INTRODUCTION

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THE URINARY CONCENTRATING MECHANISM remains a mystery despite a long history of both experiment and modeling efforts. The basic enigma centers on our inability to explain the origin of the corticopapillary gradient of osmolality in the passive inner medulla, i.e., in the absence of evidence for active transepithelial salt transport. The earliest explanation, the "passive hypothesis" (PH), was set down in 1972 by Stephenson (35) and by Kokko and Rector (17). Modeling studies based on this hypothesis predicted the permeabilities in various nephron segments in order for the PH to explain the inner medullary gradient. When it became possible to measure these permeabilities in in vitro microdissected tubules, they were found not to agree with the model's requirements. In particular, salt and urea permeabilities in the inner medullary (IM) portion of the long descending limbs (LDL) were found to be higher in species that concentrate well than in those, such as the rabbit, that cannot elaborate a concentrated urine, whereas the PH requires exactly the opposite. In the context of the PH, solute entry (especially urea entry) into the LDL compromises the inner medullary osmotic gradient (21, 36, 43). Although it is tempting to question the validity of the permeability measurements, it is also clear from micropuncture studies at the papillary tip (9) that there is considerable urea entry into long Henle's limbs somewhere between the end of the proximal tubule and the tip of the medulla, i.e., either in the outer or inner medulla or both. Therefore, we are brought to the conclusion that the PH misses some essential feature of the concentrating process.

Up to now, virtually all modeling studies testing new hypotheses or the effects of new permeability measurements or anatomical juxtapositions have judged the degree of success of new model features based on a single criterion, namely, the predicted osmolality of fluid leaving the collecting duct (CD). There have been no studies that employ model predictions of the cycling of water and solutes among the various nephron and vascular pathways and compare them with the several thoughtful discussions in the literature based on anatomy and permeability data. In particular, Lemley and Kriz (22), Bankir and de Rouffignac (3), Jamison and Robertson (13), de Rouffignac (30), and others have made detailed suggestions about probable recycling paths for water, urea, and salt founded in the general framework of the PH but enhanced by additional knowledge of the system. A recent model (44) (hereafter called the WKM model, for Wexler-Kalaba-Marsh) incorporated as much detail as possible, and it seems to be an improvement over earlier models, since it "concentrates better" (however, WKM is not without its critics, see Ref. 37). However, it remains the case that increasing salt and/or urea permeabilities along the inner medullary descending limb in these models leads to worse, not better, inner medullary osmotic concentration; so there remains a fundamental problem.

In hopes of suggesting some new directions for research, the present study explores the detailed recycling patterns predicted by our most recent incarnation of the WKM model (41), compares these patterns with those predicted by the authors mentioned above, and also compares model predictions of fractional deliveries to structures in the papillary tip and in accessible surface loops with available experimental data. Since this model is currently our best attempt to represent present hypotheses, this study is a gauge of the extent to which the suggested recycling patterns are in fact consistent with known anatomy and permeability measurements. We find that on most points the suggested patterns match those observed in the simulation, but the exceptions are interesting and may be important for improving our understanding of this system.

Glossary

	METHODS

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Model Description

The present analysis uses our most recent model (41), which, compared with the original WKM model, more faithfully represents the anatomy of the IM and OM and measured permeabilities in all segments. The architectural organization of the renal medulla is shown in Fig. 1. Figure 1, left, shows transverse cuts at each medullary level, indicating the relative positions of each tubular structure and giving the "connection strengths" used to calculate solute and volume exchanges between structures. In each medullary zone, transmural and convective connections between structures are represented by straight lines and curved arrows, respectively. Each circle represents a collection of a given type of tubules or vessels, the numbers of which used in the model are listed in Table 1. The distances between structures in the diagram are not to scale; relative anatomic distances between the respective tubes in the model are accounted for by the values of the connections strengths. Figure 1, right, gives a view of these relationships along the corticomedullary axis to illustrate the continuity from one medullary zone to the next, but since it is flattened to two dimensions, it does not faithfully represent the spatial distribution of the structures.

