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【摘要】 We used a mathematical model of the urine concentrating mechanism of rat inner medulla (IM) to investigate the implications of experimental studies in which immunohistochemical methods were combined with three-dimensional computerized reconstruction of renal tubules. The mathematical model represents a distribution of loops of Henle with loop bends at all levels of the IM, and the vasculature is represented by means of the central core assumption. Based on immunohistochemical evidence, descending limb portions that reach into the papilla are assumed to be only moderately water permeable or to be water impermeable, and only prebend segments and ascending thin limbs are assumed to be NaCl permeable. Model studies indicate that this configuration favors the targeted delivery of NaCl to loop bends, where a favorable gradient, sustained by urea absorption from collecting ducts, promotes NaCl absorption. We identified two model modes that produce a significant axial osmolality gradient. One mode, suggested by preliminary immunohistochemical findings, assumes that aquaporin-1-null portions of loops of Henle that reach into the papilla have very low urea permeability. The other mode, suggested by perfused tubule experiments from the literature, assumes that these same portions of loops of Henle have very high urea permeabilities. Model studies were conducted to determine the sensitivity of these modes to parameter choices. Model results are compared with extant tissue-slice and micropuncture studies.
【关键词】 renal medulla urine concentrating mechanism countercurrent system sodium chloride transport urea transport mathematical model
THE MEANS BY WHICH MAMMALS produce an osmolality gradient along the corticomedullary axis of the inner medulla (IM) of the kidney remains undetermined despite many decades of sustained investigation ( 27, 57 ). Most researchers believe that the IM gradient, like the outer medullary (OM) gradient, is generated by means of a countercurrent multiplication mechanism involving the renal tubules and that washout of the gradient is prevented by means of vascular countercurrent exchange.
The most frequently cited explanation for the IM urine concentrating mechanism (UCM) is the passive hypothesis proposed in 1972 by Kokko and Rector ( 31 ) and by Stephenson ( 63 ). However, this hypothesis depends on specialized loop of Henle transepithelial transport properties that appear to be inconsistent with measured values from perfused tubule experiments: when these experimental values are used in mathematical models, a significant axial 1 osmolality gradient cannot be generated ( 35, 39, 46, 66, 77 ). Attempts to salvage the passive hypothesis, by means of models using discrete, distributed loops of Henle ( 35, 39 ) or by means of representation of the preferential interactions that arise from three-dimensional medullary structure ( 67, 69, 78 ), have not been generally successful.
Recently, a new hypothesis involving the generation of an external osmolyte, namely, lactate, has been proposed ( 13, 68 ). In principle, such a mechanism can be highly effective ( 19 ), and model studies suggest that such a mechanism could contribute to the IM gradient ( 13, 68 ). Also recently, hypotheses involving pelvic peristalsis have been revived ( 27 ); functional significance for the peristalsis is suggested by major time-dependent alterations of flow patterns in loops of Henle and vasa recta (VR) and, in antidiuretic states, by bolus flow of tubular fluid along the inner medullary collecting duct (IMCD) ( 54, 62 ).
New experimental studies combining the techniques of immunohistochemical localization and computerized three-dimensional reconstruction are providing new insight into the transport properties of long loops of Henle in rat IM ( 49, 50 ). These studies, which are summarized below (see HYPOTHESES : TWO CONCENTRATING MODES ), suggest that in the IM, descending thin limbs (DTLs) have terminal segments of significant length that have limited water permeability; that these DTLs are NaCl impermeable except for a short prebend segment; and that most loop of Henle segments are nearly urea impermeable.
The goal of this study is to describe and evaluate, by means of mathematical modeling, two hypothetical concentrating modes for the IM that are based on the new studies. Both modes are similar to the passive hypothesis previously described by Kokko and Rector ( 31 ) and by Stephenson ( 63 ), in that NaCl absorption from loops of Henle is driven by interstitial urea that diffuses from the collecting duct (CD) system, and the mixing of this NaCl and urea raises the interstitial osmolality. However, both modes depend on locally high rates of NaCl absorption around the bends of the long loops of Henle, and both modes depend on active NaCl absorption from the CDs, which ensures that urea-rich fluid is absorbed from the portion of the CD system that is deep in the IM.
Both hypothetical modes appear to be capable of producing a significant corticomedullary osmolality gradient along the IM while maintaining reasonable urine flow and free water absorption rates. We call the two modes the "pipe mode" and the "solute secretion mode" (SS mode), based on differing assumptions about their urea transport properties in terminal portions of descending limbs. In the pipe mode, low solute permeabilities of the terminal descending limb allow little solute transport, and the principal solute delivered to the loop bend is NaCl. In the SS mode, substantial urea enters the terminal descending limb, and most of that urea is absorbed along the ascending thin limb (ATL); i.e., the loop of Henle acts as a countercurrent urea exchanger.
HYPOTHESES: TWO CONCENTRATING MODES
Experimental Findings on Which Our Modes Are Based
It is our objective to perform a computer-assisted, three-dimensional functional reconstruction of the rat renal IM, with emphasis on the thin limbs of Henle's loops ( 51 ). The reconstruction is based on serial transverse sections in which cross sections of tubules are labeled and distinguished by means of antibodies to appropriate transport molecules; the labeled molecules are detected by indirect immunofluorescence ( 49 ). Currently, we are using antibodies raised against 1 ) the water channel aquaporin-1 (AQP1), for DTL identification and function; 2 ) the ClC-K1 chloride channel, for ATL identification and function; 3 ) the water channel aquaporin-2 (AQP2), for CD identification and function; and 4 ) the heat shock-related protein B-crystallin, for identification of segments that do not express these channels. The initial reconstruction has involved the central region of the IM.
While this functional reconstruction was being performed, several findings involving the thin limbs of Henle's loops ( 49 ) have suggested to us the modes for urine concentration that are described herein. First, the DTLs of Henle's loops that have their bends within the first millimeter below the OM-IM border lack AQP1 and presumably are largely impermeable to water ( Fig. 1 A ). Second, all the DTLs of loops that have their bends below this first millimeter express AQP1 for about the first 40% of their length below the OM-IM border. After this point, for about the last 60% of their length, they lack AQP1 expression ( Fig. 1 B ). Thus the longer the long loop, the longer the segment that does not express AQP1. Presumably, this 60% of each DTL is impermeable to water or has a moderate water permeability as found in some perfusions of rat DTLs from the deep IM ( 5 ), although it is not certain that these measurements involved only the AQP1-null segments. Third, expression of ClC-K1 chloride channels begins abruptly with a prebend segment on the descending side of the loop and continues uniformly along the entire length of the ATL. The length of the prebend segment is nearly uniform and is thus independent of the length of the loop: the prebend segment always begins 165 µm before the loop bend ( Fig. 1 C ) ( 49 ). AQP1 and ClC-K1 are not colocalized in any segment or subsegment of the loops of Henle within the IM ( 49 ). We assume, based on previous studies with isolated, perfused rat ATLs ( 14 ), that the entire region expressing ClC-K1 and lacking expression of AQP1 (prebend segment and entire ATL) has a high permeability to Cl - (and Na + ) and virtually no permeability to water.
