVR requires that values in the interstitium and in AVR
be known, and vice versa. However, the ODEs cannot be simply integrated
simultaneously along DVR and AVR, because boundary values for flow
rates and concentrations in AVR at the papillary tip are not known
until differential equations for DVR have been integrated along the
entire axis. Hence, we used the following approach.
An initial guess was made for the profiles in AVR of the nine variables
along the entire corticomedullary axis. The set of ODEs (Eqs. 1,
4, 5, 8, 11, and 12) was then numerically integrated along DVR; at each step along the corticomedullary axis, algebraic equations were solved to determine the interstitial hydraulic pressure
as well as sodium, urea, and albumin interstitial concentrations (Eqs. 13, a-c, and 23). Once papillary tip
values were obtained, the same set of differential equations was
numerically integrated back up along AVR, and AVR flow rates and
concentration values were updated. This process was iterated until the
normalized difference between the current and previous estimates of
each variable in AVR at any x was <10
5. Tests
demonstrating mass conservation are described in the
APPENDIX.
ODEs were integrated along vasa recta by use of Gear's method, which
is a self-adaptive, multistep, predictor-corrector method for stiff
ODEs. At each value of x, the system of three or four nonlinear algebraic equations (Eqs. 13, a-c, and
23) was solved using a modified Powell hybrid method. This
algorithm, which is a variation of Newton's method, uses finite
difference approximations to the Jacobian and avoids large step sizes
or increasing residuals (13). Simulations were performed
on an Alpha PC64 workstation. Convergence was typically achieved in
5 h.
When the effects of concentration polarization are assessed, the
incorporation of Eq. 19 into the simulations of medullary microvascular transport is complicated by the fact that
vA, and hence Pea and
Ja, are themselves functions of
C
through the interstitial oncotic pressure term in
the paracellular and transcellular volume fluxes (Eqs. 2 and 3). At each integration step along DVR and AVR, we
first calculated the volume fluxes on the basis of the bulk
interstitial concentration of albumin. The albumin interstitial
concentration immediately adjacent to the walls was then determined
using Eq. 19, and the volume fluxes were calculated anew on
the basis of this value. The latter two steps were iterated until
convergence was achieved.
 |
RESULTS |
We first examined the extent to which concentration polarization
in the medullary interstitium affects flow rates and concentration profiles in vasa recta; for simplicity, the bulk interstitial concentration of albumin was assumed to be constant and known in those
calculations. We then eliminated that hypothesis and used instead
conservation equations in the interstitium to determine protein
interstitial concentrations and the mechanisms by which proteins are
exchanged between vasa recta and the interstitium. In the absence of
measurements for certain capillary wall permeabilities and reflection
coefficients, parameter sensitivity studies were performed in which a
range of possible values was explored.
Albumin Concentration Polarization
The AVR-to-interstitium albumin concentration difference is a
major determinant of fluid reabsorption into the microcirculation. To
evaluate this driving force, the effects of concentration polarization must be taken into consideration, because polarization significantly reduces the oncotic pressure gradient across AVR walls. We had previously postulated that the accumulation of albumin on the interstitial side of the AVR wall is high enough to eliminate the
oncotic pressure difference due to albumin (4). The more rigorous approach to concentration polarization developed here allowed
us to test this hypothesis as well as to examine the effects of reverse
polarization at DVR walls. During volume efflux from DVR, interstitial
concentrations adjacent to the membrane are smaller than those in the
bulk, thereby increasing oncotic pressure gradients across DVR walls.
