A
mathematical model was used to investigate how concentrated urine is
produced within the medullary cones of the quail kidney. Model
simulations were consistent with a concentrating mechanism based on
single-solute countercurrent multiplication and on NaCl cycling from
ascending to descending limbs of loops of Henle. The model predicted a
urine-to-plasma (U/P) osmolality ratio of ~2.26, a value consistent
with maximum avian U/P osmolality ratios. Active NaCl transport from
descending limb prebend thick segments contributed 70% of
concentrating capability. NaCl entry and water extraction provided 80 and 20%, respectively, of the concentrating effect in descending limb
flow. Parameter studies indicated that urine osmolality is sensitive to
the rate of fluid entry into descending limbs and collecting ducts at
the cone base. Parameter studies also indicated that the energetic cost
of concentrating urine is sensitive to loop of Henle population as a
function of medullary depth: as the fraction of loops reaching the cone
tip increased above anatomic values, urine osmolality increased only marginally, and, ultimately, urine osmolality decreased.
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INTRODUCTION |
BIRDS, LIKE MAMMALS,
can produce hypertonic urine when body water must be conserved to
maintain a stable blood plasma osmolality. However, this capability is
limited in birds compared with mammals: urine-to-plasma (U/P)
osmolality ratios seldom exceed 2 in birds (20), whereas
most mammals can produce ratios of 
4 (2).
Hypertonic urine is produced in the medullary cones of the avian kidney
(16, 66). The number of medullary cones per kidney may
range over two orders of magnitude, from tens to thousands, as a
function of body mass (20). Each cone may contain up to several hundred loops of Henle (20), and each cone
contains a coalescing system of collecting ducts (CDs)
(3). The populations of loops of Henle and CDs decrease
rapidly as a function of medullary cone length; in Gambel's quail
(Callipepla gambelii), the number of these tubules can be
well approximated by exponentially decreasing functions
(11). Only a fraction of the nephrons in each avian kidney
have loops of Henle; most nephrons are loopless, like those in
reptiles. In Gambel's quail ~10% of nephrons have loops.
Experimental and theoretical studies have supported the hypothesis that
countercurrent multiplication produces concentrated urine in the
mammalian renal medulla (26, 40, 63). According to this
hypothesis, a small osmotic pressure difference between adjacent renal
tubules is multiplied (or augmented) by countercurrent (i.e.,
antiparallel) fluid flow, resulting in a large increase in osmotic
pressure along the corticomedullary axis. The small osmotic pressure
difference, which is perpendicular to the flow directions, is called
the "single effect." The source of the single effect in the outer
medulla is the vigorous active transport of NaCl across the thick
epithelium of the ascending limbs (ALs); that transport is driven by
Na+-K+- ATPase pumps in the basolateral
membranes of the cells (21). The transepithelial osmotic
gradient so generated can be sustained because of the low permeability
of the thick AL epithelium to water and NaCl (63).
However, experiments indicate that the thin ALs of the inner medulla
can neither generate nor sustain a significant transepithelial NaCl
gradient, and no generally satisfactory explanation has been advanced
for how the inner medulla functions in the concentrating mechanism
(12).
In the avian kidney, the present evidence indicates that all ALs are
thick limbs. As in mammals, the thick AL epithelium in birds actively
transports NaCl from the tubular lumen to the interstitium, and in the
Japanese quail (Coturnix coturnix) the thick limb epithelium appears to have a low permeability to water and NaCl (49,
50). In addition, each avian loop of Henle has a prebend thick
descending segment [or, alternatively, prebend enlargement (PBE)], of
variable length, which is contiguous with the AL and is believed to
have epithelial transport characteristics similar to those of the avian AL (6, 11). Thin descending limbs (DLs) in the Japanese
quail have low water permeability, but the limbs are highly permeable to Na+ and Cl
relative to mammalian DLs
(51). CDs in the Japanese quail appear to have a low
osmotic water permeability, compared with mammalian CDs under the
influence of antidiuretic hormone, and the avian ducts appear to be
little affected by arginine vasotocin, the avian antidiuretic hormone
(52). Little is known about the morphological or transport
properties of the medullary cone vasculature.
On the basis of the counterflow configuration of the tubules in the
medullary cone and the finding of an NaCl corticomedullary gradient in
chicken and turkey, Skadhauge and Schmidt-Nielsen (66)
proposed that birds produce concentrated urine by means of
countercurrent multiplication. On the basis of the subsequent elucidation of the transport characteristics of the tubular epithelium (chiefly in the Japanese quail), Nishimura et al. (51) set
forth a specific hypothesis for the operation of the avian
concentrating mechanism as a countercurrent multiplier system. They
proposed that active transepithelial transport of NaCl from the ALs is the source of the single effect of the avian countercurrent mechanism, that thin DL fluid osmolality is increased principally by NaCl entry
via transepithelial diffusion, and that this NaCl is then, in turn,
delivered by advection (i.e., the motion of fluid, or the solute
carried by the fluid, along the tubular lumen) to thick AL lumens,
where it is again subject to active transepithelial transport. They
hypothesized that this process of single-solute cycling, in conjunction
with a transport cascade resulting from loops of Henle of various
lengths reaching to different depths within the medullary cone,
contributes to the concentrating effect. Concentrated urine was assumed
to form by the near-osmotic equilibration of CD fluid with the
medullary cone interstitium. As in the outer medulla of the mammalian
kidney, the thick ALs would carry fluid that is dilute, with respect to
blood plasma, from the medullary cone into the cortex.
In mammals, nitrogen is excreted in urea, and urea is believed to have
an important role in the mammalian urine concentrating mechanism
(17, 62). In birds, however, nitrogen is excreted mostly
in uric acid, which is incorporated in small, spherical structures that
form a stable hydrophobic suspension; that suspension is an osmotically
inactive component of avian urine (7, 10). Urea is found
in low concentrations in the avian medullary cone and is therefore
thought to have no significant role in the avian concentrating
mechanism (66).
In this study we describe a mathematical model of the avian urine
concentrating mechanism, as found in quail, and we present simulation
results based on that model. For a "base-case" simulation, we used
mostly morphological and transepithelial transport parameters measured
in Gambel's quail and in the Japanese quail. Simulations based on the
model were used to predict intratubular concentrations, intratubular
flows, and transepithelial transport rates as a function of position
along the cone. By altering the parameters of the base case, we
investigated the effects of parameter values on concentrating
capability and efficiency.
The model simulations tend to confirm that the medullary cone operates
as a countercurrent multiplier system, much as proposed by Nishimura et
al. (51). Moreover, the simulations indicate that prebend
thick limbs can significantly augment concentrating capability compared
with the hypothetical case where prebend thick limbs are absent. In
addition, simulation results suggest an explanation for the decreasing
population of loops of Henle as a function of cone depth: because of
the decrease in total CD flow (as ducts traverse the medullary cone)
and the resulting decrease in osmotic load, additional loops, exceeding
the number reported and reaching deep into the cone, would have little
effect on final urine osmolality. Thus there would be diminishing
returns for the additional solute that would be transported across
thick ALs, and additional long loops would serve only to increase the
energetic cost of producing hypertonic urine.