Fig. 1. Computer-assisted reconstruction of loops of Henle from rat inner medulla (IM) showing expression of aquaporin-1 (AQP1; red) and ClC-K1 (green); gray regions express undetectable levels of AQP1 and ClC-K1. Loops are oriented along the corticomedullary axis, with the left edge of each image nearer the base of the IM. A : thin limbs that form a turn within the first millimeter beyond the outer medulla (OM)-IM boundary. Descending segments lack detectable AQP1. ClC-K1 is expressed continuously along the prebend segment and the ascending thin limb (ATL). B : loops that form a turn beyond the first millimeter of the IM. AQP1 is expressed along the initial 40% of each descending thin limb (DTL) and is absent from the remainder of each loop. ClC-K1 is expressed continuously along the prebend segment and the ATL. Boxed area is enlarged in C. C : enlargement of near-bend regions of 4 thin limbs from box in B. ClC-K1 expression, corresponding to DTL prebend segment, begins, on average, 165 µm before the loop bend (arrows). Scale bars: 500 µm ( A and B ); 100 µm ( C ).
We have not yet examined in detail the expression of possible urea transporters along the thin limbs of Henle's loops in the IM. However, a few preliminary sections show no evidence of expression of the urea transporters UT-A1, UT-A2, or UT-A4 in thin limbs below the first millimeter of the IM (Pannabecker TL and Dantzler WH, unpublished observations). These observations appear to agree in general with the expression pattern indicated in other types of immunocytochemical studies ( 48, 71 ), although Wade et al. ( 71 ) reported colabeling of UT-A type protein and AQP1 in DTLs from the base of the IM. Preliminary studies also show no expression of the urea transporter UT-B1 in IM thin limbs (Pannabecker TL and Dantzler WH, unpublished observations). This was expected because other data indicate that this transporter is found only in descending VR, not in thin limbs ( 3 ). These preliminary observations suggest that the AQP1-null segment of the DTLs does not express any of these known urea transporters, although a full reconstruction needs to be completed. Moreover, these observations suggest that the AQP1-null segment of the DTLs has a very low permeability to urea. However, this urea permeability has yet to be measured directly, and a high urea permeability could be found in these DTL segments and in the ATLs as the result of an unidentified transporter.
Pipe Mode Hypothesis
The pipe mode corresponds to the low urea permeability limit of the loop of Henle: those portions of the loop of Henle that are AQP1 null are assumed to have nearly zero urea permeability. The key elements of the pipe mode are these:
1 ) The DTL AQP1-null segment is assumed to be nearly impermeable to NaCl and urea and to have only a low-to-moderate permeability to water.
2 ) Because permeabilities to NaCl and urea are low, their transepithelial fluxes are small, and therefore the DTL AQP1-null segment functions much like a conduit or a "pipe" with respect to NaCl and urea. However, this segment's moderate water permeability allows it to maintain an approximation to interstitial osmolality, as indicated by micropuncture experiments ( 8, 52 ).
3 ) At the prebend segment, the DTL permeability to NaCl increases greatly, and that high permeability is sustained along the ATL. The prebend segment and ATL are assumed impermeable to water.
4 ) Because of urea absorbed from the IMCD (and especially urea absorbed from the innermost IMCD), the interstitial urea concentration is large and the concentrations of electrolytes small. Thus a large transepithelial gradient favors NaCl absorption around the loop bend. A fortiori, this is the case for those loops that reach deep into the papilla.
5 ) Along the ATL, NaCl continues to be absorbed; indeed, the ATL serves as a near-equilibrating segment for NaCl, but as fluid ascends the ATL, the gradient favoring its absorption diminishes, owing to NaCl that is absorbed from loop bends that turn nearer the OM-IM boundary.
6 ) This mode concentrates principally by the vigorous net absorption of solute, namely, NaCl, from loop bends, which is unaccompanied by water absorption from loop bends.
7 ) The VR carry away solutes and water absorbed from CDs and loops of Henle. The VR form a countercurrent exchange configuration that maintains a high interstitial urea concentration and that prevents washout of the axial interstitial osmolality gradient.
8 ) Active absorption of NaCl from the IMCD, accompanied by water absorption, raises CD tubular fluid urea concentration and reduces the load presented to the concentrating mechanism in the inner portions of the IM. More generally, active NaCl absorption from the IMCD serves to promote, modulate, and spatially distribute urea absorption from the CD.
9 ) Even with low urea permeabilities ( 1 x 10 -5 cm/s), substantial urea enters the loop of Henle, perhaps sufficient to account for micropuncture data.
SS Mode Hypothesis
The SS mode corresponds to the high urea permeability limit of the loop of Henle: those portions of the loop of Henle that are AQP1 null are assumed to have very high urea permeability so that near-equilibration with the interstitium can be maintained. The key elements of the SS mode are these:
1 ) The DTL AQP1-null segment is assumed to be nearly impermeable to water and NaCl; however, in this mode, the segment is assumed highly permeable to urea.
2 ) Because permeability to water and NaCl along the AQP1-null portion of the DTL is low, NaCl concentration changes little along the DTL; however, because urea permeability is high, urea enters the DTL and tubular fluid urea nearly equilibrates with the interstitial urea concentration. Thus substantial urea is secreted into the DTL, and DTL tubular fluid osmolality increases substantially. Indeed, if interstitial NaCl concentration is less than DTL NaCl concentration, DTL osmolality may exceed interstitial osmolality.
3 ) At the prebend segment, the DTL permeability to NaCl increases greatly, and the high permeability to urea is sustained along the prebend segment and ATL. Thus the prebend segment and ATL are assumed to be highly permeable to both NaCl and urea, but they are assumed to be impermeable to water.
4 ) Because of urea absorbed from the IMCD (and especially urea absorbed from the innermost IMCD), the interstitial urea concentration will be large and the concentrations of electrolytes small. Thus a large transepithelial gradient will favor NaCl absorption around the loop bend. A fortiori, this is the case for those loops that reach deep into the papilla.
5 ) Along the ATL, NaCl continues to be absorbed; indeed, the ATL serves as a near-equilibrating segment for NaCl, but as fluid ascends the ATL, the gradient favoring its absorption diminishes, owing to NaCl that is absorbed from loop bends that turn nearer the OM. Also, as urea-rich fluid flows up the ATL, it is opposed by a decreasing interstitial urea concentration, and urea is absorbed, thus decreasing the ATL urea concentration and maintaining a near-equilibrium with the interstitial urea concentration.
6 ) As in the pipe mode, the SS mode concentrates principally by the vigorous net absorption of a solute, i.e., NaCl, from loop bends, which is unaccompanied by water absorption. However, unlike the pipe mode, the loop of Henle functions as a highly effective countercurrent urea exchanger.
7 ) The VR carry away solutes and water absorbed from CDs and loops of Henle. The VR form a countercurrent exchange configuration that maintains a high interstitial urea concentration and prevents washout of the interstitial osmolality gradient.
8 ) As in the pipe mode, active absorption of NaCl from the IMCD raises CD tubular fluid urea concentrations and reduces the load presented to the concentrating mechanism in the inner portions of the IM.
