Vol. 86, Issue 6, 1866-1880, June 1999
Detection of changes in lung tissue properties with
multiple-indicator dilution
D. L.
Roerig1,4,
S. H.
Audi2,
J. H.
Linehan2,
G. S.
Krenz3,
S. B.
Ahlf4,
W.
Lin5, and
C. A.
Dawson2,4,5
1 Department of Anesthesiology and
Pharmacology/Toxicology and 5 Department of
Physiology, Medical College of Wisconsin, Milwaukee 53226;
2 Biomedical Engineering Department and
3 Department of Mathematics Statistics and Computer
Science, Marquette University, Milwaukee 53201-1881; and
4 Department of Veterans Affairs, Zablocki
Veterans Affairs Medical Center, Milwaukee, Wisconsin 53295
 |
ABSTRACT |
We evaluated the
potential utility of a group of indicators, each of which targets a
particular tissue property, as indicators in the multiple-indicator
dilution method to detect and to identify abnormalities in lung tissue
properties resulting from lung injury models. We measured the pulmonary
venous outflow concentration vs. time curves of
[14C]diazepam, 3HOH,
[14C]phenylethylamine, and a vascular reference indicator
following their bolus injection into the pulmonary artery of isolated
perfused rabbit lungs under different experimental conditions,
resulting in changes in the lung tissue composition. The conditions
included granulomatous inflammation, induced by the intravenous
injection of complete Freund's adjuvant (CFA), and intratracheal fluid
instillation, each of which resulted in similar increases in lung wet
weight. Each of these conditions resulted in a unique pattern among the concentration vs. time outflow curves of the indicators studied. The
patterns were quantified by using mathematical models describing the
pulmonary disposition of each of the indicators studied. A unique model
parameter vector was obtained for each condition, demonstrating the
ability to detect and to identify changes in lung tissue properties by
using the appropriate group of indicators in the multiple-indicator
dilution method. One change that was particularly interesting was a
CFA-induced change in the disposition of diazepam, suggestive of a
substantial increase in peripheral-type benzodiazepine receptors in the
inflamed lungs.
diazepam; phenylethylamine; mathematical modeling; lung
inflammation; benzodiazepine receptors
| |
INTRODUCTION |
THE MULTIPLE-INDICATOR DILUTION (MID) method is used to
measure organ perfusion, cellular and chemical composition, vascular permeability, and metabolic function in a nondestructive manner in
intact organs and in vivo (1-5, 7-23, 26-29, 35, 36,
38). It involves the injection of a bolus containing two or more
indicators into the organ's arterial inlet, followed by the
measurement of the indicator concentration vs. time in the venous
effluent. One indicator, referred to as "the vascular reference
indicator," is confined to the vascular space and does not interact
with the tissue as it travels through the organ. The other
indicator(s), referred to as "test indicators," interact with the
tissue in some way related to the properties of the indicator(s) and
the tissue. As a result of these interactions, the concentration vs. time outflow curves of the test indicators are changed in amplitude and/or timing with respect to the reference indicator curve. Comparison of the reference and test indicator curves reveals the tissue interactions of the test indicator and provides the information necessary to evaluate aspects of the organ function that influence those interactions. MID methods applied to the lungs have the potential
for discriminating among lungs having different physical and chemical
properties such as occur in lung injury and disease (7, 8, 22, 23, 28,
29, 35). This is the basis for MID studies of lung water volume (8, 12,
21, 28), capillary permeability (7, 8, 14, 15, 22, 23, 36, 38),
endothelial enzyme activity (8, 9, 15, 18, 29, 35), and other aspects
of lung tissue function (1-5, 7, 8, 16, 19, 20).
In a previous study (16), the lipophilic amine
[14C]diazepam was found to provide a measure of the
perfused nonaquous lung tissue volume that was independent of the
tissue water content in edematous, but otherwise normal, lungs. Thus,
under the conditions of that study, the ratio of the extravascular
volume accessible to 3HOH to that accessible to
[14C]diazepam provided a nondestructive index of lung
wet-to-dry weight ratio (16).
The present study was carried out to evaluate the potential utility of
[14C]diazepam, when used in conjunction with other test
indicators such as 3HOH and
[14C]phenylethylamine ([14C]PEA), for
detecting and identifying abnormalities in lung tissue properties. The
3HOH was used to trace the perfused extravascular water
volume and the [14C]PEA, which is extracted by the
endothelial cells (6, 18), was used as an indicator of perfused
endothelial surface. The MID experiments were carried out on isolated
perfused rabbit lungs to facilitate control over a number of variables
(1, 2, 4, 5, 16, 28, 29). Lung tissue properties were manipulated in
several ways. One was the intravenous injection of complete Freund's
adjuvant (CFA) to produce granulomatous inflammation (11, 31). This
lung inflammatory stimulus induces complex changes in lung tissue
composition, including a substantial increase in lung weight, which
have been well defined in the rabbit (11, 31). To determine whether the
chosen group of indicators could distinguish between a change in lung
weight and changes in lung properties associated with the inflammatory
response, the airways of otherwise normal lungs were filled with a
physiological salt solution (PSS) by intratracheal instillation to
increase the lung weight to the same extent as that caused by the
inflammatory response. To manipulate the fraction of perfused tissue in
some of these fluid-filled lungs, they were also embolized with enough
glass beads to reduce the accessible extravascular water volume to that in the inflamed lungs. Thus several experimental groups were studied in
which the lungs had different properties among groups, but some
variables were also matched among groups. The differences among the
outflow concentration curves in the various study groups were
quantified by using mathematical models appropriate for each indicator.
