A recent model of
volume and solute microvascular exchange in the renal
medulla was extended by simulating the deposition of NaCl, urea, and
water into the medullary interstitium from the loops of Henle and
collecting ducts with generation rates that undergo spatial variation
within the inner medullary interstitium. To build an exponential
osmolality gradient in the inner medulla, as suggested by Koepsell et
al. (H. Koepsell, W. E. A. P. Nicholson, W. Kriz, and H. J. Höhling. Pflügers Arch. 350: 167-184, 1974), the ratio of the interstitial area-weighted generation rate of small
solutes to that of water must increase along the corticomedullary axis.
We satisfied this condition either by holding the area-weighted generation rate of water constant while increasing that of NaCl and
urea or by reducing the input rate of water with medullary depth. The
latter case, in particular, yielded higher solute concentrations at the
papillary tip. Assuming that the fraction of the filtered load
recovered by inner medullary vasa recta for water, NaCl, and urea is
1%, 1%, and 40%, respectively, papillary tip osmolality is 1,470 mosmol/kgH2O when urea generation and NaCl generation per
unit volume of interstitium increase exponentially and linearly, respectively. The inner medullary osmolar gradient also increases further when 1) medullary blood flow is reduced, 2)
hydraulic conductivity of descending vasa recta (DVR) is lowered, and
3) vasa recta permeability to NaCl and urea is maximized. The
coupling between water and small solute transport, resulting from
aquaporin-1-mediated transcellular flux in DVR, also enhances tip osmolality.
kidney; microcirculation; vascular transport; sodium; urea; mathematical model; urinary concentration.
 |
INTRODUCTION |
THE MICROCIRCULATION in the renal medulla must remove
solutes and water recovered from nephrons while preserving the
corticomedullary gradients of sodium and urea generated by the loops of
Henle and collecting tubules. This dual task is made possible by the
countercurrent arrangement of the vasa recta. Descending vasa recta
(DVR) arise from the efferent arterioles of juxtamedullary glomeruli,
travel through the outer medulla, and diverge at varying levels to be converted into ascending vasa recta (AVR), which then return to the
cortex. The structural properties of DVR and AVR differ considerably. DVR have continuous endothelial cells and intercellular tight junctions, whereas AVR have a highly fenestrated endothelium.
The precise role of the renal medullary microcirculation in regulating
water and solute excretion remains to be elucidated. Models of the
urinary concentrating mechanism have generally neglected the role of
vasa recta by assuming that the capillaries offer negligible resistance
to transport of solute and water (13, 30). However, anatomical
differences between the outer and inner medulla, ultrastructural
heterogeneities between DVR and AVR, and the existence of facilitated
transport pathways in DVR suggest that the microvasculature plays a
significant role in the trafficking of sodium and urea. We therefore
recently developed a multiunit model of water and solute exchange
between vasa recta and the interstitium (7). Both across DVR and AVR,
the paracellular pathway is shared by water and solutes, and transport
is driven by hydraulic and oncotic pressure differences across the
walls (classic Starling forces). In addition, two transcellular
pathways are present in DVR: aquaporin-1 water channels (AQP1), which
are impermeable to all solutes, and urea transporters, the presence of
which raises DVR permeability to urea by a factor of 4 (24).
Water, sodium, and urea are reabsorbed into the interstitium from the
limbs of Henle's loop and the collecting ducts. These inputs were not
accounted for explicitly in our previous approach. Instead, exponential
increases in urea and sodium concentrations were specified in the inner
medulla based on electron microprobe data obtained by Koepsell et al.
(11), and the fractional osmolality due to sodium was determined by
interpolating the data of Atherton et al. (1). Wang and co-workers (36,
37) recently developed a simple but elegant analytic model of
microvascular exchange, in which a countercurrent capillary loop is
embedded in a secretory epithelium. Their latest predictions show good
agreement with the experimental data of Koepsell et al. (11), but their
theoretical approach is constrained by the need for analytic, closed
solutions. Wang and Michel (37) do not account for the exchange of
fluid and macromolecules, thereby neglecting significant coupling
between the transport of water, small solutes, and proteins. Moreover, they specify a constant solute input rate into the interstitium per
unit axial length. Little is known about where and how much fluid and
small solutes are reabsorbed from the loops of Henle and the collecting
ducts. To determine how the spatial distribution of interstitial water
and solutes affects concentration profiles and the axial osmolality
gradient in the inner medulla, we extended our previous multiunit model
to account explicitly for water and solute input rates in the medullary interstitium.
 |
METHODS |
Microvascular transport is simulated in the entire medulla.
However, as described in Edwards and Pallone (7), we only
consider vasa recta that are destined to the inner medulla.
As a consequence, the number of DVR and AVR in the outer medulla is
taken to be constant. In addition, since water and solute exchanges can
only occur among DVR, AVR and the interstitium in the outer medullary vascular bundles, input rates are zero in the outer medulla.
Conservation equations. Mass balance and flux equations are
described in detail in previous studies (6, 7) and briefly summarized
below. If x is the axial coordinate, the total plasma flow
rate, QP, obeys the following conservation equation at
steady state
|
(1)
|
where
"+" and "
" apply to AVR and DVR, respectively. The
volume fluxes per unit membrane area (in cm/s) across the capillary wall and the red blood cell (RBC) membrane are denoted by
JPv and
JRv, respectively,
is the
ratio of cell-to-vessel surface area averaged over time, N is
the number of vessels, and D is the vessel diameter. Mass
balances for solutes to which RBCs are impermeable, such as sodium,
albumin, and other plasma proteins, can be written as
|
(2)
|
where
CPi is the molar plasma
concentration of solute i and
JPi its molar flux per unit membrane area (in
mmol · cm
2 · s
1)
from plasma to interstitium. Conservation of urea, which is also
exchanged across the RBC membrane, yields
|
(3)
|
where
JPu and
JPuc are the paracellular and
carrier-mediated transcapillary molar fluxes of urea, respectively, and
JRu is the molar flux of urea
across RBCs.
