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J Appl Physiol 84: 769-781, 1998;
8750-7587/98 $5.00
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Vol. 84, Issue 3, 769-781, March 1998

Pulmonary fluid extraction and osmotic conductance, sigma K, measured in vivo

Joseph M. Karch and Jen-Shih Lee

Department of Biomedical Engineering, University of Virginia, Charlottesville, Virginia 22908

    ABSTRACT
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
References

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 (sigma K; where sigma  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 sigma 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 · mosmol-1 · l · g-1 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

    INTRODUCTION
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
References

THE OSMOTIC CONDUCTANCE of the lung (sigma K, the product of the reflection coefficient for the solute, sigma , 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 sigma 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 sigma 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 sigma 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 sigma 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.

    MATERIALS AND METHODS
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
References

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 rho a. Its value before a hypertonic infusion is designated rho 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 rho a was recorded for later computation of the cardiac output (Q), 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 Qinf (3.8 ml/min, which is <= 2% of Q) 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 (rho 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.

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 (pi 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 beta  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 [pi pl,inf (t)]
&Dgr;&pgr;<SUB>pl,inf</SUB> (<IT>t</IT>) = &Dgr;&pgr;<SUB>max</SUB><IT>u</IT>(<IT>t</IT>) (Eq. 1)
where Delta pi max is Qinfpi h/[Q(1 - beta 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 beta  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 tau  and tau  + dtau as a group. Let h<SUB><A><AC><IT>Q</IT></AC><AC>˙</AC></A></SUB>(tau ) be the fractional flow distribution so that the blood flow passing through this stream-tube group is prescribed as Qh<SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>(tau )dtau . In a stream tube in this group, flow dispersion is characterized by the transport function hs(t,tau ). Their interrelations with the overall flow and transport function yield the following two identities
<LIM><OP>∫</OP><LL>0</LL><UL>∞</UL></LIM><IT>h</IT><SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>(&tgr;)d&tgr; = 1 (Eq. 2)
<IT>h</IT>(<IT>t</IT>) = <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>s</SUB>(<IT>t</IT>,&tgr;)d&tgr; (Eq. 3)
In the dispersion analysis of Hunter and Lee (10), tau  is the subscript i identifying the group, Qh<SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>(tau )dtau is Qi, hs(t,tau ) is hi(t), and <LIM><OP>∫</OP></LIM>[ ]dtau is Sigma [ ].

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
<IT>h</IT>(<IT>t</IT>) = <IT>h</IT><SUB>pa</SUB>(<IT>t</IT>)⊗<IT>h</IT><SUB>c</SUB>(<IT>t</IT>)⊗<IT>h</IT><SUB>pv</SUB>(<IT>t</IT>) (Eq. 4)
where otimes  is the convolution operator. Here, h(t) is CR(t) of Audi et al. and hpa(t)otimes hpv(t) is Cn(t). The first moment of hc(t) is tc and the second (central) moment is &sfgr;<SUP>2</SUP><SUB>c</SUB>. The corresponding moments of Cn(t) are tn and &sfgr;<SUP>2</SUP><SUB><IT>n</IT></SUB>, and those of h(t) are tm and &sfgr;<SUP>2</SUP><SUB>m</SUB>. The nature of convolution yields
<IT>t</IT><SUB>m</SUB> = <IT>t</IT><SUB><IT>n</IT></SUB> + <IT>t</IT><SUB><IT>c</IT></SUB> &sfgr;<SUB>m</SUB><SUP>2</SUP> = &sfgr;<SUB><IT>n</IT></SUB><SUP>2</SUP> + &sfgr;<SUB>c</SUB><SUP>2</SUP> (Eq. 5)
For the stream-tube group, the MTT of blood flowing through its capillary segments is tau  - 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, delta (t - tau  + tn). The transport function of the stream-tube group is
<IT>h</IT><SUB>s</SUB>(<IT>t</IT>,&tgr;) = <IT>h</IT><SUB>pa</SUB>(<IT>t</IT>)⊗&dgr;(<IT>t</IT> −&tgr; + <IT>t</IT><SUB><IT>n</IT></SUB> )⊗<IT>h</IT><SUB>pv</SUB>(<IT>t</IT>) (Eq. 6)
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
<IT>h</IT><SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>(<IT>t</IT> + <IT>t</IT><SUB><IT>n</IT></SUB>) = <IT>h</IT><SUB>c</SUB>(<IT>t</IT>) (Eq. 7)
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 sigma 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/sigma c are the same for the lungs studied. These assumptions allow us to calculate from tm the value of tc and then sigma c. Through Eq. 5, we determine tn and sigma 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* (tau )dtau ] has the same fractional distribution as the flow, i.e.
<IT>S</IT>*(&tgr;)d&tgr; = <IT>Sh</IT><SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>(&tgr;)d&tgr; (Eq. 8)
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<SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>(tau )dtau .

