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【摘要】 We developed a two-dimensional model of NO transport in a cross section of the inner stripe (IS) of the rat outer medulla to determine whether tubular and vascular generation of NO result in significant NO concentration (C NO ) differences between the periphery and the center of vascular bundles and thereby affect medullary blood flow distribution. Following the approach of Layton and Layton (Layton AT, Layton HE. Am J Physiol Renal Physiol 289: F1346-F1366, 2006), the structural heterogeneity of the IS was incorporated in a representative unit consisting of four concentric regions centered on a vascular bundle. Our model suggests that the diffusion distance of NO in the interstitium is limited to a few micrometers. We predict that, under basal conditions, epithelial NO generation raises the average C NO in pericytes surrounding peripheral descending vasa recta (DVR) by a few nanomoles relative to that in pericytes surrounding central DVR. The short descending limbs and long ascending limbs are found to exert the greatest effect on C NO in pericytes; long descending limbs and short ascending limbs only have a moderate effect, whereas outer medullary collecting ducts, which are situated far from the vascular bundle center, do not affect pericyte C NO. Our results suggest that selective stimulation of epithelial NO production should significantly raise the periphery-to-center DVR diameter ratio, thereby increasing the outer medulla-to-inner medulla blood flow ratio. However, concomitant increases in epithelial superoxide (O 2 - ) production would counteract this effect. This model confirms the importance of NO and O 2 - interactions in mediating tubulovascular cross talk.
【关键词】 kidney superoxide mathematical model nitric oxide transport
THE RENAL MEDULLARY MICROCIRCULATION plays a fundamental role in maintaining fluid balance and controlling blood pressure. As reviewed by Cowley and colleagues ( 24 ), a 15-30% reduction in blood flow to the medulla can lead to the development of hypertension. The structural organization of the medullary blood vessels, i.e., the descending and ascending vasa recta (DVR and AVR, respectively) that form a countercurrent arrangement, is highly specialized. The inner stripe (IS) of the outer medulla (OM) is divided into vascular bundles and the interbundle region. DVR in the bundle center are destined to the inner medulla (IM), whereas DVR on the bundle periphery give rise to capillary plexuses that perfuse the interbundle region, where medullary thick ascending limbs (mTAL) are located. DVR are surrounded by pericytes, smooth muscle-like cells that impart contractile properties to the vessels. Since NaCl reabsorption across mTAL epithelia requires significant oxygen consumption, Pallone et al. ( 26 ) suggested that preferential vasodilation of DVR on the bundle periphery or constriction of DVR in the bundle center should enhance perfusion of (and thus oxygen delivery to) the interbundle region.
NO acts as a vasodilator in the renal medullary microcirculation. Blocking NO synthesis in the medulla leads to a reduction in blood flow, salt retention, and hypertension ( 21 ). Cowley and colleagues ( 7 ) demonstrated that NO serves as a paracrine substance that mediates cross talk between the tubular epithelium of mTAL and pericytes. They found that angiotensin II (ANG II) stimulates NO release from the mTAL and diffusion to adjacent pericytes. This functional coupling between vascular and tubular units was subsequently termed "tubulovascular cross talk." Mori and Cowley ( 23 ) later observed that ANG II also stimulates superoxide (O 2 - ) production in mTAL and that interactions between NO and O 2 - determine the effectiveness of the cross talk between mTAL and DVR. There is perhaps a similar cross talk in the IM between the tubular epithelium of inner medullary collecting ducts (IMCD), which produce high levels of NO, and pericytes ( 5 ).
The objective of this study was to determine theoretically the extent to which tubular epithelial production of NO creates radial NO concentration (C NO ) gradients, which could lead to preferential vasodilation in the bundle periphery relative to the bundle center. Following the rat structural data analyzed by Layton and Layton ( 17 ), we consider all the vessels and tubules within and around one vascular bundle, and we model the production, diffusion, and consumption of NO in two-dimensional cross sections of the IS of the rat OM. We showed in a recent study ( 39 ) that, aside from shear stress-mediated effects on endothelial NO production, blood flow per se (i.e., NO convection) does not affect NO radial and axial concentration profiles within and around vasa recta (VR). These findings suggest that a two-dimensional representation of NO transport in medullary cross sections (i.e., a model that neglects axial convection) is adequate to predict C NO.
MODEL AND PARAMETERS
Geometric representation. There are hundreds of vascular bundles in the renal OM of rats, with hundreds of vessels and tubules within each bundle. Layton and Layton ( 17 ) recently compiled and analyzed medullary structural data obtained by other investigators ( 10, 12, 15, 16, 18, 28 ), and they incorporated into their mathematical model of the rat urinary concentrating mechanism the precise distribution of tubules and vessels in a medullary bundle. As described below, our geometric model is based on their work.
We examine a cross section of the OM at the midpoint of the IS (i.e., at a distance of 1.35 mm from the corticomedullary junction). Following the approach of Layton and Layton ( 17 ), we consider a representative unit (i.e, assuming periodicity) consisting of four concentric regions centered on a vascular bundle. The number of tubules and vessels in each region is taken from the same study ( 17 ) and summarized in Table 1. Located in region R1 are all the long DVR (LDV) and about a fourth of the long AVR (LAV), all of which extend into the IM; the rest of the LAV are distributed in region R2; half of the short DVR (SDV) are located in region R1 and the other half in region R2.