In a strict sense, the results of this study are applicable only to the
concentrating mechanism as found in species similar to the quail, and
thus this is a study of "an" avian concentrating mechanism. Yet,
because most avian species so far studied have medullary cone anatomy
that is substantially similar to that of the quail and because our
simulation results predict U/P osmolality ratios that range over the
values found in many birds, our results may be representative of the
avian urine concentrating mechanism in many or most avian species.
However, a species exhibiting exceptions to hypotheses adopted in this
study is known: the Anna's hummingbird, Calypte anna
(9). This nectarivorous bird has only ~0.4% looped nephrons, and these looped nephrons have no thin DLs, but instead their
loops of Henle have cells entirely like those of thick ALs. The kidneys
of the Anna's hummingbird are unable to produce a urine more
concentrated than blood plasma.
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MATHEMATICAL MODEL |
Model structure.
A schematic diagram of the mathematical model is given in Fig.
1. The model, which is structurally
similar to several previous models (42-44), uses the
central core (CC) formulation introduced by Stephenson
(70). In this formulation, the extratubular contents of
the medulla (i.e., vasculature, interstitial spaces, and interstitial cells) are merged into a single tube, the CC, through which the loops
of Henle and the CD system interact. The CC is closed at the medullary
tip but is contiguous with the cortical interstitium at the
corticomedullary boundary.
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Fig. 1.
Schematic diagram of model for avian medullary cone. The
collecting duct (CD) system is merged into a composite CD. Descending
limbs of loops of Henle (DL), ascending limbs of loops of Henle (AL),
and CD interact through a common tubular compartment, the central core
(CC). The number of loops decreases exponentially as a function of
medullary depth, as indicated by morphological investigation
(11) and suggested in this diagram by the reduced numbers
of loops of Henle as a function of medullary depth; 6 representative
loops are shown here, but the numerical formulation of the model uses
80 loops of Henle to approximate a continuously decreasing
distribution. Each DL has a prebend enlargement (PBE) that is assumed
to have the same transepithelial transport properties as ALs. In the
model, loop bends are not explicitly represented; flow from each DL
enters directly its associated AL.
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Because experimental evidence indicates that only one solute, NaCl,
plays a significant role in the avian urine concentrating mechanism
(16, 66), the model was formulated for a single solute,
represented by Cl. Osmolalities were computed by assuming
that each Cl ion was accompanied by a cation, assumed to
be principally Na+.
The CD system was represented as a single tube of varying diameter,
scaled according to the number of CDs reaching each medullary level.
Fluid flow, Cl concentration, and fluid osmolality in the
CC and CD system were represented by functions of position along the
corticomedullary axis. The loops of Henle, which reach to different
levels of the avian medullary cone, were represented in the model by a
continuous, monotonically decreasing distribution of tubes (Fig. 1).
The continuous distribution is constructed by formulating each
dependent variable (e.g., concentration or flow rate) associated with a
DL or an AL as a function of both axial position and the medullary
level at which the loop of Henle turns. The continuous distribution provides detailed information about the luminal contents of individual loops of Henle and how those contents vary as a function of loop-bend depth. Moreover, this formulation allows for the representation of
axial inhomogeneities in the transport properties of DLs; in particular, it allows the explicit inclusion of a prebend segment with
transepithelial transport characteristics similar to those of the ALs.
The model was formulated for the medullary cone only and did not
explicitly include a representation of the renal cortex. Some renal
models have included additional equations describing water flows and
solute concentrations in the cortical tubules that connect the ALs and
the CD system (72). However, since in this report we are
concerned mostly with medullary function and since knowledge of
transepithelial transport in the distal tubule is less complete than
for the medulla, the boundary conditions (flow rates and
concentrations) were prescribed for the DLs and the composite CD at the
base of the medullary cone, i.e., at the corticomedullary boundary.
Analogous conditions were not required for the ALs and CC, since the
flow from these tubules is normally into the cortex.
The fundamental equations for the mathematical model used here have
been derived elsewhere (46, 47); however, the formulation used here differs, in that a nonzero fraction of model loops of Henle
reaches the tip of the cone, in accordance with experimental findings
(11). For completeness, the model equations are given in
the APPENDIX.
Model parameters.
In the simulation studies reported here, morphological and membrane
transport parameters were varied relative to a particular base case
chosen to approximate the quail medullary cone. The base-case
morphological parameters were based on a medullary cone from a specimen
of Gambel's quail (designated cone 11 in Ref. 11). Cone 11 was a typical representative of
other cones within that animal, except it had the greatest length, as
measured from cone base to cone tip, i.e., 3.35 mm, and it therefore
provided the most detailed structural information. For the model, we
used data from the deepest 3.25 mm of that cone, where the loop and CD
counts were found to be nearly monotonically decreasing.
To represent the 98 loops of Henle and 28 CDs that entered the
particular cone, a ratio of total loops to CDs of 98:28 was assumed at
the corticomedullary boundary. The avian loops of Henle are of variable
length and turn back at various levels along the medullary cone
(11), with most turning back near the cone base. Similarly, the number of CDs decreases along the cone because of the
successive coalescences of two ducts into one duct (3). Morphological measurements indicate that, as a function of medullary depth, the loop of Henle and CD populations decrease approximately exponentially (11). Similar patterns have been found in
rat (25, 36) and rabbit (64).
A least-squares fit to the natural logarithm of the loop population
measured in medullary cone 11 in Ref. 11 showed
that the fraction, w, of loops of Henle reaching to
medullary depth x is well approximated by a function of the
form w(x) = e3.22x/L, where x = 0 and x = L correspond to the cone base and
the cone tip, respectively (percent cone depth is
x/L × 100%). Similarly, the fraction of
CDs, wcd, reaching to medullary depth
x is well approximated by
wcd(x) = e2.88x/L. The continuous curves
corresponding to these loop and CD fractions are shown in Fig.
2. The fractional loop distribution was
used in computing the composite fluxes from the aggregate of loops of
Henle, and the fractional number of CDs was used in scaling the
composite CD.
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Fig. 2.
A: fraction of loops of Henle as a
function of cone depth.  , Experimental counts of loops
of Henle, divided by total loops at cone base (data from the study
published as Ref. 11); curve, approximation used in the
model to give appropriate weight to loops of Henle of differing
lengths. B: fraction of CDs as a function of cone
depth. , Experimental counts of CDs divided by total
number of CDs at cone base; curve, approximation used in the model to
give the appropriate weight to surface area of composite CD.
C: comparison of fractional populations of loops of Henle
and CDs (curves from A and B). The pattern of
similar exponential decrease exhibited by these curves suggests that CD
osmotic load at each level is balanced with concentrating capacity
available from loops of Henle at that level.