Results based on the present model of polarization were compared with
those obtained in two cases: 1) concentration polarization and its reverse are negligible (the "no-polarization" hypothesis); and 2) the accumulation of albumin on the interstitial side
of the AVR wall is so significant that albumin oncotic pressure
differences across that barrier vanish entirely (the
no-AVR-
a hypothesis). The bulk interstitial
concentration of albumin, C
, was kept fixed, either
at 3.4 g/dl, as measured by Pallone (15) or at 1 g/dl, as
reported by MacPhee and Michel (12). To maintain high
osmolalities at the papillary tip, we varied only the spatial distributions of the interstitial area-weighted generation rate of
urea. As described in the previous section, the set of differential equations (Eqs. 1, 4, 5, 8, 11, and 12) was
numerically integrated along vasa recta to obtain flow rates and
concentration profiles in plasma; at each step, the algebraic equations
(Eq. 13, a-c) had to be solved to yield interstitial
values. When polarization is accounted for, the albumin interstitial
concentration at the wall was related to that in the bulk through
Eq. 19. Results are shown in Table
2.
Reverse polarization at the DVR wall increases the transcapillary
albumin oncotic pressure difference, thereby reducing water efflux from
DVR; the rise in sodium and urea concentrations along the
corticomedullary axis is therefore less accentuated. Polarization at
the AVR wall has the same effect: a reduced 
a limits
water influx into AVR and, hence, efflux from DVR, since the
interstitial water balance must be maintained; sodium and urea
concentrations thus remain lower. Consequently, as shown in Table 2,
the osmolality at the papillary tip is always overestimated when
concentration polarization and its reverse are neglected and
systematically underpredicted if 
a across AVR walls
is omitted.
In the former case, however, the error remains small, <2%, and the
lower the C
, the smaller the error, because
differences between interstitial concentrations in the bulk and near
the capillary walls then have less of an effect on oncotic pressure
gradients (see Eqs. 2 and 3). If the assumption that 
a can be neglected across AVR walls is employed
rather than our present approach, the discrepancy can be as high as
15%, suggesting that the no-AVR-
a hypothesis, which
we used previously (4), is an overly simplifying assumption.
Given the uncertainty in model geometry and in parameter values such as
generation rates and albumin permeability, errors on the order of 2%
are not very significant. The annular space model developed here,
although based on an idealized representation of the medulla, therefore
suggests that the effects of concentration polarization in the renal
medulla can be neglected, as they will be in the remainder of this study.
Transport Mechanisms of Plasma Proteins Across Vasa Recta
Paracellular and transcellular volume fluxes.
AQP1 water channels in DVR are impermeable to all solutes
(18). Because small solutes such as sodium and urea are
more concentrated in the medullary interstitium than in DVR, osmotic
pressure gradients drive water from DVR toward the interstitium through
this transcellular pathway (i.e., Jvt > 0). The reflection coefficient to small solutes of the paracellular
pathway (
ss), however, is close or equal to zero
(16), so that osmotic pressure gradients have little to no
effect on Jvp. Transcapillary protein
concentration differences are therefore the dominant driving force
across that route, and water moves in the opposite direction through
the paracellular pathway, i.e., from the interstitium toward DVR
(Eq. 2, a and b).
Shown in Fig. 2 are the paracellular and
transcellular water fluxes across DVR and AVR when albumin interstitial
concentration is specified and with the assumption that
ss is zero. Generation rates are those of the baseline
case, parameter values are given in Table 1, and C
is
fixed at 3.4 g/dl, as measured by Pallone (15). As
illustrated in Fig. 2, the paracellular flux of water across DVR is
positive only near the corticomedullary junction; it is negative, i.e.,
directed toward the capillary lumen, throughout most of the medulla.
With a smaller interstitial albumin concentration, in the range of 1 g/dl as measured by MacPhee and Michel (12), albumin
concentration gradients across DVR walls are even larger, resulting in
more water influx through the paracellular route.

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Fig. 2.
Transcapillary volume fluxes across descending (DVR) and
ascending vasa recta (AVR) based on the circumference of all vessels
(i.e., JvN D, as in
Eq. 1), as a function of position along the corticomedullary
axis, x. L represents the total length of the
medulla. The junction between the outer medulla (OM) and the inner
medulla (IM) corresponds to x/L = 0.24. The sharp bends
at this junction are due to anatomical changes and the sudden
reabsorption of water and solutes from the loops of Henle and the
collecting duct in the IM. The interstitial concentration of albumin is
fixed at 3.4 g/dl. The permeability to albumin of DVR
(P ) and AVR
(P ) is taken to be 1 × 10 7 and 1 × 10 6 cm/s, respectively,
and the reflection coefficient of the paracellular pathways to small
solutes ( ss) is zero. Because the paracellular flux of
volume across DVR is directed mostly toward the capillary lumen, it is
unlikely that albumin is carried to the interstitium only by solvent
drag from DVR.
