In the absence of evidence
for lymphatics in the inner medulla of the kidney, it has been proposed
that plasma proteins are cleared by convection out of the medullary
interstitial fluid (ISF) directly into the ascending vasa recta (AVR).
To clarify this hypothesis we have developed a mathematical model of
the microvascular exchange of fluid, plasma proteins, and small solutes among the descending vasa recta (DVR), the AVR, and the ISF. The model
represents the DVR and AVR as limbs of a countercurrent exchange loop
separated and surrounded by the ISF. Steady-state exchange of fluid and
solute are considered by using conservation and exchange equations. We
have used values for parameters based on experimental measurements and
investigated the effects of the properties of the vasa recta, the flow,
and the gradient of small solutes on the distribution of plasma
proteins. Results from the model agree reasonably well with
experimental measurements, suggesting that convection may account for
the clearance of plasma proteins from the renal medulla maintaining
their concentration below that of the AVR.
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ARTICLE |
THE RENAL MEDULLA HAS A
HIGHLY specialized microvascular bed. Descending vasa recta (DVR)
and ascending vasa recta (AVR) run to and from the tip of the papilla
as straight and closely packed vessels. In the outer medulla, DVR and
AVR are grouped into vascular bundles and are separated from the
parallel segments of nephron. In the inner medulla, the microvessels
are more evenly distributed. DVR and AVR are both conducting and
exchange vessels, and they connect to each other through capillaries at
different depths. In addition to their function of delivering oxygen
and other nutrients to the tissues, the medullary microcirculation
reabsorbs water that has been extracted during the concentration of the
urine and transports it back to the rest of the organism
(14).
In most microvascular beds, net fluid uptake from tissues to
capillaries occurs when the osmotic pressure of the plasma exceeds the
sum of the osmotic pressure of the pericapillary fluid and the
transcapillary hydrostatic pressure difference (30). In these tissues, there are lymphatics that clear plasma proteins and
excessive fluid from the interstitium and maintain the oncotic pressure
differences across the capillary walls (13). In the inner
medulla of the kidney, however, there is little or no evidence for the
existence of lymphatics (1, 33). When labeled
albumin is injected into the systemic arterial blood, it appears in the renal medulla in <2 min, which demonstrates that the medullary microcirculation is permeable to plasma proteins (8). If
oncotic pressure differences across the AVR are responsible for the
clearance of fluid from the inner medulla, the question arises as to
what mechanism is involved regarding the simultaneous drainage of
plasma proteins from the interstitium to keep its oncotic pressure low.
In a recent review, Michel (14) discussed three possible
routes for protein clearance from the inner medulla: 1)
proteolysis occurred in the medullary interstitium; 2)
proteins were cleared through prelymphatic channels in the
interstitium; and 3) proteins entered the AVR by convection.
He concluded that the most likely route was through convection into the
AVR. In a later paper in the same year, MacPhee and Michel
(11) reported that the reflection coefficient of the AVR
to albumin is between 0.59 and 0.72, on the basis of their measurements
using 15-day-old Sprague-Dawley rats. In the appendix of the same
paper, the mechanism of the convective transport of a solute by osmotic
flow up its own concentration gradient was presented by using a
three-compartment system. They demonstrated how the system could work
in theory if two membranes had different properties. Pallone and
colleagues (17-19, 24) reported
differences in the hydraulic permeability of the DVR (10
6
cm · s
1 · cmH2O
1)
and AVR (9.2-13.8 × 10
6 cm · s
1 · cmH2O
1) and the
reflection coefficient to serum albumin (0.9-0.99 for DVR and 0.78 for AVR). All these findings support the convective mechanism for
protein clearance by the AVR. Nevertheless, it remains to be examined
whether the proposed convective mechanism can function in the renal
medulla, where changes in solute concentration exist along the whole
length of the DVR and AVR due to fluid filtration and reabsorption.
Previous models of the urinary-concentrating mechanism have generally
neglected the vasa recta by assuming that the microvessels offer
negligible resistance to the transport of solute and water (9, 31). The medullary microcirculation and
its functions in the transport of plasma proteins, for example, are
poorly understood. In the present study, we focus on the role of the
DVR and AVR on protein clearance from the inner medulla of the kidney.
