## Abstract

The mammalian kidney is particularly vulnerable to hypoperfusion, because the O_{2} supply to the renal medulla barely exceeds its O_{2} requirements. In this study, we examined the impact of the complex structural organization of the rat outer medulla (OM) on O_{2} distribution. We extended the region-based mathematical model of the rat OM developed by Layton and Layton (*Am J Physiol Renal Physiol* 289: F1346–F1366, 2005) to incorporate the transport of RBCs, Hb, and O_{2}. We considered basal cellular O_{2} consumption and O_{2} consumption for active transport of NaCl across medullary thick ascending limb epithelia. Our model predicts that the structural organization of the OM results in significant Po_{2} gradients in the axial and radial directions. The segregation of descending vasa recta, the main supply of O_{2}, at the center and immediate periphery of the vascular bundles gives rise to large radial differences in Po_{2} between regions, limits O_{2} reabsorption from long descending vasa recta, and helps preserve O_{2} delivery to the inner medulla. Under baseline conditions, significantly more O_{2} is transferred radially between regions by capillary flow, i.e., advection, than by diffusion. In agreement with experimental observations, our results suggest that 79% of the O_{2} supplied to the medulla is consumed in the OM and that medullary thick ascending limbs operate on the brink of hypoxia.

- region-based model
- red blood cells
- medullary thick ascending limbs
- active sodium transport

renal oxygen tension (Po_{2}) in the rat kidney ranges from 40–50 mmHg in the cortex to 20 and 10 mmHg in the outer medulla (OM) and inner medulla (IM), respectively (40). Low medullary Po_{2} results from the high metabolic requirements of medullary thick ascending limbs (mTALs) and the countercurrent arrangement of blood vessels in the medulla. The countercurrent architecture, combined with low medullary blood flow [IM blood flow is <1% of total renal blood flow (12)], serves to preserve the corticomedullary tubular and vascular fluid osmolality gradient that is necessary for the production of concentrated urine, but the concomitant shunting of O_{2} between descending and ascending vessels and limbs significantly reduces O_{2} supply to the deep medulla. The production of concentrated urine is also contingent on the active transport of NaCl in mTALs, a process that is highly energy dependent. Thus the O_{2} supply to the renal medulla barely exceeds its O_{2} requirements (16).

The kidney is therefore particularly vulnerable to hypoperfusion, inasmuch as the associated hypoxia leads to localized injury. Under experimental conditions resulting in reduced medullary Po_{2}, mTALs are the most injured segments; lesions may also spread to surrounding structures such as thin limbs, interstitium, and collecting ducts (CDs) (16, 18). The mTAL is prominently susceptible to decreases in medullary O_{2} supply as a result of its high metabolic requirements, as well as its distance from descending vasa recta (DVR), i.e., the source of O_{2} delivery. Indeed, the highly structured organization of tubules and vessels in the rat OM (27, 29), in which 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, and in which CDs and mTALs are found distant from the vascular bundles, likely increases the vulnerability of mTALs to hypoxia.

In this study, we examined the impact of the structural organization of the rat OM on O_{2} distribution. We extended the region-based model of the urine-concentrating mechanism of the rat OM developed by Layton and Layton (33), which was originally formulated for two solutes, Na^{+} and urea, the principal solutes in the mammalian urine-concentrating mechanism. To investigate O_{2} distribution in the OM, we incorporated RBCs, as well as Hb and O_{2}.

In this study, we discuss our baseline results. In a companion study (8), we investigate model sensitivity to fundamental structural assumptions and to parameters whose values are uncertain.

## GLOSSARY

### Acronyms

- AVR/DVR
- Ascending vas rectum/descending vas rectum
- CD
- Collecting duct
- IMCD
- Inner medullary collecting duct
- IM/OM
- Inner medulla/outer medulla
- iRBC
- Interstitial RBC
- 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
- mTAL
- Medullary thick ascending limb
- 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
- SDVa/SDVb
- SDV in R1/SDV in R2
- SDLa/SDLb
- SDL with a prebend segment/SDL without a prebend segment

### Indexes

*i*- Tubule, vas rectum
*j*- a, b or 1, 2, 3, 4
*k*- Solute Na
^{+}, urea, O_{2}, Hb, or HbO_{2} *R*- Region R1, R2, R3, or R4

### Independent Variables

*x*or*y*- Medullary depth, in axial direction
*z*- iRBC length, in radial direction

### Dependent Variables

- C
_{i,k} - Concentration of solute
*k*in tubule*i* - C
_{R,k} - Concentration of solute
*k*in concentric region*R* - C
_{i,k}^{P} - Concentration of solute
*k*in vessel*i*plasma - C
_{i,k}^{R} - Concentration of solute
*k*in vessel*i*RBC - F
_{i,V} - Water flow rate in tubule or vessel
*i* - F
_{i,V}^{P} - Water flow rate in vessel
*i*plasma - F
_{i,V}^{R} - Water flow rate in vessel
*i*RBC or in iRBC tube *J*_{i,V}- Transmural water flux into tubule or vessel
*i* *J*_{i,V}^{P}- Transmural water flux into vessel
*i*plasma *J*_{i,V}^{R}- Transmural water flux into vessel
*i*RBC or iRBC tube *J*_{i,k}- Transmural flux of solute
*k*into tubule or vessel*i* *J*_{i,k}^{P}- Transmural flux of solute
*k*into vessel*i*plasma *J*_{i,k}^{R}- Transmural flux of solute
*k*into vessel*i*RBC or iRBC tube *J*_{R,diff}- O
_{2}flux from region*R*into neighboring regions via diffusion *J*_{R,adv}- O
_{2}flux from region*R*into neighboring regions via capillary advection *J̄*_{iRBCj,R,k}- Transmural flux of solute
*k*from iRBC tube*j*into region*R*

### Parameters

*A*_{i,cell}- Cross-sectional area of surrounding cell layer in tubule or vas rectum
*i* *A*_{R}- Area of region
*R* *Ā*_{R}- Area occupied by interstitial cells in region
*R* - F
_{i,O2}^{B} - Total amount of O
_{2}in vessel*i* - F
_{O2}^{OM,ascending} - Total amount of O
_{2}flowing back to OM via LAL and LAV *K*_{M,Na}- Michaelis-Menten constant for Na
^{+}active transport *K*_{M,O2}- Michaelis-Menten constant for O
_{2}basal consumption *k*_{1}/*k*_{−1}- Dissociation/association constant of HbO
_{2} *L*- Length of OM
*n*_{i}- Number of tubules or vessels
*i*per nephron *P*_{O2}^{C}- Overall O
_{2}permeability of capillary *P*_{O2}^{R}- O
_{2}permeability of RBC membrane *P*_{O2}^{W}- O
_{2}permeability of capillary wall *P*_{R,R′,k}- Permeability of boundary between regions
*R*and*R*′ to solute*k* *Q*_{O2}^{IM}- Total O
_{2}consumption in IM *Q*_{O2}^{OM,active}- Overall O
_{2}consumption for active transport in OM *Q*_{O2}^{OM,basal}- Overall O
_{2}consumption for basal metabolism in OM *Q*_{R,R′}- Transregion capillary flow from region
*R*into*R*′ *r*_{i}- Inner radius of tubule or vas rectum
*i* *r*_{R}- Radius of region
*R* - R
_{i,O2}^{active} - O
_{2}volumetric consumption rate for active transport in tubule*i* - R
_{i,O2}^{basal} - O
_{2}volumetric consumption rate for basal metabolism in tubule or vas rectum*i* - R
_{max,O2}^{basal} - Maximum volumetric rate of O
_{2}basal consumption - R
_{RBC,O2}^{scav} - Volumetric rate of O
_{2}scavenging by RBCs - S
_{O2}^{OM} - Overall O
_{2}supply to OM - T
_{i,Na}^{active} - Volumetric rate of Na
^{+}reabsorption in tubule*i* *V*_{max,i,Na}- Maximum transport rate of Na
^{+}by tubule*i* - α
_{SDVa} - Fraction of plasma flow from terminating SDVa into region R1
- σ
_{k} - Reflection coefficient of solute
*k* - φ
_{k} - Osmotic coefficient of solute
*k* - ω
_{SAV}/ω_{SDV} - Fraction of SAV/SDV reaching a given medullary level
- Ψ
_{i,Na}^{active} - Rate of Na
^{+}active transport in tubule*i*(in mol·m^{−2}·s^{−1}) - θ
_{cell} - Fraction of solute consumed in epithelial or endothelial cell layer that is taken from the lumen

## MODEL DESCRIPTION

The region-based model of the urine-concentrating mechanism in the rat OM developed by Layton and Layton (33) accounts for the three-dimensional architecture of the OM and preferential interactions among tubules and vessels by representing four concentric regions centered on a vascular bundle. Tubules and vasa recta that reach into the IM are represented by rigid cylinders that extend from the corticomedullary junction (*x* = 0) to the OM-IM boundary (*x* = *L*). Vasa recta that supply the OM, i.e., short descending and ascending vasa recta (SDV and SAV), are represented by continuously decreasing distributions of vasa recta that reach different levels in the OM.

