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【关键词】 medulla
Department of Mathematics, Duke University, Durham, North Carolina
ABSTRACT
In a companion study (Layton AT and Layton HE. Am J Physiol Renal Physiol 289: F1346F1366, 2005), a region-based mathematical model was formulated for the urine concentrating mechanism (UCM) in the outer medulla (OM) of the rat kidney. In the present study, we quantified the sensitivity of that model to several structural assumptions, including the degree of regionalization and the degree of inclusion of short descending limbs (SDLs) in the vascular bundles of the inner stripe (IS). Also, we quantified model sensitivity to several parameters that have not been well characterized in the experimental literature, including boundary conditions, short vasa recta distribution, and ascending vasa recta (AVR) solute permeabilities. These studies indicate that regionalization elevates the osmolality of the fluid delivered into the inner medulla via the collecting ducts; that model predictions are not significantly sensitive to boundary conditions; and that short vasa recta distribution and AVR permeabilities significantly impact concentrating capability. Moreover, we investigated, in the context of the UCM, the functional significance of several aspects of tubular segmentation and heterogeneity: SDL segments in the IS that are likely to be impermeable to water but highly permeable to urea; a prebend segment of SDLs that may be functionally like thick ascending limb (TAL); differing IS and outer stripe Na+ active transport rates in TAL; and potential active urea secretion into the proximal straight tubules. Model calculations predict that these aspects of tubular of segmentation and heterogeneity generally enhance solute cycling or promote effective UCM function.
kidney; short loops of Henle; vascular bundle; vasa recta; Na+ transport; urea transport
IN A COMPANION PAPER (23), henceforth called study I, we formulated a region-based mathematical model of the urine concentrating mechanism (UCM) in the outer medulla (OM) of the rat kidney. In that model, the complex structural organization of the OM (19, 21) was represented by means of four concentric regions, centered about a vascular bundle; the radial positions of structures were incorporated by assigning appropriate tubules and vasa recta (or fractions thereof) to each concentric region.
In this study II, we investigate model sensitivity to fundamental structural assumptions and to parameters whose values are uncertain. The model incorporates a large number of parameters that characterize regionalization, boundary conditions, structural dimensions, transmural transport properties, and relative positions and distributions of tubules and vessels. Most of these parameters were based on, or estimated from, the experimental literature. However, considerable uncertainty remains in the specification of appropriate values for some other parameters. In study I, we made assumptions that appeared reasonable to us; in this study II, we examine how variation in these uncertain parameters affects model results. Of particular importance is a prediction that regionalization has a significant impact on solute concentrations, and therefore osmolality, of collecting duct (CD) tubular fluid that enters the inner medulla (IM).
In this study II, we also investigate, in the context of the UCM, four instances of tubular segmentation or heterogeneity found (or hypothesized) in the rat OM. We assess the effects of two segments of the short descending limb (SDL): a segment that spans much of the inner stripe (IS) of the OM and that appears to be water impermeable and highly urea permeable (28, 39); and a prebend segment of SDL that may be functionally like the thick ascending limb (TAL) (33). We also assess the effects of differing IS and outer stripe (OS) Na+ active transport rates in TALs (7) and of active urea secretion into proximal straight tubules (PSTs), as hypothesized in Ref. 4.
For base-case model parameters and model results, and for a GLOSSARY of acronyms and symbols, see study I.
PARAMETER SENSITIVITY
Conditions at the OM-IM Boundary
Because we are concerned with solute distribution and concentrating effects in the OM, the model does not explicitly represent the IM. Instead, the actions of an antidiuretic IM are represented implicitly via boundary conditions imposed on luminal flows from the IM. These boundary conditions are based on mass balance and on a typical urine flow and composition for an antidiuretic kidney; long ascending limb (LAL) and long ascending vas rectum (LAV) fluid composition are computed as described in study I (MODEL FORMULATION).
Rat urine osmolality varies widely, from 60 mosmol/kgH2O in diuresis to up to 2,850 mosmol/kgH2O in antidiuresis (2, 5). In our model’s base case, urine flow rate, Na+ concentration, and urea concentration were set to 0.065 nl?min1?nephron1, 531 mM, and 457 mM, respectively; thus the model’s base-case urine osmolality is 1,420 mosmol/kgH2O (1), which corresponds to a moderately antidiuretic state. To study the sensitivity of model results to urine osmolality and composition, urine osmolality was varied by approximately ±20% by first varying urine flow rate by ±20%, while the urine solute flows were kept constant, and then by simultaneously varying Na+ and urea concentrations by ±20%, while the urine flow rate was kept constant. A highly antidiuretic state having urine osmolality of 2,500 mosmol/kgH2O was simulated by scaling urine solute concentrations by a factor of 2,500/1,420 while base-case urine solute flow rates were maintained (by scaling urine flow rate by a factor of 1,420/2,500). In another study, urine Na+ concentration was varied by ±20%, while the urine osmolality (by adjusting the urine urea concentration) and the urine flow rate were kept constant; urine urea concentration was then varied in the analogous way. These changes in urine composition had little effect on model predictions for the OM: in all cases, the CD fluid osmolality at the OM-IM boundary was changed by <2.16%, and other variables were similarly little affected. Thus model results are nearly insensitive to urine composition; this is attributable to small urine fluid and solute flows, relative to tubular flows in the OM. Indeed, the model’s base-case urine fluid flow rate is only 1.92% of the aggregate flow into the IM through the long descending limbs (LDLs), CDs, and long descending vasa recta (LDV).
In the base case, we assumed that 85% of the fluid flow entering the IM via the LDLs is returned via the LALs and that LAL Na+ and urea concentrations are 5% lower and 20% higher, respectively, than that of tubular fluid returned from the IM by LAV (23). Because these assumptions are not supported directly by experimental data, we assessed the sensitivity of model results to LAL and LAV fluid composition at the OM-IM boundary. We let Na+ and urea denote the percentage differences in Na+ and urea concentrations, respectively, between the fluid returned from the IM in LALs and LAV; thus in the base case, = 5% and = 20%. We computed model results for = 10, 5, 0, and 5%, while urea = 20% was maintained; for urea = 30, 20, 10, 0, and 10%, while = 5% was maintained; and for = urea = 0. As and urea were varied, the largest increase in CD fluid osmolality at the OM-IM boundary was 1.16% relative to base case, obtained for = 10% and urea = 20%. The largest decrease in CD fluid osmolality at the OM-IM boundary, obtained for = 5% and urea = 20%, was 2.04% relative to base case. In all of these experiments, deviations in fluid osmolality, concentrations, and flows in tubules other than CDs were also small. Thus these results suggest that CD fluid osmolality is nearly insensitive to variations in LAL and LAV fluid composition at the OM-IM boundary.