Transport and Structural Properties That May Support the Hypothesized Modes
AQP1-positive DTL segment. As noted above, our studies have shown that loops of Henle reaching beyond the first millimeter of the IM have an AQP1-positive DTL segment that makes up 40% of the IM portion of the DTL. We hypothesize that water absorbed from this segment will tend to raise the tubular fluid NaCl concentration before that fluid reaches the AQP1-null DTL segment. A resulting higher NaCl concentration in tubular fluid at the loop bend will favor more vigorous absorption of NaCl from the loop bend and ATL. Although our base-case model studies (in RESULTS ) show no significant net water absorption from the AQP1-positive DTL segment, we consider it likely that more complete experimental information and more detailed model formulations will support net water absorption from that segment.
If the AQP1-positive DTL segment is sufficiently permeable to urea, transepithelial gradients may favor urea secretion into this segment. Such secretion could support the pipe mode by contributing to a urea-cycling pathway that conveys the secreted urea to the IMCD by means of tubular fluid advection along the distal nephron. Urea secretion into the AQP1-positive DTL segment could support the SS mode by contributing to the accumulation of urea in the papilla; indeed, the DTL and ATL may sequester urea and, by participating in countercurrent urea exchange with the VR, the DTL and ATL may help maintain the axial interstitial urea gradient and thereby help promote NaCl absorption from loop bends and ATLs.
Loop of Henle distribution. The loop of Henle distribution, with bends at all levels of the IM, will tend to distribute vigorous near-bend NaCl absorption all along the corticomedullary axis, thus providing a concentrating effect distributed all along that axis. Those water-impermeable loops that turn within the first millimeter below the OM-IM boundary present no load on the concentrating mechanism if no net solute is secreted into them. However, Na + absorption from these loops may promote water absorption from longer loops, which have an initial AQP1-positive segment, and thus raise the NaCl concentration in the longer loops (as described above). Moreover, the approximately exponential decrease in loop population, as a function of depth, will balance the local load presented by the CD and VR with a degree of NaCl absorption that is sufficient to produce a concentrating effect at each medullary level.
The interstitium. The interstitium is the medium for communication among the tubules and VR; however, the extracellular gelatinous interstitial matrix and the lipid-laden cells that predominate in the interstitium of the IM may hinder axial fluid and solute movement and may thereby effectively eliminate axial advection and diffusion in the interstitium ( 41 ).
MATHEMATICAL MODEL
Our mathematical model of the IM, which is based on the central core (CC) formulation introduced by Stephenson ( 63 ), includes loops of Henle and a CD; the loops of Henle and the CD interact in a common tubular compartment, the CC. The DTLs, ATLs, CD, and CC are represented by rigid tubules that are oriented along the corticomedullary axis, which extends from x = 0 at the OM-IM boundary to x = L at the papillary tip (see Fig. 2 ). The model is formulated for three solutes: NaCl, urea, and a nonreabsorbable solute; NaCl is represented by Na +. The nonreabsorbable solute, denoted NR, is assumed to be present only in significant amounts in the tubular fluid of the CD; therefore, in the model, NR is represented only in CD tubular fluid. The model predicts fluid flow, solute concentrations, transepithelial water and solute fluxes, and fluid osmolality as a function of medullary depth, in the tubules and in the CC.
Fig. 2. Schematic diagram of IM model, showing loops of Henle, collecting duct (CD) and central core (CC). The number of loops decreases as a function of medullary depth. Six representative loops are shown, but the numerical formulation of the model uses 300 discrete loops of Henle to approximate a continuously decreasing distribution. Each DTL that reaches beyond the first millimeter of the IM is divided into the LDL2 (which makes up 40% of the DTL), LDL3, and a prebend segment. A DTL that turns within the first millimeter of the IM is divided into LDL2 S (S denoting short) and a prebend segment. Flow from each model DTL enters directly into its associated ATL. The CD system is represented by a composite CD with decreasing luminal surface area. The DTLs, ATLs, and composite CD interact with a common compartment, the CC.
Because short loops of Henle turn in the inner stripe of the OM, mostly within a narrow band near the OM-IM boundary ( 12 ), only long loops are included in our model. We assume that 12,667 long loops of Henle (one-third of a total of 38,000 loops of Henle) and 7,300 CDs ( 12 ) extend into the IM. The long loops appear to form loop bends at all levels of the IM; thus the population of loops decreases as a function of increasing medullary depth. That decreasing loop population can be represented by a model formulation having continuously distributed loops ( 34 ) (see Figs. 2 and 3 A ); in such a formulation, tubular concentration profiles in loops turning at differing levels are assumed to differ, and transmural fluxes are weighted at each medullary depth according to the number of loops of a particular length remaining at that depth (see Eq. A8 in the APPENDIX ). Measurements in the rat ( 12 ) indicate that the fraction of long loops of Henle decreases nearly exponentially along the IM; based on those measurements, we approximated the fraction w l of loops remaining at IM depth x by
Fig. 3. Spatially distributed model properties. A : fractions of loops of Henle or CDs remaining as a function of IM depth. B : CD urea permeability as a function of IM depth. C : CD maximum Na + active transport rate as a function of IM depth.
Because the CDs undergo successive coalescences along the IM, the population of CDs also decreases as a function of increasing medullary depth. In the model, all CDs are merged into a single composite tubule, and the effect of the coalescences on tubular surface area is represented by decreasing the tubular radius as a function of increasing medullary depth. That radius is decreased through multiplication by the fraction of CDs w CD remaining at medullary level x. Based on measurements in the rat ( 12 ), that fraction was approximated by
The loop of Henle and CD population fractions, as approximated by Eqs. 1 and 2, are shown in Fig. 3 A.
The model equations, which are summarized in the APPENDIX, embody the principle of mass conservation of both solute and water and represent transmural transport processes, which are described by single-barrier model equations that approximate double-barrier transepithelial transport. Transmural solute diffusion is characterized by solute permeabilities, and active transport is approximated by a saturable expression having the form of Michaelis-Menten kinetics. Transport equations for water represent osmotically driven fluxes. Boundary conditions prescribe flows and concentrations in the DTLs and the composite CD at the OM-IM boundary, i.e., at x = 0.
Base-case parameters for transtubular transport are given in Table 1. The model DTL of a loop of Henle that reaches beyond the first millimeter of the model IM was divided structurally and functionally into three segments. The first segment, which we call LDL2 and which spans the initial 40% of the DTL, was assumed to be highly water permeable but NaCl impermeable. (We reserve the notation LDL1 for the portion of a long descending limb that passes through the OM, a segment that is not represented in this model.) The second IM segment, which we call LDL3 and which corresponds to the AQP1-null segment of the DTL, was assumed to be impermeable to NaCl but to have a water permeability of 400 µm/s in the pipe mode, based on micropuncture experiments in rat DTLs from the deep the IM ( 5 ), and to have no water permeability in the SS mode, consistent with our finding of no AQP1 expression and with micropuncture experiments in chinchilla indicating a water permeability of 50 µm/s in DTLs from the deep IM ( 5 ). The third segment, which corresponds to the prebend segment, is a 166.7-µm-long terminal portion of the DTL that was assigned the transport properties of the ATL (the length of 166.7 µm, rather than the length of 165 µm found in experiments, ensures that each prebend segment begins at a numerical grid point; a transition at a grid point allows the accurate representation of an abrupt change in tubular properties). The ATL was assumed to be water impermeable but highly NaCl permeable.
