Vol. 84, Issue 3, 769-781, March 1998
Pulmonary fluid extraction and osmotic
conductance, 
K, measured in
vivo
Joseph M.
Karch and
Jen-Shih
Lee
Department of Biomedical Engineering, University of Virginia,
Charlottesville, Virginia 22908
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ABSTRACT |
The change in aortic blood density in an in vivo rabbit
preparation was measured to assess fluid movement at the pulmonary capillaries caused by infusion of hypertonic solution (NaCl, urea, glucose, sucrose, or raffinose in isotonic saline) into the vena cava
over 20 s. The hypertonic disturbance increased the plasma osmotic
pressure by 
30 mosmol/l. The density change indicates that the fluid
extraction from the lung tissue was completed within 10 s. It was
followed by a fluid filtration into the lung tissue and then an
extraction and filtration from peripheral organs. An exchange model
with flow dispersion yields two equations to estimate the osmotic
conductance (
K; where
is the reflection coefficient of the test solute and
K is the filtration coefficient including the total capillary surface area), and the tissue fluid volume from the area and first moment of the measured density change
over the extraction phase. The values of
K are 1.40 ± 0.11, 1.00 ± 0.10, 1.71 ± 0.10, 2.60 ± 0.23, and 3.73 ± 0.34 (SE) ml · h
1 · mosmol1 · l · g1
for NaCl, urea, glucose, sucrose, and raffinose, respectively. Consistent with the model prediction, the tissue fluid volume (0.28 ± 0.04 ml/g wet lung tissue) was independent of the solute used.
This value suggests that all fluid spaces in the alveolar septa
participate in the process of fluid extraction due to an increase in
plasma osmotic pressure.
filtration; low-molecular-weight solutes; tissue fluid volume; reflection coefficient; rabbit lung; blood density; flow dispersion
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INTRODUCTION |
THE OSMOTIC CONDUCTANCE of the lung
(K, the product of the reflection
coefficient for the solute, , and the filtration coefficient of the
capillary endothelium including its total surface area, K) has been estimated from the
dilution of plasma indicators in pulmonary venous blood (3), the
density reduction of pulmonary venous or aortic blood (11, 12), or the
weight change of isolated perfused lungs (20, 26, 28) after the
injection or infusion of various low-molecular-weight (LMW) solutes
into the pulmonary artery or vena cava. Because of nonuniform velocity
distribution in vessel flow, irregular flow distribution among
microvessels, and the mixing of the heart, the blood flow through the
pulmonary circulation and the heart chambers is dispersive in nature.
This dispersion effect, along with the tissue effect (the change in tissue osmotic pressure to counter the hypertonic disturbance), was not
considered in most of the above estimations. Utilizing a compartmental
model to describe the extraction process and an analytic model to
characterize dispersion in the pulmonary circulation, Hunter and Lee
(11) showed that the dispersion and the tissue effects can
significantly alter the estimation of
K .
The density-measuring technique has a much higher sensitivity to
measure changes in hematocrit (or hemoglobin content) than an optical
technique (11). As a result, Hunter and Lee (11) could still assess
K for a hypertonic disturbance at
the site of infusion approximately one order of magnitude smaller than that used by Effros (3) and Semler et al. (23). The finding of similar
density dilution results for isolated rabbit lungs perfused with blood
or autologous plasma by Hunter and Lee leads to the conclusion that the
reduced disturbance does not impede red blood cell (RBC) flow, and thus
the fluid extraction from the tissue is the only factor producing the
observed density reduction. In this study we extend the in vitro
infusion protocol of Hunter and Lee (11) to an in vivo condition and
incorporate the dispersion analysis of Hunter and Lee (10) and Audi et
al. (1) to interpret the measured density dilution in terms of
K. Our first objective is to assess
how the hypertonic disturbance of five LMW solutes (NaCl, urea,
glucose, sucrose, and raffinose) produces a transcapillary fluid
exchange with the tissue, determine the role of the tissue in the
exchange process, and assess the dependence of
K on the test solutes.
The tissue is separated from the plasma by a semipermeable membrane. In
formulating the governing equations for the extraction process, we
consider that the infused hypertonic disturbance extracts from the
tissue a fluid with a low, but constant, osmotic pressure over the
course of the experiment. It is this hypotonic extraction that changes
the tissue osmotic pressure. Under this framework, the extraction
process "sees" the change in plasma osmotic pressure, not the
test solute producing the hypertonicity. This implication leads to the
hypothesis that the tissue fluid volume estimated from the density
reductions of the five LMW solutes should be identical. Our second
objective is to evaluate and test the validity of this hypothesis.
Hypertonic solutions have been used in the treatment of hemorrhagic
shock and pulmonary edema (24, 27). The pathophysiology of lung
diseases such as acute respiratory distress syndrome has been
associated with increased pulmonary permeability (14). Improved
understanding of the hypertonic infusion therapy and characterization
of the pulmonary permeability could facilitate the development of
treatment and prevention of these diseases.
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MATERIALS AND METHODS |
Surgical preparation.
Sixteen New Zealand White male rabbits weighing 2.91 ± 0.04 (SE) kg
were sedated with chlorpromazine (6.7 mg/kg body wt im) and
anesthetized with pentobarbital sodium (20 mg/kg) injected into the
lateral vein of the ear. After the administration of lidocaine to the
areas of incision, the trachea, right common carotid artery, left
femoral artery, and both jugular veins were isolated. A tracheostomy
was performed, and the rabbit was allowed to spontaneously ventilate.
An arterial catheter was inserted 5 cm into the carotid artery to reach
the aorta, and heparin (3,000 U/kg) was infused for anticoagulation.
The remaining isolated vessels were cannulated after a 5-min waiting
period to allow steady-state distribution of heparin.
Measurement systems.
Aortic blood was withdrawn from the arterial catheter at a rate of 30 ml/min with a Masterflex roller pump. The blood was pumped through the
U tube of a density meter (model DMA 602W, Mettler-Parr) and returned
via a catheter inserted through the left jugular vein into the vena
cava. The density meter was incorporated with an IBM personal computer
to form the density-measuring system (DMS). The density of the aortic
blood flowing through the U tube is identified as
a. Its value before a
hypertonic infusion is designated
0.
The systemic arterial and venous blood pressures were measured via the
femoral arterial and left jugular catheters using saline-filled pressure transducers (model P23, Statham). A port for infusion of
hypertonic solution or injection of isotonic saline was located between
the pump and venous catheter. Another pressure transducer was used to
monitor the pressure at the input port to indicate the time course of
injections and infusions. The tracheal tube was attached to an
air-filled disposable pressure transducer (Abbott Critical Care
Systems) to monitor ventilation.
Blood was sampled at the start of the experiment and then after every
third infusion for the determination of hematocrit (H). A second DMS
was configured to discretely measure the densities of the hypertonic
solutions, plasma, and isotonic saline.
Experimental protocol.
Typically, three infusions of each test solution were administered to
each rabbit. Each infusion constituted part of an individual run, which
was randomized for each rabbit. The first eight rabbits were infused
with only NaCl, urea, and glucose. The last eight were given all five
test solutions. At the beginning of each run, an 0.8-ml bolus of
isotonic saline was injected within 0.5 s into the jugular vein.