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Luminal loop of Henle and CD diameters, based on data from the same
medullary cone, are given in Table 1. The
thin DL diameter given in Table 1 is an average over all thin DLs at
all depths. The thick AL diameter at the cone base (x = O)
was based on a least-squares fit to average diameters all along the
cone (average diameters were measured at 50-µm intervals in Ref.
11); the thick AL value at the cone tip is an average over
the loops reaching the cone tip. The CD diameter at the cone base is
the average diameter over all CDs at x = 0; the cone
tip diameter is based on the cone tip values of a least-squares fit to
average diameters along the cone. The diameters used in the model at
each numerical grid point were obtained from these values by smooth
interpolations similar to those previously described and used in Refs.
45 and 47.
Loop of Henle diameters, as incorporated in the model, are represented
by means of level curves in Fig. 4A (to facilitate comparisons, Fig. 4A is grouped with level curves of
simulated quantities in RESULTS). Level curves represent
functional values in the same way that elevations are specified on
topographical maps: the label on each curve gives the value that the
variable parameter assumes on that curve. Thus level curves provide a
means for representing a three-dimensional surface in two dimensions. Loop of Henle diameters change continuously and smoothly between the
level curves portrayed in Fig. 4A. In Fig. 4, A
and D, the lengths of DL segments are indicated as a
percentage of cone depth along the lower horizontal axis and the
lengths of AL segments are indicated along the right vertical axis.
Loop bends occur along the dashed diagonal line extending from the
upper left to the lower right. A shaded region labeled "DL" and
"AL" suggests a loop of Henle reaching about three-fourths of the
way into the medulla. (See APPENDIX for more information
about construction of Fig. 4.)
CD diameter, as a function of medullary cone depth, increased
approximately exponentially by a fractional amount given by e1.25x/L, where x
ranges from 0 to L along the medullary axis. The CD system
surface area was determined from this diameter scaling and from the
number of CDs as a function of medullary depth.
The cross-sectional area of the CC, not including the tubules enclosed
by the core, was taken to be 0.9w(x) + 0.1 (where w is the fraction of loops) times the luminal
cross-sectional area of a DL at the corticomedullary boundary. Thus the
volume of the cross-sectional area of the CC approximates that of the
aggregate DLs. However, steady-state solutions do not depend on the
cross-sectional area of the CC (or of any other tubule; see
APPENDIX).
Our base-case transepithelial transport parameters are summarized in
Table 2. To obtain transepithelial
transport parameters that could be scaled according to tubular areas
measured in Gambel's quail, we assumed that typical tubular areas per
unit length are similar in the Japanese quail and Gambel's quail.
Using this assumption, we converted transport parameter values reported
per unit tubular length in the Japanese quail to values expressed per
unit tubular area. Thus hydraulic conductivity in DLs of Japanese
quail, which has been reported to be 83.3 ± 32.4 × 109
cm2 · s1 · atm1
(51), corresponds to an osmotic water permeability of
552 ± 215 µm/s, when one assumes a luminal tubular diameter of
6.81 µm and a temperature of 310°K. Similarly, hydraulic
conductivity in ALs of the Japanese quail has been reported to be
12.8 ± 2.1 × 109
cm2 · s1 · atm1
(49); for a tubular diameter of 10.1 µm (the average
luminal thick limb diameter), that hydraulic conductivity corresponds to an osmotic water permeability of 57.2 ± 9.4 µm/s.
Osmotic water permeability in the CD was estimated from measurements in
Ref. 52, which reported an osmotic water flux
of ~0.396 nl · min1 · mm1
with 50-200 mosmol/kgH2O osmotic gradient along a
tubule ~0.430 mm long. Under normal circumstances, one expects that
the transepithelial osmotic gradient will be small. For an osmotic
gradient of 50 mosmol/kgH2O and an assumed inner diameter
of ~20 µm (typical of the initial CD of Gambel's quail), an
osmotic water permeability of 115 µm/s can be calculated.
The Cl flux coefficient (107
cm2/s) in DLs of the Japanese quail (51) has
been measured (51) to be 24.9 ± 3.6. With the assumption of a diameter of 6.81 µm, this corresponds to a
permeability in standard units (105 cm/s) of 116 ± 17. Cl permeability for the AL was based on the
Cl influx coefficient measured in Ref. 49. A
typical influx coefficient of 1.37 × 107
cm2/s and an assumed diameter of 10.1 µm indicate a
Cl permeability of ~4.32. Cl permeability
(105 cm/s) in the CD was assumed to have a value of 1, similar to the value for Na+ (~0.39) reported in the
rabbit outer medullary CD (73).
The active transport rate for Cl from the AL was based on
the average efflux rate of 370.4 ± 27.7 peq · mm1 · min1 reported
in Ref. 49. If one assumes an AL inner diameter of 10.1 µm, this efflux rate corresponds to a flux of 19.5 ± 1.5 nmol · cm2 · s1. This flux
was taken to be the maximum active transport rate; this rate is similar
to those used in models of the mammalian concentrating mechanism
(45, 78), which have generally been chosen to obtain an
outer medullary osmolality increase of a factor of ~2. Other active
transport rates in the medullary cone were assumed to be too small to
have a significant role in the concentrating mechanism; consequently,
they were set to zero. The Michaelis constant
(Km) was set to 40 mM (22), and
reflection coefficients were everywhere set to 1 on the basis of
findings in the mammalian medulla (63).
The boundary conditions for incoming flows are summarized in Table
3. Cl concentration
entering the DLs and the CD system was set to 130 mM, consistent with
measurements (130.60 ± 3.27 mM) in the proximal tubule of the
European starling (41). Fluid flow rate entering DLs at
the corticomedullary boundary was taken to be 5.53 nl/min, which is
35% of single-nephron glomerular filtration rate measured for
long-looped nephrons in Gambel's quail (5). Fluid flow rate entering the CD system per looped nephron was taken to be 0.500 nl/min on the basis of the following considerations. Total urine flow
rate from both kidneys combined, in water-deprived Gambel's quail, has
been measured to be 0.15 ± 0.1 ml/h (associated urine osmolality
was 637 ± 90 mosmol/kgH2O) (80). The
number of looped nephrons has been estimated at 4,678 per kidney
(5). Thus the rate of urine production per looped nephron
is ~0.27 nl/min. Since the avian medullary cone is reported to
achieve U/P osmolality ratios approaching 2 (5, 16, 66),
we assume that ~45% of fluid entering the CD system at the
corticomedullary axis is removed along the medullary cone, resulting in
a flow of ~0.5 nl/min entering the CD system per looped nephron. Thus the fluid load on the concentrating system is about one-tenth of the
fluid rate entering the loops of Henle at the corticomedullary boundary.
Numerical calculations.
