## Abstract

We have developed a highly detailed mathematical model for the urine concentrating mechanism (UCM) of the rat kidney outer medulla (OM). The model simulates preferential interactions among tubules and vessels by representing four concentric regions that are centered on a vascular bundle; tubules and vessels, or fractions thereof, are assigned to anatomically appropriate regions. Model parameters, which are based on the experimental literature, include transepithelial transport properties of short descending limbs inferred from immunohistochemical localization studies. The model equations, which are based on conservation of solutes and water and on standard expressions for transmural transport, were solved to steady state. Model simulations predict significantly differing interstitial NaCl and urea concentrations in adjoining regions. Active NaCl transport from thick ascending limbs (TALs), at rates inferred from the physiological literature, resulted in model osmolality profiles along the OM that are consistent with tissue slice experiments. TAL luminal NaCl concentrations at the corticomedullary boundary are consistent with tubuloglomerular feedback function. The model exhibited solute exchange, cycling, and sequestration patterns (in tubules, vessels, and regions) that are generally consistent with predictions in the physiological literature, including significant urea addition from long ascending vasa recta to inner-stripe short descending limbs. In a companion study (Layton AT and Layton HE. *Am J Physiol Renal Physiol* 289: F1367–F1381, 2005), the impact of model assumptions, medullary anatomy, and tubular segmentation on the UCM was investigated by means of extensive parameter studies.

- kidney
- countercurrent multiplication
- countercurrent exchange
- NaCl transport
- urea transport

concentrated urine, i.e., urine having an osmolality exceeding that of blood plasma, is produced by the absorption of water, in excess of solute, from the medullary collecting ducts (CDs). In the outer medulla (OM), water absorption from CDs is driven by vigorous active transport of NaCl from the thick ascending limbs (TALs) into the interstitium; at each medullary level, this transport results in an osmolality difference between the TALs and other medullary structures. This difference, frequently called the “single effect,” is augmented (or “multiplied”) by the countercurrent flow configuration of renal tubules and vasa recta, to generate a substantial osmolality gradient along the corticomedullary axis, a gradient that is believed to be common to all OM structures. In the inner medulla (IM), however, the means by which water is absorbed from CDs remains undetermined (20, 67).

Substantial effort has been directed to constructing mathematical models of the mammalian urine concentrating mechanism (UCM) (27, 44). None of these models (including the one proposed in this study) has incorporated more than one fully realized spatial dimension, viz., the dimension corresponding to the corticomedullary axis. Indeed, in some model studies interstitial solute concentrations have been assumed to be uniform at each medullary level; i.e., tubules (and blood vessels, if represented) were assumed to interact with each other through a common surrounding medium in which solute concentrations varied only along the corticomedullary axis (47, 49, 60, 72, 73, 85). However, this assumption appears to be inconsistent with anatomical studies. A number of investigators, notably Kriz and colleagues (32), have reported that the medullary organization of tubules and vessels is highly structured in a number of mammals including rats and mice (31, 34, 36). Descending vasa recta (DVR) and ascending vasa recta (AVR) form tightly packed vascular bundles that appear to dominate the histotopography of the OM, especially in the inner stripe (IS).

Throughout the OM, CDs are found distant from vascular bundles, whereas the loops of Henle are positioned nearer the bundles. The structural organization is believed to result in preferential interactions among tubules and vasa recta, interactions that may contribute to more efficient countercurrent exchange or multiplication, to urea cycling and IM urea accumulation, and to sequestration of urea or NaCl in particular tubular or vascular segments (5, 6, 26, 33, 35, 52, 77, 86). (We will say that a solute is “sequestered” when its fractional contribution to local fluid osmolality substantially exceeds its fractional contribution to blood plasma osmolality.)

Several investigators have sought to represent aspects of three-dimensional medullary structure in mathematical models of the UCM. Knepper et al. (28) and Chandhoke and Saidel (12) separated the vasculature and the tubules of the OM into two separate compartments. More recently, Wexler et al. (86, 87) developed a model (the “WKM” model) that represented a very substantial degree of structural organization by means of weighted connections between tubules and vessels. Although the WKM model was formulated primarily to investigate the UCM of the IM, OM function played a large role in both the original (86) and subsequent WKM studies (77, 81).

In the present study, we describe a new, highly detailed mathematical model for the UCM in the OM of the rat. To represent the radial distribution of tubules and vasa recta, with respect to the vascular bundle, the model uses a “region-based” configuration: the model has four concentric regions, centered on a vascular bundle, and radial structure is incorporated by assigning appropriate tubules and vasa recta (or fractions thereof ) to each concentric region. The vascular bundle is contained within the two central-most regions, most loops of Henle are assigned to a third region, and CDs are assigned to a fourth. The methodological framework, mathematical formulation, and numerical solution techniques for the region-based configuration were developed and tested in a prototype model that used two concentric regions (40).

By means of the region-based model, we aim to represent, to a high degree of accuracy, both qualitative and quantitative aspects of medullary function. Moreover, we seek to incorporate and evaluate experimental findings not previously included in models, including the implications of recent immunohistochemical localization experiments. In a companion study (41), we used the region-based model to investigate, by means of parameter studies, the impact of a number of model assumptions relating to medullary structure and tubular segmentation. The present study will hereafter be frequently called “*study I*”; the companion study (41) will be called “*study II*.”

## GLOSSARY

### Acronyms

- AVR
^{1}/DVR - Ascending vas rectum/descending vas rectum
- CD
- Collecting duct
- IM/OM
- Inner medulla/outer medulla
- IS/OS
- Inner stripe/outer stripe
- LAL/LDL
- Long ascending limb/long descending limb
- LAV/LDV
- Long ascending vas rectum/long descending vas rectum
- LAVa/LAVb
- LAV in R1/LAV in R2
- LPST/SPST
- PSTs of long loops/PSTs of short loops
- PST
- Proximal straight tubule
- R1, R2, R3, R4
- Concentric regions 1–4
- SAL/SDL
- Short ascending limb/short descending limb
- SAV/SDV
- Short ascending vas rectum/short descending vas rectum
- SAVa/SAVb
- SAV in R3/SAV in R4
- SDLa/SDLb
- SDL with a prebend segment/SDL without a prebend segment
- SDL1/SDL2
- Second SDL segment
^{2}/third SDL segment

### Indices

*i*- Tubule, vas rectum, or concentric region (e.g.,
*i*= SDL, SDV, or R1) *k*- Solute
*k*= Na^{+}or urea

### Independent Variables

*x*or*y*- Medullary depth

### Dependent Variables

- C
_{i,k} - Concentration of solute
*k*in tubule, vas rectum, or concentric region*i* - F
_{i,v} - Water flow rate in structure
*i* *J*_{i,v}- Transmural water flux into structure
*i* *J*_{i,k}- Transmural flux of solute
*k*into structure*i* *Q*_{lava}/*Q*_{lavb}- Fluid accumulation carried away from R1/R2 by LAV
*Q*_{sava}/*Q*_{savb}- Fluid accumulation carried away from R3/R4 by SAV
*Q*_{r,r′}- Transregion capillary flow from region
*R*into*R*′ *Q*_{sdv}- SDV blood source

### Parameters

*A*- Fractional area for diffusion between region
*R*and*R*′ *A*_{i,r}- Interstitial area of region
*R* *A*_{i}- Cross-sectional area of tubule or vas rec-tum
*i* *D*_{k}- Diffusivity of solute
*k*in dilute aqueous solution *K*- Michaelis constant for tubule
*i*and solute*k* *L*- Axial thickness of OM
*L*_{c}- Axial cortex thickness used for Neumann boundary condition
*L*_{im}- Axial thickness of IM
*L*_{is}/*L*_{os}- Axial thickness of IS/OS
*N*_{b}- Number of loops of Henle per vascular bundle
*P*- Permeability of boundary between regions
*R*and*R*′ *P*_{i,f}- Osmotic water permeability coefficient of tubule or vas rectum
*i* *P*_{i,k}- Permeability of structure
*i*to solute*k* *V*- Maximum transport rate of solute
*k*by tu-bule*i* - V̄
_{w} - Partial molar volume of water
- d
_{i} - Product of V
_{w}and*P*_{i, f} *l*_{i}- Thickness of AVR endothelial walls
*n*_{i}- Number of structure
*i*per loop of Henle *r*- Average radial distance between two points in regions
*R*and*R*′ *r*_{i}- Inner radius of a tubule or vas rectum
*i* *w*_{sdv}/*w*_{sav}- Fraction of SDV/SAV reaching to a given medullary level
- Ω
- Diffusion resistance of interstitium
- α
_{lava}/α_{lavb} - Fraction of net fluid accumulation in R1/R2 drained by LAVa/LAVb
- α
_{sava}/α_{savb} - Fraction of net fluid accumulation in R3/R4 drained by SAVa/SAVb
- α
_{sav}(*x*) - Fraction of SAV fluid flux entering SAVa or SAVb originating at level
*x* - α
_{sdv} - Fraction of SDV fluid flux entering R1
- κ
_{i,r} - Fraction of tubule or vas rectum
*i*in contact with region*R* - ρ
_{i} - Fraction of the AVR endothelial walls that are open pores arising from fenestration
- φ
_{k} - Osmotic coefficient of solute
*k*

## MODEL FORMULATION

In this section we describe a “base-case” model configuration and a set of base-case model parameters, including physical dimensions, transport parameters, and boundary conditions. In addition, we briefly describe the numerical methodology used to obtain model solutions. Acronyms and symbols, for the manuscript body and the appendix, are given in the glossary.

### Configuration and Radial Organization

Figure 1, *A*–*C*, provides a schematic representation of the model configuration. Tubules and vasa recta are represented by rigid tubes that extend in space from *x* = 0 (corticomedullary boundary) to *x* = *L* (OM-IM boundary). The model includes two short loops of Henle that reach to the OM-IM boundary, a long loop of Henle that extends into the IM, continuously distributed vasa recta terminating or originating at all levels of the OM, a representative LDV and two representative LAV (LAVa and LAVb) that extend into the IM, and a CD. Because this study is concerned principally with the UCM of the OM, the model does not include an explicit representation of the renal cortex or of the IM; rather, the interaction of the OM with the cortex and the IM is represented implicitly via boundary conditions imposed on tubules, vessels, and compartmental spaces.