The results provide an example of how the MID model parameters can
reveal differences in lung tissue properties.
| |
EXPERIMENTAL METHODS |
Animal Preparation (Isolated Rabbit Lung)
The experiments were performed by using an isolated rabbit lung
preparation, as previously described (1, 2, 5). New Zealand White
rabbits of either sex were given chlorpromazine hydrochloride (25 mg/kg
im), followed by pentobarbital sodium (20-25 mg/kg) via an ear
vein, and then were heparinized (1,200 IU/kg) and exsanguinated via a
carotid artery catheter. The pulmonary artery, vein, and trachea were
cannulated, and a ligature was secured around the ventricles. The lungs
were removed from the chest and attached to the perfusion system primed
with a perfusate containing a PSS (in g/l: 0.35 KCl, 0.37 CaCl2 · 2H2O, 0.29 MgSO4 · 7H2O, 0.16 KH2PO4, 6.9 NaCl, 1 glucose, and 2.1 NaHCO3) with 45 g/l of BSA (1, 2, 5). The perfusion
system included a heated perfusate reservoir and a Master Flex roller
pump, which pumped perfusate at a constant mean flow of 3.33 ml/s from
the reservoir into the pulmonary artery, with the left atrial pressure set equal to atmospheric (pleural) pressure by adjusting the height of
the venous outflow into the recirculation reservoir. Arterial and
venous pressures were referenced to the level of the left atrium. The
lung was ventilated with 95% O2-5% CO2 at 10 breaths/min under positive pressure with the use of a solenoid
respirator with end-inspiratory and end-expiratory airway pressures of
7.17 ± 0.52 and 1.62 ± 0.54 (SD) cmH2O,
respectively. The perfusate was equilibrated with the respiratory gas
mixture, which maintained the pH at 7.37 ± 0.05 (SD) at 37°C.
Before each of the bolus injections described below, the ventilator was
stopped at end expiration for the duration of the sampling period.
To produce a bolus injection, a solenoid-operated injection loop (1, 2,
5) was situated in the inflow tubing so that a 1.0-ml bolus could be
rapidly introduced into the inflow stream without changing the flow or pressure.
In the experiments involving alveolar instillation, to produce as even
a distribution of the instillate as possible, the lungs were made
atelectatic before perfusion. This was accomplished by ventilating the
anesthetized rabbit with 100% O2 for 5 min and then
clamping the trachea. Five minutes later, the chest was opened, and the
lungs were cannulated and placed in the perfusion system as described above.
MID Studies
The 1.0-ml bolus of the perfusate solution contained 2.5 mg of
FITC-labeled 40,000-mol wt dextran (FITC-Dex), and 0.5 µCi of
3H or 0.1 µCi of 14C of one or more of either
3HOH, [14C]diazepam, or
[14C]PEA. The specific activities for 3HOH,
[14C]diazepam, and [14C]PEA were 90 mCi/mol, 55 mCi/mmol, and 50 mCi/ml, respectively. Just before
injection, the venous outflow was directed into the sample tubes of a
modified (1, 2, 5) Gilson Escargot fraction collector. A total of one
hundred 2-ml samples was collected, with a sampling interval of 0.6 s.
After each experiment, the lungs were removed from the perfusion
system, and an additional bolus containing FITC-Dex was made, with the
arterial and venous cannulas connected directly together. The data from
this injection were used to obtain the moments for the passage of the
bolus through the tubing from injection to fraction collector in the
absence of the lungs. In one of these experiments, all three test
indicators were also included in a bolus to ensure that no separation
of the test indicators and the FITC-Dex occurred within the tubing alone.
The concentration of the FITC-Dex in the outflow samples was measured
spectrophometrically (494 nm). The 14C and/or
3H activities were measured by liquid scintillation
counting. Measured quantities of the solution used as the
injectate were added to sample tubes collected before the emergence of
the indicators. These samples served as internal standards for
the calculation of indicator concentrations. The fractions of
injected indicators recovered in the collected samples, calculated
based on these standards, are given in Table
1.
Conditions Studied
Control conditions.
The MID studies described above were carried out on lungs from seven
normal rabbits.
Granulomatous inflammation.
Eight rabbits were each given a 1-ml ear vein injection of CFA (8.5 ml
Bayol F, 1.5 ml Arlacel, and 5 mg Myco. Butyricum) (11, 31).
After 13.8 ± 7.9 (SD) days, these rabbits were anesthetized, and the
MID studies were carried out on the lungs.
Alveolar instillation.
In contrast to the complex changes in lung tissue composition resulting
from the inflammatory response induced by CFA, well-defined changes in
lung wet weight and tissue composition were induced by instilling 35 ml
of a solution that had the same composition as the perfusate [PSS
containing 4.5% BSA (PSS + BSA)] into the alveolar space of the
isolated lungs from normal rabbits. The volume of fluid instilled was
chosen to result in similar lung wet weight as that obtained in lungs
treated with CFA. In 6 of the 13 lungs, the instilled solution was 35 ml of PSS with no BSA. For these lungs, the instilled PSS solution
included 4.5% dextran (70,000 mol wt) to match the oncotic pressure of
the PSS + BSA. The MID studies were carried out before (atelectasis)
and after the alveolar instillation of the fluid.
Embolism.
In a subset of the fluid instillation groups (six filled with
PSS + BSA and five filled with PSS), following the last bolus injection, 240 mg of glass beads (2.6 × 104 beads, 194 µm in diameter) were slowly introduced into the pulmonary artery to
occlude a portion of the vascular bed. The number of beads was chosen,
based on previous studies (17), to reduce the
3HOH-accessible extravascular water volume of these
fluid-filled lungs to approximately that accessible in lungs with
granulomatous inflammation. The MID studies were then repeated.
