Inner medullary lactate production and urine-concentrating
mechanism: a flat medullary model
Stéphane
Hervy and
S. Randall
Thomas
Institut National de la Santé et de la Recherche
Médicale Unit 467, Necker Faculty of Medicine, University of
Paris 5, F-75015 Paris, France
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ABSTRACT |
We used a mathematical model to explore
the possibility that metabolic production of net osmoles in the renal
inner medulla (IM) may participate in the urine-concentrating
mechanism. Anaerobic glycolysis (AG) is an important source of energy
for cells of the IM, because this region of the kidney is hypoxic. AG
is also a source of net osmoles, because it splits each glucose into
two lactate molecules, which are not metabolized within the IM.
Furthermore, these sugars exert their full osmotic effect across the
epithelia of the thin descending limb of Henle's loop and the
collecting duct, so they are apt to fulfill the external osmole role
previously attributed to interstitial urea (whose role is compromised
by the high urea permeability of long descending limbs). The present simulations show that physiological levels of IM glycolytic lactate production could suffice to significantly amplify the IM accumulation of NaCl. The model predicts that for this to be effective, IM lactate
recycling must be efficient, which requires high lactate permeability
of descending vasa recta and reduced IM blood flow during antidiuresis,
two conditions that are probably fulfilled under normal circumstances.
The simulations also suggest that the resulting IM osmotic gradient is
virtually insensitive to the urea permeability of long descending
limbs, thus lifting a longstanding paradox, and that this high urea
permeability may serve for independent regulation of urea balance.
urine-concentrating mechanism; anaerobic glycolysis; lactate; mathematical model
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INTRODUCTION |
THE DRIVING
FORCE, or "single effect," behind the development of the
inner medullary osmolality gradient that serves to concentrate urine
during its final passage along the collecting ducts has still not been
adequately explained. As has so often been the case, it is worthwhile
to quote an early paper by Carl Gottschalk (and Karl Ullrich)
(11)
Solute production in the inner medulla. As first
suggested by Ullrich (43), the liberation of osmotically
active solute, as in the acidifying mechanism or anaerobic metabolism
of glucose, would contribute to the osmolality in inner medulla. It
seems unlikely, however, that this is the sole mechanism responsible for the increasing tonicity in the inner medulla, and it is even more
difficult to attribute the increase in sodium concentration in this
area to such a mechanism. Quantitative considerations make it apparent
that solute production alone could not explain the entire urinary
concentrating process, and this need not be seriously entertained in
view of the known activity of the thick ascending limb of the loop of
Henle.
Given these doubts about significant papillary lactate
accumulation, which were supported by the earlier in vivo micropuncture results of Ruiz-Guinazu et al. (30), this idea was quietly abandoned.
We know now that the active transport of the medullary thick ascending
limb (MTAL) is limited to the outer medulla (OM), leaving open the
question of the single effect in the inner medulla (IM). The present
modeling study explores exactly the possibility mentioned by
Gottschalk, illustrating a scenario by which, in answer to his doubts,
the metabolically produced osmoles do not themselves constitute the
osmotic gradient but rather serve to amplify papillary NaCl
accumulation. We have proposed that metabolic production of net osmoles
(39, 42), and in particular lactate production by
anaerobic glycolysis (AG) (41), might constitute a
significant contribution to the IM single effect. It is well
established both that the inner medulla is hypoxic (7, 32)
and that lactate accumulates within the IM (8, 31).
In a simple model of the inner medullary vasa recta (41),
we previously calculated that lactate from AG could plausibly accumulate to significant levels within the papilla, given
physiological estimates of glycolytic rate and IM blood flow. In the
present work, we use a so-called "flat" model of the full medulla
to further explore the hypothesis that this recycling of IM lactate may
help generate the IM osmotic gradient. This model includes not only vasa recta but also loops of Henle and collecting ducts. It is "flat", as opposed to three-dimensional (3-D), in that it assumes all structures at each level are bathed by a common interstitium, in
the manner of classic "central core" models [although the
descending vasa recta (DVR) are treated here as full-fledged tubes, not
grouped with the ascending vasa recta (AVR)]. It is well established
(25, 47) that such flat (or "one-dimensional" or
central core) models cannot explain the steep IM osmotic gradient
observed in antidiuretic rodents while respecting measured permeability
values, the main problem being the high measured urea permeability of
long descending limbs, P
, in the IM
(6, 25).
Here, we show that addition of glucose and lactate to such a model (in
addition to the usual NaCl and urea) and the conversion of 15-20%
of entering glucose to lactate (each consumed glucose is converted by
AG to 2 lactates, thus generating net osmoles) result in a sizeable
osmotic gradient that is essentially insensitive to the
P
.
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MODEL DESCRIPTION |
The steady-state medullary model used here, illustrated in Fig.
