Inner medullary lactate production and accumulation: a vasa
recta model
S. Randall
Thomas
Institut National de la Santé et de la Recherche
Médicale Unité 467, Necker Faculty of Medicine, F-75015
Paris, France
 |
ABSTRACT |
Since anaerobic glycolysis
yields two lactates for each glucose consumed and since it is reported
to be a major source of ATP for inner medullary (IM) cell maintenance,
it is a likely source of "external" IM osmoles. It has long been
known that such an osmole source could theoretically contribute to the
"single-effect" of the urine concentrating mechanism, but there was
previously no suggestion of a plausible source. I used numerical
simulation to estimate axial gradients of lactate and glucose that
might be accumulated by countercurrent recycling in IM vasa recta
(IMVR). Based on measurements in other tissues, anaerobic glycolysis
(assumed to be independent of diuretic state) was estimated to consume ~20% of the glucose delivered to the IM. IM tissue mass and axial distribution of loops and vasa recta were according to reported values
for rat and other rodents. Lactate (PLAC) and
glucose (PGLU) permeabilities were varied over a
range of plausible values. The model results suggest that
PLAC of 100 × 10
5 cm/s
(similar to measured permeabilities for other small solutes) is
sufficiently high to ensure efficient lactate recycling. By contrast,
it was necessary in the model to reduce PGLU to
a small fraction of this value (1/25th) to avoid papillary glucose
depletion by countercurrent shunting. The results predict that IM
lactate production could suffice to build a significant steady-state
axial lactate gradient in the IM interstitium. Other modeling studies (Jen JF and Stephenson JL. Bull Math Biol 56: 491-514,
1994; and Thomas SR and Wexler AS. Am J Physiol Renal Fluid
Electrolyte Physiol 269: F159-F171, 1995) have shown that
20-100 mosmol/kgH2O of unspecified external,
interstitial, osmolytes could greatly improve IM concentrating ability.
The present study gives several plausible scenarios consistent with
accumulation of metabolically produced lactate osmoles, although only
to the lower end of this range. For example, if 20% of entering
glucose is consumed, the model predicts that papillary lactate would
attain about 15 mM assuming vasa recta outflow is increased 30% by
fluid absorbed from the nephrons and collecting ducts and that this
lactate gradient would double if IM blood flow were reduced by
one-half, as may occur in antidiuresis. Several experimental tests of
the hypothesis are indicated.
glucose; descending vasa recta; blood flow; urine
concentrating mechanism; anaerobic glycolysis
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INTRODUCTION |
Glossary
AVR |
ascending vasa recta
|
DVR |
descending vasa recta
|
IMVR |
inner medullary vasa recta
|
IM |
inner medulla
|
OM |
outer medulla
|
IMBF |
inner medullary blood flow (=FVDVR(0))
|
IMCD |
inner medullary collecting duct
|
LDL |
long descending Henle's loop
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LAL |
long ascending Henle's loop
|
N(x) |
number of DVR at depth x
|
cGLU |
glucose concentration
|
cLAC |
lactate concentration
|
Fij |
tubular flow of i in tube j (pmol/min)
|
Ji |
transmural flux of i
(pmol · min 1 · mm 1)
|
PGLU |
glucose permeability across DVR
|
PLAC |
lactate permeability across DVR
|
ksh |
coefficient for exponential decay of N(x)
|
x |
distance from OM/IM border (mm)
|
L |
distance to papillary tip
|
i |
reflection coefficient of solute i
|
Subscripts and Superscripts
i |
glucose, lactate, or volume
|
j |
DVR, AVR, or SH (for shunt flows)
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THIS STUDY
EXPLORES the possibility that countercurrent recycling of lactate
produced continually in the renal inner medulla (IM) by anaerobic
glycolysis might lead to an axial gradient of interstitial
osmoles.1 By serving as
"external osmoles" (i.e., external to the tubular fluid), such
osmoles could contribute to the single effect for countercurrent multiplication in the IM (20).
The extent to which a metabolically produced external solute can
accumulate an axial interstitial (and vasa recta) concentration gradient by countercurrent exchange will depend on at least four factors: its rate of production within the medulla, its descending vasa
recta permeability (a high value favors recycling), inner medullary
blood flow (IMBF) (low IMBF favors accumulation by minimizing washout; Ref. 53), and volume uptake from the nephron system. It has
long been known that energy for the buildup of the IM osmotic gradient
could theoretically be supplied not only by recycling of urea and NaCl
(the crux of the "passive hypothesis"; Refs. 30 and 52) but
alternatively (or in addition) by "external," or interstitial,
osmoles (24, 31, 37, 46), that is, solutes present in
interstitial fluid but not in tubular fluid of the loop of Henle or
collecting duct. Using a detailed three-dimensional model of rat
medulla to estimate the amount of such external osmoles needed to boost
concentrating ability, we calculated (55) that 100 mosmol/kgH2O of interstitial osmolytes would considerably increase the axial IM osmotic gradient. The literature has remained mute as to the identity or possible source of such hypothetical interstitial osmoles, which would have to be produced continuously within the IM and recycled downward by the vasa recta to accumulate significantly toward the papilla.