The paracellular and transcellular volume fluxes
(JPvp and
JPvt, respectively) from plasma to
interstitium are given by
|
(4)
|
|
(5)
|
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 other
plasma proteins, respectively, and
a is the reflection coefficient of the paracellular pathway to albumin. The interstitial concentration of solute i is denoted by
CIi, and its activity
coefficient by
i. The paracellular flux of
solute i (i = albumin, sodium, urea) across
vasa recta walls is given by
|
(6)
|
where Pi and
i
are the (paracellular) permeability and reflection coefficient of the
capillary wall to solute i, respectively, and Pe is the Peclet
number. Across the paracellular pathway, small solute reflection
coefficients are taken to be zero. The transcapillary carrier-mediated
flux of urea can be written as
where
Puc is the permeability of urea transporters.
Expressions for RBC fluxes are given in Edwards and Pallone (6).
Concentration polarization. In a previous study (7), we
examined the assumption that the accumulation of protein in the interstitium at the walls of AVR may eliminate oncotic pressure differences across the AVR barrier. In that case, instead of Eq. 4, the expression for the AVR transmural volume flux becomes
|
(8)
|
where
LAp is the (paracellular)
hydraulic conductivity of AVR, PA denotes the hydraulic
pressure in the AVR lumen, and PI the hydraulic pressure in
the interstitium (Since there are no AQP1 water channels in AVR, there
is no transcellular component in the volume flux). This hypothesis
leads to occasional inconsistencies. In some simulations,
PI remains very close to PA throughout a large
part of the medulla, meaning that very little reabsorption or
filtration occurs there, that concentration polarization must then be
negligible, and that oncotic pressure differences should be accounted for.
In this work, we have adopted the intermediate approach of assuming
that nonalbumin plasma proteins, but not albumin, exert oncotic pressure across the AVR wall. This is justified
because the dominant protein available to be polarized in the
interstitium is likely to be albumin and not larger globulins
to which vessel walls are less permeable. With this approach,
significant volume uptake occurs throughout the medulla, as
observed experimentally (28), and transmural volume flux across AVR can
be written as
|
(9)
|
Interstitial hydraulic pressure and small solute
concentration. The loops of Henle and collecting ducts across which
water, sodium, and urea are exchanged with the interstitium lie in
parallel with the vasa recta. Input rates are thus given per unit axial length per unit cross-sectional area of the interstitium. At any location along the corticomedullary axis, the sum of the fluxes from
DVR and AVR, weighted according to their respective surface areas, must
be equal and opposite to the rate of generation in the interstitium
|
(10a)
|
|
(10b)
|
|
(10c)
|
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 outer medulla, where in the vascular bundles the exchange of water,
sodium, and urea can only occur between vasa recta and interstitium. At
a given depth x, Eqs. 10a-10c when solved
simultaneously yield the interstitial hydraulic pressure
(PI) as well as sodium and urea interstitial concentrations.
The cross-sectional area of the inner medullary interstitium was
calculated based upon that of the inner medulla,
Aim. The latter was determined by Becker (2) and
can be approximated by the following
polynomial
|
(11)
|
where xim, the dimensional coordinate
along the corticomedullary axis in the inner medulla, is zero at the
junction between the outer and the inner medulla and
Lim at the papillary tip. The nondimensional
coordinate xim/Lim thus runs
from 0 to 1 between these two limits. The volume fraction of the
interstitium in the inner medulla was assumed to vary linearly from 0.1 at xim/Lim = 0.2 to 0.3 at
xim/Lim = 1 (9), so that
|
(12)
|
Combining the mass balance Eqs. 1-3 with Eqs.
10a-10c, the total amount of water or solute recovered by vasa
recta (VRR) is given by (see APPENDIX)
|
(13)
|
where
i = volume, sodium, urea.
Water and solute generation rates. Parameter values for input
rates were determined by considering the fractions of filtered volume
and solute that are reabsorbed into the inner medullary interstitium.
If CSi is the systemic
concentration of solute i and GFR is the glomerular filtration
rate, then the filtered load of solute i is given by the
product
CSi · GFR.
Estimates of the fractions of filtered volume, sodium, and urea
(fv, fNa, and fu, respectively)
that are recovered by vasa recta from the inner medullary interstitium
were obtained as follows.
Water and sodium, reabsorbed from the loop of Henle and collecting
duct, must be removed from the medullary interstitium by vasa recta.
Urea is reabsorbed from the collecting duct and secreted into the loop
of Henle; vasa recta must therefore remove only the net difference.
Literature estimates of the fraction of volume (fCDv) and sodium
(fCDNa) removed from the collecting duct
are given in Table 1. The data for the loop
of Henle are both sparse and difficult to interpret. One difficulty is
that fractional deliveries of filtered loads represent only those
nephrons that reach the papilla, where micropuncture can be performed.
In the rat, only about one- third of nephrons are of the long-looped
variety that send thin descending limbs beyond the outer-inner
medullary junction (14). Of those nephrons, only a small fraction
reaches the papillary tip, making extrapolation to the entire inner
medulla uncertain. If we assume that the contributions of the loop of
Henle and the collecting duct to inner medullary interstitial water are
similar, the data in Table 1 suggest that fv should be 4%.
There are little data that helps to gauge the amount of urea that must
be absorbed by vasa recta in the inner medulla of the concentrating
kidney. Marsh (18) measured tubular fluid-to-plasma concentration
ratios in the loop of Henle of the hamster and found that 280% and
90% of the single-nephron filtered load of urea is secreted into the
thin descending limb between base and tip and thin ascending limb
between tip and base, respectively. For the reasons described above, it
is difficult to infer the total fraction of the whole kidney filtered
load of urea secreted into the loop. The inner medullary collecting
duct reabsorbs large amounts of urea, and vasa recta must take up the
difference, but even an order of magnitude of that amount is hard to
approximate. It has been estimated that 10-35% of the filtered
load of urea is reabsorbed by the inner medullary collecting duct (10,
38, 40). End distal delivery of urea must represent an upper limit of
the amount of urea available to be reabsorbed by the collecting duct.