The radial organization is incorporated by assignment of appropriate tubules and vasa recta (or fractions thereof) to each concentric region. Model positions of tubules and vasa recta are shown in Fig. 1 for outer and inner stripes. Two distinct populations of SDV are represented in the two central-most regions (R1 and R2) to allow us the potential to distinguish vasodilatory effects in different regions. Distinct populations of AVR are used to avoid straddling of region boundaries by the highly permeable AVR, which introduces excessive mixing between regions. Vasa recta that supply the IM, i.e., long descending and ascending vasa recta (LDV, LAVa, and LAVb), are assumed to form the vascular bundle, which is contained within the two central-most regions (R1 and R2). SDVa and SDVb are assumed to reside in R1 and R2, respectively, consistent with immunolabeling results, which indicate that DVR are distributed centrally within the vascular bundle (24), and also with SDV peeling off to supply the capillary plexus of the inner stripe. CDs, which are located at a distance from the vascular bundle, are assigned to R4. Distinct populations of SAV, labeled SAVa and SAVb, ascend through R3 and R4, respectively. Thick ascending limbs from short loops of Henle (i.e., SALs), which are near the CD throughout the OM, are assumed to straddle R3 and R4. At the beginning of the outer stripe, the long loops of Henle are situated near the vascular bundle, in R2 and R3, where the long ascending limbs (LALs) maintain their position throughout the OM.

The portion of each concentric region that is exterior to tubules and vessels represents interstitial spaces, interstitial cells, and capillary plasma flow (given the highly fenestrated nature of the capillaries). This exterior compartment is designated the “interstitial region.” Capillary RBC flow is modeled following the approach used by Layton (32): RBCs within capillaries [referred to as interstitial RBCs (iRBCs)] are represented by rigid tubes extending perpendicular to the medullary axis (i.e., in the *z* direction) at each medullary level *x*. The iRBC tubes at level *x* form at the end of the SDV that terminate at *x* and extend to the SAV originating at *x*. Since we consider two distinct populations of SDV (SDVa and SDVb, which descend through R1 and R2, respectively) and two distinct populations of SAV (SAVa, and SAVb, which ascend through R3 and R4, respectively), four different populations of iRBC tubes are represented at each level *x* (Fig. 2). In the model developed by Layton and Layton (33), the SDV were represented as straddling R1 and R2. In the present study, the SDVa-to-SDVb number ratio was taken as 1:99 in the base case, to account for the fact that the vessels that irrigate the interbundle region peel off from the outer edge of the vascular bundles.

Conservation equations that describe fluid flow and solute concentrations at steady state in each tubule, vas rectum, and interstitial region are summarized in the appendix. Boundary conditions (BCs) prescribe flows and concentrations in tubules and vessels at the corticomedullary junction and the OM-IM boundary. Below we describe the representation of O_{2}-related processes in the model.

## Rate of Active Na^{+} Reabsorption

The active transport rate in tubule *i* (*Eq. A41*) is usually characterized with the assumption of C_{Na}-dependent Michaelis-Menten kinetics (33, 56) (1) where C_{i,Na} is Na^{+} concentration in tubule *i*, *V*_{max,i,Na} (in mol Na^{+}·m^{−2}·s^{−1}) is the maximum rate of Na^{+} transport in tubule *i*, and *K*_{M,Na} is the Michaelis-Menten constant, taken as 70 mM (35). The critical Po_{2}, below which O_{2} consumption becomes limited by O_{2} concentration, was found to be very low, <1 mmHg (10). Hence, we assume that, at <1 mmHg luminal Po_{2}, the active transport rate is a linearly decreasing function of Po_{2}, that is (2) The results are indistinguishable if we replace luminal Po_{2} with interstitial Po_{2} in *Eq. 2*. *V*_{max,i,Na} is estimated as 10.5 and 25.9 nmol Na^{+}·cm^{−2}·s^{−1} in mTALs in the outer and inner stripes, respectively (33).

## O_{2} Consumption

O_{2} is carried to the medulla by DVR blood, in free form (O_{2}) and as HbO_{2}. In RBCs, the dissociation of HbO_{2} into O_{2} and Hb is expressed as a one-step reaction where Hb denotes one of the four heme groups in each Hb molecule. Thus the volumetric rate of O_{2} scavenging by Hb in RBCs (R) is calculated as (3) Given that the kinetics of HbO_{2} dissociation are significantly faster than O_{2} diffusion, the reaction is generally taken to be at equilibrium, such that (4) where C_{50} is the O_{2} concentration at half-saturation (equivalent to 26.4 mmHg) and *n* is the Hill equation parameter, taken as 2.6 (9). The solubility coefficient for O_{2} (i.e., the C-to-Po_{2} conversion factor) is taken as 1.34 μM/mmHg in plasma and tubular fluid and 1.56 μM/mmHg in RBCs. Following the approach of Clark et al. (9), which is suitable for low Po_{2}, the dissociation constant *k*_{1} is fixed (49 s^{−1}), and the association constant is calculated at each level as a function of local O_{2} concentrations, to yield compatibility with the Hill equation for saturation (5) We consider basal O_{2} consumption by interstitial cells, vascular endothelial cells, and tubular epithelial cells (*Eqs. A2*–*A4* and *A62*). The volumetric rate of basal O_{2} consumption in tubule or vas rectum *i* (R, in mol O_{2}·m^{−3}·s^{−1}) is calculated as (6) Since epithelial or endothelial cells are situated at the interface between the interstitium and the lumen of tubule or vessel *i*, the fraction θ_{cell} of *R* that is attributed to the lumen (*Eq. A2*) is calculated on the basis of the luminal C, whereas the remaining fraction (1 − θ_{cell}), which is attributed to the interstitium (*Eq. A62*), is calculated on the basis of the interstitial C.

The maximum volumetric rate of O_{2} consumption (R) is assumed to be the same in each compartment and taken as 10 μM/s, as discussed in the companion study (8). The Michaelis-Menten constant (*K*) is taken as 5.4 μM, or 4 mmHg, under basal conditions for nitric oxide (7). The overall basal consumption of O_{2} in the OM (*Q*) is then estimated as follows (7) where *n*_{i} is the number of tubules or vessels *i* per nephron and *A*_{i,cell} is the cross-sectional area of the epithelial or endothelial cells. Po_{2} is expected to be significantly >4 mmHg in most OM segments; hence, R≈ R, and *Eq. 7* can be simplified as (8) The term in brackets on the right-hand side of *Eq. 8* is calculated as 2.0 × 10^{−12} m^{3}/nephron. If R= 10 μM/s, then *Q*≈ 2.0 × 10^{−14} mol O_{2}·s^{−1}·nephron^{−1}.

Under maximal efficiency, moles of Na^{+} reabsorbed through the transcellular pathway per mole of O_{2} consumed [also known as the T_{Na}-to-*Q* ratio (T_{Na}/*Q*)] is 18, as suggested by Na^{+}-K^{+}-ATPase stoichiometry. Under favorable thermodynamic conditions, additional moles of Na^{+} are reabsorbed in parallel through the paracellular route, namely, when the driving force imparted by the lumen-negative transepithelial potential is not counterbalanced by too large an interstitium-to-lumen Na^{+} concentration gradient. This may explain why measurements in thick ascending limbs (TALs) yielded T_{Na}/*Q*= 36 (11). To account for paracellular transport, which is not explicitly represented in the present model, we set T_{Na}/*Q*= 24. Using baseline parameters, our model predicts a lumen-to-interstitium Na^{+} concentration ratio that is sufficiently large along most of the mTAL length to support significant paracellular reabsorption. That result is consistent with our choice for T_{Na}/*Q*. The rate of O_{2} consumption for active transport in tubule *i* (R, in mol O_{2}·m^{−3}·s^{−1}) is thus determined as (9) where T(in mol Na^{+}·m^{−3}·s^{−1}) is the rate of Na^{+} reabsorption in tubule *i*. T and Ψ are related as follows (*Eqs. 9*, *A2*, and *A41*) (10) where *r*_{i} is the inner radius of tubule *i*. The overall O_{2} consumption for active transport in the OM (*Q*) is then calculated as (11) Using baseline parameters, we estimate *Q* to be 5.1 × 10^{−13} mol O_{2}·s^{−1}·nephron^{−1}.

### BCs at the OM-IM Boundary

As described in the appendix, BCs for O_{2} in LAV and LAL at the OM-IM boundary are determined on the basis of O_{2} consumption in the IM (*Q*). The basal component of *Q* is estimated as ∼0.9 × 10^{−14} mol·s^{−1}·nephron^{−1} if *R*= 10 μM/s in the IM (as in the OM). Total Na^{+}-K^{+}-ATPase activity and active Na^{+} transport are significantly lower in inner medullary CDs (IMCDs) than in mTALs. Moreover, a significant proportion of energy is provided by anaerobic glycolysis in the IM (52). We therefore assume that O_{2} consumption in the IM is 5% of O_{2} supply to the OM.

We postulate that the architecture of the IM plays a determinant role in the distribution of O_{2} between LAL and LAV. The dominating organizing structural elements in the IM for arrangement of the tubules and vasa recta are the clearly distinguishable clusters of CDs (43, 44). Loops of Henle that turn within the first millimeter of the IM are located within the CD clusters, whereas DVR and LDL that are associated with the remainder of the loops of Henle are nearly always located outside these clusters. In contrast, LAL and AVR are arranged nearly uniformly across the IM when viewed in transverse sections (44). Recall that we distinguish between two populations of LAV in this model: a model LAV, denoted LAVa, lies within the vascular bundle in the OM and is assumed to be close to DVR in the IM; another model LAV, denoted LAVb, lies on the periphery of the bundle in the OM and is assumed to be distant from DVR in the IM. Thus the flow of O_{2} in LAVa at the OM-IM boundary is likely to be significantly greater, per vessel, than that in LAVb. In addition, LAL do not carry HbO_{2}, the concentration of which is much greater than that of O_{2}. Given the position of the two populations of LAV and also taking into consideration that 17 LAVa and 33 LAVb are represented per vascular bundle, we assume, as a baseline, that 38.8% and 60.2% of the O_{2} and HbO_{2} at the OM-IM boundary that flow back in the OM via ascending limbs and vessels (F), respectively, enters via the LAVa and LAVb at the OM-IM junction (i.e., 25% more O_{2} in each LAVa than LAVb), whereas the remaining 1% enters via the LAL.

