J Appl Physiol 95: 571-576, 2003.
First published April 18, 2003; doi:10.1152/japplphysiol.00196.2003
8750-7587/03 $5.00
Calibration of pneumotachographs using a calibrated syringe
Yongquan Tang,
Martin J. Turner,
Johnny S. Yem, and
A. Barry Baker
Department of Anaesthetics, University of Sydney, Royal Prince Alfred Hospital, Sydney, New South Wales 2050, Australia
Submitted 25 February 2003
; accepted in final form 14 April 2003
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ABSTRACT
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Pneumotachograph require frequent calibration. Constant-flow methods allow polynomial calibration curves to be derived but are time consuming. The
iterative syringe stroke technique is moderately efficient but results in
discontinuous conductance arrays. This study investigated the derivation of
first-, second-, and third-order polynomial calibration curves from 6 to 50
strokes of a calibration syringe. We used multiple linear regression to derive
first-, second-, and third-order polynomial coefficients from two sets of
6–50 syringe strokes. In part A, peak flows did not exceed the
specified linear range of the pneumotachograph, whereas flows in part
B peaked at 160% of the maximum linear range. Conductance arrays were
derived from the same data sets by using a published algorithm. Volume errors
of the calibration strokes and of separate sets of 70 validation strokes
(part A) and 140 validation strokes (part B) were calculated
by using the polynomials and conductance arrays. Second- and third-order
polynomials derived from 10 calibration strokes achieved volume variability
equal to or better than conductance arrays derived from 50 strokes. We found
that evaluation of conductance arrays using the calibration syringe strokes
yields falsely low volume variances. We conclude that accurate polynomial
curves can be derived from as few as 10 syringe strokes, and the new
polynomial calibration method is substantially more time efficient than
previously published conductance methods.
polynomial; linear regression; conductance; pneumotachometer
SINCE IT WAS INTRODUCED IN 1925 by Fleisch (3), the pneumotachograph (PT)
has been used widely in the respiratory laboratory, intensive care unit,
bioengineering laboratory, and in clinical anesthesia. Both Fleisch and screen
PTs generate differential pressures approximately proportional to the volume
flow and viscosity of the gas, but independent of the pressure. As viscosity
varies with the gas composition, temperature, and humidity,
flow-to-differential pressure conversion of a PT depends on all of these
factors, as well as on the up- and downstream geometry of the tube
(2,
5,
11). Therefore, PTs must be
calibrated routinely under conditions that are as close as possible to those
under which measurements are performed.
Several methods for describing a PT response curve have been reported, including first- (6), second-,
and third-order polynomial (4,
9) and conductance arrays
(8,
12). PTs are normally linear
over a limited range of flows, and, if they are used outside that range,
substantial errors may result. Polynomial methods have been shown to be
practical and accurate over wider flow ranges
(9), but the determination of
the coefficients requires numerous data points at constant flows by using an
accurate flow or volume standard as reference. Existing polynomial calibration
procedures can be tedious and are not suitable for frequent routine
calibration.
Yeh et al. (12) developed a simpler method for calculating PTs by using multiple strokes of a calibration
syringe from which conductance arrays are calculated by a weighted-averaging
algorithm. Yeh's iterative syringe stroke calibration method has the advantage
that a complete set of conductance values can be determined easily and with
moderate effort. A disadvantage of Yeh's conductance method is the fact that
adjacent conductance values are independent and can differ substantially if no
additional smoothing is applied, although physical considerations suggest that
the curve should be continuous and smooth.
In this study, we report a new, efficient method for determining polynomial calibration curves from a small number of syringe strokes and compare this new
method of calibration with the conductance array calibration method of Yeh et
al. (12).
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MATERIALS AND METHODS
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Apparatus. The experimental apparatus included a 3-liter precision syringe (Pulmonary Data Service Instrumentation) connected to a screen PT
(model 3700A, 0–160 l/min linear range, 5 Pa · l-1
· min resistance; Hans Rudolph) and an airway pressure transducer (HCXM100D6 Sensortechnics, Puchheim, Germany). The PT was connected to