Table 1. Base-case transtubular transport parameters
The DTL of a loop of Henle that turns within the first millimeter of the model IM was assumed to be water impermeable; it is divided functionally and structurally into two segments. The first, which we call LDL2 S (S denoting short) and which we assumed to be NaCl impermeable, extends to the second segment, the prebend segment, which was assumed to be functionally like its associated ATL, which we designate ATL S.
Our base-case urea permeabilities in loops of Henle differ substantially between the two modes. Our preliminary experimental results (see above, HYPOTHESIS : TWO CONCENTRATING MODES ) suggest that urea permeability in most segments is essentially zero. However, in the pipe mode we used a value of 1 x 10 -5 cm/s to allow some urea entry, because micropuncture studies suggest urea entry into DTL ( 52 ). Based on findings by Wade et al. ( 71 ) of apparent colocalization of AQP1 and a UT-A urea transporter, we assumed that the LDL2 segment has a moderate permeability of 13 x 10 -5 cm/s, a value suggested by microperfusion measurements ( 47 ).
In the SS mode, we assumed that all IM loop segments have at least a moderate permeability to urea and that the urea permeabilities of LDL3, the prebend segment, and the ATL are very large. The LDL2 S, ATL S, and LDL2 were assigned a permeability of 13 x 10 -5 cm/s ( 47 ). The urea permeabilities of the LDL3, the prebend segment, and the ATL were suggested to us by the high permeabilities measured in the long loops of Henle of chinchilla ( 6 ). The permeability of the prebend segment and ATL was taken to be 150 x 10 -5 cm/s, lower than the measured value of 170 x 10 -5 cm/s in chinchilla ATL, but about an order of magnitude larger than reported values in rat of 14-23 x 10 -5 cm/s ( 14, 47 ). The permeability for LDL3 was taken to be 100 x 10 -5 cm/s, about twice the value of 48 x 10 -5 cm/s reported in chinchilla for the lower DTL ( 6 ). Our value for urea permeability in LDL3 supports the effective function of the SS mode and may not be unreasonable, because results from microperfusion studies of DTLs may have been skewed by tubules that spanned more than one functional segment.
The CD urea permeability was assumed to be 1 (in units of 10 -5 cm/s) for the first half of the model IM (i.e., for x [0, L /2]); for the second half of the IM ( x [ L /2, L ]), CD urea permeability was assumed to increase exponentially, according to the formula
where P 0 and P 1 are the initial and terminal CD urea permeabilities, 1 and 110, respectively, and = 7; the CD urea permeability profile is shown in Fig. 3 B. This profile was constructed to be consistent with experiments in antidiuretic rats showing high urea permeabilities in the terminal CD ( 22, 59 ) and to ensure sufficient urea delivery to the deep medulla to support the hypothesized modes (see above).
The CD Na + maximum transport rate ( V max, Na ) was assumed to be 9 nmol·cm -2 ·s -1 for the initial three-tenths of the model IM; it linearly decreases to 2.5 nmol·cm -2 ·s -1 for the next two-tenths, and linearly decreases to 0 nmol·cm -2 ·s -1 along the remainder of the IM; the V max, Na profile is shown in Fig. 3 C. Substantial evidence, recently summarized by Weinstein ( 75 ), indicates that the IMCD is capable of brisk active Na + absorption. The profile for V max, Na was chosen to ensure that substantial urea was absorbed from the CD (by means of maintaining a sufficient transepithelial urea gradient) and that the solute load reaching the terminal CD was consistent with experimental evidence for moderately antidiuretic rats (see below). The Michaelis constant for CD Na + active transport was set to 40 mM ( 10 ). All tubules were assumed to have no active urea transport: although active urea transport has been found in the CD ( 23 ), the rate of such transport appears to be small relative to passive fluxes.
The osmotic coefficients k were set to be 1.84 for NaCl and NR and 0.97 for urea ( 74 ). The reflection coefficients i, k for all solutes were set to be 1 for all tubules ( 56 ). The partial molar volume of water w was set to 0.018136 cm 3 /mM for 37°C ( 74 ).
Axial length L of the model IM was taken to be 5 mm, an appropriate value for the rat kidney ( 26 ). Tubular diameters were assumed to vary as a function of medullary depth. Luminal diameters for loops of Henle were based on measurements by Koepsell et al. ( 29 ): DTL diameter decreases linearly, starting at the OM-IM boundary (i.e., at x = 0), from 15 to 11 µm, and then abruptly increases to 13 µm at the prebend segment and remains at that value to the point of loop bend; along the ATL, the diameter increases linearly from 13 to 20 µm at the return to the OM-IM boundary. CD diameter, based on measurements by Knepper et al. ( 26 ), increases from 20 to 25 µm. CC diameter was assumed to be 20 µm at x = 0, and CC cross-sectional area decreases with medullary depth at the same fractional rate as the CD population decreases ( Eq. 2 ).
The boundary water flows and solute concentrations that were specified for the DTL and CD at the OM-IM boundary (i.e., x = 0) are given in Table 2. DTL boundary water flow was assumed to increase linearly as a function of the length of the loop, based on evidence that juxtamedullary glomeruli have higher SNGFRs than glomeruli that are likely to give rise to short-looped nephrons ( 55 ). Because the boundary values have not been measured directly by experiment, our choices (especially those for the CD) were influenced by general considerations based on relevant measurements (e.g., the composition of urine and of tubular fluid in distal tubule) ( 1, 52 ) and were chosen to provide urea delivery rates sufficient for the concentrating modes. Indeed, CD inflow rate and concentrations at the OM-IM border, and the magnitudes and spatial distribution of Na + and urea transport along the IMCD, were chosen, in part, by means of an informal optimization procedure. Many trial calculations were conducted using various combinations of the CD boundary conditions and transport properties. Based on the experience gained, the conditions and properties were chosen to obtain model urine osmolalities and flow rates that are consistent with experimental findings. We believe that this optimization procedure is justified by the substantial regulatory control exhibited by the CD system in response to physiological needs.
Table 2. Boundary conditions at OM-IM boundary
The SLSI-Newton method, which was previously developed and tested for models of the UCM ( 32, 33 ), was used to obtain steady-state numerical solutions for this model study. A spatial discretization of 300 subintervals was used; thus 300 discrete loops of Henle were represented, each having a loop bend at a distinct numerical grid point. All calculations were performed by means of computer programs written in FORTRAN and executed in double precision on a computer system with two Intel Pentium IV 1.8-GHz processors and with 1 GB of RAM.
RESULTS
The model equations ( APPENDIX ) were solved, using the base-case model configuration, parameter sets ( Table 1 ), and boundary conditions ( Table 2 ) to obtain steady-state model solutions for the pipe and SS modes. Key results are displayed graphically in Figs. 4 and 5, and they are compared with micropuncture measurements in Table 4 (see DISCUSSION ). A quantity expressed in units "per nephron" is the total of that quantity per kidney (or the total of that quantity per kidney at a specified medullary level), divided by the number of nephrons per kidney.