In addition, sensitivity to LAL and LAV tubular fluid flow rates at the OM-IM boundary was assessed by examining cases in which 75 and 95% of the LDL tubular fluid was returned to the OM via the LALs (base-case value is 85%); the remainder was returned via the LAV. Resulting CD tubular fluid osmolality at the OM-IM boundary showed deviations of 2.43 and 1.41%, respectively, relative to base-case values.
Descending Vasa Recta Flow Rates at the Corticomedullary Boundary
The model does not include explicit representation of the cortex; rather, the actions of the cortex are represented implicitly by means of boundary conditions imposed at the corticomedullary border. Specifically, we assume that the flow rates and solute concentrations of the descending limbs and descending vasa recta (DVR) are known a priori, and CD fluid composition is computed from TAL flows at the corticomedullary boundary (see study I). Because it has been proposed that blood flow along DVR may be regulated by their intrinsic contractile capability (30), we investigated how variation in DVR flow rate affects the model’s concentrating capability. The base-case corticomedullary boundary flow rate of 8 nl?min1?DVR1 was varied by ±10 and ±20%; DVR boundary solute concentrations were kept at the base-case values. Decreasing the DVR boundary flow rate by 10 and 20% resulted in 2.98 and 5.90% increases, respectively, in CD fluid osmolality at the OM-IM boundary, whereas increasing the flow rate by 10 and 20% resulted in decreases of 2.97 and 5.91%, respectively. Thus these findings yield the intuitive result that decreasing DVR flow decreases the load on the concentrating mechanism and consequently increases the concentrating effect. (As in study I, we use "load" to mean a descending tubular or vascular fluid flow that must be concentrated by the concentrating mechanism.)
Solute Permeabilities of Concentric Region Boundaries
Our model’s region-based formulation represents the structural organization of the OM by means of four concentric regions centered on a vascular bundle: an innermost region containing the central vascular bundle (R1); a peripheral region of the vascular bundle (R2); a region neighboring the vascular bundle (R3); and the region most distant from the vascular bundle (R4). The radial organization of tubules and vasa recta with respect to vascular bundles is represented by specifying the fractions of the tubules and vasa recta assigned to each concentric region at each medullary level; see Fig. 1 in study I.
To assess the impact of regionalization on model results, we varied the solute permeabilities of the region boundaries. The solute permeabilities of region boundaries used in the base case (in study I) were based on estimates of the fractional areas of the regions that consist of interstitium, of diffusion resistance arising from interstitial cells and macromolecules, and of the tortuosity of the diffusion path around tubules and vessels (23). The degree of regionalization increases as the region boundary permeabilities decrease. In our sensitivity study, the base-case boundary solute permeabilities were scaled by factors of = 0.01, 0.1, 1, 10, and 100; = 0.01 corresponds to greatest regionalization, = 100 corresponds to least regionalization, and = 1 corresponds to the base case.
Osmolalities, Na+ concentrations, and urea concentrations of the interstitial fluid in the regions for differing degrees of regionalization are shown in Fig. 1. As regionalization was decreased by increasing from 0.01 to 100, four distinguishable regions ( = 0.01) tended to approximate three regions in which R3 and R4 differ little ( = 10), and then approximated one well-mixed compartment ( = 100). In the base case ( = 1), interstitial fluid Na+ concentration, and thus osmolality, is substantially higher in R3 and R4 (which contain most TALs), than in R1 and R2. This difference was augmented for increased regionalization, but diminished for decreased regionalization.
In the base case, urea concentration is highest in R4, except near the OM-IM boundary, owing to urea effux from the CD. Near the OM-IM boundary, fluid with elevated urea concentration entering in LAVb from the IM raised the urea concentration in the periphery of the vascular bundle, R2, above the other three regions. As regionalization increased, the difference in urea concentration between R4 and other regions increased, and for = 0.01, urea concentration in R4 exceeded that in R2 throughout the OM. As regionalization decreased, interstitial urea concentrations increasingly approximated that of a well-mixed compartment, resulting in a general increase in R1 and R2 fluid urea concentrations, and a general decrease in R3 and R4 urea concentrations.
Figure 2 exhibits Na+ and urea concentration profiles for those model flows that enter the IM, i.e., for luminal flows in LDV, LDL, and CD; and Table 1 gives fluid composition and flow rates at the OM-IM boundary. The effects of regionalization on LDV and CD tubular fluid solute concentrations are similar to the effcts on the regions containing the LDV and CD (R1 and R4, respectively). As radial organization became increasingly well-defined (smaller ), Na+ and urea concentrations in LDV fluid decreased, whereas CD fluid solute concentrations increased and fluid flow rate decreased. Thus these results suggest that regionalization may contribute to overall concentrating capability by increasing the osmolality of the CD fluid and by decreasing load on the IM concentrating mechanism. A more complex pattern was observed in Na+ and urea tubular fluid concentrations in LDL, which straddles R2 and R3 in the OS and is located in R3 in the IS: when regionalization was increased or decreased, both concentrations decreased, relative to base-case concentrations, although the decrease was larger, and significant, only in the extreme of reduced regionalization (Fig. 2, B1 and B2).
In a previous study (22) we developed a prototype model that used two concentric regions. As noted in study I, in addition to representing two instead of four regions, that study differs from the present one in several respects, and those differences make a direct comparison between the two studies complicated. However, to ascertain the impact of the more detailed representation of radial organization afforded by the four-region model, we calculated results for the present model’s base case, except that the solute permeabilities between R1 and R2 and between R3 and R4 (i.e., PR1,R2,k and PR3,R4,k) were multiplied by 1,000. In the resulting model, which we refer to as the essentially two-region (E2R) model, the differences in interstitial compositions between R1 and R2 and between R3 and R4 nearly disappeared (see Fig. 3A). Owing to the structural differences between the E2R model and the previous two-region model (22), notably the absence of the water-impermeable segment along the SDL in the previous study, CD fluid osmolality at the OM-IM boundary was 826 mosmol/kgH2O in the E2R model compared with 571 mosmol/kgH2O in the previous study.