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 (Delta pi pl,c) is the hypertonic input (Eq. 1) dispersed by the arterial dispersion, which is
&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] 
= &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] (Eq. 9)
where Delta pi 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 (pi f) as the only factor effecting an increase in the tissue osmotic pressure [pi ti(t,tau )]. Let the initial osmotic pressure in the tissue fluid compartment be pi ti,0. As prescribed earlier, the initial tissue fluid volume surrounding the capillaries of the stream-tube group is Vti,0h<SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>(tau )dtau . Suppose this volume experiences the following change: Delta Vti(t,tau )h<SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>(tau )dtau . 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 [Delta pi ti(t,tau )]
&Dgr;&pgr;<SUB>ti</SUB>(<IT>t</IT>,&tgr;) = − (&pgr;<SUB>ti,0</SUB> − &pgr;<SUB>f</SUB> ) <FR><NU>&Dgr;V<SUB>ti</SUB>(<IT>t</IT>,&tgr;)</NU><DE>V<SUB>ti,0</SUB></DE></FR> (Eq. 10)
Before the infusion, rho a remained at a constant level, indicating a zero filtration rate, i.e., Js(0,tau ) = 0. This initial condition corresponds to no change in the tissue fluid volume [Delta Vti(0,tau ) = 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<SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>(tau )dtau is the surface area of the stream-tube group. Thus its filtration coefficient is Kh<SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>(tau )dtau . Let the rate of filtration through the capillary wall be Js(t,tau )h<SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>dtau . 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 [Delta Vti(t,tau )]
<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;)] (Eq. 11)
where sigma  is the reflection coefficient of the capillary wall for the test solute and sigma f is the reflection coefficient of the abundant solute in the tissue. We use d/dt to represent the time derivative, as tau  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
<FR><NU>d&Dgr;V<SUB>ti</SUB></NU><DE>d<IT>t</IT></DE></FR> + &agr;&Dgr;V<SUB>ti</SUB>
= − &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] (Eq. 12)
where
&agr; = <FR><NU>&sfgr;<SUB>f</SUB> <IT>K</IT>(&pgr;<SUB>ti,0</SUB> − &pgr;<SUB>f</SUB>)</NU><DE>V<SUB>ti,0</SUB></DE></FR> (Eq. 13)
This expression for alpha , being independent of tau  and test solute and having the unit of seconds-1, 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>) (Eq. 14)
where g(t) = alpha exp(-alpha t)u(t). For the capillaries in the stream-tube group, the flow rate is Qh<SUB><A><AC><IT>Q</IT></AC><AC>˙</AC></A></SUB>(tau )dtau and the density of blood is rho 0. The extraction adds to the flow at the rate Js(t,tau )h<SUB><A><AC><IT>Q</IT></AC><AC>˙</AC></A></SUB>(tau )dtau and with the density rho f. The mixing of these two fluid flows and the omission of higher-order terms produce the following density change
&Dgr;&rgr;<SUB>c</SUB>(<IT>t</IT>,&tgr;) = (&rgr;<SUB>0</SUB> − &rgr;<SUB>f</SUB> ) <FR><NU><IT>J</IT><SUB>s</SUB>(<IT>t</IT>,&tgr;)</NU><DE><A><AC>Q</AC><AC>˙</AC></A></DE></FR> (Eq. 15)
With the dispersion from the capillary to the sampling site as delta [t - (tau  - tn)/2] otimes  hpv(t), we have the density change exiting the stream tube at the sampling site [Delta rho s(t,tau )] 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>) (Eq. 16)
With the flow of the stream-tube group as Qh<SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>(tau )dtau , 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 Delta rho a(t) which is also the aortic blood density change
&Dgr;&rgr;<SUB>a</SUB>(<IT>t</IT>) = <LIM><OP>∫</OP><LL>0</LL><UL>∞</UL></LIM>&Dgr;&rgr;<SUB>s</SUB>(<IT>t</IT>,&tgr;)<IT>h</IT><SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>(&tgr;)d&tgr; (Eq. 17)
With the help of Eqs. 3 and 6, the substitution of Eqs. 14-16 into Eq. 17 yields
&Dgr;&rgr;<SUB>a</SUB>(<IT>t</IT>) = − (&rgr;<SUB>0</SUB> − &rgr;<SUB>f</SUB> ) <FR><NU>&sfgr;<IT>K</IT>&Dgr;&pgr;<SUB>max</SUB></NU><DE>&agr;<A><AC>Q</AC><AC>˙</AC></A></DE></FR> <IT>h</IT>(<IT>t</IT>)⊗<IT>g</IT>(<IT>t</IT>) (Eq. 18)
The equation deduced for hypertonic NaCl infusion by Hunter and Lee (10) is a special case of Eq. 18 once sigma  is identified as sigma  of NaCl. Their derivation requires no assumption on having the input transport functions to capillaries prescribed by hpa(t).

