Advective transport of nitric oxide in a mathematical model ofthe afferent arter - 《American Journal of Physiology》美国生理学杂志干细胞之家

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发表于 2009-4-21 13:37 |只看该作者 |倒序浏览 |打印

作者：Kayne M.Smith, Leon C.Moore,  Harold E.Layton作者单位：1 Department of Mathematics, Duke University,Durham, North Carolina 27708-0320; and Department ofPhysiology and Biophysics, State University of New York, Stony Brook,New York 11794-8661
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      【摘要】
   Endothelium-derived nitric oxide (NO) isthought to be short-lived in blood because of rapid removal fromplasma, mainly by binding to Hb. The extent to which removal limits NOadvection is unclear, especially for blood flow in the renal afferentarteriole (AA), which has a transit time of 3-30 ms. Amathematical model of AA fluid dynamics and myogenic response thatincludes NO diffusion, advection, degradation, and vasorelaxant actionwas used to estimate NO advective transport. Model simulations indicatethat advective transport of locally produced NO is sufficient to yieldphysiologically significant NO concentrations along much of the AA.Advective transport is insensitive to NO scavenging by Hb because theNO-Hb binding rate is slow relative to AA transit time. Hence, plasma NO concentration near the vessel wall is influenced by both diffusion from endothelial cells and advection from upstream sites. Simulations also suggest that NO advection may constitute a mechanism to stabilize arteriolar flow in response to a localized vasoconstriction accompanied by enhanced NO release.
      【关键词】 kidney renal hemodynamics myogenic mechanism immersed boundarymethod
   INTRODUCTION1 f) h1 y: O6 \! D' a* C/ ~* G
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ENDOTHELIUM-DERIVED NITRIC OXIDE (NO) is an important regulator of renal microvascular tone.NO is a mediator of the vasorelaxant effects of several agents,including acetylcholine and bradykinin ( 27 ), and NOmodulates the effects of several vasoconstictors, including endothelinand ANG II ( 11, 35 ). As a consequence of its action todesensitize vascular smooth muscle (VSM), NO also modulates themyogenic reactivity of the preglomerular vasculature. Myogenicreactivity, an intrinsic property of arterial vessels to constrict inresponse to increased intravascular pressure, plays a major role in theautoregulation of renal blood flow and glomerular filtration rate( 9, 26 ).0 r3 k% o0 d. x

Because of its small size, NO diffuses rapidly through most biologicalmembranes and tissues, where it is consumed in various reactions. Inparticular, NO binds to Hb and other heme-containing proteins, itreacts with superoxide to form peroxinitrite radicals, and it forms S -nitroso (SNO) adducts on Hb and on other proteins ( 15, 32 ).8 O1 x) G9 Q" a7 y8 q& \

In the traditional paradigm, NO is a paracrine agent that affectsmyogenic reactivity by diffusing from endothelial cells (EC) toneighboring VSM cells, where it influences a variety of cellularfunctions ( 27 ). In contrast, the fate of NO released fromEC into the vascular lumen is less clear. Initially, it was thoughtthat such NO would diffuse into the bloodstream and rapidly bind withHb; as a consequence, free NO levels in plasma would be very low( 20 ), and NO transport in blood by means of axial advection 1 would be of littlephysiological significance. This view has been challenged in recentyears. The reaction that forms nitrosothiol residues on Hb and albuminis reversible, and that reversibility provides a mechanism for NOtransport over long distances ( 15 ). Furthermore, after NOis released from the endothelium, four stages of diffusion are requiredfor NO to reach Hb in a red blood cell (RBC): diffusion through bloodplasma from the endothelium to the central lumen (where most RBC aresuspended), diffusion to the RBC membrane, diffusion across the RBCmembrane, and diffusion within the RBC cytoplasm to reach Hb. Detailedexperimental and mathematical analyses of these transport processessuggest that the NO scavenging rate in flowing blood is about threeorders of magnitude lower than previously believed ( 37, 38 ).
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One factor that slows the scavenging of NO in the vascular lumen is thesurprisingly low NO permeability of the RBC plasma membrane, which wasrecently estimated to be ~2,000 times less than the NO permeabilityof a pure lipid membrane ( 38 ). Another factor that maylimit NO scavenging, and a focus of this investigation, is thediffusion resistance presented by the (nearly) RBC-free boundary layeradjacent to the endothelium. Evidence for such resistance to NOdiffusion comes from experiments that show that intravascular flowreduces NO consumption by RBC but does not affect NO consumption byfree Hb, implying that NO consumption is reduced by the RBC-free zonepresent in flowing blood ( 21 ). The physical basis fordiffusion resistance in the boundary layer has been thought to be NOaccumulation in the layer via diffusion from the EC, resulting inboundary layer NO levels higher than in VSM and in local concentrationgradients that favor net transport from the EC to the VSM cells( 21 ).0 T) ~/ [+ [4 K; S
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In a renal afferent arteriole (AA), the presence of such high NOconcentration in the RBC-free plasma layer suggests that significant NOmay be advected in that layer, because the layer is sufficiently thick(~2 µm) ( 4 ) to have an average flow velocity that is asubstantial fraction of whole AA average flow velocity. If so, NOadvection in the boundary layer could substantially increase the axialdistance over which locally produced NO influences vascular tone, aswell as contribute to the development of the diffusion resistance inthe boundary layer. This possibility has not been previously analyzedin model studies ( 5, 20, 39, 41, 45 ), probably becausedetailed modeling of transport dynamics involves formidabledifficulties. A detailed model requires a representation of fluiddynamics, coupled with NO diffusion and advection, within walls havinga configuration that is determined, in part, by local physical factorsand NO concentration.

The objective of this study was to investigate the influence ofadvective NO transport in the plasma boundary layer on the thedistribution of NO in an AA. To accomplish this, we used a previouslydeveloped mathematical model of solute diffusion and advection in an AAto predict the fate of NO released from the vascular wall. The modelsimulations suggest that advection of locally produced NO is ofsufficient magnitude, relative to the NO degradation rate, to influencevascular tone substantial distances downstream from the release site.Furthermore, simulations suggest that such axial NO transport may serveto stabilize segmental blood flow in response to local constrictionand/or injury in the proximal AA.  u! q9 P; P9 @2 m) {" B6 W& u# q
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MATHEMATICAL MODEL

An analysis of NO transport in an AA requires an approximation ofthe flow field within a region having dimensions and pressures similarto those in an AA; a representation of the impact of myogenic reactivity, including its modulation by NO, on AA diameter; and asimulation of the transport and metabolism of NO. The model AA used inthis study, which was previously developed and tested ( 2 ),consists of three analogous components. The first component is arepresentation of the AA wall, the fluid motion and pressure within theAA wall, and the motion of the AA wall in response to fluid pressureand myogenic forces. The second component consists of myogenicsubmodels, which are elastic-contractile elements that represent thespatially distributed myogenic force generated in response tointraluminal pressure; these submodels modulate the myogenic force as afunction of local NO concentration. The third component is arepresentation of NO release, binding, degradation, and transport (byboth diffusion and advection) in the model AA lumen and in nearbysurrounding tissue.9 R/ m3 F9 |2 M0 D8 p, j' m. u
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AA Model Configuration- }; {+ L. |) k$ n- _, u