Figure 3, B and C, shows Na+ and urea concentration profiles in the LDV, LDL, and CD, obtained for the (4-region) base case (dashed lines) and for the E2R model (solid lines). Concentrations and flow rates at the OM-IM boundary for the E2R model are given in Table 1 for the LDV, LDL, and CD. In both the base-case and E2R models, interstitial fluid urea concentration in the periphery of the vascular bundle (R2) was increased by the return of urea-rich fluid from the IM via the LAVb. However, owing to the high permeability between R1 and R2 in the E2R model, urea concentration in the central vascular bundle (R1) was higher in the E2R model than in the base case. Thus the model LDV, which is located within R1, has higher urea concentration and flow in the E2R model than in the base case (136 and 118% of base case at the OM-IM boundary). In the IS of the E2R model, the interstitial fluid surrounding LDL (i.e., R3) had a higher urea concentration, owing to increased effux from R4 to R3; this resulted in a higher LDL urea concentration and urea flow that were 104 and 101%, respectively, of base case at the OM-IM boundary. A lower urea concentration in R4 in the E2R model resulted in increased urea efflux from the CD. However, because of increased urea delivery into the IM via the LDL and LDV (see above), and increased return via LAL to the distal nephron, CD urea flow in the E2R model was 104% of base case at the corticomedullary boundary. Thus despite a 1.64% increase in urea reabsorption from the CD in the E2R model, CD urea concentration and urea flow at the OM-IM boundary were 101 and 104%, respectively, of base case.
Throughout the model OM, interstitial Na+ concentration in the vascular bundle (R1) was higher in the E2R model than in base case; thus the E2R model predicted a higher LDV Na+ concentration and flow (123 and 107% of base case at the OM-IM boundary). For most of the IS (approximately the inner 45% of the OM), Na+ concentration and osmolality of the interstitial fluid in R3 were also higher in the E2R model than in base case; thus water absorption from LDL and Na+ secretion into LDL were increased. These competing effects resulted in a negligible 0.48% decrease in LDL Na+ flow into the IM. In the E2R model, the increased Na+ effux from R4 into R3 results in a lower Na+ concentration in R4, and consequently lower Na+ concentration and flow along the CD. The aggregate Na+ and urea flows into the IM via the LDV, LDL, and CD were 1,089 and 209 pmol/min, respectively, in the E2R model compared with 1,105 and 198 pmol/min in the (4-region) base case. These results suggest that a two-region model may underestimate aggregate Na+ delivery into the IM and CD fluid Na+ concentration at the OM-IM boundary but may overestimate urea delivery into the IM and CD urea concentration at the OM-IM boundary.
SDL IS Position
In the IS of rat OM, the SDLs are found within the periphery of the vascular bundles (21). In contrast, in mammals with simple medullas (e.g., the rabbit), SDLs are found outside the vascular bundles (12). To investigate the effect of SDL position in IS, we varied the the fraction (denoted ) of the SDL that is in contact the periphery of the vascular bundle, R2; the remaining fraction (1 of the SDL) is taken to be in contact with the neighboring interbundle region, R3.
Figure 4 shows model SDL, LDV, and CD tubular fluid osmolalities for differing SDL positions (the SDL having a prebend, SDLa, is shown; the SDL not having a prebend, SDLb, has a similar profile). These results show that as smaller fractions of the SDL were contained within the vascular bundle, the SDL fluid osmolality in the IS progressively increased (Fig. 4A), whereas LDV and CD fluid osmolalities decreased (Fig. 4, B and C). The increased SDL fluid osmolality is directly attributable to the interactions of SDLs with the TALs that mostly populate R3 and R4 and thus sustain a higher interstitial fluid osmolality than in R1 and R2. The decreased LDV fluid osmolality arises from the removal of the SDL prebend segment from R2; in the base case, the Na+ actively transported from that segment raised the osmolality of the fluid in the LDV and the short descending vasa recta (SDV). The decreased CD fluid osmolality arises from the increase in load presented by the SDL fluid flow that must be concentrated within R3; SDL flow presents a load to the concentrating mechanism, because SDL fluid osmolality lags below that of the intrabundle region R3. As smaller fractions of the SDL assumed a position within the periphery of the bundle, the load outside the bundle was increased; this resulted in reduced osmolality in R3, and thus in the adjoining region R4, and, consequently, in CD fluid. LDL osmolality was also reduced (results not shown). However, the effect on both CD and LDL osmolalities at the OM-IM boundary mostly occurred as decreased from 1.0 to 0.5; the additional effect on CD and LDL was small as decreased from 0.5 to 0.0.
Ascending Vasa Recta Solute Permeabilities
Transendothelial solute fluxes for the ascending vasa recta (AVR) are assumed to arise from advection and diffusion through pores. In study I, effective AVR solute permeabilities for the base case were based on estimates of solute diffusivities in dilute aqueous solution, AVR wall thickness, and AVR fenestration fraction (the wall area fraction occupied by open pores). Because the value of a critical parameter, the ratio of the fenestration fraction to the wall thickness, is substantially uncertain, we investigated that ratio’s effect by multiplying it by factors of = 100, 10, 1, 0.1 and 0.01, in both LAV and SAV. Because effective AVR solute permeability is proportional to the fraction of the AVR wall that is occupied by open pores, a change in may be interpreted as a scaling of AVR permeability.
Figure 5 exhibits CD fluid osmolality profiles for the selected values of . As increased, the AVR became more effective participants in countercurrent exchange; i.e., not only did the AVR take up local fluid to ensure fluid conservation, but increasing amounts of solute diffused out of AVR so that AVR Na+ and urea concentrations were more nearly equilibrated with interstitial fluid. Consequently, countercurrent exchange was less dissipative of the medullary gradient, and the concentrating effect increased, although it tended to an upper limit, as indicated by the progressively increasing, but converging, CD fluid osmolality profiles. Osmolality profiles in other tubules were similarly affected by the scaling of AVR permeability (results not shown).