I recently proposed (54) that IM glycolysis may contribute
to such a pool of external osmoles, since it is known that in the
relatively hypoxic (49) IM, a large fraction of the energy for cell metabolism is supplied by anaerobic glycolysis (32, 47), which yields two lactate molecules (and two protons) for each glucose molecule; that is, anaerobic glycolysis produces net
osmoles as a matter of course. Anecdotally, this phenomenon is the
reason for corneal swelling under hypoxic conditions (27).
Having suggested that IM metabolic osmole production may contribute to
the IM single effect for urine concentration, one immediately wonders
how the effect would depend on the animal's diuretic state. It seems
unlikely that cell metabolism in the IM, which is just housekeeping
since there is no known active epithelial transport in this region
except in the collecting ducts, should vary with the organism's water
balance; that is, there is no reason to suppose a priori that IM
glycolytic rate should vary as a function of the animal's salt and
water balance, so its involvement in the concentrating mechanism
appears hard to rationalize. However, studies of the IM
microcirculation (e.g., 19, 35, 36, 50, 62) have shown that the IM
blood supply may vary in response to perturbations of the organism's
water balance, being reduced under antidiuretic conditions. Also, it is
clear from earlier modeling studies [see review by Stephenson
(53)] that net flow rate through a countercurrent exchange system determines the extent to which intrinsic osmole production will build up an axial gradient, with low flow being favorable to steeper gradients.
The crux of my conjecture is thus: first, that intrinsic metabolic
osmole production may be sufficiently high and IM blood flow
sufficiently low, especially in antidiuresis, to result in significant
osmole accumulation by countercurrent exchange, and second, that this
could contribute importantly to the urinary concentrating mechanism.
The present work addresses the first of these.
The present study uses a mathematical model of inner medullary vasa
recta (IMVR), first, to discover conditions under which lactate
produced by IM anaerobic glycolysis might accumulate a significant
axial concentration gradient and, second, to propose experimental tests
of this idea. Using a conservative estimate of IM glycolytic lactate
production, based on glycolytic rates measured in the kidney and in
other tissues, I calculate projected axial IM gradients of lactate and
glucose for a range of assumed vasa recta permeabilities to lactate and
glucose. I also briefly treat the case of species with different
percentages of long loops reaching all the way to the papillary tip. I
find it plausible that lactate accumulation could be considerable in
the deep IM, though volume uptake from the nephrons (resulting in part
from the metabolic osmole production) will limit the tendency. It
remains for one to apply this idea in medullary models incorporating
the nephrons and collecting ducts explicitly to see how this
extratubular metabolic osmole production might affect salt and urea
recycling and, thus, the IM osmotic gradient.
 |
MODEL DESCRIPTION |
Since the nephron is essentially impermeable to small sugars,
glucose and lactate are distributed only among the interstitium, the
microcirculation, and of course the cytoplasm of IM cells (i.e.,
interstitial cells, epithelial cells, and cells of the capillary
walls). If, in addition, we consider only the steady state, we can
estimate IM recycling and accumulation of glucose and lactate using a
simple model of IMVR, lumping the interstitium with ascending vasa
recta (AVR) and excluding nephrons and collecting ducts. The important
parameter with respect to the question of a single effect for the IM
axial osmotic gradient is the interstitial concentration of supposed
external osmoles, but interstitial concentrations are hard to handle
both experimentally and theoretically. I thus limit this modeling study
to consideration of steady-state glucose and lactate flows in
descending vasa recta (DVR) and AVR, with a conservative range of
estimated glycolytic rate within the IM, assumed here not to depend on
diuretic state. The "AVR" represent the interstitium as well as
ascending vasa recta. Previous models specifically investigating vasa
recta flow and exchange (14, 15, 34, 56) addressed the
relative importance of hydrostatic vs. osmotic pressure, protein
oncotic force, dissimilarity between DVR and AVR, likely roles of water
channels and urea transporters, importance of varying number of vessels
with medullary depth, and other issues. Compared with those earlier
models, the present study adopts a simpler model, since it addresses
the simpler issue of accumulation by recycling of solutes excluded from
the nephrons.
Mass balance constraints require that total glucose consumption and
lactate production in all cells within a slice of medulla at a given
depth x must, in the steady state (and independent of
considerations of reduced numbers of vessels with depth), equal the net
difference of vasa recta glucose and lactate outflows and inflows
through the slice, so by restricting the analysis to the steady state,
one sidesteps the difficulty of estimating glucose and lactate
distributions between cell cytoplasm and interstitium. As stated above,
I make the further assumption that AVR concentrations are equal to
interstitial concentrations. This model thus says nothing about
intracellular glucose or lactate concentrations. Figure
1 schematically depicts the model.

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Fig. 1.
Schematic representation of the steady-state model of IMVR. Dotted
walls for AVR indicate that it represents interstitium as well as AVR.
See Glossary for complete description of abbreviations.
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Since in vivo flows and concentrations are unknown at the OM/IM border,
I set up the baseline case according to predictions of our recent
calculations with a three-dimensional model of the whole medulla
(54, 57).
Anatomy.
DVR and AVR are assumed to diminish exponentially in number along the
IM toward the tip of the papilla according to the same relation as in
our earlier models and in conformity with reported rat anatomy
(28), i.e.
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(1)
|
with ksh = 1.213 mm
1.