standard anesthesia circuit components: biological filter and 8-mm
endotracheal tube. The PT differential pressure taps were connected to a
Validyne MP45 differential pressure transducer (Validyne) by 20 cm of 3-mm ID
Tygon tubing. This configuration has been shown to exhibit a bandwidth of
60 Hz (10).
A carrier demodulator (Validyne CD15, Validyne) converted the differential pressure to a voltage signal. Flow and airway pressure signals were filtered
by matching sixth-order, linear-phase, low-pass filters (Telectronics, Sydney,
Australia; -3 dB at 45 Hz), digitized at 320 Hz by a 12-bit analog-to-digital
converter (ADC) (PCI-MIO-16-e, National Instruments) and acquired by an
IBM-compatible computer. Barometric pressure was measured by using an
electronic barometer (Vaisala PTB100A, Helsinki, Finland). Data acquisition
was controlled by Matlab and Simulink software by using XPC Target and
Realtime Workshop toolboxes (The Math Works, Natick, MA). The airway pressure
transducer was calibrated by using a precision manometer (Airflow
Developments).
Methods. The syringe was pumped manually with alternate low, medium, and high flows. All data points from all syringe strokes were checked
for saturation of the measurement system. The mean value of the baseline
zero-flow signal was calculated by using the first and last 1,000 samples of
each sequence of syringe strokes and subtracted from the flow signal.
In part A of this study, peak flows did not exceed 120 l/min, which is within the linear range specified by the PT manufacturer. Fifty
calibration strokes and 70 additional validation strokes were made in part
A. In part B, peak flows up to 250 l/min were used, which
exceeded the linear limit specified by the PT manufacturer by 60%. The
gain of the carrier demodulator was reduced to accommodate the higher
differential pressures. Fifty calibration strokes and an additional 140
validation strokes were made in part B.
Theory. If the zero-flow offset is removed from the ADC output, a second-order polynomial for calculating the ith flow value
(
i) as a function of the
ith ADC output (ni) can be expressed as
where b1 and b2 are unknown constants. If the sample interval is Ts, the volume of gas
(V) that flows through the PT can be calculated as follows
The initial and final syringe pressures are atmospheric when the flow is zero, but the resistance of the airway or the connectors (in this study an
endotracheal tube) causes gas exiting the syringe to flow through the PT at
increased pressure, resulting in reduced volume flow in the PT. Resistance PTs
are volume-flow transducers; therefore, comparisons with a volume standard
require corrections for pressure changes. Each calculated flow sample requires
correction for pressure changes in the PT as follows
Hence
 | (1) |
where ki is given by
 | (2) |
where PB is the barometric pressure and Pi is the airway pressure associated with the ith flow sample. In cases in
which the airway geometry is such that the pressure in the PT remains at
atmospheric pressure during calibration, then the pressure correction term
ki can be omitted.
Equation 1 is linear in the coefficients b1
and b2 and applies to any syringe stroke. Unique values of the coefficients b1 and b2 can be
determined from any two different syringe strokes of known volumes. If more
than two different strokes are made, then multiple linear regression
(1) can be used to obtain
better estimates of b1 and b2 (see
APPENDIX A).
Data processing. We used multiple linear regression (see APPENDIX A) to estimate the coefficients of first-, second-, and
third-order polynomials from sets of 6, 10, 20, 30, 40, and 50 syringe
strokes. Regression software was written by using Matlab (The Math Works).
Coefficients of n2 in the second-order polynomials and
n3 in the third-order polynomials were tested for
significance by examination of the 95% confidence intervals (with Bonferroni
correction). With the use of each set of polynomial coefficients, the volume
errors of the same set of calibration strokes and the additional sets of
validation strokes were analyzed. Calibration data from parts A and
B were processed separately. Conductance values for sets of 6, 10,
20, 30, 40, and 50 calibration syringe strokes were calculated by using the
algorithm of Yeh et al. (12),
modified to include airway pressure correction and iterated four times, as
suggested by Stromberg and Gronkvist
(8) (see APPENDIX
B).
With the use of these conductance arrays, volume errors for the same set of calibration strokes and for the validation stokes were calculated.
Mean volume errors from all of the polynomial groups were compared with the mean volume errors of the conductance groups with corresponding numbers of
syringe strokes, and with the 50-stroke conductance group. The variances of
the volume errors in the second- and third-order polynomial groups were
compared with the variances of the conductance groups with the corresponding
number of syringe strokes, and with the 50-stroke conductance group. Volume
errors in the 50-stroke conductance groups calculated by using the calibration
strokes were compared with the corresponding errors obtained by using the
evaluation strokes. Results are shown as means ± SD. One-way ANOVA with
Tukey's post test was performed by using Prism V3.00 (GraphPad Software). The
comparison of variances was conducted by using the two-tailed F-test
with Bonferroni correction for multiple comparisons. P < 0.05 was
considered to be statistically significant.