View larger version (69K): [in this window] [in a new window]   Fig. 1.   Model illustration. Left: placement of each structure in cross section through each medullary region. Right: tubes in a flattened 2-dimensional projection, approximately in their correct positions. Values in left (e.g., 0.33, 0.25) indicate connection strengths from Henle's loops and from descending vasa recta to ascending vasa recta, reflecting their relative proximities in each region. Distances are not to scale. Arrows indicate convective connections.

In accordance with the assumed tubular distribution (Table 1), the ratio of SDL population to LDL population is two to one in outer medulla (3, 22); the ratio of the Henle's loop population to the collecting duct population is six to one in outer medulla (~5 to 1 in Ref. 16). At the outer-inner medullary border, the ratio of the number of long loops to collecting ducts is two to one, and the ratio becomes one to one as they approach the tip of the papilla (16). The model structure thus explicitly represents the six nephrons associated with a single OMCD, but it is scaled according to total numbers of nephrons at each medullary level in rat kidney. Thus "total GFR" (as given, e.g., in Table 3) is not here numerically equal to whole kidney GFR but rather equals a value six times greater than the SNGFR.

Boundary conditions and parameters. The model treats the renal medulla from the corticomedullary border down to the papillary tip. The boundary conditions (flow rates and concentrations) are specified for all the descending limbs and vasas recta at the corticomedullary border, for mass conservation in the descending and ascending structures at the loop bends, and for solute and fluid reabsorption at the end of the late distal tubule (entry to the CD). Proximal tubules are assumed to reabsorb two-thirds of filtered NaCl and volume flow and one-third of filtered urea, so at the corticomedullary border, single-tubule flow rates in LDL and SDL are 10 nl/min and NaCl and urea concentrations are 140 and 18 mM. The cortical portions of the distal tubules and cortical collecting ducts are not treated explicitly. Instead, fluid and solute entry into the medullary collecting ducts are calculated from values at the top of the ascending limbs, using three assumptions: 1) fluid entering the medullary CD is isosmotic to blood; 2) NaCl concentration is 35 mM; and 3) urea delivery to the CD is 85% of its delivery to the beginning of the distal tubules, i.e., the distal tubules reabsorb 15% of the urea delivered to them by the ascending limbs. The detailed formulation is given in previous work (38, 42, 44).

Numerical method. The numerical package based on collocation, COLNEW (available in the netlib archives on the internet at http://netlib.bell-labs.com/netlib/ode/), was used to solve the models. Detailed descriptions of the system equations and numerical scheme have been given previously (38, 42, 44). The computations were performed in double precision on an IBM RS/6000-590 or RS/6000-320H running AIX. The predictions converged after calculating over two meshes and one or two iterations for each mesh, using the initial set of estimations saved from previous runs to reduce numerical instability. On an IBM RS/6000-590, one iteration takes about 1 to 2 CPU seconds, and a run from a previous solution to convergence is completed in less than 10 s.

Solute and Water Cycles: Flux Calculations

A program was written to calculate net transmural flux from each tube within an arbitrary medullary slice, i.e., between any two medullary depths, from the output data.

To illustrate the description of the new cycles' calculations, Fig. 2 schematically depicts a portion of the shunted tubular structure that represents the long Henle's limbs, illustrating the early returning loops in the inner medulla. The diagram applies equally to flows of volume, urea, or NaCl. The model (for more detail, see Ref. 44) uses the following differential equations for conservation of mass within a section of lumped tubule

(1)

and

(2)

where F_d and F_a are tubular (i.e., luminal) flow rates within the descending and ascending lumped structures, respectively (flows toward the papilla are positive, and those away from papilla are negative), J_d and J_a are the fluxes across the descending and ascending tubular walls at depth x (i.e., loss from the tubular lumen; secretion into lumen if the value is negative), and F_shunt is the shunt flow at depth x, which is given by

(3)

where n (x) is the number of individual tubes represented by the lumped structure at depth x, so F_d (x)/n (x) is the single-nephron flow at x.

View larger version (15K): [in this window] [in a new window]   Fig. 2.   Schematic illustration of loop structure with shunt return, for flux calculations.

