Vol. 88, Issue 4, 1155-1166, April 2000
Magnetic resonance behavior of normal and diseased lungs:
spherical shell model simulations
Carl H.
Durney1,
Antonio G.
Cutillo2, and
David C.
Ailion3
Departments of 1 Electrical Engineering,
2 Medicine, and
3 Physics, University of Utah, Salt Lake
City, Utah 84112
 |
ABSTRACT |
The
alveolar air-tissue interface affects the lung NMR signal, because it
results in a susceptibility-induced magnetic field inhomogeneity. The
air-tissue interface effect can be detected and quantified by measuring
the difference signal (
) from a pair of NMR images obtained using
temporally symmetric and asymmetric spin-echo sequences. The present
study describes a multicompartment alveolar model (consisting of a
collection of noninteracting spherical water shells) that simulates the
behavior of as a function of the level of lung inflation and can be
used to predict the NMR response to various types of lung injury. The
model was used to predict as a function of the inflation level
(with the assumption of sequential alveolar recruitment, partly
parallel to distension) and to simulate pulmonary edema by deriving
equations that describe for a collection of spherical shells
representing combinations of collapsed, flooded, and inflated alveoli.
Our theoretical data were compared with those provided by other models
and with experimental data obtained from the literature. Our results
suggest that NMR measurements can be used to study the mechanisms
underlying the lung pressure-volume behavior, to characterize lung
injury, and to assess the contributions of alveolar recruitment and
distension to the lung volume changes in response to the application of
positive airway pressure (e.g., positive end-expiratory pressure).
alveolar air-tissue interface; lung magnetic resonance imaging; lung pressure-volume behavior; alveolar recruitment; pulmonary edema
| |
INTRODUCTION |
THE UNIQUE, FOAMLIKE STRUCTURE of the lung produces an
NMR free induction decay (FID) in inflated lungs that is shorter than that observed in fully collapsed lungs or in solid tissue (1, 32). This
shorter FID has been attributed to the effects of magnetic
susceptibility differences between tissue and air (32). Because tissue
is slightly diamagnetic, the extensive alveolar air-tissue interfaces
perturb the magnetic flux density, causing internal (tissue-induced)
magnetic field inhomogeneity (also termed inhomogeneous NMR line
broadening), which is responsible for the NMR signal loss (shortening
of the FID). Detergent foams, slurries, and shaving cream have also
been found to produce shortened FIDs (1, 32).
In previous work (5, 7, 8, 13, 17) the effects of internal magnetic
field inhomogeneity on the NMR signal (also called susceptibility
effects) have been quantified by measuring a difference signal denoted
by . The basic principles underlying the measurement of are
discussed in detail in a previous publication (13). Briefly, is the
difference between spin-echo signals produced by temporally symmetric
and asymmetric spin-echo sequences. A spin-echo sequence (Fig.
1) is symmetric if the time 
from the
center of the 90° pulse to the center of the 180° pulse is equal to the time ' from the center of the 180° pulse to
the center of the echo. The spin-echo sequence is asymmetric if 
'. The magnitude of the temporal asymmetry is expressed
by the asymmetry time a = 'a 
a.
is generally determined, by subtraction, from two NMR images
(obtained using, respectively, symmetric and asymmetric spin-echo
sequences with asymmetry times of 3-6 ms). It is expressed as the
difference between the symmetric and asymmetric signal intensities
divided by the symmetric signal intensity (see METHODS). In
inflated lungs the internal magnetic field inhomogeneity caused by the
presence of the alveolar air-tissue interface reduces the asymmetric
signal but not the symmetric signal (see METHODS).
Therefore, is a measure of the alveolar air-tissue interface
effects and reflects the state of lung inflation. For example, is
expected to be zero for fully collapsed (nonaerated) lungs and to
increase markedly as the lungs are inflated (13).
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Fig. 1.
Symmetric (A) and asymmetric (B) spin-echo pulse
sequences that produce symmetric and asymmetric signals from which
difference signal () is obtained. B1,
radio-frequency magnetic field; FID, free induction decay signal;
Gx, x component of magnetic field
gradient; A1, area of gradient pulse; , time
between 90° and 180° pulses; ', time between 180°
pulse and peak of echo signal. s = 's in symmetric sequence;
a 'a
in asymmetric sequence. Magnetic field gradients in y and
z directions are not shown. [From Durney et al.
(16).]
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|
The connection between lung structure and has been explained in
terms of "thick" and "thin" material (16, 17).
"Thickness" and "thinness" depend only on relative
dimensions, not on absolute dimensions. Thick material (e.g., spheres
and cubes) has approximately the same dimensions in three orthogonal
directions. Thin material (e.g., thin slabs and rods) has at least one
dimension that is significantly less than the other two. Thin material
produces a significantly nonzero , whereas thick material produces a
virtually zero . The characteristics of for a given structure
can be explained in terms of the relative amounts of thick and thin
material present in the structure. For example, in fully collapsed
lungs, only thick material is present, thus explaining the predicted zero . As the lungs are inflated and the air spaces pop open, thick
material is converted to thin material (the alveolar septa in aerated
alveoli), and increases rapidly. The results of measurements of as a function of the inflation level in excised rat lungs (13, 14)
agree with the above theoretical concepts. These measurements have
shown that, as predicted, is very low in degassed lungs, because
degassing virtually eliminates the alveolar air-tissue interface. The
common observation of a low, but not zero as predicted, in degassed
lungs likely reflects the presence of small, macroscopically undetectable amounts of gas (13, 14). Comparative data (45) indicate
that degassing by the oxygen-absorption method is more effective than
in vacuo degassing. However, minute amounts of gas can even be found in
lungs degassed by the oxygen-absorption method (14). When measured at
an asymmetry time of 6 ms in initially degassed normal lungs, (6 ms) increases markedly with alveolar opening and
then varies very little during the rest of the inflation-deflation cycle (13). The behavior of measured at other asymmetry times (3 ms and 5 ms) is qualitatively
similar (13).