1, includes vasa recta (DVR and AVR),
short and long Henle loops [descending (SDL and LDL) and ascending
(SAL and LAL)], and collecting ducts (CDs) and treats flows of volume,
NaCl, urea, glucose, lactate, and (only in the CD) KCl. It is thus a
system of 35 nonlinear, ordinary differential equations (5 flow
variables along 7 tubular structures). The AVR serve to represent the
interstitium surrounding all the structures. Rather than using explicit
inclusion of equations for transport along the distal tubules, inflow
to the outer medullary collecting duct (OMCD) is calculated from flows
exiting at the top of the ascending limbs (AHL), based on physiological
constraints representing the action of virtual distal tubules (see
Inputs and Boundary Conditions).

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Fig. 1.
Schematic diagram of the model. The ascending vasa recta (AVR)
represent the interstitium, which bathes all structures equally at
every level. Fv, volume flow; CNaCl,
Curea, Cglu, and Clac: NaCl, urea,
glucose, and lactate concentration, respectively; DVR,
descending vasa recta; LDL and LAL, long descending limb and long
ascending limb, respectively; SDL and SAL, short descending limb and
short ascending limb, respectively; LDL and LAL, long descending limb
and long ascending limb, respectively; CD, collecting duct; OS and IS,
outer stripe and inner stripe, respectively; UIM and LIM, upper inner
medulla and lower inner medulla, respectively; OM and IM, outer medulla
and inner medulla, respectively.
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This model corresponds closely to our 3-D models (39, 44),
with the following exceptions.
1) It is "flat" instead of 3-D; i.e., all exchange among
tubes passes via a common interstitial space instead of being
distributed among neighboring structures according to their relative
placement within each region.
2) We have added glucose and lactate as full-fledged solutes
and treated conversion of glucose to lactate (stoichiometry 1:2) within
the IM "interstitium," assimilated here to the AVR; this represents
glycolytic lactate production by all cells of the IM. The lactate thus
produced must transit by the interstitium because it is not consumed
within the IM (26). See Thomas (41) for a
comparison of our baseline glycolytic rate with available biochemical data from kidney and other tissues. Within the nephrons, we assume that
glucose and lactate concentrations are near 0 (as is generally reported
after the end of the proximal tubule). As explained further below, we
use the glucose solute within the nephron to formally represent
nonreabsorbable solutes, setting their initial concentration at 1 mM at
the entry to LDL and SDL (see Tables 3 and 5).
3) KCl is added to the fluid flowing into the collecting
ducts [a feature common to a previous model by Layton et al.
(25)].
Topology
Each type of tube is represented by a single, lumped tubular
structure, whose circumference at each depth reflects the total number
of such tubes at that depth. Within the IM, flows in the long
descending vasa recta (LDV) and in LDL are shunted directly to the long
ascending vasa recta (LAV) and LAL, respectively, in proportion to the
number of tubes that return at each depth. Here, we adopt the same
axial exponential loop distribution as in our recent 3-D medullary
models (39, 44), based on the reported anatomy of rat
kidney (13, 21). To be explicit, the number of tubes,
j, at depth x within the IM, i.e., for
x > xOM/IM, is given by
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(1)
|
with the factor describing the exponential decrease in their
number with depth (ksh) = 1.213 mm
1 for vasa recta and Henle's loops and
ksh = 1.04 mm
1 for IMCD, and
N(0) is the number entering the IM. Thus, compared with the
number of tubes entering the IM, the fraction of vasa recta and
Henle's loops reaching the papillary tip is 1/128 for an IM thickness
of 4 mm, and over the same distance, 64 OMCDs converge to a single
exiting collecting duct. Also in conformity with the 3-D models,
two-thirds of the descending vasa recta turn back within the inner
stripe of the OM (we call these the SDV), and the remaining third (the
LDV) extend at least part way into the IM, their number diminishing
exponentially as explained above. The SDV and LDV are distinct
structures in the WKM-type 3-D models, but in this flat model they are
lumped into a single structure, the DVR. For the whole system, the
basic scaling factor is NCD, 0 (=
64), the number of OMCD entering the OM. Table
1 gives the numbers of tubes at each
depth according to this scheme.
Because species other than the rat have different proportions of
tubes and vessels extending to the tip (2), everything is
scaled to the assumption of a single exiting CD. By this strategy, the
model can represent kidneys containing any number of nephrons simply by
varying the medullary length and/or the factor describing the
exponential decrease in their number with depth
(ksh).
Although it has long been recognized as a crucial parameter for
concentrating ability, the total IM blood flow relative to total flow
in the nephrons is not established in the literature due to the
difficulty of measuring it. We explore this in the parameter studies.
System Equations
The equations describing the changes in flows and concentrations
with depth in each tube are identical to those used in earlier models.
System variables are the axial tubular flows of water and solutes.
Concentrations of solutes i in tubes j are
calculated from the ratio of solute flow to volume flow,
C
= F
/F
. As in Thomas (41), shunt flows from descending tube
j to the corresponding ascending tube at depth x
are given by
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(2)
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We then have the following system of differential equations,
adopting the usual convention that descending tubule flows are positive
and ascending flows are negative
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(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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where in each case i refers to NaCl, urea, glucose, or lactate.