The transient decrease in
a was recorded for later
computation of the cardiac output (
), the
mean transit time (MTT) from the injection site to the sampling site,
the midpoint of the U tube
(tm), and the
transport function of the circulation situated between the two sites
[h(t)]
(11). After the blood density had stabilized, the hypertonic solution
was infused at a rate
inf
(3.8 ml/min, which is 2% of ) for 20 s while the blood density was measured. A typical record of the changes in blood
density and infusion pressure is depicted in Fig.
1. Approximately 15 min were allowed
between runs for the blood density to stabilize to a new steady-state
value. If the systemic arterial and central venous pressures were
altered by 
10 mmHg, then the run was neglected (~2% of all data
collected).
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Fig. 1.
A: aortic blood density
(a) change resulting from a
20-s hypertonic NaCl jugular infusion at time
0. Rapid transient density decrease indicates a quick
extraction from lung tissue. It is followed by a rapid density
increase: result of filtration of fluid to lung tissue.
tr, Appearance
time of recirculation. Pulmonary extraction is completed before
recirculation. Smooth solid curve drawn on experimental data represents
density response due to fluid extraction from peripheral organs.
B: hydrostatic pressure at infusion
site (Pinj) showing time course
of 20-s infusion.
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Hypertonic disturbance.
We prepared the hypertonic test solution by dissolving solid NaCl,
urea, glucose, sucrose, and raffinose in isotonic saline. The
osmolarities of these solutions above the isotonic saline level
(
h) were 2,100, 2,400, 2,155, 1,200, and 600 mosmol/l for NaCl, urea, glucose, sucrose, and
raffinose, respectively. With NaCl as the most abundant solute in the
plasma, the other solutes are each 2% of the plasma concentration of
NaCl (8). As a result, the small infusion rate has a negligible
dilution effect (1%) on the solutes and resident NaCl in the plasma.
The only osmotic disturbance is caused by the hypertonicity above the
isotonic level. Because of the exchanges across the RBC membrane and
the concentration dependence of the hemoglobin osmotic coefficient (25), the role of RBCs in the hypertonic disturbance could be accounted
for by excluding a 
fraction of the RBC volume from the dilution
blood volume. With these considerations, the following equation can be
derived to predict the temporal increase in the plasma osmotic pressure
due to the infused solute
[pl,inf (t)]
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(Eq. 1)
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where
max is
infh/[(1 
H)] and
u(t)
is a unit step function. The replacement of
u(t)
by
u(t) u(t 20 s) simulates the current protocol of a 20-s infusion with no
recirculation. Solomon et al. (25) and Gary-Bobo and Solomon (7)
reported that the for NaCl and glucose is ~0.54, which is used to
analyze the results of all test solutes.
Stream-tube model.
We model the flow from the injection site to the sampling site as a
bundle of parallel stream tubes (15) and gather stream tubes having an
MTT within 
and + d as a group. Let
h
() be the fractional flow distribution so that the blood flow passing through this stream-tube group is prescribed as
h
()d. In a stream tube in this group, flow dispersion is characterized by the
transport function
hs(t,).
Their interrelations with the overall flow and transport function yield
the following two identities
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(Eq. 2)
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(Eq. 3)
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In
the dispersion analysis of Hunter and Lee (10), is the subscript
i identifying the group,
h()d is i,
hs(t,)
is
hi(t),
and 
[ ]d is 
[ ].
In applying the modeling of Audi et al. (1) to the current analysis, we
consider that the dispersion on the pulmonary arterial side of all
stream tubes (from the injection site to the capillary entrance) is
represented by one single transport function,
hpa(t), and that on the pulmonary venous side (from venous exits of the capillaries to the sampling site) by
hpv(t).
With the capillary transport function as
hc(t),
the overall transport function is given by
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(Eq. 4)
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where
is the convolution operator. Here,
h(t)
is
CR(t)
of Audi et al. and
hpa(t)hpv(t)
is
Cn(t).
The first moment of
hc(t)
is tc and the
second (central) moment is 
. The
corresponding moments of
Cn(t)
are tn and
, and those of
h(t)
are tm and
. The nature of convolution yields
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(Eq. 5)
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For the stream-tube group, the MTT of blood flowing through its
capillary segments is tn. Because of
the minute lateral dimension of pulmonary capillaries, the diffusion of
indicator equalizes its concentration uniformly across the capillary
cross section and makes the transport of indicator behave as if it were a plug flow. Thus the transport function of capillary flow is the
Kronecker delta function, 
(t + tn). The
transport function of the stream-tube group is
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(Eq. 6)
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After
substitution of this expression into Eq. 3, we take the convolution out of the integration and
then use the characteristics of the delta function to derive the
following identity
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(Eq. 7)
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This
equality provides an explanation for how heterogeneous flows among
capillaries contribute to the transport function hc(t).
Because the flow systems for the in vitro rabbit experiment of Audi et
al. (1) and Hunter and Lee (11) and the current in vivo system are
different, a way to compare the dispersion under the framework of the
stream-tube model is presented. Because the transport function
h(t)
was measured for these three studies, we have
tm and
m. After subtraction of the
time delay from the injecting site to the pulmonary arteries and from
the pulmonary veins to the sampling site, an estimate of the MTT of the
pulmonary vasculature
(tlung) is
obtained. It is multiplied by the flow to yield the pulmonary blood
volume (Vlung). Because all the
experiments were done with the same animal species under physiological
conditions, we assume that the capillary blood volume fraction and the
value of
tc/c
are the same for the lungs studied. These assumptions allow us to
calculate from tm
the value of tc
and then c. Through Eq. 5, we determine
tn and
n.
Distributions of surface area and tissue volume.
Let the total surface area of the lung for exchange be
S. The more capillaries are in a
stream-tube group, the higher is its total blood flow and the larger
its total surface area available for transcapillary fluid movement.
Accordingly, we assume for later analysis that the surface area of the
stream-tube group [S*
()d] has the same fractional distribution as the flow, i.e.
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(Eq. 8)
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In
addition, we regard that the tissue thickness surrounding the capillary
and the volume fraction of the tissue participating in the extraction
process are the same for all capillaries. Let the initial tissue fluid
volume be Vti,0. With these two
descriptions of the tissue compartment and Eq. 8, the tissue volume for the stream-tube group is
Vti,0h()d.
Driving pressures.
The analysis of the hypertonic results indicates that the maximum rate
of fluid extraction is 2% of the rate of blood flow. The blood flow
along the capillary can therefore be regarded as constant. As examined
later, the permeation of test solutes across the capillary wall (i.e.,
the exchange surface) may be so slow that its role in changing the
plasma or tissue osmotic pressure of test solutes can be much smaller
than that induced by the rapid extraction. As a result, we also regard
the plasma osmotic pressure of the test solute along the capillary as
constant. Consequently, the only change in the plasma osmotic pressure
"seen" by the capillary (pl,c) is the hypertonic
input (Eq. 1) dispersed by the
arterial dispersion, which is
![= &Dgr;&pgr;<SUB>max</SUB><IT> u</IT>(<IT>t</IT>)⊗<IT>h</IT><SUB>pa</SUB>(<IT>t</IT>)⊗&dgr;[<IT>t</IT> − (&tgr; − <IT>t</IT><SUB><IT>n</IT></SUB>)/2]](/content/vol84/issue3/fulltext/769/img016.gif)
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(Eq. 9)
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where pl,inf is
change in osmotic pressure at site of infusion. One-half of the
capillary MTT is used in the delta function to time the event at the
midpoint of the capillary, as suggested by the analysis of Friend and
Lee (6).