These results have interesting physiological implications, because they
suggest that measurements at appropriate asymmetry times are very
sensitive to alveolar recruitment and are relatively insensitive to
changes in the volume of open air spaces (13). This conclusion is also
supported by the results of a comparison of and morphometric
measurements in normal excised rat lungs at two levels of inflation
(14) and by theoretical calculations (model simulations of the behavior
of 3 ms and 6 ms as a function of
lung air content) (13, 16). Therefore, measurements may provide a
means for assessing the roles of alveolar recruitment and distension in
the pressure-volume (P-V) behavior of lung. In addition, flooding of
alveoli (e.g., in pulmonary edema) would be expected to reduce significantly, because it obliterates the air-tissue interface,
converting thin material to thick material. Simulations of alveolar
flooding with use of spherical-shell models (see below) have shown that
flooding does indeed dramatically decrease (16). These predictions
have been confirmed experimentally by the results of measurements
in edematous (oleic acid-injured) or saline-filled lungs (14). The
strong dependence of on the state of lung inflation and on
pathological conditions affecting the alveolar air-tissue interface
suggests the possibility of characterizing lung injury and the response
of injured lungs to changes in inflation pressure by measurements
in conjunction with other NMR measurements (e.g., determination of lung
water content). This potential for injury characterization is
important, because the alveolar air-tissue interface is affected by
many experimental and clinical conditions. As mentioned above, a loss of air-tissue interface occurs in experimental or clinical pulmonary edema [e.g., in patients with acute respiratory distress syndrome (ARDS)] because of alveolar flooding and collapse. A clinical example of the response of injured lungs to changes in inflation pressure is the increase in lung volume due to the application of
positive end-expiratory pressure (PEEP) in ARDS patients (35). This
volume response may result from reopening (recruitment) of collapsed
and flooded alveoli and/or from further distension of already open air spaces.
In previous studies (5, 7, 8, 13, 16, 17) we used several lung models
to simulate the NMR behavior of lung as a function of the level of
inflation. These simulations are based on the assumption that all lung
units operate synchronously (i.e., all alveoli are recruited
simultaneously and then uniformly distended) and, therefore, behave
virtually as a single-compartment system. In the present study our
simulations are extended by considering the case of a multicompartment
model in which the individual units operate asynchronously. This
approach allows not only a more comprehensive evaluation of the process
of lung inflation but also the extension of our simulations to diseased
lungs. Multicompartment models can be used to simulate the NMR
properties of edematous lungs (in particular, the response of collapsed
or flooded alveoli to changes in inflation pressure, such as in the
application of PEEP). Our simulations indicate that the two basic
mechanisms potentially responsible for the increase in lung volume
induced by PEEP (alveolar recruitment and distension) may be
distinguished by measurements. This distinction is important not
only in research, but also in clinical medicine, as discussed below.
In this study a simple multicompartment model is described to predict
the response of to alveolar recruitment and distension in normal
and edematous lungs. Then calculated results are presented and discussed.
| |
METHODS |
Selection of the lung model.
The behavior of as a function of the lung inflation level
can be simulated by models consisting of a simple air-filled spherical shell or arrays of air spaces in a water medium (5, 7, 8, 13, 16, 17).
The air spaces can be spherical (spherical foam), cubical (cubical
foam), or polyhedral (Wigner-Seitz) (3). Spherical-foam, cubical-foam,
and Wigner-Seitz models are more accurate than the spherical-shell
model, because they take into account the magnetic interaction between
alveoli. A more detailed discussion of these models can be found
elsewhere (16). The Wigner-Seitz model has been used in previous
investigations (13, 14) to compare experimental measurements in
lung with theoretical predictions. However, for the present study we
selected a simple spherical-shell model, because the Wigner-Seitz model
does not allow regional variations in air space size (which are
required to simulate asynchronous and nonuniform volume changes in
normal and diseased lungs). In this study the Wigner-Seitz model is
used only for comparative purposes, as explained in greater detail below.
The present model simulates the parenchymal (respiratory) portion of
the lung and basically corresponds to air and alveolar septal tissue.
In the model the term "tissue" (corresponding to the spherical
shell of water) refers to the normal lung tissue, which is assumed to
consist of 100% water (whereas the real lung tissue contains a
nonwater component and ~80% water). The term "water" refers to
the extra water that accumulates in the interstitial tissue and in the
air spaces in pulmonary edema. Interstitial edema and alveolar flooding
are simulated by thickening the spherical water shell and by completely
filling the spherical shell with water, respectively.
Spherical-shell model derivations.
The spherical-shell model consists of a collection of noninteracting
spherical shells, each one representing an alveolus. The great
advantage of the spherical-shell model is its mathematical simplicity,
which leads to understanding that would be difficult to obtain with
more complicated models. Furthermore, previous calculations have shown
that spherical-shell results are very similar to those obtained from
more sophisticated models. The principal disadvantage of the
spherical-shell model is that it does not include the effects of
alveoli on each other in perturbing the magnetic flux density. The for a collection of noninteracting spherical shells has been described
elsewhere (16); for the convenience of the reader, that derivation is
briefly outlined here.
The derivation for begins with an expression for the y
component of the magnetization of the spin echo at any time after the
180° pulse is applied (29)
|
(1)
|
where
M0 is the thermal equilibrium magnetization,
T2 is the spin-spin relaxation time, Vt T is
the total volume (tissue and water) that produces magnetization,
1 = 
Gx0x1 (phase change between the 90° and 180° pulses),
2 = t' + Gx0x(t 2) (phase change between the 180° pulse and
the echo), t' = t (t is time
after the 90° pulse, and t' is time after the 180° pulse), = B0 + B , is the gyromagnetic ratio, B0 is
the applied direct-current magnetic field, B is
the internal magnetic field inhomogeneity induced by the tissue,
Gx0 is the magnitude of the
x gradient, x is the distance along the x-axis,
and is the time between the 90° and the 180° pulse
(s or a in Fig. 1). Other terms (e.g.,
1 and 2) are defined in Fig. 1.
