Vol. 94, Issue 2, 621-630, February 2003
Chest wall kinematic determinants of diaphragm length by
optoelectronic plethysmography and ultrasonography
A.
Aliverti1,2,
G.
Ghidoli2,
R. L.
Dellacà1,2,
A.
Pedotti1,2, and
P. T.
Macklem3
1 Dipartimento di Bioingegneria, Politecnico di
Milano and 2 Centro di Bioingegneria, Fondazione Don
Gnocchi Istituto di Ricovero e Cura a Carattere Scientifico and
Politecnico di Milano, I-20133 Milan, Italy; and
3 Meakins-Christie Laboratories, Montreal Chest
Institute, McGill University Health Centre, Montreal, Quebec,
Canada H3H 2R9
 |
ABSTRACT |
To estimate diaphragm fiber
length from thoracoabdominal configuration, we measured axial motion of
the right-sided area of apposition by ultrasonography and volumes
displaced by chest wall compartments [pulmonary, abdominal rib cage,
and abdomen (Vab)] by optoelectronic plethysmography in four normal
men during quiet breathing and incremental exercise without and with
expiratory flow limitation. Points at the cephalic area of
apposition border were digitized from echo images and mapped into
three-dimensional space, and the axial distance from the xyphoidal
transverse plane (Dap) was measured simultaneously with the
volumes. Linear regression analysis between changes (
) in
Dap and the measured volume changes under all conditions
showed that 1) Dap was linearly related more
to Vab than to changes in pulmonary and abdominal rib cage volumes;
and 2) this was highly repeatable between measures. Multiple stepwise regression analysis showed that Vab accounted for
89-96% of the variability of Dap, whereas the rib
cage compartments added <1%. We conclude that, under conditions of
quiet breathing and exercise, with and without expiratory flow
limitation, instantaneous Dap can be estimated from
Vab.
chest wall volume; diaphragm area of apposition; echography
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INTRODUCTION |
MOST OF THE AVAILABLE
INFORMATION on chest wall kinematics is based on the
two-compartment chest wall model of Konno and Mead (12)
composed of rib cage and abdomen (AB), with each behaving with a single
degree of freedom, so that changes in volume of each compartment can be
measured by a single dimension. Because the movable parts of the AB are
the anterolateral abdominal wall and the diaphragm, the volume of
abdominal contents (Vab) displaced from under the diaphragm must be
equal and opposite to the volume displaced by the anterolateral
abdominal wall (Vabw, where is change). Thus diaphragmatic
displacements are closely linked to abdominal wall
displacements, and it had been assumed that diaphragm fiber length
(Ldi) would be simply related to displacements of the AB and nearly unrelated to rib cage displacements. This assumption was supported by the studies of Goldman et al.
(8) and Grassino et al. (9), who found that
length-tension behavior of the diaphragm was much less sensitive to
changes in rib cage diameter than to displacements of the abdominal wall.
Subsequent studies have shown that displacements of the abdominal
contents under the diaphragm are more complex than those measured
simply by displacements of the anterior abdominal wall. An analytic
approach to these complexities was developed in an important paper by
Mead and Loring (14). They put forward the concept that
displacements of the lower part of the rib cage where the diaphragm is
apposed to its inner surface, along with displacements of the AB, are
important determinants of Ldi and provided
experimental evidence to support their theoretical analysis
(14). However, in preliminary studies, our laboratory
found that >80% of the variance in Ldi in the
area of apposition (Aap) during exercise was
accounted for by abdominal displacements (3).
These findings are in agreement with those of Chen et al.
(6), who found that twitch diaphragmatic pressure in
response to supramaximal, transcutaneous, bilateral phrenic nerve
stimulation was exquisitely sensitive to abdominal displacements and
relatively insensitive to rib cage displacements. These studies suggest
that Ldi is more closely linked to abdominal
displacements and less to rib cage displacements than has generally
been thought. In this paper, we develop a method to assess factors
determining Ldi using optoelectronic
plethysmography (OEP) (5) to measure volume displacements
of the rib cage (Vrc) and Vabw, combined with ultrasound
measurements of changes in Ldi in the
Aap (Dap). We hypothesized that
it would be possible to predict Dap from Vabw during
quiet breathing and exercise, with and without expiratory flow
limitation (EFL). To develop our ideas, it is necessary to review Mead
and Loring's analytic paper (14) in some detail.