The number of long limbs and vasa recta decreases exponentially within the inner medulla, falling at the papillary tip (x = 0.6 cm) to th of their value at the OM/IM border (x₀ = 0.2 cm), according to

(4)

which has the derivative

(5)

We know the values of the tubule flows F_d and F_a at every point (from the simulation output) and are interested in calculating the integral fluxes from descending tubules

(6)

and from ascending tubules

(7)

within any given slice, i.e., within a region from arbitrary depth x₁ to x₂. Illustrating the calculation first for J_d, we substitute in Eq. 6 from Eq. 1

(8)

Substitution for F_shunt (x) from Eq. 3 gives

(9)

Since n (x) and dn (x)/dx are available analytically (Eqs. 4 and 5), and F_d (x) is available at every point from the simulation output, we can approximate the integral on the right-hand-side as the sum

(10)

where N = (x₂  x₁)/x, x ¹_i = x₁ + (i  1)x, and x ²_i = x₁ + ix.

Similarly, for flux from the ascending tubule, we have

(11)

with the same approximation for the integral of F_shunt.

As a check on the accuracy of these calculations, we took advantage of the fact that water permeability along the LAL is zero in the model, so LAL volume flux calculated using this method must give values arbitrarily near zero. By this criterion, it was necessary to chop each medullary region of interest into at least 30 slices.

The same method was used for LVR, but since each LDV gives rise in the model to two LAV, we have, instead of Eq. 2

(12)

Likewise, for the SVR in the inner stripe, each SDV gives rise to four SAV, so we have

(13)

	RESULTS

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Since the following discussion is rather detailed, we first present the basic model results in three different ways, to make things more accessible.

Basic results

Standard output: Volume flows and concentrations. Table 2 presents the basic model output, namely, volume flows and solute concentrations and osmolality, plus TF/P inulin values, in each tube structure at key medullary levels (the actual simulation output table gives the results over a much finer mesh and to greater precision). Slight differences from the numbers in table 4 of Wang et al. (41) are due to stricter error tolerance here (attested by the smaller % imbalances). These numbers are used for construction of Tables 3 and 4. Figure 3A shows the profiles of urea and salt concentration and of osmolality in each structure along the corticopapillary axis.

View larger version (24K): [in this window] [in a new window]   View larger version (29K): [in this window] [in a new window]   View larger version (27K): [in this window] [in a new window]   Fig. 3.   Corticopapillary profiles along each tube predicted by the model simulation. Top graphs in each of A-C show profiles for short nephrons and short vasa recta (and for OMCD in some). Bottom graphs in each of A-C show profiles for long nephrons, long vasa recta, and CD. A: profiles of urea and NaCl concentrations and osmolality (from Table 2). B: profiles of fractional deliveries (FD) as percents of total (tot) filtered loads, i.e., 100 × Fi(x)/FLi(x), where i is urea, salt, or osmoles (from Table 3). C: profiles of single-nephron fractional deliveries (SNFD) as percents of single-nephron filtered loads, i.e., 100 × (Fi(x)/ni(x))/SNFLi(x), where i is urea, salt, or osmoles (from Table 4).

Total tubular flows as a percent of total filtered loads. Table 3 and Fig. 3B present the same results as Table 2 and Fig. 3A but in a different form, namely, as the tubular flow rates not only of water but also of urea (F_v × c_u ), salt (F_v × c_s ), and total osmoles [F_v × (c_u + 1.82 × c_s )] all scaled as a percent of their respective total filtered loads. These flows represent the sums of flows in, say, all LDL, represented in the model by the lumped LDL structure. These values make clear the reduction of flows toward the papillary tip due to the reduced numbers of nephrons and vasa recta, and permit evaluation of the contribution of each structure to global balance. For instance, if we look at just the flows entering and leaving the IM via the whole population of long loops of Henle (LDL and LAL), we see that 1) the LDL enter the IM carrying 3.14% of GFR and leave in LAL carrying 3.65%, showing a small net gain of water, which occurs in the LDL, since LAL are impermeable to water; 2) the LDL carry 21.9% of FL_u, the filtered load of urea, into the IM and LAL carry 45.8% back out to the OM, more than doubling the urea flow within the long loops within the IM; 3) the LDL enter the IM carrying 11.3% of FL_s, the filtered load of salt, but LAL carry only 9.63% back up to the OM, for a net loss of 1.67% of the filtered salt load; 4) the net osmole contribution of Henle's loops to the IM is 1% of the filtered load of osmoles (11.7-10.8%).