Substituting the expressions for 1 and
2 given above into Eq. 1 and using the
convenient definition that ' is the time from the 180°
pulse to the peak of the echo so that + ' = 1 + 2 gives
![<IT>M</IT><SUB><IT>y</IT></SUB> = <FR><NU>1</NU><DE>V<SUB>tT</SUB></DE></FR> <LIM><OP>∫</OP><LL>V<SUB>tT</SUB></LL></LIM> <IT>M</IT><SUB>0</SUB><IT>e</IT><SUP>−<IT>t</IT>/T<SUB>2</SUB></SUP> cos {&Dgr;ω(<IT>t</IT> − 2&tgr;) + &ggr;<IT>G</IT><SUB><IT>x</IT>0</SUB><IT>x</IT>[<IT>t</IT> −(&tgr; + &tgr;′)]} dV](/content/vol88/issue4/fulltext/1155/img002.gif)
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(2)
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Because the cosine functions in Eq. 2 at different
points in space will be out of phase with each other when and
Gx0x vary with space,
integrating these functions over Vt T reduces My compared with its value if all the
cosine functions were in phase. Refocusing of the dephasing caused by
occurs at t = 2, and refocusing of the dephasing
caused by Gx0 occurs at
t = + '. The spin-echo signal will be largest if refocusing of both dephasings occurs at the same instant of time, i.e.,
if 2 = + ' and, therefore, = '. Because
s = 's, the
spin-echo amplitude is not reduced by tissue-induced inhomogeneous line
broadening for the symmetric signal (Fig. 1). On the other hand, the
asymmetric sequence does produce a reduced spin-echo amplitude, because
a 'a.
Thus the difference between the symmetric
(My s) and asymmetric
(My a) spin-echo signals can
be used as a measure of tissue-induced inhomogeneous line broadening.
This is defined as
|
(3)
|
where
|
(4)
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and
|
(5)
|
and
a = 'a a. For spherical shells consisting of
homogeneous tissue or water, the perturbation of the spatially uniform
B0 will be on the order of a few parts per million.
Consequently, M0 will be spatially uniform within a
few parts per million and, to a good approximation, can be considered
spatially uniform. For homogeneous material, T2 is also
spatially uniform. Therefore, assuming that M0 and
T2 do not vary with space, we obtain
|
(6)
|
Because Vt T in Eq. 6 is the entire
volume in which magnetization is produced, for a collection of
noninteracting spherical shells of different sizes is obtained by
integrating over all the shells in Eq. 6 to obtain
|
(7)
|
where
Vt i is the tissue and/or water
volume of the ith shell (i.e.,
Vt i is the volume that produces
magnetization in the ith shell) and Vt T is the
total magnetization-producing volume of all the shells given by
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(8)
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It
is convenient to write for the entire collection of shells, as
given in Eq. 7, in terms of i ( of
the ith shell), defined as
|
(9)
|
Solving
for
cos(ia)
from Eq. 9, substituting into Eq. 7, and using Eq. 8 gives
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(10)
|
A principal advantage of the spherical-shell model is that
Bi contained in
i in Eq. 9 is easily calculated for a
spherical shell (7). Because Bi is a
function of r and 
in standard spherical coordinates, the
integral in Eq. 9 must be evaluated by some numerical method.
In the calculations described below, we integrated with respect to
r using a series expansion for the cosine function and then
integrated numerically over .
Evaluating Eq. 9 requires choosing a value for the asymmetry
time a, which is a parameter under the control of the
experimenter. For our purposes, the optimum value of
a would be one for which has a strong, but
monotonic, dependence on lung inflation. Calculated values of for
spherical-foam, Wigner-Seitz, and spherical-shell models as a function
of lung inflation with a as a parameter show that does increase monotonically with the level of lung inflation, but only
for a specific range of values of a (7, 16). Because
the variation of with lung inflation is greater for larger values
of a than for smaller values (7, 13), larger values
of a would generally be better, but above a certain critical value of a the variation of with lung
inflation is no longer monotonic. For the spherical-foam and
Wigner-Seitz models, and in actual lung, we have found the best value
of a to be 6 ms; for greater values, the dependence
of on lung inflation is no longer monotonic. For the
spherical-shell model, however, at a = 6 ms
(6 ms) does not vary monotonically with lung inflation, presumably because the spherical-shell model does not take
into account the magnetic interaction between adjacent alveoli, which
is included in the spherical-foam and Wigner-Seitz models. The best
a value for the spherical-shell model is 3 ms, for which the behavior of (3 ms) with lung inflation
is very similar to, although not numerically the same as, that
predicted for a = 5-6 ms with use of more
refined models. Thus the simulations based on 3 ms
for the spherical-shell model are also qualitatively applicable to the
prediction of the experimental behavior of 6 ms. Furthermore, because the spherical-shell model can simulate lung conditions (e.g., asynchronous behavior of parallel lung units) that
cannot be analyzed by other models, it can provide valuable understanding not obtainable otherwise. For the above reasons (see Ref.
16 for a more detailed explanation), all the spherical-shell model
calculations described in the present study were performed with the
assumption that a = 3 ms. Results of calculations
based on 6 ms and the Wigner-Seitz model were used
only for comparisons with the spherical-shell model data and with
experimental observations.
Because the level of lung inflation can be described in terms of air
fraction [FA, air content as a fraction of total
(air + tissue) volume], it is convenient for subsequent
calculations to evaluate Eq. 10 for given FA values
of individual spherical shells and the FA of a combination
of shells. The FA of the ith spherical shell
(FA i) is defined as
|
(11)
|
where
Va i is the air volume of the
ith spherical shell and Vt i
is the tissue volume of the ith spherical shell.
Similarly, FA, the air fraction for a collection of
spherical shells, is given by
|
(12)
|
Solving
for Va i from Eq. 11 and
substituting into Eq. 12 gives
|
(13)
|
Equations
12 and 13 are used subsequently to calculate for
various simulated lung conditions.
Recruitment vs. distension: models of lung inflation.
Essentially, lung inflation occurs by two basic processes:
1) recruitment of totally collapsed alveoli and 2)
further distension of already open alveoli. Additional mechanisms,
changes in alveolar shape and "accordion-like" unfolding of air
spaces with no variation in the alveolar surface area (20), are not
considered in our simulations, because they are not expected to affect
sizably (unless they are associated with air space recruitment, in
which case the predicted effect on is that of process 1).
