Abstract
Tidal ventilation gasexchange models in respiratory physiology and medicine not only require solution of mass balance equations breathbybreath but also may require withinbreath measurements, which are instantaneous functions of time. This demands a degree of temporal resolution and fidelity of integration of gas flow and concentration signals that cannot be provided by most clinical gas analyzers because of their slow response times. We have characterized the step responses of the Datex Ultima (Datex Instrumentation, Helsinki, Finland) gas analyzer to oxygen, carbon dioxide, and nitrous oxide in terms of a Gompertz fourparameter sigmoidal function. By inversion of this function, we were able to reduce the rise times for all these gases almost fivefold, and, by its application to real online respiratory gas signals, it is possible to achieve a performance comparable to the fastest mass spectrometers. With the use of this technique, measurements required for nonsteadystate and tidal gasexchange models can be made easily and reliably in the clinical setting.
 breathbybreath analysis
 enhancement of response time
 rise time
traditional continuous ventilation models in respiratory physiology (14,16) have been founded on respiratory gas measurements, which are averaged over several breaths or minutes. Mixed expired gas concentrations are measured from expirate, which is pooled from several breaths over a period in which the subject is assumed to be in the steady state. Tidal ventilation models and nonsteadystate analysis require breathbybreath measurement of airway flow and gas concentration, which, when integrated together, provide the necessary data on gas flux to enable mass balance equations to be solved breathbybreath and in the nonsteady state. To do this, both the airway flow and gas concentration data need to be aligned (i.e., contemporaneous) and both signals need to have sufficiently fast response characteristics. Flow transducers generally have response characteristics, traditionally quoted as the time required for the signal to rise from 10 to 90% of its maximum (t _{0.1–0.9}), on the order of 10 ms (10). Gas concentration signal responses vary widely among instruments. Modern mass spectrometers have quotedt _{0.1–0.9} values for oxygen and carbon dioxide that range from 70 to 200 ms (1, 17), which is usually adequate for breathbybreath measurement. By far the most widely used respiratory gas analyzers in clinical practice are sidestream infrared absorption (carbon dioxide, nitrous oxide, and anesthetic agents) and “fast” paramagnetic (oxygen) instruments. Most sidestream clinical gas analyzers, however, have rise time characteristics that are considerably poorer [t _{0.1–0.9} for oxygen up to 500 ms (2)] than those for mass spectrometers. Therefore, although sidestream clinical analyzers are perfectly adequate for monitoring endtidal gas concentrations at moderate respiratory rates, they are unsuitable for breathbybreath integration with flow signals.
The very nature of tidal breathing exaggerates any errors due to the slow rise time of a gas signal. At the beginning of normal quiet expiration, expiratory flow rises very quickly and may exceed 30 l/min within a few milliseconds, and then it decays in an exponentiallike manner throughout expiration. Unfortunately, this early, rapid increase in expiratory flow coincides almost exactly with a rapid change, or transient, in the gas concentration due to the emergence of alveolar gas after the deadspace gas from the airways has been expired. The error in the integral of concentration and flow, being proportional to the instantaneous product of the concentration error and the prevailing flow, is therefore exaggerated during this phase of expiration. It is for this reason that most clinical gas analyzers give disappointing results when used to provide data for tidal ventilation models of gas exchange.
Under normal conditions, and in the steady state, the inspiratoryexpiratory concentration transient is of the order of about 4 or 5% units (30–40 Torr). However, a number of techniques in respiratory physiology, including washinwashout techniques (15) and the forced inspiratory sinewave technique (19, 20), provide conditions in which the magnitude of the transient may be much greater than this. In this study, we describe how the combination of large concentration transients, high expiratory flow during these transients, and a slow signal response time leads to increased errors in the estimation of gas flux.
Glossary
 b
 Time constant derived empirically from Gompertz sigmoidal function (s)
 c(t)
 Measured concentration of the tracer gas (%) as a function of time (t)
 c′(t)
 Responsetimecorrected concentration (%)
 c_{measured}
 Measured concentration of tracer gas (%)
 c_{step}
 Step change in concentration (%)
 ε_{gas}
 Error in gas volume (ml)
 Fē or Fī
 Mixed expired or inspired gas concentration, respectively (fractional)
 Fi_{O2}
 Inspired oxygen concentration (fractional)
 τ_{1} and τ_{2}
 Rate constants applied to secondorder exponential model (s)
 t_{0.1–0.9}
 10–90% Rise time (s)
 τ
 Time constant of exponential process (s)
 V˙
 = dV˙/dt, airway gas flow (ml/s)
 V_{tracer}
 Volume of tracer gas (ml)
 Vt
 Tidal volume (ml)
BACKGROUND AND METHODS
Tracer Gas Flux Calculations and Errors
Errors due to finite gas concentration measurement response time.