Thus, compared with the number of tubes entering the IM, the fraction
of tubes reaching the papillary tip is 1/128 for L = 4 mm at the tip and x = 0 at OM/IM border.
Other species have different proportions of tubes and vessels extending
to the tip. I investigate possible implications of this by running
simulations with different values of ksh.
The number of entering DVR, N(0), was nominally set at 128, so the simulation represents behavior of a system that gives a single
DVR at the papillary tip. Qualitatively, the model behavior is in fact
independent of the numerical value chosen for this parameter, depending
instead on the loop distribution, determined by
ksh. Everything is scaled to this assumption, in
particular, glycolytic glucose consumption (and lactate production) is
expressed as percent of glucose inflow, and DVR volume reabsorption and net volume uptake into AVR from nephrons and collecting duct are expressed as percent of DVR inflow. By this strategy, the model can
represent kidneys containing any number of vasa recta simply by varying
the medullary length and/or the factor describing the exponential
decrease of their number with depth (ksh).
Inflows.
Baseline volume flow into DVR is set to 3.75 nl · min
1 · tube
1, the DVR
flow at the OM/IM border in (57). The glucose and lactate
concentrations entering the IM DVR are set at 10 and 2 mM,
respectively, i.e., cGLU(0) = 10 mM,
cLAC(0) = 2 mM.
Mass balance considerations.
Conservation of matter in the steady state requires that in any slice
of IM extending from depth x1 to depth
x2, with x2 > x1, we must have, taking glucose flows as an
example, the following: (rate of glucose entry into the slice)
(rate of glucose exit from the slice) = (rate of glucose
conversion to lactate within the slice), i.e., accounting for both DVR
and AVR flows through the slice and noting that ascending flows carry a
negative sign
|
(2)
|
For lactate balance, since each converted glucose molecule yields
two lactate molecules, we have the following: (rate of lactate exit
from slice)
(rate of lactate entry into slice) = 2 × (rate of glucose conversion to lactate within slice), i.e.
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(3)
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Integrating all the way to the tip from any point x,
and given the continuity condition at the papillary tip, namely
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(4)
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we have net glucose consumption from any depth x to the
tip equal to
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(5)
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and net lactate production from x to the tip is
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(6)
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In particular, for the IM as a whole
|
(7)
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From this, it is straightforward to show that the maximum rate of
osmole addition to the IM from anaerobic glycolysis, if all the glucose
were converted to lactate, in which case
FGLUAVR(0) = 0, would equal the rate of entry of
glucose into the IM in the DVR. The actual IM rate of glycolysis must
be only a fraction of this maximum.
Prediction of the gradients of glucose and lactate concentration along
the vasa recta is the purpose of what follows.
Estimate of glycolytic rate.
Although extraction of a consensus estimate for IM glycolytic rate from
the literature is subject to the usual difficulties, the following two
facts seem clear: the IM is relatively hypoxic [PO2 ~10 Torr (33)], and
anaerobic glycolysis supplies a substantial part of the energy budget,
with lactate clearly being produced in the IM, whereas it is consumed
in the cortex (3), although complete dependence of IM on
anaerobic glycolysis is a myth (10). In what follows here,
I attempt a best, albeit conservative, estimate of fractional IM
glucose consumption.
Quoting from Ross and Guder (47), p. 393: "The high rate
of glycolysis observed in [rat] papillary slices corresponds to the
highest local activity of hexokinase. It may be calculated that this
enzyme activity, expressed in terms of dry weight of tissue, exceeds by
a factor of 2 the maximum rate of glycolysis which has been recorded,
i.e., 800-1,000 µmol/hr/g dry weight lactate production under
anaerobic conditions." Their table II shows about 100 ng dry
weight/mm tubule, so this converts to 1.67 pmol · min
1 · mm of
tubule
1, which I consider an extreme upper limit on
glucose consumption.
To compare glucose consumption with its delivery rate, I consider a
minimal group, served by each DVR, to correspond roughly to seven
tubule equivalents, as follows: DVR, two AVR, long descending limbs
(LDL), long ascending Henle's loops (LAL), IM collecting ducts (IMCD),
plus some interstitial cells. I thus estimate maximal glycolytic rate
of a minimal group as roughly seven times that of one tubule, or 11.7 pmol · min
1 · mm
1. This
need must be met by the input from one DVR: given my base case
assumptions of 10 mM glucose and 3.75 nl/min inflow rate to DVR
entering the IM, the glucose supply for each group is 37.5 pmol/min.
This rough estimate of maximal metabolic glucose consumption thus
implies that all entering glucose would be consumed in a group of
length 4 mm, the IM length of the longest loops in a rat kidney.
However, in the rat kidney, most long loops turn back before reaching
the papillary tip. Integration over the whole depth of IM using our
assumed loop distribution for rat kidney (Eq. 1, above)
predicts a maximal total IM consumption of 25% of the entering
glucose. This equals the estimate made by Ruiz-Guinazu et al.
(48).