Effros (3) observed that after the hypertonic infusion the
concentration dilution of several tracers in the blood flowing out of
the lung could be explained by the extraction from the tissue of a
hypotonic fluid having a constant concentration of LMW solutes.
Together with the earlier omission of solute permeation, we consider
the extraction of a hypotonic fluid of LMW solutes at a constant
osmotic pressure (f) as the
only factor effecting an increase in the tissue osmotic pressure
[ti(t,)].
Let the initial osmotic pressure in the tissue fluid compartment be
ti,0. As prescribed earlier,
the initial tissue fluid volume surrounding the capillaries of the
stream-tube group is
Vti,0h()d. Suppose this volume experiences the following change:
Vti(t,)h()d. To simplify the formulation further, we consider that the fluid in the
tissue compartment and the hypotonic fluid contain only one LMW solute.
As reported by Hunter and Lee (10), the mass balance of this solute for
the tissue compartment yields the following relation between the change
in tissue fluid volume and the change in tissue osmotic pressure
[ti(t,)]
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(Eq. 10)
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Before the infusion, a remained
at a constant level, indicating a zero filtration rate, i.e.,
Js(0,) = 0. This initial condition corresponds to no change in the tissue fluid
volume [Vti(0,) = 0] and the balance of all Starling driving pressures before the
hypertonic infusion. Although hypertonic infusion can impede the flow
of RBCs and alter the pulmonary vascular flow resistance (3, 23), the
in vitro study of Hunter and Lee (11) indicates no impediment of RBC
flow for the low hypertonic disturbance used in the present protocol.
Accordingly, we regard no change in the hydrostatic pressure in the
capillary during the course of hypertonic infusion. On the tissue side,
the tissue hydrostatic pressure may not be affected by fluid extraction
because of a large tissue compliance (10). Accordingly, we conclude that Eqs. 6 and 10 describe the only two driving
pressures, induced by the hypertonic infusion, to produce an extraction
across the capillary wall.
Tissue volume change.
Suppose the filtration coefficient (including the total surface area)
is K. As specified earlier,
Sh()d is the surface area of the stream-tube group. Thus its filtration coefficient is
Kh()d.
Let the rate of filtration through the capillary wall be
Js(t,)hd. With these two definitions and the conclusion in
Driving presssures, the simplification
of the generalized Starling equation (Eq. 7 in Ref. 19) to
the current infusion protocol leads to the following rate of
filtration, which is the first derivative of the change in the tissue
fluid volume
[Vti(t,)]
![<IT>J</IT><SUB>s</SUB>(<IT>t</IT>,&tgr;) = <FR><NU>d&Dgr;V<SUB>ti</SUB></NU><DE>d<IT>t</IT></DE></FR> = −<IT>K</IT>[&sfgr;&Dgr;&pgr;<SUB>pl,c</SUB>(<IT>t</IT>,&tgr;) − &sfgr;<SUB>f</SUB>&Dgr;&pgr;<SUB>ti</SUB>(<IT>t</IT>,&tgr;)]](/content/vol84/issue3/fulltext/769/img018.gif)
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(Eq. 11)
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where
is the reflection coefficient of the capillary wall for the test
solute and f is the reflection
coefficient of the abundant solute in the tissue. We use
d/dt to represent the time derivative,
as is constant for the stream-tube group. Substituting Eqs. 9 and 10 into Eq. 11 and reorganizing Eq. 11, we find that the tissue fluid volume change is
governed by the following differential equation
![= − &sfgr; <IT>K</IT>&Dgr;&pgr;<SUB>max</SUB> <IT>u</IT>(<IT>t</IT>)⊗<IT> h</IT><SUB>pa</SUB>(<IT>t</IT>)⊗&dgr;[<IT>t</IT> − (&tgr; − <IT>t</IT><SUB><IT>n</IT></SUB>)/2]](/content/vol84/issue3/fulltext/769/img020.gif)
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(Eq. 12)
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where
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(Eq. 13)
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This
expression for 
, being independent of and test solute and having
the unit of seconds1,
allows us to regard it as a rate constant universal among all stream-tube groups and for all test solutes.
Density change.
Solving Eq. 12 for a zero initial
volume change and then finding
Js through the
equality in Eq. 11, we have
![<IT>J</IT><SUB>s</SUB>(<IT>t</IT>,&tgr;) = − <FR><NU>&sfgr;<IT>K</IT>&Dgr;&pgr;<SUB>max</SUB></NU><DE>&agr;</DE></FR> <IT>h</IT><SUB>pa</SUB>(<IT>t</IT>)⊗&dgr;[<IT>t</IT> − (&tgr; − <IT>t</IT><SUB><IT>n</IT></SUB>)/2]⊗<IT>g</IT>(<IT>t</IT>)](/content/vol84/issue3/fulltext/769/img022.gif)
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(Eq. 14)
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where
g(t) = exp(t)u(t).
For the capillaries in the stream-tube group, the flow rate is
h()d and the density of blood is 0.
The extraction adds to the flow at the rate
Js(t,)h()d and with the density f. The
mixing of these two fluid flows and the omission of higher-order terms
produce the following density change
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(Eq. 15)
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With
the dispersion from the capillary to the sampling site as
[t ( tn)/2] hpv(t),
we have the density change exiting the stream tube at the sampling site
[s(t,)]
as
![&Dgr;&rgr;<SUB>s</SUB>(<IT>t</IT>,&tgr;) = &Dgr;&rgr;<SUB>c</SUB>(<IT>t</IT>,&tgr;)⊗&dgr;[<IT>t</IT> − (&tgr; − <IT>t</IT><SUB><IT>n</IT></SUB>)/2]⊗<IT>h</IT><SUB>pv</SUB>(<IT>t</IT>)](/content/vol84/issue3/fulltext/769/img024.gif)
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(Eq. 16)
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With
the flow of the stream-tube group as
h()d,
the application of mass conservation principle to the mixing of flows
with variable density changes (10) yields the following density change
in the sampling site
a(t)
which is also the aortic blood density change
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(Eq. 17)
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With
the help of Eqs. 3 and 6, the substitution of
Eqs. 14-16 into
Eq. 17 yields
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(Eq. 18)
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The
equation deduced for hypertonic NaCl infusion by Hunter and Lee (10) is
a special case of Eq. 18 once is
identified as of NaCl. Their derivation requires no assumption on
having the input transport functions to capillaries prescribed by
hpa(t).
Computation of K, , and
Vti,0.
The computer program developed by Hunter and Lee (10) was used for the
computation. First, the transient density decrease from the isotonic
injection is revised by subtracting the density oscillation extended
from the ventilatory density oscillation before the injection. The
revised decrease (Fig.
2A, dashed
line) is used to determine the area
(A), , and MTT
tm. Along with the second and third moments, a lagged normal density function is
constructed as
h(t).
The solid line in Fig. 2A is
Ah(t)
so calculated. Because the infusate density
(inf) may be different from the blood density
(0), the infusion imposes on
a an infusion "artifact" described by
(inf 0)(inf/)[u(t) u(t 20 s)] h(t). The program subtracts from
a (Fig. 1A) the ventilatory oscillation and infusion artifact. The revised density change subsided
within 15 s after the infusion. We integrate the change over this
period to find an area A'
and a first moment
t'm.
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Fig. 2.