Distribution of Short Vasa Recta
In the base-case simulation of study I, the number of short vasa recta was assumed to decrease piecewise linearly with increasing OM depth. To assess the impact of the short vasa recta population distribution, we conducted simulations using alternative distributions of the form
(1)
where wSDV(x) and wSAV(x) denote the fractions of SDV and SAV, respectively, that reach to level x. In Eq. 1, the case of a = 0.1 corresponds to the base-case distribution. Population distributions corresponding to a = 0.01, 0.1, 1.0, and 2.0 are illustrated in Fig. 6A.
Figure 6B shows simulated CD fluid osmolality profiles for the differing vasa recta distributions. As the fraction of short vasa recta reaching deep into the OM decreased, the concentrating effect increased. For a = 2.0, in which only 14% of the short vasa recta reach beyond x = 0.75L (compared with 35% in the base case), the CD fluid osmolality at the OM-IM boundary was 113% that of the base case, whereas the CD fluid osmolality obtained for a = 0.01, in which 36% of the short vasa recta reach beyond x = 0.75, was 95% that of the base case at the OM-IM boundary. Thus the CD osmolality is particularly sensitive to changes near the base case. Osmolality profiles in other water-permeable tubules were similarly affected by the differing vasa recta distributions. These studies support the following intuitive principle: when a smaller fraction of the short vasa recta extend into the deep OM, the fraction of DVR flow that must be raised to the higher concentrations of the OM is reduced; consequently, the effectiveness of short vasa recta as countercurrent exchangers is enhanced and the osmolality of CD fluid at the OM-IM boundary is increased.
Axial Diffusion in Concentric Regions
Diffusion of solutes along the corticomedullary axis is represented in the interstitium of each concentric region. This diffusion is based on the diffusivities for Na+ and urea in dilute aqueous solution, on an estimate of the interstitial cross-sectional area associated with each nephron, and on a diffusion resistance arising from the cells and macromolecules of the interstitium (see study I). To assess sensitivity to axial diffusivity (or, equivalently, the area available for diffusion), we conducted simulations in which diffusivity was varied relative to base-case values. When axial diffusivities were set to 1/10 of their base-case values, CD fluid osmolality was slightly increased: a maximum increase of 0.669% was obtained at the OM-IM boundary. With axial diffusivities set to 10 times their base-case values, the CD fluid osmolality profile was decreased; a maximum decrease of 0.396% was obtained at the OM-IM boundary. For both decreased and increased diffusivities, deviations in fluid osmolality in other tubules and in vessels were also small, relative to base-case values. Thus increased diffusivities reduced axial solute concentration gradients, and reduced diffusivities increased axial solute gradients. However, for the range of diffusivities examined, the model is nearly insensitive to variation in axial diffusivity.
TUBULAR SEGMENTATION AND INHOMOGENEITY
We investigated four aspects of tubular segmentation and inhomogeneity: an IS segment of SDL that is likely to be water impermeable and highly urea permeable (28, 39); the differing IS and OS Na+ active transport rates reported for TAL (7); a short prebend segment of SDL that may be functionally like TAL and that may occur in a substantial fraction of loops of Henle (33); and potential active urea secretion into PSTs (4).
UCM efficacy may be assessed by means of the ratio of the free-water absorption rate (FWA) to the total active transport rate (TAT). FWA is the volume, per unit time, of blood plasma that could be considered to be completely cleared of solutes by the production of urine that is more concentrated than blood plasma (12, 42); FWA is given by
(2)
where Furine,V is the urine flow rate and the urine-to-plasma osmolality ratio is formed by the quotient of urine osmolality Uosm and blood plasma osmolality Posm. In this model, however, because urine flow rate and concentrations are specified a priori as boundary conditions, FWA is constant and thus cannot be used to assess concentrating mechanism efficacy. Thus we propose an alternative criterion, "outer medullary free-water absorption" (OM-FWA), defined by
(3)
where FCD,V and CCD,O are the steady-state CD tubular fluid flow rate and osmolality at the OM-IM boundary, respectively. OM-FWA measures the volume, per unit time, of blood plasma that would be cleared of solute by the production of urine having the osmolality and flow rate of CD fluid at the OM-IM boundary.
Although we propose to use OM-FWA primarily as an indicator to help assess the efficacy of the OM UCM, some insight into the role of the IM may be gained by comparing FWA and OM-FWA, and by taking into account the values of Furine,V, FCD,V, Uosm, and CCD,O. In study I, we assumed that urine flow rate and urine osmolality are 0.065 nl?min1?nephron1 and 1,420 mosmol/kgH2O, respectively. In our base case, FCD,V = 0.365 nl?min1?nephron1 and CCD,O = 851 mosmol/kgH2O. From these values, one computes that FWA = 0.229 nl/min whereas OM-FWA = 0.640 nl?min1?nephron1. Thus if the CD fluid at the OM-IM boundary were excreted as urine with no change in flow rate or composition, then the OM, in the absence of a functioning IM, would clear solute from nearly three times as much blood plasma volume than would be cleared based on our base-case assumptions for urine composition.
This result suggests that a function of the IM, in its action on the output from the OM, is to allow the clearance of solute from a smaller, but significant amount of blood plasma, through the excretion of less fluid at higher osmolality: urine flow (0.065 nl?min1?nephron1) is 17.8% of CD flow at the OM-IM boundary (0.365 nl?min1?nephron1), whereas urine osmolality (1,420 mosmol/kgH2O) is 167% of CD fluid osmolality at the OM-IM boundary (851 mosmol/kgH2O). By our base-case assumptions for urine flow rate, the action of an IM allows a urine flow (in ml?day1?kidney1) of 2.81, compared with 15.77 in the absence of a functioning IM. The difference of 13 ml?day1?kidney1 (26 ml?day1?animal1) corresponds to a significant fraction of the masses of rats that have been typically used in renal experiments (200350 g) (17, 18, 32). Indeed, in female Wistar rats having masses of 250 g, the extracellular fluid contributes 35% of total body mass (corresponding to 88 ml) (38); thus the 26 ml/day that could be lost in the absence of a functioning IM corresponds to 30% of total extracelluar fluid volume.