An alternative approach to estimation of IM glycolytic rate is to
estimate IM energy requirements. A minimal estimate (it ignores basic
cell metabolism needs) can be based on measurements of total IMCD
sodium transport, since IMCD are the only demonstrated site of IM
active transepithelial transport. These measurements were recently
summarized in association with a model of IMCD transport (table 4 of
Ref. 58). Taking a conservative estimate of reabsorption of,
say, 2% of filtered Na+ from the IMCD and glomerular
filtration rate (GFR) of 500 µl/min gives
If anaerobic glycolysis were the only source of ATP (but see
below), and given transport of 3 Na+ per ATP and production
of 2 ATPs per glucose, then this would amount to consumption of ~240
nmol glucose/min, or 1.6 µmol · min
1 · g wet wt
1
of IM, if the IM represents 10% of a 1.5-g rat kidney. This is higher
than direct measurements of lactate production in rabbit medulla
(32), which reported values of 0.2 µmol · min
1 · g wet
tissue
1 in the presence of oxygen and 0.4 µmol · min
1 · g wet
tissue
1 under N2 perfusion, but it is much
lower than measured anaerobic lactate production in guinea pig medulla,
namely 6.2 µmol · min
1 · g wet
tissue
1 (from Refs. 18 and
13, cited in Ref. 10). We can compare these
consumption rates to an estimate of IM glucose delivery rate as
follows. Estimates of papillary plasma flow (PPF) in antidiuretic rats
range from 0.3-0.5 ml · min
1 · g wet
tissue
1 (4, 16, 25, 40). If glucose
concentration in blood entering the IM is 10 µmol/ml (i.e., 10 mM),
then glucose delivery to IM is around 3 to 5 µmol · min
1 · g wet
tissue
1, or double to triple the above estimate of
minimal expenditure. Thus it would appear that glucose (via anaerobic
glycolysis) could in principle suffice as the IM energy supply.
Nonetheless, despite the relative hypoxia of the IM, anaerobic
glycolysis is in fact only responsible for perhaps one-third of the
energy supply (6).
Based on these considerations, I adopt a baseline glucose consumption
of 20% of IM delivery and investigate a wider range of values in the
sensitivity studies reported below.
System equations.
The model treats steady-state flows and exchanges of volume, glucose,
and lactate along DVR and AVR. The interstitium and cells (epithelial
and interstitial) are assimilated with AVR. I assume that glucose
consumed by cellular glycolysis is supplied from AVR and that the
resulting lactate is recovered into AVR/interstitium. Also included is
net volume reabsorption into the AVR from LDL and IMCD, designated as
JVABS(x).
The system is subject to the continuity condition at the papillary tip
(Eq. 4) and is described by the following system of six
differential equations
|
(8)
|
where
kshFiDVR(x)
is shunt transfer of i from DVR to AVR at depth x
(59). Ji(x) terms
are the diffusional transfers (pmol · min
1 · mm
1) and
osmotic volume flow
(nl · min
1 · mm
1) at
x from all DVR to AVR. Treating the capillary walls as a single barrier (i.e., no distinction here between transcellular and
paracellular transport), we have for glucose and lactate fluxes
|
(9)
|
where N(x) is the number of DVR at depth
x (Eq. 1), PGLU and
PLAC are permeabilities to glucose and lactate,
GLU and
LAC are reflection coefficients
(both set at 0.5 here), and concentrations of solute i in
tube j are
|
(10)
|
Since volume flux along the DVR
(Jv(x)) depends on forces not
represented in this model, it cannot be calculated explicitly here.
Jv(x) is thus taken to be an explicit
fraction (30% as baseline value) of entering flow, distributed over
the length of the IM in proportion to the number of DVR at each depth.
Simulations were run for various fractional volume fluxes.
To avoid unrealistic glycolytic glucose consumption in the event that
glucose concentration falls locally to zero in the course of numerical
analysis, I describe glycolytic rate simply with a first-degree
Michaelis-Menten equation, saturable as a function of AVR glucose
concentration, setting Km very low (0.1 mM) and Vmax equal to estimated local glycolytic rate
|
(11)
|
Thus glycolytic rate will be virtually equal to
Vmax except for extremely low glucose
concentrations. In practice, for exploration of model behavior, I
wanted to specify Vmax values that would result
in specified fractions of total glucose consumption (rather than
specifying glycolytic rates per unit tissue volume or per mm of
medullary depth). To this end, assuming Km
cGLUAVR for all x, substituting from
Eq. 1 for N(x), and integrating over
the whole IM, total glucose consumption is
|
(12)
|
where L is the total length of the IM. Solving this for
Vmax and expressing
JGLYTOT as a fraction GlyFract
of total baseline glucose delivery, FGDVR(0),
we obtain
|
(13)
|
In like manner, JVABS(x),
the volume reabsorbed from LDL and IMCD, was distributed in proportion
to the number of DVR (assumed equal to the number of LDL) at each depth
|
(14)
|
By analogy with the treatment of the glycolytic
Vmax, I express total IM volume absorption as a
proportion of entering blood flow, i.e.,
(JVABS)TOT = VolFract · FVDVR(0), and following
the development of Eq. 13, we obtain an expression for
kv
|
(15)
|
Parameter values.
Table 1 gives baseline parameter values
(in dimensions of most literature reports as well as in the dimensions
used for the present model).