In these equations, transmural fluxes of volume and solute i
out of tube j are given by
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(9)
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The AVR concentrations here also represent the interstitial
concentrations surrounding all the other tubes. Notice also that there
is no mention in the above equations of glycolytic conversion of
glucose to lactate. This is because the conversion is limited to the
"interstitial space," i.e., the AVR. The AVR concentrations and
volume flow were calculated from the constraint of global mass balance,
as follows.
Conservation of mass for the medulla as a whole in the steady state
(35) says simply that, at any depth x, the
algebraic sum of flows of type i in all tubes j
(taking flows to be positive toward the papilla and negative away from
the papilla) must equal the exit rate of i from the terminal
collecting duct minus the total amount of i
synthesized from x to the papillary tip,
x = L:
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(10)
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where flows at level x are taken to be 0 in tubes that
do not extend all the way to x. The
S
term is, of course, 0 everywhere
for NaCl and urea and applies only in the AVR/interstitium for lactate
and for glucose (for which it is negative, because glucose is consumed).
As in our earlier vasa recta model (41), we specify the
total IM glycolytic glucose consumption,
Jgly,tot, as a percentage of total glucose
inflow into DVR, and the rate of glycolysis at a given depth is then
scaled to the number of vasa recta at that depth. To be specific,
calling the fractional glucose consumption Jrx,fract, we have
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(11)
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which is used to calculate Kgly, the
glycolysis rate per tube and per millimeter
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(12)
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Because in this model glucose is converted to lactate only within
the IM, we have within the OM (i.e., for x
xOM/IM)
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(13)
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and within the IM (i.e., for x > xOM/IM)
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(14)
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Boundary conditions at the bottoms of loops are based on tube
connectivity, where E indicates the end of tube j, i.e., at the tips of Henle's loops and at the bottom of vasa recta
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(15)
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In general, it is considered that there are no sources or sinks
(except glycolysis, for lactate and glucose), that hydrostatic pressure
plays a negligible role compared with osmotic pressure forces, and that
axial diffusion is negligible relative to convective flow of solutes
[these last 2 assumptions were discussed in Moore and Marsh
(28)].
Baseline Parameter Values
The baseline parameter values follow those of our earlier 3-D
model (39, 44) as closely as possible and are
given in Table 2.
Km for the pump equation in Eq. 9 was
taken as 50 mM.
High-Urea Permeability Parameter Values
To further explore the impact of high urea permeability of LDL
(P
), we also used a second parameter set
(taken from Table 2 in Ref. 25), which was based mainly on
permeability measurements in Henle's loops of chinchilla (although some of the values are from the rat literature, because there is not a
complete set of measurements for chinchilla). The chinchilla has been
reported to concentrate its urine as high as 7,600 mosM (46). We will call this the
"high-Pu" parameter set. Table 4 shows the
values that are different from the baseline set. Here, we did not adopt
the high value of LDL salt permeability used by Layton et al.
(25).
Inputs and Boundary Conditions
The inputs to the system are the volume flows and solute
concentrations at the entry into the LDL and SDL and into the vasa recta. F
and F
were set at 10 nl · min
1 · nephron
1 based
on a single-nephron glomerular filtration rate (SNGFR) of 30 nl/min and
ratio of inulin concentration in tubular fluid to that in urine
[(TF/P)inulin] of 3 at the end of the proximal tubule.
Fv into vasa recta was set at 7.5 nl · min
1 · tube
1 (as in
Ref. 39). For the LDL and SDL, entering concentrations of
urea, glucose, and lactate were set at 10 mM, 1 µM, and 1 µM, respectively. For the vasa recta, entering concentrations of urea, glucose, and lactate were set at 5, 5, and 2 mM, respectively. NaCl
concentrations were calculated from these, assuming global entering fluid osmolality of 263 mosM and an osmotic activity coefficient for NaCl of 1.82 (45).
Inputs to the OMCD.
Rather than include distal tubules explicitly, the entry to the
collecting ducts is calculated from flow and concentrations at the top
of the SDL and LDL, based on constraints deduced from the literature.
To calculate the volume flow and four concentrations into the OMCD, we
need five constraints. In particular, the following was assumed.
1) Fluid entering the OMCD is isosmotic to plasma and is
assigned the value OsmCD, 0 = 263 mosM.
2) A specified fraction, ufac = 0.85, of urea is delivered to OMCD [i.e., the distal tubules
reabsorb (1
ufac) of the urea delivered to early
distal tubules].
3) NaCl concentration entering the OMCD has a fixed value,
C
(0) = 35 mM; glucose and
lactate flows are conserved along the virtual distal tubules, i.e.,
their flows into OMCD equal the sum of their flows out of the LAL and SAL.