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Because there can be no transport of albumin across AQP1, solvent drag
is effective only across paracellular routes and will therefore carry
albumin away from the interstitium in most of the medulla and toward
both DVR and AVR. Hence, it is unlikely that convective transport can
solely explain the presence of protein in the medullary interstitium.
Transport of albumin and other plasma proteins.
To understand the mechanisms by which albumin appears in the medullary
interstitium, albumin concentration in the ISF was then calculated on
the basis of interstitial mass conservation (Eq. 23) instead
of being specified. The permeability of DVR and AVR to albumin was
initially taken as 1 × 10
7 and 1 × 10
6 cm/s, respectively; we assumed that
ss = 0, and all other parameters were set to their
baseline value (Table 1). The resulting concentration profile is shown
in Fig.
3A.
Transcapillary fluxes of water and albumin are shown in Fig. 3,
B and C, respectively, and Pe values for albumin
are given in Fig. 3D. A positive flux of albumin across DVR
(or AVR) walls indicates that albumin is carried from DVR (or AVR) into
the interstitium, and vice versa. In addition, the greater the absolute
value of Pe, the greater the importance of convection relative to
diffusion.

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Fig. 3.
The interstitial concentration of albumin
(C ) is calculated on the basis of an interstitial
balance (Eq. 23), assuming that other plasma protein cannot
be exchanged across vasa recta. The permeability to albumin of DVR and
AVR is taken to be 1 × 10 7 and 1 × 10 6 cm/s, respectively, and ss = 0. Other parameters are those of the baseline case. A: albumin
concentration in DVR, AVR, and interstitium, divided by its initial
value in DVR at the corticomedullary junction. B:
transcapillary volume fluxes across vasa recta, based on the
circumference of all vessels. Note that, in the OM vascular bundles,
the sum of the fluxes is zero. C: transcapillary albumin
fluxes across vasa recta, based on the circumference of all vessels.
Because of mass conservation, the fluxes balance each other.
D: albumin Peclet number (Pe). After the sign change in the
DVR paracellular volume flux near the corticomedullary junction,
albumin enters the interstitium by diffusing out of AVR and is then
carried by convection into DVR.
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Near the corticomedullary junction, water is drawn out of the lumen
through both pathways in DVR; in that region, albumin is carried mainly
by convection out of DVR and into AVR, and the concentration of albumin
increases simultaneously in interstitium and DVR. Below that upper
region, the DVR paracellular flux of water is reversed, and diffusion
out of AVR and convection into DVR account for the presence of albumin
in the interstitium. Indeed, even though there is volume influx into
AVR, the Pe for AVR is small, and diffusion of albumin down its
concentration gradient (i.e., from AVR toward the interstitium)
dominates; solvent drag then carries albumin into DVR, as Fig.
3D suggests.
Before the OM-IM junction, as water reabsorption into AVR decreases,
there is less and less solvent drag into AVR to oppose diffusion out of
AVR, and C
thus rises (right before the boundary,
there is actually some water efflux from AVR, so that both solvent drag
and diffusion carry albumin from AVR into the interstitium). After
the OM-IM junction, conversely, the increase in water influx into AVR
(due to volume generation rate in the interstitium) leads to a decrease
in C
. Toward the papillary tip, water fluxes are much
reduced as the generation rate for water decreases to zero, and
C
increases rapidly again. The volume average
interstitial concentration of albumin is 1.21 g/dl in the entire
medulla and 0.94 g/dl in the IM only.