### Parameter Values

Parameters related to the transport of water, Na^{+}, and urea across tubules and vessels are taken from the study of Layton and Layton (33). Those that concern the transport of O_{2} across vasa recta are taken from the study of Zhang and Edwards (57). The permeability of medullary tubules and vessels to O_{2} has not been measured directly, to the best of our knowledge. The (overall) capillary permeability to O_{2} (*P*) can be expressed as (12) where *P* and *P* denote the O_{2} permeability of the RBC membrane and the capillary wall, respectively. On the basis of literature estimates (for review see Ref. 57), the *P*-to-*P* ratio is taken as 3, and *P* is taken as 0.01 cm/s (i.e., *P*= 0.04 cm/s). For simplicity, we assume that the permeability of O_{2} across vascular endothelial cells and tubular epithelial cells is the same. The baseline values of parameters related to O_{2} transport are summarized in Table 1.

The study of Rasmussen (47) suggests that blood hematocrit in the OM (30–40%) is comparable to that in the cortex and significantly higher than that in the IM. The baseline value of the OM hematocrit is taken as 35% in our simulations.

Inner diameters and other dimensions are taken from the study of Layton and Layton (see Tables 1 and 2 in Ref. 33). The thickness of the vascular endothelial layer is estimated as 1 μm (T. L. Pallone, personal communication); that of the epithelial layer is taken as 8 μm in the mTAL (26), 1.1 μm in the descending limb (the average of the 0.4- to 1.8-μm range reported in Ref. 41), and 9 μm in the outer medullary CD (OMCD) (55).

Areas occupied by interstitial cells (*Ā*_{R}) are needed to estimate basal O_{2} consumption in the regions (*Eq. A62*). We estimate interstitial cell areas by assuming that, in the outer stripe, 3.71% of region areas are occupied by interstitial cells (45). In the inner stripe, interstitial cells are sparse within the vascular bundles (36) but less sparse in the interbundle regions. We thus assume that 3.71% of the region areas are occupied by interstitial cells in the vascular bundle (i.e., R1 and R2); however, in R3 and R4, that percentage increases linearly from 3.71% at the outer stripe-inner stripe junction to 10.07% at the mid-inner stripe. The latter value (10.07%) is chosen so that, at the mid-inner stripe, 8.27% of the region areas are occupied by interstitial cells (45). In other words, for *R* = R3 and R4 (13) where *A*_{R} is the area of the whole region, *L*_{OS}, *L*_{IS}, and *L* denote the length of the outer stripe, inner stripe, and OM, respectively, and ρ_{1} and ρ_{2} are taken as 0.0371 and 0.1007, respectively.

As described above, the model assumes that capillary plasma is well mixed with the local interstitium, whereas capillary RBCs flow within iRBC tubes. In this representation, RBCs in capillary tubes exchange directly with the interstitium, whereas RBCs in vasa recta are surrounded by plasma. To account for the additional resistance of interstitial cells and capillary walls, the permeability of capillary RBCs to water, Na^{+}, and urea is taken as one-tenth of that of RBCs in vasa recta.

### Numerical Methods

The steady-state differential equations are discretized to form a system of nonlinear algebraic equations. A spatial discretization of 200–400 grid points along the medullary axis and 1,000 grid points in the radial direction (for iRBCs) is used. The system of coupled, nonlinear conservation equations at each medullary level is solved using MINPACK, a numerical package implemented in FORTRAN for solving nonlinear equations with a modification of Powell's hybrid algorithm. For computation of steady-state solutions, the following steps are repeated until the convergence criterion is satisfied: an initial guess is made for interstitial solute concentrations; concentration profiles in descending and ascending tubules and vessels, as well as in the horizontal RBC tubes, are obtained by solving the nonlinear algebraic equations along the *x* and *z* axes; and water and solute conservation equations in the four regions are solved, and interstitial solute concentrations are updated accordingly. Iterations are stopped when the largest relative difference in solute concentration from one iteration to the next is <10^{−6}.

## RESULTS

### Overall O_{2} Supply in the Rat OM

The overall supply of O_{2} to the OM (S) is determined as the total molar flow rate per nephron of O_{2} and HbO_{2} entering descending vessels and tubules at the corticomedullary junction (i.e., *x* = 0) (14) where (15) (16) (17) where F_{i,V} is the volumetric flow rate in tubule or vessel *i*, C_{i,k} is the concentration of solute *k* in tubule or vessel *i*, and the superscripts B, P, and R denote blood, plasma compartment, and RBC compartment, respectively. In *Eq. 16*, ω_{SDV}(*y*) denotes the fraction of SDV that reaches medullary depth *y* (*Eq. A22*); the negative of its derivative, −dω_{SDV}(*y*)/d*y*, is the rate at which SDV terminate along the OM, reckoned in the positive *x* direction. With the BCs specified in Table 2, S is calculated as 6.6 × 10^{−13} mol·s^{−1}·nephron^{−1}, 97% of which is delivered to the OM in the form of HbO_{2}; we thus ignore the amount of O_{2} supplied by aqueous fluids relative to that supplied by RBCs in the remainder of this study.

### Interstitial, Intratubular, and Intravascular Po_{2} Gradient

Using the base-case configuration, parameter set, and BCs, we solved the model equations to yield flow and concentration profiles. Figure 3 shows Po_{2} profiles in each class of tubules and vessels and in each concentric region. Figure 3, *A, B, C*, and *D*, shows results for tubules and vessels associated with regions R1, R2, R3, and R4, respectively; tubules and vessels are assigned to the region with which they are in contact for half or more of their inner stripe length. Figure 3*E* shows Po_{2} profiles in each concentric region. As described in model description, the SDV are represented by a continuous distribution, with SDV terminating at all depths of R1 and R2 and with SAV originating at all depths of R3 and R4. However, Fig. 3 contains only the profiles corresponding to the longest SDVa, SDVb, SAVa, and SAVb (i.e., those that reach the OM-IM boundary). Figure 4 shows O_{2} flow, per nephron, in each class of tubules and vessels. Because Fig. 4 portrays directed flows, flow toward the cortex is considered to be negative.

#### Interstitial Po_{2} gradients and their generation.

As described below, our model predicts that O_{2} availability varies greatly from one region to another. In this section, interstitial Po_{2} profiles in each region are depicted consecutively. Figure 3*E* shows a decreasing Po_{2} gradient along the model's corticomedullary axis in R1, the core of the vascular bundle, and in LDV and LAVa, both of which lie within the central-most region through the OM (Fig. 3*A*). In R1, interstitial Po_{2} decreases from 58.2 mmHg at the corticomedullary junction to 48.3 mmHg at the OM-IM boundary, as O_{2} diffuses radially outward to O_{2}-poor regions. Because interstitial Po_{2} in R1 lags behind that of the LDV, O_{2} is continually reabsorbed from the LDV, the O_{2} flow of which decreases from 0.0707 to 0.0494 pmol·min^{−1}·nephron^{−1} (Fig. 4*A*).

Similarly, the R2 interstitial Po_{2} profile exhibits a general decrease along the corticomedullary axis. The O_{2} demand in R2 is at its highest when the prebend segments (which, similar to the mTALs, actively transport Na^{+}) begin, at ∼50 μm from the OM-IM boundary. Closer to the OM-IM boundary, however, the SDL move away from R2 to occupy a position between R3 and R4. The consequent decrease in O_{2} demand results in a rise in R2 interstitial Po_{2}.

Po_{2} is predicted to be low in the interbundle regions, i.e., in R3 and R4, where Po_{2} hovers between 5.7 and 8.7 mmHg between the mid-outer stripe and the mid-inner stripe. These outer regions do not contain O_{2}-supplying vessels but, instead, contain most of the mTALs, which impose the largest O_{2} demand. (In the outer stripe, one-half of the LAL and one-half of the SAL reside in R3, and the remainder of the SAL resides in R4; in the inner stripe, one-half of the LAL and three-fourths of the SAL reside in R3, and the remainder of the SAL resides in R4.) Interstitial Po_{2} in R3 and R4 decreases in the lower inner stripe, where the mTAL Na^{+} active transport rates are the highest.

Because of the radial organization in the OM, in particular, the localization of DVR in the vascular bundles and of mTAL away from the vascular bundles, the model predicts substantial R1-R4 Po_{2} gradients. Because the vessels are tightly packed in the vascular bundle core, diffusion of O_{2} out of R1 is limited. Also, O_{2} flow via capillaries from R1 to R2 is assumed to be small (only 5% of net fluid accumulation in R1 is directed to R2 as capillary plasma flow). These factors contribute to the significant Po_{2} gradient, a difference of about two-thirds of R1 interstitial Po_{2}, between R1 and R2. Because R2, R3, and R4 contain different tubules and vessels, all of which have different metabolic needs, which may also vary axially, the Po_{2} profiles of these three regions differ significantly. In most of the inner stripe, Po_{2} does not decrease monotonically in the radial direction from R1 to R4.

