INTRODUCTION

Top Abstract Introduction Results Discussion Conclusions References

THE ABILITY OF MAMMALIAN kidneys to form concentrated urine depends on the structure of the renal medulla. Studies of the histotopography and comparative anatomy of the renal medulla (1, 4, 13, 15, 19) have provided essential information on the relative spatial distribution of the structural elements of the medulla and their functional relation to the process of urine concentration. Mathematical models have been developed to simulate concentration gradient formation in mammalian kidneys (5, 6, 11, 16, 17, 24, 30). Although all of the models are able to predict the concentration gradient observed in the outer medulla (OM), none of the models yet proposed can successfully account for the generation of the concentration gradient in the inner medulla (IM) without making idealized assumptions regarding kidney anatomy or membrane properties.

The model proposed by Wexler, Kalaba, and Marsh in 1991 (30) (referred to hereafter as the WKM model) was the first to incorporate a three-dimensional representation of the spatial heterogeneity of nephron segments and vasculature of different renal medullary regions, based on anatomic studies by Kriz and colleagues (13, 15, 19). The WKM model employs a spatial distribution of structures in the renal medulla and includes preferential interactions between certain renal tubules and vessels. Their results showed that a significant inner medullary osmolarity gradient could be explained.

Histotopographical observations and anatomic descriptions (1, 4, 7, 19) fail to support some of the anatomic and transport assumptions in WKM (30). Some modifications of the model (30) were made in our earlier work (29) in response to the critical reviews from Han et al. (7) and Stephenson et al. (25) of the WKM model. We corrected the relative positions of long Henle's loops and collecting ducts (CD) in the IM in accordance with morphometric results (7). These modifications led us to distinguish the radially separated upper IM from the more homogeneous lower IM. In addition, the measurement of Na-K-ATPase activity in upper inner medullary collecting ducts (IMCD) (26) showed that active NaCl reabsorption exists in this segment and contributes to formation of a hypertonic region around the CD. With the improved anatomic descriptions of the IM and the active transport of NaCl in the upper IMCD, the modified model (29) predicted more realistic concentrating effects in both OM and IM while employing a non-zero water permeability in the upper IMCD.

In the present article, we develop a more accurate anatomic representation of the OM that incorporates observations from a number of anatomic studies (1-4, 8, 14, 19). The modified model is used to simulate several functionally related features of urinary concentration. Hypotheses pertaining to the possible recycling routes and functionally separated domains are tested through predictions of the mathematical model and are compared with histotopographical reports. Parametric and anatomic sensitivity analyses are also performed to evaluate the quality of the model predictions.

	TOPOGRAPHY OF RAT RENAL MEDULLA

Detailed anatomy of the renal medulla and its functional implications have been described by many authors (1, 2, 4, 13, 15, 19). It is known that the relative positions of tubules and vessels are quite different in different renal medullary regions. In the outer stripe (OS), there are pars recta, thick ascending limbs, and CD, surrounded by ascending vasa recta (AVR) originating both from the IM and from the inner stripe (IS). At the bottom of the OS, vasa recta start gathering into highly organized vascular bundles (VB), which persist through the IS and disperse within the IM. The bundles contain both descending vasa recta (DVR) and AVR. The thick ascending limbs of the loop are reportedly situated nearer to the CD than to the VB in the OS.

In the IS, the AVR from the IM (LAV), come into contact with the DVR within the VB, whereas the AVR that originate in the IS (SAV), ascend in the interbundle region and perfuse tubules there. The short descending limbs (SDL) lie close to the VB, and short ascending limbs (SAL) lie closer to the CD. In rat and in other rodents to differing extents, the SDL of the loop shift from a position near the CD in the OS to the periphery of the VB in the IS. The LDL migrate away from the VB in their course down through the OM, intermingled with ascending limbs. The long descending and ascending limbs switch their radial positions near the border of the OM and IM. A clear radially separated arrangement is observed in the IS (7, 9, 19). There is considerable evidence suggesting that the SDL shift to the periphery of the VB enhances the SDL-AVR countercurrent exchange.