Radial organization with respect to a vascular bundle, as revealed in anatomic studies of the rat renal medulla (31, 36), is represented by means of four concentric regions. The portion of each concentric region that is exterior to both tubules and vasa recta represents merged capillaries, interstitial cells, and interstitial space. (When meaning is clear from context, this exterior compartment will be called a “concentric region” or, simply, a “region.”) At a given medullary level, each concentric region is assumed to be a well-mixed compartment with which tubules and vasa recta interact. To specify the relative positions or distributions of the tubules and vasa recta and to simulate the potential preferential interactions among them, each tubule or vas rectum is assigned to a particular concentric region, or, in some cases, fractions of a tubule or vas rectum are distributed to two concentric regions. Tubules and vasa recta that are in contact with different concentric regions are influenced by differing interstitial environments. However, tubules or vasa recta that do not contact the same region may still interact via interstitial diffusion of solutes around tubules and vasa recta; these diffusive fluxes are simulated by assigning nonzero solute permeabilities to the boundaries that separate concentric regions.

Model positions of tubules and vasa recta, based on descriptions in Refs. 21, 31, 36, and 52, are shown in Fig. 1, *A* and *B*, for the outer stripe (OS) and IS. Distinct populations of AVR were used to to avoid having AVR straddle region boundaries. Otherwise, unrealistic mixing between regions would be introduced, owing to the high effective solute permeabilities assumed for AVR. (This problem is not introduced by TALs, because they have relatively low permeabilities to the represented solutes and to water.) Most vasa recta that extend into the IM, i.e., those represented by the LDV and LAVa, are assumed to be situated centrally in the vascular bundle; thus they were assigned to the central-most concentric region, R1. However, a second population of long ascending vasa recta, LAVb, was assigned to R2, where, in the IS, LAV are reported to intermingle with short descending limbs (SDLs) along the bundle periphery (36). Short descending vasa recta (SDV) are assumed to straddle R1 and R2 in both OS and IS, consistent with immunolabeling results that indicate that DVR are distributed centrally within the vascular bundle (19), but also consistent with SDV peeling off to supply the capillary plexus of the IS. CDs, which are located distant from the vascular bundle, were assigned to R4. Distinct populations of short ascending vasa recta (SAV), SAVa and SAVb, ascend through R3 and R4, respectively. In the OS, the proximal straight tubules (PSTs), which are denoted SDL in Fig. 1*A* (see rationale below) and TALs from short loops of Henle (SALs) are near the CD; thus they straddle R3 and R4. This configuration changes near the OS-IS boundary: the proximal tubules become SDLs and leave their near-CD positions to take a course in the periphery of the vascular bundle, R2. In contrast, the SALs maintain a position neighboring the CD and are always distant from the vascular bundle. In the IS, both SDLs and SALs retain the position that they occupy at the transition from the OS to IS. Thus, the SDLs descend within the periphery of the vascular bundle near the SDV, whereas the SALs ascend distant from the bundle, in R3 and R4. At the beginning of the OS, the long loops of Henle are situated near the vascular bundle, in R2 and R3. At the transition of the OS to IS, the long descending limbs (LDLs) leave the near-bundle position and tend toward the CDs; they maintain a position apart from the vascular bundle, but between that bundle and the CDs, throughout the IS (36). In the model, each tubule or vessel may be considered to be a representative of a population; when a structure straddles a region, we aim to represent some individuals that may be fully in one or the other region as well as those that truly straddle a regional boundary. In particular, we note that in the IS, long ascending limbs (LALs) tend to form a ring about the vascular bundle that likely serves as a significant barrier to direct osmotic equilibration between the vascular bundle and less central structures; thus these LALs are assumed to straddle the boundary between R2 and R3; see Fig. 8 in Ref. 31.

### Tubules and Vasa Recta

In the rat kidney, loops of Henle turn at various medullary depths, with loop turns distributed from about the OM-IM boundary to the papillary tip (16). Because most short loops in the rat are believed to turn in the IS in a narrow band near the OM-IM boundary, it may suffice in a model of the OM to include one or two representative short loops and a single long loop. To represent population differences between short and long loops, their transmural fluxes into the concentric regions can be scaled by their respective number ratios.

In our model, we represent two short loops of Henle. The descending limb of one short loop (SDLa in Fig. 1) is structurally and functionally divided into four segments. The first segment corresponds to the PST, which, though strictly not a part of the SDL, we consider it so, in our figures, for notational simplicity. The second descending limb segment, which we call “SDL1,” is highly water permeable and moderately permeable to solutes, characteristics typically ascribed to SDL (67). The third segment, which we call “SDL2,” is a water-impermeable, but urea-permeable, portion of the SDL located in the IS; its transport properties are inferred from immunohistochemical localization studies (61, 80). The fourth segment, which we call the “prebend segment,” is a terminal portion of the descending limb that is given the diameter and transport properties of the TAL. The prebend segment is indicated in Fig. 1*C* by thicker lines along the terminal portion of SDLa. The other short loop of Henle (SDLb in Fig. 1) has no prebend segment and is divided into only three segments: the PST, SDL1, and SDL2. By adjusting the number ratios of the two types of short loops, the effects of a prebend segment can be predicted. Each SDL is contiguous with a SAL. Ultimately, all transmural loop solute and water fluxes are scaled relative to a single loop of Henle.

Because CDs undergo no coalescences in the OM (33), the CD system is represented by a single tubule; its transmural fluxes into R4 are scaled by the ratio of CDs to loops of Henle, i.e., per loop of Henle.

As previously noted, vasa recta that supply the IM are represented by three vessels, one for LDV and two for LAV. SDV and SAV, which are assumed to reach to differing OM levels, are represented by continuously decreasing distributions of vasa recta; distinct SAV distributions, SAVa and SAVb, are used for R3 and R4, respectively. Each distribution is constructed by formulating each dependent variable (e.g., concentration, flow rate, etc.) associated with a SDV or a SAV as a function of both the axial position and the medullary level at which the vas rectum terminates (a similar formulation has been used previously for loops of Henle) (42, 49). Transmural vasa recta fluxes are scaled by the number of vasa recta, descending or ascending, per loop of Henle. Capillary flow is represented implicitly, at each medullary level, by flows between regions that are computed on the basis of mass balance.

The transmural fluxes for each class of tubule or vessel are scaled per loop of Henle, as are transregional fluxes and flows. By means of this uniform scaling, the appropriate solute and water ratios needed for computing regional concentrations and osmolalities are preserved.

### Solutes

The model is formulated for two solutes, NaCl and urea. Na^{+} is the representative for NaCl, and in the CD, where KCl is likely present in significant concentration (84), Na^{+} represents both NaCl and KCl. The model predicts fluid flow, Na^{+} concentration, urea concentration, and fluid osmolality (arising from NaCl and urea) in the tubules, vasa recta, and concentric regions; fluid flow, solute concentrations, and osmolality are represented as functions of medullary depth along the corticomedullary axis.

### Equations, Boundary Conditions, and Numerical Method

The methodological framework for the region-based model configuration was developed and extensively tested in a prototype model that used two concentric regions (40). The equations for the present study (a 4-region model) that differ most significantly from the equations for the two-region model are summarized in the appendix;a comprehensive list of the equations for the four-region model is provided in a supplement titled “Complete Model Equations,” which is available at http://ajprenal.physiology.org/cgi/content/full/00346.2003./DC1. The model equations embody the principle of mass conservation of solute and water and represent transmural transport processes, which are described by single-barrier equations that approximate double-barrier transepithelial or transendothelial 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, except for AVR, where water flux is assumed to be pressure-driven advection through fenestrations. Boundary conditions prescribe flows and concentrations in tubules and vessels that enter or leave the OM at the corticomedullary boundary or at the OM-IM boundary.

The model equations were solved by means of the semi-Lagrangian semi-implicit (SLSI)-Newton method, which was originally developed to obtain steady-state solutions for central core models (37, 38, 39) and subsequently adapted for the prototype region-based model (40). The SLSI-Newton method has two stages: *1*) a dynamic stage, in which the SLSI method is applied to the dynamic formulation of the model to obtain an approximate steady-state solution; and *2*) a steady-state stage, in which a Newton-type solver is applied to the steady-state formulation of the model equations; the initial guess for this solver is the approximate steady-state solution computed in the dynamic stage, and the solution that is produced is a very accurate steady-state solution. Calculations were performed using FORTRAN programs, which were executed in double precision on a computer equipped with an Intel Pentium IV 3.2-GHz processor having 1 GB of RAM. A spatial discretization of 160 subintervals was used. Convergence and mass balance tests gave results similar to those reported in Ref. (40).

### Parameters for Tubules and Vessels

In mid-IS, Kriz (31) found the ratio of CDs to loops, which we denote by *n*_{CD}, to be 1/6.1, and we assume this value throughout the OM. Two-thirds of the loops of Henle are assumed to be short (31); thus, the number ratios of loop limbs, taken relative to one loop, are *n*_{SDL} = *n*_{SAL} = (n_{SDLa} = and n_{SDLb} = ) and *n*_{LDL} = n_{LAL} = . In cross-sections of mid-IS, Kriz found an average ratio of vessels to tubules of 1.5 (31); although this number may include some capillaries that generally run laterally, we assumed that they are all vasa recta. In addition, Kriz found an average of 71 nephrons and 11.5 CDs per vascular bundle, representing 153.5 tubular cross sections associated with a vascular bundle. Thus, a total number of 230 vasa recta may be inferred to be associated with each vascular bundle. Of these, an average of 84 vasa recta were in the vascular bundle, leaving 146 vasa recta external to the bundle; the external vasa recta are assumed to be all SAV. The intrabundle vasa recta were found to be ∼40% DVR and ∼60% AVR, although Kriz noted that they were difficult to distinguish. Because the AVR in the vascular bundle are LAV, we assumed 50 LAV per bundle. The remaining 34 DVR include both LDV and SDV. We assumed that of these 12 were LDV, based on a LAV/LDV ratio of 4 found at the base of the papilla (18); thus at mid-IS, we assumed 22 SDV, and, consequently, a ratio of SAV to SDV of ∼6.5. We assumed that the numbers of SDV and SAV decrease linearly with increasing IS depth, to 0 at the OM-IM boundary. From this assumption we inferred 44 SDV and 294 SAV at the OS-IS boundary.

In the OS, DVR form by arborization from larger vessels, and AVR coalesce to form larger vessels (33). Because we have no basis for the rates of these effects, the sizes of the larger vessels, or their transport properties, the vasa recta were represented as distinct, unbranched vessels throughout the OS. In addition, laterally running capillaries appear sparse in the OS (52). However, to implicitly include a small number of such capillaries, we assumed that ∼3% of the total SDV entering the OS from the cortex terminate to form capillaries within the OS; thus at the corticomedullary boundary we represented 45.4 SDV and 303 SAV.