Table 1. Number of tubules and vasa recta per vascular bundle at midpoint of inner stripe of rat outer medulla
Within each concentric region, the tubules and vessels are positioned so as to obtain a random distribution: starting from the center of R1 and going counterclockwise and outward, the type of tubule or vessel that is placed next is chosen based on the numbers of tubules and vessels that remain to be positioned. For example, at the very beginning, the fractions of LDV, SDV, and LAV remaining to be positioned are equal to 12/(12 + 11 + 12), 11/(12 + 11 + 12), and 12/(12 + 11 + 12), respectively. A random number n R between 0 and 1 is generated by computer. The vessel that is placed next is a LDV if n R is lower than 12/34, a SDV if n R is between 12/34 and 23/34, and a LAV if n R is greater than 23/34. We also perform computer calculations to position the tubules and vessels so that the distance separating adjacent tubules and VR is 0.1 µm. Given that the cylinder-like tubules and vessels are rigid in our model, the tubules or VR assigned to a given region may not all fit in; if so, they are placed in the adjacent concentric region. It was therefore necessary to slightly adjust the diameter of region R4 so that all tubules and vessels could be included in the IS cross section. The R4 radius value determined by Layton and Layton ( 17 ) is 226 µm, whereas ours varies between 230 and 235 µm. A representative distribution of vessels and tubules is illustrated in Fig. 1.
Fig. 1. Representative geometric model (corresponding to Random 1 distribution in Table 3 ) of cross section at the middle point of inner stripe of renal outer medulla. The centermost region corresponds to a vascular bundle and is surrounded by 3 other concentric regions. The distribution of tubules and vessels is based on the analysis of Layton and Layton ( 17 ) as summarized in Table 1. Vasa recta are represented by 3-layer concentric circles corresponding to a red blood cell (RBC)-rich layer, a cell-free parietal layer, and a wall/endothelial layer; nephron segments and collecting ducts (CD) are represented by 2-layer concentric circles corresponding to lumen and wall/epithelial layers. Descending vasa recta (DVR) and ascending vasa recta (AVR): smaller and larger 3-layer concentric circles, respectively. It is not possible to distinguish in the figure the long (LDV) vs. short (SDV) DVR, nor the long (LAV) vs. short (SAV) AVR. Short and long descending limb (DL): thin-wall, 2-layer circles located in R2 and R3, respectively; SAL and LAL: wide-wall, 2-layer circles with small and medium diameter, respectively. CD: wide-wall, two-layer circles with large diameter, located in R4 only.
Our previous model ( 39 ) suggested that the average C NO over the pericyte cross-sectional area differs by 0.3% from C NO at the pericyte-interstitium interface and at the endothelium-pericyte interface. Hence, for simplification, we do not explicitly consider the pericytes and the basement membrane in this model. We distinguish three layers in each VR: within the lumen the erythrocyte-rich core layer and the cell-free parietal layer that surrounds it and abluminally the vessel wall, which we assume consists simply of an endothelial barrier. C NO in pericytes, which determines the extent of vasodilation, is approximated as that at the endothelium-interstitium interface.
Diffusion model. In a recent study ( 39 ), we developed a three-dimensional model of concentric cylinders representing a single VR embedded in interstitium and predicted that axial convection of NO in medullary blood vessels has a negligible effect on C NO. As described in that study, the absence of specific velocity effects stems from the balance between NO generation and consumption. When NO is scavenged by hemoglobin and O 2 -, our model predicts that the flux of NO that diffuses into the lumen is independent of blood velocity and is equal (to within 0.6%) to the amount of NO consumed within the vascular lumen. If there were no consumption of NO, predicted C NO would depend significantly on blood velocity. In other words, under normal flow conditions, variations in blood velocity per se do not affect C NO profiles, as the latter are determined mainly by NO production and consumption rates. We therefore use a two-dimensional model of NO transport in medullary cross sections in the present study. The conservation equation for NO is written as
where D i is the diffusivity of NO in compartment i, i is the net NO generation rate of i, and u is the two-dimensional water velocity. In the vascular lumen, radial convection of NO is predicted to be negligible ( 39 ). In the interstitium, u is determined by 1 ) the net transmural water fluxes from tubules and VR and 2 ) capillary flow, as terminating SDV at a given medullary depth empty blood into capillaries (which are taken to be part of the interstitium). Calculating u locally is beyond the scope of this model. However, based on the flows calculated by Layton and Layton in their region-based model of the OM ( 17 ), we estimated upper limits for the planar interstitial water velocity and the Péclet number (Pe). In each region R i, Pe was calculated as Lu / D, where the characteristic length L was taken as the interior radius of R i. The resulting estimates of Pe are on the order of 10 -3 to 10 -4, suggesting that NO convection in the interstitium may also be neglected. Therefore, Eq. 1 is simplified as
NO is generated by vascular endothelial cells and tubular epithelial cells. It is scavenged by hemoglobin in the vascular core layer and by O 2 - in the vascular parietal layer, the vascular wall, and the interstitium. Therefore
where G en and G ep are the NO synthesis rate in vascular endothelia and tubular epithelia, respectively, and k O2 - and k Hb are the kinetic constants corresponding to the reaction of NO with O 2 - and heme, respectively, and are given in Table 2. The red blood cell (RBC) concentration of heme, denoted by C Hb, is four times that of hemoglobin. As described previously ( 39 ), we account for the resistance to NO diffusion within the RBC-rich core layer using a hindrance factor h RBC that lumps together the resistance of unstirred boundary layers, that of the RBC membrane and associated cytoskeleton, and that of the cytosol. H C, the hematocrit in the core layer, is estimated as
where H T is the tube hematocrit, r V is the radius of vascular lumen, and p is the thickness of the parietal cell-free layer. By perfusing a suspension of human erythrocytes in isotonic medium into the rabbit mesentery, Tateishi et al. ( 32 ) found that p increases significantly with increasing vascular diameter and decreasing hematocrit; however, variations in p with erythrocyte size (at a given H T ), or with flow velocity ranging between 0.2 and 2 mm/s (when H T 8%), were reported to be negligible. We use their measurements for H T = 25% (our baseline hematocrit in VR) to estimate the value of p in medullary vessels. In DVR and AVR, p is taken as 1.6 and 1.9 µm, respectively.