Fig. 4. Selected base-case results for pipe mode, as a function of IM depth. A : osmolality profiles for longest DTL and ATL, for CD, and for CC. B : osmolality profiles for loops of Henle reaching to IM depths of 1.0, 2.0, 3.5, and 5.0 mm. C and D : concentration profiles of longest DTL, longest ATL, CD, and CC. Dashed line, CD nonreabsorbable solute (NR) concentration (NR concentration is 0 in loops of Henle and in CC). E : aggregate fluid flow rate (per nephron) in DTLs, ATLs, CD, and CC. F : CD solute flow rates (per nephron).
Fig. 5. Selected base-case results for solute secretion (SS) mode as a function of IM depth. A : osmolality profiles for longest DTL and ATL, for CD, and for CC. B : osmolality profiles for loops of Henle reaching to IM depths of 1.0, 2.0, 3.5, and 5.0 mm. C and D : concentration profiles of longest DTL, longest ATL, CD, and CC. Dashed line, CD NR concentration (NR concentration is 0 in loops of Henle and in CC). E : aggregate fluid flow rate (per nephron) in DTLs, ATLs, CD, and CC. F : CD solute flow rates (per nephron).
Table 4. Comparison of model mode values with measurements in the rat
Base-Case Pipe Mode Model Results
The pipe mode model predicts a urine osmolality of 1,265 mosmol/kgH 2 O and urine flow rate of 0.0780 nl·min -1 ·nephron -1 (2.96 µl·min -1 ·kidney -1, assuming 38,000 nephrons/kidney). Model urine concentrations were 254, 597, and 118 mM for Na +, urea, and NR, respectively. These values are consistent with experimental measurements from moderately antidiuretic rats ( 52 ).
Fundamental results for the pipe mode, as a function of IM depth, are shown in Fig. 4. Figure 4 A shows that osmolality increases, with increasing depth, in the CD, CC, and longest DTL (except near the OM-IM boundary and in the prebend segment). In the CD, the osmolality increases from 673 to 1,265 mosmol/kgH 2 O, i.e., by a factor of 1.88. Along the longest DTL, osmolality increases from 673 to 1,218 mosmol/kgH 2 O at the beginning of the prebend segment, and then it decreases to 1,073 mosmol/kgH 2 O at the loop bend.
Osmolality profiles for loops of Henle lengths of 1.0, 2.0, 3.5, and 5.0 mm are shown in Fig. 4 B. The loops of Henle that turn within the first millimeter of the IM are assumed to be entirely water impermeable, and thus the osmolality in the LDL2 S of the 1-mm-long loop shows only a small increase, which is due to diffusive urea entry. Beginning at the transition to the prebend segment, osmolality decreases abruptly in all loops, and a rapid rate of decrease continues around the loop bend. Osmolality continues to decrease (as considered in the tubular flow direction) along the contiguous ATL for a portion of that ATL that is about one-third to one-half of its length. In sufficiently long loops, ATL tubular fluid osmolality eventually exceeds that of its corresponding DTL at the same level; however, because of the overapping of shorter loops that have an ATL osmolality that is lower than corresponding DTL osmolality, the osmolality of ATL fluid, taken as a whole at each medullary level, is dilute relative to DTL fluid. Thus the ATLs, taken as a whole, carry dilute fluid out of the IM. This dilute ascending flow is a necessary mass-balance requirement for concentrating urine in our modes, because the osmolalities of flows entering DTLs and the CD at the OM-IM boundary are fixed at a common boundary value and because the CC carries fluid out of the IM that has only a very slightly lower osmolality at the OM-IM boundary than DTL and CD tubular fluid at the boundary.
CC fluid is very slightly dilute at the OM-IM boundary (672 mosmol/kgH 2 O) relative to DTL and CD fluid (673 mosmol/kgH 2 O) because urea and Na + secretion into ATLs is sufficiently large near the boundary to have a net diluting effect on CC fluid; this diluting effect is apparent in Fig. 4 A along the first millimeter to the right of x = 0: CC osmolality is initially decreasing with increasing medullary depth, but ATL tubule fluid osmolality, along its flow direction, is increasing. Electron microprobe measurements performed at a corresponding medullary levels in antidiuretic rat kidneys detected a similar local nonmonotonicity in Na + and Cl - concentrations ( 30 ).
The change in the patterns of osmolality in the loops of Henle can be understood by consideration of the transepithelial Na +, urea, and water fluxes, which can be inferred from their concentrations relative to the CC and CD. The Na +, urea, and NR concentrations for the longest loop, CD, and CC are shown in Fig. 4, C and D. Because the LDL2 is assumed to be impermeable to Na +, its nearly constant Na + concentration indicates little net reduction in water flow along the LDL2. Thus its osmolality is increased by the entry of urea, which continues, although at a slower rate, along the LDL3, because the LDL3 is assumed to have a lower urea permeability than the LDL2. Water absorption from the moderately water-permeable, but Na + -impermeable, LDL3 segments contributes to the concentrating mechanism by increasing tubular fluid Na + concentration significantly before that fluid reaches the prebend segments.
The vigorous absorption of Na + around the loop bend is a consequence of the abrupt increase in Na + permeability and a transepithelial gradient that favors Na + absorption. That gradient arises because urea absorbed from the CD results in a high CC urea concentration relative to the CC Na + concentration. Indeed, the resulting CC Na + concentration is below that of the loop Na + concentration, and thus a substantial transepithelial outward-directed Na + concentration gradient is maintained around the loop bend and along part of the ATL. This effect is particularly marked for loops that reach deep into the IM, because of the high CD urea permeability and thus large CD urea absorption rate along the terminal CD. However, the secretion of some urea into the loops of Henle tends to somewhat reduce the concentrating effect generated by Na + absorption from the loops: the osmolality of core fluid is directly reduced by the urea secretion, and, in addition, this reduction has the indirect effect of discouraging water absorption from LDL2 segments, which tends to reduce loop Na + concentration and thus the gradient favoring Na + absorption from Na + -permeable loop segments.
Fluid flow rates, along the CD and CC, and composite fluid flow rates along the DTLs and ATLs, per nephron, are shown in Fig. 4 E; the flow rates are taken to be positive when flow is in the direction of increasing medullary depth. Because of loops turning back along the IM, the DTLs-to-CD flow ratio (i.e., the ratio of total flow in all DTLs to total flow in all CDs at a given medullary level) rapidly decreases, from 4.12 at the OM-IM boundary to 0.141 at the papillary tip, where the longest loops are assumed to turn back.
Solute flow rates, per nephron, along the composite CD are shown in Fig. 4 F. The CD loses 76% of its Na + in the first 2 mm of the IM, because, by our modeling assumptions, Na + is vigorously actively absorbed (with a V max, Na of 9 nmol·cm -2 ·s -1 ). This active absorption promotes water absorption from the early CD, because the low CD urea permeability (1 x 10 -5 cm/s) restricts the participation of urea flux in transepithelial osmotic equilibration. Moreover, the active Na + absorption raises the relative concentration of urea in CD tubular fluid (see Fig. 4 D ), which ultimately promotes vigorous urea absorption from the terminal CD, where urea permeability greatly increases. In the inner 60% (approximately) of the CD, active Na + transport is only sufficient to nearly balance diffusive Na + backleak.
