## 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

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*
.

## 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*).

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 > x*
_{OM/IM}, is given by
Equation 1with the factor describing the exponential decrease in their number with depth (*k*
_{sh}) = 1.213 mm^{−1} for vasa recta and Henle's loops and*k*
_{sh} = 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 *N*
_{CD, 0} (= 64), the number of OMCD entering the OM. Table1 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 (*k _{sh}
*).

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
Equation 2We then have the following system of differential equations, adopting the usual convention that descending tubule flows are positive and ascending flows are negative
Equation 3
Equation 4
Equation 5
Equation 6
Equation 7
Equation 8where 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
Equation 9
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*:
Equation 10where 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,*J*
_{gly,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*J*
_{rx,fract}, we have
Equation 11which is used to calculate *K*
_{gly}, the glycolysis rate per tube and per millimeter
Equation 12Because in this model glucose is converted to lactate only within the IM, we have within the OM (i.e., for *x ≤ x*
_{OM/IM})
Equation 13and within the IM (i.e., for *x > x*
_{OM/IM})
Equation 14
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
Equation 15In 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.*K*
_{m} 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-*P*
_{u}” 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. F_{v} 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 Osm_{CD, 0}
* = *263 mosM.

*2*) A specified fraction, u_{fac}
* =*0.85, of urea is delivered to OMCD [i.e., the distal tubules reabsorb (1 − u_{fac}) 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 Equation 16thus obtaining Equation 17We then have, for the other values entering the OMCD Equation 18 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
Equation 19where *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 (F_{v} and 4 concentrations, C_{i}) 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
Equation 20
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
Equation 21
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
Equation 22This global Newton iteration is repeated until global convergence is achieved (i.e., until global mass balance is respected to within our chosen error tolerance).

## 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*P*
_{u} along the LDL, namely, the frank absence of an IM osmotic gradient.

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.

Table3gives 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-*P*
_{u} parameter set (Table5) 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).

### 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 4
*A* 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. 4
*A*, the osmotic gradient was nearly eliminated by doubling MBF.

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. 4
*A*), but the ratio of IMBF to nephron flow increased only from 1.2 to 1.77. Taken together with the results of Fig.4
*A*, 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. Figure6
*B* shows that the profile of lactate concentration is unaffected by DVR glucose permeability.

### 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-*P*
_{u} parameter set the gradient actually increases with increasing *P*
. Detailed results from the high-*P*
_{u} 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.”

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-*P*
_{u} 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.

#### 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 (FE_{u}) 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-*P*
_{u} parameter set, with its higher *P*
and other parameter changes, FE_{u} (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-*P*
_{u} parameter set, compared with typical values of ∼50% in antidiuretic animals.

#### Concentrating work.

Another apparent improvement associated with the high-*P*
_{u} 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-*P*
_{u} 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.

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
Equation 23where *RT* = 2.5773 J/mmol at 37°C, and U_{osm} and P_{osm} 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 *× U_{osm}, where *V* is urine flow rate. Thus
Equation 24For comparison of our simulation results with experimental results from the literature in various species, we normalize this by the GFR
Equation 25where 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
Equation 26For U_{osm} 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-*P*
_{u} parameter set.

## 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-*P*
_{u} parameter set performs much better by this criterion. In addition, FE_{u} 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-*P*
_{u} 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.

## Acknowledgments

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).

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*advertisement*” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.August 27, 2002;10.1152/ajprenal.00045.2002

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