INTRODUCTION

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BY MODULATING BLOOD FLOW, vasa recta in the renal medulla can have a major effect on sodium and water homeostasis as well as on the long-term control of arterial blood pressure (3). The role of the medullary microcirculation in the urinary concentrating mechanism nevertheless remains poorly understood. Models of the urinary concentrating mechanism have generally neglected the role of vasa recta by assuming that the capillaries offer negligible resistance to transport of solute and water (10, 20). However, the ultrastructural heterogeneity of descending vasa recta (DVR) and ascending vasa recta (AVR), the anatomical differences between the outer and inner medulla, as well as the existence of facilitated transport pathways in DVR suggest that the microvasculature might play a significant role in regulating the exchange of sodium and urea.

In a previous study (5), we developed a new model of the renal microcirculation based on recent findings about the presence of water channels and urea transporters in DVR. We concluded that water channels probably reduce the efficiency of solute trapping in the medullary interstitium and that urea transporters might compensate for that effect. This model, heretofore referred to as the single unit model, was limited in that we simulated transport across a vascular unit consisting of one DVR originating at the corticomedullary junction of the kidney and extending to the papillary tip, where it gave rise to several AVR. In the present work, we extended the single unit model to account for the three-dimensional architecture of the renal medulla by considering all vessels which enter and leave the inner medulla via vascular bundles.

In simulations based upon the single unit model, volume uptake into AVR was predicted to be very large, and values of AVR-to-DVR blood flow rate ratios near the papillary tip were significantly above those reported experimentally (6, 23). These overpredictions may have stemmed from the fact that concentration polarization at the vessel walls was neglected. MacPhee and Michel (12) have shown that AVR can withstand external pressures greater than their internal pressure and have postulated that the accumulation of albumin at the AVR walls may be such that volume uptake is driven by small transmural hydrostatic pressure differences only. We examined this hypothesis in the present study.

Finally, we had previously assumed that urea transporters are expressed only in outer medullary DVR (OMDVR), because the permeability of OMDVR to urea is much greater than that to sodium, whereas inner medullary DVR (IMDVR) and AVR are equally permeable to both solutes (17). However, recent in situ hybridization experiments indicate that the urea transporter isoform UT3 is expressed in papillary DVR as well as in OMDVR (21). In vivo microperfusion experiments by Pallone et al. (17), suggesting that the urea transporter is present in OMDVR only, may have been limited by boundary layer effects. The transporter UT3 is the rat homolog of the human erythrocyte urea transporter HUT11, also found in OMDVR and IMDVR (22). We therefore assumed here that the urea transporter is present both in OMDVR and IMDVR. The overall objective of this work was to gain more insight into the factors that govern volume and solute transcapillary exchange in the renal medulla.

	METHODS

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In this model, we focused exclusively on those vasa recta that are destined for the inner medulla, i.e., those that lie in the center of the vascular bundles and do not perfuse the outer medullary capillary plexus.

Mass balance equations. As DVR descend to varying depths of the medulla and convert into AVR, the number of vessels (N) varies along the corticomedullary axis. If x is the axial coordinate, ranging from 0 at the corticomedullary junction to L at the tip, D is the vessel diameter, Q(x) is the total flow rate at a given x, and J_v is the single-vessel cross-sectional flow rate from the vessel to the interstitium, then the flow rate at (x + dx) is given by

(1)

where Q(x)/N(x) corresponds to the single-vessel flow rate and N(x + dx)  N(x) to the number of added vessels between x and (x + dx), so that the second term on the right side of Eq. 1 represents the variation in Q due to the varying number of vessels. Taking the limit of Eq. 1 as dx goes to zero, we obtain

(2)

If Q^P and Q^R denote the total plasma and red blood cell (RBC) volume flow rates, respectively, then the steady-state differential equations expressing conservation of volume are given by

(3)

(4)

where J^P_v and J^R_v are the volume fluxes across the capillary wall and the RBC membrane, respectively, and  is the ratio of cell-to-vessel surface area averaged over time (16).