Base-Case SS Mode Model Results
The SS mode model predicts a urine osmolality of 1,031 mosmol/kgH 2 O and urine flow rate of 0.0889 nl·min -1 ·nephron -1 (3.38 µl·min -1 ·kidney -1, assuming 38,000 nephrons/kidney); model urine concentrations were 110, 430, and 95 mM for NaCl, urea, and NR, respectively. These values are consistent with experimental measurements in moderately antidiuretic rats ( 52 ).
Fundamental model results for the SS mode are shown in Fig. 5. Figure 5 A shows that osmolality increases, with increasing depth, in the CD, CC, and longest DTL (except near the OM-IM boundary and in the prebend segment). In the CD, the osmolality increases from 673 to 1,031 mosmol/kgH 2 O, i.e., by a factor of 1.53. The osmolality increase along most of the DTLs is large, due to the large urea permeability of the LDL3, which results in substantial urea entry; indeed, the osmolality near the bend of the longest loop exceeds the osmolality of all other IM model structures, and even after an abrupt decrease of osmolality around the loop bend, the ATL has an osmolality that exceeds that of the CC and CD.
Osmolality profiles for loops of Henle with lengths of 1.0, 2.0, 3.5, and 5.0 mm are shown in Fig. 5 B. The loops of Henle that turn within the first millimeter of the IM are assumed to be water impermeable, and thus the osmolality in the DTL of the 1-mm-long loop shows a small increase due to urea influx, which arises from a higher urea permeability than in the pipe mode. Each of the longer loops of Henle exhibits a very substantial increase in osmolality along its LDL3, which is due to the very high urea permeability (100 x 10 -5 cm/s) assigned to the LDL3 in this mode. At the transition to the prebend segment, each loop has a substantial decrease in osmolality, and that osmolality continues to decrease, in the flow direction, along much of the ATL. Although, at some levels, the osmolality of an ATL may exceed that of its corresponding DTL, the net effect, as in the pipe mode, is that the overlapping loops, taken as a whole, at each level carry a fluid that is osmotically dilute relative to DTL, and thus the mass-balance requirement for a concentrated urine is met. (A small mass-balance difference in this mode, however, relative to the pipe mode, is that the CC carries fluid from the IM that is slightly higher in osmolality than the inflow osmolalities of the DTLs and the CD at the OM-IM boundary.)
The Na +, urea, and NR concentrations for the longest loop, CD, and CC are shown in Fig. 5, C and D. As in the pipe mode, the LDL2 is assumed to be Na + impermeable, and its nearly constant Na + concentration indicates little net change in water flow along the LDL2; indeed, water flow slightly increases (detailed results not shown). Thus tubular fluid osmolality in LDL2 is increased only by urea entry, a process that is accelerated along the LDL3 by its yet higher urea permeability.
As in the pipe mode, the vigorous absorption of Na + around the loop bend is a consequence of the abrupt increase in Na + permeability and a transepithelial gradient that favors Na + absorption, and that gradient is sustained by urea absorption into the interstitium from the CD. Also, as in the pipe mode, the effect of near-bend Na + absorption is particularly prominent for loops that reach deep into the IM. Unlike the pipe mode, however, the high urea permeabilities that are assumed in the SS mode for the water-impermeable loop segments result in the loops of Henle acting as countercurrent urea exchangers: at each IM level, much urea enters DTLs and much urea leaves ATLs, resulting in a urea cycle that is closed by advection of urea in tubular fluid around the loop bend and by transfer of urea from ATLs to DTLs through the CC. Although the addition of urea to loops is ultimately dissipative, because the ATLs carry more urea out of the IM than the DTLs carry in (detailed results not shown), urea entry into DTLs nonetheless results in the advection of fluid that is hyperosmotic, relative to fluid in other structures, toward the papillary tip. This effect is made possible by the water-impermeable segments of the DTLs, which permit a decoupling of loop tubular fluid osmolality from interstitial osmolality.
Fluid flow rates along the CD and CC, and composite fluid flow rates along the DTL and ATL, per nephron, are shown in Fig. 5 E. As in the pipe mode, the distributed loop configuration yields a small DTLs-to-CD flow ratio at the papillary tip, where the longest loops turn. Na +, urea, and NR flow rates along the CD are shown in Fig. 5 F. As in the pipe mode, a high urea concentration is maintained in the CD via a vigorous active NaCl absorption near the OM-IM boundary. However, for the SS mode, we used a higher urea inflow at the OM-IM boundary, and a lower Na + inflow, relative to the pipe mode, to ensure that sufficient urea was available to attain a urine osmolality closer to that attained in the pipe mode.
These base-case model results, for the pipe and SS modes, are generally consistent with our hypotheses (as described above in HYPOTHESES : TWO CONCENTRATING MODES ), with the exception that, in the model results, significant water is not absorbed from the LDL2.
Parameter Studies
By means of parameter studies, we investigated the impact of transmural transport properties, structural assumptions, CD inflow rate, and our CC configuration on simulated urine osmolality, urine flow, and the free water absorption rate (FWA). FWA, which 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 ( 18, 76 ), is given by
where F urine, V is the urine flow rate, and U/P is the urine-to-plasma osmolality ratio; in our studies we assumed that the plasma osmolality is 310 mosmol/kgH 2 O ( 11 ). A typical value for FWA, for a moderately antidiuretic rat having a urine osmolality of 1,200 mosmol/kgH 2 O and a urine flow of 0.060 nl·min -1 ·nephron -1, is 0.17 nl·min -1 ·nephron -1 ( 52 ). Our base-case pipe and SS modes yielded FWA rates of 0.240 and 0.207 nl·min -1 ·nephron -1, respectively.
In some instances, we assess changes from base-case model urine osmolality by means of a change in relative percent osmolality increase along the IM. The relative percent osmolality increase is defined by
where U B is the model base-case urine osmolality (1,265 or 1,031 mosmol/kgH 2 O in the pipe or SS modes, respectively, as appropriate to context); U is the model urine osmolality obtained by a change in parameter value from its base-case value; and C OM is the osmolality of tubular fluid entering the IMCD from the outer medullary collecting duct (OMCD) at the OM-IM boundary (i.e., at x = 0), which is 673.1 mosmol/kgH 2 O. The relative increase given by Eq. 5 provides a measure of osmolality change that is relative only to the base-case increase in osmolality produced along the IM; this measure is appropriate because we assume that the OM always provides a fixed CD tubular fluid osmolality at the OM-IM boundary, and the OM increase above systemic blood plasma osmolality should not contribute, in our parameter studies, to comparisons of osmolality change along the IM. In subsequent text, changes in this measure are called "relative" increases (or decreases or changes) in urine osmolality (or concentrating capability).
LDL3 water permeability. Based on perfused tubule studies in rats ( 14 ) and other mammals ( 57 ), the rat DTL has usually been assumed to be highly water permeable up to the loop bend, or until the beginning of the prebend segment (39, 64, 72; for an exception, see Ref. 66 ). Indeed, DTL water permeability was considered as an essential element of the passive mechanism ( 31, 63 ): as in our pipe mode, water absorption from the DTL was hypothesized to raise the NaCl concentration of tubular fluid entering the ATL and thus promote a larger transepithelial gradient favoring NaCl absorption from the ATL.