In the absence of measurements of DVR permeabilities to lactate and
glucose (PLAC and PGLU),
I set the baseline value for PLAC at 100 × 10
5 cm/s, which is midway between the measured DVR
permeabilities to NaCl and urea used in our recent simulations of the
whole rat medulla (57). During the simulations, it became
immediately necessary to reduce the glucose permeability well below
this value to avoid convergence problems due to near zero glucose
concentrations. Baseline PGLU was thus set
25-fold lower than PLAC. Whether this requirement reflects reality remains to be seen; certainly such low
permeability for glucose across the vasa recta wall contradicts the
general assumption that such vessels are very leaky to small solutes.
Around these baseline values, simulations were run over a wide range of
PLAC and PGLU values. It
is interesting in this context to cite Kean et al.
(26): "It should be pointed out that anaerobic
metabolism of the renal medulla in vivo, although a satisfactory
explanation for the metabolism by which energy is made available
despite probable deprivation of oxygen, still poses the significant
problem of substrate delivery to the tissue for this type of
metabolism. If oxygen exchanges across the limbs of the vasa recta,
thereby depriving the deeper portions of the medulla, why does not
glucose also exchange in a similar manner?"
JvABS baseline value is set at 30% of flow
into IM DVR. This is the value reported by two studies in antidiuretic
rats based on measurement of vasa recta protein concentration at the
base and tip of the papilla (38, 63). However, values as
high as 117% have been reported (21). This is a crucial
parameter for lactate accumulation, as the results below show, but
proper investigation must be done in a model of the full medulla that
includes not only vasa recta but also nephrons and collecting ducts.
Numerical solution.
I programmed Mathematica to solve this nonlinear system of
ordinary differential equations by simple shooting, which also amounts
to multidimensional Newton-Raphson (43). Briefly,
the system of six equations and six unknowns (Eq. 8) was
solved subject to three initial conditions (flows into DVR) at
x = 0 and three boundary conditions at L,
where L is the depth at the papillary tip. The three
boundary conditions are continuity relations on the flows at the
hairpin turn, normalized by the entering flow rate
|
(16)
|
where i is volume, glucose, or lactate. Ideally,
score equals zero. Using the initial conditions for the
three DVR inflows FiDVR(0) and
guesses for the three AVR outflows FiAVR(0),
the equations were integrated from 0 to L
(Mathematica's function NDSolve, which switches
automatically between a non-stiff Adams method and a stiff Gear method,
based on LSODE). Using Mathematica's LinearSolve function
(which uses LU decomposition), the error vector, score, from
Eq. 16 is used with a numerically determined Jacobian matrix, JAC, to solve for a corrections vector,
s, to the guesses for FiAVR(0)
according to
|
(17)
|
Improved guesses for AVR flows at x = 0 are
obtained by adding the corrections vector s to the previous
guesses. The system is again integrated from x = 0 to
L, and score is recalculated. This cycle is
repeated until the maximum of score is
10
5.
Typically only two or three iterations were needed to reach a solution.
As a post hoc check on the quality of the solutions, mass balance at
each depth is verified based on conservation of glucose equivalents,
i.e., according to
|
(18)
|
The maximum of this sum was typically less than 10
15
pmol · min
1 · mm
1.
 |
RESULTS |
Logically, the main factors influencing IM accumulation of
metabolically produced lactate should be: 1) the rate of
glycolysis; 2) PLAC, the lactate
permeability of DVR, which should be sufficiently high to efficiently
recycle lactate to the papilla instead of having it carried out in the
AVR; 3) blood flow into the IM, which should be low to favor
accumulation instead of washout, especially if glycolytic rate is
insensitive to the animal's water balance state; and 4) the
amount of volume absorption from the nephrons and collecting ducts,
which will tend to dilute the interstitial lactate and increase the
washout rate by increasing AVR flow rate.
Glycolytic rate.
Figure 2 shows baseline model behavior
for overall glycolytic glucose consumption of 0 to 40% by steps of
5%. The volume flow profiles (Fig. 2, A and B)
show the 30% net increase of volume flow due to assumed uptake from
the nephrons and IMCD and the 30% fall of single-vessel DVR flow due
to the assumed baseline rate of DVR volume flux. Note that volume flows
here are not coupled to glucose and lactate concentrations (see
MODEL DESCRIPTION, above). As glycolytic rate increases,
glucose concentration falls and lactate rises. Note that the fall of
glucose concentration even for glycolytic rate of zero is due to
dilution by incoming volume from the nephrons
(JvABS in the model equations).

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Fig. 2.
Profiles of volume flow and solute concentration along IM, for
varying glycolytic rates. Vmax was varied to
yield 0% to 40% consumption of delivered glucose, as indicated on the
curves; 20% consumption will be adopted as the baseline value for
subsequent simulations. All other parameters were set to baseline
values. For this and subsequent Figs. 3-9, solid curves show
values for DVR, dashed curves for AVR. A: total volume flow
(normalized by delivered flow rate), showing that outflow from AVR is
30% higher than DVR inflow, due to assumed volume uptake from
nephrons. B: volume flow per vessel (normalized by delivered
flow rate per vessel), showing DVR volume flux equal to 30% loss over
the length of the IM; sn, single vessel. C: glucose
concentration profiles; fall of glucose concentration at 0%
consumption is due to dilution by volume uptake from nephrons.
D: lactate concentrations.
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Volume absorption from nephrons.
The effect of varying JvABS is shown in Fig.