Assuming that RBCs are impermeable to solute i (i = sodium, albumin and other plasma proteins), the corresponding mass balance can be written as

(5)

where C^P_i is the molar plasma concentration of solute i, and J^P_i is its molar flux from plasma to interstitium. Conservation of urea, which is exchanged across the RBC membrane, yields

(6)

(7)

where C^R_u is the urea concentration in RBCs, J^R_u is the molar flux of urea across RBCs, J^P_u and J^P_uc are the paracellular and carrier-mediated transcapillary molar fluxes of urea, respectively, and f is the fractional volume of distribution of urea within RBCs. All other mass balance equations used in the single unit model must be transformed in an analogous manner (Ref. 5) by adding a derivative term to the right side to account for changes in volume or mass flow rates that are due to variations in vessel numbers.

As described previously, water and solute exchange between vasa recta and interstitium occurs through a shared paracellular pathway and is driven by the hydraulic and oncotic pressure differences between the two compartments (i.e., classic Starling forces). In DVR only, water can also be transported across an additional, transcellular pathway consisting of water channels that are impermeable to solutes; volume flux across this second route is predominantly governed by sodium and urea concentration gradients. The general expressions for paracellular and transcellular volume fluxes (J^P_vp and J^P_vt, respectively) from plasma to interstitium can be written as

(8)

(9)

where L^P_p and L^P_t represent the hydraulic conductivities of the paracellular and transcellular pathways, respectively, P is the transcapillary hydraulic pressure difference, _a and _p are the transcapillary oncotic pressure differences due to albumin and other plasma proteins, respectively, and _a is the reflection coefficient of the paracellular pathway to albumin. The concentrations of solute i (i = sodium, urea) in plasma and interstitium are given by C^P_i and C^I_i, respectively, and _i is the activity coefficient of i. The (paracellular) transcapillary flux J^P_i of solute i (i = albumin, sodium, urea) is given by

(10)

(11)

where P_i and _i are the (paracellular) permeability and reflection coefficient of the capillary wall to solute i, respectively, and Pe is the Peclet number. The carrier-mediated flux of urea can be written as

(12)

where P_uc is the permeability of the urea transporter. Expressions for fluxes across RBCs are given in Edwards and Pallone (5).

Number of vasa recta. The number of DVR and AVR versus axial coordinate has not been explicitly reported in the literature and is estimated from AVR-to-DVR number ratios and the fraction of medullary cross-sectional area that is vasa recta. Since we only considered vasa recta destined to the inner medulla, the number of AVR and DVR in the outer medulla remains constant. The cross-sectional area of the inner medulla was determined by Becker (1) and can be approximated by the following polynomial

(13)

where is zero at the outer-inner medullary junction and unity at the papillary tip, and the area is given in square centimeters. Measurements by Knepper et al. (8) of fractional volume distributions as a function of axial position in rat renal medulla indicate that the fractional area corresponding to vasa recta (F_VR) is approximately constant in the inner medulla; we therefore assumed that F_VR remains fixed. The data of Knepper et al. (8) suggest using a value of ~0.15. However, Zimmerhackl et al. (23) have reported estimates of DVR and AVR numbers and outer diameters (906 and 2,038, and 16.3 µm and 19.8 µm, respectively) at the base of the papilla that yield a cross-sectional area for vasa recta at this level of 0.82 mm². Assuming that the papillary base lies between  = 0.6 and 0.7, F_VR should be comprised between 0.3 and 0.5. The reason for this discrepancy is not clear, so that a range of possible values for F_VR was explored.

Measurements of the AVR-to-DVR number ratio (N_v) are difficult since DVR become fenestrated before they turn into AVR. At the papillary base, N_v was estimated as 2.25 (23), a value in agreement with a reported value of 2.2 near the papillary tip (2). We thus assumed that the AVR-to-DVR number ratio is constant. The respective numbers of DVR and AVR versus axial coordinate in the inner medulla are then given by

(14)

(15)

Plotted in Fig. 1 are the numbers of AVR and DVR as a function of position along the corticomedullary axis, with F_VR = 0.3 and N_v = 2.25, and assuming that the junction between the outer and the inner medulla lies at x/L = 0.24. As described above, the number of DVR and AVR is taken to be constant in the outer medulla and decreases proportionately to the surface area in the inner medulla.

View larger version (13K): [in this window] [in a new window]   Fig. 1.   Number of descending vasa recta (DVR) and ascending vasa recta (AVR) as a function of position along the corticomedullary axis (x). L, total length of the medulla. AVR-to-DVR number ratio is Nv = 2.25, and junction between outer and inner medulla lies at x/L = 0.24.