A: reduction in a due
to injection of isotonic saline (dashed line). Ventilatory oscillations
have been removed. Solid line, fit generated from a lagged normal
density function. B: transport function (dashed line) for a stream tube with same mean transit time
(MTT) as fit shown in A (solid line).
C: distribution of flow among
stream-tube groups as a function of MTT of stream tube. h(t),
Transport function of circulation between injection and sampling site;
A, area; subscripts s,
, and c: stream tube, cardiac output, and
capillary.
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The values of K, , and
Vti,0 are calculated by setting
the area of the predicted density as
A' and the first moment as
t'm. The first
moment of
g(t)
is 1/, and the first moment of
h(t) is tm. Their sum
is the first moment of
g(t)
h(t),
which is also t'm.
Solving for , we have
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(Eq. 19)
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The
area of the function
g(t)
h(t)
in Eq. 18 is unity. Therefore,
K is
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(Eq. 20)
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We
obtain from Eq. 13 this relation to
calculate Vti,0
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(Eq. 21)
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In the calculations, ti,0 is
taken as 300 mosmol/l and f as
40 mosmol/l. The latter is selected from Effros' assessment that the
sodium ion concentration in the hypotonic fluid extracted is 16 ± 7% of the plasma concentration (3). This low osmotic pressure is also
consistent with Watson's observation (30). The density of this
hypotonic fluid (f) is 994 g/l. The value of
fK
estimated from the hypertonic NaCl infusion is used in Eq. 21 to calculate
Vti,0 for all test solutes.
Once K and are calculated,
Eq. 18 is used to calculate the
expected density reduction due to the hypertonic infusion. The program
also calculates the transient density increase of the filtration phase
after the cessation of infusion with no consideration of the
recirculation of hypertonic disturbance. The density change so
calculated for the extraction and filtration phases is depicted as the
solid line in Fig. 3A.
The normalized sum of the squared differences between the calculated
and the measured density change over a period of 15 s is taken as the
coefficient of variation, an index of how well the prediction of
Eq. 18 matches the measured density
change.
Rate of fluid extraction.
The rate of extraction for each stream-tube group is
Js(t,)h()d.
Its sum over all groups is the total rate of fluid extracted for the
entire lung
![⋅ <LIM><OP>∫</OP><LL>0</LL><UL>∞</UL></LIM> <IT>h</IT><SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>(&tgr;){<IT>h</IT><SUB>pa</SUB>(<IT>t</IT>) ⊗ &dgr;[ <IT>t</IT> − (&tgr; − <IT>t</IT><SUB><IT>n</IT></SUB>)/2]} ⊗ <IT>g</IT>(<IT>t</IT>)d&tgr;](/content/vol84/issue3/fulltext/769/img031.gif)
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(Eq. 22)
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The factoring out of the convolution and the use of the
identity 2(2t) = (t) reduce Eq. 22 to
![<IT>J</IT><SUB>f</SUB> (<IT>t</IT>) = − <FENCE><FR><NU>&sfgr;<IT>K</IT>&Dgr;&pgr;<SUB>max</SUB></NU><DE>&agr;</DE></FR></FENCE> [<IT>h</IT><SUB>pa</SUB>(<IT>t</IT>) ⊗ 2<IT>h</IT><SUB>c</SUB>(2<IT>t</IT>)] ⊗ <IT>g</IT>(<IT>t</IT>)](/content/vol84/issue3/fulltext/769/img032.gif)
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(Eq. 23)
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The
integration of
Jf (t)
from 0 to 
is the maximum reduction in tissue volume. Let it be
Vti,max. Because of the
integration of
hpa(t),
2hc(2t)
and
g(t)
is all unity, we have
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(Eq. 24)
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Let the MTT of
hpa(t)
be tpa. The MTT
of the transport function in brackets in Eq. 23 is
tpa + tc /2. In
reference to the overall MTT
tm, the partition
fraction q is 1 (tpa + tc /2)/tm. The procedure to partition the overall transport function
h(t) yields an arterial transport function
ha(t),
with an MTT, ta, equal to (1 q)tm (10). For
the particular case that q = 0.5 and
hpa(t)
equals
hpv(t),
we find that
ha(t)
has a second and a third central moment slightly larger than that
within brackets in Eq. 23. Without
precise measurement of
hpa(t)
and
hpv(t),
it may be appropriate to approximate the function within brackets as
ha(t)
for computing
Jf (t).
Statistics.
Values are means ± SE. For evaluation of statistical difference, a
two-way, randomized incomplete block analysis of variance is applied.
Differences are considered significant if
P < 0.05 (analysis of variance).
When significance is determined by analysis of variance, a Newman-Keuls
multiple comparison test is used to identify the differences among the
means.
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Fig. 3.
A: revised a change
from Fig. 1 (dashed line) with model prediction of pulmonary extraction
and filtration superimposed (solid line).
B: theoretical prediction of change in
aortic plasma osmolarity (a). Its difference from a
step increase at time 0 and a step decrease at 20 s reflects effect of flow dispersion.
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RESULTS |
Table 1 lists the body weight, the lung blood volume,
the pulmonary blood flow, and the six moments in Eq. 5 for the in vitro experiments of Audi et al. (1) and
Hunter and Lee (11) and for the current in vivo experiments.
and
are indexes describing the
dispersion (or spread) of
hc and
h, respectively. We term the ratio
/ as the fractional dispersion in the pulmonary capillary flow. For the
perfusion system of Audi et al., the tabulated value for the ratio
indicates that the heterogeneity in capillaries contributes to a
fractional dispersion of 79%. Because the catheter delay formed a
larger fraction of
tm for the in
vitro lung system of Hunter and Lee, the system's capillary
heterogeneity makes up a smaller fractional dispersion (61%). The
tabulated value of /
for the current in vivo experiments indicates that the capillary
heterogeneity makes up only 30% of the total dispersion. This smallest
fractional dispersion results from the presence of the heart chambers
in the in vivo system, the dispersion induced by the mixing action of
the chambers, and the smallest fraction of
tc in
tm.
Taking as 70% of
(Table 1), we calculate
hs(t)
of a stream tube having the same MTT as the overall transport function
h(t).
Both are depicted in Fig. 2B for
comparison. The transport function for the capillaries ( being 30% of
) is depicted in Fig.
2C, with the time axis shifted by
tn. As
demonstrated earlier, this function also describes the fractional flow
distribution as a function of the MTT of the stream tubes.
A typical density response to a 20-s infusion of hypertonic saline into
the rabbit's vena cava is shown in Fig.
1A. The step change in plasma
osmotic pressure at the infusion site is depicted in Fig.
1B, with
max = 28.7 mosmol/l. In the
isotonic injection experiment done 2 min earlier,
was 435 ml/min and MTT was 4.7 s. The revised density, the
measured density with the effects of ventilation and infusion artifact
removed, is illustrated in Fig. 3A.
Because the hypertonic saline has a density of 1,041.48 g/l and the
average blood density is 1,041.67 g/l, the effect of density difference
is negligible in this particular case. The K calculated for this example is
18.2 ml · h1 · mosmol1 · l,
the rate constant is 0.44 s1, the tissue volume
participating in the extraction process
(Vti,0) is 2.7 ml, and the
volume extracted (Vti,max) is
0.3 ml.
The solid curve superimposed on the revised density change in Fig.