Table 2. Parameter values
Boundary conditions. We assume that the representative unit in this model, consisting of four concentric regions centered on a single vascular bundle, is surrounded by identical mirrors of itself. Hence, we use a homogeneous Newmann boundary condition (i.e., zero flux) at the exterior R4 boundary. At all other boundaries, i.e., at the interfaces between different compartments, we assume that both concentrations and normal fluxes are continuous.
NO generation rates. Wu et al. ( 38 ) estimated the production of NO in the renal medulla by measuring the production of L -citrulline from microdissected segments in the rat kidney. The measured NO generation rates from VR, outer medullary thin limbs (LDL and SDL), mTAL [short (SAL) and long (LAL) ascending limbs, and outer medullary collecting ducts (OMCD) were reported as 3.2, 0.6, 0.5, and 0.3 fmol·mm -1 ·h -1, respectively, based on 20-mm-long segments. Neither DVR and AVR nor the outer stripe (OS) and IS were distinguished in their study. To convert these values to units based on endothelial or epithelial volume, we make the following hypotheses.
Since the walls of tubules and collecting ducts (CD) consist mostly of epithelial cells adjacent to a thin basement membrane ( 16 ), we assume that NO generation occurs throughout the entire wall of those segments; the wall thickness has been reported as 3 µm for descending limbs and 5 µm for ascending limbs and CD ( 2, 14 ).
We also assume that the vasa recta (VR) segments dissected in the study of Wu et al. ( 38 ) come from the vascular bundle, more specifically, from the R1 region. The thickness of the VR endothelium is taken as 1 µm. Based on VR diameters given in Table 2, we calculate the number-weighted average endothelial volume per unit length of vessel. In DVR, the endothelial volume is estimated as (6.5 2 - 5.5 2 ) = 37.7 µm 3 /µm in both the OS and the IS. In AVR, it is calculated as (1 - 0.3) (13.5 2 - 12.5 2 ) = 57.2 µm 3 /µm in the OS and as (1 - 0.3) (8.5 2 - 7.5 2 ) = 35.2 µm 3 /µm in the IS; the factor 0.3 represents the fraction of the AVR wall that is perforated by fenestrations ( 22 ). The number of VR in the R1 region is 34 DVR (LDV + SDV) and 12 LAV in the OS and 23 DVR and 12 LAV in the IS ( 17 ). Therefore, the average endothelial volume is calculated as (in µm 3 /µm length)
that is, 4.02 x 10 -14 m 3 /mm length. The VR endothelial NO generation rate measured by Wu et al. ( 38 ) is thus converted to 22.1 µmol NO·m -3 endothelium·s -1 (or 1.3 x 10 -17 µmol·µm -2 endothelial surface area·s -1 ).
Similarly, the epithelial NO generation rates in tubular segments are converted to the same units by calculating the number-weighted average epithelial volume for each type of segment. The corresponding NO synthesis rate for ascending limbs (LAL and SAL), descending limbs (LDL and SDL), and OMCD is then estimated as 0.390, 0.461, and 0.163 µmol·m -3 wall·s -1, respectively.
Other parameter values are derived from our previous study ( 39 ) and are summarized in Table 2.
Numerical solution. The equations are solved with the finite element-based software FEMLAB 3.1 (Comsol, Burlington, MA). FEMLAB treats each closed geometry as a subdomain, for which an independent diffusion model is specified. There are about 1,000 subdomains in our model, consisting of the interstitium and all the layers of the 400 vessels and tubules. FEMLAB uses Cartesian coordinates to compute the solution. The software generates unstructured triangular mesh elements of variable size by specifying a size-increasing rate, with smaller elements near boundaries. FEMLAB divides the geometric boundaries into segments that correspond to the edges of the triangular elements so as to handle the discontinuity at the boundary (i.e., interface) between subdomains. To ensure convergence, we increase the number of elements until average concentrations do not vary anymore with mesh resolution; the final element number is generally comprised between 400,000 and 700,000, depending on the simulations. Given the computer memory requirements for simultaneously solving the large amount of linear equations (1 for each vertex of the triangular elements), calculations are performed in the Tufts University Linux cluster.
RESULTS
Baseline case. In the renal medulla, NO is synthesized by the endothelia of VR and by the epithelia of ascending limbs, descending limbs, and CD. Wu et al. ( 38 ) determined the NO synthesis rate of these different segments by measuring the production of converted L -citrulline from the in vitro culture of the segments with L -arginine. We first used these experimental data to determine the C NO profile across the IS section using the geometric representation shown in Fig. 1 (which correspond to Random 1 in Table 3 ). These generation rate estimates yield C NO throughout the cross section (i.e., in vascular walls, interstitium, and nephron segments) that are on the order of 0.01 nM, that is, significantly lower than experimental observations: Zou and Cowley ( 43 ) reported that C NO ranges from 57 to 139 nM in the medullary interstitium, and Stamler et al. ( 30 ) measured the C NO in plasma drawn from human subjects as 3 nM. Since C NO is roughly proportional to the NO generation rate (G en and G ep ), as indicated by Eqs. 2 and 3, we multiplied the G en and G ep estimates of Wu et al. ( 38 ) by a factor of 1,000, to obtain good agreement with measured C NO. As illustrated in Fig. 2, C NO is then predicted to range from 10 to 40 nM in the VR lumen, from 40 to 60 nM in the abluminal region of VR, and from 35 to 90 nM in the interstitium. The baseline G en and G ep values were therefore taken as 1,000 times those of Wu et al. ( 38 ), as shown in Table 2. Vaughn et al. ( 35 ) fitted their model of NO transport to the experimental data of Malinski et al. ( 20 ); similarly, they estimated the NO endothelial generation rate in smooth muscle as 6.8 x 10 -14 µmol·µm -2 ·s -1, or 1,000 times the estimate of Wu et al. ( 38 ). Possible reasons for the discrepancy between the in vitro data of Wu et al. ( 38 ) and numerical predictions are discussed further below.
Table 3. NO concentration distribution in interstitium and DVR pericytes
Fig. 2. Predicted NO concentrations for 3 random distributions of vessels and tubules (see Table 3 for concentration ranges).
