Vol. 88, Issue 4, 1295-1302, April 2000
Measurement of thoracoabdominal asynchrony: importance of
sensor sensitivity to cross section deformations
Anne
de Groote1,
Yves
Verbandt1,
Manuel
Paiva1, and
Pierre
Mathys2
1 Biomedical Physics Laboratory,
Université Libre de Bruxelles, 1070 Brussels; and
2 Department of Electronics, Microelectronics and
Telecommunications, Université Libre de Bruxelles, 1050 Brussels, Belgium
 |
ABSTRACT |
Discrepancies in the assessment of
thoracoabdominal asynchrony are observed depending on the choice of
respiratory movement sensors. We test the hypothesis that these
discrepancies are due to a different dependence of the sensors on
cross-sectional perimeter and area variations of the chest wall. First,
we study the phase shift between perimeter and area
(
PA) for an elliptical model, which is deformed
by sinusoidal changes of its principal axes. We show that perimeter and
area vary sinusoidally in the physiological range of deformations, and
we discuss how PA depends on the ellipticity of
the cross section, on the ratio of transverse and dorsoventral movement
amplitudes, and on their phase difference. Second, we compute the
relationship between perimeter, area, and the output of the inductive
sensor, and we proceed by comparing inductive plethysmography with
strain gauges for several cross section deformations. We demonstrate
that both sensors can provide different phase information for identical cross section deformations and, hence, can estimate thoracoabdominal asynchrony differently. Furthermore, the complex dependence of the
inductive sensor on perimeter and area warns against this sensor for
the evaluation of thoracoabdominal asynchrony.
respiratory movements; strain gauges; inductive plethysmography
| |
INTRODUCTION |
THORACOABDOMINAL ASYNCHRONY is known to depend on
subject age (3, 8, 11), sleep stage (23), and severity of airway obstruction (2, 21, 22). This parameter is commonly measured by the
phase shift between two signals representative of the thoracic and
abdominal movements, according to the model of Konno and Mead (14). In
these studies, strain gauges (1), inductive plethysmography (17),
magnetometers (16), and linear differential transducers (14) are used
without any detailed discussion of their specific characteristics.
However, Heldt (10) compared the use of magnetometers, mercury-in-rubber strain gauges, and inductance plethysmograph in
preterm infants. He observed that the three abdominal signals were
similar, whereas the thoracic phase measurements from the magnetometers
sometimes deviated from those of the strain gauges and of the
inductance plethysmograph. In addition, Ross Russell and Helms (19)
compared inductance plethysmographs, magnetometers, and Hall device
strain gauges in children aged 1 mo to 13 yr. Although similar
waveforms were observed during spontaneous tidal breathing, the phase
angles sometimes varied between sensors. We have also compared signals
recorded in infants by inductive plethysmography and piezoelectric
strain gauges and have observed phase differences resulting in
discrepancies between the measurements of thoracoabdominal asynchrony.
Figure 1 shows an example of data obtained
from a quietly breathing 4-mo-old baby. Thoracic and abdominal
movements were simultaneously recorded by inductive sensors and
piezoelectric strain gauges. The inductive belts were incorporated in a
pyjama and placed respectively on the umbilicus and on the nipples. Two
piezoelectric belts were superposed on the inductive belts. Figure 1,
A and C, shows the signals as a function of time.
Figure 1, B and D, shows each thoracic signal plotted
in ordinate as a function of the corresponding abdominal signal in
abscissa. This representation indicates that the thoracoabdominal asynchrony measured by strain gauges is greater than 90° (negative slope of the principal axis of the curve), whereas the asynchrony measured by inductive plethysmography is smaller than 90° (positive slope of the principal axis of the curve). The additional
x-y plot of Fig. 1F shows that the abdominal
signals obtained from both sensors are in phase (no hysteresis),
whereas the hysteresis in Fig. 1E indicates that the thoracic
signals are phase shifted. This thoracic shift accounts for the
differences in thoracoabdominal phase shift.
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Fig. 1.
Comparison of thoracic (Tho) and abdominal (Abd) respiratory movements
measured simultaneously in a 4-mo-old baby by means of piezoelectric
strain gauges (S) and inductive plethysmography (I). A and
C: thoracic and abdominal signals as a function of time.
B and D: thoracic signal as a function of corresponding
abdominal signal. E and F: inductive signal as a
function of corresponding signal obtained from strain gauge. au,
Arbitrary units.