Late distal delivery of urea determined by micropuncture of superficial
nephrons has been reported in the range of 40-110% of the
filtered load (12, 25, 29).
Estimates of fv, fNa, and fu can
also be gathered from the theoretical work of Stephenson et al. (31)
and Wexler et al. (39). Stephenson et al. (31) extended the WKM model
(named for "Wexler, Kalaba, and Marsh") developed by
Wexler et al. (39) to study the effect of vasa recta flow on the
concentrating ability of the renal inner medulla. Given their results
for volume flow and small solute concentrations, which are based upon a
single DVR and two AVR (31), the fractions of filtered water, sodium, and urea that are recovered by vasa recta between the midpoint of the
upper inner medulla and the papillary tip can be determined. Assuming
that GFR is 784 µl/min (35) and that the number of DVR at the
junction between the OM and IM is about 5,850 (7), fv is
found to be approximately 1%, fNa 2%, and fu
50-60%, depending on the rate of radial diffusion (31). More
recently, Thomas (32) calculated that DVR cross the outer-inner
medullary junction carrying ~60% of the filtered load of urea while
AVR exit with ~80%, predicting recovery of 20% of the filtered load
of urea by the inner medullary microcirculation.
Different spatial distributions for the interstitial area-weighted
generation rates of water and small solutes were considered. Since the
reabsorption of urea occurs closer to the papillary tip than that of
sodium and water (8), input rates were made dependent upon position
along the corticomedullary axis. The following four cases were examined
|
(14a)
|
|
(14b)
|
|
(14c)
|
|
(14d)
|
where
ic,
il,
ie,
id, and
are given constants. Equations 14a,
14b, 14c, and 14d correspond to a constant, a
linear increase, an exponential increase, and a linear decrease along
the corticomedullary axis, respectively.
Parameters and computational methods. Parameter values such as
morphological data and baseline permeabilities are summarized in Table
2. The GFR was taken as 784 µl/min (35),
and we assumed systemic concentrations of sodium and urea of 150 and 5 mmol/l (or µmol/cm3), respectively. Hence, the filtered
loads of water, sodium, and urea are 1.3 × 10
2 cm3/s, 2.0 µmol/s, and
6.5 × 10
2 µmol/s, respectively.
At the corticomedullary junction in DVR, the hematocrit was taken as
0.25 and the single-vessel blood flow rate
(qB0) as 10 nl/min. Reflection
coefficient values are given in Edwards and Pallone (6).
Since the initial (i.e., at x = 0) values of flow rates and
concentrations are unknown in AVR, the ordinary differential equations (ODEs) expressing mass conservation in DVR, together with the three
algebraic equations for mass balance in the interstitium (Eqs.
10a-10c), were first solved simultaneously at each
value of x along DVR by assuming flow rate and concentration
profiles in AVR. As the ODEs corresponding to AVR with the three
algebraic equations for the interstitium were then solved back up along AVR, the AVR flow rate 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 less than 10
3. Simulations were
performed on an AlphaPC64 workstation (model Durango II), and
convergence was typically obtained in 5 min.
 |
RESULTS |
The electron microprobe data of Koepsell et al. (11) indicate that
sodium concentrations increase exponentially along the corticomedullary
axis. Since the fraction of osmolality due to urea increases from about
2 to 50% between the corticomedullary junction and the papillary tip
(22), the concentration of urea must also increase exponentially. The
baseline case for our simulations was chosen so as to be consistent
with those results and to yield a papillary tip osmolality in the range
of 1,700 mosmol/kgH2O (19). The total amount
of water recovered by vasa recta in the inner medulla, fv,
was taken as 1.0% of the filtered volume (i.e., 1.3 × 10
4 cm3/s), and the
fractions of filtered sodium (fNa) and urea
(fu) recovered by the medullary microvasculature were
assumed to be 1.0% and 40.0%, respectively (i.e., 2.0 × 10
2 µmol/s and 2.6 × 10
2 µmol/s). The interstitial
area-weighted generation rate of water in the inner medullary
interstitium was decreased linearly from its maximum value at the
junction between the outer and the inner medulla to zero at the
papillary tip, whereas that of sodium was increased linearly between
these two points. The interstitial area-weighted input rate of urea was
increased exponentially, to reflect the fact that urea appears to be
secreted in the interstitium mostly toward the papillary tip;
specifically, we assumed that
u is proportional to
exp[6(xim/Lim
1)]. The corresponding values of
v,
Na, and
u are given in Table
3. Those assumptions are further discussed
below.
Based upon these parameters, variations in plasma flow rate and in
sodium and urea concentrations along the corticomedullary axis are
illustrated in Figs. 1-4. Shown in Figs.
1 and 2 are the overall and single-vessel plasma flow rates, respectively. Starling forces favor volume uptake through the paracellular pathway. However, when small solutes, to which water channels are impermeable, are much
more concentrated in the interstitium than in DVR, the transcellular volume flux across DVR water channels is directed toward the
interstitium, resulting in fluid loss. The model predicts that DVR
plasma flow decreases along the corticomedullary axis, both overall and
in each vessel, except near the papillary tip. Efflux from DVR stems from the significant sodium and urea interstitium-to-DVR concentration gradients (see below). Conversely, the AVR plasma flow rate increases as blood flows back toward the corticomedullary junction: AVR in the
inner medulla must recover both the water generated in the interstitium
(i.e., from nephron loops) and that which is driven out from DVR; in
the vascular bundles of the outer medulla, only the latter term is
present. Near the papillary tip, there is no or very little water
generated into the interstitium, and the force balance favors a slight
volume efflux from AVR, necessarily accompanied by a parallel influx
into DVR. Overall, most of the water recovered by vasa recta is taken
up by AVR, and the AVR-to-DVR blood flow rate ratio is 1.14 at the
corticomedullary junction. The interstitial hydraulic pressure
(PI) drops from about 1 mmHg at the corticomedullary
junction to
11 mmHg at the papillary tip.