We define TAT to be the total active Na+ transport rate, integrated over the areas of all tubules of the OM; thus TAT has units of nanomoles per second. The principal contribution to TAT is from TALs and the prebend segment; however, our model PSTs and SDL2 segments also have Na+ active transport, although at much lower rates (2.1 and 0.43 nmol?cm2?s1, respectively) (7, 13, 14) than TALs. We will assess the efficiency E of the concentrating mechanism by means of the ratio of OM-FWA to TAT (analogous to the ratio of FWA to TAT used in Ref. 27), thus
(4)
SDL2 Segment
The SDL2 segment, a portion of the SDL that precedes the thick limb, spans much of the IS and appears likely to be water impermeable and highly urea permeable (28, 39). We compared base-case results from study I to the case where the SDL2 segment is replaced by an extension of the SDL1 segment, which we assume to be highly water permeable and moderately urea permeable, as in the hamster (11). The TAT was computed for both cases and found to be 0.97% higher in the base case. However, when the SDL2 segment is absent, CCD,O, OM-FWA, and E are only 76.77, 82.36, and 83.16%, respectively, of base-case values. These significant differences arise from the marked increase in load that is introduced by the elimination of the SDL2 segment and the extension of the SDL1 segment.
This increase is implicit in results shown in Fig. 7A, which exhibits fluid osmolalities in SDL, SAL, and CD for the two cases. In the base case (solid curves), SDL fluid osmolality is nearly nonincreasing along the SDL2 segment and thus the fluid delivered to the SAL is more dilute, relative to the case lacking the SDL2 segment. Because concentrating capacity is not used to raise SDL fluid osmolality in the SDL2 segment, that capacity is available to raise fluid osmolalities in other structures. This capacity is made clear by its absence in the case lacking the SDL2 segment: a substantial decrease in CD osmolality, relative to the modest increase in terminal SDL osmolality, arises from the large rate of water absorption from the terminal SDL, relative to CD tubular fluid flow in the deep OM. The water so absorbed substantially increases the load of water that must be transferred by capillary flow, to maintain mass balance, from SDV to the SAV of R3 and R4, resulting in a substantial dilution of R3 and R4, relative to base case.
The role of the SDL2 segment in urea cycling was investigated by comparing the base case to three alternative cases, in which: 1) the SDL2 segment was eliminated and the SDL1 segment lengthened, as above; 2) the SDL2 segment was water impermeable but had urea permeability 10 times the base-case permeability of 20 x 105 cm/s; and 3) the SDL2 segment was impermeable to both water and urea. Figure 7B shows the resulting SDL urea flows. When the SDL2 was eliminated (which reduced urea permeability to the SDL1 value of 1.5 x 105 cm/s), or when the SDL2 segment was made urea impermeable, urea flow at the SDL bend was 7.31 and 7.21%, respectively, lower than in base case. When the urea permeability of the SDL2 segment was increased by a factor of 10, the SDL urea flow was decreased by 3.41%. In this case, although urea entry into the second half of the SDL2 segment was increased, this urea gain was offset by increased urea loss near the first half of the SDL2 segment. These results suggest that, for a moderate urea permeability of 20 x 105 cm/s, the SDL2 segment can contribute to a urea cycling pathway, but that SDL2 urea permeabilities that differ significantly from the base-case value reduce that urea cycling capability.
Prebend Segment
Thick prebend segments of the loop of Henle have been identified in Gambel’s quail (6), in hamster (3), and in a subpopulation (10%) of mouse short nephrons that turn in an "inner-most" stripe of the OM (20). In addition, thick prebend segments have been reported in neonatal rats (15), and in mature rats evidence suggests that some loop bends consist entirely of thick epithelium (33) like that of TAL. Because of uncertainty about the presence and length of thick segments in prebend descending limbs of mature rats, our model represents two short loops of Henle; one has a prebend segment, the other does not. In the short loop without a prebend segment, the SDL2 segment was extended to the loop bend. We assessed the impact of the prebend by varying this configuration.
Table 2 shows CD outflow and efficiency measures for the three cases: both loops having prebend segments; one loop having a prebend segment (the base case); and neither loop having a prebend segment. Both loops having a prebend segment increased CD fluid osmolality and reduced CD flow rate, but efficiency E was only slightly increased. However, if the ratio of CD fluid osmolality increase (along the OM) to TAT is used as an efficiency measure, then the efficiency of the system having prebend segments in both SDLs is 5.23% higher than the case with no prebend segments. The increase in this measure of efficiency is achieved with a TAT increase of only 2.61%.
TAL Na+ Transport Rate
The base-case TAL Na+ active transport rate (Vmax) is 10.5 nmol?cm2?s1 in the OS and 25.9 nmol?cm2?s1 in the IS, based on Na+-K+-ATPase activities reported by Garg et al. (7). The effect of inhomogeneous TAL Na+ active transport was investigated by comparing base-case results to three alternative cases, each of which had uniform Vmax all along the TAL (and the prebend also, for 1 of the short loops). For the three cases, the active transport rate (in nmol?cm2?s1) was set to 10.5 ("low"), 25.9 ("high"), and 19.95 ("average"). The average rate was computed so that its Vmax value, weighted by the TAL radius, equaled the (inhomogeneous) base-case values of Vmax, weighted by the TAL radius (the weighting was accomplished by means of appropriate definite integrals).
For the base case and three alternative cases, Fig. 8A exhibits the CD osmolality profiles and Table 3 reports TAT, CCD,O, OM-FWA, and E. The average case had an energy cost 2.92% lower than base case and yielded an OM-IM boundary CD fluid osmolality and an OM-FWA that were 12.98 and 6.21% lower, respectively, than base-case values. The high case produced an OM-IM boundary CD fluid osmolality and an OM-FWA that exceeded base case values (by 9.33 and 4.92%, respectively). However, because of its much higher TAT, the high-case efficiency was 14.52% less than that of the base case. The low-case efficiency was slightly higher (3.16%) than the base-case efficiency; however, in the low case, OM-IM boundary CD fluid osmolality was increased by a factor of 1.56 only, relative to blood plasma osmolality, and OM-IM boundary CD fluid flow was increased by 74.5% relative to the base case. Thus the low case, with a more dilute fluid entering the IM via the CD at a higher rate (relative to base case), would impose a substantially increased load on the IM concentrating mechanism. These results indicate that axially inhomogeneous TAL Na+ absorption, as represented in the base case, promotes efficient and effective OM concentrating function.