In the IM, measurements carried out recently on cross sections of rat renal medulla have been used to estimate average distances from descending limbs to CD (7). Combined with the observations of Kriz et al. (15), a continuous decrease in the distance between long loops of Henle and the CD is observed, and the long loops of Henle are much closer to the CD in the IM than in the OM. Although Kriz et al. (15) pointed out that the descending limbs often appeared closer to the VB and the ascending limbs closer to the CD, this tendency does not seem systematic, and in any case the separation between the two should be quite small (7). A more detailed model of the inner medullary anatomy has been presented in our previous study (29) and is preserved here.

	MODELING ASSUMPTIONS

The mathematical model of tubular flow and transport and the numerical solver are the same as in earlier work (27, 29, 30). The steady-state model is formulated in accordance with qualitative descriptions of the rat renal medullary anatomy. A schematic form of the architectural organization of the renal medulla is illustrated in Fig. 1. Compared with our earlier work, the most distinguishing feature of this description is the OS representation, which differs from the WKM model and more closely reflects reported OS anatomy. In Fig. 1, transepithelial and convective connections between structures are represented by straight lines and curved arrows, respectively. Each structure in Fig. 1 represents a number of tubules or vessels, the numbers of which are listed in Tables 1 and 2.

View larger version (21K): [in this window] [in a new window]   Fig. 1.   Detailed configuration of anatomy used in the model. Arrangements of tubules and blood vessels in outer stripe (OS) and inner stripe (IS) of outer medulla (OM) and inner medulla (IM) are shown. Left: vascular bundles (VB). Right: tubular and collecting duct (CD) regions. Each circle represents a single type of structure. Number of each structure type in each region is given in Tables 1 and 2. Permitted exchanges between structures are indicated either by straight lines for transmural connections or curved arrows for convective flows in shunts or capillaries. Fractions represent the connection strengths. LDV and LAV1-2, long descending and ascending vasa recta; SDV and SAV1-4, short descending and ascending vasa recta; NODE, interstitial capillary nodes between LDL, LAL, and CD; LDL and LAL, long descending and ascending loops of Henle; SDL and SAL, short descending and ascending loops of Henle.

According to our assumed tubular distribution given in Table 1, the ratio of SDL population to LDL population is 2 to 1 in OM (4, 19); the ratio of the Henle's loop population to the CD population is 6 to 1 in OM (~5 to 1 in Ref. 9). At the OM/IM border, the ratio of the number of long loops to CD is 2 to 1, and the ratio becomes 1 to 1 as they approach to the tip of papilla (9).

	SENSITIVITY TESTS

Sensitivity of Anatomic Variations

(1)

View larger version (14K): [in this window] [in a new window]   Fig. 2.   Illustration of the "sliding test" for sensitivity analysis by varying the position of tested structure, i, with respect to its neighboring structures. Reference positions and connection distributions are defined in text.

Two positional sensitivities were explored. First, it has been observed in the comparative anatomy literature that highly concentrating mammals integrate their SDL into the VB, whereas those that do not concentrate their urine as well have SDL located closer to the CD (1). The concentrating ability of the model is calculated for SDL positions ranging from adjacent to the VB (position 3) to adjacent to the CD (position 6).

Second, in the current model, there is a regular radial organization of the tubules and vessels in the IS (Fig. 1). The organization of the long tubules and vessels is maintained in the IS, but because of the continuous reduction in the number of tubules and vessels in the IM as one proceeds toward the papillary tip from the OM (7, 9), the current model assumes spatially homogeneity in the lower IM. Again, the location of the long ascending limb is not well elucidated in the literature, so its position and range of connections are explored.

Sensitivity of Model Performance to Variations of Membrane Parameter Values

Boundary Conditions and Parameters

Model parameters listed in Table 3 assume values of tubular and vascular transport properties based on measurements in hamster and rat. All the values used by the model were within their physiological limits. We assumed that the length of the medulla is 6 mm divided as follows: OS, 0.7 mm; IS, 1.3 mm; upper portion of the IM, 1.3 mm; transition zone, 0.2 mm; and lower portion of the IM, 2.5 mm. Other parameters were identical to those used previously (27, 29, 30).