3A is the theoretical prediction of
Eq. 18 expanded to cover the
extraction and filtration induced by the 20-s infusion under the
assumption of no recirculation. The coefficient of variation for the
record over the first 12 s is 0.07, indicating a good fit. Figure
3B shows the variation in
hypertonicity in the aortic blood passing through the DMS calculated as
max[u(t) u(t 20 s)]h(t).
The density response to an infusion of hypertonic saline in the in
vitro experiment of Hunter and Lee (11) clearly shows that the
extraction transient is followed by a filtration transient. In addition
to these two transient changes, the in vivo results show a third
density change, a slow decrease over the latter one-half of the
response (Fig. 1A). Also shown in
Fig. 1A is a recirculation time
(tr). This time
is determined from the injection of isotonic saline earlier in the run.
The coincidence of this time with the initiation of the third density
change indicates that the change results from fluid extraction in
peripheral organs after the passage of the hypertonic disturbance. For
reference, we identify the third density phase as peripheral exchange.
The extended density record indicates that the maximum decrease in
density is reached at 50 s. The density fully returns to the
preinfusion level within the next 10-15 min.
As shown later, the density responses of the four other solutes, after
revision for the infusion artifact, exhibit these three features:
pulmonary extraction, pulmonary filtration, and peripheral exchange.
Because there is no peripheral exchange over the phase of pulmonary
extraction, it is analyzed by the program to obtain the
K and tissue fluid volume
participating in the extraction.
To better understand the dynamics of extraction and filtration leading
to the observed density variation shown in Fig.
3A, we apply the partition analysis of
Hunter and Lee (10) to the stream-tube transport function shown in Fig.
2B to calculate the expected change in
capillary osmotic pressure, the tissue osmotic pressure, the rate of
fluid extraction, and the a exiting the stream tube.
Their dynamics are depicted in Fig. 4. For
this example, the maximum rate of fluid extraction is ~1% of the
blood flow through the stream tube. The extraction ceases as the
osmotic pressure of NaCl, the most abundant solute in tissue fluid,
increases to a level in balance with the elevated plasma osmotic
pressure of the test solute. In the case in which the abundant solute
is different from the test solute, their concentrations will be
equilibrated by permeation. The minimal change in
a after the cessation of fluid
extraction indicates that the permeations may not alter the net osmotic
pressure difference between the tissue and plasma to produce a fluid
filtration capable of changing the density of blood leaving the
capillary.
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Fig. 4.
Filtration dynamics for stream tube characterized by Fig. 2.
A: change in capillary plasma
osmolarity (c) and tissue
fluid osmolarity (ti)
expected with a 28.8 mosmol/l, 20-s hypertonic infusion for stream tube
with transport function in Fig. 2B.
B: transcapillary fluid flux
(Jf).
C: predicted change in
a.
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Figure 5A
shows a typical density response to a 20-s infusion of hypertonic urea.
The hypertonic disturbance
(max) is 33.6 mosmol/l.
is 423 ml/min, and
tm is 4.8 s. The
dashed curve depicts the infusion artifact due to the density
difference between the infusate and the blood. The density response
shown in Fig. 5B is revised for the
effects of ventilation and the infusion artifact. In
this example for urea, is 0.47 s1,
K is 16 ml · h1 · mosmol1 · l,
Vti,0 is 2.3 ml, and the
coefficient of variation is 0.08.
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Fig. 5.
Density change in response to a 20-s infusion of hypertonic urea
[maximum elevation in osmotic pressure during infusion
(max) = 30 mosmol/l]. A: aortic density change
(solid line) and expected density change due only to density difference
between blood (1,041.73 g/l) and hypertonic solution (1,033.33 g/l,
dashed line). B: revised density with
model prediction superimposed. Extraction parameters are given in
RESULTS.
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The density response to a 20-s infusion of hypertonic glucose is
illustrated in Fig.
6A. The
hypertonic infusion caused a max of 24.5 mosmol/l.
is 328 ml/min, and
tm is 5.7 s. The hypertonic glucose has a density of 1,105 g/l, which causes the blood
density to increase by 0.5 g/l and masks the extraction and filtration
components of the density response (solid line) by the infusion
artifact (dashed line). The revised density change (Fig.
6B) reveals distinctly the pulmonary
extraction and filtration. For this glucose result,
K is 24.5 ml · h1 · mosmol1 · l,
Vti,0 is 3.4 ml, is 0.30 s1, and the coefficient of
variation is 0.14.
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Fig. 6.
Density response to a 20-s infusion of hypertonic glucose
(max = 30 mosmol/l).
A: aortic density change (solid line)
and expected density change due only to density difference between blood (1,039.47 g/l) and hypertonic solution (1,108.60 g/l, dashed line). B: revised density with model
prediction superimposed. Extraction parameters are given in
RESULTS.
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Figures 7 and
8 show typical density responses to
hypertonic solutions of sucrose and raffinose, respectively. Similar to the response to the hypertonic glucose solutions, once the overwhelming infusion artifact is removed, the features of pulmonary extraction and
then filtration are indicated by Figs.
7B and
8B. For sucrose, max is 20.4 mosmol/l,
is 272 ml/min,
tm is 5.4 s,
K is 18.1 ml · h1 · mosmol1 · l,
Vti,0 is 3.6 ml, is 0.33 s1, and the coefficient of
variation is 0.15. For raffinose,
max is 7 mosmol/l,
is 375 ml/min,
tm is 4.7 s,
K is 27 ml · h1 · mosmol1 · l,
Vti,0 is 5.4 ml, is 0.33 s1, and the coefficient of
variation is 0.10.
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Fig. 7.
Density response to a 20-s infusion of hypertonic sucrose
(max = 20.4 mosmol/l).
A: aortic density change (solid line)
and expected density change due only to density difference between blood (1,040.97 g/l) and hypertonic solution (1,108.64 g/l, dashed line). B: revised density with model
prediction superimposed. Extraction parameters are given in
RESULTS.
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Fig. 8.
Density response to a 20-s infusion of hypertonic raffinose
(max = 8.8 mosmol/l).
A: aortic density change (solid line) and expected density change due only to density difference between blood (1,039.47 g/l) and hypertonic solution (1,085.64 g/l, dashed line). B: revised density with model
prediction superimposed. Extraction parameters are given in
RESULTS.
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The 16 rabbits used in this study weighed 2.91 ± 0.04 kg. The
aortic hematocrit of the rabbits was 36.2 ± 0.3%. The average measured in this study was 386 ± 7 ml/min. The
MTT from the infusion site to the sampling site was 5.02 ± 0.06 s.
Blood density averaged 1,040.07 ± 0.26 g/l. Arterial pressure
averaged 71.5 ± 0.9 mmHg, and the venous pressure was 2.0 ± 0.3 mmHg. Plasma osmolarity was measured in 10 rabbits before the
hypertonic infusion every third run over the course of the experiment.
The results indicate that the plasma osmolarity returned to baseline
conditions during the 15-min recovery period allowed between each
infusion. The plasma density after the 15-min recovery period also
returned to its baseline value. The
inf,
max,
A',
t'm, ,
coefficient of variation,
Vti,max,
Vti,max/max,
K, and Vti,0 for the five solutes are
tabulated in Table 2. The last four values
have been normalized by the wet lung weight estimated from an empirical
factor of 4 g wet lung/kg body wt (10).