We also assume that KCl enters the OMCD at the fixed concentration
C
(0) (= 20 mM) but that its
absolute flow rate, F
= C
(0) F
(0), then remains unchanged
along the rest of the CD, and its concentration at depth x
is then C
(x) = F
/F
(x). This is used
along with the other solute concentrations to calculate the osmotic
driving force for water flux across the CD wall.
Thus, to be explicit, we can solve for volume flow entering the OMCD,
F
(0), by rearranging the equation for total
osmolality
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(16)
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thus obtaining
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(17)
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We then have, for the other values entering the OMCD
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(18)
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where we take the absolute values of the flows exiting the
ascending limbs.
Numerical Solution
The system was solved using a method based on that described by
Stephenson et al. (35) and used by us in an earlier model with six cascading nephrons (40). The differential
equations are approximated by finite difference equations centered in
space. If we consider tube j to be divided into n
slices, then the space-centered finite difference equations between
nodes k-1 and k are
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(19)
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where i represents flows of volume, NaCl, urea,
glucose, or lactate. Thus the fluxes
J
are
evaluated at the middle of the interval ["midpoint method" (38)], on the assumption that concentrations in the
middle of the interval are the arithmetic average of the concentrations at k-1 and k.
The solution proceeds as follows. An initial guess is made for the
interstitial/AVR concentrations, then these are taken as fixed, and
given the defined input volume flow and solute flows for LDL and SDL
and for the DVR, the equations for each tube are integrated stepwise
[we used a spatial chop of 120 slices (121 nodes)] in the direction
of flow using Newton's method on the system of five finite difference
equations and five unknowns (Fv and 4 concentrations,
Ci) and using an analytically calculated Jacobian matrix. We found it advantageous to use a much stricter error
tolerance (<10
10) on these "tubular" iterations than
was necessary on the "global" iterations. Using the relative values
for tubular flows and concentrations, F
(k) is calculated to
satisfy global mass balance at each mesh node by applying
Eq.10 to volume flow and rearranging to obtain
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(20)
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Then, using these AVR volume flows, one checks for global mass
balance for each solute at each discrete depth. This gives the
following "scores," which would ideally equal 0. These are the
relative deviations from an ideal solution
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(21)
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If the maximum relative deviation is less than 10
6,
we have a solution. If not, then a global Jacobian is constructed
numerically by varying each interstitial/AVR concentration in turn (the
variation used here was 10
4 times the concentration in
question) and reintegrating the system. This Jacobian matrix and the
error vector based on Eq. 21 are then used to solve for a
corrections vector s to the interstitial concentrations by
LU decomposition
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(22)
|
This global Newton iteration is repeated until global convergence
is achieved (i.e., until global mass balance is respected to within our
chosen error tolerance).
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RESULTS |
Here, we present the results of several key simulations
demonstrating the effect of IM metabolic osmole production (glyocolytic conversion of glucose to lactate) in the flat medullary model described
above. Using the baseline parameter set (Table 2), we show that
conversion of 15% of the glucose entering the medulla suffices to
engender a sizeable IM osmotic gradient, mainly by amplifying the IM
recycling of NaCl. We also show that this simulated osmotic gradient is
essentially unaffected by raising the urea permeability of the thin
descending limbs even to values several times higher than those
reported in the microperfusion literature. Then, using a set of
parameters corresponding more closely to the chinchilla kidney, which
has an even higher value of P
than the
rat, we show that urea can accumulate to levels closer to observed
values and yet still be independent of the lactate effect on NaCl recycling.
In addition to these key results, we show some results from a partial
sensitivity analysis, concerning in particular the predicted role of IM
blood flow as the potential regulator of the importance of glycolytic
osmole production for the concentrating mechanism, and the sensitivity
to lactate and glucose permeabilities of the IM DVR.
Increasing Glycolytic Rate
As shown in Fig. 2, the model
predicts that conversion of 15% of entering glucose to lactate would
lead to the establishment of a sizeable IM osmotic gradient, whereas in
the absence of glycolytic lactate production we obtain the classic
result for flat medullary models with a passive IM and high
Pu along the LDL, namely, the frank absence of
an IM osmotic gradient.

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Fig. 2.
Effect of glycolytic rate (A and B) on the
corticopapillary osmolality profile using the baseline parameter set
(Table 2). The osmolality of long ascending limbs of Henle's loop
(LAL) is slightly less than that of all other tubes.
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Figure 3 shows the composition of the
simulated IM osmotic gradient along the AVR/interstitium. We see that a
small accumulation of lactate toward the papilla leads to greatly
increased recycling of NaCl but not of urea.

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Fig. 3.
Composition of total osmolality along the medulla in the absence
(A) and presence (B) of glycolysis.
