A mathematical model of O₂ transport in the rat outer medulla. I. Model formulation and baseline results

Abstract

The mammalian kidney is particularly vulnerable to hypoperfusion, because the O₂ supply to the renal medulla barely exceeds its O₂ requirements. In this study, we examined the impact of the complex structural organization of the rat outer medulla (OM) on O₂ 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 O₂. We considered basal cellular O₂ consumption and O₂ 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 Po₂ gradients in the axial and radial directions. The segregation of descending vasa recta, the main supply of O₂, at the center and immediate periphery of the vascular bundles gives rise to large radial differences in Po₂ between regions, limits O₂ reabsorption from long descending vasa recta, and helps preserve O₂ delivery to the inner medulla. Under baseline conditions, significantly more O₂ 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 O₂ supplied to the medulla is consumed in the OM and that medullary thick ascending limbs operate on the brink of hypoxia.

renal oxygen tension (Po₂) 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 Po₂ 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 O₂ between descending and ascending vessels and limbs significantly reduces O₂ 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 O₂ supply to the renal medulla barely exceeds its O₂ 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 Po₂, 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 O₂ supply as a result of its high metabolic requirements, as well as its distance from descending vasa recta (DVR), i.e., the source of O₂ 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 O₂ 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 O₂ distribution in the OM, we incorporated RBCs, as well as Hb and O₂.

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, O₂, Hb, or HbO₂

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

O₂ flux from region R into neighboring regions via diffusion

J_R,adv

O₂ 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 O₂ in vessel i

F_O2^OM,ascending

Total amount of O₂ 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 O₂ basal consumption

k₁/k₋₁

Dissociation/association constant of HbO₂

L

Length of OM

n_i

Number of tubules or vessels i per nephron

P_O2^C

Overall O₂ permeability of capillary

P_O2^R

O₂ permeability of RBC membrane

P_O2^W

O₂ permeability of capillary wall

P_R,R′,k

Permeability of boundary between regions R and R′ to solute k

Q_O2^IM

Total O₂ consumption in IM

Q_O2^OM,active

Overall O₂ consumption for active transport in OM

Q_O2^OM,basal

Overall O₂ 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

O₂ volumetric consumption rate for active transport in tubule i

R_i,O2^basal

O₂ volumetric consumption rate for basal metabolism in tubule or vas rectum i

R_max,O2^basal

Maximum volumetric rate of O₂ basal consumption

R_RBC,O2^scav

Volumetric rate of O₂ scavenging by RBCs

S_O2^OM

Overall O₂ 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.

Fig. 1.

Schematic representation of the region-based model: cross sections through outer stripe (A) and inner stripe (B) of rat outer medulla (OM). In the model formulation, the 4 regions (R1–R4) have coincident centers; display is intended to minimize the figure area. LDV, long descending vas rectum; SDVa and SDVb, 2 populations of short descending vasa recta; LAVa and LAVb, 2 populations of long ascending vasa recta; SAVa and SAVb, 2 populations of short ascending vasa recta; LDL, long descending limb of Henle's loop; SDL, short descending limb; LAL, long ascending limb; SAL, short ascending limb; CD, collecting duct. Relative weight of interaction between a type of vessel or tubule and a given region (i.e., the parameter κ_i,R in Eqs. A31, A32, A41, and A42) is represented by 0.25, 0.5, and 0.75.

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.

Fig. 2.

Schematic representation of interstitial RBC (iRBC) tubes that carry capillary RBC flow from the SDV (SDVa in R1, SDVb in R2) that terminate at a given level x, along the radial (z) direction, to the SAV (SAVa in R3, SAVb in R4) that originate at that level. RBCs travel a radial distance d_r before being taken up by SAV. r_Ri, outer radius of region Ri (i = 1, 2, 3, and 4). We assume that each population of short vasa recta is, on average, localized in the center of the region where it is distributed.

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 O₂-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 Po₂, below which O₂ consumption becomes limited by O₂ concentration, was found to be very low, <1 mmHg (10). Hence, we assume that, at <1 mmHg luminal Po₂, the active transport rate is a linearly decreasing function of Po₂, that is

(2)

The results are indistinguishable if we replace luminal Po₂ with interstitial Po₂ 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).