Countercurrent efficiency. An efficient countercurrent exchanger will minimize the net amount of sodium and urea removed from the medulla by vasa recta. To examine the main factors governing the amounts of water, sodium, and urea recovered from tissue between the papillary tip and a given location, volume, sodium, and urea removal indexes (I) were defined as a function of position along the corticomedullary axis as the differences between outflow and inflow to regions of the medulla below the given location, divided by the inflow at that location

(16a)

(16b)

(16c)

The higher the removal index, the greater the amount of fluid or solute removed from the interstitium by the microcirculation at a given location. A removal index equal to unity at the corticomedullary junction means that the outflow at that location is twice as large as the inflow.

Parameter values. Parameter values for the baseline case were identical to those reported for the single unit model (5) except for the following assumptions. The respective lengths of the outer medulla and inner medulla were taken as 1.9 and 5.9 mm (15). Hydraulic pressures and small solute concentrations in the outer medullary interstitium were calculated assuming that within the vascular bundles, vasa recta and interstitium can only exchange with one another, so that the sum of the fluxes for either volume, sodium, or urea is zero (5). In the inner medulla, small solute concentration gradients were determined by assuming that the total interstitial osmolality increases exponentially in the inner medulla, as suggested by the electron-microprobe data of Koepsell et al. (9). The osmolality profile was calculated based upon values of 305 mosM at the corticomedullary junction, 573 mosM at 2.3 mm from the papillary tip, and 1,011 mosM at 0.6 mm from the papillary tip (18). The fraction of osmolality due to sodium (f_Na) was determined by linear interpolation of data from Pallone et al. (18), yielding f_Na = 0.489  0.188x. The fraction of osmolarity due to urea is given by f_u = 1  2f_Na. The hydraulic pressure in AVR (P^A) was taken as 7.8 mmHg (19). Measured pressure drops in DVR from the base of the papilla to the tip are small, less than 0.5 mmHg (18). Our calculations indicate that the single vessel fluxes are similar in AVR and DVR, so that the pressure drop along AVR and DVR should be comparable; from this arises our assumption that P^A remains constant. In the inner medulla, the interstitial hydraulic pressure (P^I) was varied between 7.8 and 10.8 mmHg.

	RESULTS

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Solute transport across a membrane is partially controlled by the concentration boundary layers adjacent to the membrane, which offer additional resistance to mass transfer. The accumulation of retained solute next to a filtering membrane, referred to as concentration polarization, increases the effective osmotic pressure near the wall, and this effect is all the more likely as the filtration rate is high. We first assumed that concentration polarization was negligible at the AVR walls. Variations in total and single vessel plasma flow rates along the corticomedullary axis are shown in Figs. 2 and 3, respectively. Both interstitial and luminal hydraulic pressures were taken as constants in the inner medulla. Results are given for P = P^A  P^I ranging from 0 to 3 mmHg in Fig. 2 and for P = 2.5 mmHg in Fig. 3 (curve A), where P^A denotes the hydraulic pressure in the AVR lumen and P^I is that in the interstitium. As in the one-dimensional model, Q^P increases slightly in OMDVR, where the oncotic pressure difference favors volume uptake through the paracellular pathway; small solute concentration gradients are too small for the transcellular flux to be significant. In IMDVR, total plasma flow rate decreases sharply, parallel with the number of vessels. Volume efflux at the level of individual DVR begins halfway through the inner medulla, as small solute gradients become large and the transcellular flux is thus predominant. The value of P does not affect DVR plasma flow rate significantly, since the fluxes are governed mainly by protein and small solute concentration gradients. In AVR, total plasma flow rate increases with the number of AVR. In the inner medulla, individual AVR initially reabsorb significant volume from the interstitium due to favorable hydraulic and oncotic pressure gradients; as AVR plasma protein concentration and the transmural oncotic pressure difference are reduced by this large fluid influx, volume uptake slows down. As expected, the higher the P^I, the larger the volume influx.

View larger version (15K): [in this window] [in a new window]   Fig. 2.   Ratio of total plasma flow rate (QP) to initial DVR blood flow rate (QB0) as a function of position in DVR and AVR for different values of the transmural hydrostatic pressure difference across AVR: P = 0.0 (solid line), 1.0 (dotted line), 2.0 (dotted-dashed line), and 3.0 mmHg (dashed line). Concentration polarization is neglected.