Numerical Method

	RESULTS

Top Abstract Introduction Results Discussion Conclusions References

Simulations were performed to predict solute concentrations and tubular flow rates in both OM and IM. The sensitivity of the predictions to the anatomic configurations and to the value of several membrane permeabilities was evaluated according to the methods described above.

Simulation Results

View larger version (45K): [in this window] [in a new window]   Fig. 3.   Predicted volume flow rates in tubules and blood vessels of renal medulla. Each ridge represents a structure; height is the magnitude of flow rate. UIM and LIM, the upper and lower portion of IM, respectively; TZ, transition zone. Abbreviations for other structures are given in legend of Fig. 1 and text.

Figure 3 shows that the renal fluid flow rate greatly decreases in all structures down to the papillary tip due to fluid reabsorption and to the decreased number of loops and vessels. The final urine flow rate of 0.24 nl/min gives a urine-to-plasma inulin ratio of 750 (cf. a reported value of 690; Ref. 18). A considerable amount of water reabsorption was predicted in both the outer medullary CD and the IMCD.

The cross section area-weighted average solute concentrations and osmolarity are plotted as a function of the corticopapillary axis position in Fig. 4. Increased urea concentration and total osmolarity are observed in the medulla. The urea concentration reaches its highest value near the border of the upper and lower IM. From Fig. 4, it is clear that solute gradients predicted by the model for the innermost IM are not exponential as have been repeatedly reported experimentally. Thus even a model as detailed as this one is still missing some essential feature(s) related to the creation and maintenance of the IM gradients.

View larger version (18K): [in this window] [in a new window]   Fig. 4.   Area-weighted concentrations and osmolarity along the corticomedullary axis.

Sensitivity Results

Outer medullary anatomy. In the OS, urea transfer occurs from the ascending to descending long vasa recta. This returns urea to the IM and enhances water extraction from the descending limbs of Henle's loops in the OS. In previous models, Wexler et al. (30) have shown that removing the outer medullary countercurrent system would greatly impair concentrating ability due to the loss of solutes to the cortex.

To investigate the impact of tubular positions on the function of the OS, the positions of SDL and short DVR were varied with respect to all the ascending vasa recta. Figure 5, A and B, show the final urine osmolarity as a function of position of the tested structure for several values of the variance, . Positions are defined in Fig. 2. For  = 0.5, the maximum concentrating effect is reached for a SDL position of 5.5, corresponding to a position midway between SAV3 and SAV4 (see Figs. 2 and 5A), which is close to the base case connection strengths used in Fig. 1. The optimal concentrating ability is achieved when the short DVR are located at approximately 4.5-5.0 for   1, corresponding to a position between SAV2 and SAV3 and transport primarily to the local vessels (see Fig. 5B). The maximum concentrating effect at a higher value of  implies that radial dispersion of SDV transport in the OS increases the concentrating ability of the medulla.

View larger version (20K): [in this window] [in a new window]   View larger version (21K): [in this window] [in a new window]   View larger version (19K): [in this window] [in a new window]   View larger version (20K): [in this window] [in a new window]   Fig. 5.   A: sensitivity of model performance to variation in OS SDL positions. Numbers labeled on each line represent the variance of the normally distributed connection strengths. B: sensitivity of model performance to variation in OS SDV positions. Numbers labeled on each line represent the variance of the normally distributed connection strengths. C: sensitivity of model performance to variation in IS SDL positions. D: sensitivity of model performance to variation in the upper IM LAL positions.

Inner stripe anatomy. Many observations have demonstrated the radial organization of the IS. Our representation of the IS anatomy in Fig. 1 is similar to that used in WKM, except that we emphasize the "complex" VB in rat IS as described by the anatomic literature. In both models, SDL are adjacent to the VB. This arrangement of tubules in the IS enhances the solute exchange between the long ascending vasa recta and SDL. To test the sensitivity of the concentrating performance regarding this arrangement, we varied the position of SDL relative to the VB (Fig. 5C). In the IS, when the SDL is distant from the VB, the countercurrent exchange between the SDL and vasa recta is not as strong, and less solute is trapped by the descending structures, which reduces the urine concentrating ability. This is in agreement with a large body of comparative anatomy data (8). In the IS, lower radial transport, manifested in the model by smaller  values, results in greater concentrating ability, that is, it is advantageous to have virtually all the SDL situated adjacent to the VB in the IS.