Computation of sigma K, alpha , 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), Q, 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 (rho inf) may be different from the blood density (rho 0), the infusion imposes on rho a an infusion "artifact" described by (rho inf - rho 0)(Qinf/Q)[u(t- u(t - 20 s)] otimes  h(t). The program subtracts from rho 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 rho 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, Q, and c: stream tube, cardiac output, and capillary.

The values of sigma K, alpha , 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/alpha , and the first moment of h(t) is tm. Their sum is the first moment of g(t) otimes  h(t), which is also t'm. Solving for alpha , we have
&agr; = <FR><NU>1</NU><DE>(<IT>t</IT>′<SUB>m</SUB> − <IT>t</IT><SUB>m</SUB>)</DE></FR> (Eq. 19)
The area of the function g(t) otimes  h(t) in Eq. 18 is unity. Therefore, sigma K is
&sfgr;<IT>K</IT> = <FR><NU>&agr;<IT>A</IT>′<A><AC>Q</AC><AC>˙</AC></A></NU><DE>&Dgr;&pgr;<SUB>max</SUB>(&rgr;<SUB>0</SUB> − &rgr;<SUB>f</SUB> )</DE></FR> (Eq. 20)
We obtain from Eq. 13 this relation to calculate Vti,0
V<SUB>ti,0</SUB> = <FR><NU>&sfgr;<SUB>f</SUB><IT> K</IT>(&pgr;<SUB>ti,0</SUB> − &pgr;<SUB>f</SUB> )</NU><DE>&agr;</DE></FR> (Eq. 21)
In the calculations, pi ti,0 is taken as 300 mosmol/l and pi 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 (rho f) is 994 g/l. The value of sigma fK estimated from the hypertonic NaCl infusion is used in Eq. 21 to calculate Vti,0 for all test solutes.

Once sigma K and alpha  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,tau )h<SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>(tau )dtau . Its sum over all groups is the total rate of fluid extracted for the entire lung
<IT>J</IT><SUB>f</SUB> (<IT>t</IT>) = <LIM><OP>∫</OP><LL>0</LL><UL>∞</UL></LIM> <IT>J</IT><SUB>s</SUB>(<IT>t</IT>,&tgr;)<IT>h</IT><SUB><A><AC>Q</AC><AC>˙</AC></A></SUB>(&tgr;)d&tgr; = <FR><NU>− &sfgr;<IT>K</IT>&Dgr;&pgr;<SUB>max</SUB></NU><DE>&agr;</DE></FR>
 ⋅ <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; (Eq. 22)
The factoring out of the convolution and the use of the identity 2delta (2t) = delta (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>) (Eq. 23)
The integration of Jf (t) from 0 to infinity  is the maximum reduction in tissue volume. Let it be Delta Vti,max. Because of the integration of hpa(t), 2hc(2t) and g(t) is all unity, we have
&Dgr;V<SUB>ti,max</SUB> = <FR><NU>&sfgr;<IT>K</IT>&Dgr;&pgr;<SUB>max</SUB></NU><DE>&agr;</DE></FR> (Eq. 24)
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 rho 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 (Delta pi a). Its difference from a step increase at time 0 and a step decrease at 20 s reflects effect of flow dispersion.