The objective of this study was to investigate whether C NO in DVR at the bundle center varies significantly from that in DVR at the bundle periphery, since such variations could play an important role in blood distribution. The vasomotion properties of DVR are imparted by pericytes, smooth muscle-like cells that surround the vessel endothelium; hence our focus on average (i.e., over all DVR) C NO at the endothelium-interstitium interface of DVR in regions R1 and R2 (denoted by C ), which determine the extent of vessel contraction or dilation. Since tubulovascular cross talk is mediated by diffusion across the interstitium, we also examined average interstitial C NO in each region (denoted by C ).
Effect of distribution of tubules and vasa recta. We built five models to investigate the effects of the distribution of tubules and vessels on C NO. Given that creating a model comprising 400 tubules and vessels is very computation intensive, in three of the models ( Random 1-3 in Table 3 ), the tubules and vessels in each region (R1-R4) were positioned throughout the full circle as described in MODEL AND PARAMETERS; in the other two ( Mirror 1 and 2 in Table 3 ), we introduced some symmetry. Specifically, one-fourth of the number of tubules and vessels in each region were positioned in a one-fourth circle (i.e., a quadrant), and the full circle was constructed with mirror images of the quadrant.
Shown in Table 3 are C NO ranges and averages in the interstitium and at the DVR endothelium-interstitium interface in each region. C NO across the four concentric regions are also plotted in Fig. 2 for the three random distributions. Under normal conditions, RBC hemoglobin constitutes a much stronger scavenger of NO than O 2 - does, so that C NO is significantly lower, and spans a narrower range, within DVR and AVR than within tubules and CD. C NO are predicted to reach the highest values in areas where five or more tubules form a cluster from which VR are excluded.
As shown in Table 3, even though the range of C NO in DVR endothelium is wide, the DVR-averaged C NO at the endothelium-interstitium interface in regions R1 and R2 (C and C, respectively) varies little among the five cases; the mean values of C and C for the five distributions are 52.5 ± 1.5 nM and 56.5 ± 0.7 nM, respectively. Pericyte C NO are mostly determined by endothelial NO generation rates and to a lesser degree by epithelial NO generation rates (see below). As expected, C is greater than C in all five cases, given the closer proximity of NO-generating tubules to vessels in R2 relative to R1. Similarly, the average interstitial C NO in R1 and R2 does not change very significantly from one distribution to the next; the mean values of C and C for the five distributions are 48.9 ± 1.1 nM and 51.1 ± 3.5 nM, respectively.
A closer look at each distribution suggests that the positioning of DVR within R1 significantly affects C : the latter decreases when DVR are concentrated toward the center of the vascular bundle and increases when they are more scattered and generally closer to the R1-R2 boundary (see Random 3 in Fig. 2 and Table 3 ), where they are more likely to be reached by NO diffusing from adjacent tubules. The local distribution of DVR in R2 has a smaller effect on C since the number of DVR in this region is much lower than that of tubules, so that DVR are usually surrounded by tubules.
In general, the more regular the scattering of tubules among the vessels, the lower C NO overall, and the narrower the concentration range (see Random 2 in Fig. 2 and Table 3 ), given the dominant role of NO scavenging by hemoglobin. As described above, C NO is the highest within clusters of tubules and therefore high as well in the surrounding interstitium.
Given the complexity of calculations involving 400 tubules and vessels, we sought to determine whether examining only a portion of the entire circle would yield accurate results. The three random distributions in Table 3 were arbitrarily divided into four quadrants each, one of which was then arbitrarily selected for calculations. Our results, summarized in Table 4, indicate that predicted differences in C and C between the entire unit (be it a "random" circle or a "mirror" circle) and a quadrant from this unit range from 0 to 10%. As expected, these differences are negligible (<0.2%) when we consider only mirror circles. Arbitrarily dividing a random distribution into four quadrants generates incomplete portions of vessels and tubules within each quadrant, thereby introducing numerical inaccuracies.
Table 4. Predicted NO concentrations with different geometric representations
Given the uncertainty in other parameter values, the full unit-to-quadrant variations were deemed to be acceptable. We therefore chose to perform the rest of our simulations with a quadrant representation so as to decrease computer memory requirements. To obtain the maximum accuracy, we constructed seven quadrants with a random distribution of vascular and tubular segments in each concentric region; the number of each type of vessel and tubule in a given area was taken as one-fourth that of the whole unit (given in Table 1 ). For each of the seven distributions, we calculated the average C NO at the endothelium-interstitium interface and within the interstitium in each region (R1-R4). We then computed the mean over the seven distributions of these average C NO values and found that the standard deviations were equal to 1-7% of the mean; the lowest values were predicted in regions R1 and R2, where all DVR pericytes are located. The representative quadrant used for the rest of our calculations was chosen as that which minimizes the variations about the mean of all six concentrations, that is, the quantity
where x kj ( j = 1, 2...6) is equal to the value of C, C, C, C, C, and C, respectively, predicted using quadrant k, and j is the value of x kj ( k = 1, 2...7) averaged over the seven quadrants.
Effect of NO generation rates. With baseline epithelial and endothelial NO generation rates, we predict that the diffusion of NO from tubules to vessels raises the average pericyte C NO at the bundle periphery relative to that within the bundle; Table 3 suggests that, on average, C is 4 nM greater than C. However, there is some uncertainty regarding the NO generation rates: the baseline NO generation rate in the DVR endothelium (22 mmol·m -3 ·s -1 ) is equivalent to 1.3 x 10 -14 µmol·µm -2 ·s -1 based on the endothelial surface area and is therefore comparable to the estimates given by Vaughn et al. ( 35 ) for nonspecific microvessels (6.8 x 10 -14 µmol·µm -2 ·s -1 ), but it is three orders of magnitude higher than the in vitro measurements of Wu et al. ( 38 ) in VR and tubules. Moreover, physiological or pharmaceutical stimuli may significantly alter NO generation rates ( 7 ).