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THEORY |
In this section, we develop the relationship between abdominal and
diaphragmatic displacements following the approach of Mead and Loring
(14). To do so, we develop equations that are similar, but
not identical, to those they used. They assumed, quite reasonably, that
the Vab is constant and that it is composed of three compartments, all
of which can change their contribution to the total volume during
breathing. The three compartments are displaced during breathing by
motion of the diaphragm dome apposed to the lung, by motion of the rib
cage in Aap, and by displacement of the
anterolateral abdominal wall. Thus
|
(1)
|
and
|
(2)
|
where Vab,L is the Vab in the diaphragm dome; Vab,rc
is that part of Vab contained between the surface separating the dome from the cephalad limit of Aap and the surface
separating the costal margin from the rest of the abdominal contents,
Vabw; and is changes in the volume of the three compartments. If
xiphi-pubic distance is constant so that the cephalad border of Vabw
does not move relative to the pelvic floor, changes in the third
compartment are equal to Vabw.
Vab,rc has two components: the first is Vab in the
Aap of diaphragm to rib cage, and the second is
Vab in the part of the rib cage caudal to the insertion of the
costal part of the diaphragm (the obligatory ring). Mead and Loring
(14) estimated the fraction of the internal surface of the
rib cage at functional residual capacity occupied by
Aap (fapp) to be ~0.25, whereas the fraction occupied by the obligatory ring (fobr) was estimated to be ~0.15. Mead and Loring assumed that "in the resting tidal range fapp decreases only modestly during inspiration, and 0.41 is a reasonable average value of (fapp + fobr)." Thus they estimated Vab,rc
by
Substituting for Vab,rc in Eq. 2 yields
|
(3)
|
Equation 3 requires that the middle term, the Vab
contained within Aap, be always positive if the
rib cage expands, and ignores the consequences of a decrease in
Aap. Using this approach, they calculated that
the Vab displaced from the diaphragmatic dome during inspiration was
"more than twice the contribution based solely on anterolateral
abdominal expansion." The implication they drew is that diaphragm
shortening could be considerably greater than that estimated by
measuring abdominal wall motion alone, because the diaphragmatic
contribution to tidal volume includes the middle term of Eq. 3, which is always positive with rib cage expansion. This is not
measured by Vabw, but it can be estimated by measuring Vrc. The
problem with this conclusion is the use of an average value of fapp
that remains constant during inspiration. If one allows fapp to change
during inspiration rather than taking an "average" value, a quite
different result is obtained.
Substituting (fapp + fobr) Vrc for Vab,rc in Eq. 1
gives Vab at the beginning of inspiration
|
(4)
|
At the end of inspiration
|
(5)
|
where * indicates a new value at end inspiration. Subtracting
Eq. 4 from Eq. 5
|
(6)
|
The difference between Eqs. 6 and 3 is the
inclusion of 
fapp Vrc fapp Vrc in Eq. 6. Thus, according to Eq. 6, the Vab contained
within Aap is
This can be either positive or negative, depending on the relative
values of Vrc, Vrc, fapp, and fapp.
If one assumes a breath of 500 ml in which fapp decreases from 0.25 to
0.20, fobr remains constant at 0.15, Vrc = 375 ml, and
Vabw = 125 ml, the Vab in Aap becomes
(150 0.05 Vrc 6.25) ml. A reasonable value for the
internal volume of the rib cage measured by OEP (5) (a new
technology described in detail below) is 5 liters (A. Aliverti,
personal communication). Thus the Vab in Aap
is a negative value of 106.25 ml compared with +150 ml calculated by
using the Mead and Loring equation.
Thus the predictions arising from Eq. 6 are quite
different from those arising from Eq. 3. Equation 6 does not predict that the Vab in Aap
is necessarily positive. It can increase, decrease, or stay the same,
depending on the relative magnitudes of rib cage expansion and decrease
of the Aap. The sum of Vab,L and Vab,rc gives the Vab contained within the diaphragm, and this sum
is equal and opposite to Vabw. The assumption that the Vab contained
within Aap always increases with rib cage
expansion cannot be correct when Aap decreases.