PEA metabolites.
PEA is metabolized within the pulmonary endothelial cells to
phenylethylacetic acid (PAA) (6, 18). Thus, as time progresses during
bolus passage, a fraction of the 14C injected as
[14C]PEA returns to the perfusate as
[14C]PAA. To determine the time during bolus passage
before the appearance of significant [14C]PAA, a bolus
injection was carried out with [14C]PEA as the only test
indicator in at least one lung from each experimental group described
above. The collected samples were extracted in methanol and spotted on
thin-layer chromatography plates, which were developed in a solvent
system consisting of ethylacetate-isopropanol-25% ammonium hydroxide
(50:35:10). A maximum of two peaks was detectable, which corresponded
to PEA and PAA. A [14C]PAA peak was not detectable until
samples collected after the peak of the FITC-Dex concentration vs. time
curve. The analysis described below is based only on the
[14C]PEA concentration in samples obtained up to the peak
of the FITC-Dex curve.
After each experiment, the lungs, except those involving alveolar
instillation, were weighed and lyophilized to a constant weight. For
the seven normal lungs, the wet and dry lung weights were 9.4 ± 0.4 and 1.59 ± 0.1 g, respectively, and the wet-to-dry ratio (total lung
wet weight to lung dry weight) was 5.8 ± 0.04 (SE). For the eight
CFA-treated lungs, the wet and dry lung weights were 44.8 ± 4.1 and
8.8 ± 0.9 g, respectively, and the wet-to-dry ratio (total lung wet
weight to lung dry weight) was 5.3 ± 0.1. For comparison with the
other groups, estimates of the wet weight of the fluid-filled lungs
shown in Table 2 were obtained by adding the weight of the water in the instilled solution; i.e., the weight of
the instillate (35 ml × 1.02 g/ml) minus the weight of non-water constituents of the instillate (and glass beads if they were injected) to the average wet weight of control lungs. The dry weight was assumed
equal to the average dry weight of control lungs.
The body weights and arterial and venous pressures for each group
studied are given in Table 2. The airway pressure during the bolus
passage was 1.6 ± 0.5 (SD) cmH2O for air-filled lungs and
atmospheric at the trachea for the fluid-filled lungs.
| |
EXPERIMENTAL RESULTS |
Figure 1 shows an example of the measured
venous effluent concentration vs. time curves for FITC-Dex,
3HOH, [14C]diazepam, and
[14C]PEA from isolated rabbit lungs under each of the
conditions studied. Each condition provided a unique pattern among the
four indicator curves. Both CFA treatment and alveolar fluid
instillation resulted in a reduction in the peak and a prolongation of
the 3HOH concentration curves relative to control,
reflecting the increases in the lung water volume under these
conditions.
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Fig. 1.
Venous effluent concentration vs. time curves for FITC-Dex,
[14C]diazepam, 3HOH, and
[14C]phenylethylamine ([14C]PEA) after
bolus injection of these indicators into pulmonary artery of an
isolated perfused rabbit lung under each of experimental conditions
studied. Concentrations on this and subsequent graphs are normalized to
amount of injected indicator and are, thus, the fraction of injected
dose per milliliter of effluent perfusate. FITC-Dex, FITC-labeled
dextran; PSS, physiological salt solution; CFA, complete Freund's
adjuvant.
|
|
Filling the alveolar space with PSS resulted in almost no change in the
[14C]diazepam curves, whereas both PSS + BSA instillation
and CFA treatment resulted in marked changes in diazepam curves. When the lungs were filled with PSS + BSA, the changes in the diazepam curves were qualitatively similar to the changes in the water curves.
This reflects the rapidly equilibrating associations of the diazepam
with BSA (2, 5, 16). In CFA-treated lungs, there was a large reduction
in the recovery of diazepam relative to that under the other conditions
(Table 1), and the peak of the diazepam curve shifted to the left (Fig.
1B), resulting in a very different shape from that in the
other conditions.
The [14C]PEA curve was hardly affected by either CFA
treatment or alveolar fluid instillation, although the tail of the
14C curve in the CFA-treated lungs tended to be depressed.
Figure 1, E and F, shows that reduction in the fraction
of perfused tissue by embolizing fluid-filled lungs with glass beads
shifted the curves of all indicators upward and to the left while the
relationships among the curves were essentially maintained.
The data analysis described below is an attempt to provide a
quantitative basis for comparison among these patterns.