Tubular fractional deliveries. From the values in Table 3 alone it is not possible to know whether individual long nephrons gain or lose solutes or water along their course in the IM. Table 4 and Fig. 3C give average flows per tube as a percent of single-nephron filtered loads, which is often called the fractional delivery (given here as percentages) and is the same as reporting 100 × (TF/P)_i/(TF/P)_inulin from micropuncture data. These values are best for evaluation of net gain or loss of solutes or water along individual nephrons of a given type and can be directly compared with micropuncture data (see below). From Table 4 and Fig. 3C, one can see that luminal water and salt flows change very little in the IM along the longest LDLs, whereas luminal urea flow increases 10-fold, attaining 658% of the single-nephron filtered load of urea at the hairpin turn, though we see in Table 3 that total urea flow at the tip in the whole (small) population of longest LDLs amounts to only 1.71% of the total filtered load of urea.

Exchange and Recycling of Urea, Salt, and Water

View larger version (77K): [in this window] [in a new window]   View larger version (78K): [in this window] [in a new window]   View larger version (80K): [in this window] [in a new window]   Fig. 4.   Fluxes in each medullary region. Arrow bars represent fluxes out of/into each structure, and the relative thickness of the arrows roughly indicates the relative flux values in a given region (actual values given in Table 6). Note that the algebraic sum of fluxes at any given level is zero, by mass conservation. A: water fluxes. Note that OS water fluxes are about 100-fold greater than those in the papilla (LIM), so arrow thicknesses cannot be exactly to scale. B: urea fluxes. C: salt fluxes.

Let us define some terminology: we will use the word "recycling" to describe transfers from ascending to neighboring descending tubes, which tend to keep a substance from escaping into higher regions of the medulla; the word "short-circuit" will be applied especially to water flows effectively shunted from descending to neighboring ascending tubes, thereby reducing total volume flow in the next deeper slice; instead of referring to (re)absorption and secretion, we will use the terms "dumping" (for transport out of a tubule) and "uptake" (for transport into a tubule).

Water cycles (as percent of total GFR to all model nephrons). Starting our description in the deepest region, LIM, we see (Fig. 4A; Tables 6 and 7) that the CD and to a lesser extent the LDV dump volume, all of which is recovered by the ascending vasa recta. In the narrow transition zone, not only LDV and CD but also LDL dump a bit of volume on their way down, all of which is carried back up by the LAVs. We thus see that total volume flow at the tip is reduced due not only to the smaller number of tubes but also to a water "short-circuit" which transfers water from the downward-flowing tubes to the upward-flowing vasa recta. This behavior matches the usual notions of water cycling in the deep inner medulla, except perhaps for the lack of water loss from LDL in the deepest part of the papilla.

Urea cycles. UREA CYCLES IN THE IM. As with the above account of water cycles, we begin in the deep papilla and work upward (Fig. 4B). The discussion of inner medullary urea exchanges centers naturally around the destiny of the urea dumped there from the CDs. The CDs in this simulation dump into the LIM an amount of urea equivalent to nearly one-third of the total filtered load of urea (Table 6) (also equivalent to 7.7 times the total osmoles delivered to the papillary tip in LDLs, Table 7). This urea load is picked up by all the other structures in the region, descending as well as ascending (except for LAV1).

Salt cycles. Along the CD, although salt reabsorption is quantitatively very small, it is crucial for the organism's salt balance; absolute NaCl reabsorption along the IMCD in antidiuresis has been reported as high as 3% of FLs (34), about 10 times the rate of fractional salt excretion. Moreover, sodium concentration along the CD may rise in the OM before falling again in the IM (5). However, models of the medullary countercurrent system, the present one included, do not yet do justice to the role of the CD in salt regulation; they lump all salt into one pool labeled "NaCl," whereas renal handling of sodium and potassium (to say nothing of ammonium, phosphate, etc.) are in fact quite different, especially in the distal nephron and CD. For instance, by free-flow micropuncture at both the base and tip of IMCD, Diezi et al. (5) showed clearly that Na⁺ concentration ([Na⁺]) falls and [K⁺] rises along the IMCD. The minimal salt reabsorption included in the present model (~0.1% of FL_s) avoids some problems of earlier models, but proper treatment of this question must await more complete models.