Upper inner medullary heterogeneity. In our previous work, we posited that the anatomic heterogeneity of the upper IM is necessary for the IM to concentrate urine. To evaluate this assumption quantitatively, we varied the position of the long ascending limb relative to the CD. As shown in Fig. 5D, the IM is better able to form concentrated urine when there is more efficient exchange between LAL and the LAV.

Sensitivity to key membrane parameters. Our base case model uses the anatomy in Fig. 1 and the parameters in Table 3. Some of the parameters are different from those used previously. The adjusted parameters are listed in Table 5. We report the results of varying these parameters on urine concentrating performance in Fig. 6, A-C. The final urine osmolarity was taken as the performance measure.

View larger version (20K): [in this window] [in a new window]   View larger version (21K): [in this window] [in a new window]   View larger version (19K): [in this window] [in a new window]   Fig. 6.   A: effect of varying the descending vasa recta (DVR) NaCl permeabilities on urine concentrating ability. B: effect of varying the DVR recta urea permeabilities on urine concentrating ability. C: influence of changing the outer medullary DVR water permeability on urine concentrating ability.

Figure 6, A and B, shows that decreasing the NaCl or urea permeabilities of the DVR in the OM would increase urine osmolarity, whereas increasing the urea permeability of the DVR in the IM would improve the urine concentrating ability. On the other hand, increasing the water permeability of the outer medullary DVR also improves the concentrating ability as shown in Fig. 6C.

These results make sense in the following way. As the LDV descend through the progressively more concentrated OM, salt and urea tend to diffuse into the LDV and water tends to flow out. If salt or urea permeabilities are reduced, then solute equilibration will be less complete, and osmotic equilibration will still be good due to the very high water permeability, so the LDV would gain less solute and lose more water, with the net result that vessel flow rate into the IM would be reduced, which has long been recognized as an advantage for the inner medullary concentrating mechanism (less washout).

Results by Pallone et al. (21) lend support to this possibility. They showed not only that permeability of OM DVR to Na⁺ may in fact be quite low, but also that although Na⁺ and Cl diffusion apparently share a common, probably paracellular, path, their diffusion is not correlated to that of urea or of water, suggesting that urea and water take a different, probably cellular, route. This agrees with the evidence of water channels (20) and urea transporters (23, 28) in OM DVR cell membranes.

	DISCUSSION

Top Abstract Introduction Results Discussion Conclusions References

In many early experimental observations (1-4, 10, 13, 15, 19) and mathematical simulations (5, 6, 17, 25), an extensive countercurrent organization of the OM was observed and played the putative role of recycling solutes and maintaining hypertonicity in the IM.

The current study is based on the development of the WKM model established by Wexler et al. in 1991 (30), and is a continuation of our previous work. Several key modifications have been made: 1) we present a new representation of the OS renal anatomy, different from that used in the WKM model but in better agreement with the anatomic literature; 2) homogeneous representations of inner medullary anatomy are considered instead of the radially separated organization; 3) active NaCl reabsorption is incorporated into the upper IMCD. 4) the most recent experimental value of the upper IMCD water permeability is employed; and 5) the urea and salt permeabilities of the DVR reported by Pallone et al. (22) have been used in the model.

Although our new representation is more consistent with anatomic reports, some assumptions still need to be made because of the scarcity of quantitative measurements regarding OS and inner medullary structural organization. For example, the inner medullary structure is not yet clear. On the other hand, improvement in the numerical solvers for these models provides a powerful tool with which to investigate sensitivity to the assumptions. Our model sensitivity studies fall into two categories: anatomic and parametric. Unlike WKM, the anatomic sensitivity test in this work employs the sliding test.

Modifications to the WKM Representations of the Renal Anatomy

View larger version (27K): [in this window] [in a new window]   Fig. 7.   Difference of the OS anatomy representation between current model and WKM. A: OS anatomy used in current model. B: OS anatomy used in the WKM model (30).