The respective contributions of recruitment and distension to changes
in lung volume vary according to the experimental conditions. For
example, recruitment characterizes the inflation of nonaerated
(initially degassed) excised lungs or the in vivo response to
PEEP of groups of collapsed or flooded alveoli in edematous lungs. In
contrast, alveolar distension is generally the mechanism responsible
for lung volume changes in normal inflated lungs in vivo or in aerated alveoli in edematous lungs. On the basis of the above concepts, the
process of inflation of a completely collapsed lung can be simulated
using a model that can behave in two modes: synchronous and
asynchronous. In the synchronous mode, all alveoli are recruited simultaneously (by application of an appropriate level of inflation pressure), instantly attaining a given volume, expressed by what we
call the opening FA (which is constant for all
alveoli). Then the alveoli gradually distend as the inflation pressure
is further increased. The volume changes of the parallel alveolar units
are strictly uniform also during this stage of the inflation process. In the asynchronous mode, all the alveoli are initially collapsed, as
in the synchronous mode. When an appropriate inflation pressure is
applied, some alveoli suddenly open (are recruited), attaining a
certain opening FA. Then, as the inflation pressure is
further raised, more alveoli open while the previously opened alveoli distend further. This process, in which recruitment and distension occur in a parallel manner, continues until all alveoli are recruited and fully distended. Because the predicted for a collapsed alveolus is zero, for a collection of alveoli, some of which are collapsed, will be smaller than for a collection of alveoli that are open. We
expect, therefore, that calculated by the model in the asynchronous mode would be less than that predicted in the synchronous mode at all
lung inflation levels at which alveolar recruitment is incomplete.
| |
RESULTS |
and lung inflation.
Figure 2 shows 3 ms as a
function of FA (-FA curves) calculated using
the spherical-shell model in the synchronous (solid line) and
asynchronous (dashed line) modes. The dashed lines are based on
different values of the opening FA (50-80%, as
indicated by arrows). In addition, the asynchronous-mode simulations
presented in Fig. 2 assume that, initially, all the alveoli are
completely collapsed (FA = 0). Then 5% of the alveoli
suddenly open with an opening FA of 50, 60, 70, or 80%.
Subsequently, another 5% open, and the previously opened 5% further
distend so that their FA is now 55, 65, 75, or 85%. Then
another 5% open with an FA of 50-80%, and the
FA of the previously opened alveoli increase 5% each, so
that FA is 60, 70, 80, or 90% for the first 5% and 55, 65, 75, or 85% for the second 5%. When the recruited alveoli distend
to FA of 90%, they are assumed to remain at that
FA, distending no further. This pattern continues until all
the alveoli are open to FA of 90%.
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Fig. 2.
NMR (percentage of symmetric signal intensity) for asymmetry time
of 3 ms (3 ms) as a function of air fraction
[FA, air content as a fraction of total (air + tissue) volume], calculated using spherical-shell model in
synchronous and asynchronous modes. Solid line, synchronous mode, in
which all alveoli are recruited simultaneously and then distend as
inflation pressure is further increased. Dashed lines, asynchronous
mode, in which alveolar recruitment and distension are partly
parallel.
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As expected, in the asynchronous-mode simulation,
3 ms rises much more slowly than predicted by the
model in the synchronous mode, and the shapes of the two curves are
quite different. In contrast, the lines describing the behavior of the
model in the asynchronous mode for different opening FA
have very similar shapes, although they differ numerically. To help
explain these shapes, Fig. 3 provides an
expanded view of the bottom left of Fig. 2, showing only the 70 and 80% curves. Points A and C correspond to 5% of
recruited alveoli with opening FA of 70 and 80%,
respectively. At points B and D, 10% of the alveoli
are recruited. Point C is only slightly higher than point
A, and point D is only slightly higher than point
B, indicating that variations in the opening FA have
little effect on . Point C, however, is substantially further to the right than point A, because the opening
FA affects the total FA significantly. Thus
because the opening FA affects only slightly but
affects FA substantially, the 80% curve rises less steeply
than the 70% curve.
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Fig. 3.
Expanded view of bottom left of Fig. 2 showing only 2 lower
dashed lines. Point A, 5% of alveoli are recruited with an
individual opening FA of 70%. Point B, 5% of
alveoli are recruited with an individual opening FA of
70%, and another 5% are further distended to an FA of
75%. Point C, 5% of alveoli are recruited with an opening
FA of 80%. Point D, 5% of alveoli are recruited
with an opening FA of 80%, and another 5% are further
distended to an FA of 85%.
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The continuous line and each of the dashed lines shown in Fig. 2
circumscribe an area that encompasses a wide spectrum of possible lung
inflation behaviors, ranging from perfect synchrony in the function of
the alveolar units (spherical-shell model in the synchronous mode) to a
very high degree of asynchrony (alveolar recruitment is distributed
over the entire course of the inflation limb of the P-V curve and
occurs parallel to distension).
Theoretical predictions and experimental data.
The lack of sufficient experimental 3 ms data
in the literature prevented a direct comparison of our model
predictions with 3 ms measurements at various levels
of lung inflation. Because the spherical-shell model does not include
the magnetic interaction between alveoli, we expect calculated values
of for the spherical-shell model to correlate qualitatively with
experimental values, but not numerically. We do expect
6 ms for the Wigner-Seitz model in the synchronous
mode to correlate numerically with measured values in the lung if the
lung is behaving in the synchronous mode. However, as mentioned
previously, the Wigner-Seitz model cannot simulate asynchronous
behavior, whereas the spherical-shell model can. We have found that the
most meaningful way to compare 3 ms for the
spherical-shell model qualitatively with available experimental results
is to simulate experimental data that have approximately the same
qualitative relationship to the synchronous-mode, spherical-shell
3 ms as published experimental values of
6 ms have to the synchronous-mode
6 ms calculated from the Wigner-Seitz model. Figure
4 shows the Wigner-Seitz-calculated
6 ms as a function of inflation level (also expressed
by FA), with the assumption of perfectly synchronous
alveolar recruitment/distension. Clearly, the behavior of Wigner-Seitz
synchronous-mode 6ms is very similar qualitatively to
that of the spherical-shell synchronous-mode 3 ms
shown in Fig. 2, although the 6 ms curve is more
markedly nonlinear than the 3 ms curve.
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Fig. 4.