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Table
3
gives numerical values from these simulations for solute concentrations
and (TF/P)inulin at key points along the nephrons, using
the baseline parameter set. Actual simulations had 120 spatial chops
and were run in double precision. Complete tabulated output is
available from the authors. Two details should be noted: 1)
the solute labeled "glucose," and to which the nephron is
impermeable, was used here to represent nonreabsorbable solutes, set at
1 mM at the entry of LDL and SDL and progressively concentrated along
the nephron by water withdrawal. However, this tactic is only a partial
remedy for the problem (typical of flat models) that
(TF/P)inulin rises (i.e., flow rate diminishes) to
unphysiological values in the distal nephron and along the collecting
ducts. We contend that this problem is due to the lack, in the flat
model, of correct recycling paths that exist in real kidneys thanks to the vascular bundles, and we expect that proper handling must thus be
done in 3-D models. Note, however, that the results with the
high-Pu parameter set (Table
5) give more
physiological (TF/P)inulin values; 2)
(TF/P)inulin is a misnomer for the vasa
recta, wherein this table simply gives values for the ratio of initial
volume flow to vasa recta flow at given points along the tubes
(normalized per tube).
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Table 3.
Results, using baseline parameters, for simulations with 0 or 15%
conversions of glucose to lactate within IM interstitium
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Effect of Medullary Blood Flow and Inner Medullary Blood Flow
It has long been appreciated that the tradeoff between efficient
IM solute recycling and washout must depend on the rate of total blood
flow vs. total nephron flow in the IM, but there exists no convenient
method for experimental determination of this ratio. At least one study
did describe a videomicroscopic method for determination of papillary
blood flow (18), but the authors did not report the IM
nephron flow for comparison. We explored this relationship with our model.
Figure 4A shows the strong
role predicted for the absolute rate of medullary blood flow (MBF). In
this series of simulations, we increased total MBF up to double its
baseline value (keeping simulated GFR constant). Over this range, the
ratio of IM blood flow (IMBF) to total volume flow entering the IM in
the nephrons and collecting ducts also nearly doubled, increasing from
1.2 to 2.2. At the same time, the IMBF/MBF ratio increased from 0.126 to 0.177. As shown in Fig. 4A, the osmotic gradient was
nearly eliminated by doubling MBF.

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Fig. 4.
Effect of varying total medullary blood flow (MBF; A) or
its fractional distribution between the outer medulla (OM) and inner
medulla (IM; B). Absolute glycolytic rate was held constant.
In the right panel, the ratio of IM blood flow (IMBF)/MBF varied from
13 to 24%; the steeper gradients are for lower ratios.
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Figure 4B shows the effect of redistribution of MBF between OM and IM,
with no change in total MBF. We see that although a simple
redistribution of MBF in favor of the IM has a negative effect on the
IM osmotic gradient, this effect is rather small over the range we were
able to explore here. For these simulations, we increased the fraction
of vasa recta entering the IM from one-third to one-half of the total
number of vasa recta. As indicated in the figure, this resulted in
effective IMBF/MBF ratios from 0.126 to 0.19 (comparable to the change
in Fig. 4A), but the ratio of IMBF to nephron flow increased
only from 1.2 to 1.77. Taken together with the results of Fig.
4A, these results suggest that mere redistribution of MBF
between OM and IM is less effective than variation of absolute MBF as a
means of affecting the osmotic gradient. In the absence of experimental
data, it remains to be seen to what extent these results will carry
over to more complete 3-D models.
Note that in this series of simulations the absolute amount of lactate
production was maintained at the baseline level of 15% (i.e.,
conversion of 15% of entering glucose to lactate). This is in keeping
with our basic, conservative assumption that the IM metabolic rate is
independent of the animal's hydrosmotic state. Data on this question
are limited, especially in antidiuresis. Bernanke and Epstein
(4) found that high urea concentrations depressed IM
glycolysis, and it has been found (8, 31) that osmotic
diuresis actually increased IM lactate compared with antidiuretic controls. Also, Tejedor et al. (37) showed in dog kidneys
that papillary collecting ducts metabolize glucose to lactate
stoichiometrically (1:2) when incubated under anaerobic conditions but
that the ratio falls to 1:1.6 under aerobic conditions.
DVR Lactate Permeability
Figure 5 shows that the IM osmotic
gradient induced by IM lactate production is quite sensitive to the DVR
lactate permeability. That is, efficient lactate recycling is necessary
to obtain the effect on the osmotic gradient. The values in this series
of simulations are in the range of measured DVR permeabilities to other
small solutes such as NaCl and urea (see Table 2), suggesting one need not postulate specific DVR lactate transporters to raise lactate permeability to effective levels. However, as explained in the next
subsection, the model predicts that DVR glucose permeability must be
very low to deliver sufficient glucose to the IM. If this is the case,
one would also expect passive permeability to lactate to be low. Thus
if lactate is indeed recycled efficiently by IM vasa recta, one may
expect to find specific lactate transporters. In any case, the present
results suggest that variation in DVR lactate permeability over this
range, by whatever means, would exercise strong control over the
importance of lactate production for the IM osmotic gradient.