The volume of tracer gas that passes a certain point at which measurements are made over the time period T is given by
Errors due to malalignment of concentration and flow signals.
Flow and concentration signals are separated in time by the transport delay of the aspirated gas sample. The correct temporal alignment of these signals is essential for their correct integration to allow breathbybreath estimation of mixed tracer gas concentration. The delay between the two signals is considered to have two components: one is the (usually constant) time for mean gas wavefront to traverse the sample catheter, and the second is the additional phase delay that occurs as a result of the finite response time of the device measuring the concentration signal. The total delay time therefore depends on what concentration threshold, or rise fraction (i.e., the rise in concentration expressed as a fraction of the maximum rise), is taken as the indication of the beginning of the concentration signal (e.g., any rise >0, 0.2, 0.5, etc.). Underestimation of the transport delay results in overestimation of tracer gas volume and underestimation of deadspace. Therefore, integration errors that result from the finite concentration rise time can to some extent be offset by errors (in the opposite direction) in signal alignment. Indeed, one simple and effective means of correcting for errors in the integration of concentration and flow signals is to select empirically a value for the transport delay that gives the least error in the measured variable of interest, such as volumetric tracer gas estimation, mixed expired concentration, or deadspace. Arieli and Van Liew (3) found that the time interval from the beginning of sample aspiration to the point on the step response corresponding to a rise fraction of 0.2 was optimum.
In this study, our aim was solely to optimize the response time of the instrument; therefore, we have not considered the effects of manipulating the signal alignment, although we appreciate that this is an additional tool. In addition, we sought to avoid reliance on the assumption of a fixed delay time because transport delay not only depends on the composition of the aspirate (5) but also (with the highflow, lowresistance sample catheters used in these instruments) varies cyclically within a breath with positive pressure ventilation (unpublished observations). Our method of alignment of the flow and concentration signals is described in Evaluation of Correction Methods, below, and in the .
Exponential Models
The idea of enhancement of risetime characteristics of measuring instruments is not new. Various means have been used to achieve this, ranging from analog electronic inverse Fourier transformers (11, 12) to digital computer algorithms, in instruments ranging from bloodgas electrodes to mass spectrometers (3, 4, 13). Mitchell (13) used a single exponential model of the step response of a mass spectrometer to correct the output of the instrument, and this produced some improvement. This firstorder model is as follows
Wong et al. (21) recently used this secondorder exponential algorithm to enhance the rise time of the Datex Ultima clinical gas analyzer (Datex Instrumentation, Helsinki, Finland). This instrument has t _{0.1–0.9} values quoted by the manufacturer of 480 ms for oxygen and 360 ms for both carbon dioxide and nitrous oxide. The algorithm of Wong et al. was based on that of Arieli and Van Liew (3), as described above, and was shown to reduce the t _{0.1–0.9} value for oxygen twofold, to 240 ms.
Algorithms such as this are specific to the mathematical model, in that they require knowledge of the instrument's response to a stepchange transient. In the case of Arieli and Van Liew (3), Wong et al. (21), and others, this was assumed to be twin exponential, of the form
The Gompertz Sigmoidal Model
The response of the Datex Ultima to a step change in gas concentration appears sigmoidal. This was characterized more precisely with the use of a nonlinear regression analysis that used a Gompertz fourparameter transform (SigmaPlot v.4; SPSS, Chicago, IL). The reasons behind the selection of this particular transform are, first, because it gave the best fit and, more importantly, because the function is such that it can be characterized by its firstorder derivative. Therefore, this [unlike a Hilltype sigmoid (8)] can be written in a form that is independent of time (Eqs. 7 and 8 ).
The Gompertz function is as follows
To solve Eq. 4
, (and hence determine the value of c_{step}), one needs to know the value of t −t
_{0} (the time after the imposition of the step change), which, although easily determined for a step response, is meaningless in the context of a real, continuous physiological signal. However, the function can be expressed independently of tand in terms of its derivative (as follows). Differentiating with respect to t gives
The same function, however, can also be used to obtain a “risetimecorrected” signal, c′(t), from a real, continuous (i.e., not a step response) signal as follows
Characterization of the Step Response
A stepresponse gas concentration transient (and its derivative, the impulse response) was generated by thrusting a 19gauge needle connected to the standard 3m gas aspirating line of the Datex Ultima (Datex Instrumentation) through a membrane into a cuvette containing a gas with a fixed oxygen concentration of 87%. The stepdown response was created similarly when the needle was quickly pulled out of the membrane into ambient air. Mixing of the aspirate within the water trap (before entering the measuring cells) was reduced by packing the water trap with cotton wool.