As a result, because fobr Vrc can be estimated, and to the extent
that Vab,rc and Vab,L carry information about
diaphragmatic shortening, Vabw can potentially be used to estimate
Ldi.
Clearly, how much fapp changes with a breath is crucial to the
understanding of the diaphragm's contribution to breathing. Mead and
Loring's (14) use of a constant average fapp must
introduce substantial errors. Although it is true that their
measurements (13) tended to support their theoretical
analysis (14), not all measurements were made in all
subjects, and some questionable assumptions tend to make their data
open to reinterpretation (see Ref. 6 for further
discussion). It would seem that Mead's earlier calculations of the
effect of thoracoabdominal configuration on Ldi
[made with Goldman et al. (8) and Grassino et al.
(9)] may have been closer to the truth than his later calculations.
| |
METHODS |
Subjects
We studied four healthy normal men 42 ± 3 (SD) yr old,
with anthropometric and functional respiratory characteristics shown in
Table 1. They were all laboratory
personnel trained in respiratory maneuvers.
Protocol
Each subject was studied on two occasions while seated on a
cycle ergometer with the arms supported away from the trunk. On both,
the subjects breathed spontaneously for 3 min and then performed an
incremental exercise test starting at 0 W and increasing by 25 W every
4 min until exhaustion. The incremental exercise tests were performed
under control conditions (Ex,c) and with a Starling resistor in the
expiratory line, which limited expiratory flow to ~1 l/s (Ex,s). We
measured maximal power output under Ex,c and Ex,s. Reports of the
determinants of exercise limitation and dynamics of breathing during
these experiments have previously been published (2, 10).
During Ex,s, the subjects breathed through a mouthpiece, which was
attached to a Hans Rudolph valve, which separated inspiratory and
expiratory flow. Flow limitation was achieved by putting the Starling
resistor on the expiratory port of the valve. A 2-liter jar was placed
in parallel with the Starling resistor, which acted as a capacitance,
so that, at the beginning of expiration, flow was somewhat greater than
1 l/s, whereas, at the end, it was somewhat less (see Ref.
10, Fig. 1). Data were gathered during the
last 40 s of each workload. We randomized the order in which these
two exercise tests were performed.
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Fig. 1.
A: schematic diagram
of the measurement, data processing, and analysis system. B:
representative example of an echographic image obtained from the right
lateral abdominal rib cage. The shadowed area at the top corresponds to
the lung, whereas the lighter area at the bottom corresponds to the
liver. The arrow indicates the point P where the diaphragm reflects
from the chest wall and the lung intervenes. This point was used to
identify the position of the cephalic margin of the zone of apposition.
2D, two-dimensional; 3D, three-dimensional.
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Measurements
OEP.
Figure 1A shows a general
overview of the measurement system. Chest wall kinematics and
compartmental volumes were measured by OEP, as previously described,
and validated at rest and during exercise (1, 5, 11).
Eighty-nine reflecting markers were placed front and back over the
chest wall from clavicles to pubis. Each marker was tracked in three
dimensions (3D) by four video cameras: two in front of the subject and
two behind. A dedicated image processor measured the position of each
marker at 50 Hz. For volume computation, chest wall surface was
approximated by 182 triangles connecting the markers. Then, using
Gauss' theorem, the volume of the chest wall (Vcw) and of its
compartments was calculated. We modeled the chest wall as a
three-compartment system, comprised of the pulmonary or lung-apposed
rib cage (RCp), the abdominal or diaphragm-apposed rib cage (RCa), and
the AB (5, 15). The sum of the volume of each compartment
equaled the Vcw: Vcw = Vrc,p + Vrc,a + Vab, where Vrc,p
is pulmonary rib cage volume and Vrc,a is abdominal rib cage volume.
Ultrasonography of the diaphragm.
