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 than in normal lungs. To compare the
kinetics of tissue disposition of diazepam under all conditions
studied, we used the general model, of which the flow-limited model is
a nested version, and the same fitting procedure for all conditions.
When this was done, in several cases, multiple parameters turned out to
be identifiable for diazepam even under control conditions. The reasons
for this apparent incongruity and for the large SE values in the
estimated values of some diazepam kinetic model parameters in Table 3
and the stochastic parameters in Table 4 are developed in the
APPENDIX.
The fact that the tissue disposition of the injected indicators is
affected by multiple factors helps to complicate interpretation. However, the concept proposed herein and discussed previously (1, 7, 8,
16, 23, 29) is that with a sufficient number of indicators, each having
at least one unique or relatively selective interaction, enough of the
ambiguity can be eliminated to distinguish among lungs having different
properties. For example, Harris et al. (23) demonstrated how the ratio
of permeability surface area products for hydrophilic and amphipathic
indicators could be used to distinguish changes in the capillary
permeability from changes in perfused surface area. Merker and Gillis
(29) used the lipophilic indicator propranolol to separate the effects of changing surface area and endothelial cell metabolism on endothelial serotonin uptake in injured lungs. We had demonstrated how an index of
wet-to-dry weight ratio could be obtained by using the ratio of the
volumes accessible to 3HOH and a lipophilic indicator such
as diazepam, i.e., Qw/QR (16). In the present
study, that ratio is modified to
Qw/(QR + QS) to include the
more general representation of the lipoid volume accessible to
diazepam, (QR + QS), which also
accommodates the compositional changes that took place in the
CFA-treated lungs.
The alveolar fluid instillation was carried out on atelectatic lungs to
obtain as even filling as possible (16). Atelectasis allows for more
even filling by eliminating the effects of surface tension at the
distributed air-liquid interface and by eliminating any mixture of air-
and liquid-filled alveoli. MID injections were also made into the
alelectatic lungs before fluid filling to make sure that the
atelectasis alone did not have some unforeseen effects. The effects of
atelectasis on vascular volume, PS, and RDv are
predictable from the known effects of atelectasis on the pulmonary
vascular bed (33).
The ratio of the extravascular water accessible to 3HOH,
Qw, to the water volume of the normal lungs, measured as
the difference between the lung wet and dry weights, was 82%, as shown
in Table 2. Part of the difference was no doubt due to the fact that we made no attempt to account for the perfusate trapped in the vascular space, which contributes to an overestimation of the lung wet weight
(12). Issues regarding the fraction of the extravascular water volume
recovered by indicator dilution methods have been discussed extensively
by others (12, 13) and are not of major concern in the present study.
On the other hand, the fact that this fraction was hardly affected by
filling the lungs with fluid indicates that the instilled alveolar
fluid was accessible to about the same extent as the normal cellular
and interstitial water. Thus the longer diffusion distances that
resulted from alveolar filling did not have a substantial impact on the
accessibility of the extravascular water volume to the 3HOH
bolus. Table 2 shows that the result was different for CFA-treated lungs, wherein only ~45% of the lung water volume measured as the
lung wet dry weight was detected by 3HOH, even though
these inflamed lungs had almost as large a gravimetrically detectable
water volume as the fluid-filled lungs. This may imply a significantly
more heterogeneous access to the extravascular water volume in lungs
with granulomatous inflammation. Such an effect may be mimicked by
embolization which, as Table 2 shows, resulted in an even lower
Qw-to-lung wet minus dry weight ratio, presumably because
the water in some areas wherein the perfusion was obstructed was too
far from perfused vessels to be effectively traced by the
3HOH.
In lungs filled with fluid having the same composition as the perfusate
(PSS + BSA), assuming that all of the instillate is accessible via
diffusion, the model would predict that QR would have
increased by the volume of the instillate. The increases in
QR was on average ~32 ml in comparison to the 35 ml
instilled, which suggests that the alveolar fluid was in fairly rapid
diffusional communication with the vascular perfusate (16).