Numerical approximations to steady-state solutions of the model
equations were obtained via a previously developed, fully explicit
dynamic method (46, 47). The numerical method was programmed in FORTRAN, and computations were performed in
double-precision mode on a Sun Microsystems SPARCstation Ultra 1. In
each model simulation, the numerical approximation was computed in time
until the osmolality of the CD effluent was varying by <1 part in
1014, the maximum machine accuracy attainable in
double-precision mode. A space grid with 80 subintervals was used to
allow for the rapid transition from the transport characteristics of
thin DLs to the characteristics of thick descending prebend segments. In the numerical method, a model loop of Henle reaches to the right
endpoint of each of the subintervals; thus 80 loops of Henle are
represented in the model calculations. Test calculations with 160, 320, and 640 subintervals yielded steady-state CD effluent osmolalities that
are converging at a rate better than second order to a value that is
~0.54% less than the base-case value obtained with 80 subintervals.
Test calculations demonstrated second-order spatial convergence for net
mass and water flow through the simulated medullary cone.
Relative efficiency and relative concentrating effect.
Some of the parameter studies reported in RESULTS make use
of a measure of relative efficiency, which we now explain. The concentrating mechanism depends on several processes that require the
sustained consumption of metabolic energy. These processes include the
general maintenance of renal tissues, the pumping of fluid through the
renal vasculature and tubules, the active transport of NaCl from
proximal and distal tubules, and the active transport of NaCl from
thick limbs. Of these processes, the one most intimately connected with
the concentrating mechanism is NaCl active transport from thick limbs
of Henle. The rate of transport is directly proportional to the energy
consumed, owing to the nature of the
Na+-K+-ATPase pump. Thus, to assess the
efficiency of the concentrating effect, one may compare the net urine
concentrating effect with the total rate of active transport from all
thick limbs of Henle. Because fluid is delivered to the avian kidney at
plasma osmolality, the net urine concentrating effect is proportional
to the U/P osmolality ratio minus 1. To obtain a measure of efficiency,
we divided the net concentrating effect by the total rate of active transport. To provide a unitless measure of efficiency that is relative
to that of the base case, we normalized that quotient by the analogous
quotient corresponding to the efficiency of the base case. Thus we
defined the relative efficiency by
![relative efficiency<IT>=</IT><FR><NU>[(U<IT>/</IT>P)(<IT>v</IT>)<IT>−1</IT>]<IT>/</IT>TAT(<IT>v</IT>)</NU><DE>[(U<IT>/</IT>P)<SUB>b</SUB><IT>−1</IT>]<IT>/</IT>TAT<SUB>b</SUB></DE></FR>](/content/vol279/issue6/fulltext/F1139/img001.gif)
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(1)
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where (U/P)(v) is the simulated urine osmolality
divided by the plasma osmolality when the model is evaluated for a
value v of a stipulated parameter (all other parameters are
set to their base-case values), and where (U/P)b is the
base-case U/P osmolality ratio. The total rate of active
Cl transport (which is entirely from thick DLs and ALs)
is TAT(v); the corresponding base-case value is
TATb. The definition is formulated so that the "relative
efficiency" is 1 when a stipulated parameter v takes on
the base-case value. A precise characterization of TAT is provided in
the APPENDIX.
A second unitless quantity used in RESULTS is the relative
concentrating effect, which is the net urine-concentrating effect for a
particular parameter value (i.e., the U/P osmolality ratio 
1)
normalized by the base-case net concentrating effect, i.e.
![relative concentrating effect<IT>=</IT><FR><NU>[(U<IT>/</IT>P)(<IT>v</IT>)<IT>−1</IT>]</NU><DE>[(U<IT>/</IT>P)<SUB>b</SUB><IT>−1</IT>]</DE></FR><IT>×100%</IT>](/content/vol279/issue6/fulltext/F1139/img002.gif)
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(2)
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RESULTS |
Base case.
Fundamental base-case quantities obtained from the model are summarized
in Table 4. These values include
osmolality ratios at the cone tip and flow and absorption data for the
longest loop, the aggregate loops, the CD system, and the whole
medullary cone. Fundamental spatially distributed base-case model
results are represented in Fig. 3 and
Fig. 4, B-D.
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Fig. 3.
Profiles of base-case tubular fluid-to-plasma (TF/P)
osmolality ratios in longest loop of Henle (DL and AL), collecting duct
(CD), and central core (CC). Arrows indicate flow directions. Gray line
(AVG) gives average of osmolality in all represented structures,
weighted according to cross-sectional area. The CC profile gives
osmolality of the accumulated net absorbed fluid, which flows from
medullary cone tip to cone base. The CC profile increases rapidly to a
high plateau near cone tip, because only thick limbs of Henle are
present in deepest 255 µm of cone. In general, osmolality profiles
appear to increase linearly, but a closer examination reveals that,
with the exception of the AL, profiles tend to be concave down.
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Fig. 4.
Level curves of quantities associated with loop of Henle
distribution. Left vertical axes, DL values at cone base; top
horizontal axes, AL values at cone base; bottom horizontal axes,
longest DL; right vertical axes, longest AL; diagonal line from
top left to bottom right (gray line), loop bends.
In lower left triangles, DL length increases along horizontal axis,
from left to right; in upper right triangles, AL length increases along
vertical axis, from top to bottom. Wide gray lines suggest a
representative loop of Henle that reaches ~75% of the distance into
the medullary cone; arrows indicate normal flow direction of
intratubular fluid. Gray dashed lines in B-D correspond
to site of PBE transitions from thin to thick DLs. Variables in
A-D are constant along each level curve and take on the
value labeled on the curve. Except in A, level curves were
constructed to give equal increments of represented quantity. The level
curves permit the value of each quantity to be represented as a
function of loop length and medullary depth. A: inner
diameters of tubules based on measurements of Casotti et al.
(11). Thin limbs, i.e., limbs before PBE transitions, are
of nearly constant diameter of ~6.81 µm, corresponding to lower
triangular region having no curves. B: base-case
Cl concentration. Concentrations within thin DLs are
nearly uniform at each medullary level; concentrations within thick
limbs also tend to be uniform at each medullary level, but less so than
in thin limbs. C: base-case intratubular water flow rate.
Some water is absorbed from thin DLs, which are somewhat water
permeable; little water is absorbed from ALs, which have much lower
water permeability. D: base-case Cl advection
rate. Substantial Cl (and, hence, NaCl) enters thin DLs,
while Cl advection rate is nearly uniform in DLs at each
fixed medullary level. Different amounts of Cl are
absorbed from thick limbs as a function of loop length, leading to
differing thick limb Cl advection rates at each fixed
medullary level.
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Base-case tubular fluid-to-plasma (TF/P) osmolality ratios along the
medullary cone are shown in Fig. 3. The curves labeled "DL" and
"AL" correspond to the longest DL and the longest AL, respectively.