Diaphragm motion was visualized by a general-purpose echo camera (Aloka
Echo Camera), equipped with a linear probe (3.5 MHz, 128 mm). As the
intrapulmonary air greatly attenuates the transmission of ultrasound
waves, the resulting abrupt discontinuity in the image at the level at
which the diaphragm reflects from the chest wall and the lung
intervenes was used to identify the position of the cephalic margin of
the zone of apposition (Fig. 1B). The caudal border was
measured by markers along the costal margin.
The probe was placed in a plastic frame fixed to the skin by adhesive
tape and manually kept in a fixed position during the experiments. The
probe was aligned approximately axially and placed between the two
markers defining the right lateral border of the RCa. Three additional
markers were placed on the probe and tracked by the OEP cameras to
measure its position and orientation.
Ultrasonographic images were synchronized with the motion analyzer by
generating a trigger signal that created a symbol at a frequency of 0.5 Hz on the ultrasonic image and that was recorded on a analog-to-digital
card (RTI800, Analog Devices, Norwood, MA) synchronized with the motion
analysis system. The ultrasonic images were then recorded on a standard
video recorder and successively digitized by a frame grabber (Screen
Machine II, Fast Electronic) in motion JPEG format at 10 frames/s with
a resolution of 320 × 240 pixels. When the first symbol and the
first trigger appeared, respectively, on the image and the digitized
signal, the sequence of echographic images and OEP volumes
(down-sampled at 10 Hz) was aligned to be in synchrony and
automatically maintained synchronous for the duration of the
measurement by a specifically developed software. Small movements of
the margin of the zone of apposition could be easily detected, even if
sometimes the margin was obscured by a rib at the initial or final position.
Data Analysis
Ldi in the Aap.
Changes in Aap were estimated by measuring axial
motion of its cephalic margin in the echographic images of the right
hemidiaphragm. On each image, the cephalic extremity was manually
selected. Successively, the two-dimensional coordinates were mapped
into 3D space by using the information on the position and orientation
of the probe. For each image, the transverse plane at the level of the
xiphisternum was estimated by computing the regression plane outlined
by the markers placed at the xiphoid level. Finally, the
Dap to the upper border of Aap was
computed as reported in detail in the APPENDIX and in Ref.
4.
Regression analysis.
For each test, four to five breaths were analyzed at a sampling rate of
10 Hz. Successively, a linear regression analysis between
Dap and Vrc,p, Vrc,a, Vab, and Vcw was performed, and the
results were expressed as squared linear regression coefficient (r2) and slope of the regression line.
Analysis of variance.
To study the repeatability on different days and experiments and the
differences between subjects on r2 and slope
values, we applied a one-way ANOVA on repeated measures to four
different tests of quiet breathing. To study the effect of the presence
of Starling resistor (factor 1), of different compartments (factor 2), and of different workloads
(factor 3) on r2 and slope
values, we applied a three-way ANOVA on repeated measures to all of the tests.
For factor 3, we considered quiet breathing at rest, 25 W,
maximum workload (Wmax) during exercise with EFL, and Wmax/2
(half-maximal Wmax during Ex,c).
As r2 values are all between 0 and 1, this
parameter is not usually normally distributed. For this reason, we
applied Fisher's transform, which allows acquisition of a statistical
estimator (Y) of r2, defined as Y = arctanh (r2) = 1/2 ln(1 + r2)/ (1 r2). This estimator, calculated as the inverse
hyperbolic tangent of r2, follows the normal
distribution: all of the analyses of variance for
r2 were carried out by using Y instead of
r2. The fit to a normal distribution and
homogeneity of variances were verified by Kolmogorov-Smirnov's test
and univariate/Levene's test, respectively. Only when ANOVA was
significant post hoc tests (Scheffé's test) were made to verify
the statistical significance of the differences between all pairs of
means. For all tests, the significance level was taken as
P < 0.05. All results are reported as means ± SE.
Multiple stepwise linear regression analysis.