Total osmoles. One of the basic tenets of the classic view of the concentrating mechanism is that at the OM/IM border the descending structures, LDV, LDL, and CD, enter the IM with nearly equivalent osmolalities and that the rising LAL fluid is dilute relative to the other structures at the same level as it leaves the IM (not all workers insist that LAL fluid must be dilute as it leaves the IM; see Refs. 4, 21, 30). The present model contradicts both of these tenets, namely, we see (Table 2) that at the OM/IM border c ^LDV_osm < c ^LDL_osm < c ^CD_osm , reflecting the respective positions of LDV, LDL, and CD along the radial osmolality gradient in the IS. Furthermore, despite the considerable loss of "water-free" osmoles along the LAL within the IM, LAL is here an osmotically equilibrating rather than a diluting segment in the IM. Although these two points both seem quite straightforward consequences of the anatomic and permeability characteristics represented by the model, the matter of the profiles along the CD remains troublesome. This will be discussed below.

	DISCUSSION

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For the discussion that follows, we remind the reader that our purpose is not to justify the present model (41) but rather to explore its predictions in detail, since to date it is the most accurate representation of the available experimental measurements and anatomic data. As such, this model analysis serves two purposes: 1) it serves as a quantitative check on qualitative extrapolations that have been based on these same data; and 2) by our detailed analysis, we hope to clarify directions for future improvements and new measurements.

To facilitate the discussion in comparison with the experimental literature, Fig. 4, A-C, illustrates the results given in Tables 6 and 7, showing the basic patterns of intertubular exchanges and recycling of water, urea, and salt, respectively, for the present model.

Summary of Water Cycles

The model predicts exceptions to this water short-circuiting pattern in descending Henle's loops where water uptake accompanies high urea influx, namely, in the LDL of the deep papilla (uptake of urea dumped form the terminal IMCD), in LDL of the UIM (massive recycling of urea dumped from parallel ascending LAL and LAV), and in SDL running close to vascular bundles in the inner stripe (uptake of urea dumped from LAV in the vascular bundles and also form nearby LAL). A close look at Tables 6 and 7 reveals (see columns LVR_tot and LHL_tot, row IM_tot) that in the IM, the net volume receivers are in fact LDL due to their considerable (for the region) water uptake in the UIM, not the LVR, which despite their considerable water short-circuiting from LDV to LAV actually have a small net water loss within the IM of this model.

Summary of Salt Cycles

Within the IM, according to the passive hypothesis, passive salt loss from LAL, made possible by high interstitial urea thanks to urea dumped from the CD (17, 35), is taken to be the single effect for buildup of the IM osmotic gradient. In the present model, the LAL does in fact dump salt along its whole IM ascent; this salt is not here recycled into the parallel descending Henle's limbs but is instead carried up and out of the IM by the ascending vasa recta, which corresponds to Stephenson's analysis of the passive mechanism (36). It is sometimes suggested (2, 3, 22, 30) that some or most of the salt transported out of the ascending limbs in the OM is recycled down into the IM in the LDL, which have a moderately high measured salt permeability, but in this model osmotic equilibration along the LDL in the OM by water loss occurs fast enough to keep the salt gradient small across the LDL wall; consequently, the amount of salt diffusing into the LDL within the OM (0.3% of FL_s, see Table 6) is only a negligible fraction of that dumped into the OM by the SAL and LAL (20.6% and 8.9% of Fl_s, respectively). We thus see that in this simulation virtually all of the salt actively transported out of ascending limbs in the OM is returned directly to the general circulation via the vasa recta. The role of this active salt transport in the concentrating mechanism is thus to dilute the ascending limb fluid and concentrate the descending fluid not by adding salt but by drawing water out of the LDL and CD (but not from SDL in the inner stripe, which is concentrated by urea uptake as mentioned above).