First-, second-, and third-order polynomials were derived from 50-stroke flow data of part B without correction for airway pressure and
compared with polynomials derived with airway pressure correction to assess
the error that would be incurred if pressure correction were omitted.
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RESULTS
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The second- and third-order polynomials and conductance arrays derived from 50 syringe strokes and normalized by subtraction of the first-order
calibration curve are plotted in Fig.
1, A (part A) and B (part B).
In Fig. 1A, the
second- and third-order polynomial curves overlap over most of the flow range.
The conductance values are randomly distributed around the higher order
polynomial curves in Fig. 1.
The 95% confidence intervals of the coefficients of n2 in
the second-order polynomials exclude zero in all groups of parts A
and B, indicating significance. The 95% confidence intervals of the coefficients of n3 include zero in all groups, indicating
that these coefficients are not significant. The 95% confidence intervals of
the third-order polynomial coefficients of both n2 and
n3 in the six stroke groups of both parts A and
B included zero, indicating lack of statistical significance.
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Fig. 1. Calibration curves obtained from 50 syringe strokes normalized by subtraction of the first-order curve [part A low flow (A);
part B high flow (B)]. The thick horizontal line at zero
difference represents the first-order polynomial. The second- and third-order
polynomial curves are almost exactly superimposed and are represented by the
thick curved lines. The conductance array is represented by the dotted points
surrounding the polynomial curves. ADC, analog-to-digital converter.
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Figure 2 shows means ± SDs of the volume errors evaluated by using the same set of
6–50 syringe strokes from which the calibration curves were derived.
There are no statistically significant differences in mean errors in part
A or B. The variances of the first-order polynomial groups in
parts A and B are all significantly larger than the
variances of the corresponding conductance groups in parts A and
B. The variances of the 20-stroke second- and third-order polynomial
groups in part A and the six-stroke second- and third-order
polynomial groups in part B are significantly different from the
variances of the corresponding conductance groups in part B. In
part B, the variances of the six-stroke third-order polynomial group
and the 6-, 10-, and 20-stroke conductance groups are smaller than the
variance of the 50-stroke conductance group.
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Fig. 2. Volume errors obtained by analysis of the same syringe strokes used to derive the calibration curves [part A (A); part B
(B)]. Values are means ± SD. Variance differs significantly from *corresponding conductance group and from
+50-stroke conductance group: P < 0.05. pol,
Polynomial.
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Figure 3 shows means ± SDs of volume errors evaluated by using the separate set of 70
validation syringe strokes in part A and 140 validation syringe
strokes in part B. In part A, none of the variances of the
second- and third-order polynomial groups differed from the variance of the
50-stroke conductance group. In part B, all of the second- and
third-order polynomial groups, with the exception of the six-stroke
third-order high-flow group, exhibit variances that are significantly smaller
than both the corresponding conductance groups and the 50-stroke conductance
group. The variances in the first-order polynomial groups are all
significantly bigger than corresponding variances in the conductance groups.
The mean volume errors in all of the six-stroke second- and third-order
polynomial groups are significantly smaller than the mean volume errors in the
corresponding conductance groups.
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Fig. 3. Volume errors obtained by analysis of 70 validation strokes (A) and by analysis of 140 validation strokes in the nonlinear range (B).
Values are means ± SD. Mean differs significantly from
#corresponding conductance group and from $50-stroke
conductance group; variance differs significantly from
*corresponding conductance group and from +50-stroke
conductance group: P < 0.05.
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Hence, in parts A and B both, both second- and third-order polynomials yield volume errors with variability that is either
not different from or is significantly smaller than the volume variability
delivered by the conductance method at 50 strokes.
The variances of the 50-stroke conductance volume errors calculated by using the calibration strokes (Fig.
2) were significantly smaller than the corresponding variances
calculated by using the evaluation strokes
(Fig. 3).
Airway pressures measured in part B ranged between 0 and 10.2 kPa. The differences between first-, second-, and third-order polynomials derived
without and with airway pressure correction, as well as the rise in airway
pressure expressed as a percentage of barometric pressure, are shown in
Fig. 4. Second- and third-order
calibration polynomials derived without airway pressure correction
overestimate flow by approximately the same factor as the airway pressure
increase, whereas the first-order curve overestimates flow by a constant
2.2%.