| |
DATA ANALYSIS |
Glossary
| [bi], i = 1, ... ,N |
Concentration of the ith binding species
|
| CD(t) |
Concentration vs. time outflow curve of diazepam
|
| CF(t) |
Concentration vs. time outflow curve of 3HOH
|
 in(t) |
(q/F)  n(t)
|
| Cin(t) |
Capillary input function
|
| Cp(t) |
Concentration vs. time outflow curve of PEA
|
| CR(t) |
Concentration vs. time outflow curve of FITC-Dex
|
| CT(t) |
Tubing concentration vs. time outflow curve
|
| [D](x, t) |
Vascular concentration of diazepam at distance x from the
capillary inlet and time t
|
| [Dbi] |
Concentration of diazepam bound to the ith binding species
|
[Dei](x, t) = [Dbi](x, t)/ |
Concentration of diazepam in Qti bound to the binding
species with association and dissociation rate constants
ki and k i,
respectively
|
| [Dp](x, t) |
Vascular concentration of PEA at distance x from the capillary
inlet and time t
|
| F |
Flow
|
| hc(t) |
Capillary transit time distribution
|
| c(t) |
Transit time distribution that can account for the effect of
hc(t) in the modeling of MID data
|
| n(t) |
Noncapillary transport function (i.e., in arteries, veins, connecting
tubing, and the injection system)
|
| ka and kd |
Respective association and dissociation rate constants of diazepam with
plasma protein
|
| kf = QR(k1[b1])/ |
Association rate of diazepam with the binding species with association
and dissociation rate constants k1 and
k1 (ml/s)
|
| ki and ki |
Association and dissociation rate constants, respectively, of the
ith binding species
|
| Ki = ki/ki |
Equilibrium dissociation constant of the ith class of
associations
|
| KP = kd/ka |
Plasma protein-diazepam equi librium dissociation constant
|
| kseq = QR(k2[b2])/ |
Sequestration rate of diazepam within Qti (ml/s)
|
| Ke = ke/ke |
Equilibrium dissociation rate constant for [14C]PEA
within QF
|
| M |
Number of classes of slowly equilibrating associations
|
 |
Largest number of classes of slowly equilibrating associations
resolvable from the data
|
| m3 |
third central moment
|
 |
Third central moment of c(t)
|
| N |
Number of classes of associations having different dissociation rate
constants
|
| Np |
Number of parameters
|
| [ne] |
Concentration of the rapidly equilibrating association sites within
QF, having association and dissociation rate constants ke and ke, respectively
|
| [P] |
Plasma protein concentration
|
| PS |
Permeability-surface area product for [14C]PEA
|
| q |
Mass of the injected indicator
|
| Qc |
Capillary volume
|
| QF |
Flow-limited volume accessible to [14C]PEA
|
QR = (Qti ) |
Virtual volume including Qti and the effects of rapidly
equilibrating associations in Qc and Qti
|
QS =  kf |
Virtual volume reflective of the capacity of slowly equilibrating
classes of association (ml)
|
| Qti |
Tissue volume
|
| Qv |
Pulmonary vascular volume = 
|
| Qw |
Perfused extravascular water volume accessible to 3HOH = F × 
|
| [R](x, t) |
Vascular concentration of the reference indicator at distance x
from the capillary inlet and time t
|
 |
Vascular relative dispersion
|
| SSD |
Sum of squares differences
|
| t |
Time
|
 |
Mean transit time (first moment)
|
 |
 ]
|
 |
Extravascular mean transit time of 3HOH
|
 |
Mean transit times of CR(t),
CF(t), hc(t), and
CT(t), respectively
|
| VF |
QF(1 + [ne]/Ke)
|
| W |
Average linear flow velocity
|
| x |
Distance from the capillary inlet (x = 0)
|
| z |
z score = [(parameter for unknown  mean of control
group)/(SD of control group)]
|
 |
Mean z score of a parameter for a given experimental condition
|
 |
Mean z score for the ith parameters for the jth
experimental group
|
Greek letters
 2 |
Variance (second central moment)
|
| 2R, 2F,
2c, and 2T |
Variances of CR(t),
CF(t), hc(t),
and CT(t), respectively
|
 |
Variance of c(t)
|
= KP/([P] + KP)  (t) = k1
ek 1t |
Sojourn time distribution
|
| = 1/k1 |
Mean sojourn time (first moment) of (t)
|
FITC-Dex
The mean transit times () and
the second (2) and third (m3)
central moments of the outflow curves of FITC-Dex, 3HOH,
and the tubing outflow curve [CT(t)] were obtained by
fitting each to a shifted random walk function, the functional form of which can be specified by its first three moments (1, 2, 10).
The vascular volume (Qv) was estimated as the product of
the flow (F) and the difference between the mean transit times of the
outflow curves of the vascular reference indicator FITC-Dex, which are
denoted by CR(t) and
CT(t) from
|
(1)
|
where
are the mean transit times of CR(t) and
CT(t), respectively.
The vascular relative dispersion (RDv) was estimated from
the moments of CR(t) and
CT(t) from
|
(2)
|
where 2R and
2T are the second central moments of
CR(t) and CT(t), respectively.
3HOH
The perfused extravascular water volume (Qw) was
estimated as the product of the flow F and the difference between the
mean transit times of the outflow curves of 3HOH,
CF(t), and of FITC-Dex,
CR(t), from
|
(3)
|
where
is the mean transit time of CF(t) obtained by
fitting CF(t) to a shifted random walk function
(1, 2, 10).
[14C]Diazepam Concentration vs. Time Outflow Curves
In previous studies in which [14C]diazepam was used (16),
the analysis has been similar to that for 3HOH indicated
above. However, in CFA-treated lungs, the [14C]diazepam
behavior was clearly more complex (Fig. 1). Therefore, in this study,
data analysis was carried out by using the more general model we have
developed previously (1) as follows.
Single-capillary model.
A single-capillary element of the general model we have developed (1)
is composed of a capillary volume (Qc) and a tissue volume (Qti). The model assumes rapid equilibration
between the free and protein-bound diazepam in Qc (1) and
that the free form of the diazepam is the species having diffusional
access to Qti. Within Qti, the various
diazepam-tissue associations can have a range of rate constants and are
represented by N classes of associations with different
dissociation rate constants (1). If one visualizes the
associations as analogous to binding to a particular molecular species,
[bi], i = 1, ... , N, would be then the concentration of ith binding species and
[Dbi] the concentration of diazepam bound to that
species, with association and dissociation rate constants
ki and ki, respectively. Physically, these associations or interactions could be
the dissolution of diazepam in membrane lipid or other types of
interactions with the various chemical and cellular constituents of the
tissue (1).