The osmolality of intratubular fluid in the terminal, thick part of the
DL and in the AL is progressively reduced along the flow direction by
the active transport of NaCl. As a result, the AL fluid that returns to
the cone base is significantly hyposmotic with respect to blood plasma
(~218 mosmol/kgH2O) compared with a plasma osmolality of
239 mosmol/kgH2O. The NaCl absorbed from the thick segments
raises the osmolalities in the DL, the CD system, and the CC. The CC
exhibits a large rise in osmolality in the 0.25 mm nearest the cone
tip, because the NaCl absorbed from the near-bend thick segments of the
longest loops of Henle is not in apposition with any thin DL segments.
The curve labeled "AVG" in Fig. 3 shows the average TF/P osmolality
ratio in all structures represented in the model, weighted according to
the cross-sectional areas of all structures (this curve is comparable
to values that could be obtained from a tissue-slice experiment). To
obtain the weighted average, the TF/P osmolality ratio in each tubule,
at each medullary level, was multiplied by its cross-sectional area,
and all such products were summed together to obtain a value
SM. To obtain the appropriate weighting, the
summands corresponding to loops of Henle were weighted according to the
fraction of loops reaching each level, and, for the purposes of this
calculation, the CC was counted as a tubule. The sum
SM, with units of area, was then divided by the
total cross-sectional area of all tubules, at the corresponding level,
to obtain the values labeled by AVG. Because the CD is the dominant
structure near the cone tip in terms of cross-sectional area, the
average osmolality near the cone tip closely approximates CD osmolality.
In a CC model, the osmolality in the CD is influenced directly only by
the CC osmolality. The osmolality of the CD flow tracks the axial
increase in CC osmolality, but because of the base-case CD osmotic
water permeability of 115 µm/s [which is low relative to the value
of 445 µm/s measured in rabbit CD (28, 61)] and the
magnitude of the CD flow, CD osmolality at each medullary level lags
below that of the CC.
To a first approximation, the TF/P osmolality ratios shown in Fig. 3
increase linearly as a function of cone depth, but comparison with a
straight-edge will show that the curves tend to be concave down, except
for the AL. This concavity suggests that the load presented by flow in
the CD becomes less well matched by the concentrating capacity of the
loops of Henle as medullary depth increases. The concave-downward
property is consistent with tissue-slice studies reported in Fig. 7 of
Ref. 16 (see DISCUSSION).
Figure 4, B-D, exhibits level curves of loop of Henle
concentration, water flow, and Cl advection. (Figure
4A, described in MATHEMATICAL MODEL, represents loop of Henle diameters used in the model formulation.) The results exhibited in Fig. 4 are generally consistent with our expectations. The
level curves of Cl concentration in Fig. 4B
indicate that DLs, at each level, are in near osmotic equilibrium,
which is consistent with their high permeability to NaCl. The curves
for AL concentration (Fig. 4B) also indicate similar
osmolalities at each level, but there is less consistency at each
level. The reduced consistency can be reasonably attributed to the
varying diameters, as a function of cone depth (because diameter is
proportional to available transport area and inversely proportional to
flow speed), and to the more complete saturation of active transport at
high than at low concentrations. Nonetheless, at each level the
concentrations are sufficiently similar among all limbs of Henle that
Fig. 4B serves as an illustration of a key aspect of the
theory of countercurrent multiplication: at each fixed medullary level
the concentrations (and, hence, in this case, osmolalities) of tubular
fluid vary little from tubule to tubule relative to the difference
generated along the flow axis from the cone base to the cone tip.
The level curves of water flow, exhibited in Fig. 4C,
indicate a greater water loss from DLs than from ALs, consistent with the higher water permeability of DLs (552 vs. 57.2 µm/s for ALs). In
the diagonal strip corresponding to the PBE (the strip between the gray
dashed line and the gray line) and in the region corresponding to the
ALs, the level curves tend to be more nearly parallel to the flow
direction, which indicates an approximation to constant intratubular flow.
The level curves of Cl advection (Fig. 4D)
illustrate Cl (and, hence, NaCl) cycling from thick limbs
to thin DLs. By comparison with Fig. 4C, one can determine
that the fluid in the DLs is principally concentrated by the addition
of Cl. Indeed, numerical results from the model indicate
that, in the thin portion of the DL of the longest loop, 79.9% of the
increase in osmolality arises from solute addition, whereas only 20.1% arises from water absorption. Advection of Cl in the ALs
of longer loops is more reduced at each level than advection in shorter
loops. This difference (similar to the case of Cl
concentrations) arises because the reduced flow speed and increased surface area in longer ALs result in greater transepithelial active transport than in shorter limbs.
In summary, the results from the base case indicate that 1)
urine is concentrated by means of a countercurrent multiplier that
relies on active NaCl transport from ALs to generate the single
effect, 2) this countercurrent multiplier system
employs NaCl cycling from ALs to DLs, 3) fluid in DLs is
principally concentrated by NaCl addition, 4) the NaCl
concentrations (and osmolalities) within all DLs will tend to be nearly
equal to each other at each level, and the analogous result is
predicted for ALs, and 5) the osmolality profiles in the
tubules of the medulla tend generally to be linear, with a slight
concavity, oriented down. These results are consistent with the
countercurrent hypothesis advanced by Skadhauge and Schmidt-Nielsen
(66) and the NaCl cycling hypothesis advanced by Nishimura
et al. (51).
Parameter studies.
Extensive studies were conducted to determine the sensitivity of the
results to changes in base-case parameter values. In most studies, a
single parameter was varied systematically while all other parameters
retained their base-case values. In these studies, the stipulated
parameter was incrementally increased and the corresponding
steady-state solution was computed. The size of the increment was
determined empirically to yield smooth curves. However, some studies
involved special cases in which two parameters were simultaneously
changed from base-case values. In the results given below, those
studies will be clearly distinguished from the single-parameter studies.
Permeability to water and Cl.
Figure 5 exhibits results obtained by
varying the osmotic water permeability and the Cl
permeability of the represented tubules. Each curve represents a
numerical experiment in which only one parameter was changed in one
tubule; all other parameters remained at base-case values. In Fig. 5
and in similar subsequent figures, the gray horizontal bar indicates
the base-case value of the quantity represented as the ordinate. Each
open circle corresponds to a base-case value of a parameter that is
varied along the interval of abscissa values. A wide black curve
segment, where present, corresponds to the standard deviation (or,
alternatively, range) of a measured experimental value as reported in
the literature.
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Fig. 5.