Summary of Urea Cycles

Before summarizing the details of the urea exchanges within each region, it is useful to look at the big picture, starting with the primary event of interest, namely, the dumping of 43.4% of the filtered load of urea into the IM by the IMCD. What is the ultimate fate of this dumped urea in this simulation? From Table 3 we can calculate the net dumping or uptake of urea in each tube type over the whole medulla by subtracting urea outflow in the ascending tubes at the corticomedullary border from urea inflow in the descending tubes. Thus for LVR

(14)

remembering that upward flows are negative. Similar equations apply for SVR, LHL, and SHL. Results of these calculations are given in Table 8. From this table, we see that in this simulation 1) nearly all of the urea dumped from IMCD in the inner medulla actually ends up returning to the cortex via the SVR, and 2) Henle's loops make no net contribution of urea to the medulla, since the net amount picked up by the long loops is identical to the net amount dumped by the short loops. Since the SVR do not extend into the IM, there is clearly massive urea exchange within the outer medulla from the LAV and LAL, which carry the urea up from the IM, to other tubes.

Figure 4B shows the urea paths in this simulation. Within the IM, the two main features are urea dumping by the IMCD and massive urea recycling between descending and ascending branches of both Henle's loops and the vasa recta. To better appreciate the extent of inner medullary urea recycling, we can point out (Table 6) that the LDL take up an amount of urea equal to 36% of FL_u on their way down within the IM and the LAL take up another 8% of FL_u in the LIM and TZ before they start dumping urea further up in the IM, for a total of 44% taken up by Henle's loops. LDV take up an amount of urea equal to 122% of FL_u and LAV2 take up another 14% in the deepest medulla, for a total of 136% of FL_u taken up by vasa recta. This amounts to a grand total of 180% of FL_u taken up by vasa recta and Henle's loops, whereas "only" 43.4% of FL_u is dumped by the IMCD. The difference, or the equivalent of 137% of FL_u, is recycled within the IM, where total volume flow is only a small fraction of GFR.

Within the OM, in accord with classic predictions, urea from LAV is recycled to LDV and SDV within the vascular bundles of the IS and also within the OS. More surprising, however, is the role of the short Henle loops in this model; they pick up considerable urea as they pass near the VB in the IS, as is generally supposed, but then they dump an even greater quantity of urea in the OS before returning to the cortex (urea dumping from MTAL has in fact been suggested by some authors; see, e.g., Ref. 30); that is, the model predicts that the fate of the urea they pick up is not to be carried up and around to contribute to the urea load delivered to the collecting ducts, but rather just to be transferred from the IS to the OS, where part of it is recycled via the SDV and the rest is lost to the cortex in the SAV. This OS urea loss from the SAL is of course made possible by their relatively low but certainly not negligible measured urea permeability (15). Although recently cloned urea transporters have led to localization of their expression along the nephron (25, 27, 28, 33), none has been found in the medullary thick assending limb (MTAL), which, assuming the measured permeability value is correct, suggests either that some as yet unidentified urea transporter operates in this segment or that urea may pass between the cells across the tight junctions. A more recent report confirms this tendency using purified vesicles from apical and basolateral membranes of MTAL (29), since it suggests that urea permeability of both cell membranes is very low.

Whatever the eventual outcome of this question of the effective in vivo urea permeability of the outer stripe MTAL, the handling of urea by this model's short loops is clearly in disagreement with micropuncture data from surface convolutions. Armsen and Reinhardt (1) measured end proximal and early distal tubule fractional delivery of urea in Wistar rats by micropuncture under antidiuresis and varying degrees of water diuresis. They found that in accessible short loops of nondiuretic rats, FD_u increased from ~60% of FL_u at the end of the accessible proximal convoluted tubule (PCT) to 100% at the beginning of the distal convoluted tubule (DCT), a near doubling of tubular urea flow in superficial nephrons despite a halving of volume flow [(TF/P)_inulin increased from 3 to 6]. This is clear evidence of considerable net urea uptake in the superficial loops of Henle during antidiuresis, contrary to the model's prediction of a small net urea loss.