It is likely that endothelial NO generation rates of DVR and AVR vary simultaneously (as when stimulated by blood flow shear stress at the endothelial surface).
Shown in Fig. 3 are predicted variations in C with the rate of NO generation by VR endothelia (G en ), that is, G DVR = G en and G AVR = 0.7G en. At very low G en values, predicted C NO are at least 10-fold lower in R1 than in R2, given the absence of tubules in R1. As G en increases, the relative significance of epithelium-generated NO diminishes, so that C increases faster than C. As G en reaches 10 times its baseline value, the difference in DVR pericyte C NO between R1 and R2 disappears.
Fig. 3. Effect of vascular endothelium NO generation rates (G en ) on average NO concentrations at the DVR endothelium-interstitium interface in regions R1 and R2 (C and C, respectively). The endothelial NO generation rates in DVR (G DVR ) and in AVR (G AVR ) are varied simultaneously, with G DVR = G en and G AVR = 0.7G en.
As expected, the effect of changes in the rate of NO generation by tubular epithelium (G ep ) is the opposite. As illustrated in Fig. 4 A, at very low G ep values the differences between C and C are negligible. In fact, C is slightly lower than C because the lower density of vessels in R2 enhances the effects of NO scavenging by O 2 - in the interstitium of that region; the interstitium-to-VR cross-sectional area ratio is equal to 0.26 and 0.60 in R1 and R2, respectively. As G ep increases, predicted C NO remain constant in R1, whereas they begin to increase proportionally to G ep in R2. When G ep is equal to 10 times the baseline, the difference between C NO in peripheral and central DVR (estimated as C - C ) increases from 6 to 66 nM.
Fig. 4. Effect of tubular epithelium NO generation rates (G ep ) on average NO concentrations at the DVR endothelium-interstitium interface and in the interstitium in regions R1 and R2 ( A ) and NO fluxes across the DVR endothelium-interstitium boundary in R1 and R2 ( B ). The epithelial NO generation rates in DL, AL, and CD are increased by the same factor relative to their baseline values. The NO flux is taken to be positive when directed outwards, i.e., from the DVR lumen toward the interstitium.
Another way to visualize the effect of G ep is to examine the direction of the NO flux across the endothelium-interstitium interface, as illustrated in Fig. 4 B. When G ep is very low, the NO molecules that are released by the tubular epithelial cells are almost entirely scavenged by O 2 - in the interstitium, and they cannot reach DVR endothelial cells; hence, NO diffuses from DVR toward the interstitium, and this positive flux is practically independent of G ep. When G ep is greater than half the baseline, a fraction of the NO molecules produced by the tubular epithelium are able to escape scavenging by O 2 - and to diffuse into the DVR located in the R2 region; very few reach the R1 region, however. The NO flux across the DVR endothelium-interstitium interface is then negative, and it varies significantly with G ep in R2 but not in R1.
In the above simulations, we simultaneously varied the rate of NO generation by all tubules and OMCD. Our results suggest that the diffusion distance of NO is limited, in that epithelial NO production has a very small effect on C. Given that short and long descending (or ascending) limbs are located in different regions, we then varied separately the rate of NO generation by each type of tubule. Increasing tenfold that of SDL or LAL (both of which can be found in R2) raises the R2-to-R1 interstitium-endothelium interface C NO difference by a factor of 6.1 or 4.3, respectively. Increasing tenfold the rate of NO generation by LDL or SAL (which are located beyond R2) raises that difference by a factor of 2.3 or 1.6, respectively. Finally, increasing tenfold the rate of NO generation by OMCD (all of which are in R4) has no effect on DVR C NO. Taken together, these results suggest that tubulovascular cross talk is restricted to short diffusion distances.
Effect of O 2 - concentration. One of the main determinants of the NO diffusion distance is the rate of NO scavenging. The principal scavenger of NO in the interstitium is O 2 -. To the best of our knowledge, there are no experimental measurements of O 2 - concentration (C O2 - ) in the renal medulla; such measurements are complicated by the fact that O 2 - reacts very easily with itself to form H 2 O 2 ( 37 ). Simulations performed by Buerk et al. ( 3 ) suggest that as the O2 - synthesis rate varies from 0.02 to 10 µM/s, C O2 - increases from 0.01 to 10 nM in the vascular wall and in the adjacent region (i.e., within 50 µm from the vascular wall).
We therefore varied C O2 - between 0 and 1,000 times the baseline (i.e., 0.25 nM in interstitium). Other parameters (such as NO generation rates) were kept constant. As shown in Fig. 5, as C O2 - increases over this range, the difference in C NO at the endothelium-interstitium interface between peripheral and central DVR (estimated as C - C ) decreases from +6 nM to -1 nM. Below the baseline, changes in C O2 - have no significant effect on C NO near pericytes and in the interstitium, which suggests that the amount of NO scavenged by O 2 - is negligible relative to that scavenged by RBC hemoglobin. Conversely, when interstitial C O2 - is on the order of (or higher than) 1 nM, NO consumption by O 2 - becomes significant and O2 - effectively decreases the NO diffusion distance. When the interstitial C O2 - is 3 nM, C = C : at this concentration, the higher rate of NO consumption by interstitial O 2 - in R2 relative to R1 (a result of regional differences in vessel density, as described above) abolishes the effects of the closer proximity of DVR in R2 to epithelial sources of NO. As C O2 - is further increased beyond 3 nM, C NO remains lower in R2 than in R1, but the difference is small ( 1 nM) and not likely to result in significant diameter differences between peripheral and central DVR.
Fig. 5. Effect of superoxide concentration (C O2 - ) on NO concentrations. The baseline value of C O2 - is 50 pM in plasma, 100 pM in endothelium, and 250 pM in the interstitium.