We have until now assumed that there is negligible efflux of protein
other than albumin from plasma (4, 6), but other investigators (25) do not distinguish between albumin and
other proteins. If vasa recta are also permeable to other plasma
proteins, the interstitial concentration of protein
(C
) is likely to be higher on average. To examine
this hypothesis, we assumed, in the absence of data, that the transport
properties characterizing all plasma proteins (i.e., reflection
coefficient, permeability) were equal to those of albumin, and the
interstitial mass balance for albumin (Eq. 23) was taken to
apply to all proteins. All other parameter values were identical to
those used in the previous simulation. We also confirmed that
concentration polarization is negligible when all plasma proteins, not
just albumin, can be transported to the interstitium.
Results are shown in Fig. 4 (case
A). Variations in C
along the corticomedullary
axis are similar to those in C
when albumin is taken
to be the only plasma protein that can be exchanged across vasa recta,
and the mechanisms by which all proteins are transported to and from the interstitium are also as described above. That is, except near the
corticomedullary junction, proteins diffuse out of AVR and are carried
by solvent drag into DVR. As expected, the volume average interstitial
concentration of protein is higher, at 2.25 g/dl for the entire medulla
and 1.81 g/dl for the IM only.

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Fig. 4.
Effect of changes in the permeability of DVR and AVR to
plasma proteins on transcapillary protein fluxes across vasa recta
(based on the circumference of all vessels), when the reflection
coefficient of vasa recta to small solutes ( ss) is taken
to be zero. Protein concentration profiles follow the same trends as in
Fig. 3A. Case A: P = 1 × 10 7 cm/s, and P = 1 × 10 6 cm/s; case B:
P = 1 × 10 6 cm/s, and
P = 1 × 10 6 cm/s. A
positive (or negative) flux is directed from the lumen toward the
interstitium (or vice versa). In both cases A and
B, near the corticomedullar junction, proteins are
transported by solvent drag from DVR to AVR via the interstitium.
Farther down the OM, the DVR paracellular volume flux is reversed;
proteins then start to diffuse out of AVR and are carried by convection
into DVR. In case B only, as x/L becomes >0.45,
proteins diffuse instead out of DVR and are carried by convection into
AVR. These results suggest that the mechanisms of transcapillary
protein transport depend on vasa recta permeability values and position
along the corticomedullary axis.
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Effect of Permeability of Vasa Recta to Protein
Because many parameters related to protein transport are
uncertain, we then performed parameter sensitivity studies to determine whether the transport processes described above depend on chosen values.
We first examined the effect of changes in the permeability of vasa
recta to proteins. As described above, estimates of
Pa were obtained using pore theory, on the basis
of measurements of the reflection coefficient to albumin
(
a). We used an average value of 0.70 for
a in AVR, but experimental measurements range from 0.59 to 0.78. If
a is taken to be 0.59, the permeability of
AVR to albumin is estimated to be as high as 7.7 × 10
6 cm/s (based on slit pore theory). Conversely, with
a = 0.78 in AVR, the permeability of AVR to albumin
is calculated to be only 4.5 × 10
7 cm/s (assuming
that the pores are cylindrical).
We therefore explored a broad range of values for the permeability of
DVR and AVR to plasma proteins (P
and
P
, respectively). The endothelium lining the capillary walls is fenestrated in AVR but is continuous in DVR, so that AVR are significantly more permeable than DVR. Hence, P
was taken to be
P
. Results are summarized in
Tables 3 and
4.
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Table 3.
Effect of permeability of DVR and AVR to plasma proteins on the average
protein concentration in the renal medullary interstitium
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Increasing the permeability of AVR to plasma proteins to 1 × 10
5 cm/s results in larger interstitial protein
concentrations but does not affect the mechanisms by which proteins are
exchanged across vasa recta, even if P
also increases to 1 × 10
6 cm/s. That is, proteins
enter the interstitium mostly by diffusing out of AVR (results not shown).