View larger version (17K): [in this window] [in a new window]   Fig. 3.   Ratio of single vessel plasma flow rate (qP) to initial single DVR blood flow rate (qB0) as a function of position in DVR and AVR, with P (or Pmin) = 2.5 mmHg. Different assumptions are considered. Concentration polarization at AVR walls is either neglected (curve A) or accounted for by eliminating transmural oncotic pressure gradients across AVR and assuming that the transmural hydraulic pressure gradient remains constant (curve B) or decreases linearly from 0 to Pmin in the inner medulla (curve C).

The concentration of sodium and urea in DVR and AVR closely follows that in the interstitium, and the profiles are very similar to those obtained previously (5). The fact that the AVR-DVR urea concentration gradient, large near the corticomedullary junction, decreases significantly as blood flows toward the junction between the outer and the inner medulla is due to the presence of water channels [aquaporin-1 (AQP-1)] in DVR; the sieving of small solutes by AQP-1 near the corticomedullary junction rapidly raises their concentration. Although changes in P have almost no effect on sodium concentration, they do result in subtle differences in the urea concentration profile in the outer medulla. As illustrated in Fig. 4, as P increases, the urea concentration gradient is larger at the corticomedullary junction but is then more rapidly dissipated. Indeed, the higher the absolute value of P, the higher the AVR volume flow rate at the junction between the inner and the outer medulla and thus the smaller the decrease in C_u as plasma flows back into the outer medulla.

View larger version (22K): [in this window] [in a new window]   Fig. 4.   Ratio of plasma urea concentration (CPu) to its initial value in DVR (CPu0) as a function of position in DVR and AVR for P = 1.5 (solid lines), 2 (dotted lines), and 2.5 mmHg (dotted-dashed lines). Inset: in outer medulla only. Concentration polarization is neglected. In inset, each DVR curve is beneath corresponding AVR curve.

Since the net amount of volume absorbed by the microvasculature appears to depend highly on P, a reasonable estimate of P might be obtained by comparing our results with literature data. AVR-to-DVR blood flow rate ratios and protein concentration ratios were measured in several micropuncture experiments. Zimmerhackl et al. (23) obtained values of 1.20 and 0.78, respectively, at 1.94 mm from the papillary tip; Sanjana et al. (19) reported a very similar estimate of the AVR-to-DVR protein concentration ratio. However, our simulations indicate that even when P = 0 mmHg, the AVR-to-DVR blood flow rate ratio at that location is overpredicted by ~40% and the protein concentration ratio is underpredicted by 20%. Even if there were no hydraulic pressure in the interstitium (P^I = 0 mmHg, P = 7.8 mmHg), an unlikely condition given continuous volume reabsorption from Henle's loops and the collecting duct, then the calculated blood-flow-rate ratio would remain higher than the measured one.

The model may be predicting too large a volume uptake by AVR due to the fact that concentration polarization at the vessel walls was neglected. As fluid is reabsorbed into AVR, the accumulation of albumin on the interstitial side of the AVR wall could be such that the oncotic pressure is comparable on both sides, even if the bulk concentration of protein in the interstitium is lower than that in AVR. According to our model, the volume flux across AVR near the papillary tip is on the order of 6.5 × 10⁴ cm/s. If one assumes that the diffusivity of albumin in the interstitium is 10 times smaller than that in water (7), i.e., about 3.9 × 10⁸ cm²/s, and that the characteristic length for interstitial diffusion is 1 µm, or the thickness of the capillary wall, then the estimated interstitial Pe is 1.7. Convection is thus dominant in this region and the accumulation of albumin on the interstitial side of the AVR walls is likely to be significant.