Assuming that no radial concentration gradients of the free diazepam
exist within Qc or Qti (3-5), the spatial
and temporal variations in the concentrations of the reference
indicator and diazepam are described by the following species balance
equations. In the capillary volume
![<FR><NU>∂[R]</NU><DE>∂<IT>t</IT></DE></FR> + <IT>W</IT> <FR><NU>∂[R]</NU><DE>∂<IT>x</IT></DE></FR> = 0](/content/vol86/issue6/fulltext/1866/img007.gif)
|
(4)
|
![= <FENCE><FR><NU>Q<SUB>R</SUB></NU><DE>Q<SUB>c</SUB> + Q<SUB>R</SUB></DE></FR></FENCE> <FENCE><LIM><OP>∑</OP><LL><IT>i</IT> = 1</LL><UL><IT>M</IT></UL></LIM> <FENCE><IT>k</IT><SUB>−<IT>i</IT></SUB>[D<SUB>e<SUB><IT>i</IT></SUB></SUB>] − <FR><NU><IT>k</IT><SUB><IT>i</IT></SUB>[b<SUB><IT>i</IT></SUB>]</NU><DE>&bgr;</DE></FR> [D]</FENCE></FENCE>](/content/vol86/issue6/fulltext/1866/img009.gif)
|
(5)
|
In the tissue volume
![<FR><NU>∂[D<SUB>e<SUB><IT>i</IT></SUB></SUB>]</NU><DE>∂<IT>t</IT></DE></FR> = <FR><NU><IT>k</IT><SUB><IT>i</IT></SUB>[b<SUB><IT>i</IT></SUB>]</NU><DE>&bgr;</DE></FR> [D] − <IT>k</IT><SUB>−<IT>i</IT></SUB>[D<SUB>e<SUB><IT>i</IT></SUB></SUB>] <IT>i</IT> = 1,…, <IT>M</IT>](/content/vol86/issue6/fulltext/1866/img010.gif)
|
(6)
|
where M (M < N ) is the
number of classes with slowly equilibrating associations and
(N M ) is the number of classes with rapidly
equilibrating associations (1). [R](x, t) and
[D](x, t) are the vascular concentrations of the
reference indicator and the free diazepam at a distance x from
the capillary inlet and time t, respectively. QR = (Qti) is a virtual volume including Qti
and the effects of rapidly equilibrating associations in Qc and Qti. [Dei]
(x, t) = [Dbi](x, t)/ is the concentration of diazepam in Qti bound to the binding
species with association and dissociation rate constants
ki and ki,
respectively. The = KP/([P] + KP) is the fraction of the diazepam in the vascular
space that is not bound to plasma protein, where [P] is the plasma
protein concentration, KP = kd/ka is the plasma protein
equilibrium dissociation constant, and ka and
kd are the association and dissociation rate
constants of the diazepam to plasma protein, respectively.
The
is
a factor scaling Qti, which results from the
(N M ) rapidly equilibrating classes of
associations. Ki = ki/ki is the
equilibrium dissociation constant of the ith class of
associations, where ki and
ki are the association and dissociation
rate constants for the ith class of associations, respectively.
W is the average linear flow velocity within Qc, equal to the F divided by the capillary cross-sectional area. The model
parameters are QR (ml),
ki[bi])/ (s1), and ki
(s1), i = 1, ... , M.
Previously (1), we showed that the resolution of the MID data limits
the identifiability of the kinetic parameters for each of the
potentially large number or classes of associations M. In
addition, we showed that = 2 is the sufficient
number of classes of associations (,
< M, is the largest number of classes
of slowly equilibrating associations resolvable from the data) to fit
the data over a wide range of flows and a wide spectrum of
physicochemical properties, which reduces the number of model
parameters from 2M + 1 to five, namely, QR (ml),
k1[b1]/ (s1),
k2[b2]/ (s1),
k1 (s1), and
k2 (s1).
To account for the fact that in CFA-treated lungs a significant
fraction of diazepam was not recovered within the MID sampling time,
the dissociation rate constant for one of these two classes, k2, was set equal to zero. Hence, Eqs. 5 and 6 reduce to
![= <FENCE><FR><NU>Q<SUB>R</SUB></NU><DE>Q<SUB>c</SUB> + Q<SUB>R</SUB></DE></FR></FENCE> <FENCE><FENCE><IT>k</IT><SUB>−1</SUB>[D<SUB>e<SUB>1</SUB></SUB>] − <IT>k</IT><SUB>1</SUB> <FR><NU>[b<SUB>1</SUB>]</NU><DE>&bgr;</DE></FR>[D]</FENCE> − <IT>k</IT><SUB>2</SUB> <FR><NU>[b<SUB>2</SUB>]</NU><DE>&bgr;</DE></FR>[D]</FENCE>](/content/vol86/issue6/fulltext/1866/img014.gif)
|
(7)
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![<FR><NU>∂[D<SUB>e<SUB>1</SUB></SUB>]</NU><DE>∂<IT>t</IT></DE></FR> = <IT>k</IT><SUB>1</SUB> <FR><NU>[b<SUB>1</SUB>]</NU><DE>&bgr;</DE></FR>[D] − <IT>k</IT><SUB>−1</SUB>[D<SUB>e<SUB>1</SUB></SUB>]](/content/vol86/issue6/fulltext/1866/img015.gif)
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(8)
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and the number of model parameters reduces to four, namely,
QR (ml), k1[b1]/
(s1), k1 (s1)
and k2[b2]/ (s1).