From these assumptions we find a ratio of SDV to loops of *n*_{SDV} = 45.4/71 and of SAV to loops of *n*_{SAV} = 303/71. Similarly, *n*_{LDV} = 12/71 and *n*_{LAV} = 50/71. The fractional population weights of SDV and SAV are given by (1) where *L*_{OS} is the OS thickness, *L* is the OM thickness, and the IS thickness *L*_{IS} is *L* − *L*_{OS}. *L*_{OS}, *L*_{IS}, and *L* were assumed to be 0.6, 1.5, and 2.1 mm, respectively (25). Thus, for example, *w*_{SDV} (*L*_{OS}) = 0.971, or nearly 3% less than *w*_{SDV} (0) = 1. Other piecewise linear vasa recta distributions were examined in *study II* (41).

The diameters and transport properties of tubules and vasa recta are given in Table 1; with the exception of parameters for SDL2 (for which references are given in parentheses) and a few other cases specifically noted below, these parameters were given by, based on, or estimated from, sources cited in Ref. 40. As previously noted, we have considered Na^{+} in the CD as the representative for both NaCl and KCl. Although significant amounts of NaCl may be actively absorbed from the OMCD, this electrolyte flux may be nearly balanced by KCl secretion, as suggested by calculations in Ref. 84; therefore, we have assumed that the maximum active Na^{+} absorption rate for the CD is zero (i.e., *V*= 0 for the CD). The CD urea permeability, reported (in 10^{−5} cm/s) to be 3.5 ± 0.2 (68), was reduced to 1.0 to prevent excessive urea loss from the CD (see results).

Effective AVR Na^{+} and urea permeabilities (solutes designated by *k* = Na^{+} or *k* = urea), defined by and were computed by setting fenestration fractions ρ_{LAV} and ρ_{SAV} to 0.01; setting Na^{+} and urea diffusivities *D* and *D*_{urea} to 1.5 × 10^{−5} cm^{2}/s and 1.38 × 10^{−5} cm^{2}/s, respectively, their self-diffusion coefficients in dilute aqueous solution (54, 83); and setting the AVR wall thicknesses *l*_{LAV} and *l*_{SAV} to 0.2 μm. The fenestration fractions and the wall thicknesses were based on generic properties of vascular endothelium (71) as well as the need for a sufficiently large permeability to limit washout of the medullary gradient. Note that the critical parameter is the ratio of the fenestration fraction to the wall thickness.

As was previously described, one model SDL (designated SDLa) was structurally and functionally divided into four segments; the other (SDLb) into three. The first SDL segment, which corresponds to the PST, terminates at the OS-IS boundary (i.e., at *x* = 0.2857*L*). The second segment, SDL1, extends into the IS and terminates at *x* = 0.7*L*. The third segment, SDL2, spans ∼40% of the IS. If associated with the SDL having a prebend segment (SDLa), the SDL2 segment terminates at *x* = 0.975*L*; otherwise (SDLb), it terminates at *x* = *L*.

The osmotic coefficients φ_{k} were set to 1.84 for NaCl and 0.97 for urea (83). The reflection coefficients for all solutes were set to 1 for all tubules and DVR (66). The partial molar volume of water V̄_{w} was taken to be 0.018136 cm^{3}/mmol for 37°C (see p. B-152 and F-5 in Ref. 83).

### Parameters for Regions

The cross-sectional area of the corticomedullary boundary, in Sprague-Dawley rats of ∼250 g, has been measured to be ∼2.0 cm^{2} (25). No similar measurement appears to be available for the IS, but an estimate for average IS cross-sectional area can be obtained as follows. Wistar rats averaging 339 g were found to have an IS volume of 0.157 cm^{3} (62). If one assumes an IS thickness of 1.5 mm, as was found in rats of varying body mass (25), one obtains an average cross-sectional area of 1.05 cm^{2}. However, this area should be scaled to account for the discrepancies in body mass. In Ref. 25, Fig. 13, the thick limbs of mid-IS were found to have an outer diameter of ∼28 μm in rats of ∼250 g. If we assume that the rat kidney has 30,000 nephrons (52, 65), then all thick limbs would occupy 0.1874 cm^{2}. In Ref. 62, the thick limbs in mid-IS were found to occupy 27% of the cross-sectional area, which suggests a total cross-sectional area of ∼0.68 cm^{2} in 250-g rats. The cross-sectional area decreases substantially from OS to IS mostly because in the OS the pars recta of long loops of Henle have a high degree of tortuosity and therefore occupy a disproportionate volume relative to that occupied by descending limbs of long loops in the IS (62).

For number counts of tubules at the mid-IS, we rely on Kriz (31), who examined white rats of ∼200 g. He reported an average number of loops per vascular bundle of 71, which indicates a total number of bundles of 423, if one assumes 30,000 loops. In conjunction with the cross sections determined above, this implies that a cross-sectional area associated with a vascular bundle at the corticomedullary boundary is ∼0.473 mm^{2}, whereas the corresponding area in the mid-IS would be ∼0.161 mm^{2}. These areas, if assumed circular, suggest disks having radii of ∼388 and ∼226 μm, respectively. An IS radius of 226 μm is consistent with a representative vascular bundle and associated structure in the photograph appearing as Fig. 8 in Ref. 31; that bundle, approximately circular, is in the lower center of the figure, ∼6.5 cm from the figure bottom. An effective radius for the total cross-sectional IS area associated with that bundle was found by computing the center-to-center distances from that vascular bundle to its six nearest neighboring vascular bundles, computing the mean of those distances, and then dividing that mean by two. This yielded an estimated radius of ∼228 μm.

The cross-sectional areas for the individual regions R1–-R4 associated with the vascular bundle were found by means of the tubule and vessel assignments to the OS and IS regions (as summarized in Fig. 1) and the percentage volume distribution of tissue types according to structure reported by Pfaller in Ref. 62 (p. 22–24; 38–39). (The details, which are extensive, are described in a supplement, “Model Structure,” available at http://ajprenal.physiology.org/cgi/content/full/00346.2003./DC1). From the areas for the regions, the radii, as measured from the vascular bundle center, can be computed. The radii *r*_{R} (*R* = R1, R2, R3, or R4) are given in in Table 2. Note that these radii are taken to apply to the OS at the corticomedullary boundary and to the mid-IS. The cross-sectional areas of each region may be found by calculating the area corresponding to a region’s outer radius and subtracting from that area the area associated with the outer radius of the next smaller region.

For an estimate of transregional permeabilities, we need the distances between the region midpoints of adjacent regions. The midpoint radii are *r*_{R1}/2, (*r*_{R2} + *r*_{R1})/2, (*r*_{R3} + *r*_{R2})/2, and (*r*_{R4} + *r*_{R3})/2. The differences between adjacent midpoint radii, denoted *r _{R,R′}* are given in Table 2, where

*r*

_{R1, R2}= (

*r*

_{R2}+

*r*

_{R1})/2 −

*r*

_{R1}/2,

*r*

_{R2, R3}= (

*r*

_{R3}+

*r*

_{R2})/2 − (

*r*

_{R2}+

*r*

_{R1})/2, and

*r*

_{R3, R4}= (

*r*

_{R4}+

*r*

_{R3})/2 − (

*r*

_{R3}+

*r*

_{R2})/2.

The area portions of the regions that correspond to interstitium are needed to estimate axial diffusion within regions and to estimate lateral diffusion from one region to another. We assumed that interstitium was distributed uniformly in the OS. In the IS, where most of the interstitium has been reported to reside outside the vascular bundle (52), we assumed that interstitium is five times more plentiful external to the bundle, i.e., in R3 and R4. Moreover, in the IS, we assumed that the fractional amounts of interstitium are equal within R1 and R2 and are also equal within in R3 and R4. The interstitial areas thus obtained are given in Table 2.

From the region interstitial areas and the areas of the whole regions (which can be obtained from the region radii, as described above), one can compute the fractional areas in each region occupied by interstitium. From these interstitial fractions, we estimated the fraction of the region interfaces available for interstitial diffusion (denoted *A*_{F,R,R′}) as the average of the fractional interstitial areas in adjoining regions *R* and *R*′. The fractions of the interfaces are given in Table 2.

The Na^{+} and urea permeabilities between adjoining regions *R* and *R*′ were estimated by *P*_{R,R′,k}= *A*_{F,R,R′}*D*_{k} /(*N*_{B}Ωτ*r*_{R,R′}), where *N*_{B} is the number of nephrons per vascular bundle, Ω is a diffusion resistance that represents effects of macromolecules and interstitial cells in the interstitium, and τ represents the effect of tortuosity on diffusion path length around tubules and vessels. The scaling by 1/*N*_{B} is needed because, by our convention, all fluxes are computed per nephron. The diffusion resistance is taken to be 4, based on experimental determinations for small molecules in cytoplasm and on theoretical considerations, which indicate that diffusivities in water are reduced by factors ranging from ∼2 to ∼6, depending on cytoplasmic viscosity (8, 55, 58). The path around the periphery of a tube is about π/2 greater than a direct path, and thus τ = π/2. Using values in Table 2 and values previously given, the permeabilities between regions were computed, and they are listed in Table 2. The model permeabilities were assumed to vary linearly, as a function of medullary depth, by interpolating between the permeabilities for the OS at the corticomedullary boundary and the permeabilities at the mid-IS. Some of these parameters will likely vary significantly from one bundle to another, as a function of shape and cross-sectional size, and others, especially the diffusion resistance, carry substantial uncertainty. Therefore, parameter studies for systematic scaling of the region permeabilities were conducted [see *study II* (41)].

Axial solute diffusion within regions was based on the diffusivities *D*_{k} , the diffusion resistance Ω, and the interstitial areas given in Table 2. These areas varied linearly as a function of medullary depth, by the same means as described for the permeabilities between adjoining regions.

As in prototype study (40), the relative positions of the tubules and vasa recta, shown in Fig. 1, *A* and *B*, as a function of medullary depth *x*, a fraction κ_{i,R}(*x*) of tubule or vessel *i* was designated to fall in region *R* = R1, R2, R3, and R4. The position transitions were represented by means of piecewise cubic polynomials (40).

The distribution of capillary flow in the medulla has not been characterized in experiments, so we made assumptions that appear to us to be physiologically reasonable; a schematic diagram of the distribution scheme is given by Fig. 2. Capillary source flow from terminating SDV, *Q*_{SDV}, was distributed into R1 (5%) and R2 (95%); thus, the fraction of *Q*_{SDV} entering R1, denoted α_{SDV}, was 0.05. (A subsequent calculation indicated that model results are insensitive to α_{SDV}: when α_{SDV} was set to 0.0, so that all capillary source flow entered R2, changes in base-case model results were small: CD osmolality at *x* = *L* was reduced by <1.3%.) 95% of the net fluid accumulation in R1 was assumed to taken up by LAVa (α_{LAVa} = 0.95); the remaining 5% was assumed to flow by sparse capillaries or through interstitium into R2, a flow denoted by *Q*_{R1, R2}. In contrast, 95% of the fluid accumulation in R2 was assumed to be carried directly to R3 by capillaries originating in the vascular bundle, a flow denoted *Q*_{R2,R3}; the remaining 5% was assumed to enter LAVb (α_{LAVb} = 0.05). Of the fluid accumulation in R3, 50% was assumed to enter the SAVa (α_{SAVa} = 0.5), and the remainder entered R4 as *Q*_{R3,R4}. All fluid accu-mulated in R4 was assumed to flow into SAVb (α_{SAVb} = 1).