O₂ Consumption

O₂ is carried to the medulla by DVR blood, in free form (O₂) and as HbO₂. In RBCs, the dissociation of HbO₂ into O₂ 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 O₂ scavenging by Hb in RBCs (R) is calculated as

(3)

Given that the kinetics of HbO₂ dissociation are significantly faster than O₂ diffusion, the reaction is generally taken to be at equilibrium, such that

(4)

where C₅₀ is the O₂ 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 O₂ (i.e., the C-to-Po₂ 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 Po₂, the dissociation constant k₁ is fixed (49 s⁻¹), and the association constant is calculated at each level as a function of local O₂ concentrations, to yield compatibility with the Hill equation for saturation

(5)

We consider basal O₂ consumption by interstitial cells, vascular endothelial cells, and tubular epithelial cells (Eqs. A2–A4 and A62). The volumetric rate of basal O₂ consumption in tubule or vas rectum i (R, in mol O₂·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 O₂ 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 O₂ 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. Po₂ 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 m³/nephron. If R= 10 μM/s, then Q≈ 2.0 × 10⁻¹⁴ mol O₂·s⁻¹·nephron⁻¹.

Under maximal efficiency, moles of Na⁺ reabsorbed through the transcellular pathway per mole of O₂ 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 O₂ consumption for active transport in tubule i (R, in mol O₂·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 O₂ 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 O₂·s⁻¹·nephron⁻¹.

BCs at the OM-IM Boundary

As described in the appendix, BCs for O₂ in LAV and LAL at the OM-IM boundary are determined on the basis of O₂ 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 O₂ consumption in the IM is 5% of O₂ supply to the OM.

We postulate that the architecture of the IM plays a determinant role in the distribution of O₂ 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 O₂ 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 HbO₂, the concentration of which is much greater than that of O₂. 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 O₂ and HbO₂ 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 O₂ 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 O₂ across vasa recta are taken from the study of Zhang and Edwards (57). The permeability of medullary tubules and vessels to O₂ has not been measured directly, to the best of our knowledge. The (overall) capillary permeability to O₂ (P) can be expressed as

(12)

where P and P denote the O₂ 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 O₂ across vascular endothelial cells and tubular epithelial cells is the same. The baseline values of parameters related to O₂ transport are summarized in Table 1.

Table 1.

O₂ transport parameter values

Parameter	Value	Ref.
Half-saturation in Hill equation (C50)	26.4 mmHg	9
Hill equation parameter (n)	2.6	9
Dissociation constant (k1)	49 s−1	2†
O2 solubility coeff in plasma	1.34 μM/mmHg	23
O2 solubility coeff in RBC	1.56 μM/mmHg	9
Initial overall heme concn	20.3 mM	9
Diffusivity of O2 in tissue (DO2)	2,800 μm2/s	7
Maximum volumetric rate of basal O2 consumption (Rmax,O2basal)	10 μM/s	Present study
Michaelis-Menten constant for basal O2 consumption (KM,O2)	4 mmHg	7‡
Maximum rate of Na+ active transport (Vmax,i,Na)		33
Outer stripe	10.5 nmol·m−2·s−1*
Inner stripe	25.9 nmol·m−2·s−1*
Michaelis-Menten constant for Na+ active transport (KM,mTAL,Na)	70 mM	33
O2 permeability of capillary and tubule wall (PO2W)	0.04 cm/s	Present study

^†Assuming that k₋₁ = 3.5×10⁻⁶ M⁻¹·s⁻¹ at x = 0.

^‡Assuming a resting NO concentration of ≈ 80 nM.

^*Short and long ascending limbs.

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).

Table 2.

Boundary conditions for descending tubules and vessels at x = 0

	SDL	LDL	CD	DVR
				Plasma	RBC
Fv,† nl/min	10	12	6.1	6	2
CNa, mM	160	160	88.2*	163.7	0
Curea, mM	15	15	151.2*	8	8
CO2, mM	48.2×10−3 (36 mmHg)	48.2×10−3 (36 mmHg)	48.2×10−3 (36 mmHg)	80.4×10−3 (60 mmHg)	80.4×10−3
CHb, mM	0	0	0	0	3.0‡
CHbO2, mM	0	0	0	0	17.3‡

Boundary conditions for water, Na⁺, and urea are taken from the study of Layton and Layton (33).