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Having observed these discrepancies, we decided to test the hypothesis
that the phase shift between the electrical outputs of the two types of
sensors measuring the same respiratory movements is due to a difference
of sensitivity with regard to the variations of perimeter and area of
the cross section. To verify this assumption, our approach consisted of
three steps. First, we studied the geometrical conditions for which a
phase shift occurs between variations of perimeter and area, and then
we compared inductive plethysmography and strain gauges for the
measurement of cross section deformations and for the evaluation of
thoracoabdominal asynchrony. Inductive plethysmography is
conventionally chosen to measure area variations, whereas strain gauges
are known to be sensitive only to perimeter changes. The analysis of
the sensitivity of the inductive sensor to perimeter and area
variations is based on the work of Martinot-Lagarde et al. (15), who
have analyzed the inductance-area and inductance-perimeter dependence
on the cross section and belt characteristics. Their aim was to
evaluate the accuracy of inductive plethysmography for measurement of
area and perimeter changes. We have extended their work to evaluate the
accuracy of thoracoabdominal asynchrony measurements obtained by
respiratory inductive plethysmography, for which inconsistent results
have been found, as shown in Fig. 1.
| |
METHODS |
Basic cross section and deformations.
Figure 2A shows the model used to
compute the thoracic and abdominal cross sections: an ellipse with
minor and major axes corresponding to the dorsoventral and transverse
directions, respectively. One point of the ellipse is fixed to simulate
the spine, while the ellipse center moves during the respiratory cycle.
Figure 2B shows a mathematical equivalent model, which is
obtained by fixing the ellipse center. Respiratory movements are
simulated by sinusoidal variations of each semiaxis
x(t) and y(t) as follows (Fig.
2C)
|
(1)
|
where X and Y are the motion amplitudes,
xm and ym are the mean values
of the semiaxes, f is the respiratory frequency, t is time, and
is the asynchrony between lateral and ventral movements. The
ellipticity ratio is defined as
y(t)/x(t).
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Fig. 2.
A: elliptical model of thoracic and abdominal cross sections
during breathing. Principal axes represent transverse and dorsoventral
diameters. Fixed point corresponds to spine ( ). Three different
cross sections are represented. B: mathematical model
equivalent to A, with center of ellipse considered fixed.
X and Y are amplitudes of principal semiaxis
[x(t) and y(t); t,
time] movements. C: simulation of respiratory movements
by sinusoidal variations of each semiaxis x(t) and
y(t), with amplitudes X and Y around mean
values xm and ym. f,
respiratory frequency. , Phase shift between x and y
movements.
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The area (A) and perimeter (P) of the ellipse are given
by
|
(2)
|
and
![<IT>P</IT>(<IT>t</IT>) = 4<IT>y</IT>(<IT>t</IT>)E{1 − [<IT>x</IT> (<IT>t</IT>)/<IT>y</IT>(<IT>t</IT>)]<SUP>2</SUP>}](/content/vol88/issue4/fulltext/1295/img004.gif)
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(3)
|
Here, function E( ) is the elliptic integral of the second
kind, calculated between 0 and 
/2. Detailed computations are given
in APPENDIX A.
Sensitivity analysis.
Strain gauges generate a voltage when submitted to mechanical strain.
Respiratory motion sensors based on this principle can be, for example,
obtained by including a piezoelectric film, sensitive to elongation, in
relatively stiff belts surrounding the thorax or the abdomen. For these
sensors, the strain sensitivity of the transducer combined with the
belt elasticity determines the perimeter measurement. Inductive
plethysmography uses thoracic and abdominal belts fitted with
zigzag-shaped wires, which enable some extension. The loops are
characterized by their self-inductance and are placed in an oscillator
circuit, the frequency of which is modulated by the variations of the
inductance due to the respiratory movements. Demodulation can be
obtained by a conventional Respitrace (Respitrace, Ardsley, NY), which
provides an output voltage proportional to the self-inductance. The
dependence of this sensor on perimeter and area is complex, as can be
seen in Eq. 4, which gives the expression for the
self-inductance (L) in the case of a simple circular loop of radius
a, made with a wire of cross-sectional radius r (9, 15)
![L = &mgr;<SUB>0</SUB><IT>a</IT>[ln (8<IT>a</IT>/<IT>r</IT>) − 7/4]](/content/vol88/issue4/fulltext/1295/img005.gif)
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(4)
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The self-inductance of a wire of constant length uniformly
distributed in sinusoidal shape around a plane elliptical cross section
(principal axes 2xm and 2ym)
has been computed by using the Neumann coefficients (APPENDIX
B). The inductance-area and inductance-perimeter relationships
were calculated in the physiological range for infants (6, 13, 18).
| |
RESULTS |
Perimeter-area phase shift.
