Department of Chemical and Biological Engineering, Tufts University, Medford, Massachusetts 02155
Submitted 1 May 2003 ; accepted in final form 20 June 2003
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ABSTRACT |
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kidney; vasa recta; aquaporin-1 water channels; mathematical model; transport
In the last decade, two families of urea transporters have been identified in the renal medulla, UTA and UTB. UTA isoforms are present in the collecting duct and the descending limb of Henle's loop, whereas UTB is found in DVR and erythrocytes (17, 24, 32, 34). The erythrocyte urea transporter is most likely a complex channel with modifier sites on the extracellular surface, rather than a carrier (28).
Experimental studies with UTB knockout mice suggest that the urea transporter plays a significant role in the urinary concentrating mechanism. Urinary osmolality was measured to be 25% lower in UTB-deficient mice than in wild-type mice (35); UTB was found to contribute significantly to the capacity of the kidney to concentrate urine and even more greatly to its ability to concentrate urea itself (35). UTB allows red blood cells and vasa recta walls to rapidly exchange urea, thereby preventing it from being carried away from the medulla by the microcirculation and enhancing the corticomedullary osmolality gradients (17, 28, 29). Experimental observations suggest that some of the urea delivered to the tip of the papilla by UTAs and carried up by the blood through AVR is not recycled by DVR lacking UTB but is returned to the general circulation (35). In addition, the urea transporter is thought to play an important role in preventing large volume changes in red blood cells during their transit in the medullary microcirculation, a hypothesis first developed by Macey and Yousef (12).
Yang and Verkman (36) reported a significant osmotic water permeability in Xenopus laevis oocytes expressing UT3 (a UTB isoform), suggesting the existence of a continuous aqueous channel through the UT3 protein for both water and urea transport. Sidoux-Walter et al. (29) suggested that at physiological expression levels in erythrocytes, UTB does not transport water. Whether UTB significantly contributes to transmural fluxes of water in DVR in vivo remains unknown.
Questions also remain regarding possible differences between outer medullary (OM) and inner medullary (IM) urea transporters. Although antibodies to UTB label the continuous endothelium of rat DVR (32, 34), in vivo DVR permeability measurements suggest that inner medullary DVR may lack functional urea transporters. Whereas the permeability of OMDVR to urea (Pu) is five times higher than that to sodium (PNa) and is inhibited by addition of thiourea, phloretin, and p-chloromercuribenzenesulfonate, in IMDVR Pu and PNa are closely correlated as a straight line with a slope of 1 originating from the origin and are unaffected by thiourea or phloretin (17, 24). It is possible, albeit unlikely, that unstirred layers in the renal interstitium might have obscured the Pu and PNa of IMDVR in vivo.
The objective of this study was to use a mathematical model of transport in the medullary microcirculation to gain some insight into the specific function of UTB in descending vasa recta walls and red blood cells (RBCs).
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MODEL AND NUMERICAL METHODS |
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Greek Symbols
Subscripts and Superscripts
General Description of Model
The renal medullary microcirculation is a countercurrent exchange system with blood flowing down along DVR from the corticomedullary junction to the papillary tip and looping back to the cortical veins along AVR. During transit, water and solutes in blood can diffuse radially between vasa recta and the medullary interstitium. This countercurrent flow configuration helps to maintain corticomedullary osmolality gradients, which are essential for the formation of concentrated urine.
Our model, which has been described and applied earlier (2, 4, 39, 40), consists of steady-state conservation equations for water and solutes in vasa recta and the interstitium, coupled with expressions for fluxes across vasa recta walls and RBC membranes through paracellular and transcellular pathways. We only consider those vasa recta that are destined for the IM, i.e., those that lie in the center of the vascular bundles and do not perfuse the capillary plexus in the OM.
DVR and AVR exchange water, sodium chloride, urea, and proteins via several routes. Vasa recta walls are perforated by nonselective paracellular pathways, across which fluid transport is driven by Starling forces (i.e., transmembrane hydraulic and oncotic pressure differences). In addition, two transcellular pathways have been identified in DVR endothelium and RBC membranes: aquaporin-1 (AQP1) water channels, which allow for water movement to the exclusion of all solutes, and UTB urea transporters.
One of the functions of the medullary microcirculation is to carry away water and solutes (such as sodium chloride and urea) reabsorbed from the loops of Henle and collecting ducts. In this model, reabsorption into the interstitium is accounted for by interstitial generation rates that undergo spatial variation (2). In the OM, we assume that exchanges occur only between vasa recta and the interstitium because DVR and AVR form vascular bundles from which nephron loops are excluded, so that generation rates are taken to be zero.