PEA was chosen as the endothelial surface indicator because its
extraction on passage through the lungs has been found to be mainly
through the endothelial uptake (6, 18). In contrast to the lipophilic
amine diazepam, PEA is a hydrophilic amine, and, in the context of the
present study, a key observation was that its uptake was not
substantially affected when the water volume of the lungs was changed
by fluid instillation or by CFA treatment. This is consistent with the
dominance of the endothelial barrier over the tissue water volume in
determining PEA extraction. A potential advantage of PEA instead of, or
in addition to, some hydrophilic indicators that have been used in the
lungs, such as urea or Na (2, 22, 38), is that its extraction is
relatively high, providing more sensitivity for detecting decreases in
extraction. Its high uptake is probably partly the result of the fact
that it is metabolized within the cells, thus lessening the impact of
cellular accumulation (backdiffusion) on net uptake. A disadvantage of
metabolism is that the 14C in the effluent can be
contaminated by the metabolite. In the case of PEA, its metabolite PAA
is relatively cell impermeant, which may also help retard the
backdiffusion of 14C. A common approach to the problem of
backdiffusion and metabolite contamination has been to assume that
unidirectional uptake dominates the net extraction during the rising
portion of the effluent concentration curves (26). We took this
approach in the present study to avoid having to measure
[14C]PAA concentration in the several hundred samples
collected. However, it is possible that, in future studies, the useful
information content of the data would be increased by measuring the
metabolite concentration curves as well.
Pattern Recognition
Although each indicator was chosen to target a particular property, the
results can also be evaluated nonmechanistically in terms of the
parameter patterns generated. The numbers of indicators and variations
in lung properties in this study were small in comparison to the large
number of possible indicators selective for various tissue properties
and the wide range of tissue properties affected by different lung
diseases. Even with these relatively small numbers of MID parameters
and experimental conditions, the tabular representations such as Table
4 are complicated enough to make it difficult to quickly discern the
distinguishing features of the pattern for each experimental group by
perusing the numbers in Table 4. On the other hand, in this limited
example, a trivial pattern-recognition scheme distinguishes among
experimental groups. For a given group, a 
or a 
entry in Table 4
indicates a significant increase or a significant decrease in the
corresponding parameter, compared with that under control conditions,
respectively. Each group can be uniquely identified by simply counting
the number of and in a row of Table 4, without even addressing
which parameters deviate from normal. To provide an example of the
inverse problem, that of identifying the experimental group to which an individual lung belongs, we determined the control group mean and SD
for each parameter that was significantly affected by one or more
treatment. Then the z score [z = (parameter for
unknown mean of control group)/(SD of control group)] for each
parameter for each lung was determined. For each experimental group,
the mean z score
() for each of the
Np parameters was determined. The end result is
that each lung can be characterized by a set of Np
z scores (pattern), and each experimental condition can be
characterized by a set of Np mean z scores.
Figure 4 shows the z-score patterns
for each of the experimental conditions studied. Based on its set of
Np z scores, each lung can be classified
into one of the experimental groups studied by using the
minimum-distance pattern classification (37), which consists of
determining the sum of squares differences (SSD) between its set of
z scores and that of each of the experimental conditions using
the following equation
where zi is the z score
of the ith parameter for a given lung, and
is the
mean z score for the ith parameters for the jth
experimental group. The lung is classified as a member of the
experimental group with the minimum SSD. Using this criterion, of the
52 lungs in the experimental groups indicated in Table 4, 50 were
correctly identified. Two lungs that belong to the
(PSS + BSA + embolism) experimental group were incorrectly
identified as an atelectatic lung and a (PSS + BSA) lung. All of the
lungs from the control group were correctly identified. The fact that
this approach worked to the extent that it did on this limited sample
is not of very profound significance, since there is a fairly high
expectation that most of the lungs contributing to a particular average
pattern in Fig. 4 should have similar patterns. However, it
demonstrates the concept that, with a sufficient number of indicators
having different relative specificities for different tissue
properties, the parameter vector itself may be a phenotype useful for
detecting and identifying a diseased or injured lung.
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Fig. 4.