We show in APPENDIX A that the perimeter and area time
functions [P(t) and A(t),
respectively] are sinusoidal when the amplitudes X and
Y are small with respect to the mean principal axes
xm and ym. Furthermore, the phase
shift of P(t) and A(t), with respect to
the phase of the original movements, depend on three parameters: the
mean ellipticity ratio (ym/xm), the amplitude ratio (Y/X), and the phase shift ()
between transverse and dorsoventral movements. Hence, the phase shift
between P(t) and A(t)
(PA) also depends on these parameters. Figure 3 shows PA, in a
semilogarithmic representation, as a function of the movement amplitude
ratio (Y/X = 0.1 to 10) for different phase shifts
between x and y movements ( = 60°, 100°, 160°, 170°, 180°). Figure 3A is plotted for the
mean ellipticity ratio ym/xm = 0.5, whereas Fig. 3B corresponds to
ym/xm = 0.7. In Fig. 3,
PA equal to 0° implies that the perimeter
and the area variations of the cross section are in phase. When
subjected to this type of cross section deformation, all sensors,
whether they are sensitive to perimeter or to area, will generate
equivalent output. Hence, in this case, a detailed analysis of sensor
sensitivity is superfluous. In contrast, PA
equal to 180° corresponds to the extreme situation in which the
perimeter and area modulations have an opposite phase. In this case,
very different phase measurements can be expected from sensors with
different sensitivity to variations of perimeter and area.
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Fig. 3.
Perimeter-area phase shift (PA) calculated on an
elliptical cross section, semiaxes of which vary sinusoidaly, as a
function of movement amplitude ratio (Y/X on a
logarithmic scale), for different mean ellipticity ratios
(A: ym/xm = 0.5;
B: ym/xm = 0.7) and
different phase shifts () between movements in x and
y directions.
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Sensitivity of the inductive sensors.
The self-inductance of a 650-mm-long wire, with eight sinusoidal
zigzags, was calculated by discretization of the Neumann equation. We
assumed wires with an elliptical layout defined by ratios
ym/xm equal to 1, 0.9, 0.8, 0.7, 0.6, or 0.5, respectively. For each ellipticity, we considered five different sizes of cross section, providing an area ranging from 110 to
150 cm2. As described by Martinot-Lagarde et al. (15), we
obtained a set of inductance-area (Fig.
4A) and inductance-perimeter (Fig. 4B) curves, depending on the section shape. The same
information is shown in Fig. 5, in which we
plotted isoshape (ym/xm) and
isoinductance (L) in a perimeter-area plane.
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Fig. 4.
Inductance-area (L-A; A) and inductance-perimeter
(L-P; B) relationships for a 650-mm sinusoidal wire (8 zigzags) placed on different elliptical sections defined by their
ellipticity ym/xm.
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Fig. 5.
Perimeter-area (P-A) relationships obtained from Fig.
4. Curves of isoshape and of isoinductance have been drawn on a major
grid (bold lines) and a minor grid. Bold rectangle delimits region
examined more closely in Fig. 6.
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P(t) and A(t) are sinusoidal for the
section deformations considered in this paper (APPENDIX A).
Therefore, the respiratory movements can be described by means of
elliptical trajectories in the perimeter-area representation. Table
1 details the sets of parameters
(xm, ym, X, Y,
) used for the different simulations illustrated in Fig.
6. Three cases are possible, corresponding
to PA = 0°, 0° < PA < 180°, and PA = 180°, respectively. Figure 6A shows an example of the first
case: P(t) and A(t) are in phase
(PA = 0°). The curve describing the cross
section deformation is a straight line with positive slope (in fact, a
degenerate ellipse). The second case, where 0° < PA < 180°, is illustrated in Fig.
6B. The curve describing the cross section deformation is an
ellipse. The curve hysteresis and the sign of the slope of the major
axis indicate the phase difference for P(t) and
A(t). Figure 6C shows three examples of the
third case, where P(t) and A(t) have an
opposite phase (PA = 180°). The curves describing the cross section deformations are straight lines with negative slopes (degenerate ellipses). The difference between these
three curves lies in their slopes, which are respectively equal,
greater than, or smaller than the slope of the isoinductance curves. In
fact, the absolute slope of the perimeter-area curve varies from
infinity to zero when Y/X increases within the range defining PA = 180° (see Fig. 3).