Conversely, if P
is fixed at
1 × 10
6 cm/s, increasing the permeability of DVR to
proteins by a factor of 10 significantly reduces the corresponding DVR
Pe but does not affect water fluxes. As a consequence, the diffusive
flux of proteins out of DVR begins to overcome the convective flux into
DVR about halfway down the corticomedullary axis; that is, when
transcapillary protein concentration differences have become very
large. For x/L > 0.45, proteins enter the interstitium
by diffusing out of DVR, and they are carried away to AVR by solvent drag, as shown in Fig. 4 (case B). Overall, interstitial
protein concentrations are then higher, as illustrated in Table 3.
These results suggest that the mechanisms underlying protein transport
are very much dependent on the (unknown) value of vasa recta
permeability to proteins.
Reflection Coefficient of Paracellular Pathways to Small Solutes
If the osmotic reflection coefficient of the paracellular pathway
in DVR to sodium and urea is close to, but not equal to, zero, will
small solute concentration gradients be large enough to reverse the
direction of the paracellular volume flux in DVR? To examine how
nonzero osmotic reflection coefficients would affect albumin transport,
we estimated the reflection coefficient of the paracellular pathway to
small solutes on the basis of pore theory. With the assumption that the
pores are cylindrical, Eq. 25 yields 0.014 as the reflection
coefficient of DVR to urea and 0.009 as that of AVR (smaller values are
obtained using slit pore theory). By use of these values for
ss, protein concentrations in the interstitium were
again determined using an interstitial mass balance. Results are
summarized in Tables 4 and
5.
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Table 5.
Effect of reflection coefficient of DVR and AVR to small solutes on the
average protein concentration in the renal medullary interstitium
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If the permeability of DVR and AVR to plasma proteins is equal to
1 × 10
7 cm/s and 1 × 10
6 cm/s,
respectively, increasing
ss from 0 to 0.014 in DVR
reduces the amount of water being drawn into DVR through the
paracellular route (and thus increases that which is reabsorbed into
AVR). Even though it is not enough to reverse the direction of the
paracellular volume flux across DVR (except in a very small region
close to the papillary tip), the protein Pe for DVR becomes so low that diffusion of protein out of DVR becomes more significant than convection into DVR in part of the IM; in that region, proteins diffuse
from DVR and are carried by solvent drag into AVR, as shown in Fig.
5 (case A).

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Fig. 5.
Effect of changes in the permeability of DVR and AVR to
plasma proteins on transcapillary protein fluxes across vasa recta
(based on the circumference of all vessels) when the reflection
coefficient of DVR to small solutes ( ) = 0.014 and that of AVR ( ) = 0.009. Case
A: P = 1 × 10 7
cm/s, and P = 1 × 10 6
cm/s; case B: P = 1 × 10 6 cm/s, and P = 1 × 10 6 cm/s. Protein transport processes across vasa recta
vary according to the values of permeability to protein and of
reflection coefficient to small solutes, as well as with medullary
depth. In case A, proteins in the IM diffuse out of AVR and
are carried by convection into DVR, except for 0.65 < x/L < 0.95, where the opposite occurs. In case
B, proteins are transported by diffusion out of DVR and by
convection into AVR in most of the IM.
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If P
is taken to be 1 × 10
5 cm/s, C
increases greatly, thereby
reducing transcapillary oncotic pressure gradients, so that the
paracellular flux of water across DVR is reversed halfway through the
inner medulla. Beyond this reversal point, both solvent drag and
diffusion carry proteins from DVR to AVR via the interstitium. The
average interstitial protein concentration is then higher (Table 5).
If P
is increased to 1 × 10
6 cm/s with P
equal to
1 × 10
6 cm/s, the paracellular flux of volume
across DVR is directed predominantly from the interstitium toward the
vessel, but the diffusive flux of proteins out of DVR becomes larger
than the opposing convective flux in most of the IM. Diffusion of
proteins out of DVR thus constitutes the main source of
proteins in the IM interstitium, as shown in Fig. 5 (case
B).