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Fig. 4. Percent difference between 50-stroke calibration polynomials derived without airway pressure (Pairway) correction and polynomials
derived with pressure correction. The Pairway of the highest flow
calibration stroke, expressed as a percentage of barometric pressure, is
superimposed.
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DISCUSSION
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This study describes a calibration method in which polynomial calibration curves for a screen PT can be derived from as few as 10 different strokes of a
calibration syringe. Our results show that second- and third- order polynomial
calibration curves derived from 10 syringe strokes yield volume errors with
means and variances that are not greater than those obtained from conductance
arrays derived from 50 syringe strokes in a set of independent validation
data. When the flow is extended 60% beyond the linear range of the PT, a
10-stroke polynomial calibration yields volume variances that are smaller than
those obtained by the conductance method with 50 strokes. Our polynomial
technique is substantially more efficient than the conductance method for
flows up to 60% greater than the linear range of the PT.
Yeh et al. (12) recommended 50–100 calibration strokes for the determination of conductance arrays,
which is time consuming and tedious, especially if frequent calibration is
required. Our results (Fig.
3A) suggest that calibration with the use of the
conductance method needs at least 30 syringe strokes to achieve stable
results. If fewer calibration strokes are used to determine conductance
arrays, the accuracy decreases, and individual conductance values vary as
fewer data points are obtained at each differential pressure.
The weighted averaging process of the conductance method can result in adjacent conductance values that differ substantially, leading to
discontinuous conductance curves (Fig.
1), if additional smoothing is not used. In some cases, a large
number of syringe strokes is required to generate an acceptably smooth curve,
particularly at low and high flows, which are difficult to generate with a
syringe by hand. The continuous, smooth nature of polynomials matches the
expected physical characteristics of the PT and facilitates the determination
of smooth calibration curves from a small number of strokes.
Figure 2 shows that falsely accurate results can be obtained if calibration curves are evaluated by using
the same data set from which the curves are derived. The conductance method
appears to be particularly susceptible to this effect. Our study suggests that
calibration curves should be evaluated by using a separate data set.
We found that an assumption of a linear relationship between flow and differential pressure results in volume errors that are significantly higher
than those obtained by using polynomial relationships, even when the maximum
flow is well within the manufacturer-specified linear range. The volume errors
are, however, small physiologically (SD < 0.6%,
Fig. 3A), if the PT
used in this study is used within its linear range. Linear resistance PTs are
available from many manufacturers and cover a wide range of flows. PTs not
identical to the device calibrated in this study might exhibit stronger
nonlinearity inside the manufacturer-specified linear region. Appropriately
selected higher order polynomial calibration methods should allow most
nonlinearity errors at any flow range to be reduced. A limitation of
polynomial calibration curves is that a high-order polynomial might be
required for a PT with a strong nonlinearity, and it is more difficult to
estimate the coefficients of high-order polynomials. In our case, the minor
nonlinearity within the manufacturer-specified linear region linear range can
be corrected (Fig.
3A), and the dynamic range of the PT can be extended well
beyond the specified range, with acceptable accuracy with a second-order
polynomial (Fig.
3B).
The volume errors obtained in this in vitro study are substantially smaller than might be expected from clinical measurements in which temperature,
humidity, gas composition, and airway geometry often vary. The emphasis of
this study, however, is on the comparison of two calibration methods and not
the low absolute errors that were obtained under ideal laboratory
conditions.