3 for absorption rates from 10 to 90% of
DVR inflow. This fluid absorption is seen to significantly reduce the
accumulation of a lactate gradient. As an indication that this range
spans physiological values, it can be easily shown that IMCD volume
reabsorption of 1 or 2% of GFR corresponds to 25-50% of PPF,
assuming a filtration fraction of 25% and PPF equal to 1% of renal
blood flow.

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Fig. 3.
Effect of increasing volume absorption from nephrons and collecting
ducts on glucose and lactate concentration profiles.
JvFract was 10, 30, 50, 70, or 90% of IM
DVR inflow rate; arrows indicate increasing volume absorption. Other
parameters are at baseline values.
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Reduction of IMBF.
Figure 4 shows the effects of reducing
IMBF, under the assumption that tissue glucose consumption is not
affected by the change of blood flow. Lactate accumulation is seen to
dramatically increase as IMBF falls to one-half its baseline value. The
predicted lactate profiles clearly suggest that IMBF may play an
important role in the extent of lactate accumulation.

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Fig. 4.
Effect of reducing inner medullary blood flow (IMBF) on
glucose (cGLU) and lactate concentrations
(cLAC). IMBF was reduced from 100% to 50% of its baseline
value. Absolute glucose consumption was held constant.
Bottom: glucose and lactate concentrations at the papillary
tip (mM) vs. IMBF (expressed as percentage of baseline IMBF).
Top: glucose and lactate profiles from
x/L = 0 to 1.0.
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Lactate permeability.
The effect of increasing PLAC from 0 to its
baseline value in steps of 10% is shown in Fig.
5. As is clear from the tendency seen in
the lactate profiles, higher values lead to no further improvement of
lactate accumulation. That is, given the baseline assumptions, there is
no reason to postulate specialized lactate transport systems that would
increase its effective permeability above the values measured in IMDVR
for salt and urea. On the other hand, lactate permeability would have
to be less than half that of salt and urea before its efficient
recycling would be compromised.

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Fig. 5.
Effect of lactate permeability (PLAC) on
glucose (cGLU) and lactate concentrations
(cLAC). PLAC was increased from 0 to
100% of baseline value by steps of 10%. Other parameters are at
baseline values (except PGLU = PLAC/25). Bottom: lactate
concentrations at papillary tip vs. PLAC
(expressed as percentage of baseline value). Top: glucose
and lactate profiles from x/L = 0 to 1.0.
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|
Glucose permeability.
As mentioned above, it was necessary in the present simulations to
drastically reduce glucose permeability to assure its delivery to the
deep medulla. That is, higher values resulted in glucose shunting, much
as outer medullary oxygen shunting leads to hypoxia in the IM. Figure
6 illustrates this behavior.

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Fig. 6.
Effect of glucose permeability on glucose and lactate profiles.
Bottom: glucose concentrations at papillary tip vs.
PGLU (expressed as fraction of baseline
PLAC). PGLU was varied
from 0.01-0.085 × baseline PLAC. All other
parameters were held at baseline values, with glucose consumption at
20% of delivery rate. Top: glucose and lactate profiles
from x/L = 0 to 1.0.
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Jv along DVR.
Since this model does not include solutes other than glucose and
lactate and also ignores hydrostatic and oncotic pressures, volume flux
along the DVR could only be treated by assuming some ad hoc profile.
The results shown in Fig. 7, where volume
loss along the DVR was varied from 0 to 40% of delivery rate, indicate that this is not a crucial parameter with respect to glucose or lactate
profiles. Simulations (not shown) with reflection coefficients of 0 (instead of 0.5) for both glucose and lactate showed that the increased
solvent drag had only a minor effect on the profiles.

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Fig. 7.
Effect of fractional DVR volume loss. Volume flux out of DVRs was
varied from 0 to 40% of entering flow rate. As a function of
normalized medullary depth, top shows total
(left) and single-vessel (right) volume flows
(expressed as fraction of entering flow), and bottom shows
profiles of glucose and lactate concentrations.
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|
Percentage of early returns.
All simulations up to here have assumed a conic papilla like that of
the rat, with only a small fraction (1/128) of IM tubes reaching the
papillary tip. Many species of rodents have longer papillae with a
greater proportion of tubes extending to the region near the tip. In
Psammomys, for example, about 40% of nephrons enter the IM
and nearly all of these run essentially the whole length of the very
long papilla. In such papillae, since metabolic rate must be
proportional to the amount of tissue, it seems reasonable to assume
that glycolytic lactate production must consume a greater fraction of
entering glucose. Also, one would anticipate that, with a longer
papilla, lactate accumulation by countercurrent recycling would be
favored. To get some idea how well the present model reflects this
hunch, I ran a series of simulations with decreasing values of
ksh, the factor for exponential decrease of the
number of vessels. As a starting point, I chose a
Vmax giving 20% glucose consumption for
baseline ksh value (this percentage increases as
the proportion of long vessels increases). Figure 8 (A and B) shows
the resulting profiles for various values of ksh. It is seen that lactate accumulation is
favored by an increased proportion of long loops. Table
2 gives, for each value of
ksh, the percentage of long loops attaining the
papillary tip and the corresponding percentage of delivered glucose
converted to lactate.

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Fig. 8.