Near the cortex, interstitial Po_{2} in R2, R3, and R4 decreases sharply along the corticomedullary axis. This boundary layer effect can be attributed, in large part, to the rapid decrease in the LDL tubular fluid Po_{2} near the cortex. The model assumes that the content of the fluid entering the descending limbs at the corticomedullary boundary approximates that of plasma. The underlying assumption is that the cortex is largely homogeneous and highly oxygenated. Because Po_{2} is substantially higher at the LDL boundary than in the ascending limbs and SAV, with which LDL share a position in R3, and because of the high O_{2} permeability of LDL, there is a large efflux of O_{2} from the LDL near the corticomedullary junction and, thus, a precipitous drop of Po_{2} in LDL and a subsequent decrease in interstitial Po_{2} in R3 and in the neighboring regions R2 and R4. This boundary layer effect might be reduced by an explicit representation of the cortex, into which the regions continue their dissolution. Such a model would give a better prediction of the LDL Po_{2} at the corticomedullary boundary and would also allow O_{2} to diffuse into the upper outer stripe from the O_{2}-rich cortex, increasing the interstitial Po_{2} and smoothing any boundary layer.

#### Intravascular and intratubular Po_{2} gradients and their generation.

Basal metabolism of all tubules and vessels and active transport metabolism in mTALs consume O_{2}. Taken in isolation, this consumption should result in decreasing O_{2} flows in all tubules and vessels along their flow directions. However, tubules and vessels are assumed to be highly permeable to O_{2} (0.04 cm/s), which results in significant O_{2} diffusion across epithelial or endothelial walls when an O_{2} gradient is established. Thus tubular and vascular plasma Po_{2} profiles generally approximate those of the surrounding interstitium. Figure 3, *A–D*, shows Po_{2} profiles for each class of tubules and vessels; the corresponding O_{2} flow rates, per nephron, are shown in Fig. 4. The flow rates are taken to be positive when flow is in the direction of increasing medullary depth. Composite flow rates are given for SDV and SAV. Because short vasa recta turn back along the OM, the per-nephron SDV O_{2} flow rapidly decreases, from 0.2573 pmol·min^{−1}·nephron^{−1} at the corticomedullary boundary to 0 at the OM-IM boundary, where the longest SDV is assumed to terminate.

LDV and LAVa, which lie within the core of the vascular bundle (R1), have the highest Po_{2}. As Po_{2} in DVR plasma decreases, within RBCs O_{2} dissociates from HbO_{2} and, subsequently, diffuses down a concentration gradient into the surrounding plasma (see HbO_{2} flow profiles for LDV in Fig. 4*E*). Along the LDV, 7.4% of the HbO_{2} that enters at the corticomedullary boundary has dissociated before reaching the IM.

The Po_{2} profiles of LAVb and SDL have similar gradients in the inner stripe but diverge in the outer stripe (Fig. 3*B*). This occurs because, although LAVb stays in R2 throughout the OM, SDL occupies an outer stripe position farther from the vascular bundle, between R3 and R4, where Po_{2} is significantly lower.

Intratubular fluid Po_{2} of SAL and LAL is lowest near the OM-IM boundary. In the mTAL, basolateral Na^{+} reabsorption is carried out by Na^{+}-K^{+}-ATPase pumps, which hydrolyze ATP, thereby requiring O_{2} consumption. As a result, Na^{+} concentrations of intratubular fluid in the prebend segments and in the mTALs are progressively reduced along the luminal flow direction (Fig. 5). Taken in isolation, the active transport processes would decrease mTAL Po_{2} along the direction of their flow. However, Po_{2} is lower within the mTAL lumen than in the surrounding interstitium, so that O_{2} diffuses into the mTAL. These two competing factors result in a general increase in mTAL O_{2} flow in the SAL in the inner stripe and in the LAL along their flow direction and a decrease in SAL O_{2} flow in the outer stripe.

### Radial O_{2} Transport: Diffusion vs. Capillary Transport

We then examined the relative contribution of different O_{2} transport mechanisms. O_{2} exchange between regions can occur in two ways: advection and diffusion. Advection takes place via capillary flow. At each medullary level, a fraction of the SDV population breaks up into capillaries and releases their plasma content, including O_{2}, into R1 and R2 interstitium (see *Eqs. A52a* and *A52b*). The RBCs associated with those SDVs become iRBCs that traverse in the direction perpendicular to the corticomedullary axis toward R4 (see *Eqs. A29a*–*A30b*). These capillaries form some SAV in R3 and the rest in R4. As the capillaries traverse from R1 and R2 toward R3 and R4, some O_{2} also diffuses out of the iRBC tubes into the interstitium. Interregion diffusive O_{2} exchange takes place as O_{2} permeates out of the LDV and SDVa into R1 and out of the SDVb into R2. A fraction of that O_{2}, which depends on the interregion permeability to O_{2} (*P*), then diffuses radially toward the surrounding regions.

To compare the contributions of diffusion vs. advection to interregion O_{2} transport, we computed the fluxes of O_{2} exchanged between neighboring concentric regions that take place via interregion boundary diffusion and via capillary advection, respectively, given by (18) (19a) (19b) (19c) (19d) where J̄ is the O_{2} flux from the iRBC*j* tubes into region *R* (*Eqs. A58*–*A61*), α_{SDVa} denotes the fraction of the capillary plasma flow from terminating SDVa that is directed into R1 (10%), C is the interstitial O_{2} concentration in region R, and *Q*_{R,R′} represents the interregional flow from region *R* to *R*′.

Figure 6 shows interregion diffusive and advective O_{2} fluxes as a function of medullary depth. In all regions, advective O_{2} fluxes are significantly larger in the inner stripe than in outer stripe, since capillary flow is almost negligible in the outer stripe, where there are very few capillaries (28). [The model assumes that only 3% of SDV break up above the outer stripe-inner stripe junction (see *Eq. A22*).]

Throughout R1, O_{2} is reabsorbed mainly from LDV; the number of SDVa, and thus the flow across iRBC1 and iRBC2, is negligible in the base case. Most of that reabsorbed plasma O_{2} is carried away by LAVa, but a small fraction (5%) enters R2 via plasma capillary flow (i.e., the last term in *Eq. 19b*). Because interstitial Po_{2} in R1 is more than twice that in R2, O_{2} also diffuses into R2. Since capillary flow is small and R1 Po_{2} is high, diffusion exceeds advection throughout the OM.

Because capillary flows are assumed to be more abundant outside the vascular core, advection exceeds diffusion in R2, R3, and R4 throughout most of the OM (except in portions of the outer stripe for R4). For R3, advective and diffusive fluxes are positive throughout the OM; this indicates that O_{2} enters R3, driven by capillary flows (advection) and by a favorable interregion O_{2} gradient (diffusion). The diffusive fluxes in R1 are approximately one order of magnitude lower than the advective fluxes in R2–R4: the fact that R1-to-R2 O_{2} diffusion remains relatively low, despite a significant Po_{2} gradient, can be attributed to the small interstitial area in R1 (Fig. 7), which limits diffusion.

### Basal and Active Transport O_{2} Consumption

We distinguish in this study between the basal and active components of O_{2} consumption. As defined by Mandel and Balaban (38), the basal rate comprises *1*) the energy required for active transport processes not linked to Na^{+} transport and *2*) the energy required for biochemical reactions in renal cells (such as synthetic activity and interconversion of substrates, but not glycolysis). The active O_{2} consumption rate corresponds to the energy required for the active transport of Na^{+} across renal cells. The relative contribution of each component is discussed below.

The molar flow of O_{2} entering descending vessels and tubules at *x* = 0 (i.e., the corticomedullary junction) is calculated as 6.6 × 10^{−13} mol·s^{−1}·nephron^{−1}, or 39.6 pmol·min^{−1}·nephron^{−1}, as described above. Most of that O_{2} is consumed in the renal medulla, mostly in the OM. The molar flow of O_{2} returned to the cortex at *x* = 0 via ascending vessels and tubules (i.e., SAL, SAVa, SAVb, LAL, LAVa, and LAVb) is predicted to be 6.2 pmol·min^{−1}·nephron^{−1}. Since O_{2} consumption in the IM is taken as 5% of medullary O_{2} supply, overall O_{2} consumption in the OM is calculated as 0.95(39.6) − 6.2 = 31.4 pmol·min^{−1}·nephron^{−1}, that is, 79.3% of the supply.

Table 3 shows total basal and active transport O_{2} consumption, per nephron, along each class of tubules and vessels and in each concentric region. The basal (or active) O_{2} consumption reported for a given type *i* of tubules or vessels reflects the total basal (or active) consumption of the epithelial or endothelial cells surrounding the lumen of *i*. The basal O_{2} consumption reported for region *Ri* reflects the basal consumption of the interstitial cells in *Ri*. Basal O_{2} consumption is by far the lowest in R1, because the core of the vascular bundles has a small area and is assumed to be tightly packed, with only a small number of interstitial cells (45). In contrast, R3, which has the largest area (thus, the largest number of interstitial cells) and contains the greatest proportion of tubules and vessels, has the highest basal consumption among all regions.

Different basal O_{2} consumption rates are reported for LAVa and LAVb (0.01580 and 0.02617 pmol·min^{−1}·nephron^{−1}), in large part because the model represents almost twice as many LAVb as LAVa (there are 17 LAVa per 71 nephrons and 33 LAVb per 71 nephrons). Indeed, because LAVa resides in the O_{2}-rich R1, the basal O_{2} consumption, which depends on luminal Po_{2}, of each LAVa actually exceeds that of each LAVb. The discrepancies among basal consumption among the different populations of SDV and SAV are also attributable to their number ratios and luminal Po_{2}.