We investigated the impact of LDL3 osmotic water permeability by varying it from 0 to 1,000 µm/s; results are shown in Fig. 6, Aa - Ac. In the pipe mode, as LDL3 water permeability increased, urine osmolality increased monotonically from 1,198 to 1,298 mosmol/kgH 2 O. Thus the pipe mode does not require LDL3 water permeability, but relative osmolality change (as defined by Eq. 5 ) was reduced 11% when LDL3 was made water impermeable. Reckoned from the base case of 1,265 mosmol/kgH 2 O obtained at a water permeability of 400 µm/s, relative osmolality change was increased 5.6% from base case when water permeability was increased to 1,000 µm/s. The increased osmolality arises from increased water absorption from LDL3. Because the LDL3 tubular fluid osmolality is hyposmotic relative to the CC, more water was absorbed as the LDL3 became increasingly more water permeable. The absorption of water increased DTL tubular fluid Na + concentration and promoted increased rates of loop-bend Na + absorption. However, increased LDL3 water permeability also resulted in reduced urine flow rates: water absorbed from the LDL3 reduced CC urea concentration and increased urea absorption from the CD, and this urea absorption was accompanied by water absorption from the CD. A LDL3 water permeability of 1,000 µm/s resulted in a urine flow rate of 0.075 nl·min -1 ·nephron -1, a 3.8% decrease. However, FWA decreased little relative to base case, because of the increase in urine osmolality.
Fig. 6. Parameter studies for water permeability of LDL3 ( column A ) and for and Na + permeabilities of LDL3 ( column B ) and ATL ( column C ). Effects are evaluated in terms of urine osmolality ( row a ), urine flow rate ( row b ), and free water absorption rate (FWA; row c ). Solid curve, pipe mode; dashed curve, SS mode;, base-case value for pipe mode;, base-case values for SS mode.
In the SS mode, as LDL3 osmotic water permeability increased from 0 to 1,000 µm/s, urine osmolality decreased to 799 mosmol/kgH 2 O, a 65% relative decrease (as defined by Eq. 5 ). In contrast to the pipe mode, tubular fluid osmolality in the LDL3 is hyperosmotic to CC fluid in the base-case SS mode. Thus as LDL3 water permeability increased, water entered the LDL3, diluted its tubular fluid, and thus reduced Na + absorption near the loop bend. Consequently, urine osmolality and FWA decreased, whereas urine flow rate increased.
LDL3 and ATL Na + permeabilities. As we previously noted, in both base-case modes high rates of Na + absorption from loops of Henle are localized near loop bends ( Figs. 4 C and 5 C ). This absorption pattern arises from our specification of loop Na + permeability: to conform to evidence from our immunohistochemical localization studies, LDL3 Na + permeability was set to zero, and the Na + permeabilities of the model prebend segment and ATL were set to a high value of 80 x 10 -5 cm/s ( 14 ). Model results, as LDL3 Na + permeability varied from 0 to 20 x 10 -5 cm/s, are shown in Fig. 6, Ba - Bc. As LDL3 Na + permeability increased, more Na + was absorbed from LDL3, resulting in decreased Na + delivery to the loop bend and thus reduced tubular fluid Na + concentration near the loop bend and along the ATL. As Na + absorption from the LDL3 increased, urine osmolality and FWA decreased, and urine flow increased. These results suggest that Na + absorption from the LDL3 is less effective than Na + absorption from the deeper, loop-bend region.
The effect of variation in ATL Na + permeability is shown in Fig. 6, Ca - Cc. As permeability increased from 1 to 150 x 10 -5 cm/s, urine osmolality increased, FWA increased, and urine flow decreased, effects due to more vigorous Na + absorption around the loop bend and more nearly complete Na + concentration equilibration with the CC along the ATL. However, when ATL Na + permeability is sufficiently high (higher than the base-case value), additional increases have a diminishing effect on the rate of urine osmolality increase.
LDL3 and ATL urea permeabilities. The effect of varying urea permeabilities in LDL3 and ATL is shown in Fig. 7. Figure 7, Aa - Ac, shows the impact of increasing LDL3 urea permeability from 1 to 200 x 10 -5 cm/s while keeping all other urea permeabilities at base-case values; Fig. 7, Ba - Bc, shows the impact of increasing ATL urea permeability from 1 to 200 x 10 -5 cm/s, while keeping all other urea permeabilities at base-case values; and Fig. 7, Ca - Cc, shows the impact of simultaneously increasing LDL3 and ATL urea permeabilities from 1 to 200 x 10 -5 cm/s.
Fig. 7. Parameter studies for urea permeability of LDL3 ( column A ) and ATL ( column B ), and for simultaneous variation of urea permeabilities in LDL3 and ATL ( column C ). Solid curve, pipe mode; dashed curve, SS mode;, base-case value for pipe mode;, base-case values for SS mode. SS model base-case permeabilities are not marked in column C because no point on the dashed curve can simultaneously represent the differing values used for the SS base case.
In the pipe mode, as one of the LDL3 or ATL urea permeabilities increased while the other remained at base-case value, urine osmolality and FWA decreased sharply, whereas urine flow rate increased. When the urea permeability of the LDL3 or ATL was set to 20 x 10 -5 cm/s, relative urine osmolality decreased by 88 or 87% to 745 and 756 mosmol/kgH 2 O, respectively; further increases in urea permeability further reduced urine osmolality, but at a slower rate. The increased urea permeabilities increased urea secretion into the LDL3 and the ATL; this flux reduced CC urea concentration but increased loop urea concentration. The secretion of urea in the loop tends to reduce CC osmolality, and when LDL3 urea permeability was increased, the secreted urea tended to be sequestered in the ATL (which has low urea permeability), thus increasing the osmolality of ATL fluid, which, in turn, tends to dissipate the concentrating effect by carrying fluid toward the OM that is not sufficiently dilute. When ATL urea permeability was increased, secretion of urea into the ATL was similarly dissipative.
When LDL3 and ATL urea permeabilities were simultaneously increased, urine osmolality showed an initial dramatic reduction: a urine osmolality of 711 mosmol/kgH 2 O was obtained for urea permeabilities of 20 x 10 -5 cm/s. However, as the urea permeabilities were further increased, urine osmolality increased as the model configuration more nearly approximated the SS mode; for urea permeabilities of 200 x 10 -5 cm/s, an osmolality of 927 mosmol/kgH 2 O was attained. In this limit, the entry of urea results in loop-bend fluid having an osmolality exceeding that of the adjacent CC, and along the ATL, the high urea permeability supports the effective absorption of urea and limits its dissipative effect.
In the SS mode, when LDL3 or ATL urea permeability was reduced while the the other was kept at its base-case value, the model yielded lower urine osmolality, lower FWA, and a higher urine flow rate. Urine osmolalities of 677 and 628 mosmol/kgH 2 O were obtained for LDL3 and ATL urea permeabilities of 1 x 10 -5 cm/s, respectively, which correspond to relative osmolality decreases of 99 and 113%, reductions which indicate that the IM osmolality gradient was abolished at these low-permeability limits.