The full mathematical model (AA model) represents incompressiblefluid flow through a two-dimensional channel. That channel representsthe AA lumen, the channel boundary represents the arteriolar wall, andthe channel flow is considered to consist of blood plasma and bloodcells. To incorporate the dynamics of the interaction of channel flow(i.e., blood flow) with the channel boundary (i.e., arteriolar wall),the model employs the immersed boundary method ( 29, 30 ). Atwo-dimensional flow channel was used because the immersed boundarymethod is more readily formulated in rectangular coordinates than incylindrical coordinates and because a three-dimensional formulationwould require prohibitive numerical computational times. The use of atwo-dimensional channel rather than a cylindrically axisymmetricconfiguration introduces a systematic bias, in that flow velocitiescomputed in a channel exceed those in a cylinder, given the samepressure gradient. However, relative to the normal variations inlength, diameter, and flow in a population of afferent arterioles, thisbias is not large (see DISCUSSION ). Our model arteriole wasformulated so that pressure profiles in its flow channel are in goodagreement with physiological measurements (see Figs. 3 and 5 andassociated text).
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The forces exerted on the fluid by the channel boundary are computed bymeans of a phenomenological model that provides the relationshipbetween intravascular pressure and the myogenic response, a responsethat is modulated by NO concentrations at the channel boundary. Thatphenomenological model, the myogenic response model, or, simply,"myogenic model," is a subcomponent of the AA model: the AA modelincludes hundreds of myogenic model elements (see below).
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The AA model is illustrated in Fig. 1. Inthe bottom section of the figure, the model arteriole wall(labeled W) is contained within a rectangular fluid domain. A fluidsource (shaded square) is situated near the left end of themodel AA, and a corresponding sink is near the right end; anarrow indicates flow direction. The flow rate can be adjusted byvarying upstream and downstream pressures and resistances, that is, anupstream pressure that is separated from the source by a resistance anda corresponding downstream pressure that is separated from the sink bya resistance (resistances are not illustrated). To minimize theinfluence of end effects (arising from the source or sink) on modelinterpretation, most figures showing results will include only theinterior region between the vertical gray lines, i.e., the regionbetween x = 0 and x = 140 µm.

Fig. 1. Schematic diagram of model configuration. Bottom portionshows model renal afferent arteriole (AA) wall (W) within the fluiddomain, which is enclosed by the bold rectangle. Shaded boxes representfluid source and sink; the arrow indicates direction of fluid motion.The enlarged region of model AA ( top ) shows the wallconfiguration. The nodes ( ) that make up model AA wallare connected along the wall by elastic springs. Nodes are connectedacross the model AA by elastic-contractile myogenic elements (ME). Rightmost element is expanded to show that each ME isformulated as a spring of variable strength K M configured in parallel with a dashpot having damping coefficient.Model results in most subsequent figures show only that portion ofmodel AA for x between 0 and 140 µm.
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For some simulations, we computed AA blood flow; the calculationinvolved two steps. First, flow was calculated for a channel having areference "height" of 100 µm (height indicates distance along the z -axis, which is perpendicular to the plane that contains the vessel walls illustrated in Fig. 1 ); this calculation, which corresponds to Eq. 8 in Ref. 2,was based on the pressure drop across the upstreamresistance that is external to the model vessel. In the second step,flow through a cylindrical vessel was estimated by scalings thatincorporated the vessel diameter and its presumed circular crosssection: the value obtained from the first step was multiplied by D /(100 µm), where D was the average model AA diameter, to represent flow through a square cross section with side D, and that result was multiplied by /4 to produce arectangular cross section with an area equivalent to a circle havingdiameter D. The scaling of the channel flow by a factor of D preserves the proportionality of flow to the fourth powerof diameter that is characteristic of cylindrically axisymmetric flow.

The top portion of Fig. 1 is a schematic enlargement of aportion of the model AA. The AA wall is constructed of nodes [material points in the fluid ( in the figure)] that areconnected by elastic or elastic-contractile elements. Along the AAwall, the nodes are linked by elastic elements (essentially passivesprings with fixed rest lengths). Across the wall (i.e., transverse tothe flow direction), the nodes are connected by elastic-contractile elements that make up the myogenic model; each element is composed ofan active spring, of variable strength K M,configured in parallel with a dashpot that is characterized by dampingcoefficient (see expanded element in Fig. 1, top right ).As the model walls move within the fluid, and as the myogenic responseis activated (through K M ) in the myogenic model,the elastic and elastic-contractile elements exert forces on the nodes,and the nodes, via the immersed boundary method, exert a force density,localized along the AA wall, on the fluid domain.+ a% k' \+ f6 V  e
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The myogenic model includes both steady-state and transientresponses ( 2 ). The myogenic force applied to the boundarynodes by the model is a function of intravascular pressure, elapsed time, vessel diameter, rate of change of vessel diameter with respectto time, and local NO concentration. The parameter values for themyogenic model were obtained by fitting the model to myogenic responsesobtained in vitro from blood-perfused rat juxtamedullary AA. The AAwere subjected to approximate step-changes in perfusion pressure (from60 to 100 mmHg) with a transition time constant of ~1.1 s; resultingchanges in vessel diameter were determined from video recordings usingmethods described in Ref. 8. The measured autoregulatoryresponses included contributions from both the myogenic andtubuloglomerular feedback (TGF) mechanisms. Thus the strength of themyogenic response, as determined by the fitting, is sufficiently highto include the contribution of the TGF mechanism, which is notexplicitly represented in our model. The experimental response and ourmodel fit are illustrated in Fig. 2. TheAA model myogenic response magnitude and time course are similar tothose shown for both the experimental response and the response of asingle myogenic element.
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Fig. 2. Transient myogenic responses as percentage of initialluminal diameter. Experiment, rat AA response when perfusion pressurewas increased from 60 to 100 mmHg; ME, diameter computed by a singlemodel ME when perfusion pressure was increased from 60 to 100 mmHg; AAmodel, full simulation response at x = 120 µm afterfeed pressure was increased from 80 to 120 mmHg. Both experimental andsimulation time scales are given.! s/ n) p$ e4 s  p

To compensate for the decline in intravascular pressure along the AAsegment, a spatial inhomogeneity was incorporated into the strength ofthe myogenic model elements ( 2 ). Without this inhomogeneity, the diameter of the model AA increased as luminal pressure decreased along the length of the model AA, whereas AA in vivohave diameters that are relatively uniform along the length of thearteriole (see Fig. 5 ).
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Figure 3 illustrates the capability ofthe full AA model to autoregulate flow as feed pressure increases. Thedashed lines show steady-state results obtained without myogenicreactivity: the model vessel dilates passively, and both flow andoutflow pressure increase. With the myogenic response activated (solid lines), the vessel constricts as feed pressure increases, and bloodflow and outflow pressure are stabilized. The predicted steady-stateautoregulation responses are in reasonable agreement with experimentalmeasurements ( 7, 8 ).

Fig. 3. Diameter ( A ), outflow pressure ( B ),and flow rate ( C ) for passive (dashed lines) and formyogenically active (solid lines) model AA evaluated at feed pressuresof 80, 100, and 120 mmHg. Diameter was measured at x = 120 µm, and pressure was measured at x = 140 µm and y = 0. Outflow pressure and flow rate in myogenicallyactive vessel are relatively unchanged, compared with passive vessel,as feed pressure varies through normotensive range.