A simplified capillary loop represents the countercurrent arrangement
of DVR and AVR. Basic principles governing the transcapillary exchange of small solutes, plasma proteins, and water are used. Nonlinearity introduced by the transcapillary exchange of water and plasma proteins
makes it necessary to seek steady-state distributions of solute and
flow numerically (2). We pay particular attention to the
following questions: 1) whether the convective mechanism for
protein clearance by AVR has steady-state solutions when parameters take physiological values; 2) whether the distribution of
small solutes and proteins predicted by the model in the steady state agrees with data measured in the renal medulla; and 3) how
changes in the flow and the permeability properties of the DVR and AVR influence plasma protein concentration distribution. Some of the features in the renal medulla are purposely left out, e.g., anastomoses between the DVR and AVR and the exponential distribution of small solutes in the interstitium, and we believe that by simplifying the
model in this way we are able to focus on the fundamental questions.
Transport of Fluid and Solute in and Across Vasa Recta
In the renal medulla, DVR and AVR are a few micrometers apart.
They run parallel to each other for several millimeters and form a
countercurrent exchange system. We consider the vessels to be simple
loops bearing a constant relationship to each other and to the
neighboring nephron segments. The model includes DVR, AVR, and a common
interstitium as sketched in Fig. 1. The
length of the unit is L, and its cross-sectional area is
S. The countercurrent exchange loop runs from the junction
between the inner and outer medulla, x = 0, to the tip
of papilla, x = L. DVR and AVR have different properties, e.g., cross-sectional area
(S1 and S2), solute
permeability, and water conductivity. There are also more AVR than DVR
in the medulla and the ratio of the two,
, is between 1.7 and 2.3 (22). Protein-free fluid entering the interstitial fluid
(ISF) from neighboring nephrons is considered in the model, which, in
the steady state, satisfies volume conservation in the interstitium. We
assume a linear distribution of small solutes with the depth,
x, in the interstitium, which is maintained by the input of
small solutes from the neighboring nephron segments. The linear
distribution of small solute in the ISF is used here not only for its
simplicity but also for its good agreement with experimental data
reported by Koepsell et al. (6) over 85% of the
length of the renal medulla, i.e., between x = 0 and
x = 0.85.

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Fig. 1.
Schematic geometry of the countercurrent exchange system
of the descending vasa recta (DVR), ascending vasa recta (AVR), and
interstitial fluid (ISF). x, Length along the direction of
the flow in the DVR; S1,
S2, and S: cross-sectional areas of
the DVR, AVR, and the unit, respectively; L, length of the
unit. At the entrance of DVR, flow velocity is
U0 and concentration of small solutes and plasma
proteins are Cs0 and Cp0, respectively.
We assume a linear increase of small solute concentration with
x. AVR and DVR vary not only in their properties but also in
their numbers.
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On the basis of data from the renal medulla of the rat and hamster
(4, 5, 12), L is
between 5 and 10 mm, radii of the DVR and AVR are 7 and 11 µm,
respectively, and the cross-sectional area of the ISF is approximately
the same as that of DVR. Flow velocity in vasa recta (u) is
between 100 and 1,000 µm/s, and the value varies along the vessel as
water filtration and reabsorption occur. The diffusion coefficient of
small solutes, e.g., Na+, in plasma is 10
5
cm2/s, and, in the ISF, is slightly smaller. The diffusion
coefficient for plasma proteins, e.g., albumin, in plasma is 6 × 10
7 cm2/s and may decrease to 1-6 × 10
8 cm2/s in the ISF. Considering the
small diameters of the microvessels, d, and low flow
velocities in them, u, the Reynolds number of the flow,
ud/µ, is very small, where
is the density of the plasma and µ is its viscosity. Order of magnitude analysis reveals that (34, 35), in vasa recta, solute
diffusive transport in the axial direction is negligible compared with
that by convection; flow velocity in the radial direction of the vasa
recta is very small compared with that in the axial direction; and
changes in solute concentration in the radial direction are negligibly
small compared with those in the axial direction. As far as solute
transport is concerned, we also neglect details of the velocity profile in the vasa recta and use averaged velocities. In the interstitium, transport of fluid and solute is dominantly in the direction normal to
the axis of the unit between the neighboring DVR and AVR. Transport in
the axial direction, by comparison, is negligible, mainly because the
length of the renal medulla is several thousand times greater than the
distance between adjacent vessels. This assumption is reexamined for
its consistency later in this paper when the concentrations of solute
are solved.
Governing equations for this countercurrent exchange system are the
following:
Transcapillary exchange of water.