Conservation Equations
If x is the axial coordinate along the corticomedullary axis,
conservation of volume in plasma and RBCs can be expressed as
![]() | (1) |
![]() | (2) |
Because the RBC membrane is impermeable to sodium chloride, proteins, and
hemoglobin, conservation of sodium chloride and proteins in plasma and
hemoglobin and other nonurea solutes in RBCs, yields, respectively
![]() | (3) |
![]() | (4) |
![]() | (5) |
![]() | (6) |
Order-of-magnitude analysis suggests that axial diffusion in the
interstitium is negligible relative to radial transport
(39). If reabsorption from the
loops of Henle and collecting ducts is accounted for by interstitial
generation rates, conservation of volume, sodium chloride, urea, and proteins
in the interstitium can be written
(2) as
![]() | (7) |
![]() | (8) |
Reabsorption of Water and Solutes from Tubular System
Water and sodium chloride are reabsorbed from the loops of Henle and the
collecting ducts into the interstitium; urea is reabsorbed mostly from
terminal IM collecting ducts through UTA urea transporters
(28). The reabsorbed water and
solutes must then be removed by the medullary microcirculation to avoid
accumulation in the medulla. The fractions of filtered volume, sodium, and
urea that are recovered by vasa recta from the IM interstitium are denoted by
fv, fNa, and fu, respectively. Baseline
values for fv, fNa, and fu are taken as 1, 1,
and 40%, respectively (2). If
is the systemic
concentration of solute i and GFR is the glomerular filtration rate,
the filtered load of solute i is
. Hence,
the overall amount of water or solute recovered by vasa recta (VRR) may be
expressed (2) as
![]() | (9) |
Koepsell et al. (7) observed
that sodium concentration increases exponentially along the corticomedullary
axis. Because the fraction of osmolality due to urea increases from 2 to
50% between the corticomedullary junction and the papillary tip
(19), urea concentration must
increase even more rapidly. To yield a profile consistent with those
observations, the baseline expressions for the interstitial area-weighed
generation rates are taken (2)
as
![]() | (10) |
![]() | (11) |
![]() | (12) |
Flux Equations
As described earlier, there are three different pathways for water
transport across DVR endothelia. Hence, in Eq.1
is the sum of three
contributions, the fluxes through paracellular pathways, AQP1 water channels,
and UTB urea transporters, respectively
![]() | (13) |
![]() | (14) |
![]() | (15) |
Equations 1315 indicate that the driving force for water movement differs for paracellular and transcellular pathways. Indeed, the reflection coefficient of paracellular pathways to small hydrophilic solutes is negligible (i.e., there is no sieving of urea and sodium chloride across these pathways), whereas that of AQP1 water channels and UTB urea transporters is >0. Transmural sodium and urea concentration differences contribute significantly to water flux across AQP1 and UTB, given the large value of RT (19.3 mmHg/mM). Because small solutes are more concentrated in the interstitium than in DVR plasma, there is water efflux from DVR across AQP1 and UTB; across paracellular pathways, however, sodium and urea exert no effect, and Starling forces favor volume reabsorption into vasa recta (essentially because proteins are more concentrated in plasma than in interstitium). Previous simulations in which water fluxes across UTB were neglected suggested that there is a net efflux of water across DVR walls throughout the medulla. We found that a net amount of 1.7 x 104 cm3/s is transported from interstitium to lumen across paracellular pathways, but twice that amount (i.e., 3.4 x 104 cm3/s) is carried from lumen to interstitium across AQP1 water channels, so that the overall amount of water exiting DVR is 1.7 x 104 cm3/s (39).
The volume flux across the RBC membrane,
, is the sum of three terms,
corresponding to AQP1 water channels, the lipid membrane, and UTB urea
transporters, respectively
![]() | (16) |
![]() | (17) |
![]() | (18) |
The paracellular flux of solute i (i = sodium, protein,
urea) across vasa recta walls can be written as
![]() | (19) |
![]() | (20) |
Because UTB serves as a common channel for water and urea, the UTB-mediated
transmural and transmembrane fluxes of urea are given by, respectively
![]() | (21) |
![]() | (22) |
![]() | (23) |
Expressions for the cell-to-wall surface area ratio, the number of vasa recta, the cross-sectional area of the interstitium, and the relationship between protein concentration and oncotic pressure are summarized in the APPENDIX. Transport and morphological parameters, as well as initial values, are given in Table 1. Parameters specifically related to the transport of urea and water across UTB, also shown in Table 1, are further described below.
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UTB Transport Parameters
UTB permeability to urea. Yang et al.
(35) reported that, at
10°C, the urea permeability of the RBC membrane was 45-fold lower in UTB
knockout mice than in wild-type mice. In the absence of data at higher
temperature, we assumed that the urea permeability of UTB transporters,
(UTB), and that of the lipid
membrane,
(LM), are equal to
44/45 and 1/45 of the overall permeability of RBC to urea, respectively, taken
as 160 x 105 cm/s
(1). Hence,
(UTB) = 156 x
105 cm/s and
(LM) = 3 x
105 cm/s in our simulations. The former estimate
is close to the measured urea permeability of the human RBC urea transporter
hUT-B1 (i.e., HUT11), 1.2 x 103 cm/s
(14). In DVR, the urea
permeability of UTB transporters,
(UTB), was taken as 285
x 105 cm/s, as reported by Pallone et al.
(24).
UTB permeability to water. Yang and Verkman
(37) estimated that the water
permeability of UTB transporters in RBCs,
(UTB), is
0.145 x
102 cm/s at 10°C; because their results
suggest a weak dependence of that permeability on temperature, we assumed the
same value at 37°C.
Direct measurements of the water permeability of UTB transporters in DVR
walls, (UTB), have not been
reported until now. Yang and Verkman
(37) found that, in RBCs, the
single-channel permeability of UTB to water is similar to that of AQP1 water
channels. We assumed that the single-channel permeability of UTB to water, as
well as that to urea, is identical in RBC membranes and DVR endothelia. In
addition, we assumed that the overall water and urea permeabilities in RBC and
DVR are the products of their single-channel value and the channel density,
respectively. Comparing the overall permeabilities of DVR and RBC to water and
urea, we obtain
![]() | (24) |
The water permeability and hydraulic conductivity of UTB in DVR endothelia
[(UTB) and
, respectively] can be
related (36) knowing that
![]() | (25) |
![]() | (26) |
RBC lipid membrane permeability. Yang and Verkman
(37) estimated that AQP1 water
channels and the lipid membrane account for 90 and 2% of all the water that
passes through the RBC membrane, respectively, at 10°C; at 37°C, they
account for 79 and 15% of the water exchanged, respectively. Hence, we assumed
that the lipid membrane-to-AQP1 water channel hydraulic conductivity ratio,
,
is equal to 15/79 at 37°C. Because the combined hydraulic permeability of
AQP1 and the lipid membrane has been reported as 22.8 x
103 cm/s at 37°C
(13), and it may be converted
to the hydraulic conductivity as 2.1 x 108
cm · s1 ·
mmHg1 from Eq. 26,
and
were estimated as 0.34
x 108 and 1.8 x
108 cm · s1
· mmHg1, respectively, at 37°C. The
validity of these assumptions was confirmed by using the same approach at
10°C. At that temperature, if the ratio
is taken as 2/90,
and
are calculated as 0.46
x 109 and 2.05 x
108 cm · s1
· mmHg1, respectively. The former value is
in excellent agreement with the measured water permeability of RBC in AQP1/UTB
null erythrocytes at 10°C, reported as 4.5 x
104 cm/s
(37), that is, 0.42 x
109 cm · s1
· mmHg1 in units of hydraulic
conductivity.