Fleisch and screen PTs are volume flow transducers that generate differential pressures proportional to the product of volume flow and gas
viscosity (9). Viscosity is
independent of pressure at pressures encountered in physiological measurements
(7); therefore, at constant
temperature, the differential pressure is proportional to volume flow. It is
standard practice to correct calibration data for pressure and, if necessary,
temperature differences between a volume standard and the PT during
calibration (9). In our study,
the filter and 8-mm endotracheal tube, which were connected downstream of the
PT, caused the pressure in the PT to increase at high flow. At sea level, the
volume of an ideal gas flowing through a PT at increased pressure is reduced
by 1%/kPa. Consequently, we expect calibration curves forced to fit data
that are not corrected for airway pressure to overestimate volume flow by
1% for every kilopascal that airway pressure rises above barometric
pressure. Our results reveal that polynomials derived from uncorrected data
cause systematic overestimation of volume flow by a factor approximately equal
to the increase in airway pressure over barometric pressure
(Fig. 4). In applications in
which airway pressure remains close to atmospheric pressure (e.g.,
spirometry), pressure correction may not be necessary during calibration or
use. When airway pressure varies by >1–2 kPa, for example, during
calibration with high-resistance airway connectors or during mechanical
ventilation of subjects with stiff lungs, pressure compensation is necessary
to avoid systematic errors. In this study, we analyzed only inspiratory flow
in which gas flows from the syringe through the PT, filter, and endotracheal
tube. If a syringe is used to derive calibration curves for expiratory flow in
a similar airway geometry, then the airway pressure will be reduced as the
syringe plunger is withdrawn, and calibration curves derived from uncorrected
data will underestimate volume flow.
Conclusions. This study shows that polynomial calibration curves for PTs can be obtained by using a linear regression technique based on as few
as 10 strokes of an appropriately sized syringe. Our new polynomial
calibration method is more time efficient and produces results equivalent to,
or better than, previously published conductance methods.
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APPENDIX A: MULTIPLE LINEAR REGRESSION
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Equation 1 in the text describes a second-order relationship between the ADC output and the integrated flow in the PT. If N
strokes of a syringe of volume Vs are made and Eq. 1 is generalized to a polynomial of order p, we can write a set of
N simultaneous equations as follows
where 
1...N are the errors in the calculated volumes of each stroke. These equations can be written in matrix form
where Y is a column vector of N syringe volumes; 
is a column of p unknown polynomial coefficients; E is a column
vector of N volume errors; and X is an (N x
p) matrix
A least squares regression estimate B of the polynomial coefficient vector is obtained by evaluating the matrix
(1)
If the data set contains too many similar syringe strokes, then the matrix (X'X) might be ill-conditioned. Syringe strokes should be made
with random variations in peak flow and flow profile. Estimates of the
individual confidence intervals (ci) for the coefficient are given by
where sj is the estimated standard deviation of the jth coefficient given by the jth element of the diagonal of
the matrix (X'X)-1 es,
and es is the standard error of the regression
given by
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APPENDIX B: DETERMINATION OF CONDUCTANCE ARRAY
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The conductance array Cr of a PT (12) is defined by the
equation
where r refers to the possible values that n can take (0–2,047 for a binary 12-bit ADC and bidirectional flow), and
Cr are 2,047 conductance values associated with
the 2,047 possible ADC outputs.
Initial estimates Cq of the conductance values of the PT are obtained from N syringe strokes by using the equation
where i refers to the ADC samples covering each syringe stroke and ki are the pressure correction factors defined in
Eq. 2 in the text.
The initial conductance array is calculated by using the equation
where wr is the number of times the rth ADC value occurs in the qth syringe stroke. Conductances associated
with ADC values that are not covered in a set of syringe strokes are set to
the average of the 11 neighboring conductances.
To improve an existing set of conductance estimates, the following procedure is used.
First, using the current conductance array, the volumes (Vq) of a set of N syringe strokes are calculated
Second, a set of volume correction factors gq
are calculated for each stroke
Third, conductance correction factors er are calculated as weighted averages of the volume correction factors
Fourth, conductance values are corrected
Conductances associated with ADC values that are not covered in a set of syringe strokes are set to the average of the 11 neighboring conductances.
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DISCLOSURES
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The authors acknowledge the financial support of an Australian Research Council Strategic Partnership with Industry Research and Training grant,
Dräger Australia Pty, The Joseph Fellowship, The University of Sydney,
and the National Health and Medical Research Council of Australia.
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ACKNOWLEDGMENTS
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This study has been accepted for presentation in part at the 77th International Anesthesia Research Society Clinical and Scientific Congress,
New Orleans, LA, 2003.
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FOOTNOTES
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Address for reprint requests and other correspondence: M. J. Turner, Dept. of Anaesthesia, Royal Prince Alfred Hospital, Bldg. 89, Level 4, Missenden Rd.,
Camperdown, NSW 2050, Australia (E-mail:
mjturner{at}mail.usyd.edu.au).
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.
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Copyright © 2003 by the American Physiological Society.
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ABSTRACT
MATERIALS AND METHODS