In DVR
|
(1)
|
and AVR
|
(2)
|
where Jv represents the rate of
transcapillary fluid flux per unit surface area; p is the hydrostatic
pressure; Cs is the concentration of small solutes;
Lp is the hydraulic permeability of the vessel;
and
s and
p are reflection coefficients
of the vessel to small solutes and plasma proteins, respectively.
RT is the product of the universal gas constant and the
absolute temperature, and subscripts 0, 1, and 2 represent values for
the ISF, DVR, and AVR, respectively.
is the osmotic pressure of the
plasma proteins
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(3)
|
At 37°C, a1 = 2.1, a2 = 0.16, and
a3 = 0.009, where Cp is the
concentration of plasma protein (in g/100 ml), and
is measured in
millimeters mercury (7).
Transcapillary exchange of plasma proteins (27).
In DVR
|
(4)
|
and AVR
|
(5)
|
where Jp is the rate of transcapillary flux
of proteins per unit surface area, and Pp is the
permeability of vasa recta to proteins.
Transcapillary exchange of small solutes.
In DVR
|
(6)
|
and AVR
|
(7)
|
where Js is the rate of small solute
transcapillary flux per unit surface area, and
Ps is the permeability of vasa recta to small solutes.
Flow velocity in vasa recta.
Changes in u satisfy volume conservation.
In DVR
|
(8)
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and AVR (u2 is positive in the direction of
flow, i.e., in the x direction)
|
(9)
|
where r1 and r2 are
radii of the DVR and AVR.
Concentration of plasma proteins.
Changes in protein concentration satisfy mass conservation.
In DVR
|
(10)
|
AVR
|
(11)
|
and ISF
|
(12)
|
where
is the ratio of AVR to DVR.
Concentration of small solutes.
In DVR
|
(13)
|
and AVR
|
(14)
|
In the ISF, concentration of small solutes is assumed to increase
linearly with x
|
(15)
|
where Cs0 is the concentration of small solutes, G is
the gradient of the small solute concentration, and Cs0
is the value of Cs0 at x = 0.
Boundary conditions.
At x = 0
|
(16)
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(17)
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(18)
|
and at x = L
|
(19)
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(20)
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(21)
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Numerical Procedure
The nonlinear, multivariable, and interactive problem is solved
numerically for steady-state distributions of plasma proteins, small
solutes, and flow velocities. Two special features in our numerical
treatment should be emphasized.
1) We solve for the steady-state concentration of proteins
and small solutes in two separate iterative loops, i.e., under initial
values of small solute concentration, we solve for a steady-state distribution of protein concentration. This distribution is then used
to solve for a new steady-state concentration distribution of small
solutes and so on, until the final steady-state concentration for both
proteins and small solutes is reached.
2) Relaxation is applied when the values of the small solute
concentration are updated in each iteration to prevent overshooting. In
the system, the concentration of small solutes is much higher than that
of plasma proteins. Relatively small changes in the small solute
concentration can lead to significant changes in the osmotic pressure
across the DVR and the AVR. Relaxation is found to be a useful
technique in our calculation for results to converge rapidly.
The flow chart of the computation is shown in Fig.
2. In the calculation, initial
concentrations of plasma proteins in the DVR, AVR, and ISF are
and the initial concentrations of small solutes in the DVR and AVR
are
where
1 and
2 are concentration
differences between vasa recta and interstitium at x = 0.

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Fig. 2.
Flow chart of the computer program.
Jv1, Jv2,
Jp1, Jp2,
Js1, Js2: rate of
transcapillary fluid flux per unit surface area and flux of proteins
and small solutes in the DVR and AVR, respectively;
u1, u2: flow velocity in
DVR and AVR, respectively; Cp0, Cp1,
Cp2, Cs1, Cs2: concentration of
proteins and small solutes in the ISF, DVR, and AVR, respectively.
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The program is written in Fortran, and calculations are carried out on
a UNIX workstation Silicon Graphics O2. The length of the
unit is equally divided into 2,000 segments. Under normal conditions,
it takes between 5 and 10 min for results to reach a steady state.
Results and Discussion
We have investigated effects of different parameters on plasma
protein distribution in the DVR, AVR, and ISF. These parameters include
the flow velocity into the DVR, U0; the
concentration gradient of small solutes in the ISF, G; the reflection
coefficient of the DVR and AVR to protein,
p1
and
p2, respectively; and the
hydraulic permeability of the DVR and AVR,
Lp1 and
Lp2. As stated
earlier, we have paid particular attention to the
convective mechanism for the clearance of plasma proteins from the ISF
and examined whether such a mechanism functions in a countercurrent system when parameters take physiological values. We have also tested
the sensitivity and limits of the system when these parameters are changed.