This suggests, of course, that the model might "work better," i.e., might produce a higher urinary osmolality and steeper IM gradient, if P_u of SAL were reduced to lessen or prevent urea dumping in the OS and thus increase urea delivery to the CD. The results of testing this idea with the present model bear witness to the difficulty of second-guessing this complicated system. In a series of simulations (partial results given in Table 9), P_u(SAL) was gradually reduced to th of its control value. The result was a modest decrease rather than the expected increase of U_osm, from 1,434 down to 1,396 mosmol/kgH₂O, which is explained as osmotic diuresis, i.e., the short loops did in fact deliver slightly more urea to the CD, but instead of resulting in greater urea recycling, the CD dumped the same amount of urea into the IM as before, so FE_urea increased from 13.1% to 18.5% of FL_u, resulting in greater urine flow [(TF/P)_inulin decreased from 729 to 535]. These results suggested that the model's IMCD urea permeability is too low to permit effective recycling of the higher urea load. Subsequently increasing P_u(CD) in TZ and LIM to higher measured values (7, 32) did in fact lead to increased U_osm, but only if L_p(CD) was also increased to higher measured values (19, 31), permitting water to follow the urea dumping at the higher rate. This set of parameter changes (indicated by an asterisk in Table 9) effectively increased the model's urea recycling (IMCD dumps 57.4% of FL_u instead of 43.4%) and increased U_osm to 1,630 mosmol/kgH₂O. It is not our purpose here to develop this idea systematically, but Table 9 shows some of the interesting functional parameters in comparison with the basic simulation results from the earlier tables, and Fig. 5 shows the area-weighted slice profiles of urea concentration, salt osmoles, and total osmolality, similar to measurements in whole slices from each medullary depth according to

(15)

where i is urea, NaCl, or osmolality, c_{i , j} is the concentration of i in structure  j, n_j is the number of structures  j at the given medullary level, and A_j is the cross-sectional area of the individual structures (42). The inset to Fig. 5 shows the ratio of urea to total slice osmolality along the medulla. This simulation (indicated by an asterisk in Table 9) gives results more closely in accord with experimental results on several important points, namely, increased U_osm, steeper and deeper inner medullary osmolality profile, and better agreement of SNFD_u values with micropuncture results, especially concerning the significant addition of urea to the short Henle's loops, which was, after all, the impetus for making these changes. Table 9 also presents results of simulations in which CD urea and water permeabilities were changed alone, with or without the reduction of SAL urea permeability. It nonetheless remains the case (results not shown) that increasing urea permeability of LDL and/or LAL in the IM in this new scenario still results in poorer performance.

View larger version (27K): [in this window] [in a new window]   Fig. 5.   Comparison of corticopapillary slice profiles from test simulation (thick curves) with those of the baseline simulation (thin curves). Inset: profile of the ratio of urea to total osmoles.

"Osmole Cycles"

Collecting Duct Profile

When it has been mentioned in the literature, it is generally believed that the CD profile conforms essentially to that observed in medullary slices, and three studies are most often cited in support of this belief, namely Wirz (45), Gottschalk and Mylle (8), and Gottschalk et al. (9). However, a close look at these reports reveals that they do not in fact provide direct evidence for an inner medullary osmotic gradient along the CD; the spatial resolution of the cryoscopic slice technique of Wirz (45), though ingenious, was probably too poor to give clear evidence of osmotic differences between neighboring structures at the relatively high temperatures they used (10°C); Gottschalk and Mylle (8) made measurements only in structures close to the papillary tip, not further up the CD near the papillary base; and Gottschalk et al. (9) did obtain evidence for a considerable osmolality gradient along the IMCD, but only in Psammomys, a species that cannot be easily compared to other rodents for both anatomic (100% long loops) and dietetic reasons (no urea problem because they eat only succulent plants). So, based only on these reports, it would be possible to suspect the truth may lie with the model's predictions. There are other data, however, some of which is in favor of and some against an osmotic gradient along the IMCD. These studies are briefly summarized as follows.