Effect of NO scavenging by RBC hemoglobin. At steady state, the overall amount of NO generated must equal the amount of NO scavenged in our unit cell. Hence, the fact that under baseline conditions decreases in C O2 - do not significantly affect C NO in R1-R4 signifies that RBC hemoglobin scavenges most of the NO generated by both VR endothelium and tubular epithelium. We investigated the effect of changes in the rate of NO scavenging by RBC hemoglobin ( K RBC = k Hb C Hb H C /h RBC ) by varying the kinetic constant ( k Hb ), the heme concentration (C Hb ), the hematocrit (H C ), or the hindrance factor characterizing the RBC diffusion barriers to NO (h RBC ). As expected, decreasing K RBC significantly raises C NO throughout all regions (see Fig. 6 ). As the rate of NO scavenged by hemoglobin decreases, the rate of NO consumption by O 2 - increases in compensation (as dictated by NO conservation at steady state). Given the differences in the fractional area occupied by the interstitium in each region, differences in C NO between R1-R4 thus become more pronounced. In particular, C < C when K RBC is lower than one-fifth its baseline value. Conversely, increasing K RBC reduces both absolute C NO as well as regional differences in C NO. A 100-fold increase in K RBC lowers C NO by 50%.
Fig. 6. Effect of first-order kinetic constant of reaction between RBC hemoglobin and NO ( K RBC ) on NO concentrations. The baseline value of K RBC is 587 and 538 s -1 in DVR and AVR, respectively.
DISCUSSION
In the present study, we developed a two-dimensional model of NO transport in cross sections of the rat OM, to determine whether tubulovascular cross talk could result in significant differences in C NO between the center of vascular bundles and their periphery. As described in the introduction, given the architecture of the medullary microcirculation, such differences in C NO could result in preferential perfusion of the IM or the OM.
This work represents the third part of our investigation of NO transport in the renal medulla. In the first study ( 40 ), we used our full representation of medullary vessels so as to identify the main determinants of C NO under normal conditions and to ascertain certain parameter values, such as vessel permeability to NO and endothelial NO generation rate. This approach was limited in two important ways: we assumed that each compartment (i.e., DVR, AVR, and the interstitium) is well mixed at a given depth, and we did not account explicitly for the production of NO by tubular epithelia. In the second study ( 39 ), we developed a three-dimensional model of a single VR surrounded by interstitium to investigate the effects of blood flow on NO transport. Within the blood vessel, we distinguished the cell-free plasma boundary layer and the RBC-rich core layer. The production of NO by tubular epithelia was incorporated as an influx boundary condition. Our results suggested that RBCs act as a sink because of NO scavenging by hemoglobin, that C NO is controlled by kinetic and diffusion rates, and that convection per se does not significantly affect C NO, aside from the effects of blood flow shear stress on endothelial NO production. We predicted that for given generation and consumption rates, C NO is a function of radial position but not depth beyond short entrance lengths. Thus the two-dimensional representation in the present work should be adequate to investigate C NO gradients at a given medullary level. As opposed to our previous approaches, the present model incorporates the detailed arrangement of VR and tubules in outer medullary cross sections.
Our representation of the medullary architecture in the OM is based on the extensive study of Layton and Layton ( 17 ). Following their approach, we modeled the vascular bundle as a circle surrounded by three other concentric regions, and we used their data to specify the number of vessels and tubules in each region. As summarized in Table 1, in the middle of the IS, the vascular bundle (that is, region R1) consists of all the LDV ( 12 ), one-fourth of all LAV ( 12 ), and half the SDV ( 11 ). The other LAV ( 38 ) and SDV ( 11 ) are distributed in the adjacent R2 region and thus perfuse the outer medullary interbundle zones. The SAV, which originate from the outer medullary capillary plexuses, are scattered over the two outer regions along with descending and ascending limbs, whereas the CD are located in the outermost region only. We assumed that within a given region the vessels and tubules are distributed randomly, and we found that the average C NO in the interstitium and DVR pericytes does not vary much with the chosen distribution.
The highly structured arrangement of VR and tubules was investigated more than three decades ago by Kriz and colleagues (see Refs. 15, 16 ). Since then, several investigators have incorporated the three-dimensional architecture of the renal medulla in their models of the urinary concentrating mechanism. Wexler et al. ( 36 ) accounted for the relative position of tubules and vessels with an exchange weight coefficient, assuming that each segment interacts only with the most adjacent other segments. As described above, Layton and Layton ( 17 ) developed a region-based model of the urinary concentrating mechanism in the rat OM. They assumed that all tubules and VR in a given region interact with a common interstitium (which represents merged capillaries, interstitial cells, and interstitial space) that is well mixed, and exchanges between the regions were simulated with interregion capillary fluxes across the boundaries that separate the concentric regions. Our model differs from that of Layton and Layton ( 17 ) in several important ways: 1 ) we only examined radial transport in a given cross section and neglected axial effects; 2 ) however, we solved solute conservation equations locally at every point, that is, we did not assume well-mixed compartments; and 3 ) we did not investigate the exchanges of water, NaCl, and urea but focused instead on the transport of NO.
Our model suggests that the more uniform the dissemination of tubules among vessels, the lower the C NO throughout the OM cross section. We estimated the difference in C NO between the pericytes surrounding DVR that supply the IM and those surrounding DVR that perfuse the interbundle region as the R2-to-R1 difference in the average (i.e., over all DVR in a given region) C NO at the endothelial-interstitial interface. With baseline parameter values, that is, with comparable endothelial and epithelial NO generation rates, the predicted difference is 4 nM. As expected, C NO is higher in R2 than R1 given the shorter distance between pericyte and the epithelial sources of NO in R2. When the epithelial NO generation rate is increased by a factor of 10, the R2-to-R1 difference rises to 66 nM. Conversely, when the endothelial generation rate is increased by a factor of 10, the R2-to-R1 difference vanishes. When baseline C O2 - are separately increased by a factor of 10, the predicted C NO becomes lower in R2 than in R1. Indeed, NO scavenging by O 2 - then becomes significant relative to that by RBC hemoglobin; since there is more interstitial space surrounding the VR in R2 than in R1, the consumption of NO is faster in the former region and cancels out the effects of NO diffusion from tubular epithelia.