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Fig. 1.
Ratio of total plasma flow rate (QP) to initial descending
vasa recta (DVR) blood flow rate as a function of position along the
corticomedullary axis (x). L represents the total
length of the medulla. Junction between outer medulla and inner medulla
corresponds to x/L = 0.24. Effects of increasing the
initial blood flow rate in a single DVR from 10 to 20 nl/min
(B) and hydraulic conductivity of DVR from 1.4 × 10 6 to 3.4 × 10 6
cm · s 1 · mmHg 1
(C) are shown relative to the baseline case (A).
Generation rates for water, sodium, and urea are those of the baseline
case.
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Fig. 2.
Ratio of single-vessel plasma flow rate (qP = QP/N) to initial single DVR blood flow rate as a
function of position in DVR and AVR. Effects of increasing the initial
blood flow rate from 10 to 20 nl/min (B) and hydraulic
conductivity of DVR from 1.4 × 10 6
to 3.4 × 10 6
cm · s 1 · mmHg 1
(C) are shown relative to the baseline case (A).
Generation rates for water, sodium, and urea are those of the baseline
case. Discontinuity at the papillary tip is due to the fact that each
DVR gives rise to 2.25 AVR.
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Sodium and urea concentrations are plotted as a function of medullary
depth in Figs. 3 and 4, respectively. In
both cases, the increase in DVR plasma concentrations follows the
increase in interstitial concentrations, albeit with some delay, since permeabilities are finite and the countercurrent exchanger is not
ideal. As plasma flows back up AVR, the decrease in CNa and Cu is parallel to that in the interstitium but also slower
(curves corresponding to interstitial concentrations are not shown for clarity; they would be in between those for DVR and AVR). As described in Edwards and Pallone (6, 7), the concentration rise in outer
medullary DVR (OMDVR) is accentuated by the sieving of small solutes by
AQP1 water channels. In the inner medulla (IM), starting from the
junction between the outer and inner medulla, concentrations increase
slowly at first and then more rapidly, parallel with the rise in
interstitial area-weighted solute generation rates (as described above,
Na and
u increase linearly and
exponentially, respectively). This effect is enhanced by the fact that
the amount of water generated into the IM interstitium is progressively
smaller, as
v decreases linearly along the
corticomedullary axis. Near the papillary tip, concentrations reach a
small plateau due to the slight water influx into DVR (see above). The
total osmolality at the papillary tip is 1,470 mosmol/kgH2O, 47% of which is due to urea. Because
Cu must increase from 5 mmol/l initially in DVR to about
700 mmol/l at the papillary tip (vs. 150 to about 400 for
CNa), the AVR-to-DVR concentration ratio at the
corticomedullary junction is six times as high for urea as it is for
sodium. Indeed, when axial (corticomedullary) gradients are steep,
transmural gradients must also increase to give rise to the large
solute fluxes that are needed to concentrate and then dilute plasma.

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Fig. 3.
Ratio of plasma sodium concentration
(CPNa) to its initial concentration in
DVR (CPNa0) as a function of position,
assuming that the area-weighted generation rate of water in the inner
medullary (IM) interstitium decreases linearly with x
(A, baseline case), remains constant (B), or increases
linearly (C). Fraction of filtered load recovered by vasa recta
is equal to 1% for water, 1% for sodium, and 40% for urea.
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Because of the large permeability of vasa recta to sodium and urea,
diffusion is the predominant mechanism by which these small solutes are
transported across the vessels. The Peclet number for sodium and urea
ranges from 10
4 to
10
2, and Eq. 6 could be
approximated by the classic Kedem-Katchalsky equations.
Convection plays a more significant role in the transport of albumin,
for which the Peclet number is between 5 × 10
2 and 15.
Water generation rate. To assess the validity of our baseline
case assumptions, we first modified the distribution of the area-weighted generation rate of water (
v), without
changing the overall amount of water recovered by vasa recta (i.e.,
fv = 1.0%). The input rates for sodium and urea were
exactly as in the baseline case (see Table 3 for values).
If the area-weighted generation rate of water in the inner medullary
interstitium is kept constant with depth, then plasma concentrations
increase much less than in the baseline case, as shown in Figs. 3 and
4, even though the total amount of water that has to be recovered by vasa recta remains the same. Since water is
generated at a constant rate throughout the inner medullary interstitium, urea and sodium are more diluted toward the papillary tip, where the osmolality is only 715 mosmol/kgH2O.
Assuming a linear increase in
v, these effects are even
more accentuated, as illustrated in Figs. 3 and 4; the osmolality at
the papillary tip is then only 565 mosmol/kgH2O.

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Fig. 4.
Ratio of plasma urea concentration (CPu)
to the initial sodium concentration in DVR
(CPNa0) as a function of position,
assuming that the area-weighted generation rate of water in the IM
interstitium decreases linearly with x (A, baseline
case), remains constant (B), or increases linearly (C).
Fraction of filtered load recovered by vasa recta is equal to 1% for
water, 1% for sodium, and 40% for urea.
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Sodium generation rate. We then varied the spatial distribution
of
Na, keeping the overall amount of sodium recovered by vasa recta constant (i.e., fNa = 1.0%). The input rates
for water and urea were those of the baseline case (see Table 3 for
values). As expected, if
Na remains constant along the
inner medullary corticomedullary axis, the rise in plasma sodium
concentration is much smaller than in the baseline case (Fig.