Variations of Ldi in the
Aap are related to Vcw, which are determined
by the displacement of the different compartments. To quantify the
contribution of the different chest wall compartmental volumes to the
Dap, we hypothesized the following linear relationship
|
(7)
|
where B0 is the intercept and B1,
B2, and B3 are the linear coefficients. With
the use of stepwise multiple regression, the relative contributions of
each component in Eq. 7 were determined for the different
subjects during the different exercise workloads (from quiet breathing
to Wmax) with and without EFL. The goodness of fit,
r2, was used to quantify the percentage of
variation in Dap that is explained by Vab, Vrc,a, and
Vrc,p. Results obtained from all of the subjects were then averaged in
three different situations: quiet breathing, Ex,c, and EFL exercise.
| |
RESULTS |
Linear Correlation Analysis
Figure 2A shows tracings
of Vcw and its three compartments (Vrc,p, Vrc,a, and Vab) during
quiet breathing. The bottom trace is the Dap.
The time variations of the volumes of the different chest wall
compartments were approximately in phase, and the inspiratory decrease
of Dap was in the order of 2.5 cm. Figure 2B
shows the results of the linear regression analysis (slope and
r2), computed between Dap and
Vrc,p, Vrc,a, Vab, and Vcw, corresponding to the same data reported in
Fig. 2A. Dap was well correlated with the
volume variations of each chest wall compartment and with those of the
total chest wall. However, the highest value of
r2 was obtained for Vab. The slopes of the
linear regressions simply reflect the volume of each compartment
because Dap is common to all.
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Fig. 2.
A: representative example of tracings obtained during
quiet breathing. , Change; Vrc,p, volume of lung-apposed rib cage
compartment; Vrc,a, volume of diaphragm-apposed rib cage compartment;
Vab, volume of abdomen; Dap, axial motion of cephalic
margin of the diaphragm in the area of apposition. B:
correlations between Dap and Vrc,a, Vrc,p, Vab, and
chest wall volume (Vcw) for the data shown in A.
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Analysis of Variance
Figure 3 confirms the
importance of abdominal wall displacements in predicting
Dap. The r2 values on the
ordinate reveal how much of the variance in Dap is
accounted for by the volume displacements of each of the three compartments and the whole chest wall during quiet breathing and as a
function of exercise workload. Each bar contains pooled control and EFL
breaths in all subjects. Mean r2 values were
again highest for the AB and were lower for the two rib cage
compartments. The post hoc Scheffé test showed no significant differences between the r2 of the two rib cage
compartments (P > 0.95) or between the AB and the
chest wall (P > 0.80), with significant differences
for all other comparisons (P < 0.01 for all).
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Fig. 3.
The r2 values for correlations
between Dap and Vrc,a [abdominal rib cage (RCa)],
Vrc,p [pulmonary rib cage (RCp)], Vab [abdomen (AB)], and Vcw
[chest wall (CW)] as function of workload combining both control and
flow-limited exercise. Values are means ± SE. QB, quiet
breathing; Wmax/2, half-maximal control exercise workload; Wmax,s,
maximal exercise workload with expiratory flow limitation.
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Table 2 gives the results of the
one-factor analysis of variance in which we looked for possible
differences among repeated measures and different subjects. The values
of P level reported in Table 2 (r2
and slope) demonstrate the repeatability of this method and the independence of r2 from confounding variables,
such as weight, height, age, etc., with the possible exception of
Dap vs. Vab, where the data approach statistical
significance. In contrast, each subject had a characteristic slope as
shown by the P values, all of which showed statistically significant differences in slope between subjects. In addition, differences between repeated measures approached statistical
significance for Dap vs. Vab.
As shown in Table 3, there were two
significant effects for r2: the main effect for
compartment and the interaction between compartment and workload. In
contrast to the analysis of r2, Table 3 shows
that there are many significant factors that affect slope: the main
effect for compartment and workload, the interaction between
compartment and workload, the interaction between presence of EFL and
workload, and the interaction among presence of EFL, workload, and
compartment.
The results of the post hoc Scheffé's test for the effect
of workload on slope are shown in Fig. 4.
As shown in this figure, the two rib cage compartments and the chest
wall had higher slopes during exercise compared with quiet breathing,
and the slope for the chest wall was significantly greater at maximal
power output with EFL than at 25 W. Only the abdominal compartment
showed no change in slope from quiet breathing to exercise.
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Fig. 4.
Slopes of the linear correlations between
Dap, the chest wall, and its compartments as a function
of exercise workload for both control and flow-limited exercise. Values
are means ± SE.