Numerical Methods
Ordinary differential equations (ODEs) described by Eqs. 16
were solved along DVR and AVR to obtain the plasma and RBC volume flow rates
(QP and QR) and solute concentrations
( and
), with initial values
(i.e., in DVR at the corticomedullary junction) for all variables as specified
in Table 1. Solving Eqs.
16 requires determination of the fluxes (as expressed in Eqs.
1319), which are themselves a function of solute concentrations in
DVR, AVR, and the interstitium. Hence, the conservation equations cannot be
simply solved along DVR first and AVR afterward. Thus we first assumed values
for all variables throughout AVR, integrated Eqs. 16 along DVR
and then looped back along AVR, where "guess" values were replaced
with new calculated values as integration proceeded. This integration process
was iterated until the values for all variables along AVR converged. At every
step, hydraulic pressure and solute concentrations in the interstitium were
obtained by solving interstitial conservation equations (Eqs. 7 and
8).
The ordinary differential equations were integrated along vasa recta using Gear's method, which is a self-adaptive, multistep, predictor-corrector method for stiff ODEs. At each value of x, the system of four nonlinear algebraic equations (Eqs. 7 and 8) was solved using a modified Powell hybrid method, as described more fully in our previous work (38).
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RESULTS |
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Changes in the expression of UTB urea transporters and AQP1 water channels in vasa recta will likely affect the entire urinary concentrating mechanism, including the reabsorption of water and solutes into the interstitium. Nevertheless, given the absence of specific experimental data and the constraint that our model does not explicitly take into consideration tubular transport, the amount of filtered load that is reabsorbed into vasa recta was assumed to remain the same with and without UTB in the simulations described below.
In our baseline case, both AQP1 water channels and UTB urea transporters are expressed in DVR walls and RBC membranes. Illustrated in Fig. 1a are the baseline urea concentrations in RBCs, plasma, and interstitium near the papillary tip. As blood flows along DVR, it encounters regions of increasing osmolality. Urea thus diffuses from the interstitium into DVR (there is also a smaller contribution from convection, as the model predicts that water is driven into DVR across paracellular pathways by transmural oncotic gradients) (38, 39), and from DVR into RBCs, raising concentrations in both compartments, albeit with a lag because permeabilities are not infinite. As blood ascends back to the corticomedullary junction, interstitial urea concentrations decrease, so that urea diffuses in the opposite direction, from RBCs to AVR lumen, then to interstitium.
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Maximal Urea Transport Capacities
The maximum permeability of the RBC membrane to urea, in the limit of zero solute concentration, has been reported as 1.6 x 103 cm/s (14). In addition, measurements of the maximum urea flux through UTB in the RBC membrane range from 0.8 to 2.5 x 107 mol · cm2 · s1, as reviewed by Sands et al. (28). Using Eq. 24, we estimated that the maximum urea flux through UTB in the DVR wall is between 1.4 and 4.4 x 107 mol · cm2 · s1. In our baseline results, the urea fluxes through UTB in RBC membranes and DVR walls are <0.5 x 108 and <0.2 x 108 mol · cm2 · s1, respectively, that is, at least 10 times lower than saturated fluxes. Hence, urea transport through UTB does not appear to reach its maximal capacities under the conditions we considered.
Asymmetry of Urea Transport by UTB Channels
Several investigators have reported that the net efflux of urea across
erythrocytes at a given concentration difference is greater than the net
influx for an equal, but oppositely directed, gradient
(14,
28). While there could be
asymmetry in the RBC membrane, transport asymmetry across UTB in the DVR wall
is unlikely, because urea moves through both the apical and abluminal
membranes of endothelial cells, and the two sides are mirror images of each
other with respect to the urea diffusion pathway. In vitro measurements of the
permeability of DVR to urea based on diffusion from bath to lumen and lumen to
bath were indeed found to be similar
(17). We therefore simulated
transport asymmetry only across the RBC membrane. In the absence of specific
measurements of UTB permeability to urea according to the direction of
transport, we investigated the effect of higher urea efflux permeabilities by
either increasing (UTB)
10-fold during urea efflux [with
(UTB) remaining equal to its
baseline value during urea influx] or decreasing
(UTB) 10-fold during urea
influx [with
(UTB) equal to
its baseline value during urea efflux]. Although there was a slight decrease
in the contribution of urea to osmolality at the papillary tip in the latter
case (from 52 to 49%), the effects of such variations in
(UTB) were otherwise
negligible.
If the direction of asymmetry is reversed, that is, if influx
(UTB) is increased 10-fold,
or if efflux
(UTB) is
decreased 10-fold, the effects on medullary transport remain negligible.
Because there seems to be no significant effect of UTB transport asymmetry on
transport across vasa recta and erythrocytes, and given the lack of specific
experimental data, we did not take this asymmetry into account in the
remainder of our simulations.
Function of UTB as an Urea Transporter
To examine the effect of UTB on urea transport in the renal medullary microcirculation, we first investigated the effects on osmolality of eliminating the urea transporter from DVR endothelia, RBC membranes, and both. As stated above, we assumed that the interstitial generation rates of water, sodium, and urea remained unaffected by deletion of UTB or AQP1. Shown in Table 2 are plasma osmolality and u% (that is, the fraction of osmolality attributable to urea) at the papillary tip, with and without UTB. Our results suggest overall that by making transport barriers more permeable, urea transporters significantly increase urea concentrations throughout the medulla. However, the UTB-mediated decrease in transmembrane urea concentration differences is also predicted to reduce water efflux from AQP1 water channels, thereby increasing blood flow, decreasing plasma sodium concentrations, and possibly reducing osmolality.
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Eliminating UTB from DVR walls. Our simulations indicate that if urea transporters are eliminated from DVR walls, urea can only enter DVR through the more restrictive (i.e., less urea-permeable) paracellular pathway, urea influx is significantly reduced, and plasma urea concentrations decrease. Because the interstitial-to-plasma urea concentration gradient is significantly greater, water efflux from DVR to the interstitium through AQP1 water channels increases, thereby reducing blood flow and raising plasma concentrations. If reabsorption from the loops of Henle and the collecting duct remains unaffected, the net result is that the plasma concentration of sodium chloride increases whereas that of urea decreases, so that the contribution of urea to osmolality at the papillary tip is significantly reduced (from 51 to 38% if fu is 40%).