Statistically, our results show no significant difference among the
values of and among Vti,0,
indicating their independence of the molecular size of the test solutes
(Fig. 9). This independence is consistent
with the theoretical consideration used to formulate Eq. 12, which specifies as a rate
constant characterizing the extraction of a hypotonic fluid containing
the LMW solutes in the tissue fluid. The relationship between
K and the molecular size of the
test solute is shown in Fig. 10. The
positive slope of the relationship indicates that more fluid, for the
same osmotic disturbance, is extracted with larger solutes.
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Fig. 9.
A: extraction rate constant ()
obtained from each test solute ( , from left to
right: NaCl, urea, glucose, sucrose, and raffinose) as a
function of solute radius. B: initial
tissue fluid volume (Vti,0)
calculated from density response of each solute (symbols same as in
A). These results demonstrate that and
Vti,0 are independent of solute
causing extraction, which is consistent with hypothesis in text.
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Fig. 10.
Osmotic conductance (K) as a
function of molecular radius for NaCl, urea, glucose, sucrose, and
raffinose (, left to right).
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DISCUSSION |
Density change during the extraction phase.
The density of the infused hypertonic solution is different from the
density of blood. This infusion artifact has been corrected from the
measured change in a. A 30 mosmol/l increase in plasma NaCl osmotic pressure causes fluid
extraction from the RBCs and reduces the RBC volume by 6% (25). This
redistribution of fluid between RBCs and plasma reduces the hematocrit
by 2% but has no effect on the blood density. Because the transit of
plasma through the pulmonary vasculature is only slightly longer than
the transit of RBCs, the small fluid redistribution contributes
minimally to the observed change in
a.
Hypertonic infusions can impede RBC flow through the pulmonary
vasculature (3). Suppose the impediment occurs upstream of pulmonary
capillaries. The application of current dispersion analysis to the
infusion protocol of Effros (3) indicates that the RBCs are exposed to
an osmotic pressure ranging from 360 mosmol/l (at the infusion site) to
160 mosmol/l (at the microvascular extraction site). Because RBC
density is higher than plasma density, the impediment can lead to a
transient decrease in the density of blood flowing out of the lung. The
impediment could significantly increase the pulmonary flow resistance
(22, 23). If the impediment were accompanied by an
8-cmH2O decrease in the capillary
pressure, the resulting reduction in capillary blood volume can produce a transient density decrease similar in magnitude to the density reduction observed over the extraction phase (17). Estimated from the
results of Read et al. (22), a <75 mosmol/l osmotic disturbance at
the site of infusion produces no change in the pulmonary flow
resistance. When hypertonic saline was infused to produce an osmotic
disturbance of 40 mosmol/l to the isolated rabbit lung, similar
transient density changes (revised for infusion density difference)
were observed by Hunter and Lee (11) for the rabbit lung perfused with
blood or autologous plasma. The results of Read et al. and Hunter and
Lee indicate that the density changes observed in current in vivo
hypertonic experiments are not produced by flow impediment of RBCs or
change in capillary blood volume.
Transcapillary fluid movement.
Accordingly, we conclude that the transient density decrease observed
over the first 15 s of hypertonic infusion as shown in Figs. 1, 3, and
5-8 is the result of extracting from the pulmonary tissue a fluid
with a density lower than the blood density. The return of the blood
density to its preinfusion level indicates the cessation of fluid
extraction. Our analysis indicates that the cessation results from the
increase in the tissue osmotic pressure to a level in balance with the
constant elevation in plasma osmotic pressure.
For the small total volume of hypertonic solution used in this
experiment, the blood density returned to the preinfusion level within
0.5 h. Measured in 10 rabbits, the plasma osmolarity also returned to
the preinfusion level. It could be calculated that the mixing of the
infused hypertonic saline (1.3 ml) with 200 ml of blood [the
blood volume of the rabbit (16)] at a hematocrit of 36% would
elevate the plasma osmotic pressure by 17 mosmol/l {1.3 ml × 2,100 mosmol/l / 200 ml / [1 (0.54 × 0.36)]}. If the hypertonicity in plasma is
equilibrated with all tissue fluid, e.g., at 10 times the blood volume,
then the elevation in osmotic pressure is 1.7 mosmol/l. The recovery in
plasma osmolarity after 15-30 min is consistent with the theory
that the infused solute has fully permeated the entire tissue fluid
space of the rabbit.
Response time of fluid extraction.
Suppose a step hypertonic disturbance is presented to the capillary.
Then the extraction process is governed by Eq. 12 with the arterial dispersion term replaced by a step
function. The new solution of the modified Eq. 12 is proportional to
u(t)exp(t), which indicates that the change in tissue fluid volume reaches 63% (1 1/e) in a time span of 1/
(~3 s for the current experiment).
When the effect of flow dispersion through the pulmonary vasculature is
taken into consideration, the time between the initiation of tissue
volume change and 50% of the value, estimated from Fig. 3, is 2.4 s.
The time to reach 95% of the change is 8.6 s. All these characteristic
times for fluid extraction are shorter than the half time
of 18 s found for the equilibrium of sucrose tracer between the tissue
and plasma (29). This supports the conclusion that the permeation of
test solute into the tissue changes the tissue osmotic pressure at a
pace much slower than that due to the extraction of hypotonic fluid
from the tissue. The compliance reported for the pulmonary interstitium
has been considered in the extraction analysis of Hunter and Lee (10).
They estimated that the rate constant is increased by 0.7% of the
value calculated from K. This small
percentage indicates that the change in the tissue hydrostatic
pressure, negligible in the 1st min, may contribute to the long-term
equilibration between the vascular and the tissue compartment.
After hypertonic bolus injections, the dilution of a plasma protein
indicator and the dilution of osmotic disturbance at the sampling site
were measured (3). Regarding these dilutions as those exiting the
"extraction" site and analyzing them through an extraction model
with a negligible role for the tissue, Effros (3) estimated that
K of urea, NaCl, and sucrose in the
dog lung is 0.55, 0.78, and 0.66 ml · h1 · mosmol1 · l · g1,
respectively. Physically, the two dilutions at the extraction site are
dispersed by the dispersive blood flow from the extraction to the
sampling site. However, because the dilutions are not identical, K estimated from the aortic blood
would differ slightly from that of the capillaries. The
K reported by Effros (Fig. 2 in Ref. 3) indicates a reduction to one-half of the first estimated value
in 2.7 s. This time is comparable to the response time on fluid
extraction (1/), implying that the role of tissue osmotic pressure
may not be negligible and extrapolation of
K to the appearance time of
dilution is needed to remove the tissue factor. Hunter and Lee (10)
estimated that the tissue and dispersion factor may make the
"actual" K ~2.27 times the
value reported by Effros. The inclusion of this underestimate factor
makes the K of the dog lung
obtained by the bolus injection method comparable to the
K of the rabbit lung presented in
Table 2.
The time course of hypertonic infusion for the first 20 s is similar to
that used in the gravimetric experiments of Taylor and Gaar (26) and
Wangensteen et al. (28). However, the weight changes they measured
exhibit a much slower time course than the currently observed density
change. The extraction of fluid, before it leaves the lung, represents
only a redistribution of fluid from the tissue to the blood and should
have no effect on the total lung weight. In other words, the weight
measurement may not fully assess the rapid fluid extraction. For the
experiments of isolated dog lobes done by Taylor and Gaar, the
identification of the rapid extraction can be further complicated by
the fact that the weight change was made relatively later and at 6-s
intervals. Wangensteen et al. indicate an elevation in the perfusion
pressure (Fig. 3 in Ref. 28). The weight change associated with this elevation may obscure the weight change related to the rapid
extraction.