Based on the histotopography of the rat renal medulla reported by Lemley and Kriz (19), our new representation hypothesizes a countercurrent arrangement of tubules and vessels in the OS. The descending and ascending vasa recta do not form the VB in the upper part of this region; they distribute with long ascending vasa recta close to the center and short ascending vasa recta dispersed in the intertubular region. Only at the bottom of the OS do they form highly organized VB. We correct the OS positions of short and long descending limbs. Both short descending and ascending limbs run through the OS near the CD, and in the IS the SDL shift their position from near the CD to a position near the VB.

In the IM, as the interstitial space becomes tapered, the separation distance between the long loops of Henle and the CD gradually decreases (9). In the lower portion of the IM, all the long structures are homogeneously distributed, and the distances between them are very small. Based on the morphometric studies, we model three regions as in our previous work (29), instead of one unique region as in WKM, to simulate the continuous changes of the interactions among the long structures, as shown in Fig. 8, A and B.

View larger version (18K): [in this window] [in a new window]   Fig. 8.   Difference of inner medullary anatomy representation between current model and the WKM. A: anatomic representations of different regions in IM used in current model. B: anatomic representations of the IM used in WKM model (30).

Sensitivity of Model Performance to Variations in Structure Positions

The sliding test was also applied to the position of OS short DVR. The best position of the short DVR is within the tubular region (Fig. 5A), whereas the long DVR are predicted to run through the OS near the radial center, suggesting that before they form the compact VB in the IS the DVR are scattered among ascending vasa recta and tubules in the OS.

According to the literature, the long ascending limbs are near the long descending limbs throughout the IM, but in the upper IM their relation to the CD is not clearly reported. In our previous work (29), we suggested that the long ascending limbs in the upper IM would be separated from the CD as in the IS, and we modeled a change to a homogeneous distribution in the lower IM. In this study, the sensitivity test for the position of long ascending limbs supports this hypothesis, showing that heterogeneity of the upper inner medullary tubular distribution along with active NaCl reabsorption from the CD helps to concentrate urine.

Urea, NaCl, and Water Pathways

According to the membrane parameters sensitivity test, increasing the urea and NaCl permeabilities in the outer medullary DVR would not increase the urine concentrating ability of the model, whereas increasing the water permeabilities in these structures could greatly improve urinary concentration.

Evaluation of recycling patterns is not straightforward in model regions where multiple tubules of varying lengths are represented by a single structure with shunts, because reduction in the flow is due to both transmural flux and shunt flow. In these cases one cannot calculate the amount of, for example, urea secreted into or absorbed from, for example, the inner medullary interstitium by all LDL simply by subtracting the amount delivered at the tip from the amount delivered at the OM/IM border, since a large part of the difference is due to the reduction of the number of LDL. For example, in our model, the ratio of the number of long Henle's loops traversing the OM to the number arriving at the tip is 128, reflecting the fact that, in the rat, for 10,000 LDL, only ~75 arrive at the tip (7, 9, 12).

Failing this level of detail, we can nonetheless calculate the total amount of water or solutes removed from or added to a given loop structure between some level x and the tip of the papilla by subtracting the flow rate in the ascending branch from that in the descending branch at the same level. In regions where structures are shunted, we can thus evaluate the net transfers of solutes or volume from one tube structure to another (e.g., from Henle's loop to or from vasa recta) but not the recycling patterns among ascending and descending branches.

The following recycling features in this model were calculated this way (for results, see Table 6).

Volume (in terms of total glomerular filtration rate to all model nephrons). In this model, the CD loses 1.55% and 0.6% of glomerular filtration rate (GFR) in the OS and IS, respectively, and then 0.28% in the IM; but because of the low volume flow into the CD (only 2.6% of GFR), these losses suffice to raise tubular fluid-to-plasma inulin concentration ratio from 39 at the top of the OM to 750 at the exit from the papilla. The LDL, which carry 11.1% of total GFR into the medulla, lose 5.7% and 2.24% of GFR in OS and IS, respectively, then actually gain 0.5% in the IM in this model. So the whole osmolality increase in the IM LDL is due to urea entry (see below). The SDL, which carry 22.2% of GFR into the medulla, lose 14.3% of GFR in the OS and then gain 0.5% in the IS. The net volume load from the tubules is of course recovered by the vasa recta in all regions.