Modeling NO Transport, Removal, and Vasorelaxant Effect
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NO is released from the arteriolar walls, advects with the localfluid velocity, diffuses, and degrades as a function of time andlocation within the model arteriole. NO can also diffuse into theregion outside the vessel; that region represents the interstitium andtissues surrounding the arteriole.( Q9 {/ v( w- P& t3 ]

As is generally characteristic of flowing blood, we assume that thehematocrit will be highest at the AA center and essentially zero nearthe vessel wall ( 10 ). Thus to represent the spatial distribution of NO binding to Hb and of the loss of NO in reactions with superoxide and sulfhydryl residues on Hb, the rate of NO removalfrom the model fluid was scaled by a decreasing function of distancefrom the center axis y = 0 of the model AA.Specifically, the removal rate was scaled through multiplication by acurve obtained from a symmetrical pair of hyperbolic tangent functions, as illustrated in Fig. 12 A; the distance from the vesselwall to the inflection point of the rise varied locally and dynamically along the vessel as vessel diameter changed.

NO-mediated VSM relaxation was represented by making the myogenicresponse strength, K M, a function of local NOconcentration ( 2 ); the response was scaled in accordancewith a dose-response curve relating NO concentration to rat aortadiameter ( 16 ), resulting in a curve that related NOconcentration to the percentage of myogenic relaxation (see Fig. 4 ).; Y8 g5 f. R  r+ D* L

Fig. 4. Dose-response curve for nitric oxide (NO) based on datafrom Ref. 16. Percent relaxation is presented as afunction of AA wall NO concentration: 0% relaxation corresponds tofully myogenic (active) behavior; 100% relaxation corresponds to fullypassive behavior (see Fig. 3 ).1 ?) ~9 U" ]2 ?0 r; B; o/ ~
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Basal NO release was not represented explicitly in the model AA;instead, basal NO release was implicitly incorporated by assuming thatthe model configuration was determined, in part, by basal NO releaseand that the configuration therefore included the effects of basal NOrelease. Because this investigation principally involves a hypothesisabout the effects of advection and diffusion on NO distribution, itshould be sufficient for most purposes to determine the fate of arepresentative increment of NO release from a localized site. However,in one simulation a low, uniformly distributed NO release rate wasintroduced to ascertain the likely effect of advection on diffusionresistance all along the AA (see Fig. 13 ). The NO distribution soobtained should be indicative of the distribution of basally releasedNO, provided that basal release is uniformly distributed along thevessel wall.8 e! x( M4 E2 n/ C" K/ ?& F
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Model Parameters% V& o  e- E- q$ Z! ]
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AA model dimensions, used in all simulations, were chosen to besimilar to those observed in short juxtamedullary AA ( 6 ). We chose to represent a relatively short AA (140 µm, excepting thedistal segment) to reduce the computational time required to solve themodel (see below); we chose a target diameter of 16 µm. We did notrepresent the most distal segment of the arteriole, which is theprincipal effector site of TGF and which, in juxtamedullary AA, tapersby ~25% in diameter.. G& `8 v8 d  q" `. e; ?5 q

Model parameters, summarized in Table 1,characterized the fluid, the fluid pressures at the source and sink,and the release, diffusion, and removal of NO. We specified a set ofparameters that yielded a "base-case" steady-state solution to theAA model, assuming no NO release above the basal rate. In addition, wespecified a "standard" release rate for incremental NO releaseabove the basal rate, a standard that we used for comparisons andparameter studies. Along with that standard rate, we assumed abase-case parameter to characterize NO binding, a base-caseconfiguration for the spatial distribution of Hb (see above), and abase-case dose-response curve for the vasorelaxant effect of NO (Fig. 4 ).
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Table 1. Model parameters- a! X7 W3 Q6 W9 @2 v1 A$ ?

To keep the problem tractable, blood was simulated as a Newtonian fluidof uniform density 1.055 gm/cm 3 ( 1 ) and withan apparent dynamic viscosity of 1.75 × 10 2 g · cm 1 · s 1,based on measurements in an arteriole (with a diameter of 23 µm) fromcat mesentery ( 22 ). The pressures and fixed resistances atthe source and sink were chosen so that the base-case pressures at thefluid source and sink, 95 and 50 mmHg, respectively, would beconsistent with pressures found in rat juxtamedullary AA( 8 ).  V  y/ i: B* L
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We used a standard NO release rate of 3.275 × 10 15 µmol · µm 2 · s 1;this rate should be interpreted as the increment of elevation above thebasal base-case NO release rate (see above) from the luminal AA wall.This rate is more than an order of magnitude smaller than the NOrelease rate of 5.2 × 10 14 µmol · µm 2 · s 1 estimated for rabbit aorta ( 39 ).* R1 A$ f% Z# p' t- V
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NO diffusivity has been measured to be 3.3 × 10 5 cm 2 /s at 37°C in rabbit aorta ( 24 ), a valueconsistent with rapid diffusion in aqueous solutions. We assumed thatNO diffuses readily across the plasma membranes of EC and VSM cells( 24 ). The base-case half-life for NO, taken as constantthroughout the fluid domain, represents the removal of NO due tobinding with substances found outside the lumen; it is based on ameasured half-life of ~1 s for NO in blood-free perfused guinea pigheart determined in Ref. 3 from data in Ref. 17. Within the lumen, NO is removed at a much faster rate;the base-case rate of NO-Hb binding, which corresponds to a half-lifeof 20 ms, is based on the determination that 2 with NO ( 16 ).- w4 n5 I8 X/ P8 f

Numerical Calculations
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The immersed boundary method, which was used to simulate theinteraction between the fluid and the boundary (i.e., AA wall), uses aLagrangian 2 representationfor the boundary; the boundary is coupled by approximate Dirac deltafunctions to an Eulerian grid, on which the fluid calculations areconducted ( 29 ). The procedure for advancing one time step( 2 ) involves the following steps. 1 ) From the elastic and elastic-contractile elements, determine the force density(force per unit volume) on the boundary nodes and transfer thatdensity, via approximate delta functions, to the Eulerian grid. 2 ) Solve the incompressible Navier-Stokes equations for thefluid velocity and pressure throughout the fluid domain; include termsin these equations for the pressure source, the pressure sink, and theforce density arising from the immersed boundary. 3 ) Solvethe equations for solute advection and diffusion, with source terms forNO release and degradation, to obtain the concentration of NOthroughout the fluid domain. 4 ) Transfer the fluid velocity from the Eulerian grid to the nodes that make up the Lagrangian boundary by means of approximate delta functions; then, advect theboundary at the local fluid velocity.: M7 Z' e3 {$ D' B' l  _
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Because we previously demonstrated that a spatial resolution of 1 µmor less is needed to track fluid and solute motion with sufficientaccuracy for physiologically meaningful simulations ( 2 ), aspatial grid of 128 × 512 subintervals was used, resulting in adiscretization step of h = 1/512 × 0.02 cm(0.390625 µm). The model arteriolar wall was constructed of 1,920 nodes, connected by 1,920 elastic springs; 865 node pairs wereconnected by transverse elastic-contractile myogenic elements (the 190 nodes without transverse connections enclose the ends of the model vessel).) U" N% w* X, X0 @3 X5 e# |
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Although some of the numerical techniques used for solving the AA modelare implicit and thus do not restrict the numerical time step, othersare explicit (because we know of no practical alternatives) andtherefore require a small time step, relative to key physiological timescales that are relevant to our study. In particular, a numerical timestep of ~0.3 µs is required to solve the full AA model, and,consequently, ~315 h of computational time are required for 0.1 s of elapsed model time on a desktop computer equipped with a 1-GHz CPUand 256 MB of RAM. Because the full myogenic response in vivo requires~10 s, a time-scale adjustment was introduced to make modelsimulations practical: the speed of the simulated myogenic response wasincreased by a factor of 175 relative to in vivo responses.
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The adjusted time scale allows us to closely approximate steady-statemodel solutions (absolute steady states cannot be attained because thefinal phase of the approach to a steady state, which involves onlysmall changes, is slow). Moreover, depending on the object of one'sinvestigation, the change of scale may not significantly affect dynamicsimulation results. For example, the fluid transit time through themodel arteriole (~4 ms) is much faster than the characteristic timeof the myogenic response, whether a comparison is made with the in vivoresponse (~10 s) or the response in the accelerated time-scale, whichis ~60 ms. Thus even with the adjusted time scale, the acceleratedmyogenic response acts on a time scale that exceeds the AA transit time by more than an order of magnitude. Because the intravessel information is renewed rapidly relative to the myogenic response, the dynamic behavior of the AA model may still approximate qualitative in vivobehavior, although accelerated by more than two orders of magnitude.