The total OM O_{2} consumption due to basal metabolism, by all epithelial, endothelial, and interstitial cells, is computed to be 0.8789 pmol·min^{−1}·nephron^{−1}, whereas the consumption due to active transport is 30.63 pmol·min^{−1}·nephron^{−1}. The latter accounts for 97%, a majority, of the total OM O_{2} consumption. Of the active transport O_{2} consumption, 54% and 45% are attributable to SAL and LAL, respectively. Although the metabolic needs of each SAL are less than those of each LAL (see above), there are also twice as many SAL as LAL; thus SAL O_{2} consumption is greater.

### mTAL Active Na^{+} Transport

The amount of ATP produced by glycolysis in mTALs is a small proportion of that produced by aerobic metabolism (53), even though mTALs have a high capacity for glycolysis (54). Studies suggest that only 2–3% of the ATP produced in the kidney originates from glycolysis and that only in the OMCD and IMCD can glycolysis maintain a substantial fraction of tubular function (11, 21). In the base case, anaerobic metabolism is not represented.

To simulate the effects of anaerobic metabolism on Na^{+} concentration profiles in the OM, we assume that as Po_{2} decreases between 1 and 0 mmHg, anaerobic metabolism supplies an increasingly larger fraction of the energy needed to actively transport NaCl across mTALs, so that, in the absence of any O_{2} supply, anaerobic metabolism supplies enough energy to sustain an active Na^{+} transport rate that is half of the maximum rate when O_{2} supply is abundant. With this assumption, *Eq. 2* becomes (20)

The rate of active O_{2} consumption is then calculated as (21) The model predicts that anaerobic metabolism could significantly increase the concentrating capability of the OM. Figure 8 shows transmural Na^{+} fluxes driven by aerobic and anaerobic active transport along the SAL and LAL. In the early OM (for *x* < 0.3*L*), luminal Po_{2} in SAL and LAL is sufficiently high that anaerobic active Na^{+} transport does not take place. Closer to the OM-IM boundary, where interstitial and mTAL luminal Po_{2} are very low, anaerobic active Na^{+} transport becomes significant. At *x* = 0.95*L*, where luminal Po_{2} in SAL and LAL is near its minimum, anaerobic active Na^{+} transport accounts for 66.8% and 70.0% of the mTAL Na^{+} reabsorption in LAL and SAL, respectively. The trough in the LAL profile corresponds to an analogous trough in the R2 interstitial Po_{2} profile (Fig. 3*E*).

With the inclusion of anaerobic active Na^{+} transport, 848 pmol·min^{−1}·nephron^{−1} of Na^{+} is reabsorbed from the mTAL compared with 735 pmol·min^{−1}·nephron^{−1} in the base case without anaerobic active transport. This 15.4% increase in total Na^{+} reabsorption has the effect of raising the tubular and vascular fluid osmolality gradient along the corticomedullary axis. With anaerobic active transport, CD, LDL, and LDV tubular fluid osmolalities are predicted to be 665, 666, and 488 mosmol/kgH_{2}O, respectively, at the OM-IM boundary compared with baseline values of 580, 581, and 450 mosmol/kgH_{2}O, respectively. These results suggest that because the mTALs operate near the brink of hypoxia, anaerobic active Na^{+} transport along the mTAL, if it does occur, may contribute significantly to generation of the OM osmolality gradients.

To assess the degree to which mTAL active Na^{+} transport is limited by its luminal Po_{2}, we computed the total mTAL active Na^{+} transport (in pmol·min^{−1}·nephron^{−1}) under different scenarios. For the base case without anaerobic transport, total mTAL active transport is calculated to be 735 pmol·min^{−1}·nephron^{−1}. For the idealized case in which *V*_{max,mTAL,Na} is assumed to be constant (i.e., independent of mTAL luminal Po_{2}), total mTAL active transport is calculated to be 961 pmol·min^{−1}·nephron^{−1}, which corresponds to a 30.7% increase. With the inclusion of anaerobic transport, total mTAL active transport is computed to be 848 pmol·min^{−1}·nephron^{−1}, which is 15.4% higher than the base case and 11.8% less than the idealized case.

## DISCUSSION

### Summary of Main Results

We have developed a highly detailed mathematical model for O_{2} transport in the OM of the rat kidney. The model uses a region-based configuration (33) to represent radial organization of renal tubules and vessels with respect to the vascular bundles. In addition to renal tubules and vessels, the model also explicitly represents RBCs. The model predicts, at steady state, in all represented structures the concentrations of Na^{+}, urea, and O_{2} (and also Hb in RBCs), intratubular or intravascular flow rates of water and solutes, and transmural fluxes of water and solutes.

Our results suggest that the structural organization of the OM results in significant Po_{2} gradients in axial and radial directions. Because almost four-fifths of the DVR turn within the OM, ∼80% of the O_{2} supplied to the medulla bypasses the IM. The decreasing DVR population, together with the large metabolic requirements of Na^{+} active transport across mTAL cells, results in a Po_{2} drop along the corticomedullary axis in most structures. Moreover, the segregation of the DVR (i.e., the main supply of O_{2}) at the center and immediate periphery of the vascular bundles creates large regional differences in Po_{2} in the radial direction. In the context of a urine-concentrating mechanism, the vascular bundles promote countercurrent exchange by bringing LDV and LAV in close proximity and enhance the concentrating capability of the OM by shielding the relatively dilute LDV from other structures, thereby reducing the osmotic load on the concentrating mechanism (33). In the context of O_{2} transport, by isolation of LDV within the relatively O_{2}-rich vascular bundle core, O_{2} reabsorption from LDV is reduced. As a result, 92.4% of the O_{2} supplied to LDV at the corticomedullary junction reaches the IM.

In addition, our model predicts, in agreement with experimental observations (5, 17), that the mTALs operate on the brink of hypoxia. The mTALs are localized far from the vascular bundles, an arrangement that improves the OM concentrating effect, inasmuch as the Na^{+} that is actively transported out of the mTALs can be used more effectively to raise the osmolality of the CD (33). However, because of the significant distance between the mTALs and the O_{2}-carrying DVR, together with the high O_{2} consumption of the mTALs, mTAL luminal Po_{2} is very low (<10 mmHg) near the OM-IM boundary, so that mTALs are particularly vulnerable to hypoxic injury.

### Comparison With Experimental Results

Measurements of Po_{2} in the renal OM generally range from 20 to 30 mmHg. Po_{2} was measured as 20 mmHg at a depth of 3.7 mm below the cortex, that is, ∼0.7 mm below the corticomedullary junction (13), 21 mmHg at a depth of 4.0–4.5 mm (4), and 23 mmHg at depth of 4 mm (42). Higher values, between 30 and 36 mmHg, were obtained by other investigators (14, 37) at depths of 3–4 mm.

Our predictions in the base case are in agreement with the data in the lower range. As shown in Fig. 3, within the first millimeter below the corticomedullary junction (i.e., at 3–4 mm below the cortex), interstitial Po_{2} is predicted to vary between 35 and 14 mmHg in the periphery of the vascular bundles (R2) and between 17 and 5 mmHg in the interbundle regions (R3 and R4). Po_{2} is predicted to be significantly greater (>48 mmHg) in the core of the vascular bundles (R1), but it is unlikely that measurements would have been performed in the very center of the vascular bundle, where very few interstitial cells are present (36).

Furthermore, the model predicts that the O_{2} consumption-to-delivery ratio is 79.3% in the base case (Fig. 3), in excellent agreement with the 79% estimate reported by Brezis et al. (5). The basal metabolic rate has been estimated as 3–18% of the O_{2} consumption rate in the mammalian kidney (11). Our model predicts a corresponding percentage of 3% in the OM, that is, at the lower end of the reported range. It is possible that basal O_{2} consumption was underestimated in our simulations.

O_{2} distribution affects the urine-concentrating mechanism, inasmuch as the active transport rate of Na^{+} may decrease when O_{2} availability becomes limited, which indeed occurs in the interbundle regions in the late OM. Because the OM urine-concentrating mechanism is driven by the active transport of Na^{+} by the mTAL, when that transport is reduced, the concentrating effect of the model OM is reduced. Nonetheless, the model predicts a CD fluid osmolality of 580 mosmol/kgH_{2}O at the OM-IM boundary, which is a 1.9-fold increase over the inflow osmolality (309 mosmol/kgH_{2}O). Such an increase is consistent with experimental measurements (49).

### Comparison With Previous Models

O_{2} transport in the microcirculation has been the focus of mathematical modeling for several decades following the pioneering work of Krogh (30). Kinetic expressions for the formation of HbO_{2} were developed by Adair (1) and Moll (39). The model of O_{2} transport developed by Hellums (23) and further extended by Baxley and Hellums (2) was among the first to consider RBCs and plasma separately, instead of treating the blood as a continuum. In contrast with previous approaches, these studies showed that the resistance to O_{2} transport in the capillary is a significant fraction of the overall resistance. The discrete cell model was further advanced by the work of Federspiel and Popel (19) and Groebe (20). The history and present challenges of O_{2} transport simulations are reviewed in greater detail by Hellums et al. (22) and Popel (46).

In our previous model of O_{2} transport across vasa recta (57), the tubular system was not explicitly included; instead, we specified the volumetric O_{2} consumption rate along the corticomedullary axis. The model was used to predict the overall O_{2} consumption rate on the basis of measurements of the medullary interstitial Po_{2} gradient. That model predicted an upper estimate of overall consumption, 5 × 10^{−6} mmol·s^{−1}·kidney^{−1} (i.e., 1.7 × 10^{−13} mol·s^{−1}·nephron^{−1}, with the assumption of 423 bundles per kidney and 71 nephrons per bundle), which is comparable to the predictions of the present model, 5.3 × 10^{−13} mol·s^{−1}·nephron^{−1}. In the previous study, the fraction of O_{2} supplied to DVR at the corticomedullary junction that reaches the OM-IM boundary was predicted to be approximately equal to the relative number of DVR assumed to reach that medullary depth, a prediction that is consistent with the present study. However, in the previous study, we assumed that fewer DVR reach into the IM. Thus the present model predicts a significantly higher O_{2} delivery rate into the IM, at 1.3 × 10^{−13} mol·s^{−1}·nephron^{−1}, compared with 4.8 × 10^{−14} mol·s^{−1}·nephron^{−1} in the previous model.