On the basis of bulk concentrations and a reflection coefficient of AVR to albumin of 0.7 (13, 14), the effective oncotic pressure (_aa + _p, cf. Eq. 8) near the papillary tip is calculated to be 8.5 mmHg in the interstitium and 17.8 mmHg in the AVR lumen. However, a one-dimensional, stagnant film model indicates that if the boundary layer thickness is as large as 1 µm, then the interstitial concentration of albumin at the wall is about four times that in the bulk, which is taken to be 3.4 g/dl (5). Even if the boundary layer is only 0.3 µm thick, the wall-to-bulk concentration ratio is 1.5, resulting in an effective oncotic pressure of 13.9 mmHg next to the outer surface of AVR; boundary layer thicknesses of 0.4 and 0.5 µm yield values of 16.9 and 20.8 mmHg, respectively. In addition, the large volume influx will dilute protein concentration near the wall on the luminal side of the vessels and thus lower the oncotic pressure within AVR, thereby reducing the oncotic pressure gradient across the wall. To estimate the importance of the effects of reverse polarization, we used the results of Deen et al. (4). These authors calculated the extent of polarization for ultrafiltration in a cylindrical tube using two different models. Their study indicates that in the worst-case scenario, the concentration of albumin at the wall could differ from that in the bulk by 6 to 7%. Near the papillary tip, the bulk concentration of albumin is about 3.9 g/dl, and a 7% reduction in the concentration at the wall due to reverse polarization will lower the luminal oncotic pressure from 17.8 to 17.0 mmHg. The assumption that concentration polarization eliminates oncotic pressure differences across AVR would thus appear to be justified.

MacPhee and Michel (12) showed that AVR are able to remain open when the external pressure is higher than the internal pressure, and they postulated that the uptake of fluid by AVR could be driven by small hydrostatic pressure gradients only. Small solute driving forces across the thin descending limb and the collecting duct drive volume into the interstitium, thereby increasing the interstitial hydraulic pressure (P^I). We thus examined the hypothesis that P^I alone drives fluid uptake into AVR, i.e., that transmural albumin concentration gradients are ineffective due to polarization, which represents a limiting case. The transmural volume flux across AVR is then given by

(17)

where L^A_p is the AVR hydraulic conductivity.

The single vessel plasma flow rate obtained with this assumption is plotted in Fig. 3 (curve B), for P = 2.5 mmHg. The latter value was chosen because it yields the best agreement between calculated and measured AVR-to-DVR ratios of blood flow rate and protein concentration at 1.94 mm from the papillary tip; the predicted values are 1.24 and 0.83, respectively (vs. 1.20 and 0.78, respectively). MacPhee and Michel (12) reported that AVR collapse at a mean pressure of 4.0 cmH₂O, or 2.9 mmHg; our estimate of 2.5 mmHg is thus above this lower limit.

The unexpected finding, however, was that the net amount of volume reabsorbed by AVR is the same regardless of whether oncotic pressure gradients are considered (Fig. 3). Indeed, with our first assumption, the massive influx of fluid into AVR near the papillary tip soon sharply decreases the albumin concentration in AVR; the oncotic pressure gradient is then reversed, thereby decreasing the rate of volume uptake. On the other hand, when oncotic pressure gradients are entirely neglected, the transmural flux across AVR remains constant in the inner medulla (Eq. 17), and AVR reabsorb volume at a steady rate. The overall uptake is thus the same in both cases.

In other words, concentration polarization may render oncotic pressure gradients ineffective when the rate of volume influx into AVR is very large, but once massive reabsorption has occurred, these gradients cannot be omitted anymore since they actually serve to reduce volume influx. One way, short of solving complex partial differential equations, in which we can account both for this decrease in the driving force for reabsorption into AVR and for concentration polarization is to decrease the absolute value of P as blood flows back from the papillary tip to the junction between the inner and outer medulla, while still neglecting transmural oncotic pressure differences across AVR walls. In our third hypothesis, P was thus made to increase linearly from a minimum value, P_min, at the papillary tip, to zero at the junction between the inner and outer medulla, whereas the volume flux across AVR remained proportional to P (Eq. 17). Differences between the three assumptions are summarized in Table 1.

Variations in single vessel plasma flow rate along the corticomedullary axis, based upon assumption 3, are shown in Fig. 3, for P_min = 2.5 mmHg (curve C). The predicted AVR-to-DVR ratios of blood flow rate and protein concentration at 1.94 mm from the papillary tip are 1.20 and 0.85, respectively, in very good agreement with the measured values of 1.20 and 0.78. As illustrated in Fig. 3, the net amount of water uptake into AVR is significantly reduced when the driving force for reabsorption is made to decrease linearly from the tip to the junction. The efficiency of small solute trapping in the medulla is thus expected to be the highest in that case.