Analog data from the Ultima were digitized at 100 Hz (National Instruments DAQ 700 PCMCIA card) and logged with Labview Software v.5 (National Instruments). This was repeated 20 times, and the output was averaged.
Filtering.
Raw concentration data from the gas analyzer were filtered by simple moving boxcar averaging. This was done in a 70ms window around a central point to minimize phase shift. These processes were executed within the Matlab code.
Data Fitting
The Gompertz model.
The data were fitted to a Gompertz fourparameter sigmoid with SigmaPlot v.4 (SPSS), and the value of a constant (b) was derived from a least squares technique. It was found that the values ofb obtained from the averaged response were very close to those obtained from each individual response, as was the correlation coefficient. By inversion of this function, a risetimecorrected stepresponse signal was constructed, as described in Eq. 7 . This process was performed for oxygen, carbon dioxide, and nitrous oxide and executed in Matlab code (The Mathworks). Further details of this recipe are given in the .
The secondorder exponential model.
The stepresponse data were also used to determine the optimum values of τ_{1} and τ_{2} of the secondorder model by the method of Arieli and Van Liew (see ). Values obtained by this means were called the ArieliVan Liew parameters. With the use of the correction described in Eq. 2 , the “risetimeenhanced” step response signal was constructed.
Evaluation of Correction Methods
For brevity, the remainder of this paper will deal only with the correction of the oxygen signal, since this had the slowest response time. Both algorithms were employed to enhance the response times of both real and simulated step responses. Both algorithms were also used to enhance the response time of real, continuous signals as described in Eqs. 2 and 8 .
A simulated measured signal was produced by convolving an artificially created “true” signal (shown in Fig. 3 ) with the impulse response (see Fig. 1 B), using the method of Bates et al. (4) and summarized in the of this paper. The efficiency of the algorithms was checked in a number of ways.
To quantify the accuracy of both correction methods for the real and simulated step responses, the sum of the square of the differences (SSD) was calculated between the true and corrected signals and expressed as a percentage of the SSD for the uncorrected signal.
A more relevant evaluation of the responsetime enhancement is the effect it has on the measurement of mean concentration with respect to volume or on measured airway deadspace. To evaluate the efficacy these algorithms have on real expiratory data, series airway deadspace was measured by Bohr's technique (14) and by the singlebreath technique of Fowler (7), using oxygen as the tracer gas. This was done by filling the cylinder of a sine wave pump (which had a volume of 1 liter) with a mixture of expired gas containing carbon dioxide, oxygen, and nitrogen in physiological proportions. A deadspace comprising a tube of known volume (165 ml by water capacity) and containing room air was attached to the outlet of the cylinder. The deadspace tube was isolated from the gas within the cylinder of the sine pump by means of a tap that was only opened immediately before the pump was switched on. When the pump was switched on, “alveolar” gas was exhaled and room air was inhaled cyclically with a frequency of 15 min^{−1}. Measurements were made for three cycles as carbon dioxide was washed out and O_{2} washed in, but only the first cycle was analyzed. This process was repeated five times. Flow and oxygen concentration were measured by the Datex Ultima at the distal end of this tube. In addition, carbon dioxide was measured at the exact same point with a fast mainstream capnometer (COSMO+, Novametrix), which has a quoted response time of 60 ms and an inpractice delay time that is sufficiently small to be ignored.
The oxygen signal was also “inverted”; that is to say that a mirror image was constructed [c_{mirror} ^{′}(t) = Fi _{O2} − c′(t)] with an amplitude that was normalized to the fast capnogram derived from the Novametrix mainstream capnometer, with which it was then compared. Although, for a breathing human subject, the mirrorimage oxygram is not completely equivalent to a capnogram (due to nonunity of respiratory quotient and withinbreath nonsteadystate conditions), where no gas exchange is taking place, as in this bench model, the mirror image oxygram and the capnogram should be of identical form. Therefore, in this sense, the fast capnogram can serve as a proxy comparator for the enhanced oxygen signal.