Model Verification
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Although the model framework used in this study was previouslytested ( 2 ), significant changes were made in several of the model parameters to more closely simulate AA dimensions and flows.Hence, for the base-case parameters used in this new study (but with noelevated NO release), we verified thatfundamental characteristics of the modelstill agree well with expected behaviors. These tests are summarized in Figs. 5-8.

Fig. 5. Model AA diameter ( A ), average flow velocity( B ), and pressure ( C ) as a function of AAposition x for spatially homogeneous (dashed lines) andspatially inhomogeneous (solid lines) myogenic responses. Because thespatially inhomogeneous response is more physiologically accurate, thatresponse was incorporated in the base-case model formulation and wasused in all other simulations reported in this study.- s9 T  x/ l2 }) `- l0 c3 [
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Fig. 6. Base-case pressure as a function of position on fluid domain.Intersections on wireframe diagram correspond to every 4thdiscretization value along axes of computational grid.

Fig. 7. Comparison of simulated velocity and shear stressprofiles with profiles for planar Poiseuille flow. A :velocity profiles. Solid line, fluid velocity profile along y -axis computed by AA model at the cross section where x = 70 µm; dots, computational grid locations.Velocity profiles for Poiseuille flow are shown for 3 diameters:diameter D of model AA based on node locations, D 2 h, and D 4 h, where h is the numerical grid subinterval.Horizontally striped boxes above profiles show locations of boundarynodes (heavy lines) and the wall thicknesses (in h -increments), based on extent of approximate deltafunctions. B : shear stress profiles. Solid line, shearstress calculated from local pressure gradient computed by AA model;dashed line, shear stress for Poiseuille flow based on diameter D; vertical bars, extent of approximate delta functions thatform AA wall boundary (total width is 4 h ); first 3 values at top right : analytical shear stress magnitudes at wall center(based on node locations), at the interior edge of the wall withthickness 2 h (light gray sections of vertical bars), and atinterior edge of the wall with thickness 4 h (full verticalbars), respectively; last value at top right is maximumshear stress magnitude computed by model across model AA lumen.8 G1 P" h' D4 y* ^( h

Fig. 8. Advection of solute bolus in steady-state flow. Dashed yellowlines, centers of vessel walls; bar at right, concentrationscale (C; in µM). Topmost panel corresponds to t = 0; each successive panel is time-advanced by 0.45 ms. At t = 0, the bolus has a 2-dimensional Gaussiandistribution; in successive panels, the bolus is deformed by the nearlyparabolic velocity profiles of approximate Poiseuilleflow.8 l/ W& }( f" ?+ E3 k
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Figure 5 shows the diameter, average velocity, and pressure along thelength of the model AA for the case of increasing (i.e., inhomogeneous)myogenic response (solid curves) as a function of distance x and for the case of spatially uniform (i.e., homogeneous) myogenicresponse (dashed curves). The average velocity is computed across thediameter of the model vessel by dividing the planar flow rate by thediameter; the pressure is measured along the centerline of the AA flow,i.e., along the line y = 0. One should recall that thediameter is determined by the local intravascular pressures and by themyogenic response characteristics, which vary with position. For thecase of inhomogeneous myogenic response (which is the base-caseconfiguration), the diameter and average velocity vary somewhat alongthe model vessel, although less than for the homogeneous response case,and the decrease in pressure is more nearly linear than for thehomogeneous case. For the homogeneous myogenic response, theprogressive dilation of the AA is a consequence of the myogenicresponse as intravascular pressure decreases; the dilation results inan average velocity that is decreasing and in pressure with adecreasing slope magnitude. Figure 5 illustrates two points: thebase-case vessel, with inhomogeneous myogenic response, maintains adiameter that is within 6.5% of the target diameter, and therefore thediameter appears nearly uniform; and, as we pointed out in Ref. 2, in vivo the AA must provide an inhomogeneous myogenicresponse to maintain a nearly uniform diameter in the presence ofdecreasing intravascular pressure.
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Figure 6 shows pressure as a function of position on the entire fluiddomain for the base case (cf. Fig. 1 ). The fluid source and sink aremanifest as local pressure deviations at each end of the modelarteriole. Away from the source and sink, and at each cross section(i.e., at each fixed x ), the pressure is nearly constant inthe intravascular region and (with a different constant) outside thevessel wall. Along the flow direction the pressure decreases nearlylinearly inside the model vessel. The grid line that curvessubstantially along the side of the tubular wall is a consequence ofthe small change in diameter indicated by the solid line in Fig. 5 A. Figure 6 demonstrates that the pressure distributionbehaves as expected for Poiseuille flow inside the model vessel and,indeed, throughout the fluid domain.

Figure 7 A compares the simulated fluid velocity profile(solid line) along the cross section at x = 70 µmwith exact velocity profiles for planar Poiseuille flow. Excellentagreement was found between the simulated profile and the (exact)Poiseuille flow profile, when that flow profile was based on the localpressure gradient and on the diameter D 2 h, where D is the diameter that corresponds tothe distance between the centers of the opposing nodes in the modelvessel walls, and h is the spatial discretization distancein the numerical method. Because of the spatially distributed approximate delta function used by the immersed boundary method, theexact width of the model wall is not prescribed a priori but arisesfrom the interactions of the components of the numerical method.However, the investigation in Ref. 2 indicated a wall width of 2 h, which is consistent with the effective diameterfound here.