DVR Glucose Permeability
As shown in Fig. 6 (and values in
Table 2), this model predicts that glucose delivery to the deep IM
would be compromised unless DVR glucose permeability is very much lower
than that measured in capillary beds of other tissues. In other words,
the papilla will starve due to glucose shunting unless DVR permeability
is limited. This was anticipated by Kean et al. (20)
and suggests surprising selectivity of an epithelium that has
long been considered to be essentially perfectly leaky to small
solutes. This prediction calls for experimental verification. Figure
6B shows that the profile of lactate concentration is
unaffected by DVR glucose permeability.

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Fig. 6.
Sensitivity of total osmolality (A) and
glucose (B) and lactate (C) profiles to DVR
glucose permeability. Osmolality is in mosM; concentrations are in mM.
Baseline glucose permeability of DVR was multiplied by factors
0-1.0, as indicated. In B and C, solid lines
show flow down the DVR, and dashed lines show flow up the AVR.
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High P
We explored the role of P
in this
model using both the baseline parameter set of Table 2 (based on measurements for the rat kidney and also chosen to facilitate comparison with earlier 3-D models) and a parameter set (see Table 4)
based on values reported for the chinchilla kidney [as reported in
Layton et al. (25)], which has an an even higher
P
than the rat.
Figure 7 shows, for both parameter sets,
that the gradient engendered by IM lactate production is affected only
to a small extent by the value of P
. For
the baseline parameter set, raising P
leads to a slight decrease in IM osmolality gradient, and for the
high-Pu parameter set the gradient actually
increases with increasing P
. Detailed
results from the high-Pu simulation are given in
Table 5. This relative insensitivity of the IM osmolality gradient to
P
is a key
result, because the high measured value of
P
(6, 13,
15) has long been recognized as a major incompatibility with the
requirements of the classic "passive hypothesis."

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Fig. 7.
Effect of LDL urea permeability (P )
on the profile of total osmolality. Glycolytic glucose consumption was
15%. A: baseline parameter set. B: high-urea
permeability (high-Pu parameter set).
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Table 5.
Results, using high-Pu parameters, for simulations with 0 or 15% conversion of glucose to lactate within IM interstitium
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Figure 8 shows the constitution of the
interstitial osmolality in the absence and presence of glycolytic
conversion of 15% of entering glucose using the
high-Pu parameter set. By comparison with
results in the baseline model (Fig. 3), urea here constitutes a much
greater fraction of IM osmolality, and although the main effect of
lactate production is still seen on the NaCl gradient, urea
accumulation is also increased.

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Fig. 8.
Effect of glycolytic lactate production on interstitial osmolality
with high-Pu parameter set (Table 4). Glycolytic
glucose consumption was 0 or 15%, as indicated. A and
B: profiles of total osmolality in all structures.
C and D: contributions of urea, NaCl, and lactate
to interstitial osmolality.
|
|
Fractional excretion of urea.
Urea excretion in the rat ranges from ~20-60% of the filtered
urea load (1). Failure to reproduce this observed level of urea excretion while accumulating urea to the high levels observed in
the IM has been a longstanding problem in medullary modeling studies.
The present simulations show that the introduction of glycolytic
lactate production does not solve this problem in the case of the rat
parameters of our baseline simulation, because one can calculate from
the values in Table 3 (using our assumption that half of the filtered
urea is reabsorbed by the proximal convoluted tubule) that fractional
excretion of urea (FEu) is only 4% without IM glycolysis
and falls to 2% when 15% of entering glucose is converted to lactate.
However, in the case of the high-Pu parameter set, with its higher P
and other
parameter changes, FEu (calculated from results of Table 5)
is 19% in the absence of glycolysis and 15% when simulated glucose
conversion is raised to 15%, values that are much closer to
the physiological range. We also see from Tables 3 and 5 that urea
constitutes only ~10% of the osmoles at the papillary tip in
simulations with the baseline parameters but reaches 30% with the
high-Pu parameter set, compared with typical
values of ~50% in antidiuretic animals.
Concentrating work.
Another apparent improvement associated with the
high-Pu parameter set is an increase in
effective concentrating work (Fig. 9). For the case of 15% glucose
conversion, urine flow rate increases by 227% using the
high-Pu parameter set compared with the baseline simulation [i.e., (U/P)inulin = 1,409/620.7],
whereas urine osmolality falls by only 22%. We can relate these values
to the net osmotic concentrating work as follows.

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Fig. 9.