5), since sodium is being generated at the
same rate throughout the IM rather than preferentially toward the tip.
Smaller interstitium-to-DVR sodium concentration gradients result in
less volume efflux from DVR, so that urea concentrations increase
slightly less than in the baseline case (results not shown). With a
constant
Na, the osmolality at the papillary tip is
1,155 mosmol/kgH2O, with a 51% contribution from urea.
Conversely, if the area-weighted generation rate of sodium increases
exponentially along the corticomedullary axis like that of urea, then
CNa rises very steeply in the inner medulla as illustrated
in Fig. 5; there is then more water efflux from DVR, urea
concentrations increase slightly more relative to the baseline case,
and the tip osmolality is 2,205 mosmol/kgH2O, 35% of which
is due to urea. With this assumption, however, PI is found
to be lower than
20 mmHg at the papillary tip.

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Fig. 5.
Effect of changes in distribution of interstitial sodium on sodium
concentration, assuming that the interstitial area-weighted generation
rate of water decreases linearly with x in the IM. Ratio of
plasma sodium concentration (CPNa) to its
initial concentration in DVR (CPNa0) is
plotted as a function of position. Area-weighted generation rate of
sodium in the IM interstitium is assumed to be constant (A), to
increase linearly (B, baseline case), or to increase
exponentially (C). Fraction of filtered load recovered by vasa
recta is equal to 1% for water, 1% for sodium, and 40% for urea.
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Results corresponding to a constant area-weighted generation rate of
water are shown in Fig. 6. Even when
Na increases exponentially along the corticomedullary
axis, plasma sodium concentrations remain lower than in the baseline
case, and the osmolality at the papillary tip is only 915 mosmol/kgH2O, with a 40% contribution from urea.

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Fig. 6.
Effect of changes in distribution of interstitial sodium on sodium
concentration, assuming that the interstitial area-weighted generation
rate of water remains constant in the IM. Ratio of plasma sodium
concentration (CPNa) to its initial
concentration in DVR (CPNa0) is plotted
as a function of position. Area-weighted generation rate of sodium in
the IM interstitium is assumed to be constant (A), to increase
linearly with x (B), or to increase exponentially
(C). Fraction of filtered load recovered by vasa recta is equal
to 1% for water, 1% for sodium, and 40% for urea.
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Urea generation rate. Assuming an exponential increase in the
area-weighted generation rate of urea as in the baseline case, we then
examined the effect of changes in the rate of increase by varying the
multiplying factor
in the expression for
u (Eq. 14c). The assumptions regarding input rates for water and sodium were those of the baseline case, and the fraction of filtered urea that
is recovered by vasa recta (fu) was maintained at 40%. Results for urea are shown in Fig. 7. As
expected, the steeper the increase in
u, the greater the
rise in inner medullary urea concentration and the higher the tip
osmolality. Assuming that
u =
uc
exp[2(xim/Lim
1)], the osmolality at the papillary tip is only 1,110 mosmol/kgH2O, 36% of which is due to urea. With
u =
uc
exp[10(xim/Lim
1)], the tip osmolality is as high as 2,055 mosmol/kgH2O, with a 55% contribution of urea. In the
latter case, however, PI is about
15 mmHg at the
papillary tip. As
is increased, interstitial-to-DVR urea
concentration gradients become larger so that there is more water
efflux from DVR; plasma sodium concentrations are thus slightly raised
as well (results not shown).

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Fig. 7.
Effects of changes in distribution of interstitial urea on urea
concentration, assuming that the interstitial area-weighted generation
rate of water decreases linearly with x in the IM. Ratio of
plasma urea concentration (CPu) to the
initial sodium concentration in DVR
(CPNa0) is plotted as a function of
position. Area-weighted generation rate of urea in the IM increases
exponentially as u = ue
exp[ (xim/Lim 1)], and the factor is varied. Fraction of filtered load
recovered by vasa recta is equal to 1% for water, 1% for sodium, and
40% for urea. Results are given for = 2, 4, 6, 8, and 10. Corresponding values of ue are 2.73, 4.26, 5.88, 7.53, and 9.20 µmol · cm 3 · s 1,
respectively.
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Variations in the functional form of
u were then
examined. The overall amount of urea recovered by vasa recta was kept
constant (i.e., fu = 40%), and the input rates for water
and sodium were those of the baseline case. As expected, if
u increases linearly rather than exponentially in the
inner medulla, then the rise in Cu is more moderate, as
illustrated in Fig. 8. Because of smaller interstitium-to-DVR urea concentration gradients, water efflux from DVR
is reduced and plasma sodium concentrations are also slightly lower
than in the baseline case. The osmolality at the papillary tip is then
only 1,100 mosmol/kgH2O, with a 35% contribution from
urea. These effects are even more pronounced when
u
remains constant (Fig. 8). In this case, the tip osmolality is 955 mosmol/kgH2O, 28% of which is due to urea.

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Fig. 8.
Effect of changes in distribution of interstitial urea on urea
concentration, assuming that the interstitial area-weighted generation
rate of water decreases linearly with x in the IM. Ratio of
plasma urea concentration (CPu) to the
initial sodium concentration in DVR
(CPNa0) is plotted as a function of
position. Area-weighted generation rate of urea in the IM interstitium
is assumed to be constant (A), to increase linearly with
x (B), or to increase exponentially (C,
baseline case). Fraction of filtered load recovered by vasa recta is
equal to 1% for water, 1% for sodium, and 40% for urea.