The fluid entering SAV was distributed among those originating at a level *x* and those SAV passing through *x* that originated at a level deeper than *x*. Thus, at the corticomedullary boundary (*x* = 0), 90% of the capillary flow was assumed to be drained by the SAVa (in R3) or SAVb (in R4) that originates at *x* = 0, and the remainder was evenly distributed among the SAVa or SAVb that originate at levels *x* such that 0 < *x* ≤ *L*. The fraction of the capillary flow drained by the SAVa or SAVb that originates at *x*, denoted α_{S}_{A}_{V}(*x*) was assumed to increase linearly, with increasing medullary depth, from 0.9 to 1 at the OM-IM boundary.

### Boundary Conditions

The boundary concentrations and water flows for descending limbs and DVR at the corticomedullary boundary (*x* = 0) are given in Table 3, as are urine solute concentrations and the urine flow rate. The osmolality of descending limb and DVR flow was assumed to have a typical value of about 309 mosmol/kgH_{2}O (15). Urea concentration in descending limb flow was taken to be 15 mM at *x* = 0, based on extrapolation from late proximal fluid urea concentration (1, 14) (which has urea concentration ∼ of plasma concentration), and NaCl was assumed to make up the remainder of descending limb flow osmolality. Urea concentration in DVR flow was taken to be 8 mM, similar to blood plasma urea concentration (1), and NaCl was assumed to make up the remainder of DVR flow osmolality. Boundary flow for SDL, based on late proximal tubule flow, was taken as of SNGFR (1), which was assumed to be 30 nl/min, whereas for LDL, SNGFR was taken to be 36 nl/min (64). Flow in individual vasa recta, 8 nl/min, was based on measurements in long vasa recta of the rat IM (9) and is similar to the 7.5 nl/min used in the WKM model (86).

Boundary conditions for CD inflow were based on four assumptions. First, the CD water inflow rate at the corticomedullary boundary was prescribed at 6.1 nl·min^{−1}·CD^{−1}, or 1.0 nl·min^{−1}·nephron^{−1}, a middle ground between measurements in hydropenic rats, which indicate that accessible late distal tubule flow may be as little as 2.3 nl/min (1), and measurements that show that urine flow may be as little as ∼0.065 nl·min^{−1}·nephron^{−1} (see below). Second, the osmolality of CD fluid at the corticomedullary boundary is equal to blood plasma osmolality. Third, a fixed fraction of the urea that is delivered to the early distal tubule by the cortical TAL is absorbed in the cortex (an assumption also used in Ref. 86); this fraction was taken to be 0.25, based on a typical difference between early and late distal tubule urea flow as measured in hydropenic rats (1). Fourth, the water flow rate to early distal tubule equals the water flow rate in the corresponding TAL at the corticomedullary boundary. Fifth, no urea is absorbed from the cortical CD, which has been reported to have a low urea permeability (24). These assumptions suffice to determine the CD inflow Na^{+} and urea concentrations (40).

Urine outflow concentrations were based on an average of many antidiuretic states (1); NaCl was assumed to make up all nonurea solute. Urine flow per animal was taken to be 4 μl/min (1–3), or ∼0.065 nl·min^{−1}·nephron^{−1}.

Neumann boundary conditions, needed because of the inclusion of diffusion terms, were used for the concentric regions (see the appendix, the supplement “Complete Model Equations” at http://ajprenal.physiology/cgi/content/full/00346.2003./DC1, and the footnote near the end of section 2 in Ref. 40).

The LAL and LAV flow rates and concentrations at the OM-IM boundary (*x* = *L*) were computed by simulating the actions of an antidiuretic IM. The urine flow and solute concentrations were assumed known. By subtracting the water and solutes excreted in urine from the amounts entering the IM via tubular advection (in LDL, LDV, and CD) and via axial diffusive fluxes through the interstitial regions at the OM-IM boundary, water and solutes returned to the OM through the LAL and LAV were calculated.

The distribution of water and solutes between the LAL and LAV has not been well characterized. Based on measured permeability properties of long loops of Henle (67), we assumed that net water is absorbed in the IM from long loops of Henle, perhaps as much as 50% or more from the longest loops. However, most long loops of Henle extend only a short distance into the IM (16), where the rate of increase in the corticomedullary osmolality gradient is likely small relative to the rate of increase in the remaining deeper portion of the medulla (see data from Ref. 15 redrawn in Ref. 67). Thus it seems likely that the average fluid absorbed from an IM long-loop is 10–20% of LDL flow at *x* = *L*, and, in the model, a substantial percentage, 85%, of the LDL tubular fluid flow at the OM-IM boundary was returned to the OM via the LAL and the remainder via the LAV.

We assumed that LAL Na^{+} concentration is 5% lower at the OM-IM boundary than that of fluid returned via LAVa and LAVb, and that LAL urea concentration is 20% higher at the boundary than the LAVa and LAVb. From these assumptions, the LAL and LAV water flow rates and solute concentrations at the OM-IM boundary can be computed by means similar to that used in Ref. 40 (see supplement “Complete Model Equations” at http://ajprenal.physiology/cgi/content/full/00346.2003./DC1). Because urea makes up a smaller fraction of the solute than does NaCl, the relationship between LAL and LAV solute concentrations is consistent with a long-standing (but unverified) belief that the osmolality of LAL tubular fluid ascending through the IM is reduced relative to fluid osmolality in other structures at the same medullary level (67). The sensitivity of model results to these boundary conditions was evaluated in *study II (*41).

## RESULTS

Using the base-case configuration, parameter set, and boundary conditions, the model equations were solved to obtain steady-state solutions. Key results are displayed graphically in Figs. 3–5. Figure 3 shows axial osmolality, Na^{+} concentration, and urea concentration profiles in the concentric regions and in each class of tubule and vessel. Figure 4 shows corresponding water, Na^{+}, and urea luminal flow profiles; because this figure set portrays directed flows, flow toward the cortex is considered to be negative. Figure 5 is a schematic representation of water and solute fluxes from, or into, tubules and vessels.

In Figs. 3–5, the designation *A*, *B*, or *C* is used in a roughly analogous way: *A* designates either osmolality or water, *B* designates Na^{+}, and *C* designates urea. In Figs. 3 and 4, *0* corresponds to the regions themselves (i.e., to the portion of the region lying outside of tubules and vasa recta, but including capillaries and interstitium), and *1–4* indicate the principal region in which a structure is situated (a dashed line is used when a structure that is not from an indicated region is included in a figure panel for the purpose of comparison, e.g., the curve labeled “SALa” in Fig. 3*A2*, which is included for comparison with the curve labeled “SDLa”). For comparison of flow magnitudes in Fig. 4, a negative directed flow is sometimes shown, e.g., “-SALa” in Fig. 4*A2*. As described in model formulation, the SDV are represented by a continuous distribution, with SDV terminating at all depths of the R1-R2 boundary, and with SAV originating at all depths of R3 and R4. However, Figs. 3 and 4 contain only the profiles corresponding to the longest SDV, SAVa, and SAVb (i.e., those that reach to the OM-IM boundary).

In each panel of Figs. 3 and 4, the horizontal axis corresponds to the corticomedullary axis of the OM; the OS and IS subregions are separated on each panel by a vertical gray line. A horizontal gray bar corresponds to the extent of the SDL2 segment, and a black bar (which is contiguous with the gray bar) corresponds to the extent of the prebend segment that is included in one (SDLa) of the two SDLs. The curves marked “SDLa” in Figs. 3 and 4 represent results from the SDL that has a prebend segment (i.e., from SDLa).

### Osmolality and Concentration Gradients and Their Generation

Figure 3*A* shows an increasing osmolality gradient along the model’s corticomedullary axis in all regions, tubules, and vasa recta (except in the prebend segment); in the CD, the osmolality increases from 309 to 851 mosmol/kgH_{2}O, i.e., by a factor of 2.69. In R1, the core of the vascular bundle, osmolality increases from 311 to 716 mosmol/ kgH_{2}O, i.e., by a factor of 2.30. When active Na^{+} transport from the TALs and the prebend segments was eliminated, the osmolality gradients were also eliminated, except for small localized deviations, near the OM-IM boundary, which arise from the OM-IM boundary conditions (results not shown). Thus, the model, consistent with well-established evidence (67), indicates that the axial osmolality gradient of the OM is generated and maintained by active, outwardly directed transmural transport of NaCl from thick segments (recall that, in the model, the transport of each Na^{+} cation is assumed to result in the secondary active transport of an anion, assumed to be Cl^{−}).

In the model, the Na^{+} concentration of intratubular fluid in the prebend segment (represented in SDLa) and in the TALs (i.e., SALs and outer medullary LAL) are progressively reduced along the luminal flow direction by active Na^{+} transport. Luminal fluid osmolality is also reduced, because urea secretion into these segments is small, relative to Na^{+} absorption. Consequently, the luminal fluid that returns to the cortex via the TALs is significantly hyposmotic with respect to blood plasma (see profiles in Figs. 3 and 4, *rows 2* and *3*).

The absorbed Na^{+} (implicitly, absorbed NaCl) raises the osmolality of the interstitial fluid of the concentric regions. As a consequence, at most medullary levels water is continuously withdrawn from the luminal fluid of the water-permeable tubules and vessels, excepting AVR and a small portion of SDLs in the upper IS. (In AVR, osmolality is decreased, along the flow direction, by a combination of water secretion and diffusive solute absorption, processes that are integral to vascular countercurrent exchange; see below.) Thus the active transport of Na^{+} produces the essential single effect, i.e., the transverse osmolality gradient across the TAL epithelium, relative to other structures, that is multiplied by the counterflow configuration. These results are consistent with tissue slice and electron microprobe experiments, which show (or suggest) a typical increase in osmolality of about a factor of 2 or 3 from the corticomedullary boundary to the OM-IM boundary in the rat and rabbit (15, 23, 29, 88). These results are also consistent with other modeling studies, which have obtained axial osmolality increases of a factor of about 2 or 3 in most OM structures (43, 47, 60, 73, 75, 76, 77, 85, 86).