The variation in papillary tip osmolality itself depends on which of these two competing effects dominates, the reduction in urea influx into DVR or the increase in water efflux, as illustrated in Table 2. If fu remains equal to 40 or 60%, osmolality increases with the removal of UTB in DVR walls, because urea concentrations are high overall and the increase in transmural concentration gradients has a significant effect on water efflux and thus on plasma sodium concentration. If fu is taken as 20% both before and after deletion of the transporter, the reduction in urea influx into DVR more than compensates for the increase in vasa recta sodium chloride concentration, and osmolality at the papillary tip slightly decreases.
Eliminating UTB from RBC membranes. Our model indicates that if UTB is selectively removed from erythrocytes, RBC urea concentrations are significantly reduced given the lower membrane permeability. Due to diffusional (radial) equilibration between all compartments, plasma and interstitial concentrations decrease as well, as illustrated in Fig. 1b. However, there is a competing effect. Because DVR-to-RBC urea concentration gradients increase significantly, there is initially more water efflux from RBCs to DVR (Fig. 2A), which in turn translates into a higher efflux from DVR to interstitium to preserve the water balance. The net result is a more significant volume efflux from DVR, as shown in Fig. 2B, which raises plasma concentrations. Overall, the increase in the plasma concentration of sodium chloride is greater than (or equal to, if fu = 20%) the decrease in that of urea, so that osmolality at the papillary tip increases (or remains unchanged, if fu = 20%,), whereas the contribution of urea is significantly reduced, as shown in Table 2. With fu = 40%, papillary tip osmolality rises slightly from 1,077 to 1,106 mosmol/kgH2O when UTB is removed from RBC membranes, whereas u% decreases from 51 to 42%.
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Even though the value of
(LM) remains uncertain,
sensitivity analysis suggests that this parameter has a very small effect on
results.
(LM) was measured
as 1/45 of the overall urea permeability of the RBC membrane at 10°C
(35), but there are no
reported measurements at 37°C. Because the permeability of the RBC lipid
membrane to water increases about sevenfold from 10 to 37°C
(35), it is possible that the
permeability to urea increases by a similar factor. We found that in the
presence of UTB, multiplying the baseline lipid membrane-to-overall urea
permeability ratio by 10 does not affect papillary tip osmolality or the
fraction due to urea. If UTB is eliminated from RBCs, variations in
(LM) have a small effect on
osmolality. As described above, removing UTB in erythrocytes raises the plasma
concentration of sodium chloride and lowers that of urea. However, the larger
the
(LM), the smaller these
effects because the contribution of the RBC lipid membrane to urea transport
is comparatively larger. Hence, if the lipid membrane-to-overall urea
permeability ratio is 10 times the baseline value, osmolality at the papillary
tip is 1,082 mosmol/kgH2O (with u% = 50%) without UTB in RBCs, and
1,077 mosmol/kgH2O (with u% = 51%) with UTB.
Eliminating UTB from both RBC membranes and DVR walls. Not surprisingly, when UTB is removed from both DVR walls and RBC membranes, the combined effects are predicted to lead to a further decrease in u% at the papillary tip. Whether osmolality increases or decreases depends on the urea reabsorption ratio, as described above. With our baseline value (fu = 40% before and after deletion), eliminating UTB urea transporters leads to a 4% increase in papillary tip osmolality, whereas the fraction that is due to urea decreases more significantly, from 51 to 33%.
Our results suggest that UTB transporters greatly increase the diffusive
exchange of urea across vasa recta and erythrocytes and therefore raise
medullary urea concentrations in RBC, plasma, and interstitium, assuming that
reabsorption from the loops of Henle and the collecting ducts remains
unaffected; because the high permeability that UTB confers to membranes lowers
transmural urea concentration gradients, the transporter also reduces water
efflux from DVR through AQP1 water channels, thereby lowering medullary sodium
concentrations. If fu is at least equal to 40%, the increase in
imparted by UTB is
predicted to be smaller than the decrease in
, so that
osmolality at the papillary tip decreases overall.
Function of UTB Without AQP1 Water Channels
As our previous simulations indicated, AQP1 water channels in DVR favor the shunting of water from DVR to AVR in the OM, therefore reducing blood flow rate to the deep medulla and increasing osmolality (18). More specifically, without AQP1 in DVR walls there appears to be no water efflux from DVR into the interstitium (except near the corticomedullary junction) because the balance of forces (i.e., oncotic vs. hydraulic pressure differences) favors water influx across the paracellular pathway throughout most of the medulla (38). Plasma concentrations thus remain lower. Eliminating AQP1 water channels from RBC membranes also decreases osmolality, because reducing water efflux from RBCs to DVR lowers solute concentrations in RBCs, and therefore in plasma and interstitium as well. Indeed, given the countercurrent arrangement of vasa recta and the diffusive transport of urea, the concentration of urea in the interstitium (almost) always lies between that in AVR plasma and that in DVR plasma, whereas plasma urea concentrations are themselves bounded by RBC urea concentrations (see Fig. 1).
Yang and Verkman (37)
generated mice lacking both UTB and AQP1 and reported that the single-channel
water permeability of the urea transporter is similar to that of AQP1,
suggesting that UTB could play a role in facilitated water transport. Our
simulations also make it possible to selectively eliminate AQP1 water channels
and/or UTB urea transporters from DVR walls, RBC membranes, or both. Results
are summarized in Table 3. In
the absence of AQP1 water channels, small-solute concentration differences
have no direct effect on volume fluxes in this model, so that there is no
UTB-mediated reduction in water efflux, and therefore no decrease in sodium
concentration. On the assumption that interstitial generation rates remain
unaffected, our model predicts that without AQP1, UTB transporters increase
medullary urea concentrations in vasa recta and interstitium by enhancing the
diffusive (radial) transfer of urea. As a result, both the osmolality at the
papillary tip and the fraction due to urea increase. In the complete absence
of water channels, the expression of UTB results in a 35% increase in
papillary tip osmolality, and u% increases from 30 to >50%, assuming
that fu = 40% (Table
3).