Mean z score () patterns
representing a normalized change in estimated model parameters for each
of experimental conditions studied. Ratio = Qw/(QR + QS), where
QR and QS represent virtual volumes that are
reflective of the capacities of 2 classes of diazepam-tissue
associations, referred to as rapidly and slowly equilibrating classes,
respectively; Qv, RDv, and Qw are
vascular volume, vascular relative dispersion, and perfused
extravascular water volume accessible to 3HOH,
respectively; kseq and
are sequestration rate of diazepam within
tissue volume and mean sojourn time, respectively; and PS and
VF are permeability-surface area product and a scaled
flow-limited volume accessible to [14C]PEA, respectively
(see text).
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In conclusion, this study demonstrates some aspects of the feasibility
of using the MID method, with an appropriate group of test indicators
and parameterization via MID models, to detect and to quantify changes
in lung tissue properties resulting from lung injury and disease. One
of the key observations was that CFA treatment resulted in qualitative
changes in the lung disposition of diazepam, which distinguished the
inflammatory response from a simple change in lung mass or change in
tissue accessibility to indicators delivered via the pulmonary
circulation. This suggests that one may be able to take advantage of
the many different types of tissue interactions that various lipophilic
amines can participate in (1, 30, 34) to characterize lung tissue properties.
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APPENDIX |
Previously, we (3-5) developed a method for estimating the
pulmonary capillary transit time distribution,
hc(t), based on the use of "flow-limited"
indicators. In that study, diazepam was found to be nearly flow limited
by the criterion 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
than in normal lungs. The shape of the concentration curve was
different, and a substantial fraction of the injected diazepam was not
recovered within the sampling period. Hence, diazepam could not be used to estimate hc(t) in inflamed lungs in which
the required flow-limited behavior no longer exists. Because in the
present study we have at least one condition wherein diazepam is not
flow limited, to compare the tissue disposition of diazepam under all
conditions studied, we used the general model, of which the
flow-limited model is a nested version, and the same model-fitting
procedure for all conditions. When this was done, additional parameters were commonly identifiable (as revealed by the sensitivity analysis exemplified in Fig. 3) and significant (as indicated by the
F-test) for diazepam, even under control conditions. At least
two related factors contribute to this result and to the large SEs in
the estimated values of some diazepam kinetic model parameters in Table
3 and in the stochastic parameters in Table 4.
One is that the F-test is an objective means of demonstrating
whether adding parameters improves the fit, but it has virtually nothing to do with the robustness of the parameters that it adds to the
model. Very small improvements in the fit can pass the F-test
when the concentration curve, such as for diazepam in normal and
fluid-filled lungs, approaches that of a flow-limited indicator. For
instance, Fig. 5 shows two model fits to
the diazepam concentration curve from a normal lung. One fit was
obtained with the full model, i.e., with all four parameters free
(dashed line). The other was obtained with the flow-limited model,
i.e., with QR as the only free parameter (solid line) and
the other three parameters set equal to zero. Based on the
F-test for nested models, the fit to the diazepam outflow curve
with the full model is superior to the fit with the flow-limited model,
even though the superiority is difficult to discern by visual
observation. One result is that the number of parameters varies from
lung to lung, contributing to the large SEs in the estimates of some of
the diazepam model parameters under conditions wherein diazepam is
nearly flow limited.
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Fig. 5.
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 normal rabbit lung. Lines
superimposed on the data are the result of fitting Eqs. 7 and 8 to diazepam data with either all 4 model parameters free
(dashed line, full model), or with QR as the only free
parameter and the other 3 parameters set equal to zero (solid line,
flow-limited model).
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The second related factor is the sensitivity of the model fit to very
small variations in the capillary transport function as the pattern of
the outflow curve approaches that of a flow-limited indicator. This
sensitivity to hc(t) is, in fact, one of the
reasons that the flow-limited indicators can be used to estimate
hc(t) (3-5). In our previous work (4), we
demonstrated that the effect of hc(t) on the
estimated kinetic model parameters for test indicator-tissue
interactions from MID data 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 a
flow-limited indicator and a vascular reference indicator by using
Eq. 13, a-c. In the present study, the outflow curve of
3HOH was chosen to estimate
c(t), since it does not participate in any slowly equilibrating tissue binding under any experimental condition. As discussed previously (4), the sensitivity of the
estimates of the kinetic model parameters to small errors in
c(t) decreases as the behavior of
the test indicator deviates from that for a flow-limited indicator.