Figure 3, *A* and *B*, shows that at each medullary level, the Na^{+} concentration and osmolality in the concentric regions are highest in R3 (OS) or R4 (IS) and lowest in R1. The osmolality is higher away from the vascular bundle (i.e., higher in R3 or R4 than in R1 and R2) because the majority of IS TALs (all SALs and 50% of the LALs) are assumed to be situated away from the vascular bundle. Thus most Na^{+} transported from TALs is directed into R3 and R4. Although the majority of TALs are in R3, the osmolality in R3 is lower than in R4 in the IS. This paradoxical result arises because in the IS LDL fluid flow in R3 presents a higher load to the single effect than does fluid in R4, which principally contains the CD. We will use the word “load” to mean a descending tubular or vascular fluid flow that must be concentrated by the concentrating mechanism.

The urea profiles in Fig. 3*C* show a contrasting pattern in which the urea concentrations in R1 and R2 ultimately exceed the urea concentration in R3, and the urea concentration in R2 ultimately exceeds that in R4. These higher urea concentrations in R1 and R2, which are permitted by the isolation of the vascular bundle, arise from elevated urea concentrations entering in LAV from the IM, and from the transfer of much of that urea to the LDV, as is clear from Fig. 3, *C1* and *C2*, and from Fig. 4, *C1* and *C2*.

### Countercurrent Exchange by Vasa Recta

Because of sufficiently high DVR water and solute permeabilities, and because of a sufficiently large AVR fenestration fraction, vasa recta osmolality tends to track the osmolality and concentrations of the local interstitium (for LDV and LAV, compare Fig. 3, *rows 0*, *1*, and *2*; for SDV and SAV, compare Fig. 3, *rows 0*, *2*, *3*, and *4*). This tracking allows the vasa recta to serve the traditionally ascribed roles as countercurrent exchangers: at each level of the medulla, DVR flow (LDV or SDV) tends to have an osmolality or a solute concentration that is slightly below that of the surrounding region. Ascending flow (LAV or SAV) tends to have an osmolality or concentration that is slightly above that of the surrounding region. Thus the transport properties and the countercurrent anatomic arrangement of vasa recta allow them to return the solutes and water that are absorbed from the tubules back to the general circulation while at the same minimizing the impact of vascular flow on local medullary osmolality and concentrations, and, more generally, on the medullary osmolality and concentration gradients.

A comparison of Fig. 3, *row 1*, and Fig. 4, *row 1*, shows how the osmolality and concentrations in the long vasa recta are affected by both water and solute fluxes. Along the LDV flow direction, water is absorbed, whereas both Na^{+} and urea are secreted; along the LAV flow direction, water is secreted while both Na^{+} and urea are absorbed (except for Na^{+} in LAVa near the OS-IS boundary). Thus water and solute transport function cooperatively, through countercurrent exchange, to sustain the osmolality and concentration increases along the OM, as has been illustrated in qualitative descriptions (see, e.g., Fig. 18–4 in Ref. 21). However, a comparison of Fig. 3, *column C*, with *columns A* and *B* indicates that the urea concentrations increase more rapidly as a function of medullary depth than do the osmolality or Na^{+} concentrations and that this difference arises from more rapid changes in transmural urea fluxes deep in the IS (see Fig. 4*C1*). Indeed, the change in LAV urea flow is striking: from the OM-IM boundary to the corticomedullary boundary, urea flow decreases from −114 to −35.5 pmol/min in LAVa and to −21.7 pmol/min in LAVb, decreases of 69 and 81%, respectively. These results lend support to the hypothesis that the vascular bundles promote countercurrent exchange between LDV and LAV by bringing them in close proximity and by shielding them from other structures (35, 52). Moreover, these results exhibit urea sequestration in the long vasa recta and they exhibit urea cycling from the IM to the OM, and then back to the IM, via LDV.

As can be seen in Fig. 4, *row 0*, axial solute flows that are within the concentric regions, but external to tubules and vessels, have little impact on model results. Axial water flow external to tubules and vessels is prohibited by model design, and diffusive Na^{+} and urea flows, which are shown in Fig. 4, *B0* and *C0*, are typically two or three orders of magnitude smaller than other solute flows illustrated in Fig. 4. The external diffusive flows, which are proportional to the negative of the slopes of region concentration profiles, are generally toward the cortex, and they exhibit extrema where the concentration slopes change abruptly. Thus, extrema are found at the corticomedullary boundary, at the OS-IS boundary, at the onset of the SDL2, and at the OM-IM boundary. These results suggest that, in vivo, axial solute flows arising from diffusion do not contribute to a significant dissipation of axial osmolality and solute gradients. Additional support for this conclusion is supplied in *study II* (41).

### SDL Function: SDL2 Segment May Reduce IS Load on the Concentrating Mechanism

As shown in Fig. 3*A2*, SDL tubular fluid osmolality in the OS generally increases along the flow direction; although water and solute flows are both decreasing (Fig. 4, *A2*, *B2*, and *C2*), water flow decreases more rapidly than solute flows. In the upper IS, where regionalization is more complete, SDL fluid osmolality changes little, because the SDL and DVR present a significant load to the vascular bundle (R1 and R2), whereas the impact of NaCl absorption from TALs has more influence on R3 and R4, which have higher IS osmolalities.

In the lower IS, where the water-impermeable SDL2 segment prevents transepithelial osmotic equilibration, the SDL luminal fluid osmolality is nearly constant (see Fig. 3*A2*), owing to little net absorption of Na^{+} and urea (Fig. 3, *B2* and *C2*). The load that would have been presented by SDL fluid to the concentrating mechanism, if the SDL were significantly water permeable, has been removed, which helps promote a significant osmolality increase in R1 and R2 and the vascular flows contained therein. In *study II* (41), we find that, relative to an extension of the water-permeable SDL1 segment, a water-impermeable SDL2 segment results in significantly increased simulated osmolality in all regions near the OM-IM boundary. This finding suggests that, in vivo, a water-impermeable SDL2 segment will tend to promote the analogous effect.

A number of investigators (6, 26, 52, 79, 80), noting the descent of SDL in the periphery of the vascular bundles of the IS, have proposed diffusive urea secretion from vasa recta into SDL. Secreted urea would first flow with SDL fluid toward the OM-IM boundary; the urea would then flow up SALs (which have a low urea permeability of 1 × 10^{−5} cm/s) and return to the medulla by way of the distal convoluted tubules and cortical CDs. By means of immunohistochemical localization techniques, Wade et al. (80) have shown that the UT-A2 transporter is expressed in the SDL2 segment, and they have therefore hypothesized that the SDL2 segment is equipped for such urea cycling. Indeed, our results support urea cycling (see Fig. 4*C2*). Substantial urea entry is observed in the second half of the SDL2 segments (i.e., near the SDL2-prebend boundary), although 24% of that urea gain is balanced by a urea loss near the first half of the SDL2 segments (i.e., near the SDL1–SDL2 boundary). This loss arises because the DVR have lower urea concentration at the corticomedullary boundary than do SDLs: the DVR are assumed to have a urea concentration that equals blood plasma concentration, whereas the SDL urea concentration is based on experiments that indicate a tubular fluid-to-plasma urea concentration ratio in cortical late proximal tubule can range from ∼1.5 to 2.0 (1, 14, 15). A urea concentration difference resulting from our boundary assumptions at the corticomedullary boundary (see Table 3) of 15 mM urea in SDL and 8 mM in DVR (Table 3) is preserved through much of the OM as osmolality increases in both SDLs and vasa recta. Thus, when the SDLs become more urea permeable at the onset of SDL2, the transepithelial gradient favors urea absorption from SDL2. The preceding considerations suggest that the SDL2 segment contributes to the OM concentrating mechanism principally through a transepithelial osmotic disequilibration attributable to an absence of water transport capability.

### TAL Transport

In accordance with experiments by Garg et al. (13), the model TAL active Na^{+} transport rate is higher in the IS (25.9 nmol·cm^{−2}·s^{−1}) than in the OS (10.5 nmol·cm^{−2}·s^{−1}). As a result, the Na^{+} concentrations in the SAL and LAL decrease more rapidly along the flow direction in the IS than in the OS, despite a smaller TAL luminal diameter in the IS that increases luminal fluid flow speed (see Fig. 3, *B2*, *B3*, and *B4*); the corresponding Na^{+} flows also change more rapidly in the IS than in the OS (see Fig. 4, *B2* and *B3*).

Figure 3, *A2*, *B2*, *A3*, and *B3*, shows that the osmolalities and Na^{+} concentrations of SAL and LAL luminal fluid at the corticomedullary boundary (*x* = 0) are substantially below the boundary inflow values for the PSTs of the SDL and LDL at the same boundary. The osmolality of SAL outflow is 254 mosmol/kgH_{2}O, which is 18% less than in PST inflow, 309 mosmol/kgH_{2}O; Na^{+} concentration in SAL outflow is 122 mM, which is 24% less than PST inflow, 160 mM. LDL inflows are same as PST inflows for SDL, but LAL outflow osmolality and Na^{+} concentration are 227 mosmol/kgH_{2}O and 93.5 mM, which are 27 and 42% below inflow values, respectively. Differences of similar or greater magnitude have been reported in other modeling studies (47, 50, 60, 75, 77, 85).

### CD Urea Flow

CD osmolality, concentrations, water flow, and solute concentrations are shown in *row 4* of Figs. 3 and 4, along with corresponding profiles for SAVb. CD osmolality closely tracks R4 osmolality owing to sufficiently high transepithelial osmotic water permeability, but its Na^{+} and urea concentrations differ substantially from R4: CD tubular fluid urea concentration significantly exceeds Na^{+} concentration throughout the OM due to the corticomedullary boundary conditions (Table 3) and sufficiently low CD Na^{+} and urea permeabilities. Nonetheless, as indicated in Fig. 4, *B4* and *C4*, Na^{+} is secreted into the CD and urea is absorbed all along the OM. The secreted Na^{+} is principally supplied by the SALs and SAVb; the absorbed urea principally enters the SAVb.

The filtered load of urea per nephron in our model (weighted to include filtration into both short- and long-looped nephrons) is 256 pmol/min [computed from the boundary conditions (Table 3) and the assumption that one-third of SNGFR reaches the corticomedullary boundary]. Base-case model urea delivery to the distal convoluted tubule (DCT), which assumes negligible urea absorption from cortical TALs, is 186.7 pmol·min^{−1}·nephron^{−1}, or a fractional delivery 0.73. This delivery is somewhat below the range of 0.79–1.12 found in moderately antidiuretic rats by Armsen and Reinhardt (1), a range that was presumably based on early DCT flow in short-looped nephrons only.