Ullrich (39) measured (TF/P)_inulin, [Na], [K], pH, and [NH⁺₄ ] and total osmolality (but not urea) in fluid samples obtained by retrograde microcatheterization at various levels along the IMCD of golden hamsters in antidiuresis. That study showed clearly that along the IMCD there was considerable water and sodium reabsorption, virtually no potassium reabsorption, and a decrease of pH (dramatic increase of [NH₄⁺]), but that the osmolality remained virtually constant from the OM/IM border to the papilla. This evidence thus tends to agree with the prediction of the present model, but it must be pointed out that their animals had urinary osmolality ranging from 400 to 1,200 mosmol/kgH₂O, not frankly antidiuretic.

Jamison (11) used a similar technique in rats with urinary osmolalities ranging from 800 to 1,450 mosmol/kgH₂O and found an average gradient of 226 mosmol/kgH₂O per mm along the final 1.5 mm of the IMCD.

Sonnenberg (34), also using retrograde microcatheterization in rat collecting ducts, also found a considerable osmolality gradient within the IMCD under nondiuretic (U_osm up to 1,400 in the experimental kidneys, and 1,800 in the control kidney of the same animals), but not under diuretic conditions, and confirmed many of the results of Ullrich (39).

Marsh and Martin (23) used microcatheterization into the papillary CD of the hamster. They presented only (TF/P)_inulin and fractional deliveries (or fractional excretion rates) at the end of the CD (duct of Bellini), in the ureter, and at a point 4 mm higher upstream in the CD system, which in the hamster is about two-thirds of the way up to the OM/IM border. Osmolality and urea concentration in CD fluid calculated from the data they report showed an osmolality increase from 1,550 mosmol/kgH₂O at the upstream point to 2,300 mosmol/kgH₂O at the papillary tip, i.e., a gradient of 200 mosmol/kgH₂O per mm (if linear) along the IMCD of the hamster in antidiuresis. For comparison with the present model of the rat kidney, the analogous position would be situated 2.7 mm from the tip, or just at the bottom of the UIM. In the model, CD osmolality at this point is 1,229 mosmol/kgH₂O and is 1,496 mosmol/kgH₂O at the papillary tip, giving a gradient of 100 mosmol/kgH₂O per mm, or only half as steep as that found by Marsh and Martin (23). Existing data thus mostly support a steeper IMCD osmolality gradient than that predicted by the present model.

Conclusions

I suggest several possible directions for further investigation.

1) Thanks principally to the recent work of Pallone and collaborators (26), it is becoming clear that the vasa recta may play a role much more interesting than that attributed to them up to now. To mention just one possibility stemming from their work, Pallone et al. (26) found that, contrary to the notion that vasa recta allow essentially free passage to small solutes and water, the sodium permeability of OMDVR may in fact be quite low and may be principally paracellular, whereas water and urea permeabilities are high and they probably pass through the cells, In our recent presentation of the present model (41), we showed that reduction of P_s in OMDVR could dramatically increase concentrating ability by favoring osmotic equilibration in the OM in these vessels by water removal instead of solute addition, thereby reducing IM blood flow.

2) Our favored hypothesis is that there may be interstitial osmolytes other than salt and urea in the inner medulla that contribute to the concentrating mechanism by drawing water out of the descending structures. This idea was formalized by Jen and Stephenson (14) and was tested by us in a model similar to the one analyzed here (38) without specification of the possible origin of the extra osmoles. Those studies made it clear that if such osmolytes are present in concentrations on the order of around 100 mosmol/kgH₂O or more, then they could play a very important role in the concentrating mechanism. One possible source of such osmolytes, which remains to be tested experimentally, is the metabolism of glucose to lactate, since it is known that anaerobic glycolysis is the main source of ATP in the relatively hypoxic inner medulla. Without engaging in a detailed analysis here, we postulate that it could be the case that a relatively constant rate of metabolic osmole production (no a priori reason to suspect that cells in the medulla consume more ATP during antidiuresis) might be ineffective during diuresis but that it could become important to the concentrating mechanism when volume flow to the inner medulla is reduced during the onset of antidiuresis (invoking adjustment of permeabilities in the OMDVR to reduce IM blood flow as mentioned just above), allowing a buildup by countercurrent exchange in the reduced IM volume.

3) Solutes found in relative abundance in the medulla and urine but not traditionally included in models of the medulla, such as potassium and NH ⁺₄ , whose recycling paths are interesting in their own right, may turn out to play a significant role in the concentrating mechanism.