Another noteworthy difference between the present study and the work of Zhang and Edwards (57) is that they did not represent the complex radial organization of the OM. The interstitial Po_{2} profile that was assumed in the previous model was based on OM measurements that likely correspond to averages of intra- and interbundle Po_{2}. Thus the previous model predicted a DVR-interstitium Po_{2} difference that remained <10 mmHg throughout the medulla. In contrast, in the present model, DVR are surrounded by the O_{2}-rich R1 interstitium; thus the present model predicts larger DVR-R4 Po_{2} differences of 44 mmHg at *x* = 0 to 46 mmHg at *x* = *L* but significantly smaller DVR-R1 Po_{2} differences of 2 mmHg at *x* = 0 to 0.1 mmHg at *x* = *L*.

The present model's representation of the OM structural organization and the transport of Na^{+} and urea is based on the region-based urine-concentrating mechanism model of Layton and Layton (33) (hereafter referred to as the L & L model). In the L & L model, the active Na^{+} transport is approximated by a saturable expression having the form of Michaelis-Menten kinetics, with the maximum transport rate *V*_{max,i,Na} assumed constant. In the present model, *V*_{max,i,Na} is assumed to decrease when luminal Po_{2} dips below 1 mmHg. Thus the total active Na^{+} transport is predicted to be less in the present model than in the L & L model: the present model predicts that SAL and LAL tubular fluid enters the cortex with an Na^{+} concentration of 144 and 107 mM, respectively, whereas in the L & L model (33), the analogous Na^{+} concentrations were predicted to be 122 and 93 mM, respectively. In part because of the reduced Na^{+} reabsorption in the present model, a somewhat lower osmolality gradient is predicted: at the OM-IM boundary, CD, LDL, and LDV tubular fluid osmolalities are predicted to be 580, 581, and 450 mosmol/kgH_{2}O, respectively, compared with 851, 822, and 586 mosmol/kgH_{2}O, respectively, in the L & L model (33). Nonetheless, the 1.9-fold increase in fluid osmolality along the OMCD is consistent with micropuncture measurements (49). When anaerobic metabolism is included, the model predicts higher mTAL Na^{+} active transport, resulting in more concentrated fluid entering the IM: CD, LDL, and LDV tubular fluid osmolalities are predicted to be 665, 666, and 488 mosmol/kgH_{2}O, respectively; those values somewhat better approximate the predictions of the L & L model, but remain significantly lower. The remaining difference can be attributed, in large part, to the different blood flow representation of the two models. In the L & L model, which does not explicitly represent RBCs, 95% and 5% of the net fluid reabsorption in R1 and R2, part of which arises from the break-up of SDV and, thus, contains RBCs, is taken up by LAVa and LAVb, respectively, leaving only a small amount of fluid to be shunted to the neighboring regions. In contrast, the present model assumes that 95% and 5% of the reabsorbed plasma, but not RBCs, in R1 and R2, respectively, are taken up by LAV; the remainder of the plasma and all the RBCs are shunted to R2 and R3, respectively. As a result, water reabsorption from the RBC in R3 and R4 lowers the interstitial fluid osmolality in those outer regions and in tubules that reside in those regions.

### O_{2} Consumption by mTAL Cells

Na^{+}-K^{+}-ATPase stoichiometry indicates that 1 mol of ATP is needed to actively carry 3 mol of Na^{+} across the pump, which corresponds to T_{Na}/*Q*= 18 under maximal efficiency (38). However, measurements in the kidney and in various segments of the nephrons have yielded higher Na^{+}-to-ATP ratios, which implies higher T_{Na}/*Q*(21). In particular, measurements in the thick ascending limbs yielded T_{Na}/*Q*= 36 (11). The discrepancy in these measurements may be attributable to the transport processes of Na^{+} (or, in some cases, Cl^{−} or its counterions), which are facilitated by the Na^{+}-K^{+}-ATPase pump but do not consume O_{2} (see discussion in Ref. 48). Such transport processes, which may proceed through transcellular (via transporters other than the Na^{+}-K^{+}-ATPase pump) or paracellular pathways, are not explicitly discussed in this model because of its simple single-barrier representation of transepithelial transport processes. To account for these additional routes of Na^{+} reabsorption, we use a higher T_{Na}/*Q*= 24.

Na^{+}-K^{+}-ATPase pumps must hydrolyze ATP to export Na^{+} and import K^{+}. As reviewed by Gullans and Mandel (21), in all renal segments, most of the ATP appears to originate from mitochondrial oxidative metabolism. Inhibition of mitochondrial oxidative metabolism markedly stimulates glycolysis in mTALs; however, even under anaerobic conditions, the amount of ATP generated by glycolysis is a small fraction of that produced by oxidative metabolism (21). Thus only aerobic active Na^{+} transport is represented in the base case. In a separate simulation, we included anaerobic active Na^{+} transport along the mTAL. Our results suggest that if anaerobic transport were able to account for ∼38.9% of the total mTAL active Na^{+} transport, the osmolality gradient along the corticomedullary axis would be significantly enhanced, and CD fluid osmolality at the OM-IM junction would increase by 14.7% from the baseline value of 580 to 665 mosmol/kgH_{2}O.

### Regulation of O_{2} Supply and Demand

The high rate of active NaCl reabsorption and the low availability of O_{2} in their vicinity render the mTALs particularly susceptible to anoxic damage. As described by Brezis et al. (6), perfusion of isolated rat kidneys with cell-free medium produces an anatomic lesion sharply localized to mTAL cells; the lesion appears first in mTALs most removed from vascular bundles and near the IM. Other studies (for review see Ref. 3) have shown that inhibition of NaCl transport with furosemide or ouabain reduces tubular necrosis, whereas increases in Na^{+}-K^{+}-ATPase activity markedly enhance hypoxic damage. To protect itself against hypoxic injury, the kidney uses two types of mechanisms: medullary vasodilation to augment blood supply and reduction of transport work to lower O_{2} consumption (3). Schurek and Johns (50) suggested that tubuloglomerular feedback, which reduces the glomerular filtration rate, may serve as a tool to adapt the tubular energy demand to its actual O_{2} supply.

### Comparison With Other Tissues

As discussed by Cohen (10), the mammalian kidney, which constitutes ∼0.5% of the total body weight, consumes ∼10% of the body's O_{2} uptake. Thus, on a tissue weight basis, O_{2} consumption by the kidney (5 ml O_{2}·min^{−1}·100 g^{−1}) is second only to that of the working heart (8 and 70 ml O_{2}·min^{−1}·100 g^{−1} at rest and during heavy exercise, respectively). O_{2} consumption in the brain, another work-intensive tissue, is slightly lower (3 ml O_{2}·min^{−1}·100 g^{−1}), whereas that in the skin is on the order of 0.2 O_{2}·min^{−1}·100 g^{−1} (25). Yet, since the renal blood flow rate is so high [678 ml·min^{−1}·100 g^{−1} (31)], the extraction of O_{2} from each millimeter of blood is significantly lower in the kidney than in other tissues. Blood flow rate per 100 g of tissue is 5 times lower in the heart (10), 15 times lower in the brain, and 55 times lower in the skin (31).

In conclusion, our new model of O_{2} transport in the rat OM suggests that the structural organization of the OM results in significant Po_{2} gradients in the axial and radial directions. The segregation of DVR, the main supply of O_{2}, at the center and immediate periphery of the vascular bundles gives rise to large radial differences in Po_{2} between regions, limits O_{2} reabsorption from long DVR, and helps preserve O_{2} delivery to the IM.

## APPENDIX

The general framework for the region-based model is that developed by Layton and Layton (33). We expanded their model to include RBCs, O_{2}, Hb, and HbO_{2}, as well as metabolic reactions.

### Water and Solute Conservation in Tubules and Vessels

Steady-state conservation equations are used to predict the volumetric flow rate (F_{i,v}) and the concentration of solute *k* (C_{i,k}) in a tubule or vessel *i* (A1) (A2) where *x* is the position along the OM, ranging from 0 at the corticomedullary junction to *L* at the OM-IM boundary; *J*_{i,V} and *J*_{i,k} denote the transmural line flux of water and solute *k* into tubule or vessel *i*; and *A*_{i} designates the cross-sectional area of the lumen of *i* (i.e., based on its inner diameter), whereas *A*_{i,cell} denotes that of the surrounding cell layer (i.e., epithelium for tubules and endothelium for vessels). When *Eqs. A1* and *A2* are applied specifically to distributed short vasa recta (i.e., SDV and SAV), the number of which changes along the corticomedullary axis, variables must be expressed as a function of *x* and *y*, where *x* denotes the medullary depth at which the flow or flux is evaluated and *y* denotes the medullary depth reached by the terminus of the short vas rectum being considered (0 ≤ *x* ≤ *y* ≤ *L*).