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Results corresponding to a constant area-weighted input rate of water
are shown in Fig. 9. If the interstitial
area-weighted generation rate of urea is also assumed to remain
constant along the corticomedullary axis, then urea concentrations
actually decrease slightly in the inner medulla. Indeed, since water is
generated throughout the inner medullary interstitium and since
interstitium-to-DVR urea concentration gradients are lower, the driving
force balance favors volume influx into DVR in the IM, diluting both
sodium and urea. When
u is taken to increase linearly
along the corticomedullary axis, the concentration of urea overall
increases slightly in the inner medulla, as shown in Fig. 9. With a
constant
u, the osmolality at the papillary tip is found
to be 535 mosmol/kgH2O; with a linear
u, it
is 600 mosmol/kgH2O (the contribution of urea is then 24 and 34%, respectively).

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Fig. 9.
Effect of changes in distribution of interstitial urea on urea
concentration, assuming that the interstitial area-weighted generation
rate of water remains constant in the IM. The ratio of plasma urea
concentration (CPu) to the initial sodium
concentration in DVR (CPNa0) is plotted
as a function of position. Area-weighted generation rate of urea in the
IM interstitium is assumed to be constant (A), to increase
linearly with x (B), or to increase exponentially
(C, baseline case). Fraction of filtered load recovered by vasa
recta is equal to 1% for water, 1% for sodium, and 40% for urea.
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Blood flow rate variations. We then examined the sensitivity of
our results to perturbations in flow rates and permeabilities. Assumptions regarding input rates were those of the baseline case. Effects of changes in the initial blood flow rate are illustrated in
Figs. 1 and 2. The single DVR blood flow rate at the corticomedullary junction was increased to 20 nl/min, that is, to twice its value in the
baseline case (10 nl/min). With a higher initial plasma flow rate,
changes in solute concentration are slower, so that CNa and
Cu remain lower and transmural volume efflux from DVR is
reduced, as shown in Fig. 2. As expected, the total osmolality at the
papillary tip varies inversely with the efferent blood flow rate (Table
4).
Permeability variations. DVR (paracellular) hydraulic
conductivity was determined by Pallone et al. (24) as 1.4 × 10
6
cm · s
1 · mmHg
1.
However, the authors indicated that this value represents a lower
bound, because Starling forces may have been overpredicted; an upper
limit estimate of 3.4 × 10
6
cm · s
1 · mmHg
1
was suggested. Results corresponding to this upper bound for the
paracellular permeability of DVR
(LDp) are shown in Figs. 1 and 2.
In the baseline case, as in this case, the DVR paracellular flux is
actually positive (i.e., directed toward the interstitium) near the
corticomedullary junction, because the transcapillary hydraulic
pressure difference is greater than the oncotic pressure gradient (as
volume efflux proceeds, proteins will become more concentrated in the
lumen, and the balance will be reversed all the way toward the
papillary tip). Hence, an increase in
LDp first results in a greater
fluid loss from DVR. Then, as the paracellular flux changes sign,
volume influx into DVR is much more significant than in the baseline
case. Single-vessel plasma flow rates thus remain consistently higher
in the inner medulla, as illustrated in Fig. 2, thereby reducing small
solute concentrations and the osmolality at the papillary tip (Table 4). Increasing the paracellular hydraulic conductivity of AVR (LAp) in a similar way does not
produce significant effects, however, as the amount of water taken up
by AVR is primarily determined by the interstitial volume generation
rate. Rises in LAp are essentially
translated into reductions in
P, i.e., into higher interstitial
hydraulic pressures.
We then varied the permeability of vasa recta to sodium or urea,
without altering inner medullary input rates. Since the amount of
solute recovered by vasa recta remains the same when generation rates
are fixed, the smaller concentration difference between AVR and DVR
resulting from a higher permeability must be accompanied by an overall
increase in concentration or in plasma flow rate. When the permeability
of DVR to sodium is doubled, interstitium-to-DVR sodium concentration
gradients decrease, thereby reducing volume efflux from DVR through
water channels. Plasma flow rates are then higher, thus compensating
for the smaller AVR-to-DVR sodium concentration differences.
Conversely, if the permeability of AVR to sodium is multiplied by two,
then the larger interstitium-to-DVR sodium concentration gradient
results in slightly more efflux in DVR, thereby increasing both sodium
and urea concentrations.
Varying the paracellular permeability of DVR to urea does not affect
volume efflux from DVR as much as a similar change in the permeability
to sodium does, since a large fraction of urea is transported by
carriers in DVR. When the permeability of DVR or AVR to urea is raised,
Cu increases in AVR and DVR to compensate for the smaller
transmural gradients. As expected, increases (2-fold) in vasa recta
permeability to small solutes yield higher osmolality at the papillary
tip (Table 4).
Corticomedullary variations of transport properties. We assumed
in this study that permeability values are the same in outer and inner
medullary vasa recta (Table 2). Although the average permeability to
sodium (PNa) of OMDVR and IMDVR is 75 × 10
5 cm/s by in vitro and in vivo
microperfusion (24), some DVR have a very low PNa.
It is also likely that PNa measurements in IMDVR
were underestimated due to boundary layer effects on isotope efflux
(24). To investigate the effect of axial variations in the permeability
of DVR to sodium, we increased PNa from 10 × 10
5 cm/s at the corticomedullary
junction to 150 × 10
5 cm/s at the
papillary tip along DVR, then compared the results to simulations in
which DVR PNa remains uniform and equal to 75 × 10
5 cm/s throughout the medulla.
Increasing the permeability of DVR to sodium along the corticomedullary
axis results in more water efflux from OMDVR and less from IMDVR.
Indeed, the more permeable the vessels, the smaller the
interstitium-to-DVR concentration gradients, the smaller the driving
force for volume efflux through AQP1 water channels, and vice versa. At
the papillary tip, the plasma flow rate is the same as in the baseline
case, but sodium concentrations are higher and the total osmolality is
about 1,640 mosmol/kgH2O (vs. 1,470 in the baseline case).
Hence, a gradual increase in sodium permeability along the
corticomedullary axis seems to enhance the osmolar gradient in the
inner medulla.