As a consequence, in part, of our assumption that 25% of the urea reaching the DCT is absorbed from the DCT, the fractional urea delivery to the OMCD is 0.55, and the fractional delivery to the IMCD is 0.42. Our model’s delivery to the OMCD is consistent with the WKM model, which predicts a fractional delivery of 0.57 (77). In our model, urea absorption from the OMCD is a significant percentage (29% of urea entering the CD at the corticomedullary boundary) despite our use of a OMCD urea permeability of 1.0 × 10^{−5} cm/s that is smaller than a measured value in rat 3.5 × 10^{−5} cm/s (68). Reduced urea permeabilities have also been used in other investigations (43, 47, 75, 85, 86) [WKM used 0 (81, 86)], because the CD has been considered the principal conduit of urea delivery to the IM and urea delivery is essential for the effective operation of the passive mechanism proposed by Stephenson (72) and by Kokko and Rector (30). Weinstein’s (84) highly detailed, double-barrier epithelial model of the OMCD (84), which used parameters that approximated the measured urea permeability, predicted substantial urea absorption under a range of serosal urea concentrations. No explanation is offered here for the discrepancy of measured urea permeability with respect to the perceived need for substantive urea delivery to the IM, nor is it clear why model fractional urea delivery to the DCT appears to be subphysiological. However, in *study II* (41) we examine the consequences of active urea secretion into the PST of long-looped nephrons, as hypothesized by Bankir and Trinh-Trang-Tan (7).

### Transmural Fluxes, Mass Exchange, Mass Cycling, and Solute Sequestration

Figure 5 is a schematic representation of the predicted transmural water, Na^{+}, and urea fluxes from, or into, tubules and vessels; the arrow sizes are roughly proportional to the flux magnitudes. These fluxes can be related to (indeed, inferred from) the slopes of the flow profiles shown in Fig. 4; for example, a decreasing slope, relative to the flow direction, indicates absorption, and the larger the slope magnitude, the larger the absorption rate. In this subsection, we summarize the predicted fluxes, and their participation in cycling and sequestration, in a qualitative way, leaving quantitative terms such as “significant” or “small” undefined; this summary is best considered in conjunction with Figs. 4 and 5. In Fig. 4, the flows in regions (*B0* and *C0*) are scaled per nephron (assuming 30,000 nephrons); the flows in tubules and vessels are scaled per individual tubule or vessel.

The PSTs of both long and short loops lose a substantial fraction of their water flows, and much smaller fractions of their NaCl and urea flows; the water and solute are largely taken up by the SAV and returned to the general circulation. In the SDL1 segments, however, water flow is little changed, while solutes continue to be slowly absorbed. The SDL2 segments, as already noted, have no water fluxes, a small NaCl efflux, and a small net urea influx. The prebend segment differs from the SDL2 segment mostly by its larger NaCl absorption rate, arising from active transport. As LDLs traverse the IS, they lose a significant fraction of water and a small fraction of urea, but due to their proximity to TALs, significant NaCl enters by diffusion. Most of the water and solute absorbed in the IS from descending limb segments enter SAV.

The significant water absorption from the PSTs reduces the load presented to the concentrating mechanism and ultimately tends to reduce the load presented to the OMCD and IMCD. The concentration of LDL fluid, in part, by NaCl secretion, results in a degree of NaCl sequestration and not as much decrease in load as would be attained by water reabsorption. This seems contrary to the concentrating function of the IM, because more fluid must be concentrated, and it suggests that IM function may be aided by a higher LDL NaCl concentration, relative to urea concentration.

The TALs (LAL and SALs) are assumed to have no transepithelial water flux, but NaCl is vigorously absorbed, and small fractions of their urea flows are absorbed. Owing to our assumption of more vigorous NaCl transport in IS, more NaCl is absorbed, per unit length, in IS than in OS. Some of the absorbed NaCl enters LDLs and CDs, but most is taken away by SAV. The retention of urea in TALs is attributable to low tubular urea permeability (0.6 × 10^{−5} cm/s in IS and 1.4 × 10^{−5} cm/s in OS). Because the proportion of urea in TAL flow is raised substantially, relative to NaCl flow, urea can be said, by our definition, to be sequestered in TAL. Indeed, in the SALa the urea contribution to total osmolality increases from 5.3 to 9.3%, along the flow direction, from the OM-IM boundary to the corticomedullary boundary; in the LAL, the analogous urea contribution increases from 8.8 to 25%. These values are substantially larger than the 2.5% contribution of urea to blood plasma osmolality that we have assumed. [Some studies using the WKM model (77, 81) have shown substantial urea absorption from OM TALs, a result that is apparently attributable to the use of a TAL urea permeability of 4.5 × 10^{−5} cm/s, which significantly exceeds the values ranging from 0.6 to 1.4 × 10^{−5} cm/s that were found by Knepper (24) and which we have used.]

Along the OM, CDs lose ∼60% of their water flow, increase their NaCl flow by ∼20%, and reduce urea flow by more than one quarter. Because fluid entering the CD has a high urea concentration, compared with fluids in other descending structures, substantial urea is absorbed from the CD, despite its low urea permeability (1 × 10^{−5} cm/s). Absorbed water and urea enter SAVb, secreted NaCl comes from TALs. Notwithstanding the NaCl influx and urea efflux, substantial urea is sequestered in CDs, owing to effects in upstream segments, viz., the large amount of NaCl absorbed from TAL fluid and to the assumption that cortical segments of the distal nephron have only limited urea permeability.

Although only the longest SDV is shown in Fig. 4, all SDV have the same quantitative pattern, regardless of the depth reached: they lose substantial water, take up some NaCl, and take up substantial urea, especially in the inner portion of the IS, where the source of the most urea are the LAV.

The SAV have more heterogeneity, with respect to medullary depth reached, than do than SDV because SAV arise from the confluence of local capillary flow at each depth; a SAV is not merely the continuation of a SDV, and no constraint forces a SAV and a SDV that reach the same medullary depth to have matching flow magnitudes or matching solute concentrations. The longest SAVa and SAVb, in R3 and R4, respectively, show substantial water gain (along the flow direction) but substantial NaCl and urea loss, as shown in Fig. 4 (note differing scales in *rows 3* and *4*). The water that enters SAV as they originate, and as they traverse the OM, comes from descending limbs, CD, and SDV (through capillary fluxes). The NaCl that is absorbed from SAV enters CD and IS LDL. Urea absorbed from SAV is largely passed to DVR in the OS. The cycling of fluid, via capillaires, from SDV to SAV tends to reduce load on deeper structures. The cycling of solutes from SAV to DVR in the OS tends to trap solute in the OM and preserve the solute and osmotic gradients. Particularly noteworthy is the urea trap deep in the OM, a trap which may help confine urea to the IM. Transendothelial LDV and LAV fluid and solute fluxes have the same directions as do corresponding SDV and SAV fluxes. However, LDV and LAV fluxes consist mostly of exchange within regions R1 and R2.

The region portions exterior to the tubules and vessels (i.e., the interstitia of the regions) hardly have enough volume to speak of solute sequestration. However, the water that flows radially outward (at steady state) from region to region, at each level, will carry with it the solute concentration from the region of origin; in addition, solute can diffuse across region boundaries. Thus R3 and R4 may lose some NaCl to R1 and R2, and deep in the OM, R1 and R2 may lose some urea to R3 and R4, but these losses do not prevent the emergence of differing osmolality and concentration profiles in the regions.

## DISCUSSION

We have developed a highly detailed mathematical model for the UCM in the OM of the rat kidney. The model represents radial organization of renal tubules and vessels, with respect to vascular bundles, by means of a region-based configuration. Transmural transport by tubules and vessels is approximated by single-barrier expressions that summarize experimental results for osmotically driven water fluxes, solute diffusion, and active solute transport. The model, which was solved to steady state, predicts, in all represented structures, the concentrations of sodium and urea, the osmolality arising from NaCl and urea, intratubular (or intravascular) flow rates of water and solutes, and transmural fluxes of water and solutes.

In most structures, the model attains an osmolality increase of about a factor of 2.5 along the OM (see Fig. 3, *A0*–*A4*), an increase consistent with tissue slice experiments in the rat (15, 88), rabbit (23), and dog (78), and with electron microprobe experiments in the rat (29). Moreover, as predicted by many investigators (6, 21, 26, 32, 35, 53, 63, 77, 79, 86), the model indicates that the marked radial organization of the OM, which is centered on the vascular bundles, has a significant impact on the interstitial concentrations seen by the various tubules and vessels; most notably, the sequestration of urea in and near the vascular bundles, and a focusing of TAL-concentrating capacity on LDLs and CDs (Fig. 3).

Although our model makes a number of predictions that differ from that of the WKM model (see below), the most marked effects of regionalization in the two models are similar and consistent. In both models, CD and LDL tubular fluid osmolality in the OM is significantly elevated relative to osmolalities in tubules and vessels in the vascular bundle. In both models, SDL osmolality increases relatively little in IS, although urea is secreted in the terminal SDL in both models. In addition, in both models, urea is sequestered in the vessels of the vascular bundle, especially near the OM-IM boundary (77).

### Context

Our model formulation owes much to conceptual region-based frameworks proposed by Lever and Kriz (35, 53), who illustrated concentric cylinders around the vascular bundle. Our model formulation is similar to, but distinguishable from, the mathematical models developed by Knepper et al. (28) and by Chandhoke and Saidel (12), which were among the first efforts to quantify regional heterogeneity in the renal medulla. The model by Knepper et al. (28) includes two “parallel systems” in the renal medulla, whereas that by Chandhoke and Saidel (12) includes two “subsystems” in the OM and a well-mixed IM. In both models, tubules and vasa recta were assigned to differing parallel systems or subsystems depending on their radial positions with respect to the vascular bundle. A given tubule or vas rectum interacted with the parallel system or subsystem to which it was assigned, and the interactions between structures assigned to differing parallel subsystems were mediated by blood flow between regions. In our model formulation, four regions allow for a more complete representation of regionalization, direct interaction between adjoining regions is represented by means of solute diffusion, tubules or vessels may straddle regions, the vasa recta of differing lengths terminate in a continuous distribution along the corticomedullary axis, and transmural transport properties are based on current morphological and physiological knowledge.

Recent studies of the effects of regionalization (77, 81) have been based on the highly innovative and influential WKM model, proposed in 1991 by Wexler et al. (86) to represent three-dimensional medullary structure. In the WKM model, radial organization is represented by means of weighted, direct connections between each tubule and nearby vessels, and solute concentrations of AVR fluid are assumed to equal local interstitial concentrations. The connection network varies as a function of medullary depth, and at a particular medullary level, the connection-based representation allows each tubule to be influenced by the differing concentrations in adjoining vessels, which are in turn influenced by other tubules and vessels. The local nature of the interactions allows this model configuration to simulate preferential interactions among tubules and vessels that are anatomically nearby, and implicit regionalization emerges from the localized interactions (86, 87).