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The presence of EFL did not significantly affect either
r2 or slope within a compartment, as shown in
Fig. 5. Table
4 shows, as does Fig. 3, that there were
no significant differences between the two rib cage compartments, or
between the AB and the chest wall, but both r2
and slope were significantly less for the rib cage than for the chest
wall and AB. Therefore, Figs. 3-5 demonstrate that the correlation between Dap and the change of compartmental volumes is
significantly better for the AB and total chest wall than for the rib
cage compartments, and, considering both r2 and
slope, the AB is somewhat better than the chest wall in predicting Dap. Presumably, this is because the chest wall contains
both rib cage compartments, which diminishes its ability to predict Dap.
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Fig. 5.
The r2 values (A) and slopes
(B) of the linear regressions between Dap,
the chest wall, and its compartments shown separately for control and
flow-limited exercise as a function of exercise workload. EFL,
expiratory flow limitation. Values are means ± SE.
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Multiple Stepwise Linear Regression Analysis
Table 5 gives the results of the
multiple stepwise linear regression analysis on all subjects at the
different workloads. Between 89 and 96% of the variance in
Dap was accounted for by the Vabw. Adding the
displacements of the two rib cage compartments improved this by only
1%.
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DISCUSSION |
Ldi is determined by thoracoabdominal
configuration. Our results demonstrate that, under the conditions in
which we measured Dap near the midaxillary line,
abdominal displacement was its best predictor. Is Dap a
measure of change of Ldi? It may be possible
with rib cage expansion for the diaphragm to peel away from the rib
cage without a change in fiber length. However, Gauthier et al.
(7) showed in humans that changes in the axial dimension of the Aap were excellent estimates of fiber
length in the supine position. No similar data exist in the upright
posture in humans. Assuming that determinants of
Ldi are the same upright and supine, our
measurements of Dap are estimates of changes in costal
Ldi.
As illustrated in Fig. 2, we found substantial shortening of
diaphragmatic fibers during quiet breathing, so that Mead and Loring's
(14) use of an average value of the
Aap during quiet breathing does not seem
justified either theoretically (as developed above) or from an
experimental point of view.
One might expect that fiber length would be well-correlated with
displacements of the RCa from which the costal fibers originate and
with which they are in direct contact in the
Aap. In fact, Loring et al. (13)
have provided evidence that this is the case, although we were unable
to confirm their findings. However, it should be realized that we only
measured the determinants of Ldi when there was
prominent abdominal motion. Before the rib cage can be excluded as an
important determinant, it would be necessary to show that the diaphragm
is quasi-isometric during an inspiration without any abdominal
displacement. We have not studied such breaths.
To further examine the relationship between the contributions of the
RCp, RCa, and AB to tidal volume and their relationship to
Dap, we correlated the fractional contributions of these
chest wall compartments to tidal volume on one hand and
Dap on the other. The results are shown in Fig.
6. The r2 values
for all chest wall compartments are small, and neither relationship is
significant, although statistical significance is approached for the
RCp (P = 0.54).
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Fig. 6.
Relationship between the percent contributions to
tidal volume of the chest wall compartments and Dap.
A: Vrc,p. B: Vrc,a. C:
Vab.
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Loring et al. (13) showed that, when abdominal motion was
minimal, the diaphragm could shorten. They studied three subjects to
evaluate the dependence of Ldi on lung volume
and thoracoabdominal configuration and suggested that
Ldi was closely coupled to rib cage
displacement, as well as to ventral abdominal wall displacement. In
their analysis, they considered the whole rib cage as a determinant of
the Ldi. They argued that, because the costal
fibers originate at the costal margin, a diaphragmatic contraction
should displace the rib cage cranially, entailing diaphragm shortening
independent of abdominal motion. Nevertheless, the diaphragm and
abdominal wall form a compartment of nearly constant volume. The volume swept by the abdominal wall must be equal and opposite to the volume
swept by the diaphragm. If diaphragm fibers can shorten at constant
Vab, then the diaphragm must decrease its surface area at constant
volume. This can only occur if the diaphragm's configuration changes
from a less spherical to a more spherical shape.