We consider two types of consumption rates: R is the volumetric rate of metabolic consumption of solute *k* by the epithelial or endothelial cells surrounding tubule or vessel *i*, and R is the volumetric rate at which solute *k* is scavenged by other solutes (such as the reaction between Hb and O_{2}). By convention, all the rate terms are taken to be positive. Since we do not explicitly distinguish solute concentrations in the endothelial or epithelial layer surrounding the lumen of tubule or vessel *i*, we assume that a fraction θ_{cell} (taken as 0.5) of the amount of solute consumed in that layer is taken from the lumen, with the rest (1 − θ_{cell}) being taken from the interstitium (*Eqs. A2* and *A62*).

O_{2} is consumed by endothelial, epithelial, and interstitial cells for their basal metabolism and by epithelial cells in ascending and descending limbs for the active transport of Na^{+}. Thus (A3) (A4) where R and R are given in *Eqs. 6* and *9*, respectively. The rate of reaction between O_{2} and Hb (R), which is nonzero in RBCs only, is given in *Eq. 3*.

#### Conservation equations and BCs for individual tubules.

For *i* = SDL, LDL, SAL, LAL, or CD, the solute conservation equation (*Eq. A2*) can be rewritten as (A5) R is given in *Eqs. A3* and *A4*, and R is set to 0 for all other solutes.

The BCs at *x* = 0 for water flow and solute concentrations in SDL and LDL are summarized in Table 2. The BCs at *x* = *L* for SAL are obtained by continuity with SDL (A6a) (A6b) The BCs at *x* = *L* for LAL are as follows for water flow, Na^{+}, and urea (33) (A7a) (A7b) (A7c) O_{2} BCs are based on the following quantities: S is the overall supply of O_{2} (in the form of O_{2} and HbO_{2}) per nephron to the (outer) medulla, *Q* denotes the O_{2} consumption rate in the IM per nephron, taken to be 5% of S, and F is the amount of O_{2} (including that bound to Hb) per nephron that is returned to the OM from the IM via ascending limbs and vessels (A8a) (A8b) where the superscripts B, P, and R denote blood, plasma, and RBCs, respectively. With this notation, the O_{2} BC for LAL is given by (A9) where *n*_{i} denotes the number of tubules of vessels of type *i* per nephron and f is the fraction of O_{2} returning from the IM to the OM that flows within LAL (taken as 1%).

The BCs for water flow and Po_{2} in CDs are given in Table 2. Conservation of urea in the cortex, with the fractional absorption δ_{urea} (taken as 0.25) taken into account, can be written as (33) (A10a) We assume that the osmolality of the CD fluid at *x* = 0 is equal to that of plasma, so that the Na^{+} concentration in the CD at *x* = 0 is calculated as (A10b) where C_{plasma,O} denotes plasma osmolality (taken to be 309 mosmol/kgH_{2}O) and φ_{k} is the osmotic coefficient of solute *k* (taken as 1.84 for NaCl and 0.97 for urea).

#### Conservation equations and BCs for individual vessels.

In the model blood vessels, plasma and RBCs are treated as two separate compartments. F and F denote the plasma and RBC water flow rate, respectivelly, in vessel *i*, so that the total water flow rate in vessel *i* is F_{i,V} = F+ F. Similarly, C and C denote the respective plasma and RBC concentration of solute *k* in vessel *i*.

The transmural line flux entering the plasma from the interstitium is designated by the superscript “wall” and that entering the RBCs from the plasma is designated by the superscript “R”; therefore, the net water and solute fluxes entering the plasma of vessel *i* are given by (A11) (A12) Water conservation in plasma and RBCs can be expressed as (A13) (A14) Similarly, solute conservation in plasma and RBCs is expressed as (A15) (A16) The cross-sectional area of the RBC compartment is calculated as (A17) The BCs at *x* = 0 for water flow and solute concentrations in SDV and LDV are summarized in Table 2.

#### BCs for LAV.

The BCs at *x* = *L* for the two populations of LAV (LAVa and LAVb) are as follows.

##### water.

(A18a) (A18b) where F_{urine,V} denotes the urine flow per nephron (taken to be 0.065 nl·min^{−1}·kidney^{−1}) and F_{LAL,V}(*L*) is given in *Eq. A9*. Once the total blood flow F_{LAVj,V}(*L*) is calculated, the RBC flow is determined assuming that the LAV-to-LDV hematocrit ratio at *x* = *L* (f_{hem,L}) is known (A19c) On the basis of baseline results from the study of Layton (32), f_{hem,L} is taken as 10.8:16.9.

##### sodium and urea.

We assume that plasma (and RBC) concentrations of Na^{+} and urea are equal in LAVa and LAVb at *x* = *L*; that is, C(*L*) = C(*L*) and C(*L*) = C(*L*) (*k* = Na^{+} and urea). We also assume that the RBC-to-plasma concentration ratio is equal in LAV and LDV at *x* = *L*, so that C(*L*)/C(*L*) = C(*L*)/C(*L*). The plasma concentrations of Na^{+} and urea at *x* = *L* can then be found by solving the following conservation equation (A20) where F_{urine,k} denotes the urine flow of solute *k* per nephron. The ratios C_{LAL,Na}(*L*)/C_{LAVa,Na}(*L*) and C_{LAL,urea}(*L*)/C_{LAVa,urea}(*L*) are given in *Eqs. A7b* and *A7c*.

##### oxygen.

The fractions of O_{2} (O_{2} + HbO_{2}) returning from the IM to the OM that flows within LAVa (f) and LAVb (f) are taken to be fixed (A21a) where F is the total amount of O_{2} in blood in vessel *i* (*Eq. A8b*) and F is the total amount of O_{2} that flows back into the OM via ascending limbs and vessels (*Eq. A8a*). As noted above, we set f= 38.8% and f= 60.2%. In addition, conservation of Hb and reaction equilibrium at the OM-IM boundary can be expressed as (A21b) (A21c) These nonlinear equations are solved simultaneously to yield C(*L*), C(*L*), and C(*L*). We also assume that O_{2} concentration in LAV*j* plasma at *x* = *L* is equal to that in the surrounding region.

#### BCs for SAV.

The fractional population of SDV (ω_{SDV}) and SAV (ω_{SAV}) that reaches medullary depth *x* is given by (33) (A22) where *L*_{OS} is the outer stripe thickness. The plasma water flow of SAV*j* originating at *y* = *x* is given by (A23) where *j* = *a* or *b*, *Q*_{SAVj} is the total fluid accumulation carried away by SAV_{j} at a medullary level *x*, and α_{SAV} is the fraction of *Q*_{SAVj} that enters the lumen of the SAV*j* that originate at *x*. The remaining fraction enters through the endothelial pores of the SAV*j* that originate at *y* > *x*. The plasma concentration of solute *k* in SAV*j* originating at *y* = *x* is taken as that of the region in which the SAV*j* are located, i.e. (A24a) (A24b) Water and solute flows in the RBCs associated with the SAV are obtained by continuity with the iRBC tubes at medullary level *x* that empty into the SAV*j* originating at that level (A25a) (A25b) (A25c) (A25d) where dr_{mn} (*m* = 1, 2 and *n* = 3, 4) denotes the length of the corresponding iRBC tube (Fig. 2) and is given by (A26) where *r*_{Ri} is the outer radius of region *Ri* and the factors ε_{1} and ε_{2} account for the tortuosity of the capillaries in the outer regions of inner stripe, where the interbundle capillary plexus was found to be significantly denser than in the outer stripe and the vascular bundles in the inner stripe (28).

#### Conservation equations and BCs for iRBC tubes.

At a given medullary level *x*, water and solute conservation equations in the iRBC tubes that form at the end of SDV terminating at *y* = *x* can be written as (A27) (A28) where *z* is the radial distance from the position of terminating SDV. The BCs at *z* = 0 are obtained by continuity with the terminating SDV (A29a) (A29b) (A30a) (A30b)

### Water and Solute Flux Calculations

#### Water flux into tubules and vessels.

The transmural flux of water into tubule *i* is given by (A31) where d_{i}(*x*) is the product of the water partial molar volume and the water permeability of *i*, κ_{i,R} is the fraction of tubule *i* in contact with region *R*, and σ_{k} is the reflection coefficient for solute *k* (taken to be 1 for all solutes). Similarly, the flux of water flowing from the interstitium into DVR plasma is given by (A32) When *Eq. A32* is applied to SDV, variables must be expressed as a function of *x* and *y*.

Since AVR are fenestrated, we assume that their transmural water flux is driven by hydraulic pressure and enters by advection; that is, we neglect osmotic and oncotic pressures. Hence, the transmural flux of water into the plasma of LAV*j* is given by (A33) where *Q*_{LAVj} is the fluid accumulation carried into LAV*j* (and is calculated as shown below). The composite transmural flux of water into the plasma of SAV*j* originating at *y* > *x* is given by (A34) Following the approach of Layton (32), the equivalent RBC radius that gives the total surface area of the RBCs as a cylinder is calculated as (A35) where *r*_{i} is the inner radius of vessel *i*; the surface area (s_{RBC}) and volume (υ_{RBC}) of a single RBC are taken as 129 μm^{2} and 61 μm^{3}, respectively (15). Similarly, the equivalent radius for iRBC tube *j* formed by termination of SDV*j* is given by (A36a) (A36b) *Equation 35* implicitly assumes that the entire surface of all the RBCs is exposed to plasma. Since RBCs are tightly packed within a central core in the vessel lumen, it is possible that *Eq. 35* overestimates the surface area available for exchange between the RBCs and the surrounding plasma.

The flux of water from plasma into RBCs is then expressed as (A37) where Δπ_{h} and Δπ_{p} are the oncotic pressures due to Hb in RBCs and to albumin and other proteins in plasma and interstitium, respectively (see below), *N* is Avogadro's number, *k*_{B} is the Boltzmann constant, and T is the absolute temperature (the product *Nk*_{B}T equals 19.3 mmHg/mM). When *Eq. A37* is applied to SDV and SAV, variables must be expressed as a function of *x* and *y*.