^*Collecting duct (CD) inflow concentrations are obtained as described in appendix (Eq. A10).

^†Flow rates (F_V) are given for per individual tubule or vessel. We assume that hematocrit (i.e., RBC-to-blood flow ratio) is 35%.

^‡

At x = 0, having specified and assuming kinetic equilibrium, and can be obtained by solving the following equations:

, whereC_Hb^total is the total concentration of the heme species, taken as 20.3 mM at x = 0.

Areas occupied by interstitial cells (Ā_R) are needed to estimate basal O₂ 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 ρ₁ and ρ₂ 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 O₂ Supply in the Rat OM

The overall supply of O₂ to the OM (S) is determined as the total molar flow rate per nephron of O₂ and HbO₂ 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)/dy, 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 HbO₂; we thus ignore the amount of O₂ supplied by aqueous fluids relative to that supplied by RBCs in the remainder of this study.

Interstitial, Intratubular, and Intravascular Po₂ Gradient

Using the base-case configuration, parameter set, and BCs, we solved the model equations to yield flow and concentration profiles. Figure 3 shows Po₂ 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 3E shows Po₂ 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 O₂ 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.

Fig. 3.

Po₂ in tubules, vasa recta, and concentric regions. A, B, C, and D: regions R1, R2, R3, and R4, respectively; tubules are assigned to the region with which they are in contact for ≥50% of their inner stripe (IS) length. E: Po₂ profiles in the interstitium of the 4 regions. Vertical dotted lines mark boundary between outer stripe (OS) and IS; x/L, ratio of axial coordinate to total length of OM.

Fig. 4.

A–D: O₂ flow in regions, tubules, and vasa recta. Notation is analogous to that in Fig. 3. E: HbO₂ flow in LDV RBCs.

Interstitial Po₂ gradients and their generation.

As described below, our model predicts that O₂ availability varies greatly from one region to another. In this section, interstitial Po₂ profiles in each region are depicted consecutively. Figure 3E shows a decreasing Po₂ 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. 3A). In R1, interstitial Po₂ decreases from 58.2 mmHg at the corticomedullary junction to 48.3 mmHg at the OM-IM boundary, as O₂ diffuses radially outward to O₂-poor regions. Because interstitial Po₂ in R1 lags behind that of the LDV, O₂ is continually reabsorbed from the LDV, the O₂ flow of which decreases from 0.0707 to 0.0494 pmol·min⁻¹·nephron⁻¹ (Fig. 4A).

Similarly, the R2 interstitial Po₂ profile exhibits a general decrease along the corticomedullary axis. The O₂ 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 O₂ demand results in a rise in R2 interstitial Po₂.

Po₂ is predicted to be low in the interbundle regions, i.e., in R3 and R4, where Po₂ hovers between 5.7 and 8.7 mmHg between the mid-outer stripe and the mid-inner stripe. These outer regions do not contain O₂-supplying vessels but, instead, contain most of the mTALs, which impose the largest O₂ 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 Po₂ 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 Po₂ gradients. Because the vessels are tightly packed in the vascular bundle core, diffusion of O₂ out of R1 is limited. Also, O₂ 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 Po₂ gradient, a difference of about two-thirds of R1 interstitial Po₂, 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 Po₂ profiles of these three regions differ significantly. In most of the inner stripe, Po₂ does not decrease monotonically in the radial direction from R1 to R4.

Near the cortex, interstitial Po₂ 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 Po₂ 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 Po₂ 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 O₂ permeability of LDL, there is a large efflux of O₂ from the LDL near the corticomedullary junction and, thus, a precipitous drop of Po₂ in LDL and a subsequent decrease in interstitial Po₂ 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 Po₂ at the corticomedullary boundary and would also allow O₂ to diffuse into the upper outer stripe from the O₂-rich cortex, increasing the interstitial Po₂ and smoothing any boundary layer.

Intravascular and intratubular Po₂ gradients and their generation.