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Function of UTB as a Water Channel
Verkman and colleagues (36,
37) have shown that UTB urea
transporters also function as water channels. Whether the contribution of UTB
to transmural water fluxes is significant under physiological conditions has
been a matter of debate among investigators
(29,
36,
37). Our simulations indicate
that the water flux across UTB in DVR walls and in RBC membranes is negligible
relative not only to the transcellular flux across AQP1 water channels but
also compared with the water flux across the paracellular pathway in DVR walls
(Figs. 2A and
3). This is not surprising
given that the hydraulic conductivity of UTB in DVR endothelia is estimated as
2.46 x 109 cm ·
s1 · mmHg1,
as described above, whereas that of the paracellular pathway has been measured
as 1.8 x 106 cm ·
s1 · mmHg1
(23). Across DVR walls, from
the corticomedullary junction to the papillary tip, the total amount of water
transported through UTB is calculated to be only 2 and 3% of that transported
through AQP1 and the paracellular pathway, respectively. As a result,
eliminating the water transport property of UTB urea transporters in DVR walls
(i.e., setting to 0 in
Eq. 15) has a negligible effect on papillary tip osmolality, even in
the absence of water channels.
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Our model predicts that across the RBC membrane, the fraction of water transported through UTB transporters, the lipid membrane, and AQP1 water channels is 6, 15, and 79%, respectively, in close agreement with the values calculated by Yang and Verkman (37). Thus eliminating the water transport property of UTB in RBC membranes also has an insignificant effect on water flux through the RBC membrane in the presence of AQP1 water channels. Without AQP1, our simulations suggest that nearly 30% of the water carried across erythrocytes goes through UTB transporters and 70% through the lipid membrane, so that setting the water permeability of UTB to zero significantly decreases water fluxes across RBCs. However, the effect on papillary tip osmolality is small, as the latter decreases by <1%, and the fraction due to urea increases slightly. Hence, our results indicate that eliminating water transport through UTB in erythrocytes has a negligible effect on osmolality, with or without AQP1 water channels.
The simulations above were based on a reflection coefficient of UTB to urea
(u) equal to 0.3, as estimated by Yang and Verkman
(36). A zero reflection
coefficient would indicate that urea has no effect on water flux across UTB
(see Eq. 15), whereas
u = 1 would mean that UTB is
impermeable to urea. Because this parameter is uncertain, we varied it between
0 and 0.6.
If u increases from 0.3 to 0.6, the net amount of water
transported across UTB from RBCs into the lumen throughout DVR increases from
5.2 x 106 to 5.5 x
106 cm3/s, and that transported from
the lumen into the interstitium throughout DVR increases from 5.5 x
106 to 7.1 x
106 cm3/s; the latter figure
represents only 2 and 4% of the net amount of water transported across AQP1
and the paracellular pathway throughout DVR, respectively. It is not
surprising that changes in
u have small effects because the
high permeability to urea imparted by UTB results in small transmural osmotic
pressure gradients due to urea; most of the driving force for water transport
across UTB stems from other solute concentration differences. Thus if
u is either increased twofold or set to zero, variations in
the amount of water transported through UTB in DVR walls have little effect:
the osmolality at the papillary tip and u% remain the same (1,078
mosmol/kgH2O and 51%, respectively, with fu = 40%).
We assumed in our baseline case that the reflection coefficient of UTB to nonurea small solutes (i.e., sodium chloride and other RBC nonurea solutes) is equal to 1; that is, the transporter is impermeable to these other solutes. It is possible, however, that UTB constitutes a shared pathway for water, urea, and other solutes. If the reflection coefficient of UTB to nonurea small solutes is taken as 0.3 (that is, equal to that to urea), our simulations indicate that the net amount of water transported across UTB from lumen to interstitium throughout DVR decreases significantly, from 5.5 x 106 to 2.5 x 106 cm3/s. The direction of water movement through UTB is even reversed across the RBC membrane, because small-solute concentration differences then play a lesser role and oncotic pressure differences constitute the main driving force; whereas the net amount of water transported across erythrocyte UTB throughout DVR is calculated to be 5.2 x 106 cm3/s from RBC into the lumen in the baseline case, it is 2.8 x 105 cm3/s from the lumen into RBC if the reflection coefficient of UTB to nonurea solutes is taken as 0.3. The overall amount of water transported across RBC membranes is predicted to drop by only 1% nevertheless, as nonurea solutes in RBCs exert a smaller osmotic pressure and more water is then carried out across AQP1 as a result; osmolality at the papillary tip and the contribution of urea are found to remain the same as in the baseline case. It should be noted that varying the reflection coefficient of the transporter to nonurea solutes has a greater effect on water fluxes than varying that to urea because DVR walls and RBC membranes are much less permeable to these other solutes, so that corresponding transmural concentration differences are significantly larger and more important as a driving force.
UTB and the RBC Osmotic Balance
Several investigators have proposed that erythrocyte UTB urea transporters play an important role in balancing the osmotic pressure on each side of the barrier (i.e., RBC and plasma), thereby limiting the extent to which RBCs shrink along DVR and swell along AVR (11, 12).
To examine this assumption, we predicted relative RBC flow rate variations along the corticomedullary axis in a single DVR and AVR, with and without the presence of UTB in RBC membranes. Our results suggest that UTB indeed reduces the magnitude of RBC volume changes along vasa recta, as illustrated in Fig. 4. In the presence of AQP1, as blood flows from the corticomedullary junction to the papillary tip along DVR, RBC volume decreases to 63 and 55% of its initial value with and without UTB, respectively. It then increases back to 100 and 104% of its initial value, respectively, as blood returns to the cortex along AVR. The effects of UTB on RBC volume are slightly smaller in the absence of AQP1 in the RBC membrane, because water efflux across erythrocytes is reduced without the high osmotic water permeability imparted by AQP1 water channels. In this case, we found that along DVR, RBC volume decreases to 73 and 65% of its initial value with and without UTB, respectively; it then returns to 100 and 102% of its initial value, respectively, as blood flows back to the cortex.