Because the WKM model is the only recent and modern model, to our knowledge, that aspires to represent the role of three-dimensional structure in a comprehensive simulation of the OM, we compare our model results mostly with results from the WKM model. Our model differs from the WKM model in that tubules and vessels do not interact directly but, rather, through the merged capillaries and interstitial fluid that make up the portions of the concentric regions that are exterior to tubules and vasa recta. Moreover, within a region, tubule-tubule interaction is on a common footing with tubule-vessel interaction. Tubules or vessels may straddle boundary walls and interact with two regions simultaneously. Adjoining regions interact via capillary flows and direct diffusion. By varying the solute permeabilities of region boundaries, differing degrees of regionalization can be simulated. Also in our model, AVR solute concentrations are allowed to differ from local interstitial concentrations, thus allowing us to investigate the dependence of the corticomedullary osmolality gradient on AVR permeability [see *study II* (41)].

We make no claim about whether the WKM formulation or the region-based model formulation provides (or has the potential to provide) the better representation of in vivo function. Although interaction between adjoining structures may be nearly direct, as in the WKM formulation, homogeneity within regions, as in our model, may arise through the multiplicity of interacting structures, sufficiently rapid diffusion across distances on the order of tubular diameters, and mixing via capillary blood flow. Moreover, photographs of medullary cross sections showing plentiful instances of close tubule-tubule contact (see, e.g., Fig. 9a and others in Ref. 31), suggest that direct tubule-tubule interaction is just as likely as direct tubule-vessel interaction (only the latter was used in the WKM model); however, the connection-based WKM methodology could be modified to represent direct tubule-tubule interactions.

### Region-Based Model: Advantages and Innovations

We believe that the region-based model formulation is more intuitive than the WKM model and that its fundamental principles may be easier to understand. Moreover, relative to the connection-based WKM model, the region-based model may allow one to more easily study the impact of preferential interactions, because the degree of regionalization can be varied by changing solute permeabilities in region boundaries or by introducing additional concentric regions. Furthermore, even though our region-based formulation differs fundamentally from a connection-based formulation, our formulation has the potential to incorporate connection-based elements, in that it can be easily modified to also represent direct exchange, of varying degree, among tubules and vessels.

New computational methodology, developed in a series of studies (37, 38, 39, 40), was used to solve the region-based model. This methodology uses a combination of dynamic and steady-state methods to rapidly compute steady-state model solutions while maintaining numerical stability. The methods have been extensively tested to confirm mass balance and numerical convergence to solutions. Moreover, second-order convergence to model solutions is preserved even when transmural transport properties change abruptly and discontinuously (39). Such abrupt changes are suggested by instances where cell type changes abruptly, as in the change from (thin) SDL cells to (thick) TAL cells and in the transition to the aquaporin-1 (AQP1) null descending limb segment (called SDL2 by us) found in the IS in some short loops of Henle (61, 80).

Our model is also distinguishable from previous models of the OM (e.g., Refs. 43, 47, 59, 74, 75, 76, 85, 86) in its representation of the SDL2 segment and of spatial inhomogeneities in TAL luminal radius and in active transport. The TAL lumen is assumed to increase in radius, with decreasing medullary depth, and to decrease in active transport rate, as suggested by experimental studies (13, 25). Moreover, the region-based OM model is made independent of any specific hypothesis for the IM concentrating mechanism function by means of boundary conditions that are based on the fundamental principle of mass balance and on literature values for urine flow and content. As is well known, the IM concentrating mechanism has not been definitively identified (27, 67); nonetheless, the effect of the IM on OM function can be investigated by varying the IM boundary conditions over plausible ranges [see *study II* (41)].

### Magnitude of TAL Active Transport

The values of the three principal parameters that quantify TAL active NaCl transport in our model (*V*_{max} , the Michaelis constant, and Na^{+} permeability) were given in Table 1. The values for *V*_{max} were based on in vitro determinations of ATP hydrolysis rate by the Na^{+}-K^{+}-ATPase transporter (13) and on the assumption that three Na^{+} ions are pumped per ATP hydrolized. The backleak permeability was based on experimental measurements that were analyzed by means of a simple mathematical model (57). The Michaelis constant value had been used before by us (50, 51) to obtain TAL concentration profiles consistent with tubuloglomerular feedback (TGF) function.

The TAL Na^{+} transport rates used in the WKM model were chosen to achieve TAL luminal Na^{+} concentrations of ∼30 mM at the corticomedullary boundary (77, 81, 86). This target concentration, which was correctly characterized in the first WKM study (see, e.g., Refs. 57, 69, 70) as an appropriate concentration for the late distal tubule (page F374 in Ref. 86), was apparently mistakenly applied to the corticomedullary boundary (pages F372 and F378 in Ref. 86), where WKM model boundary concentrations of 37.1 (LAL) and 31.1 mM (SAL) were reported. Our model parameters produced TAL luminal Na^{+} concentrations, at the corticomedullary boundary of our region-based model, of 93.5 (LAL) and 126 mM (SALa), concentrations that are somewhat less than, but consistent with, a corresponding concentration of 138 mM (SAL) arising in our model studies of TGF, which used a differing parameter ensemble (50).

We regard reasonable agreement between our region-based model and our TGF model to be of fundamental importance, because indirect experimental evidence and theoretical considerations indicate that the NaCl luminal concentration profile, from loop of Henle bend to the macula densa (MD), is monotonically decreasing to a concentration of ∼30 mM (57, 69, 70). Indeed, a marked decreasing slope in the NaCl concentration profile at the MD is required to ensure transmission of a TGF signal. For if NaCl transport in the OM were so vigorous that active NaCl absorption equaled diffusive NaCl backleak in TAL flow approaching the MD, thus resulting in no net transepithelial NaCl flux, then the luminal concentration profile near the MD would have reached “static head” (11), a condition of nearly constant NaCl concentration along a TAL portion near the MD. In such a case, variation in TAL luminal fluid flow would have no significant impact on Na^{+} concentration at the MD, and, consequently, no TGF signal would be generated. In general, one might reasonably expect that the corticomedullary boundary NaCl concentration in a LAL would be lower than that in a SAL, because a long-looped nephron would more likely have a shorter cortical TAL segment, inasmuch as such a long-looped nephron likely arises from a juxtamedullary glomerulus (33). Although a previous model study from our laboratory (47) reported NaCl concentrations in TAL luminal flow at the corticomedullary that appears to be unrealistically low, most other recent studies (43, 59, 74, 75, 85) are more nearly in agreement with the concentrations that we report for the region-based model (excepting results reported in Ref. 17 and in Fig. 4 in Ref. 75).

The TAL transport rates used in the WKM model, coupled with preferential interactions between TAL and CDs, results in CD luminal fluid osmolalities of 1295 and 1487 mosmol/kgH_{2}O at the OM-IM boundary in the first (86) and most recent versions (77) of the WKM model, respectively; as a consequence, the model CD carries into the IM a fluid having a high osmolality and a high urea concentration, relative to values predicted by our model and relative to those that are implied by likely physiologic TAL Na^{+} transport rates. WKM model results showing IM concentrating capability may be attributable, in part, to the high osmolality and solute concentrations (relative to upper IM values in other structures) conveyed by the CD to the IM.

### Role of Late SDL

Our model includes representation of a late SDL segment (SDL2) that appears, according to immunohistochemical localization experiments, to be water impermeable but significantly urea permeable (61, 80). For our base-case model parameters, we have assumed zero osmotic water permeability, a substantial urea permeability of 20 × 10^{−5} cm/s, and a Na^{+} permeability (as in our model TALs) of 1.1 × 10^{−5} cm/s. In addition we have represented, in one of two SDLs, a terminal prebend segment that has the same diameter and transport properties as do the TALs. The presence of such a prebend segment in short loops has not been established in mature rats, although the segment has been found in neonatal rats (22), hamster (4), in some short loops in mice (34), and in Gambel’s quail (10).

In our model, because the SDL2 and prebend segments are assumed water impermeable, no water is absorbed from them, although, as in the WKM model, net urea is secreted, but less than in that model. Moreover, in the SDL having a prebend segment (SDLa), net solute is absorbed from the prebend segment via Na^{+} active transport. As a consequence of these transport characteristics, luminal fluid flow in the SDL2 segment exhibits no significant osmolality increase with increasing medullary depth, and osmolality decreases in the luminal fluid of the prebend segment.

We hypothesize that the SDL2 and prebend segments reduce the load on the concentrating mechanism that would otherwise be presented if osmotic equilibration were allowed. The reduction in load would increase the general concentrating efficacy of NaCl absorption from TALs, an efficacy that tends to be limited for the vascular bundle and its periphery because of a relative paucity of nearby TALs relative to descending flows. Moreover, NaCl absorption from prebend segments would reinforce the absorption of NaCl from the deepest portion of the OM. We have previously shown, in other model contexts, that such reinforcement can be an effective and efficient means to increase the axial osmolality gradient (45, 46, 56); pertinent parameter studies are conducted in *study II* (41).

Neither a SDL2 segment nor a prebend segment is represented in the WKM model. Nevertheless, the current WKM model predicts no water loss from the SDL in the IS; rather, it predicts a gain of water (77), apparently due to the relatively lower osmolalities of LAV in the vascular bundle. The WKM model, in contrast to our model, also predicts an increase in IS SDL fluid osmolality, and the LAL and LAV carry fluids from the IM having high urea concentrations of 80–113 mM at the OM-IM boundary (higher than our values of 63.6 and 53.0 mM, respectively). In the WKM model urea diffuses out of the LAL and LAV enters into the nearby SDL (via vascular nodes), thus increasing SDL fluid osmolality. These differences between the WKM model and our model may arise from differences in configuration and parameters; from differing boundary conditions at the OM-IM boundary; or from the higher TAL Na^{+} active transport rates used in the WKM model (see above), which may have promoted high urea concentrations in the vasa recta of IM near the OM-IM boundary. A judgment on the correctness of the differing predictions made by the two models awaits additional experimental evidence and a more comprehensive understanding of IM function.

### Mass Fluxes, Exchange, Cycling, and Sequestration

Our model exhibits exchange, cycling, and sequestration patterns similar to those that have been predicted based on qualitative considerations (5, 6, 26, 33, 35, 52) and to those that were observed in the WKM model (77, 81, 86). These include urea secretion in SDLs deep in the OM, preferential urea exchange by long vasa recta, urea sequestration by these same vasa recta, the preferential elevation of LDL and CD osmolalities by NaCl absorption from nearby TAL, water absorption from water-permeable tubules and vessels that carry descending flows, and the progressive reduction in load, as a function of medullary depth, presented by the decreasing population of short vasa recta.