In these model breath experiments, risetime enhanced concentration and flow signals were aligned temporally. There are a number of ways to achieve this, but for these experiments we have used the technique of aligning both to the beginning of inspiration (6,9). The flow signal is delayed in time such that the point on the flow signal that marks inspiration is aligned temporally with the point on the concentration signal at which the sudden change from endtidal to inspired concentration begins. Further details of this alignment algorithm are given in the . There was no appreciable inspiratory deadspace. Because no attempt was made to manipulate the transport delay in any other way, errors in derived parameters such as deadspace and mixedexpired concentration would be largely, if not solely, due to the finite response time.
RESULTS
Figure 1 A shows a typical response from the Datex Ultima to an imposed step change in oxygen concentration from air (20.7%) to 86% (balance nitrogen). The data have been fitted to a Gompertz function whose b value was found to be 0.175 (R ^{2} = 0.999976). The plot of this function (Eq. 4 ) and the real data are superimposed and together form the (inseparable) solid curve.
Figure 1 B shows the impulse functions (dc/dt) of the data in Fig. 1 A. The real data and the theoretical Gompertz function are superimposed. Also plotted is a theoretical impulse function derived from the twin exponential (Eq. 3 ) in which the values ofτ _{1} and τ _{2} are the ArieliVan Liew parameters (see ), which were found to be 0.134 and 0.157 respectively. The plots are drawn such that their peaks are aligned. Figure 1 C shows the effect of selection of values of τ_{1} and τ_{2} (0.16 and 0.18, respectively), which correspond to a leastsquares fit of the twin exponential model to the peak of the impulse response. These are what Bates et al. (4) termed the “bestfit parameters.”
Figure 2 shows the results of risetime correction of the actual measured stepresponse for oxygen, using both the secondorder exponential model and the Gompertz model. The real signal and the Gompertz correction are shown (bold and solid lines, respectively). The effect of corrections using the secondorder exponential model is also shown for values of τ_{1} and τ_{2} corresponding to the ArieliVan Liew (dotted line) and bestfit parameters. The bestfit parameters appear to produce a slightly faster response than the ArieliVan Liew parameters but at the expense of greater undershoot and overshoot.
The t _{0.1–0.9} values for the raw and corrected signal are shown in Table 1along with those for carbon dioxide and nitrous oxide.
The SSD values for the true signal vs. the enhanced signals are shown in Table 2.
Analysis of a Simulated Signal
Figure 3 shows an artificially contrived true signal, the front of which is a square wave, with values ranging from 0 to 1 arbitrary units. The falling edge begins as a square wave from 1 to 0.5 and thereafter decays exponentially. Convolution of this signal with the measured impulse response for the Ultima produces the simulated “measured signal” (bold line in Fig.3). The results of enhancing this latter signal using the methods of Arieli and Van Liew and the Gompertz function are also shown. The SSD between the enhanced and true signals are shown in Table 2.
Analysis of Real Respiratory O_{2} Signals
Figure 4 shows a more noisy “mirrorimage” expiratory oxygram measured from a breath generated by a sine pump. The result of the correction using both models is shown. These are compared with the fast capnogram from the same breath. A volumetric capnogram is shown in Fig.5. The series (Fowler) and Bohr deadspace values are shown in Table 3. The actual volume of the artificial airway deadspace was 165 ml. Inspiratory deadspace was calculated from the rapid downstroke of the signal at the beginning of inspiration. The actual volume in this experimental setup was ∼1.5 ml. The measured values are also shown in Table 3.
DISCUSSION
The utility of measuring breathbybreath indexes of gas exchange extends beyond the physiological laboratory. Physicians and anesthesiologists in the critical care and operating theatre environments cannot easily make measurements of oxygen consumption rate, respiratory quotient, or deadspace without recourse to mass spectrometry or collection of mixed expired gas. The stimulus that led to the development of this algorithm was the requirement to develop the sinewave inspiratory forcing technique for the measurement of alveolar volume, pulmonary blood flow, volumeventilation, and ventilationperfusion relationships (18, 19) in the clinical environment, using nonspecialized monitors. This perturbation technique cannot be applied to a tidal ventilation gasexchange model without breathbybreath volumetric tracer gas analysis, and clinical gas analyzers gave disappointing results even after previously described enhancement algorithms were used (3, 21).