This aspect of the sensitivity of the model to
c(t) is revealed in Fig.
6, which shows the model fit to diazepam
concentration vs. time outflow curve from a normal lung with two
different c(t) values obtained by
using the moments of the outflow curve of either diazepam (i.e.,
assuming that diazepam is flow limited under normal conditions) or
3HOH in Eq. 13, a-c. Although the two
estimated c(t) values shown in Fig.
6 are not very different, they resulted not only in different values
for the model parameters but also in different numbers of identifiable
parameters. With the use of the F-test, only two parameters
were identifiable with the c(t)
obtained by using the moments of the diazepam outflow curve in Eq. 13, a-c, whereas four parameters were identifiable with the
c(t) obtained by using the moments
of the 3HOH outflow curve in Eq. 13, a-c. For
the CFA comparison, it could not be assumed that diazepam could give a
reasonable approximation to c(t).
Therefore, to simulate the impact of choosing the wrong c(t) in the CFA case, we used the
c(t) estimated by using the moments
of 3HOH in Eq. 13, a-c and an
c(t) the moments of which deviated from the moments of that obtained by using the moments of
3HOH by the same ratio as for diazepam in the normal lung.
In the resulting simulation, similar differences between the two
c(t) values had no effect on the
number of kinetic model parameters for diazepam from CFA-treated lungs
and little effect on the estimated values of these parameters, and the
fits were indistinguishable on the scale of the figures. The key point
is that the ability of the general model to fit the diazepam data from
normal lungs better than the flow-limited model in some cases does not
imply that diazepam is not virtually flow limited under these
conditions; rather, it is a reflection of the high degree of
sensitivity to small variations in hc(t).
View larger version (23K):
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|
Fig. 6.
Venous effluent concentration vs. time curves for FITC-Dex and
[14C]diazepam after bolus injection of these indicators
into pulmonary artery of isolated perfused rabbit normal (top
left) and inflamed (top right) rabbit lungs. For each
of 2 experimental conditions, lines superimposed on the data are result
of fitting Eqs. 7 and 8 to diazepam data for the 2 different c(t) values, shown in the
respective bottom panels. Estimates of model parameters for the
normal lung were QR = 11.8 ml,
k1 [b1]/ = 0.24 (s1), k1 = 0.63 (s1), and k3[b3]/ = 0.019 (s1), with a coefficient of variation (CV) of
8.7% for c1(t); and QR = 16.2 ml, k1 [b1]/ = 0 (s1), k1 = 0 (s1), and k3[b3]/ = 0.017 (s1), with a CV of 10% for
c2(t). Estimates of model parameters
for CFA-treated lungs were QR = 12.4 ml,
k1[b1]/ = 0.54 (s1), k1 = 0.17 (s1), and k3
[b3]/ = 0.25 (s1), with a CV of
10.5% for c1(t); and QR = 16.5 ml, k1 [b1]/ = 0.46 (s1), k1 = 0.17 (s1), and k3
[b3]/ = 0.20 (s1), with a
CV of 9.8% for c2(t).
|
|
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ACKNOWLEDGEMENTS |
This study was supported by the Department of Veterans Affairs, the
Whitaker Foundation, the Falk Trust, and National Heart, Lung, and
Blood Institute Grant HL-24349.
| |
FOOTNOTES |
Address for reprint requests and other correspondence: S. H. Audi,
Research Service 151, Zablocki VA Medical Center, 5000 W. National
Ave., Milwaukee, WI 53295-1000 (E-mail: audis{at}vms.csd.mu.edu).
Received 31 August, 1998; accepted in final form 15 February 1999.
| |
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J APPL PHYSIOL 86(6):1866-1880
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Copyright © 1999 the American Physiological Society
TOP
ABSTRACT
INTRODUCTION
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![&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)