    RESULTS
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Abstract
Introduction
Materials & Methods
Results
Discussion
References

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. &sfgr;<SUP>2</SUP><SUB>c</SUB> and &sfgr;<SUP>2</SUP><SUB>m</SUB> are indexes describing the dispersion (or spread) of hc and h, respectively. We term the ratio &sfgr;<SUP>2</SUP><SUB>c</SUB>/&sfgr;<SUP>2</SUP><SUB>m</SUB> 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 &sfgr;<SUP>2</SUP><SUB>c</SUB>/&sfgr;<SUP>2</SUP><SUB>m</SUB> 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 &sfgr;<SUP>2</SUP><SUB><IT>n</IT></SUB> as 70% of &sfgr;<SUP>2</SUP><SUB>m</SUB> (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 (&sfgr;<SUP>2</SUP><SUB>c</SUB> being 30% of &sfgr;<SUP>2</SUP><SUB>m</SUB>) 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.

                              
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Table 1.   Transport parameters for the isolated perfused lung and the central circulation

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 Delta pi max = 28.7 mosmol/l. In the isotonic injection experiment done 2 min earlier, Q 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 sigma K calculated for this example is 18.2 ml · h-1 · mosmol-1 · l, the rate constant alpha  is 0.44 s-1, the tissue volume participating in the extraction process (Vti,0) is 2.7 ml, and the volume extracted (Delta 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 Delta pi max[u(t- u(t - 20 s)]otimes 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 sigma 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 rho 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 rho 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 (Delta pi c) and tissue fluid osmolarity (Delta pi 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 rho a.

Figure 5A shows a typical density response to a 20-s infusion of hypertonic urea. The hypertonic disturbance (Delta pi max) is 33.6 mosmol/l. Q 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, alpha  is 0.47 s-1, sigma K is 16 ml · h-1 · mosmol-1 · 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 (Delta pi 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.

The density response to a 20-s infusion of hypertonic glucose is illustrated in Fig. 6A. The hypertonic infusion caused a Delta pi max of 24.5 mosmol/l. Q 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, sigma K is 24.5 ml · h-1 · mosmol-1 · l, Vti,0 is 3.4 ml, alpha  is 0.30 s-1, and the coefficient of variation is 0.14. 


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Fig. 6.   Density response to a 20-s infusion of hypertonic glucose (Delta pi 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.

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, Delta pi max is 20.4 mosmol/l, Q is 272 ml/min, tm is 5.4 s, sigma K is 18.1 ml · h-1 · mosmol-1 · l, Vti,0 is 3.6 ml, alpha  is 0.33 s-1, and the coefficient of variation is 0.15. For raffinose, Delta pi max is 7 mosmol/l, Q is 375 ml/min, tm is 4.7 s, sigma K is 27 ml · h-1 · mosmol-1 · l, Vti,0 is 5.4 ml, alpha  is 0.33 s-1, and the coefficient of variation is 0.10. 


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Fig. 7.   Density response to a 20-s infusion of hypertonic sucrose (Delta pi 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 (Delta pi 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.

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 Q 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 rho inf, Delta pi max, -A', t'm, alpha , coefficient of variation, Delta Vti,max, Delta Vti,max/Delta pi max, sigma 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).

                              
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Table 2.   Experimental and computed parameters for extraction phase of density response

Statistically, our results show no significant difference among the values of alpha  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 alpha  as a rate constant characterizing the extraction of a hypotonic fluid containing the LMW solutes in the tissue fluid. The relationship between sigma 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 (alpha ) obtained from each test solute (bullet , 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 alpha  and Vti,0 are independent of solute causing extraction, which is consistent with hypothesis in text.


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Fig. 10.   Osmotic conductance (sigma K) as a function of molecular radius for NaCl, urea, glucose, sucrose, and raffinose (bullet , left to right).

    DISCUSSION
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
References

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 rho 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 rho 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.