Figure 7 B compares the magnitude of the simulated shearstress profile (solid line) with the magnitude of the (exact)Poiseuille shear stress profile (dashed line), based on the localpressure gradient. The shear stress was computed, in both cases, along the same cross section used in the calculation of the velocity profile.The width of the nodes that make up the boundary wall, 4 h,is indicated by the vertical gray bars; the more lightly shaded innerportion of the bar has width 2 h. As can be seen in thefigure, excellent agreement was found up to the interior edge of thewalls. The simulated shear stress decreases to near zero as it passes through the layer containing the approximate delta functions.' F; M" S3 z! b4 t& e
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The advection of NO is illustrated in Fig. 8. In this test, NOdiffusion, NO binding/degradation, and myogenic reactivity weredisabled, and a simulated bolus of NO, distributed as a two-dimensional Gaussian, was placed near the source end of the model vessel and allowed to advect with the fluid. Each successive frame istime-advanced by 0.45 ms. The parabolic shape of the expected planarPoiseuille flow is easily seen as are substantial amounts of NO thatremain near the walls of the model arteriole within the boundary layer near the initial location of the bolus. The nearly stationary solute inthe boundary layer illustrates that the model flow has essentially zerovelocity at the vessel wall.
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RESULTS8 r% J0 C! K! p8 x- f/ N+ `$ Z
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NO Transport in the Boundary Layer
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To estimate the extent of NO transport within the RBC-freeboundary layer, NO was released from the wall at a site near the beginning of the model vessel. Thus at t = 0, NO wasreleased from both sides of the AA model channel at the standard rate(as characterized in MATHEMATICAL MODEL ) of 3.275 × 10 15 µmol · µm 2 · s 1;the release sites, centered at x = 19.2 µm, were 8.2 µm long. The initial vessel diameter and intravascular pressureprofile were those given by the base case (shown as Inhomogeneous inFig. 5 ). The resulting advective-diffusive transport of NO isillustrated in Fig. 9; the topmost panel corresponds to t = 0.3 ms, andeach successive panel is time-advanced by 0.9 ms. The vessel boundary is indicated by dashes.
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Fig. 9. Diffusive and advective transport of NO in the model AA. Dashedyellow lines, centers of vessel walls; bar ( right ),concentration scale (in µM). At t = 0, NO was released atthe standard rate (see text) from a short proximal segment of the modelAA. Each successive panel after the topmost panel is time-advanced by 0.9 ms.

In the first millisecond, the results show rapid diffusion of NO intothe abluminal space occupied in vivo by VSM cells. In the model AA,where the myogenic response time of the VSM has been accelerated, alocal dilation occurs around the release site. NO also diffuses intothe lumen, where it is advected by the fluid stream. With increasingtime, NO diffuses further into the abluminal space and is transportedfurther down the vessel, leading to additional downstream vasodilation.The intravascular NO concentration downstream from the release site ishighest just within the vessel, but not at the vessel wall, a patternthat reflects the essentially zero fluid velocity at the wall.Downstream from the release site, NO diffuses from the boundary layertoward the wall and into the abluminal compartment. Direct diffusionfrom the release site cannot account for the NO in that compartment sofar downstream, because the distances are much greater than thediffusional spread into the abluminal space in the directionperpendicular to the x -axis of the model vessel.
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This point is illustrated in Fig. 10,which shows the concentration profiles along the line that is centeredat the release site and is perpendicular to the flow axis of thevessel. Profiles for three time points after the onset of release areshown: 3, 6, and 9 ms. (At the onset of release, the NO concentrationarising from incremental NO release is 0 throughout the fluid domain; i.e., at the onset of release, NO release is at the basal rate.) Twopoints are noteworthy. First, at 9 ms, the distance that NO diffusessignificantly (at a concentration of at least 0.005 µM) into theabluminal space is 5-6 µm, a much smaller distance than theaxial distance NO is advected down the vessel (30 µm). Second, withinthe lumen, the NO concentration is lowest at the center axis( y = 0) of the vessel, owing to the continuous arrivalof fluid with low basal NO content. This low concentration results in asteep concentration gradient directed from the wall into the lumen, agradient that persists well downstream as NO enters the flowing fluidand is bound or degraded. Unless NO diffusivity through the luminal ECplasma membrane differs significantly from the diffusivity through theabluminal EC plasma membrane, the asymmetry in the diffusion gradientswill drive the majority of locally released NO into the lumen. Thisresult shows that advective NO transport has a substantial effect onthe distribution of NO. (The NO concentration near the centeraxis of the vessel is also suppressed by NO-Hb binding; see below.)3 t4 i: z5 H2 f0 V: S  ?" B. d

Fig. 10. NO concentration cross section from model AA at x = 19.2 µm for simulation illustrated in Fig. 9.Lumen of model AA is indicated by gray bar. Concentration profiles for3 time points after onset of release are shown: 3 (solid line), 6 (grayline), and 9 ms (dashed line). Fourth panel from top in Fig. 9 corresponds to t = 3 ms.

Sensitivity Studies

Figure 11 shows results from threesimulations in which the NO release rate was varied by two orders ofmagnitude: the standard rate multiplied by 0.10, 1.0, and 10. For theconfiguration in which NO was locally released (Fig. 9 ), Fig. 11 showsNO concentration profiles along the wall of the vessel at 9 ms afterthe onset of NO release. As expected, NO concentration varied directlywith the release rate, whereas the shape of the profile changed little. NO release rate and tissue sensitivity to NO are related, in that increases in either factor will extend the distance over which thereleased NO will influence local vascular tone, whereas reciprocal changes (of similar magnitudes) in these parameters should have littleeffect." R8 I) c) `2 R* p6 l

Fig. 11. Sensitivity of wall NO concentration to NO release rate.NO concentration measured along model AA wall at 9 ms after initialrelease for indicated factors of standard NO release rate is shown. Ineach simulation, NO was released from 8.2-µm model AA wall segmentscentered at x = 19.2 µm.7 l. V/ i7 ~# r7 L1 M
) v% K7 n. h9 l. w6 u# n' l2 y& t
Figure 12, A and B, illustrates sensitivityto the effective width of the RBC-free zone in the boundary layer.Figure 12 A shows four different spatial distributions ofNO-Hb binding that vary the cell-free zone from ~0 to 25% of AAdiameter. The four curves in Fig. 12 B, which are nearlysuperimposed, correspond to the differing distributions of NO-Hbbinding shown in Fig. 12 A. At the base-case binding rate,changes in the effective width of the cell-free zone had almost noeffect on the NO concentration profile. In part, the insensitivity ofthe distribution of NO to the width of the RBC-free layer results froma base-case removal rate (i.e., a reaction velocity) that is smallcompared with AA blood flow velocity.
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Fig. 12. Sensitivity of NO distribution to location andmagnitude of NO-Hb binding. A : relative strength of bindingcoefficient across AA model cross section. At each spatial location,NO-Hb binding rate is product of NO-Hb binding coefficient and amultiple of base-case NO-Hb binding rate. The binding coefficient curveis a function of wall location in space and time; as diameterdecreases, so does relative size of nonzero region; D corresponds to local diameter of model AA, whereas D /2corresponds to model AA wall locations; solid curve, base-case NO-Hbbinding coefficient curve used in most simulations. The 3 othercurves correspond to parameter studies illustrated in B. B : sensitivity of AA wall NO concentration toNO-Hb binding coefficient. At t = 0, NO wasreleased at standard release rate from 8.2-µm model AA wall segmentscentered at x = 19.2 µm. NO concentration along modelAA wall at 9 ms of elapsed time for 4 NO-Hb binding coefficient curvesfrom A was computed. The concentration profiles are nearlyinsensitive to differing binding coefficient curves. C :sensitivity of AA wall NO concentration to variation in NO-Hb bindingrate in NO release experiments analogous to those in B usingbase-case NO-Hb binding coefficient curve ( A, solid line).Solid curve, binding at base-case rate, which corresponds to bindinghalf-life of 0.02 s; short dashed line, NO-Hb binding rate 10 times base-case rate; long dashed line, NO-Hb binding rate 100 timesbase-case rate. For comparison, simulation without NO-Hb removal (NO-Hbbinding coefficient equals 0 everywhere) is shown by medium dashedline, and simulation with constant NO-Hb binding (NO-Hb bindingcoefficient equal to 1 everywhere) is shown by gray line.9 }- v! I# O" j1 Q
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Figure 12 C shows surprising effects of NO-Hb binding withinthe lumen: aside from small changes in the magnitude of the peak at therelease site, neither a complete elimination of NO-Hb binding nor aconstant distribution of NO-Hb binding within the domain had a majoreffect on NO distribution along the vessel. To estimate the NO-Hbbinding rate that might be required to markedly reduce NOconcentration, the binding rate was increased from the base-case valueby one and two orders of magnitude. The results suggest that a 100-foldincrease in the removal rate is needed to reduce the magnitude of thepeak by ~50%, whereas a 10-fold increase significantly reduces NOconcentrations downstream from the release site.! U  U7 d- j- r* G6 M4 Z! `- T
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Effect of Distributed NO Release