Experimental measurements of NMR obtained at an asymmetry time of 6 ms (6 ms) in normal (initially degassed) excised
unperfused rat lungs. Solid curve, relationship between
6 ms and FA, as predicted by Wigner-Seitz
model. Model assumes that alveolar recruitment occurs initially and
simultaneously in whole lung, to be followed by regionally uniform
alveolar distension. Data points represent measurements at different
levels of inflation in 4 excised lungs, each identified by a different
symbol (+, *,  , ×). Lungs were initially degassed in
vacuo. FA was estimated from lung volume measurements by a
magnetic resonance imaging technique (13) as FA = (Vin Vdeg)/Vin, where
Vin is volume at a given level of inflation and
Vdeg is volume of degassed lung. Experimental data were
obtained from measurements performed in a previous study (13).
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Figure 4 also shows four sets of experimental measurements of
6 ms obtained in a previous investigation (13) from
normal excised unperfused rat lungs, each studied at multiple levels of
inflation. The corresponding FA values were estimated from lung volume measurements by a magnetic resonance (MR) imaging technique
(Fig. 4). Figure 5 shows four sets of
corresponding simulated experimental data that have approximately the
same qualitative relationship to the spherical-shell synchronous-mode
3 ms-FA curve as the experimental data in
Fig. 4 have to the Wigner-Seitz synchronous-mode
6 ms-FA curve. No simulated experimental
values were included at FA = 0, because the corresponding to this experimental point can be affected by the
presence of minute (macroscopically undetectable) amounts of gas (see
the introduction). To determine the type of lung inflation behavior
that might be represented by the simulated experimental data, we used a
computer program that tries various combinations of the following four
parameters: 1) the FA at which the recruited
alveoli open, 2) the percentage of collapsed alveoli that are
recruited at each step, 3) the FA at which the
recruited alveoli distend at each step, and 4) the alveolar
FA at the onset of the inflation process, if the alveoli are not completely collapsed. The program searches for a combination of
these four parameters that produces the best fit of the calculated 3 ms-FA curve to the simulated
experimental data. The resulting curves that fit the four sets of
simulated experimental data are shown in Fig.
6; corresponding combinations of the above
four parameters are described in the legend of Fig. 6. For three
excised lungs (Fig. 6, A-C), the results of our model
calculations are consistent with an asynchronous lung inflation
behavior, characterized by a wide distribution of alveolar recruitment
over the span of lung inflation. In contrast, in one lung (Fig.
6D), the results of our simulations are consistent with
virtually all alveolar recruitment occurring at an earlier stage of
inflation. In two of the sets of data shown in Fig. 6 (asterisks and
open circles in B and C), the
3ms-FA curve fitted to the simulated
experimental data indicates an initial nonzero (although
physiologically insignificant) value for FA. This nonzero
value likely reflects the presence of minute amounts of gas in
conventionally degassed lungs, as discussed above and in the
introduction.
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Fig. 5.
Solid line, NMR for asymmetry time of 3 ms (3 ms)
as a function of FA, calculated using spherical-shell model
in synchronous mode. Symbols (+, *, , ×) indicate
simulated experimental data corresponding to 4 sets of actual
experimental data presented in Fig. 4. Therefore, relationship of
simulated experimental data to spherical-shell, synchronous-mode
3 ms-FA curve is similar to relationship
of actual experimental data (Fig. 4) to Wigner-Seitz synchronous-mode
6 ms-FA curve.
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Fig. 6.
NMR for asymmetry time of 3 ms (3 ms) as a
function of FA, according to spherical-shell model. Solid
lines, relationship between 3 ms and FA
calculated using this model in synchronous mode; dashed lines,
3 ms-FA lines fitted to simulated
experimental data shown in Fig. 5. In A, dashed line is
calculated with assumption that all alveoli are initially collapsed
(FA = 0). Then 14% of alveoli are recruited with opening
FA = 49.9%. Subsequently, another 14% of alveoli are
recruited, and previously recruited 14% of alveoli are further
distended so that FA = 53.9%. Pattern continues until all
alveoli are open and distended to FA = 90%. In B,
dashed line is calculated with the assumption that initially all
alveoli are almost totally collapsed with individual FA = 4.7%. Then 10% of alveoli are recruited with opening FA = 34.8%. Subsequently, another 10% of alveoli are recruited, and
previously recruited 10% of alveoli are further distended to
FA = 41.7%. Pattern continues until all alveoli are open
and distended to FA = 90%. In C, dashed
line is calculated with assumption that initially all alveoli are
almost totally collapsed with individual FA = 8%. Then
12% of alveoli are recruited with opening FA = 27%.
Subsequently, another 12% of alveoli are recruited, and previously
recruited 12% of alveoli are further distended to FA = 34%. Pattern continues until all alveoli are open and distended to
FA = 90%. In D, dashed line is calculated with
assumption that all alveoli are simultaneously recruited with an
individual opening FA = 35%. Then 3% of alveoli are
further distended to FA = 80%. Subsequently, another 3%
of alveoli are further distended to FA = 80%, and 3% of
alveoli previously distended to FA = 80% are further
distended to FA = 85%. Pattern continues until all alveoli
are distended to FA = 90%.
|
|
Response to PEEP in lung injury.
Here, lung injury is modeled as causing alveolar collapse and/or
flooding, and the spherical-shell model is used to simulate the
response of injured lungs to changes in inflation pressure, such as in
the application of PEEP. Alveolar recruitment and alveolar overdistension are two mechanisms potentially responsible for the lung
volume changes occurring with PEEP. Because is affected so
differently by recruitment and distension, as shown by the results of
our previous studies (see above and the introduction), measurements
might distinguish between these two mechanisms.
Figure 7 shows the for a simulation of
the response of a lung with collapsed alveoli to the application of
PEEP. Consider a possible lung condition (point A in Fig. 7) in
which 40% of the alveoli are collapsed and 60% are inflated to an
FA of 70%. Point A shows and FA
before PEEP is applied. Then, because the tissue volume of the alveoli
is assumed to be unchanged when the alveoli are inflated, we can
calculate FA for this lung condition. FA for
point A is calculated from Eq. 13. Let N be the
total number of alveoli and Vt be the tissue volume of one
individual alveolus (Vt is the same for a collapsed and an
inflated alveolus). The FA i for all
the collapsed alveoli is zero. For all the inflated alveoli,
FA i is 70%. Also,
Vt T = NVt. Because all the inflated
alveoli are identical and because the total number of inflated alveoli
is 0.6N, the summation in Eq. 13 reduces to
and
The
for point A is calculated from Eq. 10 to be = 0.6i. From Fig. 7, when FA is 70%
for the normal lung, we find that i is 86.91%.