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Urea vs. Sodium Chloride DVR Permeability
Pallone et al. (22) suggested that a tradeoff may have evolved in the medullary microcirculation. The authors noted that whereas the urea gradient across the DVR wall is probably small due to the high permeability imparted by expression of the UTB transporter, the permeability of at least some DVR to sodium chloride is low. Relatively low DVR sodium chloride permeability would favor the bypassing of water from DVR to AVR via AQP1, the purpose of which may be to lower blood flow rate toward the papillary tip. A reduced blood flow rate to the deepest portions of the medulla is expected to enhance the efficiency of microvascular exchange in the IM by reducing solute washout.
Our model indicates that both sodium chloride and urea contribute significantly to water efflux across DVR. As illustrated in Fig. 5, interstitial-to-DVR concentration gradients are higher for sodium chloride than for urea in parts of the IM but lower in the OM. Indeed, because the initial concentration of urea is 50-fold lower than that of sodium, because the fraction of filtered urea that is reabsorbed by vasa recta is 40 vs. 1% for sodium in our baseline case, and given that the interstitial area-weighted generation rate in the IM is assumed to increase exponentially for urea but only linearly for sodium (2), the concentration of urea increases much faster than that of sodium in the OM and transmural gradients are correspondingly higher. In the IM, the high DVR permeability to urea plays a more dominant role and reduces interstitial-to-DVR urea concentration gradients.
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Thus the contribution of sodium to water efflux across AQP1 water channels appears to be more significant than that of urea in the IM, but the opposite is true in the OM. Decreasing the urea reabsorption ratio to 20% reduces transmural urea concentration differences (Fig. 5), but the latter still play a larger role in driving water out of DVR in the OM.
If the permeability of DVR to sodium were as high as that to urea, our simulations indicate that water efflux from DVR to the interstitium through AQP1 would be lowered, blood flow at the papillary tip would significantly increase as shown in Fig. 6, and plasma urea concentration would decrease. However, because a higher permeability to sodium increases transmural (i.e., radial) transport of sodium and thereby significantly raises sodium concentration, the osmolality at the papillary tip would increase from 1,077 to 1,134 mosmol/kgH2O if fu remained equal to 40% (and from 756 to 830 mosmol/kgH2O with fu = 20%). The contribution of urea to papillary tip osmolality would fall from 51 to 45% with fu equal to 40% (and from 34 to 29% with fu equal to 20%). Although it may first appear surprising that the presence of UTB (i.e., a large DVR permeability to urea) slightly lowers osmolality at the papillary tip if fu is at least 40% whereas increasing the DVR permeability to sodium has the opposite effect, the results here again depend on the fraction of filtered solute that is reabsorbed by vasa recta. If the fraction corresponding to sodium is decreased from 1 (i.e., our baseline value) to 0.5%, increasing DVR permeability to sodium as described above would have an insignificant effect on osmolality at the papillary tip. Our model shows that increasing the permeability of DVR to a given solute i gives rise to two competing effects: higher plasma concentration of solute i due to more efficient radial transport, and lower concentration of all other solutes due to reduced transmural gradients and thus less water efflux from DVR. Which effect dominates depends on the amount of solute reabsorbed into the microcirculation.
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In summary, our results confirm the hypothesis of Pallone et al.
(22) that the low permeability
to sodium chloride measured in some DVR may serve to enhance water transport
from DVR to AVR across AQP1, thereby lowering blood flow to the papillary tip.
Assuming that interstitial generation rates remained unchanged, we found that
if permeability of DVR to sodium were equal to that to urea, blood flow rate
at the papillary tip would indeed increase slightly (by 10% compared with
our baseline case). It is likely, however, that varying permeability of DVR to
sodium affects reabsorption from the loops of Henle and the collecting ducts,
so that the in vivo effects of changes in sodium permeability are difficult to
predict.
UTB Urea Transporters in OM vs. IM
As discussed by Pallone et al.
(22), in vivo DVR permeability
measurements suggest that IMDVR may lack a functional urea transporter. We
thus examined the effect of selectively removing UTB from IMDVR or OMDVR,
assuming here again that deletion of the transporter does not affect
reasborption from the loops of Henle and the collecting ducts. As shown in
Table 4, our model predicts
that papillary tip osmolality is lowest when UTB is only present in OMDVR and
highest when the transporter is only present in IMDVR. These intriguing
results stem from the competing effects of the urea transporter on sodium and
urea concentrations. As illustrated in Fig.
7A, in the OM,
is calculated to
be the same whether UTB is present throughout DVR or in OMDVR only, and it is
also the same (but lower) whether UTB is present in IMDVR only or not at all.
In the IM, however, the rate of increase in
is reduced when
UTB is present in OMDVR only, because the permeability of vasa recta to urea
is suddenly decreased. Conversely, the rate of increase in
is augmented in
the IM when UTB is present in IMDVR only. As a consequence, urea concentration
at the papillary tip is highest when UTB is present in IMDVR only and lowest
when it is only found in OMDVR, with intermediate values if the urea
transporter is either present throughout the medulla or entirely absent from
DVR walls (Fig. 7A).
Plasma sodium concentration, meanwhile, increases steadily as the urea
transporter is removed first from OMDVR and then from IMDVR as well
(Fig. 7B), due to the
progressively larger water efflux from DVR. The net result is that, relative
to the baseline case, osmolality at the papillary tip decreases if UTB is
present in OMDVR only and increases if UTB is present in IMDVR only
(Table 4).
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Our simulations thus suggest that the effects of the urea transporter are as significant in the IM as in the OM. Assuming that fu remains equal to 40%, selectively removing UTB from IMDVR reduces osmolality at the papillary tip by 5.5% (from 1,077 to 1,018 mosmol/kgH2O), whereas eliminating UTB from OMDVR only increases it by 5.2% (from 1,077 to 1,133 mosmol/kgH2O).
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DISCUSSION |
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In this study, we used our mathematical model of the renal medullary microcirculation to gain more insight into the role of UTB. Our approach is limited in two important ways. First, to obviate the need to fully simulate the urinary concentrating mechanism, the model specifies interstitial generation rates of water, sodium, and urea to account for reabsorption from the loops of Henle and the collecting ducts. Because it is not known how changes in the expression of UTB and AQP1 in DVR affect tubular transport, we assumed that the fraction of filtered load that is reabsorbed into the medullary microcirculation remains unaffected by deletion of the transporters. However, given that the entire countercurrent system of the medulla acts in an integrated manner, changes in transmural fluxes are bound to have secondary effects on transtubular gradients in vivo and therefore alter supply to the interstitium.