The foregoing results appear to suggest that the thickness of theboundary layer and the NO removal rate have little influence on thedistribution of NO along the arteriole. However, in these simulationsof localized incremental NO release, steep NO concentration gradientsare established between the site of release and the lumen upstream fromthe release site; therefore, the arriving fluid presents littleresistance to the diffusion of NO into the lumen. To investigate theestablishment and impact of NO accumulation in the boundary layer, asmight occur during basal NO release, additional simulations wereconducted where the incremental NO release was distributed along thevessel wall. Figure 13 shows theresults of a simulation with uniform NO release from x = 12 to x = 127 µm. To simulate a very smallincrement in basal NO release, the release rate was reduced to 0.01 thestandard rate. (Note that the scaling of the concentration key in Fig. 13 differs from that in previous figures.) As in the simulations oflocalized release, the steep concentration gradients at the proximalend of the model AA favor NO diffusion into the lumen. As the blood moves along the AA, more NO diffuses into the boundary layer and its NOconcentration rises. As this occurs, the NO gradient from EC into thelumen is progressively diminished, which establishes a diffusionresistance in the fluid adjacent to the vessel wall, as described byVaughn et al. ( 40 ). The net result is enhanced diffusionfrom the EC into the tissue surrounding the distal AA where the highesttissue NO concentrations are established.

Fig. 13. Distributed NO release along walls. Dashed yellow lines,centers of vessel walls; bar, concentration scale (in nM). At t = 0, NO was released, at 0.01 the standard releaserate, from a long segment of model AA wall. Elapsed time is 8.3 ms.9 {0 v5 E1 X; j: g/ i  Y/ }' l% h
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Response to Focal Vasoconstriction
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Finally, simulation results suggest that NO release may stabilizesegmental blood flow in response to a local vasoconstriction byreducing downstream vascular resistance. Increased shear stress at thesite of vasoconstriction should result in a local increase in NOrelease ( 27 ). Furthermore, focal vascular injury is often associated with vasoconstriction and increased NO production; indeed,NO can be produced in large amounts by the inducible isoform of NOsynthase (iNOS) at sites of vascular injury ( 14 ). If the vasoconstriction substantially reduced the intravascular pressure belownormal, then a compensatory myogenic vasodilation would also occur andreinforce the vasorelaxant action of elevated NO levels.( i! U/ u) N2 p1 A7 k

To examine the potential role of NO, a focal constriction was simulatedin the model AA, as illustrated in Fig. 14. After the gradual imposition, for~30 ms, of a constriction centered at x = 19.2 µm,NO was released at the standard rate on both sides of the AA modelchannel from 8.2-µm segments, also centered at x = 19.2 µm. To distinguish between local and downstream effects, theconstriction was maintained by making the myogenic elements in theconstricted region insensitive to NO. The released NO was advected withthe fluid and diffused: as it reached the model arteriolar walls, thewalls relaxed, based on the base-case dose-response curve in Fig. 4.Figure 14 shows NO distribution and the model arteriole 30 ms after theonset of NO release: note the dilation downstream from thevasoconstriction. (One should recall that the myogenic response of thearteriolar wall was accelerated in these simulations.)1 P+ m& c7 Y7 Q
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Fig. 14. NO distribution in constricted model AA 30 ms afterinitial release of NO. Dashed yellow lines, centers of vessel walls;bar, concentration scale (in mM).9 ^% i( y8 q0 l3 d" h

Figure 15 provides a quantitativecomparison of myogenic and NO-mediated compensation; Fig. 15 A shows pressure profiles along the model AA, whereas Fig. 15, B and C, shows wall position and diameter,respectively. Four cases, at near steady state, are shown: 1 ) base-case (long dashed lines), which corresponds to thepressure profile (and diameter) before imposition of the constriction (as in Fig. 6 ); 2 ) constriction (thick gray line), whichshows the effect of the constriction in the absence of either myogenic compensation or NO advection (this curve was constructed by using thespatially distributed vessel resistances from the base case, and, forthe constricted interval, from the constriction with myogenic case,then solving for the constant flow rate, and then solving for thepressure along the model AA using the constant flow rate and theresistance at each x ); 3 ) constriction with myogenic compensation alone (constriction myo; short dashed lines); and 4 ) the additional compensation provided by the advection of NO released at the site of constriction (constriction myo NO; solidline), as in Fig. 14. The percentage of base-case blood flow is givenat the right of each pressure profile.% `; l. h- n: x: S8 R& k  V
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Fig. 15. Pressure ( A ), wall position( B ), and diameter ( C ) along model AA that arenonconstricted (base-case), constricted without myogenic responseand without NO release (constriction), constricted with myogenicresponse but without NO release (constriction myo), andconstricted with myogenic response and with NO release (constriction myo NO). In A, flow rates, as percentage of base case, aregiven at right of each curve. Wall relaxation caused byresponse of wall to NO concentration downstream of constriction resultsin increased model AA flow.6 `5 S" b! p( y0 [

The constriction reduced flow through the model AA to 83% of base-case(430 nl/min) and increased the segmental resistance (pressure dropdivided by flow rate) by 41% above base case. The effect of themyogenic response on the reduced downstream pressure results in adilation along the remainder of the AA sufficient to return flow to96% of its base-case value and to return the segmental resistance to112% of its base case. Because we have employed a high myogenic gainto compensate for the absence of TGF, this degree of myogeniccompensation is likely to exceed that achievable in vivo. When theeffect of NO release is included, a further compensatory dilationoccurs in the midportion of the vessel. The result is a furtherincrease in outflow to 103% of the preconstriction value and a returnto 100% of the preconstricted segmental resistance. These resultsillustrate the principle that advective transport of NO, together withdownstream myogenic relaxation, may serve to stabilize segmental bloodflow by reducing downstream vascular resistance in response to a focalconstriction in the proximal AA.
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It is noteworthy that in other simulations of this potentialregulatory mechanism (not shown), the extent of compensation was highlydependent on the NO release rate from the constriction site, andcompensation values well above and below 100% could be generated. Thishigh sensitivity is a consequence of the feed-forward nature of thisprocess as we have modeled it. However, in vivo, this autoregulatorymechanism would be embedded within the TGF system, and some degree ofnegative feedback control would be exerted. For example, if NO releasefrom the site of constriction were insufficient to normalize AA flow,TGF would act to reduce AA resistance, thereby contributing to therestoration of normal flow. Conversely, if NO release wereinappropriately high, TGF would act to restrain an increase in bloodflow above normal.