The values of the parameters are given in Table
1. They are based on data reported in the
literature and on measurements made in our own laboratory
(11). Table 1 also shows the range of reported values for
each parameter and relevant references. Where there are no reported
values for a parameter (e.g., Pp1 and Pp2), the
criteria for choosing a particular value are given. Values for
Na+ are used as typical values for small solutes and those
for albumin for plasma proteins. The use of Na+ may be
questioned on the ground that, although the permeability of the AVR to
Na+ and urea is very similar, the permeability of the outer
medullary DVR to urea greatly exceeds that to Na+
(25). As we did not wish to overcomplicate the present
model with this feature, we did not incorporate it. In some
calculations, we have deliberately chosen a wide range of values for
certain parameters, e.g., flow velocity, reflection coefficient, and
hydraulic permeability, to examine their effects on the protein
distribution in the system.
In all results, normalization is carried out by using the value of that
variable at the entrance of the DVR, i.e., U0
for velocity, Cp0 for protein concentration,
Cs0 for small solute concentration, and
U0
r12 for
volume flux. Length x is normalized by using L,
which is 7 mm in the model.
In Fig. 3A,
concentration of small solutes in the DVR lags behind that in the ISF.
It follows a very similar linear increase with distance, x,
as in the ISF. In the AVR, small solute concentration overtakes that in
the ISF almost immediately after the turn at the tip of papilla and
decreases linearly toward the base of the capillary loop. The
concentration difference between the ISF and vasa recta is bigger in
the DVR than that in the AVR because of higher solute permeability and
bigger surface area of the AVR. It is also noticed that the
concentration difference between the AVR and ISF increases in the
direction of flow, i.e., from x = 1 to
x = 0. The ratio of the product of solute permeability
(P) and surface area (A) to flow rate (F),
(P × A)/F, determines the equilibration of
small solutes between microvessels and their surrounding ISF
(28). As water is reabsorbed from the ISF to AVR, the flow
rate in the AVR increases, decreasing (P × A)/F and hence the degree of equilibration between the AVR
and the ISF.

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Fig. 3.
The normalized distribution of solute concentrations
and flow velocities of the countercurrent exchange system.
A: concentration of small solutes. B:
concentration of plasma proteins. C: flow velocity. In the
figure, flow velocity at the entrance of the DVR
(U0) = 500 µm/s; the concentration
gradient of small solutes (G) = 4; the reflection coefficient of
proteins in the DVR and AVR ( p1 and
p2, respectively) are 0.9 and 0.6;
of small solutes in the DVR and AVR ( s1
and s2, respectively) are 0.025 and 0.01; and hydraulic
permeability of the DVR and AVR (Lp1
and Lp2, respectively) are
10 6 cm · s 1 · cmH2O 1 and of 9 × 10 6 cm · s 1 · cmH2O 1. The depth, x, is
normalized by the length of the renal medulla, L.
Thin-dashed lines, DVR; thick-dashed lines, AVR; solid lines, ISF.
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In Fig. 3B, we find a steady increase in the plasma protein
concentration in the DVR from the base to the tip of papilla. This is
mainly due to water filtration from the DVR to ISF, combined with the
high protein reflection coefficient of the DVR. From the base to the
tip of the capillary loop, there is an ~27% increase in the protein
concentration. Although there are no measurements of protein
concentrations in the plasma entering the DVR, it is reasonable to take
this as being 1.25 times greater than the concentration in systemic
arterial plasma as a result of glomerular filtration. The further
increase in protein concentration as the blood flows through the DVR
would mean that in the papilla (from x = 0.8 to x = 1), it would be 1.5-1.6 times greater than
that in systemic arterial plasma. This degree of concentration is
consistent with experimental measurements (26,
29), where the protein concentration in the DVR is
1.4-1.7 times that in the arterial plasma. In the AVR, protein
concentration decreases in the direction of flow as water is reabsorbed
from the ISF. Near the tip of the capillary loop, x = 1, we observe a rapid decrease in protein concentration. Measurements
of plasma proteins in the AVR of the papilla suggest the fall in
concentration is on the order of 20-25% rather than the
50-60% that our model predicts. The discrepancy is not
unexpected, because, in our model, there is not only a sudden change in
the properties of the vessels as plasma enters the AVR and direction of
flow reverses but also a sudden fall in the hydrostatic pressure. In
reality, these changes occur much more gradually, with a less rapid
fall in AVR plasma protein concentration. The concentration of protein
leaving the AVR (x = 0 in Fig. 3B) may
appear to be low, but this is dictated by mass balance, i.e., by the
volume of fluid that is recovered from the medulla under steady-state conditions and medullary blood flow. At present, there are no measurements against which we can compare these predictions.