Although there is evidence that the permeability to sodium of OMDVR and
IMDVR may be different (24), the data concerning urea are more
difficult to interpret. Whereas permeability measurements by Pallone et
al. (24) suggested the presence of a facilitated transport pathway for
urea in OMDVR only, more recent in situ hybridization experiments have
shown that the UT3 urea transporter is expressed in IMDVR as well (33).
Boundary layer effects may have limited the in vivo microperfusion
experiments of Pallone et al. (24), and it is difficult to speculate on
the axial variations of DVR permeability to urea. As for water, the
paracellular hydraulic conductivity of DVR appears to be the same in
the outer medulla (34) and the inner medulla (23); there is no evidence
that AQP1 expression (and hence the transcellular hydraulic
conductivity of DVR) varies with axis either. Finally, measurements of
the paracellular hydraulic conductivity of AVR have been limited to the
inner medulla.
 |
DISCUSSION |
This model of medullary microvascular exchange incorporates variable
water and solute generation rates in the medullary interstitium as a
means of accounting for deposition of solutes and water by nephrons and
the collecting duct. Complete transport equations are employed to
simulate the coupling between microvascular transport of solutes and
water. The current model follows the lead of Wang and co-workers (36,
37), who developed a model of microvascular exchange in which a
countercurrent capillary loop is embedded in a secretory epithelium.
They obtained elegant analytic solutions that reproduced the
exponential corticomedullary gradient described by Koepsell et al.
(11). The desire to obtain a closed solution and thereby avoid
numerical integration necessitated several constraints. In contrast, we
have accepted the need for extensive numerical computation to enable
exploration of the coupling of solvent and solute transport, water
channels, facilitated urea transport, RBC and paracellular fluxes,
variations in the number of vessels, and complex spatial profiles of
solute and water generation rates.
Interstitial generation rates. The principal finding of this
study is that the smaller fv and the higher
fNa, the greater the osmolality predicted at the papillary
tip. We have chosen a value of 1% for fv, which is also
consistent with the theoretical work of Wexler et al. (39) and
Stephenson et al. (31). Our choice of fNa was restricted by
the condition that the contribution of urea to papillary tip osmolality
be close to 50% (22). A value of 1% was found to be the maximum value
that satisfied that requirement. The fraction of filtered urea that is
recovered by vasa recta, fu, was chosen as 40% to achieve
a papillary osmolality of about 1,500 mosmol/kgH2O,
assuming that the interstitial area-weighted generation rate for water
decreases linearly along the corticomedullary axis while those for
sodium and urea increase linearly and exponentially, respectively. This
value for fu is well within the range of numbers used in
other modeling studies (see METHODS), but is possibly close
to upper limits in the antidiuretic rat kidney. Reducing fu
in this model to 20%, as obtained by Thomas (32), would reduce the
predicted papillary tip interstitial concentration and osmolality (the
latter by about 30%). If fu is only 20%, we conclude that
v needs to fall more sharply or that
u
has to rise more rapidly along the medullary axis than specified by
Eqs. 14c-14d for high interstitial osmolalities to exist
at the papillary tip of the antidiuretic kidney.
Table 1 suggests that the fraction of filtered sodium that is recovered
between the base and the tip of the papilla (i.e., about a third of the
inner medulla) by vasa recta from the collecting duct alone is about
0.5%. Besides, Henle's limbs contribute an additional, unknown
amount. Hence our baseline estimate of 1% for the overall amount of
sodium recovered by vasa recta may be too low. The effects of
increasing fNa up to 2% are summarized in Table
5. Multiplying the fractional recovery of
filtered sodium by a given factor increases the papillary tip
concentration of sodium by roughly the same factor. Since the amount of
urea recovered by the microcirculation is kept constant, the fraction
of the papillary tip osmolality that is due to urea falls sharply, from 47% (fNa = 1%) to 36% (fNa = 1.5%) to 30%
(fNa = 2%). It is therefore difficult to reconcile the
experimental data in Table 1, which suggest a higher recovery of sodium
by vasa recta, with measurements of the relative contribution of urea
to the osmolar gradient, about 50% at the papillary tip (22). Using
our baseline case assumptions, the osmolality at the papillary tip is
predicted to be 1,470 mosmol/kgH2O. With
fNa = 2%, it increases to about 2,300 mosmol/kgH2O. If the fraction of filtered water
recovered by vasa recta is simultaneously increased to 2%, then
the papillary tip osmolality drops significantly, to 1,040 mosmol/kgH2O; with fv = 2% and
fNa = 1%, it is found to be 700 mosmol/kgH2O.
Our baseline case predicts that the fraction of filtered water
recovered by the microcirculation between the papillary base and tip
ranges from 0.02% when
v decreases linearly to 0.05% when
v is constant (per unit cross-sectional area of the
interstitium). Higher values of fv are more consistent with
experimental data obtained from micropuncture and
microcatheterizations of the collecting duct (Table 1). It
should be noted, however, that the majority of those studies were
performed after ureteral excision. Excision of the ureter
in the rats lowers urinary concentrating ability from a maximum of
~3,000 to less than 900 mosmol/kgH2O (4, 22, 26, 40),
well below the predictions of our baseline case simulations. Assuming
that the fraction of filtered water recovered by vasa recta
(fv) is 1%, the AVR-to-DVR blood flow rate ratio is found
to be 1.1 at the corticomedullary junction and also 1.1 at the base of
the papilla, i.e., about 2 mm from the papillary tip. Zimmerhackl et
al. (41) found a somewhat higher blood flow rate ratio of 1.2 at the
base of the papilla, implying greater volume uptake by the
microcirculation within the papilla. Those studies employed
videomicroscopic observation and micropuncture of the vessels on the
surface of the papilla, necessitating excision of the ureter, and
likely yielded papillary tip osmolalities lower than predicted by this model.