Water and Cl permeability sensitivity
studies. Horizontal gray bars, base-case urine-to-plasma (U/P)
osmolality ratio, ~2.26; curves, model U/P osmolality ratios obtained
by varying permeabilities in indicated tubule types;  ,
base value of each parameter; wide black curve segments, experimental
range for varied parameter, if known. A: sensitivity of U/P
osmolality ratio to osmotic water permeability of DL, AL, and CD. U/P
osmolality ratio is nearly insensitive to DL permeability, sensitive to
AL permeability, and very sensitive to CD permeability for values below
~200 µm/s. B1: U/P osmolality ratio is insensitive to
permeability of CD to Cl (but urine flow can be
significantly affected; see text). B2: sensitivity of U/P
osmolality ratio to AL and DL Cl permeability. U/P
osmolality ratio is insensitive to DL permeability for values above
~50 × 105 cm/s, but U/P ratio is sensitive to
increasing AL permeability, which diminishes net absorption of
Cl from ALs.
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The results in Fig. 5A indicate that concentrating
capability is nearly independent of DL osmotic water permeability.
However, a low AL permeability and a sufficiently high CD permeability are essential for a significant concentrating effect. The results in
Fig. 51 indicate that concentrating capability is nearly
independent of CD Cl permeability. In cases in which CD
Cl permeability exceeded the base-case permeability, the
osmolality of CD fluid more closely approximated CC osmolality than in
the base case, but DL and CC osmolalities were reduced relative to the
base case. However, for high Cl permeability, CD
equilibration was mostly by Cl entry, so that a much
larger amount of concentrated urine was produced per looped nephron,
viz., 0.488 nl/min, which is only slightly diminished from the
base-case CD inflow of 0.500 nl/min and is 191% of the base-case urine
flow of 0.255 nl/min. This result suggests that the modulation of CD
Cl permeability in vivo might provide a mechanism for the
regulation of the rates of electrolyte and water excretion that would
not compromise concentrating capability.
The results in Fig. 52 indicate that concentrating
capability is not much affected by variation in DL Cl
permeability, provided that its value exceeds ~50 × 105 cm/s. Even when DL Cl permeability is
reduced to zero, the U/P osmolality ratio is reduced only to 2.02 from
the base-case value of 2.26. This indicates that NaCl cycling from AL
to DL is not required for the system to generate a significant
concentrating effect. Results in Fig. 52 indicate that even
small increases in AL Cl permeability substantially
decrease concentrating capability. The system is sensitive to AL
Cl permeability because Cl backleak
directly opposes the Cl active transport (the source of
the single effect) that is required for countercurrent multiplication.
Two cases were examined in which two parameters were simultaneously
changed from the base case. In the first, water and Cl
permeability were set to zero for all thin segments of DLs. The resulting U/P osmolality ratio was 2.01, a decrease in relative concentrating effect of 20% compared with the base case (see Eq. 2 for definition of relative concentrating effect). However, in this case, the TF/P osmolality ratio of fluid entering the cortex via
the CC was 1.76 (for the base case it was 1.16), whereas the TF/P
osmolality ratio of flow-weighted AL fluid entering the cortex was
0.820 (for the base case it was 0.913). Moreover, 21% more fluid was
absorbed from thick limbs than had been absorbed from thin and thick
limbs of Henle combined in the base case, owing to the higher
osmolality in the CC near the medullary cone base. The increased solute
and fluid load presented to the vasculature by absorption from loops of
Henle would likely have reduced the effectiveness of vascular
countercurrent exchange, if the vasculature had been explicitly
represented in our model. These hypothetical results for thin DLs
lacking water and NaCl transport underline the importance of the likely
normal role of these tubules in supporting a countercurrent multiplier
system with small transverse osmotic gradients between the lumens of
the different types of tubules.
In the second case, water and Cl permeability were
simultaneously set to zero for the entire CD system. In this case,
there was essentially no (useful) load on the concentrating mechanism. T/P osmolality ratios increased to 3.33 and 4.25 in the bend of the
longest loop of Henle and in the CC at the cone tip, respectively. These values and the results reported below in Fig. 7A
indicate that the maximum theoretical U/P osmolality ratio for a nearly vanishing CD flow ranges from ~3 to 4.
Rate of active Cl transport from thick limbs.
The curve in Fig. 6A
indicates, as expected, that concentrating capability increases as the
maximum rate of active Cl transport from thick limbs
(Vmax) is increased. Moreover, the curve is
concave upward, which indicates that osmolality increases more rapidly
as the transport rate increases. Since Fig. 6B shows that
total active transport from all thick limbs increases nearly linearly
with increasing Vmax, the increasing sensitivity
of concentrating capability as Vmax increases
cannot be attributed to incomplete saturation of the (assumed)
Michaelis-Menten transport. Rather, the explanation may be found in the
expressions derived for simple CC models, which show that the U/P
osmolality ratio tends to increase nonlinearly with increased
absorption from thick limbs (44, 70). Consistent with the
effect noted in Fig. 6A, the relative efficiency reported in
Fig. 6C shows that the efficiency of concentrating capability increases with increasing Vmax. (The
definition for relative efficiency was given by Eq. 1.)
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Fig. 6.
Model sensitivity to maximum active transport rate from
thick limbs (Vmax). A: U/P osmolality
ratio as a function of Vmax for active
Cl transport from thick limbs. Gray horizontal bar,
base-case osmolality; , base-case value for
Vmax. U/P osmolality ratio increases with
increasing active transport rate, as expected; however, the rate of
increase of U/P osmolality ratio also increases with increasing
transport rate. B: total active transport rate across all
thick limbs as a function of Vmax expressed per
looped nephron. Gray horizontal bar, base-case total active transport
rate from all thick limbs; , base-case value for
Vmax. Nearly linear increase indicates that
total transport is nearly proportional to Vmax.
C: relative efficiency as a function of
Vmax. Gray horizontal bar, base-case efficiency,
which is 1 by the measure of efficiency used; ,
base-case value for Vmax. Wide black curve
segment corresponds to experimental range of
Vmax. The relative efficiency of the
concentrating mechanism increases with increasing values of
Vmax.
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An analogous study for the Michaelis constant
(Km) (data not shown) resulted in monotone and
parallel decreases in the U/P osmolality ratio, total active transport,
and relative efficiency as the Km was increased
from 0 to 600 mM. The decreases were more rapid for smaller values of
Km (and near the base-case value of 40 mM) than
for larger values. This pattern, which is roughly inverse to that of
the study for Vmax, is explained by the
reciprocal effects of Km and
Vmax (see Eq. A4 in
APPENDIX).
DL and CD input flow.
The inflow rates into the DLs and the CD system at the corticomedullary
boundary are important parameters whose values have been inferred
from other measurements of related quantities (see Model
parameters). The effects of varying these parameters are shown in
Fig. 7. In Fig. 7A, CD input
flow was increased far beyond its base-case value (per nephron) of
0.500 nl/min, which resulted in a marked decrease in the U/P osmolality
ratio. Indeed, a doubling of the base-case value to 1.00 nl/min
decreased the U/P osmolality ratio from the base-case value of 2.26 to
1.64, which is a 49% decrease in relative concentrating effect.
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Fig. 7.