The values predicted for the concentration of plasma proteins in the
interstitium are very low. Near the base of the medulla, ISF protein
concentration is <10% of that in the plasma of the DVR. Protein
concentration declines slowly toward the tip of the papilla, the
gradient being ~0.1. In contrast to this small gradient, the protein
concentration difference between the plasma in the vasa recta and the
surrounding ISF is between 0.2 and 1.2 over a radial distance of
10-15 µm (~1/50 of L). Thus the average protein concentration gradient in the radial direction is 100-600 times greater than that in the axial direction. This is consistent with our
earlier assumption that, in the ISF, the protein gradient in the axial
direction is much smaller than that in the radial direction. The
concentration of protein in the ISF that is predicted by the model,
however, is very much lower than experimental estimates. Thus Pallone
(20) reported protein concentrations that were 60-70% of those in the plasma of neighboring AVR, and MacPhee and
Michel (11) estimated interstitium albumin
concentrations that were 25% of those in systemic arterial plasma. The
distribution of protein in the ISF and plasma of the vasa recta that is
predicted by the model depends on the values of the parameters used in
the calculation. The consequences of varying these values are
considered later in this discussion. Here we note that if we start with
a set of parameters in the physiological range, the countercurrent exchange system reduces an initially high concentration of plasma proteins (value used as initial condition for the calculation) to a
steady-state protein level in the interstitium fluid that is very low,
despite the lack of drainage of the ISF by lymphatics.
Velocities of flow in the DVR and AVR are shown in Fig. 3C.
The values have been normalized by the flow velocity at the entrance of
the DVR, U0. Following the direction of flow, we
observe a steady decrease in velocity in the DVR as fluid is filtered
from it, and a more rapid increase in velocity in the AVR as fluid is
reabsorbed. Direct observations of red cell velocity in the papilla
report that velocity in AVR is well below that in DVR (4),
which is consistent with our model prediction regarding velocity near
the tip of the capillary loop (i.e., between x = 0.5 and x = 1.0). The discontinuity in velocity at
x = 1 is caused by the sudden changes in the number and
size of the ascending vasa recta. In reality, changes will be more
gradual. If we examine the velocity in the DVR carefully, we find a
very small increase near the entrance, x = 0. Similarly, there is a very small decrease in the plasma protein
concentration of the DVR close to its entrance (Fig. 3B),
where, near x = 0, concentration of proteins in ISF approaches zero. These are artifacts caused by the boundary
condition at x = 0, where the same concentrations of
small solutes in the DVR and the ISF are assumed. Different conditions
could be used that would eliminate these artifacts. However, we are
satisfied with the simplest one used here, which has negligible effect
on solute distribution away from this boundary.
Figure 4 summarizes how the mean ISF
concentration of proteins is influenced by the values of some of the
different parameters of the model. In Fig. 4A, the mean
concentration of plasma proteins in the ISF,
p0, is
presented for different values of small solute concentration gradient,
G, at two values of U0. At
U0 = 500 µm/s and G = 4, which are
the values we used in previous calculation,
p0 is
below 3% of the concentration of plasma proteins at the entrance of
the DVR. When G increases, we find an increase in
p0, which peaks at ~G = 6 and is then
followed by a decrease. This is caused by changes in water filtration
and reabsorption in the DVR and AVR. As G increases, the small solute
concentration difference between the vasa recta and ISF increases, as a
steeper gradient of solute in the ISF makes it harder for the values in the vasa recta to catch up. When G < 6, increases in G promote more filtration of fluid from the DVR, carrying more plasma proteins into the interstitium. When G > 6, the balance between the
transport of proteins by filtration and reabsorption tilts toward
reabsorption into AVR.

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Fig. 4.
The mean concentration of plasma proteins in the ISF,
p0, normalized by the protein
concentration at the entrance of the DVR.
A: effects of G on the mean concentration of plasma
proteins in the ISF with s2 = 0.01 and U0 at 500 and 1,000 µm/s. B:
effects of U0 on p0,
with G remaining constant at 4 and s2 at 0.01 and
0.005. In the figure,
p1 = 0.9, p2 = 0.6, s1 = 0.025, Lp1 = 10 6 cm · s 1 · cmH2O 1, and Lp2 = 9 × 10 6 cm · s 1 · cmH2O 1.