Basal metabolism of all tubules and vessels and active transport metabolism in mTALs consume O₂. Taken in isolation, this consumption should result in decreasing O₂ flows in all tubules and vessels along their flow directions. However, tubules and vessels are assumed to be highly permeable to O₂ (0.04 cm/s), which results in significant O₂ diffusion across epithelial or endothelial walls when an O₂ gradient is established. Thus tubular and vascular plasma Po₂ profiles generally approximate those of the surrounding interstitium. Figure 3, A–D, shows Po₂ profiles for each class of tubules and vessels; the corresponding O₂ 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 O₂ 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 Po₂. As Po₂ in DVR plasma decreases, within RBCs O₂ dissociates from HbO₂ and, subsequently, diffuses down a concentration gradient into the surrounding plasma (see HbO₂ flow profiles for LDV in Fig. 4E). Along the LDV, 7.4% of the HbO₂ that enters at the corticomedullary boundary has dissociated before reaching the IM.

The Po₂ profiles of LAVb and SDL have similar gradients in the inner stripe but diverge in the outer stripe (Fig. 3B). 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 Po₂ is significantly lower.

Intratubular fluid Po₂ 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 O₂ 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 Po₂ along the direction of their flow. However, Po₂ is lower within the mTAL lumen than in the surrounding interstitium, so that O₂ diffuses into the mTAL. These two competing factors result in a general increase in mTAL O₂ flow in the SAL in the inner stripe and in the LAL along their flow direction and a decrease in SAL O₂ flow in the outer stripe.

Fig. 5.

Na⁺ concentration (C_Na) profiles in tubules (A) and vessels (B) along the corticomedullary axis. Fluid C_Na increases in all tubules and vessels along the corticomedullary axis, except along the prebend segments of the SDL and along the SAV near the OM-IM boundary.

Radial O₂ Transport: Diffusion vs. Capillary Transport

We then examined the relative contribution of different O₂ transport mechanisms. O₂ 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 O₂, 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 O₂ also diffuses out of the iRBC tubes into the interstitium. Interregion diffusive O₂ exchange takes place as O₂ permeates out of the LDV and SDVa into R1 and out of the SDVb into R2. A fraction of that O₂, which depends on the interregion permeability to O₂ (P), then diffuses radially toward the surrounding regions.

To compare the contributions of diffusion vs. advection to interregion O₂ transport, we computed the fluxes of O₂ 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 O₂ flux from the iRBCj 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 O₂ concentration in region R, and Q_R,R′ represents the interregional flow from region R to R′.

Figure 6 shows interregion diffusive and advective O₂ fluxes as a function of medullary depth. In all regions, advective O₂ 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).]

Fig. 6.

Diffusive and advective interregion O₂ fluxes, taken positive into a region. A–D: regions R1–R4, respectively.

Throughout R1, O₂ 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 O₂ 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 Po₂ in R1 is more than twice that in R2, O₂ also diffuses into R2. Since capillary flow is small and R1 Po₂ 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 O₂ enters R3, driven by capillary flows (advection) and by a favorable interregion O₂ 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 O₂ diffusion remains relatively low, despite a significant Po₂ gradient, can be attributed to the small interstitial area in R1 (Fig. 7), which limits diffusion.

Fig. 7.

Interstitial area (×10⁻⁶ cm²) in regions R1–R4.

Basal and Active Transport O₂ Consumption

We distinguish in this study between the basal and active components of O₂ 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 O₂ 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 O₂ 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 O₂ is consumed in the renal medulla, mostly in the OM. The molar flow of O₂ 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 O₂ consumption in the IM is taken as 5% of medullary O₂ supply, overall O₂ 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 O₂ consumption, per nephron, along each class of tubules and vessels and in each concentric region. The basal (or active) O₂ 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 O₂ consumption reported for region Ri reflects the basal consumption of the interstitial cells in Ri. Basal O₂ 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.

Table 3.