Compared with the pipe mode base-case urine osmolality, however, the SS mode base case is less sensitive to variations in LDL3 and ATL urea permeabilities, inasmuch as small changes in urea permeability resulted in less change in osmolality than that observed in the pipe mode's dramatic decrease. In the SS mode, the concentrating effect is generated, in part, by urea secretion into the LDL3, which, owing to the decoupling of the osmolality of the water-impermeable LDL3 from that of adjacent tubules, raises the osmolality of LDL3 fluid. A high ATL urea permeability, coupled with high Na + permeability, results in a rapid osmotic equilibration of ATL tubular fluid with the CC (ATL fluid was hyperosmotic relative to the CC at the loop end). Thus lower LDL3 or ATL urea permeabilities resulted in less concentrated DTL tubular fluid at the loop bend, or in a more concentrated ATL tubular fluid, both of which reduced concentrating capability.
In the SS mode, when both LDL3 and ATL urea permeabilities were reduced simultaneously, osmolality showed an initial decline (to 765 mosmol/kgH 2 O for urea permeabilities of 20 x 10 -5 cm/s), but further reductions in these permeabilities resulted in increased urine osmolality (to 1,398 mosmol/kgH 2 O for urea permeabilities of 1 x 10 -5 cm/s), as the model configuration closely approximated pipe mode permeabilities. Indeed, for urea permeabilities of 1 x 10 -5 cm/s, urine osmolality (and the corresponding FWA) was higher than the base-case values for both modes, because of the more favorable boundary conditions used in the SS mode (see Table 2 ).
LDL3 length. For the base case, based on our tubular reconstructions, we assumed that the LDL3 and prebend segments made up the lower 60% of the those DTLs reaching 1 mm or more into the IM. To determine the effect of this configuration on model results, we varied the normalized LDL3 length from 0 to 1. At normalized length 0, the LDL2 was lengthened to replace the LDL3 and join the prebend segment; at normalized LDL3 length 1, the LDL3 was assumed to make up the whole IM portion of those DTLs (except for prebend segments) reaching at least 1 mm into the IM.
The effect of varying normalized LDL3 length is shown in Fig. 8, Aa - Ac. As normalized LDL3 length was increased from 0 to 1, concentrating capability increased in both modes. In the pipe mode, urine osmolality increased from 752 mosmol/kgH 2 O to the base-case of 1,265 mosmol/kgH 2 O at normalized length 0.6, and then to 1,527 mosmol/kgH 2 O, i.e., from a relative 87% decrease in concentrating capability to a relative 44% increase. However, as normalized LDL3 length increased from 0 to 1, urine flow decreased from 0.143 to 0.0615 nl·min -1 ·nephron -1. Combined with the effect of increasing osmolality, this resulted in an initial increase in FWA to a relatively unchanging value nearly equal to the base-case FWA, 0.240 nl·min -1 ·nephron -1. In the SS mode, as normalized LDL3 length increased from 0 to 1, urine osmolality increased from 698 to 1,248 mosmol/kgH 2 O, urine flow decreased from 0.109 to 0.0734 nl·min -1 ·nephron -1, and FWA increased from 0.136 to 0.222 nl·min -1 ·nephron -1.
Fig. 8. Parameter studies for LDL3 segment length ( column A ), maximum length of DTLs having LDL2 S segments ( column B ), and prebend length ( column C ). Solid curve, pipe mode; dashed curve, SS mode;, base-case value for pipe mode;, base-case values for SS mode. Negative prebend length indicates that the portion of the ATL that follows the bend has the physical characteristics of the LDL3 segment.
These results indicate that the LDL3 plays an important role in both modes. In the pipe mode, urea permeability is higher in LDL2 (13 x 10 -5 cm/s) than in LDL3 (1 x 10 -5 cm/s). Thus a longer LDL2 (i.e., a shorter LDL3) promotes more urea secretion into the DTL, which results in a near-osmotic equilibration with the CC by both water absorption and substantial urea secretion. As a consequence, the DTL Na + concentration increases less than it would if osmotic equilibration were mostly by water absorption, and urea is removed from the CC and sequestered in the ATL, which has a low urea permeability. Ultimately, the gradient favoring Na + absorption is reduced, and the urea retained in the ATL tends to be highly dissipative.
In the base case for the SS mode, osmolality of the LDL2 fluid is close to that of the interstitial fluid (see Fig. 5 A ) because of the high water permeability of the LDL2, whereas osmolality of the LDL3 fluid increases significantly above the adjacent CC, because the LDL3 is water impermeable but highly urea permeable. A shorter LDL3 resulted in the DTL carrying relatively less concentrated fluid toward the loop bend and in reduced urea entry. These effects tend to reduce urea entry in the DTL and delay equilibration with the CC urea concentration. When the LDL3 is completely absent, urea entry will occur only into the loop bend and the ATL; such entry is highly dissipative because it tends to raise the osmolality of ATL flow.
Maximum length of pure AQP1-null DTLs. In both modes, we have assumed that the DTL of a loop of Henle that turns within the first millimeter of the IM is entirely AQP1 null and that its DTL, before the prebend (i.e., the LDL2 S ), is impermeable to Na + and has either low (pipe mode) or moderate (SS mode) urea permeability. We investigated the effect of these pure AQP1-null DTLs by varying this maximum length (or "cutoff" length) of the AQP1-null DTLs from 0 (no pure AQP1-null DTLs) to 5 mm (all DTLs are AQP1-null). The results are shown in Fig. 8, Ba - Bc.
For the pipe mode, the percentage range of relative variation in urine osmolality (based on the measure given by Eq. 5 ), as the cutoff length of the AQP1-null DTLs was varied from 0 to 5 mm, was 23%; the ranges of the variations in urine flow rate and FWA were 25 and 12%, respectively. These changes were nonmonotonic and too complex to easily interpret. As the cutoff length increased from 0, more loops had low urea permeability along their entire length [LDL2 S urea permeability is low (1 x 10 -5 cm/s) compared with LDL2 urea permeability (13 x 10 -5 cm/s)]. With less urea entering the DTLs, a high interstitial urea concentration (and a corresponding low interstitial electrolyte concentration) was sustained. However, the water-impermeable LDL2 S allows no water absorption that would raise the Na + concentration before tubular fluid reaches the prebend segment. These two competing factors resulted in maximum urine osmolality (and minimum urine flow rate) at a cutoff length of 3.75 mm; a further increase in the cutoff length resulted in decreased urine osmolality, but despite this decrease, urine flow rate increased sufficiently to result in increased FWA.
For the SS mode, the percentage range of relative variation in urine osmolality (based on the measure given by Eq. 5 ) was 37%; the ranges of variations in urine flow rate and FWA were 10 and 11%, respectively. As the cutoff length increased from 0, more DTLs had moderate permeability (13 x 10 -5 cm/s) before their prebend segments, which resulted in lower urea concentrations at the loop bends compared with base case. Thus urea continued to enter along the ATLs, and this entry produced a dissipative effect that decreased urine-concentrating capability.
We conclude from these results that our model study does not indicate, in the context of UCM function, a clear rationale for those loops of Henle having water-permeable segments in the DTL.
作者单位：1 Department of Mathematics, University of North Carolina, Chapel Hill 27759-3250; 3 Department of Mathematics, Duke University, Durham, North Carolina 27708-0320; and 2 Department of Physiology, College of Medicine, University of Arizona, Tuscon, Arizona 85724-5051