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The fall in
p0 at higher levels of G is dependent on
s2 > 0. The initial value 0.01, which we chose for
s2, is probably unreasonably high, and a figure of
0.004-0.006 would be consistent (on pore theory) with
p2 = 0.6-0.7 if both small and large solutes share the same pathway through the walls of the AVR as water (i.e., no
aquaporin channels here). Furthermore, our initial value of U0 = 500 µm/s is also probably on the low
side. Measurements for red cell velocity in papillary DVR are in the
range of 500-1,100 µm/s (22). According to Fig.
3C, velocity in the vessels might be expected to be
40-50% of U0. Thus it seems reasonable to
consider values of U0 higher than 500 µm/s. We
have therefore examined the effects of varying
U0 on
p0 with
s2 at 0.01 and 0.005 and G remaining constant at 4. The
results of these calculations are summarized in Fig. 4B. It
is seen here that with U0 in the range of
1.5-2.0 mm/s (consistent with papillary velocity of 0.6-0.8 mm/s) and
s2 = 0.005,
p0 rises
to 15% of the Cp0, i.e., ~19% of the concentration
in the arterial plasma. Although this is still considerably less than
the estimates by Pallone (20), it is only slightly less
than the values reported by MacPhee and Michel
(11). Further decreases in
s2 bring
p0 well into the range of values reported by the
latter authors.
The reflection coefficients of the DVR and AVR to plasma proteins also
have significant effects on the mean concentration of plasma proteins
in the ISF. In Fig. 5, we present changes
in
p0 with different values of
p1 and
p2. At a given value of
p1, increases in
p2 result in a smaller proportion of proteins being
carried into the AVR with reabsorption; therefore, there are more
plasma proteins accumulating in the interstitium. It is seen that
p0 increases more rapidly at higher values of
p2. In all cases, when
p2 rises toward
p1, the system ceases to function and the concentration
of plasma proteins in the ISF increases rapidly to infinity. On the
other hand, increases in the reflection coefficient of the DVR,
p1, result in decreases in
p0 due to less plasma protein leakage into the ISF. At greater values of
p1, i.e.,
p1 = 0.95, the
countercurrent exchange system can have steady-state solutions over a
wider range of
p2, i.e., up to values of
p2 max approaching 0.84. For protein to be cleared from
the ISF and for steady-state concentrations to be maintained there,
p1 must be significantly greater than
p2. From this it seems that if
p2 is as high as 0.78 (19), then
p1 is probably 0.95 or more.

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Fig. 5.
Effects of p1 and p2 on
p0 . In the figure, p0 is
normalized by the protein concentration at the entrance of the
DVR. U0 = 500 µm/s, G = 4, s1 = 0.025, s2 = 0.01, Lp1 = 10 6 cm · s 1 · cmH2O 1, and
Lp2 = 9 × 10 6 cm
· s 1 · cmH2O 1.
p1 = 0.90 and 0.95.
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Hydraulic permeability of the vasa recta is one of the key parameters
that determine the transcapillary water flux. We expect changes in
Lp1 and
Lp2 to have a significant effect on
the drainage of plasma proteins from the ISF. In Fig.
6, it is seen that as Lp2 increases,
p0
falls. The increased permeability of the AVR promotes the uptake of
fluid into these vessels, and more protein is cleared from the
ISF. For the values of parameters used in the calculation, the system
no longer functions (i.e., does not converge to a steady state) when
Lp2 is <7 × 10
6
cm · s
1 · cmH2O
1.
In those cases, rapid accumulation of plasma proteins in the interstitium occurs. When Lp1
decreases from 1.0 to 0.5 × 10
6 cm · s
1 · cmH2O
1, protein
concentration in ISF decreases because of less leakage of proteins from
the DVR.

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Fig. 6.
Effects of Lp1 and
Lp2 on p0. In the
figure, p0 is normalized by the protein
concentration at the entrance of the DVR.
U0 = 500 µm/s, G = 4, p1 = 0.9, p2 = 0.6, s1 = 0.025, and s2 = 0.01. Lp1 = 0.5 and
1.0 × 10 6 cm · s 1 · cmH2O 1.
Lp2 varies between 7 and 15 × 10 6 cm · s 1 · cmH2O 1.