Basal and active transport O₂ consumption

	O2 consumption, pmol·min−1·nephron−1
	Basal	Active	Total
R1	0.003121	0	0.003121
R2	0.03219	0	0.03219
R3	0.05676	0	0.05676
R4	0.03539	0	0.03539
SDL	0.01878	0.2292	0.2480
SDL2	0.06558	0	0.06558
SAL	0.01906	8.2740	8.2931
SAL2	0.06556	8.2947	8.3603
CD	0.1219	0	0.1219
LDL	0.03571	0	0.03571
LAL	0.1133	13.8334	13.9467
LDV	0.007109	0	0.007109
LAVa	0.01580	0	0.01580
LAVb	0.02617	0	0.02617
SDVa	1.6615e-4	0	1.6615e-4
SDVb	0.01526	0	0.01526
SAVa	0.1047	0	0.1047
SAVb	0.03515	0	0.03515

Different basal O₂ 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 O₂-rich R1, the basal O₂ consumption, which depends on luminal Po₂, 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 Po₂.

The total OM O₂ 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 O₂ consumption. Of the active transport O₂ 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 O₂ 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 Po₂ 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 O₂ supply, anaerobic metabolism supplies enough energy to sustain an active Na⁺ transport rate that is half of the maximum rate when O₂ supply is abundant. With this assumption, Eq. 2 becomes

(20)

The rate of active O₂ 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 < 0.3L), luminal Po₂ 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 Po₂ are very low, anaerobic active Na⁺ transport becomes significant. At x = 0.95L, where luminal Po₂ 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 Po₂ profile (Fig. 3E).

Fig. 8.

Na⁺ reabsorption along SAL (A) and LAL (B) driven by aerobic (solid line) and anaerobic (dashed line) metabolism, with the assumption that in medullary thick ascending limbs (mTALs), glycolysis provides significant energy at low Po₂ (Eqs. 20 and 21). In the outer stripe, where mTAL luminal Po₂ is sufficiently high, all Na⁺ active transport is oxidative.

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/kgH₂O, respectively, at the OM-IM boundary compared with baseline values of 580, 581, and 450 mosmol/kgH₂O, 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 Po₂, 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 Po₂), 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 O₂ 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 O₂ (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 Po₂ gradients in axial and radial directions. Because almost four-fifths of the DVR turn within the OM, ∼80% of the O₂ 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 Po₂ drop along the corticomedullary axis in most structures. Moreover, the segregation of the DVR (i.e., the main supply of O₂) at the center and immediate periphery of the vascular bundles creates large regional differences in Po₂ 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 O₂ transport, by isolation of LDV within the relatively O₂-rich vascular bundle core, O₂ reabsorption from LDV is reduced. As a result, 92.4% of the O₂ 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 O₂-carrying DVR, together with the high O₂ consumption of the mTALs, mTAL luminal Po₂ 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 Po₂ in the renal OM generally range from 20 to 30 mmHg. Po₂ 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 Po₂ 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). Po₂ 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 O₂ 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 O₂ 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 O₂ consumption was underestimated in our simulations.

O₂ distribution affects the urine-concentrating mechanism, inasmuch as the active transport rate of Na⁺ may decrease when O₂ 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/kgH₂O at the OM-IM boundary, which is a 1.9-fold increase over the inflow osmolality (309 mosmol/kgH₂O). Such an increase is consistent with experimental measurements (49).

Comparison With Previous Models

O₂ 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 HbO₂ were developed by Adair (1) and Moll (39). The model of O₂ 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 O₂ 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 O₂ transport simulations are reviewed in greater detail by Hellums et al. (22) and Popel (46).

In our previous model of O₂ transport across vasa recta (57), the tubular system was not explicitly included; instead, we specified the volumetric O₂ consumption rate along the corticomedullary axis. The model was used to predict the overall O₂ consumption rate on the basis of measurements of the medullary interstitial Po₂ 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 O₂ 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 O₂ 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 Po₂ profile that was assumed in the previous model was based on OM measurements that likely correspond to averages of intra- and interbundle Po₂. Thus the previous model predicted a DVR-interstitium Po₂ difference that remained <10 mmHg throughout the medulla. In contrast, in the present model, DVR are surrounded by the O₂-rich R1 interstitium; thus the present model predicts larger DVR-R4 Po₂ differences of 44 mmHg at x = 0 to 46 mmHg at x = L but significantly smaller DVR-R1 Po₂ 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 Po₂ 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/kgH₂O, respectively, compared with 851, 822, and 586 mosmol/kgH₂O, 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/kgH₂O, 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.