Changes in sodium and urea mass flow rates along the corticomedullary axis are illustrated on Figs. 5 and 6, for P_min = 2.5 mmHg. The profiles would exhibit the same qualitative trends if we were to use instead our first or second assumption. The shape of the mass flow rate curves is dictated by that of the concentration profiles in the outer medulla and that of the blood flow rate in the inner medulla (see Eqs. 5-7). Although outflow-to-inflow ratios are smaller for sodium, in dimensional terms there is less net removal of urea by vasa recta, since the initial DVR concentration of sodium is 30 times that of urea.

View larger version (17K): [in this window] [in a new window]   Fig. 5.   Plasma mass flow rate of sodium relative to its initial value in DVR as a function of position in DVR and AVR, for different values of the AVR-to-DVR number ratio, Nv. Transmural oncotic pressure gradients across AVR are eliminated, and hydraulic pressure gradient across AVR increases linearly from 0 to Pmin = 2.5 mmHg in inner medulla.

Summarized in Table 2 are the removal indexes for volume, sodium, and urea at the corticomedullary junction, for each of the three hypotheses we examined. Since the AVR-to-DVR blood flow rate ratio at the corticomedullary junction is 2.3 with the first two and 1.6 with the third one, the model when based upon the latter assumption yields a greatly improved countercurrent exchanger, as expected. The removal index for urea is about four times smaller than in the other two cases examined, and that for sodium is more than twice smaller.

We then examined the effect of uncertainties in vessel numbers upon flow rate and concentration profiles. We chose as our baseline case the third assumption, with P_min = 2.5 mmHg, but results were independent of the chosen hypothesis. The AVR-to-DVR number ratio, N_v, was varied between 1 and 4 [with the latter being the value reported by Holliger et al. (6)]. Since N_v does not affect sodium concentration significantly, changes in sodium mass flow rate with N_v directly reflect those in plasma flow rate (Fig. 5). An increase in N_v results in more volume uptake by AVR in the inner medulla; although the volume flux for a single vessel is determined by P and is thus the same in all cases, there are many more AVR.

Variations in the urea mass flow rate with N_v are shown in Fig. 6. As N_v increases, the concentration of urea in OMDVR rises more rapidly, since the AVR volume flow rate is then higher and the decrease in C_u in AVR consequently slower as plasma flows back from the junction between the inner and outer medulla. This effect is similar to that observed when P decreases, for a given value of N_v (see above).

View larger version (18K): [in this window] [in a new window]   Fig. 6.   Intracapillary mass flow rate of urea relative to its initial value in DVR as a function of position in DVR and AVR, for different values of the AVR-to-DVR number ratio, Nv. Transmural oncotic pressure gradients across AVR are eliminated, and hydraulic pressure gradient across AVR decreases linearly from 0 to Pmin = 2.5 mmHg in inner medulla.

Changes in the fractional vasa recta area (F_VR) do not affect in any way the nondimensional quantities that have been plotted. As shown by Eqs. 14 and 15, they only affect the overall number of vessels, not their ratio; a twofold increase in F_VR would thus result in a twofold increase in all dimensional quantities (the initial quantities, such as blood flow rate, were based on single-vessel values and multiplied by the initial number of DVR).

The effect of the two transcellular routes, the water channels and the urea transporters, was investigated by simulating their absence. We first compared results for a given value of P_min. When water channels are eliminated, there is no mechanism for water efflux in IMDVR, so that the total DVR plasma flow rate decreases only because the number of vessels decreases (individual DVR actually absorb water throughout the medulla), and the plasma flow rate at the papillary tip is thus higher than in the presence of AQP-1. In the absence of polarization, the driving force for volume influx into AVR is then smaller, since albumin is more diluted in the lumen, and the overall volume uptake by the microcirculation is also smaller. If concentration polarization is such that oncotic concentration gradients are not effective, however, then the driving force for volume uptake remains the same as when water channels are present, and the AVR plasma flow rate at the corticomedullary junction is therefore higher. Consequently, to yield a similar net influx of fluid as when water channels are present, the interstitial hydraulic pressure would have to increase with assumption 1 and to decrease with assumptions 2 and 3.