Similarly, the transmural flux of water from the interstitium into iRBCs can be expressed as (A38) where *R* in C_{R,k} depends on *j* and *z*.

For *j* = 1, *z* varies between *r*_{R1}/2 and *r*_{R2} + ε_{1}(*r*_{R3} − *r*_{R2})/2, so that *r*_{R1}/2 ≤ *z* ≤ *r*_{R1}, *R* = R1; *r*_{R1} ≤ *z* ≤ *r*_{R2}, *R* = R2; and *r*_{R2} ≤ *z* ≤ *r*_{R2} + ε_{1}(*r*_{R3} − *r*_{R2})/2, *R* = R3.

For *j* = 2, *z* varies between *r*_{R1}/2 and *r*_{R2} + ε_{1}(*r*_{R3} − *r*_{R2}) + ε_{2}(*r*_{R4} − *r*_{R3})/2, so that *r*_{R1}/2 ≤ *z* ≤ *r*_{R1}, *R* = R1; *r*_{R1} ≤ *z* ≤ *r*_{R2}, *R* = R2; *r*_{R2} ≤ *z* ≤ *r*_{R2} + ε_{1}(*r*_{R3} − *r*_{R2}), *R* = R3; and r_{R2} + ε_{1}(*r*_{R3} − *r*_{R2}) ≤ *z* ≤ *r*_{R2} + ε_{1}(*r*_{R3} − *r*_{R2}) + ε_{2}(*r*_{R4} − *r*_{R3})/2, *R* = R4.

For *j* = 3, *z* varies between (*r*_{R1} + *r*_{R2})/2 and *r*_{R2} + ε_{1}(*r*_{R3} − *r*_{R2})/2, so that (*r*_{R1} + *r*_{R2})/2 ≤ *z* ≤ *r*_{R2}, *R* = R2; and *r*_{R2} ≤ *z* ≤ *r*_{R2} + ε_{1}(*r*_{R3} − *r*_{R2})/2, *R* = R3.

For *j* = 4, *z* varies between (*r*_{R1} + *r*_{R2})/2 and *r*_{R2} + ε_{1}(*r*_{R3} − *r*_{R2}) + ε_{2}(r_{R4} − r_{R3})/2, so that (*r*_{R1} + *r*_{R2})/2 ≤ *z* ≤ *r*_{R2}, *R* = R2; *r*_{R2} ≤ *z* ≤ *r*_{R2} + ε_{1}(*r*_{R3} − *r*_{R2}), *R* = R3; and *r*_{R2} + ε_{1}(*r*_{R3} − *r*_{R2}) ≤ *z* ≤ *r*_{R2} + ε_{1}(*r*_{R3} − *r*_{R2}) + ε_{2}(*r*_{R4} − *r*_{R3})/2, *R* = R4.

On the basis of the theoretical framework developed by Sidel and Solomon (51), the oncotic pressure due to Hb is calculated as (15) (A39) (A40) where C and C are the molal and molar concentrations of Hb (i.e., one-fourth of the total concentration of heme groups), respectively, and ν̄ is the partial specific volume of Hb (0.75 ml/g). The oncotic pressure due to albumin and other proteins is calculated as (15) where C_{pr} is the total protein concentration (in g/dl). Since we do not explicitly include the transport of plasma proteins in this study, C_{pr} is fixed at 6.8 g/dl (32).

#### Solute flux into tubules and vessels.

The transmural flux of solute *k* into tubule *i* is given by (A41) where Ψ is the rate of active transport of solute *k* in tubule *i* (*Eq. 1*). That term is zero everywhere, except for Na^{+} in SAL and LAL.

The transmural flux of solute *k* flowing from the interstitium into DVR plasma is given by (A42a) (A42b) The AVR transmural solute fluxes are assumed to arise from solute advection passing through fenestrations and from solute diffusion through fenestrations, so that (A43a) (A43b) where Pe_{i,k} ≡ *J*/*P*_{i,k} is the Péclet number for solute *k* associated with *i*, and *P*_{i,k} is the effective permeability of *i* to solute *k*.

The flux of solute *k* from plasma into RBCs in vessel *i* is expressed as (A44a) (A44b)

The transmural flux of solute *k* from the interstitium into iRBCs is expressed as (A45) where *R* in C_{R,k} is defined after *Eq. A38*.

### Water Conservation Equation in the Concentric Regions

#### Water fluxes into regions.

The water flux from tubules and LDV into region *R* is calculated as (A46) The composite water flux from SDV terminating at *y* > *x* is given by (A47) Water flux from iRBC tube *j* into region *R* is assumed to be osmotically driven and depends on the distance over which the iRBC tube traverses through region *R*. In R1, SDVa is assumed to terminate and give rise to iRBC1 and iRBC2 midway between the center and perimeter of R1 (Fig. 2). Thus (A48) The iRBC3 and iRBC4 tubes, which arise from SDVb that is located in R2, contribute no water flux into R1. In R2, iRBC1 and iRBC2 traverse through the whole length, whereas SDVb terminates and gives rise to iRBC3 and iRBC4 midway between the perimeters of R1 and R2 (Fig. 2) (A49a) (A49b) In R3, iRBC1 and iRBC3 combine to form SAVa midway between R2 and R3, whereas iRBC2 and iRBC4 traverse through R3 into R4 (A50a) (A50b) In R4, iRBC2 and iRBC4 combine to form SAVb (A51) iRBC1 and iRBC3 have no contribution.

#### Interstitial water conservation equations.

Following the approach of Layton and Layton (33), fluid accumulation in each region *Rj* (*j* = 1–4) is calculated by solving the water conservation equation in that region (A52a) (A52b) (A52c) (A52d) In *Eqs. A52a* and *A52b*, the first term represents the composite water flux from SDV*j* terminating at *y* > *x* into R1 and R2, respectively. The second term represents the water flux from SDV*j* terminating at *y* = x (α_{SDVa} denotes the fraction of the capillary plasma flow from terminating SDVa that is directed into R1, taken as 10%). These two terms are absent from *Eqs. A46c* and *A46d*, because there are no SDV in regions R3 and R4. The third term corresponds to the water flux from iRBCs formed by the terminating SDV*j* into region *R*. The fourth term is the total water flux from the vessel or limb *i* into *R* (*i* = LDL, LAL, SDL, SAL, and CD). The remaining terms represent the disposition of fluid accumulation by means of fluid entry into AVR and by interregional flow; *Q*_{R,R′} represents the interregional flow from region *R* to *R*′. A fraction β_{LAVa} (taken as 95%) of the net fluid accumulation in region R1 is taken up by LAVa through its fenestrations, and the remainder enters R2 as capillary flow. Similarly, a fraction β_{LAVb} (taken as 5%) of the net fluid accumulation in R2 is taken up by LAVb, and the remainder enters R3. A fraction β_{SAVa} (taken as 80%) of the net fluid accumulation R3 is taken up by SAVa, and the remainder enters R4. Hence (A53a) (A53b) (A53c)

### Solute Conservation Equation in the Concentric Regions

#### Solute fluxes into regions.

The solute flux from tubule *i* into region *R* is given by (A54)

The solute flux from LDV into region *R* is given by (A55)

For SDV terminating at *y* > *x* and *R* = R1 and R2, we have (A56)

For SAV originating at *y* > *x* and *R* = R3 and R4, we have (A57) The solute flux from iRBC tubes into region *R* depends on the region being considered. In R1 (A58) In R2 (A59a) (A59b) In R3 (A60a) (A60b) In R4 (A61)

#### Interstitial solute conservation equation.

Interstitial concentrations in each region *R* can be obtained by solving the solute conservation equation in *R* (A62)

The first term represents the diffusive solute into region *R* from adjacent regions *R*′. The second term is the sum of solute fluxes from tubules and long vasa recta into *R*. The third and fourth terms denote the composite solute fluxes at level *x* from all SDV and SAV, respectively, that are present in region *R* and that reach to medullary level *y* > *x*. The first term in the first pair of square brackets represents the solute flux from SDV terminating at level *y* = *x* into region *R*; the coefficients α_{SDVj,R} are given by α_{SDVa,R1} = α_{SDVa}, α_{SDVa,R2} = 1 − α_{SDVa}, α_{SDVb,R1} = 0, α_{SDVb,R2} = 1, and α_{SDVj,R3} = α_{SDVj,R4} = 0 (*j* = a, b). The next term in the first pair of square brackets is the solute flux from iRBC tubes into *R*. The term C_{R′,k}*Q*_{R′,R} − C_{R,k}(*Q*_{AVR} + *Q*_{R,R′}) represents the solute that is carried by flow at the local concentration into AVR or into an adjoining region *R*′. The next term involving (1 − θ_{cell}) represents the fraction of the net amount of solute produced by endothelial and epithelial cells that is released into the interstitium (*Eq. A2*). The last two terms denote the consumption rates of solute *k* by interstitial cells in region *R*, by metabolic and scavenging reactions, respectively; *Ā*_{R} is the area, per nephron, occupied by interstitial cells in region *R*.

We neglect interstitial diffusion in the axial direction, inasmuch as its effects on water flow and solute concentration are predicted to be small (34).

## GRANTS

This work was supported by National Institute of Diabetes and Digestive and Kidney Diseases Grant DK-53775 to A. Edwards and A. T. Layton and National Science Foundation Grant DMS-0701412 to A. T. Layton.

- Copyright © 2009 the American Physiological Society