Our finding of a rapid balance between the plasma and tissue osmotic
pressure is expected to occur in these gravimetric experiments. Consequently, the weight changes monitored in the gravimetric experiments over a longer time period may reflect events that are slower than the increase in tissue osmotic pressure to balance the increase in plasma osmotic pressure. The slower events could be
related to 1) the permeation of test
solute (on the plasma side) and the abundant solute (on the tissue
side) across the permeable capillary wall,
2) the change in tissue hydrostatic pressure for a compliant tissue, and
3) the transcapillary fluid movement
through less permeable pathways. It is likely that the net driving
pressure producing the extraction and, hence, the weight change for
these slower events may be smaller than that introduced by
hypertonic infusion, e.g.,
max. Then the
K calculated by dividing the slope
of weight change by max may
be significantly smaller than the actual
K. This may be one reason that the
K values for urea and sucrose
(0.057 and 0.14 ml · hr1 · mosmol1 · l · g1,
respectively) as reported by Taylor and Gaar (26) are much smaller than
the estimates obtained from dilution measurements of Effros (3) and our
dilution experiments.
Permeable capillary wall.
A three-pathway model as sketched in Fig.
11 is proposed to interpret the current
observation. It is a refinement of the exchange model proposed by
Effros (3). The first pathway is formed by the cellular membrane of the
endothelium facing the plasma or the interstitium. This membrane may be
impermeable to major intracellular ions, and their values are close
to unity. During the passage of the hypertonic disturbance through the
capillary, fluid with extremely low solute concentration is extracted
from the intracellular fluid of the endothelium. The increase in
intracellular osmotic pressure can also extract fluid from the
interstitium. The endothelium is dehydrated as more fluid is extracted
into the plasma than that coming from the interstitium. The second
pathway directly connects the interstitium with the plasma. This
pathway may have a dependent on the sizes of LMW solute. To
accommodate the likely permeation of proteins, a third pathway is
hypothesized to play no role in current hypertonic experiments.
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Fig. 11.
Conceptual model of 3 pathways for transcapillary fluid movement of
lung. First pathway is impermeable to low-molecular-weight solutes,
whereas second pathway is permeable to NaCl and similar low-molecular-weight solutes. Four arrows indicate fluid movements between various components of alveolar septa and plasma induced by
plasma hypertonic disturbance. In third pathway, which is permeable to
proteins, there is no fluid movement produced by current hypertonic protocol.
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Aquaporins have been found on capillary endothelial cells, type I and
II pneumocytes, and airway epithelial cells to act as integral membrane
pores that are osmotically driven and water selective (13). Folkesson
et al. (5) used a stop-flow, light-scattering technique to investigate
the dynamics of osmotic shrinkage in suspended rat type II pneumocytes.
Under control conditions, they observed a rapid transient volume
decrease that was completed within 2 s and was followed by a slow
constant volume decrease. HgCl2
completely and reversibly abolishes the rapid transient without
affecting the slow volume decrease. The short timescale suggests that
aquaporins could act as the first pathway for the rapid extraction of
water with no permeation of other molecules after the hypertonic
infusion.
The fluid extracted through the second pathway may contain considerable
LMW solutes. A low hypotonicity for the fluid passing through the first
and second pathways implies that the fluid through the second pathway
is diluted significantly by the no-solute fluid extracted through the
first pathway. Because both pathways face the same plasma osmotic
pressure, this dilution is equivalent to regarding the
K of the first pathway,
1K1,
as being larger than that of the second pathway,
2K2.
The overall for two pathways connecting two well-mixed compartments
is
(1K1 + 2K2)/(K1 + K2). If
K1 is smaller
than K2, then the
overall for the two pathways can be smaller than unity. Thus the
dependence of 2 on molecular
sizes of LMW solutes may produce the dependence of K shown in Fig. 10. Fluid movement
stops when the intracellular osmotic pressure (primarily from
potassium, magnesium, and phosphate ions) and the interstitial osmotic
pressure (primarily from sodium and chloride ions) are in balance with
the elevated plasma osmotic pressure.
Tissue and blood compartment.
We find that the Vti,0
participating in the extraction process is ~28% of the wet lung
volume. If we estimate the parenchymal tissue volume to be about
one-half of the wet lung tissue volume (2, 31), we find that 56% of
alveolar septal tissue is composed of the tissue fluid participating in
the fluid exchange. For the lung of rats and dogs, the fractional
volumes of endothelium, interstitium, and epithelium in the alveolar
septa have been estimated to be 23, 46, and 28%, respectively (2).
When RBCs are exposed to higher NaCl osmotic pressure, only 54% of the
RBC volume participates in the process of fluid extraction (25). If the
cells in the alveolar septa have similar participation characteristics,
then a fractional value of 56% estimated from the hypertonic results may imply the possible participation of the entire septa in the fluid
exchange process.
When RBCs are exposed to high osmotic pressure in urea, equilibrium is
reached by the permeation of urea into the cell to increase its urea
osmotic pressure, and not by the extraction of hypotonic solution from
the cells. Hypotonic extraction is the mechanism that increases the
intracellular osmotic pressure of the RBCs to balance the increase in
the plasma NaCl osmotic pressure. In either case, Eq. 1 describes the overall change of osmotic pressure for
the blood. If the urea could permeate into the endothelial cells as
into RBCs, no fluid is extracted from the endothelium to produce a
density reduction. The permeation of plasma urea into the endothelial
cells is too small to change the density of blood. The observation of
similar density changes (revised for the infusion artifact) for
hypertonic solution in urea and NaCl indicates that both solutes act in
a similar way to extract fluid from the tissue.
The permeation of indicator into the tissue exhibits an exponential
decay in the indicator concentration along the capillary. However, this
permeation process of indicator differs from the process of fluid
extraction. First, the fluid extraction, as examined previously, is
completed before any appreciable permeation of test solute into the
tissue. Second, the presence of a concentration difference in the test
solute and NaCl may still cause them to permeate across the capillary
wall in the absence of a net fluid flux due to osmotic equilibrium. In
addition, the extraction of hypotonic solution from the tissue, which
is two orders of magnitude smaller than the capillary blood flow, is
too small to dilute the hypertonic disturbance in the plasma. All these
factors indicate that it is appropriate to approximate the blood
compartment as one with a uniform solute concentration. The thickness
of the pulmonary tissue compartment around the pulmonary capillaries may at most be a few micrometers. The diffusion of LMW solutes over
such a small thickness will rapidly equalize its concentration in the
radial direction. With a constant K
along the capillary wall and a constant concentration in the blood
compartment, no axial variation in the concentration can be produced
for the tissue. Thus we can also consider the pulmonary tissue
space as one compartment.
Flow dispersion and heterogeneity.
hc(t)dt
is the fraction of capillary blood with transit time across the
capillary between t and
t + dt. The transit time varies among
stream-tube groups because of heterogeneities in capillary flow and
volume. As indicated by Eq. 3, the
transport functions of the capillary are weighted by the flows to
obtain the transport function of all capillaries. This weighting makes
h(t + tn)
identical to
hc(t),
except
h(t + tn) refers
to the MTT of the entire stream tube, whereas
hc(t) refers to the transit time over the capillary portion
(Eq. 7).