To model a bolus injection, the solution to Eqs. 4, 7 and 8 is constrained by the initial (t = 0) conditions,
[D](x,0) = [De1](x, 0) = [R](x,0) = 0, and boundary (x = 0) conditions
[De1](0, t) = 0, [D](0, t) = Cin(t)/(1 + [P]Kp),
and [R](0, t) = Cin(t), where Cin(t) is the capillary input function.
The above deterministic model provides a conceptual basis for the
evolution of the data, but the model parameters can involve several
terms that are not separately identifiable. In addition, there is no
obvious reason to expect that the ith kinetic parameter characterizes the same physicochemical phenomenon for more than one set
of experimental conditions (1). Therefore, for making comparisons, it
is convenient to express the model parameters as stochastic parameters
as follows (1). Integrating Eq. 8 in time results in
 = <IT>k</IT><SUB>f</SUB> <LIM><OP>∫</OP><LL>0</LL><UL><IT>t</IT></UL></LIM> <IT>e</IT><SUP>−<IT>k</IT><SUB>−1</SUB>(<IT>t</IT> − &tgr;)</SUP>[D](<IT>x</IT>, &tgr;) d&tgr;](/content/vol86/issue6/fulltext/1866/img016.gif)
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(9)
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where kf = QR(k1[b1])/ (ml/s)
is the effective association rate of diazepam with the binding species,
with association and dissociation rate constants k1
and k1. Substituting Eq. 9 into Eq. 7 reduces Eqs. 7 and 8 into the
following
![− <FENCE><FR><NU>1</NU><DE>Q<SUB>c</SUB> + Q<SUB>R</SUB></DE></FR></FENCE> <IT>k</IT><SUB>seq</SUB>[D]](/content/vol86/issue6/fulltext/1866/img020.gif)
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(10)
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where kseq = QR(k2[b2])/ (ml/s)
is the sequestration rate of diazepam within Qti and
(t) = k1ek1t
is the sojourn time distribution (for the slowly
equilibrating classes of interactions) (1). The mean sojourn time is
the first moment, (1), of
(t)
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(11)
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The terms on the right-hand side of Eq. 10 represent three
possible classes of diazepam-tissue interactions. Physically,
QR (ml) and QS = 
(ml) represent
virtual volumes that are reflective of the capacities of two classes of
associations referred to as rapidly and slowly equilibrating classes,
respectively (1). The rapidly (relative to the capillary mean transit
time) equilibrating associations of diazepam within Qti and
Qc, which are not mathematically distinguishable from each
other, are all represented by QR (ml). The slowly
equilibrating associations are quantified by QS (ml) and by
the mean sojourn time (s). A third class
of diazepam-tissue interactions with dissociation rate constants that
are so small that there is virtually no return to the perfusate within
the sampling period is described by the sequestration rate
kseq (ml/s).
[14C]PEA
The model used to interpret the uptake of PEA by the pulmonary
endothelial cells was developed in Ref. 2. Again, each capillary element includes a vascular volume Qc. The PEA also has
access to a flow-limited volume QF, within which it can
participate in rapidly equilibrating associations with the tissue or it
can be transported into the endothelial cells via passive diffusion
(18) or some other linear transport mechanism (6) having a
permeability-surface area product (PS). This transport is
assumed to be unidirectional, as discussed below. The transit of PEA
through such a capillary element can be described by the following
equation
![<FR><NU>−PS</NU><DE><FENCE>Q<SUB>c</SUB> + Q<SUB>F</SUB> <FENCE>1 + <FR><NU>[n<SUB>e</SUB>]</NU><DE><IT>K</IT><SUB>e</SUB></DE></FR></FENCE></FENCE> </DE></FR> [D<SUB>p</SUB>]](/content/vol86/issue6/fulltext/1866/img025.gif)
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(12)
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where [Dp](x, t) is the
vascular concentration of PEA at distance x from the capillary
inlet and time t; [ne] represents the
concentration of the rapidly equilibrating association sites within
QF, having association and dissociation rate constants ke and ke, respectively,
such that Ke = ke/ke is the
equilibrium dissociation rate constant. The model parameters are
PS (ml/s) and VF = QF(1 + [ne]/Ke) (ml).
To model a bolus injection, the solution to Eq. 12 is
constrained by the initial (t = 0) condition,
[Dp] (x,0) = 0, and the boundary condition
[Dp](0, t) = Cin(t),
where Cin(t) is the capillary input function.
Organ Models
Equations 4, 7, 8, and 12 are for single capillary
elements. To construct an organ model, the distribution of pulmonary
capillary transit times, hc(t), needs to be
taken into account (1-5, 7, 8). Previously (4), we demonstrated
that the effect of hc(t) on the estimated
kinetic model parameters for test indicator-tissue interactions can be
accounted for by a function c(t) the
mean transit time and first two central moments of which can be
specified from the moments of the concentration vs. time curves of a
flow-limited indicator such as 3HOH,
CF(t), and a vascular reference indicator
CR(t), by using Eq. 13, a-c, which
relates the mean transit time 
c,
the variance (second central moment)
, and the third central moment
of
c(t) to those of
CF(t), CR(t), and
CT(t)
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(13 a-c)
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where
and the subscripts c, F, R, and T refer to
c(t), CF(t),
CR(t), and CT(t),
respectively; is the
extravascular part of 3HOH mean transit time;
2T is the variance of the measured tubing
concentration vs. time outflow curve CT(t); and
c(t) was represented by a shifted
random walk function, the functional form of which can be specified by
its first three moments (1, 2, 4, 10).