Urea. The CD carry 56.7% of total filtered load of urea (FL_u) into the medulla (which represents only 1.9% of the total filtered load of osmoles). The IMCD loses a total of 43.5% of FL_u into the inner medullary interstitium. Of this total IM urea load from the CD, 55% (which is 23.8% of FL_u) is recovered by Henle's loops and 45% (which is 19.6% of FL_u) by the long vasa recta, all of which is carried back up to the OM. In the OM, the LAL lose 10.6% of FL_u into the IS and 8% into the OS, whereas the long vasa recta lose a net amount of urea equal to 15.6% of FL_u. The LDL have only slight urea movements in the OM as a result of their lower permeabilities to urea.

The SDL lose 3.3% of FL_u in the OS and, in keeping with the usual idea of their reason for being close to the VB in the IS, they gain 6% of FL_u in the IS, but then on the way back up in the interbundle region, the SAL gain less than 1% of FL_u in the IS but dump 7.8% in the OS. Thus the short limbs recycle urea from the IS to OS and make a net contribution of 5.1% of FL_u into the OM.

The short vasa recta recover all the "leftover" urea (some 39% of FL_u) in the OM and carry it back to the general circulation.

NaCl. The CD plays no quantitative role in salt recycling, though the 0.13% of filtered load of salt (FL_s) that is pumped out of the CD in the uppermost IM is crucial for the system. Along the descending limbs of Henle, there is very little salt flux in this model. This is clear from a calculation of the single-nephron fractional delivery of salt (snFD_s) (i.e., salt delivery per tube at a given point as a fraction of single-nephron filtered load of salt). Since two-thirds of filtered salt are absorbed in the proximal tubule, snFD_s is 0.333 in SDL and LDL at the top of the OS. In this model, snFD_s = 0.336 in SDL and 0.341 in LDL at the bottom of the IS and 0.318 in LDL at the tip of the papilla, indicating only slight net salt fluxes. The ascending limbs, of course, absorb salt in all regions. In the OM, SAL and LAL actively absorb 20.6% and 8.9% of total FL_s, respectively. In the IM, LAL passively absorb a total of 1.6% of FL_s, in keeping with the classic "single effect" in the IM.

Total osmolar transfers. A clear picture of the elusive mechanism for maintenance of the inner medullary osmotic gradient will require knowledge of the contributions of recycling of individual solutes to total transfers of osmoles in the three IM regions. Lacking such detailed information, we can, for the moment, look at the model's predictions concerning the relative osmolar contributions of salt and urea transfers in the IM. The numbers, as percentage of total filtered load, are small, but keep in mind that the sum of total osmolar flow in vasa recta and Henle's loops at the tip of the papilla is only 0.34% of the filtered load of osmoles (FL_osm). (For these calculations from the results in Tables 4 and 6, we also use the fact that in this model, GFR = 3 nl/s, FL_u = 27 pmol/s, and FL_s = 420 pmol/s; so FL_osm = 800 posmol/s.)

In this vein, it is interesting to note that not only the CD but also Henle's loops are net contributors of osmoles to the IM, which are of course carried up to the OM by the vasa recta. The CD dumps 1.6% of FL_osm into the IM, 92% of which is urea. In Henle's loop, the salt loss equals 1.7% of FL_osm, whereas the loop's urea uptake equals only 0.8% of FL_osm, for a net contribution of 0.88% of FL_osm.

We can thus see that in this model the dilution of LAL relative to LDL at the OM/IM border is due mainly to greater loss of salt osmoles than gain of urea osmoles.

	CONCLUSIONS

Top Abstract Introduction Results Discussion Conclusions References

In summary, we have investigated the heterogeneity of the OS and the upper inner medullary anatomy using a modified urine concentrating model that includes an improved OS anatomy and membrane parameters that are all within the range measured. The new anatomic representation of the OS reflects a more isomorphic histotopography than do previous descriptions (29, 30), with respect to the reported anatomic observations and measurements, and is supported by modeling results and sensitivity tests. The sensitivity tests show that the functional consequences of the locally segregated arrangement in the OM and the continuation of the VB segregation into the upper IM facilitate the kidney's concentrating ability.