Since the first-order kinetic constant of NO consumption by hemoglobin (including the RBC hindrance factor) is 500 times greater than that of NO consumption by O 2 - under baseline conditions, all VR constitute NO sinks in the medulla. The diffusion distance of NO in a given area may therefore be estimated as the average distance between the closest tubules and blood vessels in that area. Since AVR are scattered over the entire mid-IS cross section, the diffusion distance of NO in the interstitium is short in all the concentric regions, that is, 2, 3, and 4 µm in R2, R3, and R4, respectively.
Given this, and the fact that the rates of NO generation by various types of tubules (in units of fmol·mm length -1 ·h -1 ) are close to one another, the tubules that are nearest to DVR exert the greatest effect on pericyte C NO. The significant fraction of SDL and LAL that are located in the R2 region are the closest to descending vasa recta; since there are many more SDL in R2 than there are LAL (47 vs. 12), the SDL are the tubules that have the largest effect on pericyte C NO. Our model predicts that the tubules in the R3 region, such as SAL and LDL, have a moderate effect, whereas OMCD, all of which are in the R4 region, have no effect. Recent observations by Pannabecker and Dantzler ( 27 ) indicate that in the upper IM DVR are excluded from CD clusters, whereas AVR are distributed uniformly within and around the clusters. A mathematical model of NO transport in inner medullary cross sections is beyond the scope of this study; however, our present results suggest that although IMCD produce more NO than any other tubular and vascular segment ( 38 ), they may not have a large effect on pericyte C NO.
Cowley and colleagues have shown experimentally that NO serves as a paracrine substance that mediates cross talk between the tubular epithelium of mTAL and VR pericytes. They found that ANG II stimulates NO release from the mTAL and subsequent diffusion to the adjacent pericytes and endothelium ( 7 ). In addition, they observed that ANG II significantly increases C O2 - in the isolated mTAL, but not in isolated pericytes, and that tissue O 2 - reduction by the superoxide dismutase mimetic TEMPOL increases the diffusion of NO from mTAL to the pericytes, indicating that cross talk of NO from the mTAL to the VR is also inhibited by O 2 - ( 23 ). Overall, their results suggest that interactions between O 2 - and NO ultimately determine the effectiveness of in situ free radical cross talk between the tubule and the VR.
To investigate the effects of ANG II stimulation on C NO differences between R1 and R2 based on these findings, we performed simulations mimicking the fact that ANG II raises the epithelial generation rates of both NO and O 2 -. In the absence of quantitative data, the NO generation rate of all tubules was increased by a factor of 10 relative to the baseline, and the interstitial C O2 - was also increased ten-fold in the R2-R4 regions (but not in R1, where there are no tubules). We found that these simultaneous increases raise C NO in R2-R4 by 70% to 140% relative to the baseline case, whereas increasing only G ep by a factor of 10 raises C NO in R2-R4 by 110% to 220%. Without specific data on the effects of ANG II stimulation on G ep and C O2 -, it is difficult nevertheless to conclude as to the physiological effects of ANG II on local DVR contraction.
Perfusion of the renal medulla is principally derived from the efferent arterioles of the juxtamedullary glomeruli ( 25 ). In the OS of the OM, the vessels give rise to DVR that coalesce into vessel bundles. The DVR in the bundle center extend into and perfuse the IM, whereas the DVR at the bundle periphery peel off to form the capillary plexus that perfuses the tubules. Since NO activates guanylate cyclase in the pericytes surrounding DVR, thereby inducing vasodilation, differences in C NO between the bundle center and periphery may result in differences in DVR diameters and may affect the distribution of blood between the OM and the IM.
To the best of our knowledge, DVR diameters have not been directly correlated with pericyte C NO. Pallone and colleagues ( 41 ) examined the effects of chronic ANG II infusion on DVR bathed in ANG II solution. In chronically infused DVR, DVR constriction was measured as 10% and NO generation was 2.8 times greater than the baseline; without chronic ANG II infusion, DVR constriction was 30% and NO generation was 1.6 times greater than the baseline. Since the investigators did not observe a significant effect of ANG II infusion on O 2 - synthesis, the reduced constriction associated with chronic ANG II infusion is possibly the direct result of increased NO generation by endothelia. Our model predicts that if G en is increased by a factor of 1.6 and 2.8 relative to the baseline, the average pericyte C NO increases from 50 nM to 80 and 140 nM, respectively. These results suggest that a difference of 60 nM in C NO could lead to a 20% DVR diameter variation. If we use Poiseuille's law as a first approximation and assume that blood flow is proportional to the fourth power of the diameter (that is, Q D 4 ), a 20% increase in DVR diameter should increase blood flow by a factor of (1.20) 4 = 2. Kakoki et al. ( 13 ) found that intravenous infusion of the nonspecific NO synthesis inhibitor nitro- L -arginine methyl ester into rat reduces medullary interstitial C NO from 86 to 48 nM and decreases medullary blood flow by 37%, an effect of smaller but comparable magnitude.
Taken together, these two experimental studies suggest that variations of C NO on the order of 10 nM may affect blood flow. As described above, the predicted pericyte C NO is 6 nM higher in R2 than in R1 with baseline parameter values and may be 66 nM higher in R2 than in R1 if epithelial NO production is increased 10-fold. Given these results, we postulate that preferential stimulation of tubular NO synthesis (without a concomitant increase in O 2 - production) significantly increases the diameter of DVR at the periphery of the vascular bundle relative to that of DVR in the center, thereby increasing the OM-to-IM blood flow ratio. Conversely, preferential stimulation of vascular NO synthesis is not likely to significantly affect the OM-to-IM blood flow rate ratio, given that a 10-fold increase in G en abolishes C NO differences between R1 and R2.