The n(t), which accounts for the
system dispersion outside of the capillaries (i.e., in arteries, veins,
connecting tubing, and the injection system) is related to
c(t) and
CR(t), the organ reference indicator outflow
curve, by CR(t) = (q/F)
c(t) * n(t), where * is the
convolution operator, q is the mass of the injected indicator, and
F is the total flow through the organ. As described previously (1, 2,
4, 10), n(t) was also represented by
a shifted random walk function the parameters of which were specified
by iteratively convolving in(t) = (q/F) n(t) with
c(t), until the optimal least square
fit to CR(t) was obtained. Because tracer
concentrations were used for all test indicators, all kinetic processes
are first order, and neither the actual magnitude of the organ input
concentration curve in(t) nor the
anatomic sequence of dispersing components of the system needs to be
specifically considered (1-5, 10).
For given initial and boundary conditions, Eqs. 4, 7, and 8 or Eqs. 4 and 12 were solved numerically by
using the finite-difference method (1, 4). The solution is for a
single-capillary element with in(t)
as the capillary input concentration curve. As previously described
(1, 4), the model solution for a single capillary having the
maximum capillary transit time also provides the output for all
capillary transit times between the minimum and maximum capillary
transit times (1, 2, 4). To provide the whole organ output for vascular
reference indicator CR(t) and test indicator CD(t) for diazepam or
Cp(t) for PEA, the outputs for all transit times are summed, each weighted according to
c(t) (1, 2, 4).
Parameter Estimation
For each of the conditions studied, the first three moments of
c(t) were estimated from the moments
of the outflow curves of 3HOH, FITC-Dex, and the tubing
concentration vs. time outflow curve CT(t) by
using Eq. 13, a and b.
Given c(t) and
n(t) for each of the conditions
studied, the kinetic model parameters descriptive of diazepam-tissue
interactions were obtained by fitting Eqs. 7 and 8 to
the outflow curve of diazepam. The number of model parameters
identifiable from the diazepam data was determined by using the
F ratio for nested models (25, 32). The concentration vs. time
outflow curve of diazepam was first fitted to the model with one
parameter, namely, QR, with the other three parameters set
to zero. The number of model parameters was then increased stepwise,
and the superiority of the sequential fits was evaluated by using the
F-test for nested models. To minimize the instability due to
the high correlation between QR and the class of slowly
equilibrating associations when the values of k1
and k1[b1]/ are very large, an upper bound
of 2/
was placed on
k1 (s1) and
k1[b1]/ (s1), above which
they are considered rapidly equilibrating.
For each of the conditions studied, given
c(t), the kinetic model parameters
descriptive of PEA-tissue interactions were obtained by fitting Eq. 12 to the outflow curve of [14C]PEA, up to the peak
of the FITC-Dex concentration curve.
Statistical Analysis
Parameter values are given as means ± SE. For each parameter,
statistically significant differences among the different groups studied were determined by using one-way analysis of variance, followed
by the Dunnett's method for multiple comparisons vs. the control
group. P < 0.05 was considered statistically significant.
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MODEL RESULTS |
Table 3 shows the kinetic model parameter
estimates from the model fits to the diazepam outflow curves and the
measures of precision of these estimates under the conditions studied
(1, 25). The stochastic parameters for diazepam and the other MID parameters given in Table 4 reflect the
condition-specific patterns in the concentration curves for the
experimental groups studied.
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Table 3.
Kinetic parameters and measures of precision in the estimates of their
values obtained by fitting Eqs. 7 and 8 to the outflow curves of
diazepam for each experimental condition studied
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Table 4.
Estimated stochastic parameters for diazepam and other
multiple-indicator dilution parameters for each experimental
condition studied
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FITC-Dex
Atelectasis and embolism reduced Qv, whereas neither
alveolar fluid instillation nor CFA treatment had a significant effect on Qv. The vascular relative dispersion RDv
(Eq. 2) increased in atelectatic lungs in CFA-treated lungs and
in fluid-filled lungs after embolism.
3HOH
Qw was significantly increased in CFA-treated and in
fluid-filled lungs, and it returned toward normal values after
embolization of the fluid-filled lungs.
[14C]Diazepam
Figure 2 exemplifies the model fits to the
[14C]diazepam data under each of the conditions studied.
The coefficients of variation for the model fits were on average 9.7 ± 0.5 (SE) %. The example sensitivity functions plotted in Fig.
3 provide a graphic representation of the
sensitivities of the kinetic model parameters, and the correlation
matrix shown in Table 3 quantifies the correlation between these
parameters (1, 25, 32).
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Fig. 2.
Venous effluent concentration vs. time curves for FITC-Dex and
[14C]diazepam after bolus injection of these indicators
into pulmonary artery of an isolated perfused rabbit lung under each of
experimental conditions studied. Solid line superimposed on the data
represents results of fitting Eqs. 7 and 8 to diazepam
data under each of experimental conditions studied.
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Fig. 3.
Normalized sensitivity functions (S(t) for the 4 model
parameters, namely, QR, k1
[b1]/, k1 and
k3 [b3]/ using parameter values
similar to those estimated for diazepam in a normal (top) and
an inflamed (bottom) lung. S(t) was normalized to its
peak value. For a given parameter, S(t) is approximated as
the change in CD(t), resulting from changing
the parameter by 1%, divided by the change in parameter value (see
Ref. 1). See Glossary for symbol definitions.
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The calculated stochastic parameters given in Table 4 show that CFA
treatment resulted in an increase in kseq and
QS, whereas simply increasing the water content of the
lungs by alveolar instillation of PSS resulted in no significant
changes in any of the diazepam parameters. Instillation of the PSS + BSA increased QR. Embolization of fluid-filled lungs
resulted in a return toward normal values in QR, with
virtually no change in
Qw/(QR + QS).