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Fig. 6.
Set of trajectories in P-A representation (detail of
Fig. 5) due to respiratory movements. Because P(t) and
A(t) are sinusoidal, the corresponding curve is
elliptical. Depending on PA (A:
PA = 0°, B: 0° < PA < 180°, C:
PA = 180°), the curve can be a straight line
with positive slope, an ellipse, or a straight line with negative
slope. Values chosen for each curve are given in Table 1.
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DISCUSSION |
Conditions inducing a phase shift between perimeter and area
variations.
Figure 3 defines the conditions on shape and deformation of the
elliptical section that induce a phase shift between perimeter and
area. It shows that to obtain a PA different
from zero, there must be a significant (>60°) phase shift between
lateral and dorsoventral movements. In addition, the ratio of lateral and dorsoventral movement amplitudes must fall within a well-defined range, which increases when ellipticity decreases. Actually, the more
the cross section tends to a circle
(ym/xm tends to 1), the
narrower is the range of movement amplitude Y/X for
which PA is significant and the less we risk
obtaining a PA different from zero, because
PA is smaller for equivalent and
Y/X. The implications of these three conditions will
now be analyzed in detail.
To observe a significant value of PA, lateral
and dorsoventral movements must be desynchronized and the ratio of movement amplitudes must be inside an interval extending from Y/X = ym/xm to
a value between 1 and 2. This implies well-defined complex deformations
of the cross section. Up to now, only a few studies have measured in
detail the respiratory movements. Furthermore, all were achieved on
healthy adult subjects (4, 5, 7, 20). Nevertheless, some preliminary
unpublished results obtained in our laboratory by means of a
three-dimensional optoelectronic motion analyzer (Elite, BTS, Milan,
Italy) indicate that some section deformations in infants could fulfill
these conditions. In fact, we observed the deformations of thoracic sections in three sleeping infants and found that, in half of the
recordings, Y/X ranged from 0.5 to 1.5 in conjunction
with > 100°. Our preliminary results indicate that complex
cross sections do occur and that they depend on the craniocaudal
position of the studied section. The section deformations probably
depend both on the chest compliance and on the respiratory pattern.
Hence, a significant phase shift between perimeter and area is expected to be observed in babies younger than 1 yr, showing high compliance (12), or in obstructive pathologies where even more complex respiratory
patterns may occur.
Previous studies (13, 18) have shown that the thoracic cross section is
more circular in early infancy and becomes more elliptical with aging.
Results obtained by Openshaw et al. (18) indicate a thoracic index
(dorsoventral diameter divided by transverse diameter) that evolves
from 0.8 at 3 mo to 0.6 at 1 yr. A stabilization of the thoracic index
occurs by the age of 2-3 yr. Hence, the risk of being in a
physiological situation in which PA is different
from 0 will increase from birth to 2-3 yr.
Sensitivity of the inductive sensor and its influence on phase
measurement.
From the slope of the isoinductance curves of Fig. 5, we can see the
simultaneous dependence of the inductive sensor on the perimeter and on
the area. Curves parallel to the perimeter axis (infinite slope) would
indicate that, whatever the perimeter, an inductive sensor depends
selectively on area variations. On the other hand, curves parallel to
the area axis (slope equal to 0) would indicate that, whatever the
area, only the perimeter determines the value of L. Because the
isoinductance curves of Fig. 5 have a finite nonzero slope, an
intrinsic, simultaneous dependence on the perimeter and on the area is
always present in the inductive sensor. This difference of sensitivity
can be of great importance during the evaluation of thoracoabdominal asynchrony if a nonzero PA is present (Fig. 6).
In Fig. 6A, the curve describing the cross section deformation
is a straight line with positive slope (case of a degenerate ellipse):
P(t) and A(t) are in phase
(PA = 0°). This line does not a
priori coincide with an isoshape curve, depending on whether the
deformation occurs at constant ellipticity or not. Because this curve
is closed for one respiratory cycle and shows no hysteresis, each
isoinductance curve is intercepted twice, for the same values of
P(t) and A(t). Hence, the inductance
evolution will be in phase with P(t) and
A(t). In this case, the inductive sensor measures
perimeter and area variations without distinction; strain gauges and
inductive sensors provide the same phase information.