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The DVR are much less permeable to plasma proteins than the AVR, given
their lower solute and hydraulic permeabilities and high protein
reflection coefficients. Changes in their protein concentration reflect
fluid filtration from the DVR. In Fig. 7, distribution of Cp1 with depth, x, is plotted
for different values of G. From Eq. 1, the difference in
small solute concentration between the ISF and the DVR is the driving
force for water filtration. This works against the osmotic pressure
imposed by differences in protein concentration between the DVR and
ISF. Larger values of G, as explained earlier, increase the small
solute concentration differences between the DVR and ISF and cause a
higher water filtration from the DVR. They result in increases in
plasma protein concentration inside the DVR. In Fig. 7, it is seen that
when G increases from 4 to 8, for example, the rise in protein
concentration from the base to the tip of the capillary loop increases
from <30 to >60%. Most of the increase in Cp1,
particularly when G is >6, occurs between 0 < x < 0.6. Although there are no experimental data making direct
comparisons between the concentration of plasma proteins in the DVR as
these vessels enter the medulla and their concentration at subsequent
points along the vessels within the medulla, there are several
comparisons of plasma protein concentrations in the papillary DVR and
AVR with those in the systemic arterial blood. Values in the range of
1.5 (1.38-1.76) have been reported from DVR plasma at the base of
the papilla (x = 0.8). If the protein concentration of
the plasma entering the DVR is raised 1.25 times above that in systemic
arterial blood as a result of glomerular filtration, at the base of the
papilla it is raised a further 20%. Figure 7 shows that such an
increase would be achieved if G = 4, justifying our selection of
this value in the model.

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Fig. 7.
Distribution of plasma proteins in the DVR at
different values of G. In the figure, protein concentration is
normalized by its value at the entrance, and x is
normalized L. U0 = 500 µm/s,
p1 = 0.9, p2 = 0.6, s1 = 0.025, s2 = 0.01, Lp1 = 10 6 cm
· s 1 · cmH2O 1 and
Lp2 = 9 × 10 6 cm · s 1 · cmH2O 1.
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In nephrons, urea and other end products of body metabolism are
concentrated before they are discharged. There is a net protein free
flux of fluid from the nephron segments to the ISF. In the steady
state, this volume flux has to be reabsorbed by the circulation in the
renal medulla. The ability of our model system to reabsorb fluid,
therefore, is one of the criteria that determine whether the system as
a whole is physiologically reasonable. In Fig.
8, the net fluid reabsorption by the
system, Je (total fluid reabsorption by the AVR
total filtration by the DVR), is plotted against different values of G
when U0 = 500 and 1,000 µm/s. Here,
Je is normalized by the flow rate at the entrance of the
DVR,
U0
r12,
when U0 = 500 µm/s. It is shown that at
higher values of G, the AVR are capable of reabsorbing fluid at higher
rates. Higher values of U0 also increase the net
reabsorption of water into the AVR. In antidiuresis, there is a
reduction in flow through the vasa recta (32,
36). Although a reduction in flow will tend to reduce
Je, this potential reduction will be tempered, if not
reversed, by the accompanying increase in G.

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Fig. 8.
Effects of G on the net reabsorption of fluid by the
countercurrent system at U0 = 500 and 1,000 µm/s. p1 = 0.9, p2 = 0.6, s1 = 0.025, s2 = 0.01, Lp1 = 10 6 cm
· s 1 · cmH2O 1 and
Lp2 = 9 × 10 6 cm · s 1 · cmH2O 1. Results are normalized by the flow
rate at the entrance of DVR,
U0 r12,
when U0 = 500 µm/s.
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Effects of the diffusive permeability of the DVR and the AVR to plasma
proteins, Pp1 and Pp2,
respectively, on protein clearance have also been investigated. In
Table 2, we present the effects of
Pp1 and Pp2 on the mean
concentration of proteins in the ISF. The bold value of
p0 in Table 2 corresponds to the values of Pp1 and Pp2 used in the
previous calculation. It is seen that the protein diffusive
permeability has negligible effects on the transport of plasma
proteins. When Pp1 increases 10 times from 2 to
20 × 10
8 cm/s, there is <5.4% increase in
the mean concentration of the plasma proteins in the
interstitium. Similarly, when we increase Pp2 by
an order of magnitude, there is a <6% increase in
p0. This is consistent with our assumption that
transcapillary exchange of plasma proteins in the renal medulla is
largely by convection.