We also investigated how model results would be affected if the diameters and active transport parameters were changed to those used by Wexler et al. (43) in the "WKM" model. In that model, TAL diameters are 20 μm throughout the OM; Na+ maximum active transport rates (Vmax, in nmol?cm2?s1) are 18 in the OS for both SAL and LAL and, in the IS, 36 for the SAL, and 29 for the LAL; the Michaelis constant is 35 mM. The WKM Vmax values are substantially higher than those used in our model, viz., 10.5 in the OS and 25.9 in the IS. Figure 8B shows CD and TAL Na+ concentration profiles (solid curves) obtained when the WKM diameters and transport parameters were used in our model; also shown are our base-case results (dashed curves). The higher WKM transport rates substantially increased the concentration (and hence, osmolality) gradients in the OM; but concentrations in the three tubules shown in Fig. 8B were reduced near the corticomedullary boundary (x = 0). Indeed, at that boundary, the LAL and SAL Na+ concentrations obtained in our model by using the WKM parameters were 26.08 and 23.77 mM, respectively. The analogous concentrations obtained in the WKM model were both 30 mM (37, 40, 43). The differences in these boundary Na+ concentrations arise from many differences between the models, e.g., transport parameters in SAL and other tubules, the representation of radial organization of the tubules, boundary conditions, etc. Nonetheless, in both the WKM model and in our model using the WKM TAL diameters and transport rates, TAL Na+ concentrations at the corticomedullary boundary are substantially lower that the values obtained in our base-case model (93.46 mM for the LAL and 121.86 mM for the SAL).
Lower SAL Na+ concentrations near the corticomedullary boundary will tend to significantly reduce net transepithelial Na+ transport by increasing diffusive Na+ backleak into the SAL. In our model using WKM transport values, the increased Na+ backleak results in more cost, per net NaCl molecule transported, and thus a less efficient concentrating effect: the efficiency E of our model using WKM TAL diameters and transport parameters is 73.16% that of our base-case efficiency, even though the CD osmolality at the OM-IM boundary obtained by using the higher TAL transport rates in our model is increased to 204.6% of base-case osmolality.
Active Urea Secretion into PSTs of Long Loops of Henle
Although only 4060% of filtered urea is thought to reach PSTs of the loops of Henle (1), fractional urea excretion varies over a broad range of values in rat and other mammals and has sometimes been found to exceed 100% (4). Bankir and Trinh-Trang-Tan (4), who note that several investigations have suggested urea secretion into proximal tubules, have hypothesized that active urea secretion into terminal portions of the proximal tubule, extending from the deep cortex and into the OS, may account for the discrepancy between the rate of urea delivery to PSTs via tubular flow advection and the rate of urea excretion.
To investigate the effects of urea secretion, we compared base-case results to cases in which urea was actively secreted into the PSTs of long loops of Henle. These PSTs, which have marked tortuosity in the OS (19, 21), and which are contiguous with LDL having low urea permeability in the IS (29), are good candidates for urea delivery to the IM. The inwardly directed urea active transport was assumed to have the form of Michaelis-Menten kinetics, with the Michaelis constant set to 15 mM and maximum transport rate set to 10 nmol?cm2?s1. These parameters were chosen to produce a urea secretion rate that is 50% of the urea filtration rate (summed over all nephrons, including short ones), as suggested in Ref. 4. Model results were obtained for two cases, case 1 and case 2. In case 1, base-case CD urea permeability was used (1 x 105 cm/s); in case 2, a CD urea permeability of 3.5 x 105 cm/s, as reported in Refs. 34 and 41, was used.
Figure 9, which exhibits simulated urea concentrations and urea flows in LDL, SDL, and CD, shows that active urea secretion can have a significant impact on medullary urea distribution. Because LDL urea concentration and urea flow were increased along the OS (Fig. 9, A1 and A2), LDL urea flows at the OM-IM boundary were 329 and 321% of base case, in cases 1 and 2, respectively. Thus the rate of urea return to the cortex via the LAL was substantially higher than in base case [recall the low LAL urea permeability, 0.6 x 105 cm/s (16), which tends to ensure that urea carried by LAL flow will be delivered to the cortex]. Because of increased urea diffusion into the SDL2 segment (see below), the rate of urea delivery to the cortex via the SALs was increased to 147 and 115% of base case in cases 1 and 2, respectively. Thus CD urea flow at the corticomedullary boundary, which is given by a fraction of the LAL and SAL urea flows, was increased to 210 and 149% of base case in cases 1 and 2, respectively (Fig. 9C2). The fractional filtered load of urea in CD flow at the corticomedullary boundary in cases 1 and 2 was 1.15 and 0.815, respectively, compared with 0.547 in the base case and measured late distal convoluted tubule fractional delivery ranging from 0.65 to 0.93 in moderately antidiuretic rats (1).
Using base-case CD urea permeability (which was reduced relative to measured values), the rate of urea reabsorption from the CD remained small in case 1, and CD urea flow into the IM was 215% of base-case flow (Fig. 9C2). In contrast, case 2 used the measured CD urea permeability, which is 3.5 times that of base case; this resulted in a 72.5% urea loss from the CD, and consequently, a smaller CD urea flow into the IM than in base case.
In both cases 1 and 2, urea flow into the IM via the LDV was higher than base case (results not shown). Indeed, the aggregate urea flows into the IM, via the LDL, LDV, and CD, in cases 1 and 2 were 245 and 153%, respectively, of base case. Thus the rates of urea return to the OM via the LAL and LAV were also increased by PST urea secretion. Owing to the higher LAV urea flow and concentration, a larger amount of urea diffused out of the LAV and into the SDL2 segment, which occupies a nearby position and which is assumed to be moderately urea permeable. Thus SDL urea flow showed a significant net increase along the IS, an increase that is significantly greater than in base case, particularly in case 1 (Fig. 9B2). This increased urea flow was returned to the IM via the SAL and CD.