Figure 6B shows an example for which the curve describing the
cross section deformation is an ellipse, the width and major-axis slope
of which indicate a phase difference for P(t) and
A(t) (0° < PA < 180°). This section deformation implies necessarily a variable
section shape as shown by the evolution in the isoshape network. The
curve is always closed during a respiratory cycle but shows hysteresis:
each isoinductance is intercepted twice at points with different
(perimeter, area) values. Hence, the inductive signal is shifted by
comparison to P(t) and A(t). In this
case, the inductive sensor does not measure selectively either the
perimeter or the area. The signal obtained from a strain gauge, which
measures only the perimeter, is phase shifted with respect to the
inductive sensor. Hence, the respiratory phase information obtained
from the two sensors is different.
Figure 6C shows an example in which the curves describing the
cross section deformations are straight lines with negative slopes.
This indicates a phase inversion between P(t) and
A(t) (PA = 180°). The
section deformation implies necessarily a variable section shape.
Depending on whether the absolute slope is greater than, equal to, or
smaller than the isoinductance slope, L will be, respectively, in phase
with perimeter (while in phase inversion with area), constant during
the deformation, or in phase with area (while in phase inversion with
perimeter). These three cases are illustrated by the three different
curves in Fig. 6C. Hence, depending on the situation, the
inductive sensor measures either perimeter or area. In these extreme
situations, the signal obtained from a strain gauge can be in
opposition with the inductive signal, inducing a completely different
phase interpretation. Nevertheless, it must be noted that the amplitude
of the inductive signal (measured by the number of isoinductance curves
crossed by the curve describing the cross section deformation) is
smaller in Fig. 6C than in Fig. 6, A and B.
This study shows that, if the perimeter and the area variations of the
cross section are phase-shifted (PA 
0°), the respiratory phase information obtained by means of
inductive sensors or strain gauges is different. This can explain the
discrepancies observed in the example of Fig. 1. In Fig. 1A,
the signals obtained from the strain gauges are by definition
proportional to the perimeter of the thoracic and abdominal cross
sections. In Fig. 1C, the inductive signals are proportional to
the wire inductance of the thoracic and abdominal sensor. Hence,
the x-y representations ThoI-ThoS and
AbdI-AbdS in Fig. 1, E and F,
are equivalent to LTho-PTho and
LAbd-PAbd, respectively. They can be
drawn in an L-P representation similar to Fig. 4B.
As the ThoI-ThoS curve shows hysteresis, its
drawing in the perimeter-area representation of Fig. 5 corresponds to
the simulated curve with hysteresis (case 2 of Fig.
6B). This implies a complex deformation of the section, which
accounts for the phase shift observed between signals from sensors
sensitive in a different way to perimeter and area. On the other hand,
the AbdI-AbdS curve shows no hysteresis: it
corresponds to the situation of Fig. 6A. There is a uniform
deformation of section, which explains the identical phase information
obtained from the sensors.
Implications on thoracoabdominal asynchrony measurement.
In the study of thoracoabdominal asynchrony, an additional degree of
complexity is introduced because two different cross sections are
monitored and their deformations are compared. Couples of inductive
sensors and couples of strain gauges can provide different
thoracoabdominal asynchrony information because of their respective
sensitivity on perimeter and area and the respective deformations of
the thoracic and abdominal cross sections. It is difficult to estimate
the probability of this situation occurring and the magnitude of the
discrepancy. At present, we can only speculate that the discrepancies
between inductive sensors and strain gauges will most probably be
observed in infants (<1 yr) because of their high thoracic compliance
and in obstructive pathologies because these involve complex
respiratory patterns. Because inductive sensors are sensitive to both
perimeter and area variations in a complex way and because even two
identical belts can be subjected to different thoracic and abdominal
cross section deformations, we recommend that caution should be taken
when this type of sensor is used for the evaluation of thoracoabdominal
asynchrony in young infants and in patients with obstructive patterns.
In summary, we tested the hypothesis that commonly used sensors for
measuring chest and abdominal wall motion can show a different sensitivity to perimeter and area, which could explain the observed differences in the measurement of thoracoabdominal asynchrony. After
defining the conditions that induce a phase shift between perimeter and
area variations, we demonstrated the possibility of experimental
situations for which strain gauges and inductive belts provide
different phase information for identical cross section deformations
and thus differently evaluate thoracoabdominal asynchrony. From the
complex dependence of the inductive sensors on perimeter and area, we
conclude that caution should be taken when this kind of sensor is used
for the measurement of thoracoabdominal asynchrony.
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APPENDIX A |
Area and perimeter calculations for the section deformation described
in Fig. 2 are given below.
Area calculation.