Effect of different loop distributions. The factor
ksh was divided by 1, 2, 3, 4, or 5, changing
the fraction of loops that extend into the deepest papilla (see text
and Table 2). Arrows indicate increasing number of loops reaching the
papilla. A: total (left) and single-vessel
(right) flows as a function of IM depth. B:
glucose and lactate profiles: glucose consumption was proportional to
number of loops at each depth, i.e., it increased as
ksh decreased. C: glucose and lactate
profiles: glucose consumption held constant at 20% for all values of
ksh.
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Table 2.
Effect of various ksh values on percentage of loops
reaching papillary tip and percentage of delivered glucose conversion
|
|
Alternatively, one may ask how glucose and lactate profiles would
differ in kidneys having identical overall glucose consumption but
different loop distributions. Figure 8C shows the results of
such a simulation in which glucose consumption was held constant at
20% of delivery rate. The main effect one sees is better delivery of
glucose to the papilla when more vessels extend deeper. Lactate profiles are unaffected by the changing loop distribution.
Reflection coefficients.
To test the sensitivity to assumed reflection coefficient values, I
varied
GLU and
LAC independently from 0 to 1.0. Figure 9 shows the results. The
glucose profile is seen to depend somewhat on
GLU, but
the lactate profile is essentially independent of
LAC,
mainly because baseline lactate permeability is sufficiently high to
assure nearly identical DVR and AVR lactate concentrations, i.e.,
solvent drag of lactate is negligible compared with lactate diffusion.

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Fig. 9.
Effect of GLU and LAC on glucose and
lactate profiles. Glucose (top) and lactate
(bottom) reflection coefficients were varied from 0 to 1.0. Other parameters at are baseline values.
|
|
 |
DISCUSSION |
Our earlier modelling study (55) suggested that 100 mosmol/kgH2O of external osmoles would suffice to
significantly boost the concentrating mechanism. Jen and Stephenson
(24) suggested that an even smaller amount, around 20 mosmol/kgH2O, might suffice. The present results suggest
that IM lactate production could conceivably furnish substantial levels
of such external osmoles but that lactate accumulation to
concentrations as high as 100 mM seems unlikely. To my knowledge, Dell
and Winters (12), working with dog kidney, did the only
experimental study aimed at evaluation of a possible IM gradient of
lactate concentration. In their thoughtful study, they not only
demonstrated a corticomedullary lactate gradient (lactate concentration
doubling from base to tip in normal and diuretic dogs) but also
attributed it to "countercurrent exchange between afferent and
efferent limbs of the vasa recta" and speculated that the most likely
source was anaerobic glycolysis. More thorough theoretical evaluation
of this possibility will necessitate its inclusion in a model of the
full medulla, including flows not only of glucose and lactate but also
of urea and NaCl in both nephrons and blood vessels, to account
explicitly for enhanced volume absorption from descending limbs and
collecting ducts, which will in turn affect the recycling of salt and
urea among nephrons and vessels.
To put things in scale, it is useful to compare total IM glucose
delivery to the amount of urea dumped into the papilla from IMCD. Based
on numbers from our three-dimensional models (54, 57), the
present assumed delivery of glucose to the IM amounts (in osmolar
equivalents) to about 9% of the filtered load of urea (FLu). If about 40% of FLu is dumped into the
interstitium from the IMCD, conversion of 20% of total delivered
glucose to lactate would represent generation in the IM of only about
4% as many lactate osmoles as urea osmoles. Nonetheless, it must be
kept in mind that the sugars exert their full osmotic force across nephron walls (intratubular concentration near zero and tubular reflection coefficient of 1.0), whereas urea is highly permeable (thus
only a small concentration difference), so the relative contribution of
each milliosmole of interstitial lactate to an "external osmole"
single effect will be much greater than a milliosmole of interstitial urea.
Acid production.
In addition to lactate, anaerobic glycolysis also produces protons, one
for each lactate molecule, so it is natural to wonder whether the
steady IM glycolysis poses an acid-base problem. I suggest that it does
not, since the small production of protons would presumably be buffered
immediately by ambient HCO3
or ammonia. Even if
papillary lactate accumulation by recycling is increased during
progressive onset of antidiuresis by reduction of blood flow or by
increased DVR lactate permeability, this will presumably have no effect
on the rate of local production of protons and lactate.
This does, however, raise the issue of lactate and proton exit from
cells in the papilla, whose surroundings may have high lactate
concentration. If the cells are equipped with one-to-one coupled
lactate-proton transporters [MCT family of transport proteins (44)], then high external concentrations imply high
cellular concentrations (and acidity?) as well, since these
transporters are passive and reversible. This issue warrants investigation.
Experimental tests.
In the end, of course, the relevance or irrelevance of metabolically
produced osmoles to the concentrating mechanism is an experimental
question. To my knowledge, only one attempt has been made to measure
papillary glucose and lactate concentrations. In 1961, Ruiz-Guinazu et
al. (48) enzymatically measured glucose and lactate
concentrations in micropuncture samples of vasa recta blood collected
at the tip of golden hamster papillae. They found glucose concentration
diminished by about one-third and lactate concentration doubled
compared with arterial blood (aorta), but to obtain sufficient sample
volume they had to collect for up to 30 min. Perhaps because of
perturbations due to the long collection times, osmolality in their
vasa recta samples was only about threefold plasma osmolality (based on
their freezing-point depression measurements). These data thus support
IM lactate production from glucose, but the values seem too low to
contribute significantly to the single effect. It would be interesting
to repeat these measurements (in frankly antidiuretic animals), since
there are now micro-enzymatic methods for measuring glucose and lactate
in nanoliter samples.