Interestingly, a low value for fv (i.e., 1%) leads to high
papillary tip osmolality but yields computations of negative papillary interstitial pressure. Negative interstitial pressures are obtained because transport rates of water across the highly conductive AVR wall
are low when fv is 1%, so that negative pressures are required to limit transmural water influx to that which preserves local
mass balance. Higher values for fv yield predictions of positive interstitial pressures. Attempts to experimentally measure pressure in the interstitium of the exposed papilla after ureteral excision yielded positive values, close to the hydraulic pressure in
the AVR lumen (21). MacPhee and Michel (16) have suggested that
interstitial hydraulic pressure in excess of that in AVR might be
required to drive renal medullary AVR volume uptake.
The negative values predicted by our model may be to some extent the
result of our simplified hypothesis regarding
concentration polarization. If we were to assume that the accumulation
of protein during fluid uptake completely eliminates all oncotic
pressure differences across AVR walls (i.e., use Eq. 8 rather
than Eq. 9), then hydraulic interstitial pressures would be
positive, close to the hydraulic pressure in AVR which is fixed at 7.8 mmHg. This assumption, however, leads to occasional inconsistencies as
discussed above, as when AVR fluxes are too small to render
polarization significant. A more rigorous treatment of concentration
polarization, which would include taking into consideration changes in
the AVR reflection coefficient to albumin along the corticomedullary
axis, would be needed to clarify this issue. Another possible
explanation involves the fact that the ureter undergoes rhythmic
contractions that intermittently squeeze the papilla. Based on the
predictions of this model, we are led to speculate that ureteral
contractions may serve to drive water flux into the AVR lumen by
raising interstitial pressure, followed by a period of negative
interstitial pressure that arises during the relaxation phase. Such
periodic reabsorption would allow time for interstitial-to-AVR
gradients of albumin, which accumulates on the interstitial side of the
AVR wall, to dissipate. Based on this study, it also seems probable
that excision of the ureter might in some manner lead to a shift in the
distribution of water by the loop of Henle and collecting duct such
that a larger fraction is present toward the papillary tip.
Concentration profiles. Our results indicate that sodium and
urea concentrations increase exponentially in the inner medulla, as
observed by Koepsell et al. (11), whenever the ratio of the interstitial area-weighted generation rate of small solutes to that of
water increases steadily along the corticomedullary axis. This
requirement is satisfied when
v is constant while
Na and
u increase from the junction
between the outer and inner medulla toward the papillary tip, or when
the area-weighted generation rate of water is the only one that
decreases in the inner medulla. In addition, for a given amount of
filtered water and solutes that is recovered by vasa recta overall, the
more evenly the solutes are distributed in the interstitium, the
smaller the rise in capillary concentrations along the corticomedullary
axis, and the lower the osmolality at the papillary tip. Sands and
Knepper (27) have reported that the inner medullary collecting duct is
more permeable to urea in its distal two-thirds, thus supporting the hypothesis that urea is preferentially reabsorbed toward the papillary tip.
Water and solute transport coupling. The existence of water
channels that are totally impermeable to solutes results in significant coupling between the transport of water, sodium, and urea. Although the
reflection coefficient of the paracellular pathway to sodium and urea
is zero, that of the transcellular pathway is unity, so that
concentration gradients across DVR walls will induce water fluxes.
Hence, an increase in the permeability of AVR to sodium will also
affect urea, because the larger interstitium-to-DVR sodium
concentration difference in the outer medulla leads to more volume
efflux from DVR through water channels and thus to a more rapid
increase in urea concentrations. Conversely, if both water and urea are
recovered into AVR at a constant rate throughout the inner medulla,
then transcapillary oncotic pressure differences become the dominant
driving force for water transport across DVR, and volume influx into
DVR occurs throughout the inner medulla, thereby reducing plasma sodium
concentrations significantly as well. The presence of a transcellular
pathway for water thus makes the concentration profiles of sodium and
urea highly interdependent.
Limitations. The model described above takes into account
volume and small solute generation rates in the inner medullary interstitium. Its main limitation concerns the transport of albumin and
concentration polarization. To rigorously account for the latter,
radial variations in fluid velocity and concentrations would need to be
considered, which would greatly increase the complexity of the model.
Instead, we assumed in this study that polarization is only significant
at the AVR wall, where fluid uptake from the interstitium can be very
large, and that the accumulation of albumin on the interstitial side of
the barrier results in canceling albumin oncotic pressure differences
across AVR (Eq. 9). We did not attempt either to elucidate the
source of albumin in the medullary interstitium and instead fixed the
interstitial concentration of albumin. Several studies have shown that
albumin is present in the interstitium in significant concentrations
(15, 20), but the mechanisms by which this extravascular pool of albumin is generated and maintained remain to be understood. Pallone (20) suggested that DVR are the source of interstitial proteins and
that convective rather than diffusive processes govern the accumulation
of albumin in the interstitium. It is possible in principle to maintain
steady fluxes of albumin from DVR through the interstitium to AVR (17),
and further mathematical simulations are needed to address this issue.
In summary, our results suggest that the ratio between the interstitial
area-weighted generation rate of small solutes to that of water must
increase along the corticomedullary axis to build an exponential
osmolality gradient in the inner medulla. Predicted concentration
gradients are especially steep if the input rate of water per unit
interstitial area is the only one that decreases along the
corticomedullary axis. The inner medullary osmolar gradient can also be
improved by reducing medullary blood flow, decreasing the hydraulic
conductivity of DVR, and increasing the permeability of vasa recta to
sodium and urea.
This work was supported by American Heart Association Grant
9730088N (to A. Edwards) and National Institutes of Health Grants HL-62220 and DK-42495 (to T. L. Pallone) and was performed during the
tenure of an Established Investigatorship of the American Heart
Association (to T. L. Pallone).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: A. Edwards,
Dept. of Chemical Engineering, Tufts Univ., 4 Colby St., Medford, MA
02155 (E-mail: aedwar01{at}tufts.edu).