PEA
The PS for PEA was not significantly affected by CFA treatment
or by alveolar fluid instillation but was decreased when the lungs were
embolized. It was also decreased in atelectatic lungs.
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DISCUSSION |
Each of the indicators used in this study was chosen to target a
particular tissue property. However, that does not imply that the lung
disposition of each indicator is expected to be affected only by the
targeted property. For example, the extravascular water volume
accessible to 3HOH, Qw, is affected not only by
the lung extravascular water volume but also by the fraction of the
lung tissue that it can reach by diffusion. This is revealed by the
experiments wherein the tissue water volume and the fraction of
perfused lung tissue were manipulated by filling the lungs with saline
solution and by embolizing these lungs, respectively. Disappearance of
PEA from the perfusate is via uptake into the endothelial cells (6, 18). Its uptake is expected to be sensitive to changes in the number of
endothelial cells exposed to flowing perfusate and, possibly, to
alterations in the endothelial cell PEA permeability mechanism (6, 18).
Diazepam was chosen because it accesses the lipoid fraction of the lung
tissue relatively independently of the water volume (3, 5, 16). It is
expected to also be affected by the fraction of the lung tissue that it
can reach by diffusion. An unexpected finding was that the diazepam
disposition was also altered qualitatively in the CFA-inflamed lungs.
This is revealed by the marked changes in the shape of the effluent concentration vs. time outflow curve of diazepam and by the large fraction of the diazepam that did not return to the perfusate within
the 60-s sampling period. The mechanism(s) responsible for these
changes in the lung disposition of diazepam are not certain. However,
it appears likely that the lung tissue level of "peripheral" or
"mitochondrial" benzodiazepine receptors (24) was increased in
the inflamed lungs. Whatever the mechanism, the sequestration
phenomenon may reflect invasion by monocytes or some other cell type or
a change in the function of the resident cells, and it may be
worthwhile to determine whether it has specificity for inflammatory
responses of different types.
Previously (3, 5, 16), we found that under control conditions diazepam
was nearly flow limited by the criteria that, over a wide range of
flows, its venous effluent concentration curves were nearly congruent
on a time scale normalized to the lung mean transit time. In the
present study, we discovered that in inflamed lungs the behavior of
diazepam was clearly different
TOP
ABSTRACT
INTRODUCTION
![<FR><NU>∂[D]</NU><DE>∂<IT>t</IT></DE></FR> + <IT>W</IT> <FENCE><FR><NU>Q<SUB>c</SUB></NU><DE>Q<SUB>c</SUB> + Q<SUB>R</SUB></DE></FR></FENCE> <FR><NU>∂[D]</NU><DE>∂<IT>x</IT></DE></FR>](/content/vol86/issue6/fulltext/1866/img008.gif)
![&bgr; = 1 + <LIM><OP>∑</OP><LL><IT>i</IT> = (<IT>M</IT> + 1)</LL><UL><IT>N</IT></UL></LIM>N[b<SUB><IT>i</IT></SUB>]/<IT>K</IT><SUB><IT>i</IT></SUB>](/content/vol86/issue6/fulltext/1866/img011.gif)
![<FR><NU>∂[D]</NU><DE>∂<IT>t</IT></DE></FR> + <IT>W</IT> <FENCE><FR><NU>Q<SUB>c</SUB></NU><DE>Q<SUB>c</SUB> + Q<SUB>R</SUB></DE></FR></FENCE> <FR><NU>∂[D]</NU><DE>∂<IT>x</IT></DE></FR>](/content/vol86/issue6/fulltext/1866/img013.gif)
![<FR><NU>∂[D]</NU><DE>∂<IT>t</IT></DE></FR> + <IT>W</IT> <FENCE><FR><NU>Q<SUB>c</SUB></NU><DE>Q<SUB>c</SUB> + Q<SUB>R</SUB></DE></FR></FENCE> <FR><NU>∂[D]</NU><DE>∂<IT>x</IT></DE></FR>](/content/vol86/issue6/fulltext/1866/img017.gif)
![= − <FENCE><FR><NU>1</NU><DE>Q<SUB>c</SUB> + Q<SUB>R</SUB></DE></FR></FENCE> <IT>k</IT><SUB>f</SUB>[D]](/content/vol86/issue6/fulltext/1866/img018.gif)
 d&tgr;](/content/vol86/issue6/fulltext/1866/img019.gif)
![<FR><NU>∂[D<SUB>p</SUB>]</NU><DE>∂<IT>t</IT></DE></FR> + <IT>W</IT> <FENCE><FR><NU>Q<SUB>c</SUB></NU><DE>Q<SUB>c</SUB> + Q<SUB>F</SUB> <FENCE>1 + <FR><NU>[n<SUB>e</SUB>]</NU><DE><IT>K</IT><SUB>e</SUB></DE></FR></FENCE></DE></FR></FENCE> <FR><NU>∂[D<SUB>p</SUB>]</NU><DE>∂<IT>x</IT></DE></FR> =](/content/vol86/issue6/fulltext/1866/img024.gif)
![&bgr;=1+<LIM><OP>∑</OP><LL><IT>i</IT>=(<IT>M</IT>+1)</LL><UL><IT>N</IT></UL></LIM>[b<SUB><IT>i</IT></SUB>]/<IT>K</IT><SUB><IT>i</IT></SUB>](/content/vol86/issue6/fulltext/1866/img062.gif)