DISCUSSION. A2 O$ @1 q8 V
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The fate of NO released from EC and the extent of NO scavenging inthe lumen of arterial vessels are issues that have generated much studyand debate. A common notion is that free NO is short-lived: it israpidly removed from the blood by binding to Hb, forming SNO adducts,or forming peroxinitrite. The consumption of NO in the vascular lumenis important, because rapid NO scavenging can greatly diminishdiffusive delivery of NO to VSM cells.7 D$ F' A4 `' ]' O2 {- p  b7 g

We have used a mathematical model of fluid dynamics and solutetransport in a renal afferent arteriole to assess NO advective transport to downstream VSM. Our model study extends the diffusive NOtransport models developed by others ( 5, 20, 39, 41, 45 ).In each case, the fate of NO entering the lumen is an important factor,and the relative NO fluxes in the luminal and abluminal direction fromthe EC depend on the relative magnitude of the concentration gradients,under the assumption that EC membrane permeability is homogeneous. Inthese models, an important determinant of the intraluminal NO diffusiongradient, and thus NO distribution, is the rate of NO scavenging.However, our results suggest that advective transport of NO can alsohave a major impact on NO distribution., c( ?- S# }; B. V2 _
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Three interrelated findings arise from our study. First, modelsimulations predict that significant NO is advected in AA by the bloodstream and that NO concentrations along the vascular wall aresufficiently large to influence downstream vascular tone as NO diffusesfrom the lumen into the vascular wall. Furthermore, advective NOtransport is not limited to the RBC-free boundary layer. The secondfinding provides an explanation for the first: with physiologicallyreasonable rate coefficients, the luminal rate of NO-Hb binding is tooslow, relative to the transit time of blood in a renal arteriole, toeffect a major reduction in NO advection. Thus the principaldeterminants of NO concentration near the vascular wall are diffusiveNO transport from adjacent EC into the lumen and advection of NO inblood plasma from upstream sites. The third finding, a consequence ofthe first two, is the potential for NO advection to stabilizearteriolar flow in response to a localized vasoconstriction associatedwith enhanced local NO release.7 c8 v. l7 r7 d

Model Limitations

Before discussing these findings, we examine the strengths andlimitations of the complex model used in this study. By means of theimmersed boundary method, detailed representations of fluid dynamicsand resultant solute transport were obtained within and near astructure that exhibits dynamics similar to those of an AA.Specifically, the flow profile, shear stress rates, and diffusive andadvective transport agree well with theoretical predictions. Furthermore, the AA model, which includes hundreds of embedded myogenicelements, exhibited autoregulatory behavior similar to that observed invivo, suggesting that the model provides a reasonable qualitativerepresentation of an AA., L; C; s" ~& A8 O7 P6 o5 B

However, the AA model has some disadvantages and limitations inmodeling arteriolar blood flow. First, the high velocities and steeppressure gradients characteristic of AA flow require high spatialresolution. This resolution, combined with the small time steps thatare required to maintain numerical stability, resulted in longcomputational times. Owing to this, simulation times were limited to key features of luminaladvective transport of NO but insufficient to evaluate fully the extentof NO diffusion into the space external to the AA. In addition, we didnot explicitly model structures in the extravascular space, such asVSM, that may scavenge NO at a rate higher than the base-case rate usedexternal to the vessel (half-life, ~1 s).) c6 x9 |& N# p" @" q4 g; t

A second limiting factor is that the immersed boundary method, topreserve computational tractability, was implemented in two-dimensionalrectangular coordinates rather than in cylindrically axisymmetriccoordinates, which would be more natural for a vessel's cylindricalgeometry. Thus the model represents a channel of fluid with a widthequal to the model arteriolar diameter and with an infinite extent inthe z direction, which is perpendicular to the x - y plane of Fig. 1. Frictional forces areexerted on model AA flow only at the boundary of the channel; if thechannel is considered to extend from the plane determined, e.g., by z = 0, to the plane z = 16 µm, thenat the interface of the fluid at those planes, the interaction is frictionless.1 X  ^3 r8 y) A. f

A reasonable question is, To what extent do the flow velocitiescomputed in our planar model differ from those that would be obtainedin a model having cylindrically axisymmetric flow? An estimate of thedifference can be obtained by comparing the velocity expression forPoiseuille flow (fully developed steady-state flow) through a cylinderwith the corresponding expression for flow between parallel plates(i.e., a channel of infinite height). Let D be the diameterof the cylinder or the diameter (i.e., width) of the channel used inthis study, and let the length of the cylinder or channel be given by L. Then, in both cases, the explicit solutions to theNavier-Stokes equations show that flow velocities are proportional to D 2 / L ( 19 ). However, flowvelocity in a cylinder is reduced, relative to a channel, by a factorof 1/2, provided that the diameters, lengths, viscosity, andpressure drops are the same in the two configurations.# W$ b2 o2 ]$ n4 o- A5 {# x; b

Although overestimation of flow velocity by a factor of two may seem tointroduce significant error, our simulated flow velocities are notunreasonable when considered in the context of physiological flowvelocity variation. For example, a maximal AA autoregulatory responseto elevated perfusion pressure can reduce luminal diameter by~30-40% relative to the diameter at the lower pressure limit ofthe autoregulatory range. With essentially constant blood flow, themean flow velocity at the elevated pressure is increased two to threetimes over that at the lower pressure. Furthermore, with our base-caseparameters, our planar channel has the same velocity profile (as afunction of y ), for example, as would arise in a cylinder(as a function of radius) with a diameter that is 25% larger than ourbase case and a length that is ~20% shorter (owing to the D 2 / L dependence). These variations indiameter and length are within the physiological range of dimensionsreported for afferent arterioles (see below). Moreover, our fundamentalconclusion that advection is an important mode of NO transport alsodepends on the value of NO half-life in the blood used in oursimulations. In light of recent experimental studies ( 37, 38 ), the half-life value that we used may be substantiallyshorter than in vivo (perhaps by more than an order of magnitude).Hence, our fundamental conclusion is unlikely to be affected by thebias in model flow velocity introduced by the channel flow configuration.