Comparison of osmotic work calculated using Eq. 26 for
our simulations (baseline and high-Pu) and for
data from several micropuncture studies in rats, hamsters, and
Psammomys. The labels refer to the literature references
(see Table 6).
|
|
Considering the kidney as a black box that does purely osmotic
concentrating work, the free energy change associated with excretion of each milliosmole of concentrated urine is
|
(23)
|
where RT = 2.5773 J/mmol at 37°C, and
Uosm and Posm are urine and plasma
osmolalities, respectively. The absolute osmotic work accomplished (the
actual energy cost will of course be higher; see Ref. 36)
is obtained by multiplying this by the osmolar excretion rate,
N = V × Uosm, where V is
urine flow rate. Thus
|
(24)
|
For comparison of our simulation results with
experimental results from the literature in various species, we
normalize this by the GFR
|
(25)
|
where the overbar indicates the normalized value. This can be made
more convenient, in terms of experimentally measured parameters, by
incorporating the (U/P)inulin as follows. Substituting the definition of N from above and because
(U/P)inulin = GFR/V, we have GFR = V
× (U/P)inulin. Thus Eq. 25 becomes
|
(26)
|
For Uosm in milliosmoles per liter, Eq. 27 gives the normalized concentrating work in joules per liter.
Table 6 presents calculations of the
normalized work of concentration for our results from Tables 3 and 5 at
15% glucose consumption along with some literature values for
antidiuretic animals. These are plotted in Fig. 9. The simulation
results are below all the literature values, indicating that although
incorporation of glycolytic lactate production in this flat model can
explain the generation of an IM osmotic gradient, it does not
accomplish a comparable amount of concentrating work, even using the
high-Pu parameter set.
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|
Table 6.
Osmotic work calculated using Eq. 26 for our simulations (baseline and
high Pu) and for data from several micropuncture studies
in rats, hamsters, and Psammomys
|
|
 |
DISCUSSION |
Our results show that if the glycolytic rate is set to 0, this
model, like all previous models whether flat or 3-D, does not develop
an IM osmotic gradient using reported permeability values and a passive
IM. Adding glucose-to-lactate conversion builds an osmotic gradient
within the IM, and this gradient is only marginally sensitive to the
urea permeability of the terminal IMCD.
When Hargitay and Kuhn (14) introduced the countercurrent
multiplication hypothesis in 1951, they carried out their formal analysis using a hydrostatic pressure difference but carefully explained that in the kidney the actual driving force was more likely
to be "electroosmotic." Later in the 1950s, Kuhn and Ramel (23) settled on active salt transport from ascending to
descending limbs as the most feasible single effect, and then Niesel
and Röskenbleck (29) briefly considered the idea
that interstitial "external" osmoles might also supply a single
effect; also, the idea that IM glycolysis might participate was
investigated once by in vivo micropuncture (30), but the
idea was abandoned in favor of active transport from the ascending
limbs. During the 1960s, it gradually became clear that although
vigorous active salt transport occurs from the MTAL in the OM, this is
not the case in the IM. Thus was posed the enigma that the steepest and major portion of the medullary osmotic gradient is established in the
IM with no apparent means of support.
The "passive" or "SKR" hypothesis, introduced in 1972 by
Stephenson (34) and by Kokko and Rector (22),
astutely proposed that the metabolic effort spent in the outer
medullary MTAL could serve indirectly for the establishment of the IM
osmotic gradient if not one but two solutes were recycled, namely, NaCl
and urea. Permeabilities of individual nephron segments were unknown at the time, but the SKR hypothesis made specific predictions that must
obtain if urea in fact serves the proposed external osmole role. In
particular, IM LDL must have very low urea and salt permeabilities and
high water permeability and LAL must be more permeable to NaCl than to
urea. Under these conditions, they predicted that the urea that enters
the deep medullary interstitium from the collecting ducts will draw
water from LDL, thereby concentrating their luminal solutes, especially
NaCl, which will then diffuse passively out of the water-impermeable
ascending limbs on the way back up, thus providing an osmotic single
effect with no local expenditure of metabolic energy. Subsequent
measurement of tubular permeabilities by in vitro microperfusion was in
direct conflict with these predictions; e.g.,
P
was found to be low in the rabbit,
which does not develop a highly concentrated urine, but quite high in
species with well-concentrated urine, such as the chinchilla
(6) and the rat (15).
The model proposed here is the first to reconcile these permeability
data with an appreciable IM NaCl gradient, although it still gives no
satisfactory explanation for the observed IM urea gradient. The central
new feature is that metabolically produced osmoles play the role
previously attributed to urea. Because the loops of Henle and
collecting ducts are essentially impermeable to glucose and lactate
(their permeabilities have not been measured, but their normal
concentrations in the urine are very low and there is no evidence for
their reabsorption in segments past the proximal tubule), the external
osmoles contributed by lactate production can exert their full osmotic
effect across the epithelium of the descending limb and collecting
duct. The effective accumulation of lactate in the deep IM will be
favored by reduced IMBF [known to be the case in antidiuresis
(3)] and high DVR lactate permeability. Concerning the
latter, it remains to be seen whether there are specific lactate
transporters in DVR and, if there are, whether they are regulated by
local or systemic signals. Specific transporters of the MCT family are
responsible in other tissues for one-to-one coupled exit of lactate and
protons from cells undergoing anaerobic glycolysis (12),
and the MCT-2 isoform has been localized to basolateral membranes of
outer MTAL (9), but their localization and the regulation
of their expression in IM structures remain to be characterized.