Reflection Coefficient of Paracellular Pathways to Albumin
Another parameter that may play an important role in determining
the direction of the paracellular water flux across the DVR wall is the
reflection coefficient of vasa recta to albumin. The higher the
reflection coefficient of vasa recta to albumin, the larger the effect
of transcapillary oncotic pressure differences on the paracellular flux
(Eq. 2a) and the greater the fluid reabsorption into the
lumen through the paracellular route. To draw more water out of DVR and
into AVR so as to maximize convective transport, the reflection
coefficient of DVR to albumin (or to all plasma proteins) therefore has
to be decreased and that of AVR increased. A value as high as 0.78 has
been measured for the reflection coefficient of AVR to albumin
(14); therefore, we chose an upper bound of 0.80 for the
reflection coefficient of AVR to all plasma proteins (
) and a lower bound of 0.85 for that of DVR
(
).
If the reflection coefficient of vasa recta to small solutes is taken
to be zero, decreasing 
and increasing

as described above has a small effect on
paracellular water fluxes but does not alter significantly the
transport mechanisms of proteins (Table 4).
However, if the reflection coefficient of DVR and AVR to sodium and
urea is assumed to be equal to 0.014 and 0.009, respectively, transcapillary osmotic pressure gradients due to small solutes favor
water efflux from DVR; decreasing 
, then,
sufficiently reduces the effect of oncotic pressure gradients across
DVR walls to change the direction of the paracellular flux of water
along the corticomedullary axis in the IM. At large depths, there is
volume efflux from DVR for all values of vasa recta permeability to
proteins. As a result, DVR and AVR protein concentration values near
the papillary tip increase by a factor of ~1.3. As blood flows down
along the corticomedullary axis, proteins first diffuse out of AVR into
the interstitium and are carried by solvent drag into DVR. When the
paracellular flux of water is reversed, convection and diffusion from
DVR become the mechanisms by which proteins enter the interstitium.
Results are summarized in Table 4.
The Effect of Water Channels
Experimental observations strongly support the hypothesis that
NaCl gradients drive water flux exclusively across a water-only pathway
such as that provided by AQP1. Pallone et al. (16) have shown that prolonged incubation of OMDVR with pCMBS, a mercurial agent
that inhibits AQP1 water channel activity, eliminates the osmotic
volume flux induced by transmural NaCl gradients; this inhibition can
be reversed by the addition of dithiothreitol. In separate
microperfusion studies of molecular sieving of 22Na and
[3H]raffinose by DVR, Pallone and Turner
(18) concluded that the collectate-to-perfusion ratios of
22Na and [3H]raffinose are best simulated by
a small solute reflection coefficient of 1.0 when transmural gradients
of NaCl drive water flux. If AQP1 water channels are impermeable to
small solutes, they will, a fortiori, be impermeable to larger proteins.
In modeling the exchange of plasma proteins between the
microcirculation and interstitium, Wang and Michel (25)
did not take into explicit consideration the presence of AQP1 water
channels in DVR. The authors did not distinguish between the
paracellular and the transcellular pathways, and the reflection
coefficients for the joint route that they considered instead were an
average over the two separate pathways (18, 25), which
amounts to lumping Jvp and
Jvt together. Because there is overall volume efflux from DVR (i.e., Jvp + Jvt > 0) and because this joint route is
permeable to proteins, it is not surprising that the authors found that
the convective flux of proteins from DVR into the interstitium can
balance their clearance by convection into AVR throughout the medulla.
To compare our approach with that of Wang and Michel, we performed
similar simulations in which the paracellular and transcellular pathways were lumped together. The water permeability of the joint route was taken as 1.8 × 10
6
cm · s
1 · mmHg
1 and the
average reflection coefficient of DVR to small solutes as 0.05 (18). Other parameter values were those given in the baseline case (Table 1). Results are shown in Figs.
6 and 7.

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Fig. 6.