Conclusions
In this study, we have investigated the hypothesis that, in the
absence of lymphatics in the renal medulla, plasma proteins in the
interstitium are cleared by the ascending vasa recta through fluid
reabsorption. A model of the countercurrent exchange system of the DVR,
AVR, and ISF has been built with basic equations governing the
transcapillary exchange of solutes and water. The focus of the study is
on the function of the system when parameters take values from
experimental measurements. We have also investigated whether the
steady-state distribution of solute and flow velocity predicted by the
model agree with available data measured in the renal medulla. The
countercurrent exchange system has been found to reach a steady state
when employing physiological data for its parameters, which confirms
that the leakage of plasma proteins from the DVR into the ISF can be
balanced by their clearance into the AVR. Indeed, with our initial
choice of parameters, the model reduced the ISF protein concentration
to values that were very much lower than those that have been estimated experimentally.
Although our model is greatly simplified (e.g., assumption of linear
gradient of small solutes; omission of anastomoses and capillary beds
between DVR and AVR), the rise in protein concentration that it
predicts in the plasma flowing in the DVR agrees well with experimental
estimates. By contrast, the predicted rapid fall in plasma protein
concentration at the beginning of the AVR (i.e., between
x = 1 and x = 0.8) is much greater than
experimental measurements indicate. We have already noted that this is
a consequence of the sudden change in the permeability properties of
vessels as they change from being DVR to AVR, together with the
(unrealistic) step change in intravascular pressure. In addition, our
model has assumed a step change in the number of vessels at the turn of
the loop. The presence of large numbers of anastomosing capillaries between the DVR and AVR that we have omitted from our model, together with more realistic pressure gradients in the vessels, should lead to a
more gradual reversal of fluid filtration into fluid uptake as blood
flows from the DVR to the AVR. These should give rise to a much slower
fall in Cp2 with the depth over the initial part of the
AVR. To incorporate these features into our model, however, would
require developing a multiunit system that would have taken us beyond
the aims of the present investigation. Nevertheless, it should be noted
that when mean Cp2 is estimated over the initial segment of
the AVR (i.e., between x = 1 and x = 0.8), its value is not much less than those in published data for
protein concentration in AVR plasma. The very low steady-state values
of ISF protein that the model predicted by using the initial set of
parameters were a surprise. It appears that a major reason for this was
the value of
s2 that we initially selected. If its real
value is <0.005, then higher values of
p0 will be
predicted, and these should fall well into the range of measured values.
A further omission from our model is the consideration of a radial
gradient of protein concentration in the ISF. Edwards and Pallone
(2) have pointed out that influx of fluid into the AVR may
lead to unstirred layers of protein around these vessels. These
gradients will reduce the oncotic pressure difference across the walls
of the AVR, limiting fluid uptake. Edwards and Pallone draw attention
to the apparent "safety mechanism" for fluid uptake in such
circumstances. A reduction in fluid uptake into the AVR, in the face of
steady influx of fluid into the medullary ISF from the nephrons, will
lead to an increase in ISF hydrostatic pressure. This may rise above
that in the AVR without the AVR collapsing (10). A rise in
ISF pressure of only 1-2 cmH2O above the AVR pressure
should ensure adequate fluid clearance by the vasculature. As noted
previously (11), such conditions would greatly favor the
clearance of protein from the ISF into the AVR. The predictions of the
present model, however, suggest that this method of protein clearance
may occur only occasionally.
It is possible that in this paper we may have overestimated the
convective influx of protein into the ISF from the DVR. The movement of
water from the DVR into the ISF occurs largely through aquaporin
channels (21). Although this efflux of water will concentrate the
protein in the DVR, it will not be coupled to a protein efflux. It
will, nevertheless, steepen the protein concentration gradients across
the walls of the DVR and should promote the transport of proteins from
these vessels into the ISF by other pathways. Furthermore, protein is
likely to be lost from the intervening capillaries, particularly if
these vessels are in regions where the efflux of fluid from the
circulating plasma is gradually reduced and reversed into influx from
the ISF.
Despite its shortcomings, the model does indicate that the clearance of
plasma proteins from the medullary ISF into the AVR plasma can occur
efficiently by a convection mechanism. This mechanism is
possible only because of the different permeabilities of the DVR and
AVR (most critically that
p1 >
p2 and
Lp2 > Lp1) and is facilitated by the
continual addition of protein-free fluid to the ISF as a result of
reabsorption by the nephrons.
Address for reprint requests and other correspondence: C. C. Michel, Cellular and Integrative Biology, Div. of Biomedical Sciences, Imperial College School of Medicine, London SW7 2AZ, UK.
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