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Fig. 7.
NMR for an asymmetry time of 3 ms (3 ms) as a
function of FA: effect of recruitment of collapsed alveoli
and distension of aerated alveoli in response to positive
end-expiratory pressure (PEEP) predicted using spherical-shell model in
asynchronous mode. In this case, 40% of alveoli are collapsed and 60%
are inflated before PEEP is applied. Point A, before
application of PEEP; normal (aerated) alveoli (n) are inflated to
FA n = 70%, so that FA for entire lung is
58.33%. Points B-E, effect on of recruitment of
collapsed alveoli. Percentage of recruited alveoli is shown in
parentheses; in each case, recruited alveoli are inflated to
FA = 70%, and normal alveoli are further distended so that
FA of entire lung is 70%. FA n values for
normal alveoli for points B, C, D, and E are 79.55, 76.77, 73.13, and 70.00%, respectively. Solid line, which describes
relationship between 3 ms and FA for
normal lung (spherical-shell model, synchronous mode), is shown for
reference.
|
|
We see that although the normal alveoli (n) are inflated to
FA n (air fraction for the normal aerated alveoli) of
70%, the presence of the 40% collapsed alveoli lowers
FA to 58.33% and lowers from 86.91% of normal lung at
FA of 70% to 52.15%. Point B indicates the
response to PEEP if none of the collapsed alveoli are recruited and the
60% normal alveoli distend in response to PEEP to
FA i of 79.55% (see Eq. 13), so that FA of the entire lung is 70%. Because
distension has a much smaller effect on than recruitment (see the
introduction), point B is only slightly higher than point
A. Point C corresponds to partial recruitment, with 15% recruited
alveoli opening with an FA of 70% and the 60% normal
alveoli distending to FA i of
76.77%, so that the FA for the entire lung is 70%. In
contrast to the previous case, point C is much higher than
point A. Thus recruitment increases much more than
distension of open alveoli. Points D and E show that
increased recruitment results in correspondingly greater changes in
. Recruitment has such a significant effect on , because it
increases the relative amount of thin material and decreases the
relative amount of thick material, while the total volume of tissue
remains the same.
Figure 8 shows similar calculated data for
the response to PEEP of an edematous lung with flooded alveoli. In this
simulation, when the alveolus is recruited, the water flooding the air
space before recruitment is assumed to be redistributed over the
surface of the inflated alveolus and/or moved into the interstitial
space (see DISCUSSION). Thus the total amount of NMR
signal-producing volume remains the same before and after recruitment,
similar to the conditions for recruitment of collapsed alveoli. Figures 7 and 8 show that the presence of the 40% flooded alveoli lowers more than the 40% collapsed alveoli. This lower occurs because the
water flooding the alveoli is additional thick water. Furthermore, recruitment of flooded alveoli causes greater changes in than recruitment of collapsed alveoli. This greater change occurs because recruiting flooded alveoli converts the thick water flooding the air
space to thin water in (or lining) the alveolar walls, thus converting
more thick water to thin water than occurs in recruitment of collapsed
alveoli. Moreover, the presence of the 40% flooded alveoli also lowers
FA more than the presence of the 40% collapsed alveoli
(cf. Figs. 7 and 8), because the extra volume of water flooding the air
spaces reduces the total FA.
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Fig. 8.
Calculated NMR for an asymmetry time of 3 ms
(3 ms) as a function of FA in
spherical-shell model simulating pulmonary edema: effect on
3 ms of recruitment of flooded alveoli and distension
of aerated alveoli in response to PEEP simulated using model in
asychronous mode. In this example, 40% of alveoli are flooded and 60%
are normal (aerated) before PEEP is applied. Point A, before application of PEEP; normal alveoli are inflated to
FA n = 70%, so that FA for entire lung is
42%. Points B-E, effect of recruitment of flooded alveoli
(percentage of recruited alveoli is indicated in parentheses); in each
case, recruited alveoli are inflated to FA = 70%, and
aerated alveoli are further distended so that FA of entire
lung is 70%. FA n values for normal alveoli for
points B, C, D, and E are 88.26, 84.79, 78.40, and
70.00%, respectively. Solid line, which describes relationship between
3 ms and FA for normal lung
(spherical-shell model, synchronous mode), is shown for reference.
|
|
Although there are no published experimental data regarding the effect
of PEEP on in pulmonary edema, our theoretical predictions can be
compared with the results of measurements in edematous excised rat
lungs at multiple inflation pressures (14). In these lungs,
6 ms rose markedly as airway pressure was increased;
on the average, 6 ms was 60 ± 6% at 5 cmH2O and 80 ± 5% at 30 cmH2O (at the same
pressures, it was 84 ± 3 and 82 ± 4% in normal lungs). FA could not be systematically determined from lung volume
measurements (because no absolute lung volume data were obtained in
this study). However, in some edematous lungs fixed at 25-30
cmH2O, FA could be estimated roughly as
FA = 1 Vv (where Vv is the
morphometric lung tissue fraction) and was found to be 0.75-0.82.
If these morphometric estimates adequately reflect the true values of
FA, then the values of 6 ms observed at
25-30 cmH2O are not too far below the theoretical
curve shown in Fig. 4. The above data, which are consistent with the
behavior of our models, suggest that the volume response of the
edematous lungs to the application of higher airway pressures included
substantial recruitment of collapsed or flooded alveoli, with little
alveolar flooding and atelectasis remaining at the highest level of
inflation. This interpretation is consistent with histological results,
which showed little evidence of alveolar flooding and collapse in the
edematous lungs fixed by airway instillation at 25-30
cmH2O fixation pressure.