Although we introduced a channel height scaling to ensure that flow isproportional to the fourth power of the diameter, as is the case incylindrically axisymmetric Poiseuille flow (see MATHEMATICAL MODEL ), the magnitude of the diameter changes needed to regulateflow in our planar model AA will exceed changes required by acylindrical model, owing to the two frictionless sides of the channel.However, because the viscosity of blood in an AA in vivo is notconstant, in that it tends to decrease with decreasing diameter( 10 ), the inverse fourth-power relationship between diameter and segmental resistance is only approximate in thephysiological setting. Indeed, the bias introduced by the channelconfiguration may counterbalance the bias arising from our assumptionof spatially homogeneous viscosity./ x9 i$ U7 {2 X% L6 |* l- C
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Another potential limitation of our study is that we have notexplicitly represented basal NO release, but, rather, we have represented a local increment of basal release. Because this increment is a small perturbation of the system, it should elicit a first-order accurate response that closely approximates a true response. This assumption avoids the complex issue of the interaction of incremental release with basal release, the inclusion of which would involve aformidable additional technical challenge. The parameters for our modelAA (including pressure and diameter) and the parameters for themyogenic submodels were based on physiological measurements performedin intact AA under experimental conditions wherein normal basal NOrelease presumably occurred. For an explicit representation of basalrelease, a spatially distributed model for that release, includingrelease stimulated by shear stress, would be required. This model forbasal release would have to be integrated into the existing model bymodifying model parameters so that basal release interacting withmyogenic elements would yield a configuration corresponding to anintact AA. Moreover, additional model development would be required toenable independent tracking of both basal and incremental NO releaseand distribution and to distinguish their respective effects.
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Why Model NO Advection Is Significant and NO Scavenging Is Not" X! I# ]! L: }- _) N
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Our simulations predict that NO advection can influence downstreamvascular tone and that this influence is relatively insensitive to therate of NO-Hb binding in the lumen, at least in the neighborhood of ourbase-case parameter values. This prediction arises from the hemodynamicstate of a typical renal AA and the relationship between advectivetransport and NO scavenging.
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A comparison of the NO binding rate coefficient with blood transit timethrough the arteriole helps to explain why NO scavenging may have aminimal impact. AA length is variable, with most AA lengths rangingfrom ~100 to 400 µm; the mean value in rats, including nonjuxtamedullary AA, is ~140 µm ( 6 ). AA luminaldiameter is 10-20 µm ( 8 ), and single nephron AAblood flow is 200-400 nl/min ( 33, 34 ). With thesevalues, the characteristic transit time is on the order of 5 ms (basedon length, 140 µm; diameter, 15 µm; flow, 300 nl/min), with alikely range of 3-30 ms. Thus the average transit time is 25% ofthe half-life of luminal NO with our base-case removal coefficient,which is 20 ms. Hence, although the NO half-life is short, the lifetimeof NO is long compared with transit time through an AA. In othervascular beds, with substantially lower flow velocities or longerlengths, the contribution of advective NO transport may be of lesser importance.
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NO Scavenging, Release, and VSM Response
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The accuracy of the foregoing analysis depends critically on ourbase-case rate of NO-Hb binding: the results in Fig. 12 show that a 10- or 100-fold increase in binding rate can significantly diminish axialadvection of free NO. There is sufficient Hb in blood (2.3 mM), and thecoefficient of NO binding by Hb is sufficiently large, to result in ahalf-life of ~2 µs, if NO had free access to Hb ( 39 ).In that case, NO would be an ineffective paracrine agent, becauselittle NO would reach VSM. However, NO does reach VSM, and a number ofinvestigators have sought to determine why NO is not rapidly scavengedby Hb. Three important factors have been identified by means ofphysiological experiments and mathematical models of radial NOdiffusion. First, the RBC-free boundary layer at the vessel wallappears to present a significant barrier to NO diffusion into the lumen( 21 ); our simulations verify this behavior when the fluidis loaded with NO released from upstream. Second, the boundary layer ofrelatively unstirred fluid surrounding each RBC can impede NO diffusion( 23 ). Finally, Vaughn et al. ( 38 ) haverecently shown that RBC have an intrinsic barrier to NO that greatlyimpedes NO-Hb binding; although the mechanisms are unclear, theexperimental data suggest that the rate coefficient for free NO bindingto Hb within RBC is some 2,000-fold lower than for free Hb. Thus ourbase-case rate coefficient for NO-Hb binding is likelysupraphysiological. On the other hand, we have not represented a secondmode of NO scavenging, the formation of SNO adducts. However, thisprocess is slower than Hb binding and, like Hb binding, involves someNO entry into RBC ( 15, 32 ). Hence, in the kidney, most SNOgeneration from NO released in the AA likely occurs in downstream segments.. J9 I0 z* U% p4 ?; `

In addition to the questions surrounding NO scavenging in the lumen,nearly every parameter concerning NO has uncertainty associated withit. For example, the rate of NO release can vary widely because it issubject to short- and long-term regulation ( 42 ).Furthermore, expression of iNOS by EC can substantially enhance thelocal release of NO ( 27 ). Similarly, the sensitivity oftissues to NO is uncertain. Responses to NO obtained from in vitrostudies are typically expressed in terms of bath concentrations ratherthan local concentrations within tissue. In vivo, uncertainty arisesbecause measurements of NO concentrations in blood or tissue, either insamples or in situ, may represent averages that differ substantiallyfrom local concentrations near NO sources or sinks. Thus NOdose-response curves are somewhat variable in relation to the range ofrelevant concentrations. In our model, we have used incremental releaserates and dose-response curves that result in local concentrations of1-100 nM and responses within this range ( 24 ). Otherdata suggest that the biologically relevant concentration range is anorder of magnitude higher ( 12 ). However, because the NOconcentrations scale linearly with the release rate magnitude, ourresults do not depend critically on our choice of NO release rate andVSM sensitivity; if the NO dose-response curve is adjusted accordingly,the patterns of NO distribution and transport will remain unchanged.
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Physiological Implications

Our simulation studies provide insight into the physiology of NOwithin the renal microvasculature. Because the transit time of bloodthrough an AA is much less than the half-life of NO, Hb binding andperoxinitrite formation will occur downstream in the glomerular andperitubular capillaries where fluid velocities are lower and transittimes longer. Furthermore, the NO concentration of the glomerularfiltrate is likely to be similar to that of blood leaving the AA.Because NO is known to influence a variety of transport processes( 43 ), advective NO transport may link arteriolarhemodynamics and proximal tubular transport. In addition, efferentarteriolar tone may be influenced by NO loaded into blood passingthrough the AA.
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The renal vasculature is more sensitive to NO than other vascular beds( 43 ), and our simulations suggest that tissue and bloodconcentrations of NO will be highest at the end of the AA. Thisassertion is based on the results shown in Fig. 13, in which the fateof incremental NO release distributed along the vessel can be assumedto approximate the fate of NO basal release. In the boundary layer, NOconcentration increased along the vessel owing to both diffusion fromthe wall and advection from upstream. At the distal end of the vessel,NO diffusion into the abluminal region was enhanced because of the highNO concentration in the boundary layer. The distal AA is a site ofimportant processes that are highly sensitive to NO, including reninsecretion and TGF ( 13, 36 ). Because NO is also produced inmacula densa and thick ascending limb cells ( 31, 44 ), theconvergence of
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3. Beckman, JS. The physiological and pathological chemistry of nitric oxide.In: Nitric Oxide: Principles and Actions, edited by Lancaster J.. San Diego, CA: Academic, 1996, p. 1-82.- G2 |5 C0 D) z6 g. h% R

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5. Butler, AR,Megson IL,andWright PG. Diffusion of nitric oxide and scavenging by blood in the vasculature. Biochem Biophys Acta 1425:168-176,1998 .) S' |- X0 [0 c9 [! c0 r

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