Despite the possibility of diaphragmatic shortening during a breath
with no abdominal displacement, we have shown that abdominal displacements alone, measured by OEP, are sufficient to estimate changes in Ldi in healthy subjects during quiet
breathing and exercise, with and without externally applied EFL.
Before attempting such measurements, it would be necessary to calibrate
the relationship between Dap and abdominal displacement in each subject on each occasion that measurements are made. The results of the ANOVA shown in Table 2 reveal highly significant differences in slope of the Dap-Vab regression between
subjects. Therefore, the relationship between the two variables in each subject must be known before Vab can be used as a measure of Dap. Furthermore, the differences in Dap vs.
Vab between repeated measures approaches statistical significance. This
probably reflects slight differences in posture and marker placement in
different experiments. Nevertheless, it indicates the need to calibrate the Dap-Vab regression each time a subject is studied.
Finally (again as shown in Table 2), the between-subject difference in r2 also approached statistical significance.
Therefore, we cannot exclude the possibility that there are individuals
in whom r2 of the Dap-Vab regression
is not sufficiently high to allow good estimates of
Ldi.
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APPENDIX. Dap Estimation |
For each acquired frame, the position of the point P of
coordinates xP,yP,
corresponding to the cephalic margin of the zone of apposition (see
Fig. 1B), was selected in the echographic image (plane of
axes xy) and then geometrically transformed by a
rototranslation into the point P' of coordinates
XP', YP',
ZP' in the laboratory reference system (3D space
of axes XYZ), in which the markers placed both on the chest
wall surface and on the probe were acquired (Fig.
7).
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Fig. 7.
Geometrical transformation between the coordinate system of the
echographic probe (axes x, y, and z)
and the coordinate system of the laboratory where the markers placed on
the chest wall and on the probe are measured (axes X,
Y, Z); M1, M2 and
M3: points corresponding to the markers placed on the
probe; d, distance between the markers M1 and
M2 and the skin.
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The coordinate system of the probe was defined by using the 3D
coordinates of the markers placed on the probe (M1,
M2, and M3), and the three axes (x,
y, z) were defined as follows: x-axis parallel to the straight line passing through points M1 and
M2 at a distance d, equal to the distance
between the markers M1 and M2 and the skin; the
z-axis as the vector product between x and the
straight line passing through M1 and M3;
finally, y-axis was obtained as the vector product between
x and z.
Points P were then rototranslated into the 3D space by the following
equation
|
(8)
|
where P' is the coordinate vector (3 × 1)
of the point P'; P is the coordinate vector (3 × 1) of
the point P (zP = 0); R is the
rotation matrix (3 × 3), whose components are the direction cosines of the three axis x, y, and z;
T is the coordinate vector (3 × 1) of the origin of
the probe coordinate system, defined by the point M1 and
the distance d; and Zs is a row vector (1 × 3) of zeros.
The distance between the point P and the Dap was computed
as follows: 1) by estimating the regression plane 
among
the markers placed at the xiphoid level, of equation
|
(9)
|
|
(10)
|
|
(11)
|
where b1, b2, and b0 are
identified in Eqs. 10 and 11;
Xm, Ym, and
Zm are the mean values of the coordinates of the
body markers at the xiphoid level; and Xi,
Yi, and Zi are the values
of their different coordinates; and 2) by computing the
distance Dap between the point P' (of coordinates
XP', YP',
ZP') and the xiphoid plane as
|
(12)
|
| |
ACKNOWLEDGEMENTS |
We gratefully acknowledge E. Orsi for software
development and G. Scano, R. Duranti, I. Iandelli, G. Misuri,
and B. Kayser for valuable help in preparing and performing the experiments.
| |
FOOTNOTES |
Address for reprint requests and other correspondence:
A. Aliverti, Dipartimento di Bioingegneria, Politecnico di
Milano, Piazza Leonardo da Vinci, 32, I-20133 Milan, Italy (E-mail:
andrea.aliverti{at}polimi.it).
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. Section 1734 solely to indicate this fact.
First published October 4, 2002;10.1152/japplphysiol.00329.2002
Received 12 April 2002; accepted in final form 28 September 2002.
| |
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