### Comparison to Prototype Model

In a previous study (40), we developed a prototype model that represented the radial structural inhomogeneity in the OM using two concentric regions. In addition to using fewer concentric regions, that two-region model also differs from the present model in several other aspects. The water-impermeable, but urea-permeable, segment of the SDL (i.e., the SDL2 segment) was not represented in the two-region model. In addition, the tubule and vas rectum number ratios and the boundary conditions at the OM-IM boundary conditions also differ for the two models. Thus, differing results of the two models cannot be directly attributed to the differing numbers of concentric regions. Nonetheless, some of the differences between the two sets of model results may be noteworthy. At the bend of the short loop of Henle possessing a prebend segment, fluid osmolality was found in the two-region model to be 549 mosmol/kgH_{2}O, which substantially exceeds that in the present model, 447 mosmol/kgH_{2}O. We attribute this difference to the absence of a SDL2 segment in two-region model, which resulted in the progressive concentration of SDL tubular fluid along OM, except for the prebend segment. The additional osmotic load may presented by the SDL in the two-region model may, in part, explain the lower osmolality gradient achieved by that model: at the OM-IM boundary, CD fluid osmolality was 571 mosmol/kgH_{2}O in the two-region model, but 851 mosmol/kgH_{2}O in the present model. In *study II* (41), we assess in detail the direct effects of using additional concentric regions by comparing results of our four-region base-case model with a model that approximates a two-region model by increasing the permeabilities between the two inner regions (R1 and R2) and between the two outer regions (R3 and R4) of our four-region model.

### Model Limitations and Potential Extensions

Because the IM is implicitly represented via boundary conditions, a limitation of this model may be that urine composition, used in the OM-IM boundary conditions, is assumed to be known a priori and is independent of the actions of the OM. However, we show in *study II* (41) that model results are nearly insensitive to variations in urine composition.

Another limitation of the region-based configuration is that it is not truly three-dimensional, but, like the WKM model, uses a form of coarse discretization to represent three-dimensional effects. (A truly three-dimensional model would have model equations with 3 spatial dimensions and employ a numerical formulation having a dense grid for all three spatial dimensions; cells, cell membranes, local intratubular fluid velocity boundary layers, and many other structures or features would plausibly need to be represented in such a formulation. A simplification to two fully-realized spatial dimensions might be obtained by means of a model formulation having cylindrical symmetry with respect to a corticomedullary axis through the center of the vascular bundle.) It may well be that further spatial discretization would be desirable in our model formulation, and it could be introduced by means of additional concentric regions. However, we believe it likely that little would be gained by concentric regions that are separated by less than a typical tubule diameter (i.e., ∼20 μm), owing to rapid diffusive equilibration, in the radial direction, for relevant length and time scales (44).

The region-based model framework can be extended to include additional solutes, e.g., potassium, chloride, and bicarbonate ions, all of which may have an impact on the UCM. Moreover, detailed epithelial transport can be represented and non-ideal expressions for osmolality (82) can be employed.

### Summary

The principal results and predictions of this study are the following.

The region-based model configuration provides a new framework for analyzing and predicting UCM function;

the model, with base-case configuration and parameters, produces an axial gradient along the OM that is consistent with the experimental record and with the paradigm of countercurrent multiplication;

the IS configuration contributes to the sequestration of urea in vascular bundles (especially near the OM-IM boundary) via countercurrent exchange;

the IS configuration promotes preferential increases in CD and LDL tubular fluid osmolality, relative to the flows of the vascular bundle, owing to the intermingling of the CDs and LDLs among TALs;

simulated TAL NaCl concentration at the corticomedullary boundary is sufficiently high to be consistent with indirect experimental evidence and theoretical considerations regarding TGF function that predict decreasing TAL tubular fluid NaCl concentration, in the tubular fluid flow direction, all along the cortical TAL; and

the AQP-1 null segment of the SDL may reduce the load on the IS and thereby increase the axial osmolality gradient in other structures, and some urea may enter this segment and thus contribute to intrarenal urea cycling.

## APPENDIX

### Model Equations

Model equations represent the conservation and transmural transport of solutes and water. Detailed justification for the model equations can be found elsewhere (40, 48, 89). The equations given below are a reduced set that includes only those that are needed to explain the most significant ways in which the present study differs from the two-region prototype model (40). A complete list of model equations is given in the supplement “Complete Model Equations,” which is available at http://ajprenal.physiology.org/cgi/content/full/00346.2003.DC1.

### AVR Transmural Fluxes

Because AVRs are fenestrated, their transmural water fluxes are assumed to arise only from advection, which is implicitly assumed to be driven only by hydraulic pressure and not by osmotic or oncotic pressure; thus, advection balances net fluid accumulation in the concentric regions. Net fluid accumulation in R1 and R2 is drained in part by the two populations of LAV: LAVa and LAVb, respectively. Accumulation in R3 and R4 are drained in part by the two populations of SAV: SAVa and SAVb, respectively (see below).

The transmural water flux into an LAV (LAVa or LAVb) is given by (A1) where *n*_{LAVj} is the number of LAVj per nephron, and *Q*_{LAVj}, the fluid accumulation carried from Rj by LAVj, is calculated below. The total fluid accumulation carried away by SAVj at a medullary level is given by *Q*_{SAVj} , which is also calculated below. A fraction α_{SAV} of *Q*_{SAVj} is drained by the SAVj that originates at *x*; thus (A2) where -*w*′_{SAV}, the derivative of -*w*_{SAV} with respect to *x*, denotes the rate at which the SAV originate at level *x*. The remainder of the fluid accumulation, (1 − α_{SAV})*Q*_{SAVj}, is drained by the SAV that originate at *y* > *x*. Thus the composite transmural water flux *J*_{SAVj,V} is given by (A3)

The AVR transmural solute fluxes, which arise from both solute diffusion and advection by water, are given by (A4) and (A5) where are the Péclet numbers for the solutes *k* associated with *i* = LAVa, LAVb, SAVa, or SAVb. is effective solute permeability associated with AVR *i* for solute *k* .

### Water and Solute Conservation in Concentric Regions

A schematic representation of water conservation at steady state is given in Fig. 2. Water conservation in the concentric regions is represented by (A6) (A7) (A8) (A9)

The second terms in *Eqs. A6* and *A7* are the vasa recta source terms. At axial level *x*, SDV break up at a rate of −*w*′_{SDV} and empty into capillaries. The SDV blood source is given by *Q*_{SDV}(*x*) = −n_{SDV}*w*′_{SDV}(*x*)F_{SDV,V} (*x*,*x*). As shown in Fig. 2, a fraction (α_{SDV}) of SDV empty their fluid in R1, whereas the remainder (1 − α_{SDV}) branch out into R2 before breaking up into capillaries. For a given concentric region, the sum of the capillary source flow and water fluxes arising from tubules, vessels, and its neighboring regions gives the net fluid accumulation in that region.

A fraction α_{LAVa} of the net fluid accumulation in R1 is taken up by LAVa through its fenestrations, and the remainder is shunted to R2 as capillary flow. A fraction α_{LAVb} of the net fluid accumulation in R2 is taken up by LAVb, and the remainder is shunted to R3. A fraction α_{SAVa} of the net fluid accumulation in R3 is taken up by SAVa, and the remainder is shunted to R4 as capillary flow. Net fluid accumulation in R4 is taken up solely by SAVb. The terms *Q*_{LAVa}, *Q*_{LAVb}, *Q*_{SAVa}, and *Q*_{SAVb} can be computed from *Eqs. A6*–*A9* and the relations Q_{LAVa}/Q_{R1,R2} = α_{LAVa}/(1 − α_{LAVa}), *Q*_{LAVb}/*Q*_{R2,R3} = α_{LAVb}/(1 − α_{LAVb}), *Q*_{SAVa}/*Q*_{R3,R4} = α_{SAVa}/(1 − α_{SAVa}), using known values of *J*_{R,V} (*R* = R1, R2, R3, R4) and *Q*_{SDV}.

Steady-state solute conservation for the concentric regions is given by (A10) where *R* = R1, R2, R3, and R4; and AVR,*R* = LAVa, LAVb, SAVa, or SAVb, for *R* = R1, R2, R3, and R4, respectively; and where *Ā*_{R} denotes the interstitial area per nephron of region *R*, i.e., *A*_{I,R} divided by the number of nephrons per vascular bundle. The first term on the right side of the equality represents transmural solute from tubules, vasa recta, and the other regions; the second term represents capillary solute sources (i.e., terminating SDV), with α_{SDV,R1} = α_{SDV}, α_{SDV,R2} = 1 − α_{SDV}, and α_{SDV,R3} = α_{SDV,R4} = 0; and the third term represents the sum of the solute that is carried by flow at the local concentration into AVR and the solute that is carried by capillary flow into an adjoining region *R*. The fourth term, a diffusion term characterized by diffusivity *D*_{k} for solute *k*, divided by diffusion resistance Ω, represents axial diffusion with respect to the corticomedullary axis.

Neumann boundary conditions for the diffusion term in *Eq. A10* are specified as follows: we assume that the interstitial solute concentrations in the cortex at a distance *L*_{C} (200 μm) from the corticomedullary boundary are equal to blood plasma concentrations, and we assume that the interstitial solute concentrations at the papillary tip are known and are equal to C_{urine,k}. Furthermore, we assume that the solute concentrations vary linearly within the IM and between *x* = −*L*_{C} and *x* = 0. With these stipulations, the interstitial solute concentration gradients at *x* = 0 and at *x* = *L* are given by (A11) (A12) for *R* = R1, R2, R3, and R4, and *k* = 1 and 2; *L*_{IM} denotes the length of the IM, which was taken to be 5 mm (25).

## GRANTS

This research was principally supported by the National Institute of Diabetes and Digestive and Kidney Diseases Grant DK-42091 (to H. E. Layton). Additional support was provided by National Science Foundation Grant DMS-0340654 (to A. T. Layton).

## Acknowledgments

The authors are grateful for the dedicated and meticulous efforts of the referees, which led to substantial improvements in this study (*study I*) and its companion, *study II* (41).

Portions of this work were presented at Experimental Biology 2002, April 20–24, 2002, New Orleans, LA (abstract no. 105.6; *FASEB J* 16: A51, 2002) and at the American Society of Nephrology Annual Meeting 2004, August 14–18, New York, NY (abstract no. F-PO053; *J Am Soc Nephrol* 15: 79A, 2004).

## Footnotes

↵1 This acronym, and the others involving vessels, may be used to indicate the plural (i.e., ascending vasa recta) or the singular, as appropriate for a given context.

↵2 For notational simplicity, the PST is considered to be the first SDL subsegment.

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- Copyright © 2005 the American Physiological Society