It is likely that deleting UTB affects not only fu (and the spatial variations of the interstitial generation rate of urea) but also water and sodium reabsorption, as they are closely linked. As reviewed by Sands (26), hyperosmolality in the IM collecting duct can raise facilitated urea permeability and increase urea reabsorption; water diuresis also appears to enhance urea reabsorption; and urea is actively transported from the IM collecting duct to the interstitium by "sodium-urea cotransporters" in rats on a low-protein diet. In addition, even the baseline value of fu is difficult to estimate based on experimental data, as discussed previously (2). Theoretical studies (30, 31) suggest that fu is comprised between 20 and 60%, hence the range examined in this work and our choice of 40% for the baseline case.
The predictive ability of our model is further restricted by the nature of the relevant experimental data. Whereas most of the morphological and transport parameters used in this model come from measurements in rats, experiments in which the expression of UTB or AQP1 is deleted are often performed in mice. Observations by Verkman and colleagues (35) suggest that, in mice, UTB-dependent countercurrent exchange of urea in the renal medulla may contribute to one-third of the total capacity of the kidney to concentrate urine and even more greatly to the ability of the kidney to concentrate urea itself. In studies with UTB knockout mice, urine osmolality was 25% lower, plasma urea concentration was 30% higher, and urine urea concentration was 35% lower than in wild-type mice (35). The medullary architecture of mice is different from that of rats, as mice do not have short-looped nephrons, but it has not been thoroughly described in quantitative terms in the literature. Thus defects in the concentrating ability observed in UTB knockout mice cannot be directly compared with the predictions of this model.
Despite these limitations, our approach can help in gaining an understanding of the function of UTB. AQP1 knockout mice have been shown to manifest a severe urinary concentrating defect associated with defective medullary interstitial osmolality (10). Our model, with parameters derived from measurements in rats, accurately predicted that deletion of AQP1 leads to a substantial reduction of interstitial osmolality. Simulations suggested that DVR expression of AQP1 enhances medullary osmolar gradients by providing a route for volume efflux that shunts blood flow from DVR to AVR, secondarily reducing blood flow to the IM (18). The present study, while impaired by the lack of experimental data regarding the effects of UTB deletion on fv, fNa, and fu in rats, may also provide some insights into the role of UTB as a urea transporter.
Our simulations suggest that, by greatly facilitating transmembrane urea diffusion, UTB significantly increases urea concentrations throughout the medulla. However, by decreasing radial urea concentration gradients, UTB also reduces volume efflux from DVR through AQP1 water channels and thereby lowers the plasma concentration of other solutes such as sodium chloride. The presence of UTB therefore appears to increase the contribution of urea to the corticomedullary gradient. Whether the UTB-mediated increase in the concentration of urea compensates for the decrease in that of sodium chloride depends on the fraction of filtered urea that is reabsorbed by vasa recta, fu; the net effect on overall concentrating ability (as measured by osmolality at the papillary tip) is not expected to be very significant. In the absence of AQP1, however, the reduction in transmembrane urea concentration differences imparted by UTB has no direct effect on water fluxes, and our model predicts that the urea transporter significantly increases both papillary osmolality and the fraction of total osmolality that is due to urea.
AQP1 water channels, expressed by DVR endothelia (16), have been shown to be a transport pathway across which small hydrophilic solutes such as sodium chloride and urea drive water flux (20). UTB also appears to function as a water channel, with urea and water sharing a common aqueous pathway. Yang and Verkman (36) first reported a significant osmotic water permeability in X. laevis oocytes expressing UT3 (a UTB isoform), suggesting the existence of a continuous aqueous channel through the UT3 protein that passes both water and urea. Sidoux-Walter et al. (29) later found that at physiological expression levels, the HUT11A transporter in humans (whose rat homologue is UT3) confers urea permeability to RBCs but not water permeability. They proposed that water transport activity in HUT11A-expressing oocytes occurs when the transporter takes another conformation at high density in the oocyte membrane, allowing for water movement.
In subsequent studies with knockout mice, Yang and Verkman (37) reported that the single-water channel permeability of UTB, 7.5 x 1014 cm3/s, is similar to that of AQP1. The authors found that, at 10°C, the erythrocyte osmotic water permeability was significantly reduced in AQP1-UTB-deficient mice compared with AQP1-deficient mice (0.045 x 102 vs. 0.19 x 102 cm/s). There was, however, no significant difference at 35°C; at that temperature, 79% of water was transported through AQP1, 6% through UTB, and the rest through the lipid membrane. The investigators also found that urine osmolality in double knockout mice was similar to that in AQP1 knockout mice. They concluded that UTB functions as an efficient water transporter, but its absolute contribution to total water transport in normal erythrocytes is small because RBCs express many fewer UTB urea transporters than AQP1 water channels (37).
Our theoretical predictions regarding the function of UTB as a water channel confirm the experimental results of Sidoux-Walter et al. (29) and Yang and Verkman (37), namely, that the absolute contribution of UTB transporters to water transport in normal erythrocytes is not significant. In our baseline case (i.e., both AQP1 and UTB are present in erythrocytes and vasa recta), the total amount of water transported across DVR walls through UTB is calculated to be only 2 and 3% of that transported through AQP1 and the paracellular pathway, respectively. Along DVR, the fraction of water transported across RBCs through the UTB transporters, lipid membrane, and AQP1 water channels is predicted to be 6, 15, and 79%, respectively, in close agreement with the values reported by Yang and Verkman. It is thus not surprising that eliminating water transport across UTB should have a negligible effect on small-solute concentrations and osmolality, even in the absence of water channels.
Yang and Verkman (36)
speculated that the UTB-mediated solvent drag of urea could provide a way for
urea to exit from vasa recta to balance the osmotically driven water exit.