Transcapillary volume fluxes based on the circumference
of all vessels, as a function of position. In these simulations,
P = 1 × 10 7 cm/s,
P = 1 × 10 6 cm/s, and
 = 0.009. Similar trends are obtained for
different values of vasa recta permeability to proteins. A:
 = 0.014; B:  = 0.05; and no distinction is made between the paracellular and the
transcellular pathways across DVR.
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Fig. 7.
Effect of changes in the permeability of DVR and AVR to
plasma proteins on the normalized concentration of proteins in vasa
recta and the interstitium as a function of position. In these
simulations,  = 0.05,  = 0.009, and no distinction is made between the paracellular and the
transcellular pathways across DVR. Case A:
P = 1 × 10 7 cm/s, and
P = 1 × 10 6 cm/s
(nearly identical results are obtained with
P = 1 × 10 6 cm/s);
case B: P = 1 × 10 7 cm/s, and P = 1 × 10 5 cm/s (nearly identical results are obtained with
P = 1 × 10 6 cm/s).
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As expected, if no distinction is made between the paracellular
and transcellular pathways in DVR, there is water efflux across this
joint route almost throughout the medulla, except around the OM-IM
junction, as shown in Fig. 6. With the exception of that
region, proteins are carried by convection (and to a lesser extent by
diffusion) from DVR into the ISF and then into AVR. In contrast with
the results obtained by Wang and Michel, however, we found that changes
in the permeability of AVR to plasma proteins do affect mean
C
values, as illustrated in Fig. 7, because
around the OM-IM junction, proteins enter the interstitium by diffusing
out of AVR. Average interstitial protein concentrations are given in
Table 5.
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DISCUSSION |
In this study, we have taken into consideration the effect of
protein concentration polarization and its reverse at AVR and DVR walls
by assuming that the interstitium can be represented as an annular
space around each vas rectum and by determining the radial changes in
water fluxes and solute concentrations. This model suggests that
concentration polarization has a small effect on variations in fluxes
and concentrations along the corticomedullary axis; in all the cases
that we examined, the difference in osmolality at the papillary tip
between this model and the no-polarization hypothesis was <2%.
Conversely, assuming that oncotic pressure differences due to albumin
are negligible across AVR walls because of polarization, as we had
previously done (4), yields more significant errors.
The most common hypothesis to explain the presence of a
significant pool of albumin in the ISF has been that plasma proteins are transported by convection from DVR to AVR through the interstitium. The recent theoretical work of Wang and Michel (25)
appears to confirm this assumption. The study by these authors,
however, does not take into consideration the different
properties of the paracellular and transcellular routes in DVR. Because
there is overall volume efflux from descending vasa recta, they
conclude that solvent drag can indeed carry proteins into the
interstitium. A careful analysis of transcapillary volume fluxes across
each pathway, however, reveals that, across the paracellular route, water can be reabsorbed into DVR. Indeed, transcapillary oncotic pressure gradients drive water into the lumen, and the reflection coefficient of the paracellular pathway to sodium and urea may be too
small for small solute concentration differences to counterbalance that
effect. Because the transcellular route (i.e., AQP1) is impermeable to
solutes, convection from DVR cannot be the sole mechanism by which
proteins are transported into the interstitium.
Our results suggest that both diffusion and convection play a role in
carrying proteins to the interstitium and that whether proteins come
from DVR or AVR depends on certain parameter values and can vary with
depth along the corticomedullary axis. In the absence of experimental
data, permeability of vasa recta to proteins and reflection
coefficients were varied over a broad range.
If the reflection coefficient of vasa recta to small solutes is
zero, as suggested by Pallone et al. (16), the
paracellular flux of water across DVR (Jvp) is
directed toward the lumen (except for a small region near the
corticomedullary junction), so that solvent drag carries proteins both
into DVR and AVR. The only possible source of interstitial proteins is,
therefore, diffusion. If the permeability of DVR to proteins
(P
) is <1 × 10
6
cm/s, proteins diffuse out of AVR throughout the medulla. If P
is
1 × 10
6 cm/s
(and if P
is of the same order of magnitude), halfway down the corticomedullary axis, diffusion out of
DVR becomes more favorable.