Although our simulation results with this flat model provide support
for the possible contribution of metabolically produced osmoles in the
urine-concentrating mechanism, it is still clear that this model falls
short of being a definitive explanation. Comparison of the results in
Tables 3 and 5 for simulations with the two different parameter sets
indicates that the problem remains complicated. Although a thorough
sensitivity analysis to explain the differences is beyond the scope of
the present study (we believe this would be more approriate in the
context of a 3-D model treating the vascular bundles and other
anatomical details), some indications are possible.
Several symptoms are visible in the numerical results given in Table 3,
the most notable being the high (TF/P)inulin value in the
terminal CD. It reaches 1,400 here, whereas reported physiological values above several hundred are uncommon. This problem is typical of
flat, central core-type models. Nonetheless, as seen in Table 5, the
high-Pu parameter set performs much better by
this criterion. In addition, FEu increases here to 15%,
whereas it is only 2-4% in the baseline case.
Inspection of the model's behavior suggests that this and other
problems stem from the impossibility, in such flat models, of
accommodating the additional recycling paths available in real kidneys
thanks to the vascular bundle arrangement of the inner stripe. Our
inclusion of nonreabsorbable solutes (represented as "glucose" in
the nephrons) only partially addresses this problem. It is also
interesting to note in this context that the
high-Pu parameter set gives more physiological
levels of flow [(U/P)inulin = 620, and end distal
(TF/P)inulin = 39] while still attaining a
considerable osmotic gradient. This issue thus awaits implementation in
a 3-D model for further clarification.
Suggestions for experimental tests.
1) Given modern micromethods for enzymatic analysis of
lactate (and urea and glucose) concentrations in nanoliter samples, it
would be worthwhile to repeat the in vivo papillary vasa recta micropuncture experiments of Ruiz-Guinazu et al. (30).
Collection of the microliter volumes of fluid required by them for
enzymatic analysis required long collection times that necessarily
compromised the medullary gradient. It should now be possible to do the
measurements in frankly antidiuretic animals. 2) Our results
strongly suggest that the glucose permeability of the DVR must be
uncharacteristically low (compared with vessels in other tissues) to
efficiently deliver glucose to the deep medulla, i.e., to avoid IM
"hypoglycemia" by the same countercurrent-exchange effect that is
responsible for the IM hypoxia (19). Measurement of DVR
glucose and lactate permeabilities would require in vitro
microperfusion. 3) Our results (Fig. 5) suggest that IM
accumulation of lactate would be optimal only if DVR lactate
permeability is considerably higher than measured DVR permeabilities to
NaCl and urea. This opens the possibility that there may be specific
lactate transporters in DVR epithelium. It would be interesting to
search for such transporters and, if any are found, to see whether they
are sensitive to local autocrine or paracrine factors or to the
hormones involved in antidiuresis and regulation of IMBF.
In conclusion, this flat-model exploration of a possible role for
IM metabolic osmole production in the urine-concentrating mechanism
further confirms the feasibility of the idea that we first explored in
a simple vasa recta model (41). Not only is this the first
scenario to reconcile the high measured
P
with the establishment of an IM
osmotic gradient, but it also suggests a role for the previously
paradoxical high P
; that is, by allowing Henle's loops to participate in urea recycling, it favors the papillary accumulation of urea. Thus in this scenario, the regulation of urea balance may be uncoupled from a primary role in
salt or water balance. This would make sense from a comparative physiological standpoint, considering that many of the rodents having
the highest concentrating ability have a vegetarian diet (2), so their urea load is less than that of omnivorous
species like the rat. What's more, the papillae of such species are
typically much longer and have a higher fraction of nephrons extending
deep into the papilla than does the rat kidney. These two features seemed paradoxical in the context of the urea-centered SKR hypothesis based on the rat kidney, but they make sense for the present
hypothesis, because the additional tissue mass should provide more
metabolic osmoles, and the greater papillary length should improve
lactate trapping by recycling (41). These issues and the
shortcomings of the present flat model should be further explored in
3-D models of the medulla to explore the advantages of the additional
recycling pathways afforded by the vascular bundles.
 |
ACKNOWLEDGEMENTS |
This study was financed by the general operating funds of Institut
National de la Santé et de la Recherche Médicale Unit 467 and the Necker Faculty of Medicine, University of Paris 5.
 |
FOOTNOTES |
Address for reprint requests and other correspondence: S. R. Thomas, Institut National de la Santé et de la Recherche
Médicale U467, Necker Faculty of Medicine, Univ. of Paris 5, 156, rue de Vaugiard, F-75015 Paris, France (E-mail:
srthomas{at}necker.fr).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
August 27, 2002;10.1152/ajprenal.00045.2002
Received 1 February 2002; accepted in final form 23 August 2002.
 |
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