The vascular network other than the capillaries has only 21% of the
total dispersion of the in vitro system of Audi et al. (1). If the
current hypertonic disturbance were applied to their in vitro system,
our analysis predicts that the plasma osmotic pressure at the capillary
would rise somewhat like a step increase to the elevated state instead
of the slower in vivo response shown in Fig.
4A. The fluid extraction and, hence,
the density decrease at the capillary site or the sampling site would
have a smaller spread than the in vivo result shown in Fig. 4. However,
the heterogeneity in capillary flows causes the density decrease to
arrive at the sampling site at different times. By summing the
stream-tube responses with a wide distribution in the arrival time for
the hypothetical in vitro experiment, the density change of pulmonary
venous blood becomes much wider than that of an individual stream tube.
Because the second moments, , of the
in vivo and in vitro experiments happen to be similar,
Eq. 18 indicates that the in vitro
density change in pulmonary venous blood would have dynamics similar to
the in vivo change in a.
However, because of a slower in vitro flow rate, the in vitro peak
density change due to extraction would be larger than that in vivo.
The previous considerations suggest the lack of a method capable of
directly assessing the extraction of fluid from the tissue at the
capillary level. In addition, the weight measurement may contain
responses other than transcapillary fluid movement (9). Consequently,
the incorporation of dispersion in an exchange model to analyze the
dilution of tracer or density of the blood flowing from the lung may be
the only viable option for assessing the transcapillary fluid
extraction induced by hypertonic disturbance.
The current model deduces that the tissue response to the extraction is
characterized by the function
g(t).
Its convolution with the arterial transport function of each stream
tube yields the filtration flux for that stream tube and, hence, its
density change (Eq. 15). These are
further dispersed by the pulmonary venous flow. Because of the additive
property of the convolution, the change in
a, the summation of the density
changes weighted by stream-tube flows, is proportional to the
convolution of
g(t) and the overall transport function (Eq. 18). Although flow heterogeneity and dispersion will
make the density change for each stream tube different, our analysis
indicates the same overall density change if the overall transport
function is not altered. This conclusion may be compatible with the
result of linear extraction of multiple indicator dilution. For the
case of insignificant buildup of tissue indicator concentration, the
concentration output of a permeable substance with an extraction ratio
of E1 is (1 E1) times the concentration of
the nonpermeable indicator (18). (In the case of nonlinear transport,
the ratio of the two concentrations becomes a function of time.) The
dispersion and flow heterogeneity will alter the temporal dependence of
the concentration of the nonpermeable indicator. Because of the
proportionality, the overall concentration of the permeable indicator
remains (1 E1) times the
concentration of the nonpermeable indicator. Then the extraction ratio
calculated from these two concentrations is still
E1, indicating that internal redistributions of flow heterogeneity have no effect on the extraction ratio of a capillary system with linear permeation.
Model assumptions and sensitivity analysis.
The permeation of indicator is nonlinear, if it is involved in chemical
reactions (such as metabolism) or its transport is facilitated. For LMW
solutes such as glucose, they are not likely metabolized during the
rapid extraction process. In the kidney and RBC, urea permeation could
be facilitated by urea transporters. However, Northern analysis has
demonstrated no measurable amount of RNA for the urea transporters in
the rat lung (21). Effros et al. (4) found that treatment with thiourea
to block the urea transporters in the lung had no effect on the
extraction of urea as determined by multiple indicator dilution,
indicating no facilitated transport for the urea in the lung. The
observation of similar revised density changes for hypertonic
disturbance in urea and NaCl indicates that both solutes act in a
similar way to extract fluid from the tissue. The extraction results of Hunter and Lee (11) for two different levels of hypertonic
NaCl disturbance suggest that the process of fluid extraction due to NaCl is linear. In addition, the osmotic pressure of the LMW solute is
linearly proportional to its molar concentration. Therefore, we
conclude that the linear analysis is appropriate to analyze the
extraction process produced by the low hypertonic disturbance of the
test solutes.
Derivation of Eqs. 11 and 18 requires the assumption that the
distribution of surface area among the stream-tube groups be the same
as that of flow distribution. The rationale for the equality is based
on the consideration that the surface and flow of a stream-tube group
are related to the number of capillaries in the group. We also treat
the tissue fluid compartment as one compartment, although the
endothelial cells and septal interstitium may be parts of the tissue
compartment participating in the extraction process. The ability to
obtain the matches as shown in Figs. 3 and 5-8 with a coefficient
of variation of ~7-10% indicates that these two assumptions may
be appropriate. For runs with larger coefficients of variation, the
revised density, because of the slight irregularity in the ventilation
frequency, still has considerable ventilatory density oscillations.
This precludes our refinement of the values of
K and to search for a smaller
coefficient of variation.
As specified by Eqs. 1 and 20,
K is calculated with assumed values
of and f. A smaller corresponds to less fluid volume in the RBCs available for dilution. If
is reduced by 20%, the estimate for
max is decreased by 5%, and
K is increased by 5%. The density
of extracted fluid is determined primarily by the value chosen for
f. Suppose it is reduced from
40 to 32 mosmol/l (i.e., a 20% change in
f). If this decrease results
primarily from a decrease in NaCl concentration, then the fluid density declines from 994 to 993.83 g/l, which causes a 0.4% decrease in the
estimate of K. For a 20% reduction
in f, Eq. 21 indicates that the estimate for
Vti,0 is higher by 3%. These
percent changes indicate relative insensitivities of
K and
Vti,0 on and
f.
Conclusion.
We have used the density change after a 20-s infusion, at a constant
rate, of five LMW hydrophilic solutes to assess
K, the lung tissue volume
participating in transcapillary fluid movement, and the time course of
fluid extraction and filtration from the lung and peripheral organs.
Our result indicates that the fluid extraction phase for the pulmonary
tissue is completed in ~10 s. Our analysis suggests that the
completion is due to a rapid increase in the tissue osmotic pressure to
balance the elevation in plasma osmotic pressure produced by the
hypertonic infusion. We also find that the time course of the
extraction and the pulmonary tissue fluid volume participating in the
extraction process are independent of the molecular size of the solute
infused. The estimated tissue fluid volume indicates that it may be
composed of all fluid spaces in the alveolar septa. The differences
estimated for K values of LMW
solutes suggest that there are two pathways offering different degrees
of separation on transcapillary exchange based on the molecular size of
the solutes.
| |
ACKNOWLEDGEMENTS |
This research is partially supported by National Heart, Lung, and
Blood Institute Grants HL-40893 and HL-07284.
| |
FOOTNOTES |
Address for reprint requests: J. S. Lee, Dept. of Biomedical
Engineering, Box 377, University of Virginia Health Sciences Center,
Charlottesville, VA 22908.
Received 10 September 1996; accepted in final form 16 September
1997.
| |
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Abstract
Introduction
![&Dgr;&pgr;<SUB>pl,c</SUB>(<IT>t</IT>,&tgr;) = &Dgr;&pgr;<SUB>pl,inf</SUB>(<IT>t</IT>)⊗<IT>h</IT><SUB>pa</SUB>(<IT>t</IT>)⊗&dgr;[<IT>t</IT> − (&tgr; − <IT>t</IT><SUB><IT>n</IT></SUB>)/2]](/content/vol84/issue3/fulltext/769/img015.gif)