Pore theory suggests that the reflection coefficient of vasa recta to
small solutes is on the order of 0.01. With this assumption, osmotic
pressure differences due to sodium and urea can possibly reverse the
direction of the paracellular volume flux across DVR in some regions
along the corticomedullary axis, especially if the reflection
coefficient of DVR to protein is low, thereby reducing the effect of
transcapillary oncotic pressure gradients across DVR walls. At small
depths, proteins always diffuse out of AVR and enter DVR by solvent
drag. After the point where Jvp changes sign,
the source of interstitial proteins becomes DVR; the relative importance of convection and diffusion in the total protein flux depends on the value of the permeability of vasa recta to proteins. Even when Jvp is never reversed, in certain
parts of the medulla where transcapillary protein concentration
differences are very large, diffusion out of DVR can become more
favorable than that out of AVR (because protein concentrations are
lower in AVR than DVR). In those regions as well, the net flux of
protein is then directed from DVR to AVR via the interstitium.
Our results thus indicate that the values of vasa recta permeability to
proteins and reflection coefficient to both small solutes and
macromolecules determine the mechanisms by which plasma protein are
carried to and cleared from the medullary interstitium.
Our calculations also suggest that the concentration of proteins in the
interstitium undergoes significant variations along the
corticomedullary axis, which may explain the wide range of protein
concentration measurements in the ISF (12, 15). Given both
the magnitude of the spatial variations in C
and the
broad range in the experimental data, we did not attempt to match our
calculated values with measured ones to infer the value of transport
parameters that have not been measured, such as the permeability of
vasa recta to albumin and the reflection coefficient of the
paracellular pathways to sodium and urea.
Nevertheless, some agreement between our predictions and other
experimental data can be noted. In all cases shown in Figs. 4 and 5,
plasma protein concentrations in DVR are 7-50% higher in the
papilla than at the corticomedullary junction. Because we assumed that
the concentration of plasma proteins entering DVR is 1.4 times that of
arterial plasma (i.e., that the glomerular filtration fraction is 0.3),
DVR concentrations of plasma proteins are predicted to be 1.5-2.1
times higher in the papilla than in systemic blood, which is consistent
with experimental observations (19, 21).
We have not considered any active mechanisms for protein transport in
this study. Although there is no direct evidence that albumin is broken
down or taken up by cells in the medulla, active transendothelium
transport of albumin has been demonstrated in vivo. Cultured porcine
pulmonary artery endothelial cells actively transport albumin from
interstitium to lumen (22). After absorption onto specific
binding sites, albumin-gold complexes are carried in transcytotic
vesicles across the capillary endothelium of the mouse lung, heart, and
diaphragm (7). The counterconvective transport of albumin
across venular endothelium has also been observed in rat lung
(24). Hence, it is possible that active mechanisms may
also be involved.
Along the corticomedullary axis, DVR terminate and give rise to
capillaries that either join existing AVR returning from deeper regions of the medulla or form new AVR. Terminal DVR and this interposed capillary plexus are known to be fenestrated
(17). It is thus possible that proteins are also exchanged
across the intervening capillaries, as suggested by Wang and Michel
(25).
In summary, our results suggest that proteins are carried to the
medullary interstitium by diffusion from AVR or by either diffusion or
convection from DVR, depending on medullary depth and the values of key
parameters such as vasa recta permeability to proteins and reflection
coefficients to small solutes and proteins.
As indicated above, the difference between the analytical and the
numerical results is always <1%, more often <0.5%. Similar results
are obtained with differential spatial distributions of generation
rates (results not shown).
This work was supported by National Institute of Diabetes and
Digestive and Kidney Diseases Grant DK-53775.
Address for reprint requests and other correspondence: A. Edwards, Dept. of Chemical and Biological Engineering, Tufts Univ., 4 Colby St., Medford, MA 02155 (E-mail:
aurelie.edwards{at}tufts.edu).
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. Section 1734 solely to indicate this fact.