| |
DISCUSSION |
Our results indicate that the relatively simple model developed in the
present study can predict the effects of changes in the level of
inflation on the NMR properties of normal and injured (edematous)
lungs. Although more advanced models can be used to analyze the
behavior of the NMR as a function of the level of inflation, the
spherical-shell model offers a simple and convenient method for
simulating regional nonuniformities in the lung P-V behavior. In normal
lungs, this model provides a more realistic simulation of the process
of inflation in initially degassed lungs. Although previous comparisons
have shown good general agreement between theoretical predictions and
experimental measurements (see Fig. 7 in Ref. 13), the present
simulations allow a more precise interpretation of the observed
discrepancies. The results of our qualitative comparison of theoretical
and simulated experimental data suggest that the distribution of
alveolar recruitment over the inflation limb of the lung P-V curve may
vary markedly from lung to lung. The data obtained from lungs indicated
by plus signs, asterisks, and open circles in Figs. 4, 5, and Fig. 6,
A-C, are consistent with the assumption, underlying the
spherical-shell model in the asynchronous mode, that alveolar
recruitment is widely distributed over the inflation limb of the P-V
curve and occurs, at least in part, parallel to distension. These
results are in agreement with observations of the P-V behavior of
excised lungs (37) and with experimental evidence obtained from
morphometric and physiological studies (27, 41). However, the data from the lung indicated by × in Figs. 4, 5, and 6D suggest that,
in some lungs, alveolar recruitment may be virtually complete at an
earlier stage of the inflation curve. Measurements of were conducted under truly static conditions (i.e., each lung inflation level was maintained for several minutes before measurements were
performed) (13); this procedure can be expected to maximize recruitment
at lower inflation pressure levels (37).
The simple, but more flexible, spherical-shell model was used in the
present study because of the limitations of the Wigner-Seitz model with
respect to the specific purposes of the study (i.e., its inability to
simulate lung regional nonuniformity, see METHODS). Because
this model does not account for the magnetic interactions between
spherical shells (see METHODS), its predictions cannot be
compared numerically with experimental values, but they can be compared
qualitatively. Qualitative comparisons between spherical-shell asynchronous-mode calculations and simulated experimental data (see
RESULTS) have provided a basis for evaluating the
physiological significance of the differences between the calculated
synchronous-mode Wigner-Seitz 6 ms data and the
experimental measurements shown in Fig. 4 (assuming various patterns of
alveolar recruitment and distension). The use of the spherical-shell
model with an asymmetry time of 3 ms is justified by our previous
theoretical studies and experimental measurements of at various
asymmetry times and lung volumes (7, 13), which suggest that
predictions made using 3 ms can be qualitatively
compared with experimental results for a a range of
at least 3-6 ms. Therefore, the spherical-shell model presented in
this study provides virtually all the essential theoretical foundation
for future experimental studies of the behavior of as a function of
the inflation level (in particular, the response of to recruitment
and distension). It can be expected that future systematic comparisons
between theoretical and experimental data will indicate whether the
development of more sophisticated models is warranted.
In the present study the spherical-shell model has been used to
calculate theoretical -FA curves on the basis of four
parameters, i.e., the FA at which the recruited alveoli
open, the percentage of collapsed alveoli that are recruited at each
step, the FA at which the recruited alveoli distend at each
step, and the alveolar FA at the beginning of the inflation
process, if the alveoli are not completely collapsed (see
RESULTS). Although there is no unique combination of the
above parameters that will fit the experimental data points, the
spherical-shell model can clearly assess the roles of alveolar
recruitment and distension in the lung P-V behavior. For example, our
data suggest that the model can distinguish whether alveolar
recruitment is widely distributed over the range of the inflation
process (Fig. 6, A-C) or occurs mostly at an earlier stage
of this process (Fig. 6D). It can be expected that the analysis of the characteristic changes in the -FA curve produced
by varying the individual parameters will provide valuable information
on the inflation behavior of normal and injured lungs. The present preliminary qualitative comparison of theoretical and simulated experimental data remains to be extended by systematic prospective studies specifically designed to characterize various possible patterns
of lung inflation and the experimental conditions under which they may occur.
The models used in the present study simulate the NMR behavior of the
parenchymal (respiratory) portion of the lung. Therefore, it is
possible that these models overestimate , which is likely lower in
the nonparenchymal than in the parenchymal lung tissue. Simple
preliminary calculations indicate that correction for the effect of the
nonparenchymal tissue would cause a downward shift of the curve
describing the relationship between and FA. However, the agreement usually observed between predicted and experimental 6 ms values at high levels of inflation (13, 14)
suggests that the overestimation is not substantial in excised
unperfused lungs. Furthermore, it can be expected that our imaging
technique will allow the selection of predominantly parenchymal lung
regions in in vivo measurements of .
Although the level of lung inflation depends on transpulmonary pressure
and experimental data can be easily plotted as a function of
inflation pressure (13, 14), in our models is described as a
variable dependent on lung volume. This approach is justified by
experimental data (13, 14) that clearly indicate that the applied
pressure does not affect unless it results in alveolar recruitment
(independently detected as a change in lung volume). Therefore, the
role of inflation pressure in our simulations is purely instrumental,
because the applied pressure is implied to vary to produce the volume
changes assumed for the calculations. This approach simplifies our
simulations and serves the main purpose of demonstrating the different
effects of alveolar recruitment and distension on (and, therefore,
the ability of measurements to distinguish these two mechanisms of
lung inflation). On the other hand, the application of the
spherical-shell model can be further refined by assuming changes in
inflation pressure and calculating the corresponding lung volume
changes on the basis of the well-known nonlinear lung P-V relationship.
In this case, the changes in FA undergone by the parallel
alveolar units during inflation will vary according to the P-V curve
(instead of being constant, as in the simulations presented above).
Because of the nonlinearity of the lung P-V curve and of the
relationship between FA and conventional spirometric lung
volume (see below), the FA changes at higher levels of lung
inflation will be smaller than those assumed for the present
simulations, thus further reducing the effect of alveolar distension on
. However, as shown by our calculations, the difference between the
results obtained by these two approaches is minimal, because the effect
of alveolar distension on is very small compared with the
effect of recruitment. From a general point of view, the inclusion of
pressure among the factors potentially affecting the behavior of our
models is convenient, because it allows the evaluation of the effects
of pathophysiologically and clinically relevant manipulations,
such as the application of PEEP to promote alveolar recruitment
in ARDS patients.