Without urea transporters in DVR, AVR-to-DVR urea gradients increase greatly to compensate for the reduction in permeability, thereby increasing the magnitude of the transcellular volume flux in DVR and that of net volume efflux in DVR. The total plasma flow rate at the tip is thus smaller than in the baseline case and remains so in AVR throughout the medulla. When both water channels and urea transporters are assumed to be present in OMDVR only, the results for volume uptake are almost the same as when water channels are eliminated, since the latter exert an effect in the inner medulla mostly, and urea transporters cannot affect plasma flow rate in the absence of transcellular volume fluxes.

To determine how transcellular pathways affect the efficiency of small solute countercurrent exchange, the comparison should be made on the basis of identical net volume removal by AVR. The baseline case we chose was the same as above, i.e., assumption 3 with P_min = 2.5 mmHg; the two other assumptions gave similar results (not shown). In each of the cases described below, we calculated the value of P_min that would yield the same overall uptake of volume at x = 0 as that obtained in the baseline case. Sodium concentration gradients in the outer medulla are negligible, so that none of the transcellular pathways appear to affect the efficiency of Na trapping in the renal medulla, as illustrated in Table 2.

When water channels are present, there is some volume efflux from DVR through the transcellular pathway near the corticomedullary junction, and the sieving of small solutes by AQP-1 results in a rapid increase in DVR, AVR, and interstitial concentrations (the sum of all urea fluxes must be zero in the vascular bundles to maintain mass balance, see METHODS). In the absence of a transcellular route for water, the increase in urea concentration with position in the outer medulla is much more progressive; the AVR concentration of urea at the junction is thus significantly smaller, and so is the overall removal of urea by the microcirculation, as illustrated in Table 2. When urea transporters are eliminated, AVR-to-DVR urea concentration gradients are much greater in the medulla (see above), thereby significantly increasing the amount of urea carried away by AVR and greatly reducing the efficiency of countercurrent exchange (Table 2). Both of these trends had also been observed in the single unit model (5). Finally, with the assumption that the two transcellular pathways are available in OMDVR only, removal indexes at the corticomedullary junction are equal to those in the baseline case (Table 2). Indeed, with the interstitial pressure in the inner medulla adjusted to yield the same fluid uptake, flow rate and concentration profiles are identical in the outer medulla.

	DISCUSSION

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The model presented here incorporates variations in the number of vasa recta along the corticomedullary axis, as well as the presence of water channels and urea transporters in OMDVR and IMDVR. In addition, we varied the extent to which concentration polarization may serve to limit volume uptake by AVR. As shown above, the first two features by themselves do not yield single vessel flow rates and concentration profiles different from our previous single-unit model.

Predicted variations in DVR plasma flow rate may serve to explain why measured DVR protein concentrations at the base of the papilla tend to be lower than expected from calculations based on systemic protein concentrations and juxtamedullary filtration fraction. Pallone et al. (18) reported DVR protein concentrations of 4.8 g/dl systemically, 5.2 g/dl at the papillary base, and 6.8 g/dl at the tip. Our model shows that there is first significant fluid uptake into DVR, up to about two-thirds of the medulla, followed by volume efflux from that point of the medulla onward. Since protein diffusion across the wall is negligible, DVR protein concentrations first decrease as a result of water influx and then start to increase only when efflux occurs.

If concentration polarization is entirely neglected, then the amount of volume influx into AVR is predicted to be significantly higher than that measured experimentally (23), since the bulk concentration of albumin in the AVR lumen is much higher than that in the interstitium toward the papillary tip. It is very likely, however, that the accumulation of protein at the wall when convection is very significant is sufficiently large to greatly reduce AVR transmural oncotic pressure gradients. Since polarization cannot be treated rigorously with our model, we examined its extreme consequence and entirely neglected the latter gradients, varying the interstitial hydraulic pressure in the inner medulla. This simplifying hypothesis was supported by recent experimental findings (12) showing that AVR can remain open when the external pressure is as much as 3 mmHg greater than the internal pressure. It is thus theoretically possible that fluid uptake in AVR be driven by hydrostatic pressure gradients only. Our simulations indicate that, based upon this assumption, predicted AVR-to-DVR blood flow rate ratios near the papillary tip can match experimental values reported by Zimmerhackl et al. (23).