Inadequacies in Measurement and Sampling
Our aim was to reduce the response time of the Datex Ultima to 100 ms or less. It is unlikely that we would be able to improve this figure because of the limitations of the instrument itself. The output signals of the Ultima have an update rate of 84 Hz (2). The Novametrix mainstream capnograph has a measuring rate of 80 Hz and analog output update rate (presmoothing) of 50 Hz, so there is little point in digitizing these signals at a rate greater than 100 Hz. Consequently, it is not appropriate to express improvements in signal response to a greater degree of resolution than this allows. Our analysis is therefore in a sense less precise than that of Bates et al. (4). Our purpose was not to chase response time to the nearest millisecond but to be confident in our ability to reduce even the slowest signal (oxygen t _{0.1–0.9} = 480 ms) to less than 100 ms.
Step Responses
Nonetheless, the improvement in signal response for both real and simulated steps (Figs. 2 and 3 and Tables 1 and 2) exceeds that of the twin exponential model for both the ArieliVan Liew parameters and bestfit parameters, the latter of which causes some overshoot. Why should this be the case? The impulse function for the real data matches the Gompertz model much more closely than the twin exponential model (Fig. 1), even after the values of τ_{1} and τ_{2} are optimized in the least squares sense. It should be noted that the model functions shown in Fig. 1 are shifted in time such that they best fit the experimental data. In other words, the values oft _{0} in Eqs. 3 and 4 are not zero but bestfit values. However, this is irrelevant to the process of correction of real signals because Eqs. 3 and 4 can be expressed in terms of their derivatives and independently of t and t _{0} (Eqs.2 and 8 , respectively). Correction can therefore be done without consideration of an appropriate time shift. This is distinct from the “time shift correction” described by Arieli and Van Liew (3) and Bates et al. (4). This latter technique is used to compensate for (as distinct from correcting) a finite response time. In this way, the concentration signal is shifted in time such that it begins to rise before the nominated onset of the step response. In other words, the onset of the transient is defined as a rise fraction of 0.2 or 0.5 or other arbitrary value. Bates et al. (4) showed that the time shift correction alone was a useful tool in compensating for errors in measured gas flux. However, we have not included it in our analysis because we were primarily interested in improving response time. The measurement errors, which we therefore encounter, are likely to be solely due to the suboptimal response time.
Real Expiratory Signals
Our comparison with the fast mainstream capnograph (response time of <60 ms) shows that, despite these limitations, our correction produces a signal that approaches reasonably well that of a fast “gold standard” (albeit proxy) clinical instrument. The acid test, however, is the effect our correction has on the measurement of net gas flux, mixed expired concentration, or deadspace, which amounts to much the same thing. For an expiratory signal generated from the sine pump “exhaling” via a tube of known volume, the improvement in accuracy in estimation of the expiratory deadspace over the twin exponential model is modest (40 ml). This is likely because the actual wavefront of gas (dispersing and mixing in the expiratory deadspace) is far from “square” but blunted and smoother and consequently less demanding on the correction algorithms. In addition, the sinusoidal nature of the expiratory flow is such that the volume of gas passed during the time course of the concentration transient is modest. However, the fall in concentration seen at the end of expiration, when room air is drawn over the sensor with very little inspiratory deadspace, more closely approximates a square wave. This step is tracked faithfully by the mainstream capnograph but less so by the twin exponential correction model. The measured inspiratory deadspace is greater for the twin exponential model than for the Gompertz model, and this effect is exaggerated if the inspiratory flow is set (theoretically) to a constant value of 500 ml/s, as shown in Table 3.
It can be shown that the relationship between the relative (fractional) deadspace volume error (εVd) and the relative (fractional) mixedexpired gas concentration error (εFē) is as follows
This study has introduced a new and efficient means of “tuning” a standard clinical gas analyzer, which uses a simple algorithm executed in a readily available software package, to give a performance similar in speed to that of a mass spectrometer. Consequently, measurement techniques that were hitherto limited to the physiological laboratory or specialized clinical settings can now be undertaken in the clinical environment without the replacement of existing equipment.
Although our technique is based on a specific instrument, namely the Datex Ultima, it is generic and therefore should be applicable not only to other sidestream analyzers but also to other dynamic measuring instruments whose step response can be characterized by a Gompertz function.
Acknowledgments
A. D. Farmery is supported by grants from the Association of Anaesthetists of Great Britain and Ireland, The British Journal of Anaesthesia, and the National Health Service (UK) Research and Development Fund.
Footnotes

Address for reprint requests and other correspondence: A. D. Farmery, Nuffield Dept. of Anaesthetics, Univ. of Oxford, Radcliffe Infirmary, Woodstock Rd., Oxford OX2 6HE, UK (Email:andrew.farmery{at}nda.ox.ac.uk).