In the present simulations, lung air content is expressed as a fraction
of total lung volume, which includes tissue volume. As noted above, the
lung tissue volume is assumed to consist of 100% water, thus
representing the NMR signal-producing component of the spherical-shell
model (when the model simulates a normal lung). Experimentally, for
example, in excised lung studies, the FA is a convenient
index of lung air content, because it can be obtained, with sufficient
approximation, from conventional measurements of lung weight (43) and
volume (water-displacement technique) (39). The experimental
FA values used for the comparison with our theoretical
predictions (Fig. 4) were obtained by an MR imaging technique that has
been shown to agree very closely with the standard water-displacement
method (13). However, when the model is used to simulate clinical
conditions (e.g., the effect of PEEP on in ARDS patients), it may
be of interest to relate the FA to the much more common
ratio of spirometric lung volume (VL) to total lung
capacity (TLC). This relationship is defined by the expression VL/TLC = (1 a)FA/a(1 FA), where VL corresponds approximately to the air volume, Va (i.e.,
Va i in Eq. 12), ignoring the dead space volume, and a is the value of FA at
TLC. On the basis of estimates obtained from published data for lung
tissue volume (2, 4, 42, 44) and conventional spirometric lung volume
values calculated using current prediction equations (6, 11, 12,
22-24, 36), the value of FA at TLC in normal subjects is ~90% and is only minimally affected by substantial variations in
lung tissue volume and by using spirometric lung volume data from
various sources or for different genders. Our estimates assume lung
tissue volumes of 450-900 ml (values in women being ~75% of
those in men) (4, 42), an age of 30 yr, and average anthropometric characteristics according to several published series (1.75 and 1.60 m
height and 75 and 62 kg body wt for men and women, respectively). According to the above calculations, it can be estimated that FA values of 50-80% (i.e., in our models the values
of FA that collapsed alveoli are assumed to attain
instantaneously when recruited) correspond to VL/TLC of
~10-60%. In living humans, these VL/TLC values
correspond to inflation levels encompassing residual volume and
functional residual capacity. In patients with pulmonary edema, the
value of FA at TLC is expected to be lower than in normal subjects, mainly depending on the severity of lung water accumulation. As discussed above (see Selection of the lung model), in our
simulations the extra water accumulated in the interstitial space and
in the alveoli increases the nongas component of the total volume of the lung.
Our simulations were performed by assuming different values for the
opening FA, because the volume attained by recruited
alveoli can be expected to vary depending on mechanical conditions. For example, studies in experimental animals and in humans (21, 31)
indicate that the level of inflation varies throughout the lung.
Radioactive gas measurements in normal humans (31) have shown that
regional functional residual capacity and residual volume decrease
substantially from the top to the bottom of the lung. However, the
present results (Figs. 2 and 3) suggest that variations in the opening
FA do not markedly affect the relationship between and
the level of lung inflation.
Because the spherical-shell model does not take into account the
magnetic interactions between spherical shells, the calculated values
of are obtained by simple summation of the individual values for
each spherical shell and, therefore, are not dependent on the spatial
characteristics of alveolar collapse and flooding. Consequently, no
assumptions were made, in the present simulations, regarding the
spatial distribution of the collapsed or flooded units. However,
regional differences in the behavior of the spherical-shell model can
be simulated by separately displaying the values of calculated for
different groups of spherical shells (an approach that would closely
simulate the results of regional measurements by MR imaging). Although
the simulation of the regional behavior of was not considered in
the present study, it may be of interest, because in many experimental
models (e.g., oleic acid-induced pulmonary edema), as well as in ARDS,
lung injury is characterized by marked spatial nonuniformity (15, 18,
25, 26, 33, 40). As implied above, the distribution of alveolar
flooding might significantly affect the behavior of in more refined
lung models that take into account the magnetic interactions between alveoli.
In our simulations of the response of edematous lungs to PEEP, we have
assumed that alveolar recruitment occurs because, as a result of this
treatment, the edema fluid filling the air space is redistributed over
the surface of the inflated alveolus and/or moved from the air space
into the interstitial space (see RESULTS). Therefore, the
total water content (including the edema fluid) of the lung model does
not vary (although the average NMR signal-producing volume of water per
unit model volume decreases as the level of air inflation increases).
The assumption regarding the mechanisms of recruitment of flooded
alveoli by PEEP is consistent not only with our histological
observations (see RESULTS), but also with the results of
other theoretical and experimental studies. Staub (43) and Malo et al.
(28) calculated that, when flooded alveoli are inflated with air, the
edema fluid will redistribute over the alveolar surface to form a thin
layer of only a few micrometers. This redistribution is expected to
affect , because it creates an air-water interface and transforms
thick material (the water filling a flooded alveolus) to thin material
(the thin water layer lining a reinflated alveolus). In addition,
several studies conducted on liquid-filled excised lungs have
documented the transfer of water from the flooded air spaces to the
interstitial space (9, 10, 19). Other published data suggest that the
application of PEEP in experimental models of pulmonary edema results
in inflation of flooded air spaces by moving alveolar water into the
interstitial lung space (28, 34).
From a practical point of view, the results of the present study are of
interest, because they provide the basic principles for a new
nondestructive, and potentially noninvasive, approach to the study of
the mechanisms underlying the P-V behavior of the lung. This approach
is of particular relevance to the characterization of lung injury. When
extended to the analysis of diseased lungs, the spherical-shell model
can be used to simulate certain important elements of the lung response
to injury, namely, alveolar collapse and edema, and to predict the
characteristic effects of these abnormalities on the lung NMR
properties. Our simulations of recruitment of collapsed and flooded
alveoli in edematous lungs predict that measurements may be used to
distinguish recruitment from distension in the lung volume response to
changes in airway pressure (e.g., to the application of PEEP). This
differentiation is important clinically, because, for example, alveolar
recruitment results in improved arterial oxygenation and, therefore, is
the desired response to PEEP in patients with ARDS. In contrast,
alveolar overdistension has little effect on arterial oxygen and may
cause lung damage in these patients (30, 35).
| |
ACKNOWLEDGEMENTS |
We thank Sandra Lowe for secretarial assistance.
| |
FOOTNOTES |
This research was supported by National Heart, Lung, and Blood
Institute Grant HL-31216.
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: A. G. Cutillo,
Div. of Pulmonary, Critical Care and Occupational Pulmonary Medicine,
University of Utah Health Sciences Center, 50 North Medical Dr., Salt
Lake City, UT 84132 (E-mail: acutillo{at}hsc.utah.edu).
Received 30 July 1998; accepted in final form 27 October 1999.
| |
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