Besides this technically demanding measurement of glucose and lactate
concentrations in papillary interstitium, it will be relevant to search
for specialized transport paths favoring lactate transport. These
should be located in plasma membranes of interstitial cells and on
basolateral membranes of epithelial cells of the nephrons and
collecting ducts. Obvious first candidates are members of the recently
cloned family of monocarboxylate transporters (MCT1, MCT2, ...)
(44), which serve in other tissues to couple lactate and
proton exit from glycolyzing cells (e.g., 9, 17, 23, 29, 39, 41, 45, 60). Although some of these have been reported in kidney tissue, their
precise locations and regulation have not yet been studied.
It will also be interesting to look further into the correlation of
IMBF with urinary concentrating ability. Although some studies have
shown that IMBF is reduced by almost 50% in antidiuresis (4), others showed only slight sensitivity of urinary
concentrating ability to medullary blood flow (11).
Conclusion.
The picture that emerges here is that IM glycolytic lactate production
in the range of reported values is probably sufficiently high and vasa
recta recycling sufficiently efficient to result in an osmotically
significant corticomedullary lactate gradient. The extent to which this
external osmole production amplifies concentrating ability remains to
be explored in full medullary models. It is hoped that the present work
will serve as a guide for such studies as well as a stimulus for
experimental tests of this idea.
I conclude with a few more general remarks. The countercurrent
arrangement of nephrons and blood supply was an evolutionary innovation
that permitted animals to move into arid ecological niches (20,
51, 61), but we see now that it also posed problems for the
supply of nutrients for the cells in the deep medulla. First, plasma
skimming and shunting of O2 from descending vessels to the
avid salt pumps of the outer medullary thick ascending limbs lead to
hypoxia below a certain depth (42, 49), necessitating reliance on anaerobic glycolysis for metabolic maintenance of cells
deeper in the medulla (32). By nature's serendipity, this may have provided an alternative single effect (the proposition in the
present work), namely, a source of metabolically produced osmoles,
thereby favoring continued lengthening of the papilla for progressively
more concentrated urine. That said, the suggestion must not be taken
too narrowly, since the relationship across species among papillary
length, percentage of long loops, and number of nephrons per unit body
weight is far from straightforward (2).
Nonetheless, for very long papillae (as in certain desert species) or
for high IM metabolic rates, I do predict that DVR glucose permeability
must be low in order not to starve the deepest regions by glucose shunting.
This suggested role in the concentrating mechanism for metabolically
produced osmoles suggests a nuanced interpretation of the role of urea.
In omnivores like the rat, and in carnivores [such as cat and dog,
both with 100% long loops (2)] maintenance of urea
balance and maintenance of water balance are arguably intimately
linked, in the manner suggested by the classic passive hypothesis
(according to which, the dumping of a relatively small amount of urea
from the terminal collecting ducts serves efficiently as external
osmoles in the drastically reduced volume of the rat papilla) and in
relation to dietary protein intake (1). However, most of
the desert rodents that have been studied eat a low-protein diet of dry
seeds (Mongolian gerbil, jerboa, pocket mouse, spiny mouse) or
succulent plants (Psammomys) and have very long papillae in
which the number of IM nephrons and vessels remains nearly constant
almost to the tip. Although these species can concentrate their urine
better than the rat, at least one study showed that papillary urea
concentration is not correlated with urine osmolality (22). Even in the rat, it has been reported
(5) that NaCl concentration was not correlated with urine
osmolalities above 1,500 mosmol/kgH2O. It is thus tempting
to speculate that although the kidneys of species that elaborate the
most concentrated urine are somewhat paradoxical in the context of the
classic hypothesis, their long, thick papillae, with presumably
correspondingly abundant cell metabolism throughout their length, would
appear to be consistent with an important single effect for metabolic
osmoles such as lactate. Relevant to this point is the observation,
pointed out by Beuchat (7, 8) in her exhaustive review of
the relationships between kidney size, metabolic rate, and urinary
concentrating ability, that small animals have a higher mass-specific
metabolic rate than do larger animals, although the relationship varies among individual organs.
As a final thought, I suggest that if metabolic osmole production does
turn out to participate importantly in the urinary concentrating
mechanism (through regulation of IMBF or by some other means), then it
also adds another mode of separation of renal regulation of water
balance, salt balance, and urea balance.
 |
ACKNOWLEDGEMENTS |
The manuscript was considerably improved thanks to suggestions of
two diligent referees.
 |
FOOTNOTES |
Parts of this work were previously presented in abstract form.
Address for reprint requests and other correspondence: S. R. Thomas, INSERM U.467, Necker Faculty of Medicine, 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. §1734 solely to indicate this fact.
1
I am not speaking here of the intracellular osmolytes
that protect IM cells from the high external urea and salt concentrations.
Received 17 September 1999; accepted in final form 15 May 2000.
 |
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