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
Alignment of Flow and Concentration Signals
To align the flow and concentration signals, we have adopted the technique of Cochrane et al. (6) and Jacobson and Breen (9). The point on the concentration signal corresponding to the onset of inspiration is identified as the point at which the signal begins to either fall (carbon dioxide) or rise (oxygen). This is done by identifying when the signal derivative becomes negative or positive, respectively, for 20 successive sample points. The first of these points is nominated end expiratory. Likewise, the point on the flow signal corresponding to the onset of inspiration is taken as the first point in a series of 20 successive points in which flow is inspiratory (negative in our system). For nitrous oxide, which is measured by the same cell as that for carbon dioxide, with the same transit time and response time, the endexpiratory point is taken as being contemporaneous with the endexpiratory point for the CO_{2} signal.
Numerical Recipe for ResponseTime Enhancement
Filtering.
Data were sampled at 100 Hz and smoothed with a simple moving averaging technique. The width of the averaging window was 70 ms, and the smoothed point was centered within this window to minimize phase shift.
Differentiation.
After data were smoothed, the signals were differentiated digitally around the central point of five samples. Any smaller tangent than this produced significant noise in our system.
Inverting the Gompertz function.
Equation 8
describes the relationship between the measured and “real” signals in terms of a Gompertz growth function, which is itself a complex exponential function. This can be used to determine the real value of the signal from information derived from the measured signal and its derivative. However, this form of the equation can only be used for signals that are rising because it is derived from a risingstep response. Therefore, for parts of the signal that are rising, the algorithm is applied as is. The value c_{0}, although usually zero for carbon dioxide, is always nonzero for oxygen and is determined from the minimum value of c(t) before the onset of the rise. However, for parts of the signal that are falling (i.e., the derivative is negative), the signal is conditioned to resemble a rising signal. This is achieved by subtracting the signal from its maximum value at the start of the fall (c_{max}), thus producing a signal that rises from 0. Equation 8
then becomes
The algorithm switches between one that uses Eq. 8 and one that uses Eq. EA1 , depending on whether the signal is rising or falling, i.e., depending on the sign of the prevailing derivative. Note that the derivative may oscillate around zero, having positive and negative values, during the alveolar plateau due to noise or cardiogenic oscillation. To prevent the unnecessary “chatter” in the algorithm that causes it to switch between Eqs. 8 and EA1 , it may be necessary to alter the logical operator from “if dc/dt < 0” to “if dc/dt < −0.5.” We have found this latter value to be satisfactory.
Instabilities in the Gompertz Function
Inspection of Eq. 8
reveals that there is potential instability in the exponential term. This depends on the ratio of the values of the signal derivative and the signal itself. At the start of a rise (or fall), it is possible under circumstances in which the rise is very rapid that the value of c(t) − c_{0}may be small and near zero, whereas the value of dc/dt is small but finite. This will cause the function to be unstable for a few samples until the value of c(t) − c_{0} has risen. This problem can be overcome by adding a small offset (c_{off}) to the value c(t) − c_{0}. The exact value needs to be determined for any particular system, but we have found that a value of 0.1 (units are percent gas concentration) is satisfactory. Equation 8 then becomes
Determination of ArieliVan Liew Parameters
A straight line is fitted to ln[1 − c(t)] for the 50–90% portion, where c(t) is the measured timevarying concentration value after a step response. Twothirds of the positive reciprocal of the slope of this line gives τ_{1}. The step response is corrected according to Eq.1 . Another straight line is fitted to ln[1 − c′(t)] for the 50–90% portion, where c′(t) is the onestagecorrected signal from the previous maneuver. The positive reciprocal of the slope of this line gives τ_{2}.
Convolution of a True Step Response With the Measured Impulse Response to Produce a Simulated Measured Step Response
This is fully described in the study by Bates et al. (4) and is summarized here. A true signal (Fig. 3), g(u), was contrived to resemble that which might occur in practice. It has a step front end, and, after 1.5 s, it initially falls in a stepwise manner to onehalf its value. Thereafter, it decays exponentially [g(u) = 0.5e
^{−5u}]. The measured signal [c(t)] is related to the true signal by the convolution integral, thus
With the use of Matlab's builtin convolution tool, the true signal was convolved with the impulse function to produce a simulated measured signal (bold line in Fig. 3), which could then be subjected to the various correction algorithms.