We also examined the case analogous to case 1 but differing only in having urea secretion into the PSTs of both long and short loops of Henle. The results were similar to those obtained in case 1: although the increase in SDL2 tubular fluid urea flow at the OM-IM boundary attributable to urea secretion was 300% more than the increase obtained in case 1 (relative to the base case), CD urea concentration and flow at the OM-IM boundary differed from case 1 by 6% of correspondiong differences observed between case 1 and base case. In summary, these studies predict that active urea secretion would increase the magnitude of medullary urea cycling and the degree of medullary urea sequestration and that it has the capacity to significantly increase urea delivery to the CD. (By solute "sequestration," we mean that the fractional contribution of that solute to local fluid osmolality substantially exceeds its fractional contribution to blood plasma osmolality.)
DISCUSSION
Methodology
The studies described herein generally involved parameter variation over large ranges; in some cases, parameters were varied over several orders of magnitude. By such means, a comprehensive assessment of a parameter’s effects was obtained, including nonlinear dependence of results on parameter values and limiting model behaviors.
Several measures of UCM efficacy were used, including the osmolality of CD outflow at the OM-IM boundary and a concept of the free-water absorption rate (OM-FWA) that was modified for application to the OM. Two measures of UCM efficiency were used, including ratio of OM-FWA to the total rate of energy expenditure for active NaCl transport (TAT) and the ratio of CD fluid osmolality increase along the OM to TAT.
In most model studies of the UCM, the efficacy of the concentrating mechanism has been assessed by the urine osmolality attained by the model (26, 36, 37, 40, 43). However, if the urine flow rate is sufficiently low when urine osmolality is high, the UCM will have little impact on an organism’s systemic blood plasma osmolality. Alternatively, UCM efficacy may be assessed by means of FWA, which is the volume, per unit time, of blood plasma that could be considered to be completely cleared of solute, and returned to the blood, by the production of urine that is more concentrated than blood plasma (12, 42). However, equal values of FWA can be attained by a large urine flow that is slightly more concentrated than blood plasma, or by a small urine flow that is much more concentrated than blood plasma. Consequently, a high FWA could correspond to an excretion rate that would be physiologically undesirable because it would lead to unacceptable fluid and solute loss that could result in fatal hypovolemia. Thus it may be that concentrating efficacy can be best assessed by taking into account both urine flow and FWA. For assessment of OM function, we have modified the concept of FWA for our particular context by computing the clearance obtained if an organism depended on the OM only for urine production, a measure which we called OM-FWA (Eq. 3).
Because the OM concentrating mechanism is driven by active Na+ transport, concentrating mechanism efficiency can reasonably be considered to depend on the ratio of a measure of concentrating mechanism efficacy, e.g., OM-FWA, to TAT, where TAT is the total Na+ active transport rate in all tubules; that ratio, which we called E, is given by Eq. 4. TAT is directly proportional to the costs of Na+ transport, which is proportional to the corresponding rate of ATP hydrolysis by the Na+-K+-ATPase transporter. Although other costs are also involved (e.g., cell maintenance, transporter synthesis, cell replacement, and viscous dissipation in tubular and vascular flows), such costs are difficult to quantify and would be present in the kidney even if no concentration gradient were generated.
Boundary Conditions
Because the IM is not explicitly represented in our model, boundary conditions, corresponding to a moderately antidiuretic state, were imposed on the water and solute flows of LAL and LAV at the OM-IM border; these conditions required assumptions about urine composition and flow, and about the relative allocation of solutes between LAL and LAV. Sensitivity studies showed that these assumptions embodying IM influence on the OM did not significantly affect the solutions of our OM model. This insensitivity is attributable to the small urine water and solute flows, relative to tubular flows in the OM. Thus even though changes in various parameters investigated in this study could significantly impact urine composition, e.g., TAL active transport rates, our use of a uniform urine composition to specify OM-IM boundary conditions (in those studies not specifically directed to understanding the effects of urine composition on the model) is unlikely to have significantly affected results of our parameter studies.
Regionalization
To investigate the impact of regionalization on model results, we varied the degree of radial organization and the degree of inclusion of the SDL in the vascular bundle of the IS. Our results, for both the (4-region) base-case mdoel and the (2-region) E2R model (which effectively merged region R1 with R2, and region R3 with R4), predict that regionalization increases the osmolality of LDL and CD tubular fluid at the OM-IM boundary and limits urea absorption from the OMCD, which increases the urea concentration in the CD tubular fluid entering the IM (see Figs. 1 and 2). These predictions for the base-case and E2R models are surprisingly similar, with respect to concentrating capability, but both make predictions that differ markedly from the case of a single merged region (as approximated in row 5 of Fig. 1). However, the base case, with its four regions, should provide a more realistic representation of the transepithelial concentration gradients affecting the tubules and vessels and it likely portrays more realistic incremental increases in osmolality as a function of increasing distance from the center of the vascular bundle. These increments would be reduced if the four regions were further subdivided. The predictions of studies are consistent with those based on the WKM model (37, 43), which also predicted increased osmolalities in tubules and vessels as a function of increased radial distance from the vascular bundle center.
When the fraction of SDL contained within the vascular bundle was reduced, the osmolalities of luminal flows into the IM, via the LDL, CD, and LDV, were decreased (Fig. 4). This result is consistent with that obtained by Wang et al. (40): using the WKM model, they found that the highest urine osmolality was obtained when the SDL was entirely contained within the periphery of the vascular bundle. The inclusion of the water-permeable IS portion of the SDL in the vascular bundle will tend to shield the SDL from R3 and R4 and thus prevent its presenting a load to the concentrating mechanism in those regions (see below, under SDL2 Segments, Prebend Segments, Short Vasa Recta Distribution, and DVR Flow, for further consideration of the relationship of load to concentrating capability).
AVR Solute Permeabilities
Our model calculations predict that the CD osmolality profile and CD outflow osmolality increase as AVR solute permeabilities increase (Fig. 5). This is an intuitive result, because the concentrating mechanism can use AVR flow that is concentrated relative to its local interstitium to increase local osmolality, whereas concentrated AVR flow that is carried out of the OM without interacting with local surroundings (e.g., interstitium, tubules, and DVR) tends to dissipate the axial osmolality gradient, because that flow does not participate in countercurrent exchange. Indeed, as AVR permeabilities are reduced below our chosen base-case values, CD osmolality decreases significantly, whereas increases in AVR permeabilities above the base-case values result in osmolality increases of progressively diminishing magnitude, owing to our choice of sufficiently high permeabilities to assure effective countercurrent exchange.