O₂ 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 O₂ (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/kgH₂O.

Regulation of O₂ Supply and Demand

The high rate of active NaCl reabsorption and the low availability of O₂ 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 O₂ 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 O₂ 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 O₂ uptake. Thus, on a tissue weight basis, O₂ consumption by the kidney (5 ml O₂·min⁻¹·100 g⁻¹) is second only to that of the working heart (8 and 70 ml O₂·min⁻¹·100 g⁻¹ at rest and during heavy exercise, respectively). O₂ consumption in the brain, another work-intensive tissue, is slightly lower (3 ml O₂·min⁻¹·100 g⁻¹), whereas that in the skin is on the order of 0.2 O₂·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 O₂ 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 O₂ transport in the rat OM suggests that the structural organization of the OM results in significant Po₂ gradients in the axial and radial directions. The segregation of DVR, the main supply of O₂, at the center and immediate periphery of the vascular bundles gives rise to large radial differences in Po₂ between regions, limits O₂ reabsorption from long DVR, and helps preserve O₂ delivery to the IM.

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.

APPENDIX

The general framework for the region-based model is that developed by Layton and Layton (33). We expanded their model to include RBCs, O₂, Hb, and HbO₂, 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 O₂). 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).

O₂ 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 O₂ 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)

O₂ BCs are based on the following quantities: S is the overall supply of O₂ (in the form of O₂ and HbO₂) per nephron to the (outer) medulla, Q denotes the O₂ consumption rate in the IM per nephron, taken to be 5% of S, and F is the amount of O₂ (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 O₂ 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 O₂ returning from the IM to the OM that flows within LAL (taken as 1%).

The BCs for water flow and Po₂ 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/kgH₂O) 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 O₂ (O₂ + HbO₂) 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 O₂ in blood in vessel i (Eq. A8b) and F is the total amount of O₂ 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 O₂ concentration in LAVj 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 SAVj 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 SAVj that originate at x. The remaining fraction enters through the endothelial pores of the SAVj that originate at y > x. The plasma concentration of solute k in SAVj originating at y = x is taken as that of the region in which the SAVj 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 SAVj 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 ɛ₁ and ɛ₂ 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 LAVj is given by

(A33)

where Q_LAVj is the fluid accumulation carried into LAVj (and is calculated as shown below). The composite transmural flux of water into the plasma of SAVj 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 μm² and 61 μm³, respectively (15). Similarly, the equivalent radius for iRBC tube j formed by termination of SDVj 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_BT 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 + ɛ₁(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 + ɛ₁(r_R3 − r_R2)/2, R = R3.

For j = 2, z varies between r_R1/2 and r_R2 + ɛ₁(r_R3 − r_R2) + ɛ₂(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 + ɛ₁(r_R3 − r_R2), R = R3; and r_R2 + ɛ₁(r_R3 − r_R2) ≤ z ≤ r_R2 + ɛ₁(r_R3 − r_R2) + ɛ₂(r_R4 − r_R3)/2, R = R4.

For j = 3, z varies between (r_R1 + r_R2)/2 and r_R2 + ɛ₁(r_R3 − r_R2)/2, so that (r_R1 + r_R2)/2 ≤ z ≤ r_R2, R = R2; and r_R2 ≤ z ≤ r_R2 + ɛ₁(r_R3 − r_R2)/2, R = R3.

For j = 4, z varies between (r_R1 + r_R2)/2 and r_R2 + ɛ₁(r_R3 − r_R2) + ɛ₂(r_R4 − r_R3)/2, so that (r_R1 + r_R2)/2 ≤ z ≤ r_R2, R = R2; r_R2 ≤ z ≤ r_R2 + ɛ₁(r_R3 − r_R2), R = R3; and r_R2 + ɛ₁(r_R3 − r_R2) ≤ z ≤ r_R2 + ɛ₁(r_R3 − r_R2) + ɛ₂(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 SDVj terminating at y > x into R1 and R2, respectively. The second term represents the water flux from SDVj 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 SDVj 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′,kQ_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).