However, our simulations suggest that the convective efflux of urea from DVR
across UTB represents only 1% of the diffusional influx across the same
transporter (i.e., 8.9 x 107 vs. 8.5
x 105 mmol/s). The net amount of urea that
enters DVR from the interstitium through UTB is 8.4 x
105mmol/s, which is also more than twice the
amount that enters the lumen across the paracellular pathway, 4.1 x
105 mmol/s. The ability of UTB to transport
water, therefore, does not appear to significantly affect urea fluxes across
DVR walls. Water movement through UTB is unlikely to be physiologically
important in the kidney.
As discussed by Macey and Yousef
(12), shrinkage of RBCs to
<60% of their original volume leads to irreversible damage, whereby the
membrane becomes leaky to sodium; conversely, swollen erythrocytes are less
deformable and more prone to destruction. Our results confirm that the UTB
urea transporter plays a significant role in erythrocytes by reducing the
magnitude of RBC shrinkage and swelling along the corticomedullary axis. We
found that with UTB, RBC volume decreases to 63% of its initial value along
DVR and increases back to 100% of its initial DVR value along AVR; without the
transporter, volume would be reduced to 55% of its initial value along DVR and
would slightly exceed its initial DVR value on leaving medullary AVR. Those
trends agree with the predictions of Macey and Yousef. These investigators
estimated that the RBC volume at the papillary tip is 65 and 5560%
of its initial DVR value with and without UTB, respectively, and that as blood
flows back up, RBC volume returns to 100 and 130145% of its initial
value, respectively. The quantitative differences between the two studies may
stem in part from the fact that their estimate of the urea permeability of the
RBC lipid membrane is one order of magnitude lower than ours; Macey and Yousef
used reported values for artificial bilayers on the order of 1 x
106 cm/s, whereas our estimate of 3 x
105 cm/s was based on measurements in
UTB-deficient mice (37). The
relative contribution of UTB to urea transport across RBCs is therefore higher
in their study.
Whether the effect of UTB in reducing swelling and shrinking of RBCs is significant in vivo remains uncertain. Erythrocytes in individuals lacking the Kidd (Jk) antigen also lack UTB expression (5, 27) because the human Kidd blood group and the UTB urea transporter proteins are encoded by the same Jk gene (8, 9). Whereas Woodfield et al. (33) found that some Jk-null individuals had mild hemolytic diseases, other investigators report that Jk-null individuals do not suffer a clinical syndrome except for a reduced capability to concentrate urine (27); similarly, individuals lacking AQP1 water channels do not suffer from hemolytic anemia (25).
Measurements by Pallone and colleagues
(17,
18,
24) have shown that the
apparent permeability of OMDVR to urea is reduced by the addition of thiourea,
methylurea, phloretin, or p-chloromercuribenzenesulfonate, and that
in the presence of thiourea at maximal inhibitory concentrations, the
permeability of OMDVR to urea and sodium is strongly correlated. In IMDVR,
however, the investigators found a close correlation between the estimated
permeability to urea and sodium in vivo that is unaffected by thiourea or
phloretin (24). Pallone et al.
(22) therefore speculated that
urea transporters may not be functional in IMDVR. To examine how this would
affect transport in the medullary microcirculation, we performed simulations
in which UTB was selectively removed from parts of DVR. Our model predicts
that eliminating UTB transporters from IMDVR only reduces osmolality at the
papillary tip by 5% if interstitial generation rates remain unaffected;
conversely, eliminating UTB transporters from OMDVR increases osmolality by a
similar percentage. These results suggest that UTB can have as significant an
effect in the IM as in the OM. However, it is likely here again that deletion
of UTB affects tubular transport and reabsorption rates. Moreover, no
conclusion from these simulations can be drawn as to whether IMDVR urea
transporters are truly functional in vivo.
What role does asymmetry play in the countercurrent exchange of urea? As described above, investigators have observed that the net efflux of urea across the RBC membrane at a given concentration difference is greater than the net influx for an equal but opposite directed gradient (14, 28). In addition, UTB urea transporters have been found in the continuous DVR endothelium, but they have not been identified in the highly fenestrated AVR endothelium. Our model suggests that a 10-fold increase in the urea efflux permeability of UTB in RBCs (or a 10-fold decrease in the urea influx permeability) has a negligible effect on urea transport in the medullary microcirculation. We also examined how the expression of UTB in AVR endothelium would affect urea countercurrent exchange. If the overall permeability of AVR to urea was identical to that of DVR, our model predicts that the higher efflux of urea from AVR into the interstitium would be compensated for by a higher influx into DVR, so that urea concentrations would increase everywhere. Osmolality at the papillary tip would then increase from 1,077 to 1,337 mosmol/kgH2O and u% from 51 to 58%. As expected, these results indicate overall that the efficiency of countercurrent exchange would be significantly enhanced if transport rates were maximized in both directions. Whereas the absence of UTB in AVR endothelium may stem from the fact that it is fenestrated, a better understanding of the purpose of asymmetric transport across the transporter itself would require more precise measurements of UTB permeability to urea in specific directions.
In summary, our results suggest that the presence of UTB in DVR walls and RBC membranes increases the contribution of urea to the corticomedullary osmolality gradient but not necessarily the magnitude of the gradient itself. In addition, we found that UTB significantly reduces the swelling and shrinking of RBCs as they are carried through the medullary countercurrent exchanger. The role of UTB as a water channel appears to be negligible. Although some experimental reports suggest that UTB may not be functional in the IM, our model predicts that the urea transporter may have as significant an effect in the IM as in the OM.
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APPENDIX |
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The oncotic pressure due to proteins is given
(3) by
![]() | (A1) |
![]() | (A2) |
![]() | (A3) |
Interstitial Cross-Sectional Area and Number of Vasa Recta
The cross-sectional area of the IM (in cm2) is calculated
(3) as
![]() | (A4) |
![]() | (A5) |
In the OM, we assume that Aint is constant and equal to its value at the OM-IM junction (i.e., at xIM = 0).
The AVR-to-DVR number ratio (Nv) is assumed to remain
constant along the corticomedullary axis and equal to 2.25
(41). Thus the number of vasa
recta in the IM is given by
![]() | (A6) |
![]() | (A7) |
Cell-to-Wall Surface Area Ratio
The cell-to-wall surface area ratio for DVR and AVR is given by,
respectively (40)
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DISCLOSURES |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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REFERENCES |
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