In Peskin’s highly schematic model of the UCM (10) and in Layton’s distributed-loop version of that model (24), vasa recta were not explicitly represented. Instead, water and solute absorbed from tubules into the interstitium were assumed to enter AVR directly, at each medullary level, and after entering AVR, fluid was assumed to have no further interaction with the medulla. Consequently, relatively concentrated ascending fluid does not equilibrate with progressively less concentrated surrounding interstitium. In our region-based model, this configuration roughly corresponds to the limit of zero effective AVR solute permeabilities, which is approximated by the case of = 0.01 (see Fig. 5). In contrast, the WKM model (43) assumes that AVRs are so highly fenestrated that the composition of AVR fluid can be assumed equal to local absorbate; thus in our region-based model, the WKM assumption roughly corresponds to the limit of infinitely large AVR solute permeabilities (approximated by = 100). The greatly reduced CD tubular fluid osmolalities that were obtained for the case of greatly reduced AVR permeability ( = 0.01) suggest that the models used by Peskin (10) and by Layton (24) are unrealistically dissipative of the axial osmolality gradient. We believe that our base-case assumption, which approximates the WKM configuration, more likely resembles in vivo behavior.
Axial Diffusion
Variation of axial diffusion over two orders of magnitude in the interstitial portions of the concentric regions had little impact on model results. This indicates that the degree of interstitial diffusion resistance used in our base case, a poorly quantified parameter, has no significant influence on OM concentrating capability. In addition, the degree of axial diffusion may also represent the influence of the tortuous paths of capillaries with respect to the corticomedullary axis, much as an effective diffusion may be used to quantify the impact of nonideal countercurrent exchange in a central core model (35). The insensitivity of model results to axial diffusion suggests that such tortuosity does not significantly reduce concentrating capability.
SDL2 Segments, Prebend Segments, Short Vasa Recta Distribution, and DVR Flow
We investigated the impact of several aspects of tubular inhomogeneity on the UCM. The first of these concerned the effect of the likely water-impermeable but urea-permeable SDL segment found in the IS (our SDL2 segment). Based on immunolocalization studies, Wade et al. (39) hypothesized that the SDL2 segment would possess a significant urea permeability and that its proximity to LAV would facilitate urea cycling within the medulla by providing a urea pathway from the IM to the LAV, then to the SDL, and thus ultimately to the distal nephron and CD system, from which it could reenter the IM. Indeed, urea cycling by this pathway was predicted by our base-case model. However, if the urea permeability of the SDL2 segment is significantly higher than the base-case value, then urea cycling attributable to transepithelial urea transport by the SDL2 is reduced. In that case, although urea enters the SDL near the OM-IM boundary, the urea concentration difference with respect to the LAV does not sustain urea entry into the SDL throughout the SDL2 segment, and sufficiently near the OS-IS boundary, urea is lost from the SDL.
In addition to urea cycling, model simulations support an important role for the water impermeability of the SDL2 segment: CD outflow osmolality, OM-FWA, and efficiency E are all increased by a reduction in load presented to the OM concentrating mechanism in the base case relative to a case in which the SDL2 segment was eliminated and replaced by an extension of the SDL1 segment. These findings are consistent with a theoretical study (25), which, by means of a central core model of the renal medulla, predicted that prebend segments increase concentrating capability by eliminating the load required to raise the concentration of fluid in near-bend descending limbs.
A SDL prebend thick segment could also affect OM concentrating capability, by reducing load on the concentrating mechanism and enhancing NaCl absorption in near-bends deep in the medulla (25), but our model’s base-case results predict that its impact would be small relative to that of the SDL2 segment. Our model comparison of cases having differing number ratios of the two types of SDL (one with prebend segments in both SDL, and one with no prebend segment) predicts that the presence of prebend segments would result in modest increases in OM-FWA and efficiency, an increase in CD fluid osmolality, and a decrease in CD flow.
The effect of short vasa recta distribution is somewhat analogous to the effect of the SDL2 segment. DVR flow tends to decrease concentrating capability by presenting a load to the UCM. By conducting simulations using alternative vasa recta distributions, we found that when a larger fraction of short descending vasa recta terminate in the portion of the OM near the cortex, the load is reduced in the deep OM and concentrating capability increases, as assessed by CD osmolality at the OM-IM boundary. Similarly, increases in DVR flow entering the OM increase load on the whole medulla, by increasing the amount of fluid that must be concentrated, per unit time, and by increasing AVR outflow and thus the dissipative effect of that outflow. Consequently, CD outflow osmolality, at the OM-IM boundary, is decreased.
Inhomogeneous TAL Transport
A comparison of our base-case results to a simulation having approximately equal TAT but uniform maximum active Na+ transport rate all along the TAL indicates that concentrating efficiency is increased by transport rate inhomogeneity along the TAL: consistent with theoretical predictions (25), the concentrating effect, for a fixed energy cost, assessed by CD fluid osmolality at the OM-IM boundary and by OM-FWA, can be increased by a higher transport rate deeper in the OM.
However, if the transport rate is excessively high, and the TAL luminal Na+ concentration becomes so low such the outward-directed Na+ transport is offset by backleak into the TALs, effiiency is reduced. This is evident in the results obtained from our model with the TAL active transport rates used in the WKM model. Even though our model then generates a much higher CD osmolality than in the base case, the efficiency obtained using the WKM parameters is only 73% of that obtained using our base-case transport parameters. We emphasize that because of the many differences between our model and the WKM model, this comparison is not a comparison of the two models, but rather a comparison between our model with our base-case TAL active transport rates and our model with the much higher transport rates of the WKM model.
Urea Secretion into PSTs
Our results predict that urea flow into the IM, carried by LDL tubular fluid, can be substantially increased by active urea secretion into the PSTs of long loops. Such an increase may have several implications. First, it may account for, or contribute to, the significant urea concentrations that have been measured at the loop bends of long loops of Henle (31). Second, increased LDL urea flow may promote urea accumulation in the IM, raising the interstitial urea concentration and thus reducing urea reabsorption from the CD; consequently, urea excretion rate may be increased. For example, Wade et al. (39) identified the urea transporter UT-A2 in portions of IM LDLs near the IM base; but they found no evidnce of urea transporters i