Both our previous and present models indicate that to yield renal medullary interstitial concentration in the 57-139 nM range, as measured in vivo by Zou and Cowley ( 43 ), the rate of endothelial NO generation must be on the order of 1 pmol·mm length -1 ·h -1 (or 10 -14 µmol·µm -2 ·s -1 ), that is, 10 3 times higher than in vitro estimates obtained by Wu et al. ( 38 ). If the rate of NO generation by VR was on the order of 1 fmol·mm -1 ·h -1 (or 10 -17 µmol·µm -2 ·s -1 ), predicted interstitial C NO would be lower than 10 nM, even if the rate of epithelial generation was as high as 1 pmol·mm -1 ·h -1, given that RBC act as such a potent sink for NO. Vaughn et al. ( 35 ) also developed a theoretical model of NO transport in nonspecific microvessels and estimated the NO generation rate (G NO ) as 6.8 x 10 -14 µmol·µm -2 ·s -1. The reasons for the discrepancy between the measured and predicted values of G NO remain unclear. It is possible that in vitro measurements of G NO significantly underestimated in vivo values; Shibata et al. ( 29 ) recently showed that in vitro measurements of oxygen consumption rates of arterioles in rat skeletal muscle were 100-1,000 times lower than values derived from in vivo studies. It is equally possible that neglected sources of NO ought to be taken into consideration, such as mitochondrial or neuronal forms of NOS, and NO adducts such as N -nitrosamines and S -nitrosothiols ( 1 ).
There is also some uncertainty regarding the NO scavenging rate, the main determinant of C NO in the model. The relative contributions of the extracellular boundary layer that is adjacent to RBCs ( 19 ) and that of the RBC intrinsic barrier to NO diffusion toward the hemoglobin molecules ( 34 ) have not been completely determined; if we underpredicted the NO diffusion barrier from plasma to hemoglobin in RBCs, we may have overestimated the scavenging potential of RBCs. The discrepancy between measured and predicted NO generation rates persists, however, even if we assume no NO consumption in RBCs. We performed simulations in which K RBC was set to zero and both vascular and tubular NO generation rates were set to the values measured by Wu et al. ( 38 ), on the order of femtomoles per millimeter per hour. We found that predicted C NO in the interstitium were still lower than 10 nM when the first-order kinetic constant of NO consumption by superoxide ( K SO ) was kept equal to its baseline value. Only when K SO was reduced by a factor of 10 did we predict interstitial C NO values on the order of 50 nM. Under these conditions, pericyte C NO was slightly higher in R1 than in R2 because the ratio of interstitium-to-VR cross-sectional area is lower in R1.
Our model is limited by the absence of data on the production and concentration of O 2 - (and other reactive oxygen species) in the medulla. O 2 - is produced by medullary epithelia, endothelia, and pericytes, and ANG II stimulates both NO and O 2 - production by mTAL ( 23 ). Without experimental measurements of O 2 - parameters, it is difficult to predict the effects of NO and O 2 - interactions.
Our previous study ( 39 ) suggested that variations in shear stress at the lumen-endothelium interface could significantly affect endothelial NO generation rates. This could constitute a feedback loop that increases the effectiveness of NO regulation of regional blood flow distribution: a higher blood flow in peripheral DVR will stimulate local endothelial NO generation, which in turn will enhance the difference in C NO between DVR at the periphery and those at the center of vascular bundles. However, without correlations between C NO, vessel diameter, and blood flow rate, and between shear stress and NO generation rates, examining such complex relationships is beyond the ability of our model.
Our results suggest that the production of NO by OMCD has no effect on pericyte NO concentrations. In the IM, IMCD have been found to generate much more NO than any other vascular and tubular segments ( 38 ). The recent experimental observations made by Pannabecker and Dantzler ( 27 ) suggest that DVR are excluded from CD clusters in the upper IM. Furthermore, pericytes become increasingly sparser along the cortico-medullary axis and even disappear in the deep IM ( 31 ). It is therefore difficult to predict the effect of epithelial NO generation on pericyte C NO in the IM and on IM resistance to blood flow. A complete, three-dimensional representation of the medullary architecture will be needed to assess the full effects of epithelial NO generation and tubulovascular cross talk on regional blood flow distribution.
In summary, our model of NO transport in OM cross sections suggests that NO production by tubular epithelia results in C NO differences between pericytes surrounding DVR at the periphery of vascular bundles and those surrounding DVR in the bundle center. Preferential stimulation of epithelial NO production should significantly raise the periphery-to-center DVR diameter ratio, thereby increasing the OM-to-IM blood flow ratio. Concomitant increases in epithelial C O2 -, however, would reduce, if not reverse, this effect. Our results confirm the importance of NO and O 2 - interactions in mediating tubulovascular cross talk.
Glossary
C Concentration
D Diffusivity of NO
G k Volumetric NO generation rate by vascular endothelial ( k = en) or tubular epithelial ( k = ep) cells
H C Tube hematocrit in core layer
H T Tube hematocrit in vessel
h RBC Hindrance factor characterizing NO diffusion into red blood cells (RBC)
k Hb Second-order kinetic constant of reaction between NO and heme
k O2 - Second-order kinetic constant of reaction between NO and O 2 -
K RBC First-order kinetic constant of reaction between NO and RBC, with respect to NO (i.e., K RBC = k Hb C Hb H C /h RBC )
K SO First-order kinetic constant of reaction between NO and O 2 -, with respect to NO (i.e., K SO = k O2 - C O2 - )
R1-R4 Concentric regions centered around vascular bundle
p Plasma layer thickness
Subscripts and Superscripts
en Endothelium
ep Epithelium
EI Endothelium-interstitium interface
I Interstitium
GRANTS
This work was supported by National Institute of Diabetes and Digestive and Kidney Diseases Grant DK-53775.
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作者单位：Department of Chemical and Biological Engineering, Tufts University, Medford, Massachusetts