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 crosssectional perimeter and area variations of the chest wall. First, we study the phase shift between perimeter and area (Φ_{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 Φ_{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
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, mercuryinrubber 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 4moold 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 additionalxy plot of Fig. 1 F shows that the abdominal signals obtained from both sensors are in phase (no hysteresis), whereas the hysteresis in Fig. 1 E indicates that the thoracic signals are phase shifted. This thoracic shift accounts for the differences in thoracoabdominal phase shift.
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 MartinotLagarde et al. (15), who have analyzed the inductancearea and inductanceperimeter 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 2
A 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 2
B shows a mathematical equivalent model, which is obtained by fixing the ellipse center. Respiratory movements are simulated by sinusoidal variations of each semiaxisx(t) and y(t) as follows (Fig.2
C)
The area (A) and perimeter (P) of the ellipse are given by
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 zigzagshaped wires, which enable some extension. The loops are characterized by their selfinductance 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 selfinductance. The dependence of this sensor on perimeter and area is complex, as can be seen in Eq. 4
, which gives the expression for the selfinductance (L) in the case of a simple circular loop of radiusa, made with a wire of crosssectional radius r (9, 15)
RESULTS
Perimeterarea phase shift.
We show in appendix that the perimeter and area time functions [P(t) and A(t), respectively] are sinusoidal when the amplitudes X andY are small with respect to the mean principal axesx _{m} and y _{m}. 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 (y _{m}/x _{m}), the amplitude ratio (Y/X), and the phase shift (ϕ) between transverse and dorsoventral movements. Hence, the phase shift between P(t) and A(t) (Φ_{PA}) also depends on these parameters. Figure3 shows Φ_{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 (ϕ = 60°, 100°, 160°, 170°, 180°). Figure 3 A is plotted for the mean ellipticity ratio y _{m}/x _{m} = 0.5, whereas Fig. 3 B corresponds toy _{m}/x _{m} = 0.7. In Fig. 3, Φ_{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, Φ_{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.
Sensitivity of the inductive sensors.
The selfinductance of a 650mmlong wire, with eight sinusoidal zigzags, was calculated by discretization of the Neumann equation. We assumed wires with an elliptical layout defined by ratiosy _{m}/x _{m} 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 cm^{2}. As described by MartinotLagarde et al. (15), we obtained a set of inductancearea (Fig.4 A) and inductanceperimeter (Fig.4 B) curves, depending on the section shape. The same information is shown in Fig. 5, in which we plotted isoshape (y _{m}/x _{m}) and isoinductance (L) in a perimeterarea plane.
P(t) and A(t) are sinusoidal for the section deformations considered in this paper (appendix ). Therefore, the respiratory movements can be described by means of elliptical trajectories in the perimeterarea representation. Table1 details the sets of parameters (x _{m}, y _{m}, X, Y, ϕ) used for the different simulations illustrated in Fig.6. Three cases are possible, corresponding to Φ_{PA} = 0°, 0° < Φ_{PA} < 180°, and Φ_{PA} = 180°, respectively. Figure 6 A shows an example of the first case: P(t) and A(t) are in phase (Φ_{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° < Φ_{PA} < 180°, is illustrated in Fig.6 B. 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) andA(t). Figure 6 C shows three examples of the third case, where P(t) and A(t) have an opposite phase (Φ_{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 perimeterarea curve varies from infinity to zero when Y/X increases within the range defining Φ_{PA} = 180° (see Fig. 3).
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 Φ_{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 welldefined range, which increases when ellipticity decreases. Actually, the more the cross section tends to a circle (y _{m}/x _{m} tends to 1), the narrower is the range of movement amplitude Y/X for which Φ_{PA} is significant and the less we risk obtaining a Φ_{PA} different from zero, because Φ_{PA} is smaller for equivalent ϕ andY/X. The implications of these three conditions will now be analyzed in detail.
To observe a significant value of Φ_{PA}, lateral and dorsoventral movements must be desynchronized and the ratio of movement amplitudes must be inside an interval extending fromY/X = y _{m}/x _{m} to a value between 1 and 2. This implies welldefined 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 threedimensional 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 ϕ > 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 Φ_{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 Φ_{PA} is present (Fig. 6).
In Fig. 6 A, 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 (Φ_{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 ofP(t) and A(t). Hence, the inductance evolution will be in phase with P(t) andA(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 6 B shows an example for which the curve describing the cross section deformation is an ellipse, the width and majoraxis slope of which indicate a phase difference for P(t) andA(t) (0° < Φ_{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 6 C 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) andA(t) (Φ_{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. 6 C. 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. 6 C than in Fig. 6, A and B.
This study shows that, if the perimeter and the area variations of the cross section are phaseshifted (Φ_{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. 1 A, the signals obtained from the strain gauges are by definition proportional to the perimeter of the thoracic and abdominal cross sections. In Fig. 1 C, the inductive signals are proportional to the wire inductance of the thoracic and abdominal sensor. Hence, the xy representations Tho_{I}Tho_{S} and Abd_{I}Abd_{S} in Fig. 1, E and F, are equivalent to L_{Tho}P _{Tho} and L_{Abd}P _{Abd}, respectively. They can be drawn in an LP representation similar to Fig. 4 B. As the Tho_{I}Tho_{S} curve shows hysteresis, its drawing in the perimeterarea representation of Fig. 5 corresponds to the simulated curve with hysteresis (case 2 of Fig.6 B). 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 Abd_{I}Abd_{S} curve shows no hysteresis: it corresponds to the situation of Fig. 6 A. 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.
Acknowledgments
We acknowledge Fernand Colin for critical review and helpful suggestions on the manuscript.
Footnotes

Address for reprint requests and other correspondence: M. Paiva, Biomedical Physics Lab., cp 613/3, Route de Lennik, 808, Brussels, Belgium (Email: mpaiva{at}ulb.ac.be).

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.
 Copyright © 2000 the American Physiological Society
Appendix
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
Hence, it can be seen that A(t) varies sinusoidally around the mean area (A
_{m})
Perimeter calculation.
Introducing the sinusoidal movements (Eq. 2 ) in the perimeter (P) formula (Eq. 4 ), we get
Hence, we see that P(t) also varies sinusoidally around the mean perimeter P
_{m}
Appendix
Calculation of selfinductance by the Neumann coefficients is illustrated below.
The selfinductance (L) of a closed lead (length λ) 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 (N
_{ji}) with some approximations (9,15). These coefficients allow the calculation of the mutual inductance between the leads j and i(M
_{ji})