A: sensitivity of U/P osmolality ratio to CD
input flow at corticomedullary boundary. Gray horizontal bar, base-case
U/P osmolality ratio; , base-case input flow of 0.5 nl/min. Osmolality ratio rapidly declines for values near base-case
input flow. B1-B3: model study of sensitivity to DL
input flow at corticomedullary boundary. Gray horizontal bars,
base-case values. For flow less than ~2.2 nl/min, intratubular flow
direction reverses at sites along some of the loops of Henle, because
transepithelial flux from loops exceeds the input flow (dashed lines).
, Maximum U/P osmolality ratio of 2.8 and corresponding
total active Cl transport and relative efficiency, at
input flow of 2.7 nl/min; , U/P osmolality ratio, total
transport, and relative efficiency at base-case input flow of 5.53 nl/min. As input flow increases through the interval containing the
base-case value, osmolality decreases with increasing flow
(B1), active transport is not much affected (B2),
and relative efficiency decreases (B3).
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The sensitivity to DL input flow is shown in Fig. 7, B1-B3.
The dashed lines indicate an interval in which input flow below the
base-case flow of 5.53 nl/min results in intratubular flow reversal in
at least some portions of some loops of Henle. It is doubtful that such
an effect could arise in vivo, unless tubuloglomerular feedback were
disabled. Therefore, model results corresponding to the dashed lines
should probably be disregarded, except as indicating the need for
adequate NaCl from DL inflow to drive the concentrating mechanism and
the important role of regulatory processes that were not included in
the model. The filled circle in Fig. 71 indicates the
maximum U/P osmolality ratio, 2.80, for the range of input flows
examined. Corresponding filled circles have been placed on Fig. 7,
B2 and B3. The open circle indicates the
base-case values. Over most of the valid range of the applicability of
this study, the U/P osmolality ratio is decreasing, reaching a value of
1.78 at a DL input flow of 10 nl/min. The rate of total active
Cl transport is nearly unaffected, which suggests
compensation for decreased Cl concentrations in the
Michaelis-Menten transport term (in Eq. A4), arising from
the diluting effects of higher flow, by the increased availability of
Cl, which is carried into DLs in amounts proportional to
DL input flow at the cone base. Because total active transport is
nearly unaffected, the decrease in U/P osmolality ratio may be
attributed to the increasing osmotic load presented to the
concentrating mechanism by increased flow in thin DLs. Also, because
the total active transport is nearly unaffected, relative efficiency
follows the same pattern as the decrease in osmolality ratio.
Rate of active Cl transport from CDs.
For the base case, we assumed that Cl transport from the
CD system is negligible, and we therefore took
Vmax for the CD system to be zero. To assess
parameter sensitivity and to determine whether this is a reasonable
assumption, we conducted studies where Vmax for
the CD ranged from 0 to 5 nmol · cm2 · s1. Because a
sufficiently large active transport rate can result in significantly
increased water absorption from the CD system accompanied by
unrealistically low urine flow, we also considered the effects of
increasing CD input flow at the cone base while simultaneously varying
the Vmax for the CD system.
The results of these studies are summarized in Table
5. The base-case CD input flow of 0.500 nl/min per looped nephron is our estimate based on measured urine flow
of 0.27 nl/min per looped nephron and the assumption that CD fluid is
concentrated principally by water absorption from the CD (see
Model parameters). For base-case CD input flow, an increase
of Vmax to 1 nmol · cm2 · s1 results in
a U/P osmolality ratio that is little changed and a urine flow of 0.213 nl/min per looped nephron, a value that remains consistent with the
experimental value, 0.27 nl/min. However, as
Vmax is increased to 5 nmol · cm2 · s1, urine flow
falls significantly to 0.0479 nl/min, while the U/P osmolality ratio
increases to 2.72, a value that exceeds the concentrating capability of
most birds.
For a CD input flow of 0.750 nl/min per looped nephron in Table 5, a
Vmax of 5 nmol · cm2 · s1 produces a
U/P osmolality ratio and urine flow rate that are most consistent with
experiments: 1.64 and 0.247 nl/min, respectively. However, in the case
of this CD input flow, there is the paradoxical result that as solute
absorption from the CD increases, the U/P osmolality ratio decreases
well below the base-case value of 2.26, even though for a
Vmax of 5 nmol · cm2 · s1 the central
core TF/P osmolality ratio is 2.44 at the cone tip. The vigorous active
Cl transport tends to dilute CD contents (relative to
flow in the CC and loops of Henle), because the base-case osmotic water
permeability is not large enough to allow near-osmotic equilibration of
CD fluid with CC fluid when the input CD flow and Cl
absorption from that flow are sufficiently large. This disparity is
inconsistent with a key principle of countercurrent multiplication: for
efficient and effective operation, at each level the flows in all
tubules should vary little in osmotic pressure.
For a CD input flow of 1.00 nl/min per looped nephron (as in the case
of 0.750 nl/min) in Table 5, the CD does not attain near-osmotic
equilibration with the CC, which has a cone tip TF/P osmolality ratio
of 2.27 for a Vmax of 5 nmol · cm2 · s1. Moreover,
urine flows remain superphysiological throughout the examined range of
Vmax. Thus, for a CD inflow of 1.00 nl/min, the
concentrating mechanism is simply overwhelmed.
Because of its general consistency with urine flow, urine osmolality,
measurements of CD osmotic water permeability, and countercurrent multiplier theory, a reasonable conclusion from the results in Table 5
is that the most likely parameter combination is a base-case CD input
flow rate of ~0.5 nl/min per looped nephron with a
Vmax for CD Cl transport that is
not significantly larger than 1 nmol · cm2 · s1.
Cone length.
The effect of cone length on concentrating capability is illustrated in
Fig. 8. In these studies, the length of
the PBEs was unchanged; the added length was given to thin DLs and to
ALs. Thus, when cone length decreased below 0.255 mm, loops of Henle consisted entirely of thick limbs. As shown in Fig. 8A,
increasing cone length to 12 mm increased the U/P osmolality ratio by
344% over the base case, but additional increases in length led to concentrating capabilities that were substantially below the maximum achievable osmolality. A potential explanation for the decline in
concentrating capability for sufficiently long cones is suggested by
Fig. 8B, which shows that the total active Cl
transport rate increases more slowly for sufficiently long cones, presumably because only limited Cl is available to be
pumped from thick limbs.
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Fig. 8.
Model sensitivity to cone length. A: U/P
osmolality ratio as a function of cone length increases to a maximum
value of 5.3 () at length of 12 mm and then decreases.
Gray horizontal bar, base-case U/P osmolality ratio; ,
base-case cone length, 3.25 mm. B: total active Cl
transport increases monotonically as a function of loop length. Gray
horizontal bar, base-case total transport. C: relative
efficiency as a function of loop length. Gray horizontal bar, base-case
efficiency of 1; wide black curve segment, experimental range for cone
length. Some efficiency is gained by increasing cone length beyond
experimental values, but sufficiently long cone length results in
efficiency being substantially reduced below base-case efficiency.