Introducing the sinusoidal movements (Eq. 2) in the area
(A) formula (Eq. 3), we get
When
X/xm << 1 and
Y/ym << 1, neglecting the second order
terms, it yields
where
A(t) represents the area variations.
C1 is a constant, and A
is the phase shift of A(t) with respect to the phase of
the original movement x(t). A
depends on X/Y,
xm/ym, and .
Hence, it can be seen that A(t) varies sinusoidally
around the mean area (Am)
Perimeter calculation.
Introducing the sinusoidal movements (Eq. 2) in the
perimeter (P) formula (Eq. 4), we get
E( ) is the elliptic integral of the second kind, calculated
between 0 and /2, given by
Taylor
expansion limited to the first order for
(X/xm, Y/ym)
around (0,0) yields
where
P(t) represents the perimeter variations.
C2, C3, and
C4 are constants that depend on
xm/ym. P is the phase shift of P(t) with respect to the phase of
the original movement x(t). It depends on
X/Y, xm/ym, and .
Hence, we see that P(t) also varies sinusoidally around
the mean perimeter Pm
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APPENDIX B |
Calculation of self-inductance by the Neumann coefficients is
illustrated below.
The self-inductance (L) of a closed lead (length 
) with circular
cross section of negligible radius (r) compared with the curvature radius (R) along the axis of the loop (r
<< R) is obtained by using the Neumann coefficients
(Nji) with some approximations (9,
15). These coefficients allow the calculation of the mutual inductance between the leads j and i
(Mji)
The self-inductance is obtained when the leads j
and i are
merged
Here, dli and
dlj represent length elements of wire, separated by
a distance Rij. The integrals must be calculated on
the outline of the lead. Figure 7 shows the
sinusoidal belt used in this work to model the inductive sensor.
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Fig. 7.
Three-dimensional view of sinusoidal inductive belt used to evaluate
inductive sensor dependence on perimeter and area. Lead is uniformly
distributed around an elliptical section with semiaxes
xm and ym. Rij,
distance between two length elements of wire (dli
and dlj). r, Radius of the lead.
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ACKNOWLEDGEMENTS |
We acknowledge Fernand Colin for critical review and helpful
suggestions on the manuscript.
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FOOTNOTES |
This study was supported by the Belgian Federal Office for Scientific
Affairs (PRODEX contract). During this work, A. De Groote was a fellow
of Fonds pour la formation à la Recherche dans l'Industrie et
dans l'Agriculture.
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: M. Paiva,
Biomedical Physics Lab., cp 613/3, Route de Lennik, 808, Brussels,
Belgium (E-mail: mpaiva{at}ulb.ac.be).
Received 30 August 1999; accepted in final form 15 December 1999.
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J APPL PHYSIOL 88(4):1295-1302
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ABSTRACT
INTRODUCTION
![= &pgr;<IT>x<SUB>m</SUB>y</IT><SUB><IT>m</IT></SUB>[1 + X/<IT>x</IT><SUB><IT>m</IT></SUB> sin (2&pgr;f<IT>t</IT>)][1 + Y/<IT>y</IT><SUB><IT>m</IT></SUB> sin (2&pgr;f<IT>t</IT> + ϕ)]](/content/vol88/issue4/fulltext/1295/img007.gif)
![&Dgr;<IT>A</IT>(<IT>t</IT>) = &pgr;Y<IT>y</IT><SUB><IT>m</IT></SUB>[X/Y sin (2&pgr;f<IT>t</IT>) + <IT>x<SUB>m</SUB>/y</IT><SUB><IT>m</IT></SUB> sin (2&pgr;f<IT>t</IT> + ϕ)]](/content/vol88/issue4/fulltext/1295/img008.gif)
![<IT>P</IT>(<IT>t</IT>) = 4<IT>y</IT>(<IT>t</IT>)E{1 − [<IT>x</IT> (<IT>t</IT>)/<IT>y</IT>(<IT>t</IT>)]<SUP>2</SUP>}](/content/vol88/issue4/fulltext/1295/img011.gif)
![= 4<IT>y</IT>(<IT>t</IT>)E{1 − (<IT>x</IT><SUB>m</SUB>/<IT>y</IT><SUB>m</SUB>)<SUP>2</SUP>[1 + X/<IT>x</IT><SUB>m</SUB> sin (2&pgr;f<IT>t</IT>)]<SUP>2</SUP>/[1 + Y/<IT>y</IT><SUB>m</SUB> sin (2&pgr;f<IT>t</IT> + ϕ)]<SUP>2</SUP>}](/content/vol88/issue4/fulltext/1295/img012.gif)