An important observation is that although concentration polarization may be such that transmural oncotic pressure gradients are zero at some location along the corticomedullary axis where volume influx is very large, they cannot remain null afterwards, since dilution of protein within the lumen will then favor efflux instead and thus limit volume uptake and polarization. To capture that effect, the driving force for AVR volume flux in our simulations was varied linearly in the inner medulla, by making the interstitial hydraulic pressure decrease in that manner. Predicted volume and small solute removal indexes at the corticomedullary junction were then significantly smaller and may be closer to physiological values. Unfortunately, no data are available at this date to confirm the validity of this assumption.

Concentration polarization was neglected at DVR walls, since volume fluxes are much smaller across DVR than across AVR. If we were to assume that volume influx into DVR through the paracellular pathway raises the interstitial concentration of albumin in the vicinity of the DVR wall such that there is no transmural oncotic pressure gradient, then volume influx would be entirely suppressed, thereby contradicting the very basis of our hypothesis. Conversely, if volume efflux from DVR through the transcellular route was significant enough to reduce the interstitial concentration of albumin to zero in the vicinity of the DVR wall, then the net pressure gradients across DVR would favor volume influx even through the water channels. It therefore appears reasonable not to account for concentration polarization at the DVR walls.

The issue of concentration polarization is key in understanding the microvascular exchange of albumin. Several studies have shown the existence of an extravascular pool of albumin in the medullary interstitium (11, 14), the origin of which remains to be elucidated. Radiolabeled albumin injected in the interstitium is rapidly cleared (13), but it is unclear how. Lymphatics are absent in the inner medulla and sparse in the outer medulla, and the possibility of clearance of albumin by drainage via prelymphatic channels is not supported by experimental evidence (13). The most likely mechanism is that of protein clearance by the microcirculation itself. We circumvented the issue here by assuming a fixed interstitial albumin concentration; further mathematical simulations aimed at establishing whether albumin is indeed carried into and cleared from the interstitium by convective transcapillary fluxes will require accurate estimates of the extent of concentration polarization.

Our results appear to confirm previous conclusions about the effect of water channels and urea transporter on the transport of urea in the medulla (5). The availability of either of these transcellular routes was varied, and comparisons between different cases were based on the same net volume uptake into AVR. Since the osmolality data we interpolated (18) yielded sodium concentration gradients that are insignificant in the outer medulla, the presence of the two pathways does not affect Na transport. Water channels appear to decrease the efficiency of urea countercurrent exchange, as a result of increased sieving of small solutes in the outer medulla. Conversely, urea transporters increase transcapillary exchange and thus reduce the amount of urea removed from the interstitium by the microcirculation. Finally, whether both pathways are present in the inner medulla does not seem to have a significant effect on removal indexes at the corticomedullary junction. It is possible, as we suggested previously, that DVR AQP-1 does not play a role specific to the concentrating mechanism. Urea transporters may then be present to compensate for the increase in luminal concentrations that water channels impart.

The model developed here only accounts for those DVR that are destined to the inner medulla; within the scope of our approach, to account for those DVR that peel off from the vascular bundles to supply the capillary plexus would require oversimplified assumptions that may not be physiologically accurate. The model is also limited by the absence of a rigorous theoretical treatment of concentration polarization at the vessel walls. Transcapillary fluxes are bidirectional, and transport across the paracellular and transcellular routes in DVR occurs simultaneously in opposite directions; an accurate estimation of the extent of polarization, namely of the difference between concentrations in the bulk medium and at the wall, would thus require complex simulations, in which the presence of RBCs would have to be taken into consideration as well. The countercurrent exchange of albumin also remains to be explored. The presence of significant amounts of albumin in the medullary interstitium was accounted for in our model by the assumption of a fixed interstitial concentration, but further studies are needed to understand its origin.

In summary, our multiunit model of the medullary microcirculation shows that significant volume uptake can occur in AVR even in the absence of transcapillary oncotic pressure gradients. The hydraulic pressure difference across AVR required to match predicted and measured values of AVR-to-DVR blood flow rate ratios toward the papillary tip is smaller than the maximum difference AVR can sustain (12). In addition, the presence of urea transporters in DVR appears to counterbalance that of water channels which would otherwise decrease the efficiency of small solute trapping in the renal medulla.