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Although the concentrating capability can be dramatically increased by
lengthening the cone, the relative efficiency, shown in Fig.
8C, increases only to a maximum of ~123% at cone length of 8 mm and then decreases as cone length increases further. Thus a
gain in efficiency of only 23% is obtained by the 246% increase in
cone length from 3.25 to 8 mm.
Experimentally measured cone lengths in Gambel's quail correspond to
the thick black curve segment in Fig. 8C. As cone length ranges from 1.0 to 3.25 mm, the U/P osmolality ratios increase from
1.37 to 2.26. These values are consistent with maximum U/P osmolality
ratios found in birds, ratios that seldom exceed 2.0-2.5 (4,
20). The decrease in relative efficiency to a local minimum of
~0.76 near a length of 1 mm (Fig. 8C) may be attributable
to the appearance of thin descending segments, which, at first, may reduce the concentrating capacity afforded by the presence of thick
limbs, which are the only type of limbs present in the model when
medullary cone length is <0.255 mm.
Length of PBE.
Results collected in Fig. 9 indicate that
concentrating capacity is sensitive to the length of the PBE, i.e., to
the length of the thick, terminal portion of the DL, which is assumed
to have the same transport properties as the AL. The relative location of the PBE is illustrated in Fig. 9A. The dependence of
concentrating capability on the length of the PBE is shown in Fig.
9B1. As the prebend length increases from zero to its
base-case value of 255 µm, the U/P osmolality ratio increases from
1.74 to 2.26 (open circle), which corresponds to a 70% increase in
relative concentrating effect. As the prebend length increases further
to 905 µm (filled circle), the net concentrating effect increases to
266% of the no-PBE case.
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Fig. 9.
Model sensitivity to length of PBE. A:
schematic diagram showing loop of Henle with PBE. B1: U/P
osmolality ratio as a function of PBE length increases to 2.98 () at PBE length 0.905 and then decreases. Gray
horizontal bar, base-case osmolality; , base-case value
of PBE length, 255 µm. As the PBE length increases from zero to the
base-case value, the U/P osmolality ratio increases from 1.74 to 2.26, which corresponds to a 70% increase in relative concentrating
capability. B2: total active Cl transport by
all thick limbs increases monotonically as the total length of thick
limb segments increases; however, the rate of increase of total
transport decreases, presumably because progressively less
Cl is available from solute cycling from thick limbs to
thin DLs. Gray horizontal bar, base-case total transport;
, base-case PBE length. B3: relative
efficiency as a function of PBE length. Wide black curve segment
corresponds to the experimental range of PBE length. As PBE length
increases beyond experimental values, the decrease in U/P osmolality
ratios (B1) results in a rapid decrease in relative
efficiency.
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Additional increases in PBE length, however, result in decreasing
concentrating capacity. Examination of level curves for this case,
analogous to those in Fig. 4, indicates that the decrease can be
attributed to high absorption rates at early sites along DLs, reduced
absorption at sites near loop bends, enhanced water absorption from
loops of Henle (owing to gradients arising from enhanced tubular
dilution), and limits in the amount of Cl that can be
absorbed. Indeed, Fig. 92 shows that increasing PBE length
beyond ~1 mm results in little additional total Cl
transport. The amount of Cl that can be transported by a
loop of Henle depends principally on the amount of NaCl advected into
its thick segments from the thin DL. A lengthened PBE reduces the
length of the thin DL segment available for diffusive Cl
entry. Results in Fig. 93 show that the relative efficiency
decreases substantially for PBE lengths that exceed the measured range
(the measured range is indicated by the thick black curve segment).
Loop distribution.
Figure 10A shows curves for
the fraction of loops of Henle as a function of percentage of medullary
cone depth. The gray exponential curve corresponds to our base case and
closely follows the fractional distribution measured in a particular
cone (same cone as represented in Fig. 2). The other curves correspond
to other rates of exponential decrease; they are labeled according to
the fraction of loops reaching to (and turning at) the cone tip. Thus,
in the base case, ~4 of 100 of the loops reach to the cone tip.
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Fig. 10.
Model sensitivity to loop distribution. A:
fraction of loops of Henle as a function of cone depth for 4 different
exponential decrease rates. Labels (0.01, 0.04, 0.5, and 0.9)
correspond to fractions of loops reaching cone tip at those decrease
rates. Gray curve (labeled 0.04) corresponds to the base case.
B1: U/P osmolality ratio increased to a maximum of 2.44 () as the fraction of loops reaching the cone tip
increased to 0.27. Gray horizontal bar, base-case osmolality ratio;
, fraction of loops reaching cone tip in the base case,
0.04. B2: total active Cl transport as a
function of fraction of loops reaching the cone tip increased
monotonically, since the total length of thick segments increases as
the fraction of loops reaching the cone tip increases. Gray horizontal
bar, total active transport in the base case. B3: relative
efficiency as a function of the fraction of loops reaching the cone
tip. Efficiency declines, more rapidly for smaller fractions, as the
fraction of loops reaching the cone tip increases to 1.
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Figure 10, B1-B3, illustrates the result of parameter
studies in which osmolality ratios and rates of total active transport were computed for exponential curves that corresponded to a particular fraction of loops reaching the cone tip. As that fraction increased from near zero, the U/P osmolality ratio increased rapidly, to a
maximum value of 2.44. That increase arises because, in the deepest
part of the cone, the increasing fractions of loops provide increasing
concentrating capacity to the remaining fluid flowing in the remaining
CDs. If there are too few loops, then the fluid in the CDs presents a
load that cannot be much affected by those loops.
However, as the fraction of loops reaching the cone tip increased
further, the osmolality ratio decreased gradually. The decrease may be
understood as follows. The osmotic load presented to the concentrating
mechanism comes not only from CD flow, but also from flow in thin DLs,
which (in the model) is concentrated principally by Cl
entry. As more of the loops reach further in the cone, DL flow becomes
the principal load on the concentrating mechanism and CD flow becomes
an increasingly less significant factor. Moreover, there is less
apposition of PBEs with thin DLs (and no apposition when all loops
reach the cone tip). So the opportunity for a cascade, in which bends
at almost all levels concentrate thin DL fluid but are not part of the
osmotic load at the bend level, is reduced (44).
Total active transport (Fig. 10B2) increases by 318% from
the base case to the case where all loops reach the cone tip. The increase arises from the increase in the combined length of thick limb
segments (mostly thick AL segments).
The relative efficiency (Fig. 10B3) is monotonically
decreasing: it decreases rapidly near the base case but more slowly as the fraction of loops reaching the tip increases. Figure
10B3, combined with the results in Fig. 10, B1
and B2, makes a persuasive case that the exponentially
decreasing loop distribution found in vivo is consistent with an
energetically efficient concentrating mechanism.
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DISCUSSION |
TOP
ABSTRACT
INTRODUCTION
