INTRODUCTION

TOP ABSTRACT INTRODUCTION BACKGROUND AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

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 non-steady-state analysis require breath-by-breath 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 breath-by-breath and in the non-steady 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 quoted t_0.1-0.9 values for oxygen and carbon dioxide that range from 70 to 200 ms (1, 17), which is usually adequate for breath-by-breath 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 end-tidal gas concentrations at moderate respiratory rates, they are unsuitable for breath-by-breath 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 exponential-like 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 inspiratory-expiratory concentration transient is of the order of about 4 or 5% units (30-40 Torr). However, a number of techniques in respiratory physiology, including washin-washout techniques (15) and the forced inspiratory sine-wave 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)	Response-time-corrected concentration (%)
cmeasured	Measured concentration of tracer gas (%)
cstep	Step change in concentration (%)
gas	Error in gas volume (ml)
F or F	Mixed expired or inspired gas concentration, respectively (fractional)
FIO2	Inspired oxygen concentration (fractional)
1 and 2	Rate constants applied to second-order exponential model (s)
t0.1-0.9	10-90% Rise time (s)
	Time constant of exponential process (s)
	= d/dt, airway gas flow (ml/s)
Vtracer	Volume of tracer gas (ml)
VT	Tidal volume (ml)

	BACKGROUND AND METHODS

TOP ABSTRACT INTRODUCTION BACKGROUND AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

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
Given that there is a time-dependent discrepancy between the actual and measured concentrations, the error in the measurement in the volume of tracer gas is given by
However, after a step change in concentration, c(t) becomes a constant (say, equal to c_step) and c_measured(t) remains a function of time. Most of the error accrues during the t_0.1-0.9 time period. Therefore, the following approximation can be made
From here it can be seen that the error in tracer gas volume estimation is increased as c_step, , and t_0.1-0.9 (which defines the limits of the integration) increase. A slow response time results in the underestimation of tracer gas volume and overestimation of deadspace volume.

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 breath-by-breath 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.

Exponential Models

(1)

(2)

Wong et al. (21) recently used this second-order 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 step-change 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

(3)

This function, unlike the single exponential function, is sigmoidlike and more closely resembles the sigmoidal output of a gas analyzer in response to a step change in gas concentration. However, it is also likely that, in addition to the contribution of the twin exponential processes, at least some part of the sigmoidal response results from Taylorian dispersion of tracer gas within the sampling catheter (18); therefore, a second-order correction alone may be insufficient to correct the instrument's output. We have examined the step-response characteristics of the Datex Ultima to establish how well this can be represented by a twin exponential model. We have also characterized this response empirically in terms of a Gompertz growth function, which forms the basis of our rise-time enhancement algorithm.

The Gompertz Sigmoidal Model

The Gompertz function is as follows

(4)

where c₀ is the initial concentration value, c_step is the steady-state step value, and b is a constant determined empirically from "least-squares" curve fitting.

To solve Eq. 4, (and hence determine the value of c_step), one needs to know the value of t  t₀ (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 t and in terms of its derivative (as follows). Differentiating with respect to t gives

(5)

Combining Eqs. 4 and 5 gives

(6)

Let
therefore Eq. 4 becomes
Hence Eq. 6 becomes
and hence
which simplifies to

(7)

Equation 7 enables c_step to be determined from measurements of c(t) and its derivative at any point in time after the imposition of a step-concentration transient.

The same function, however, can also be used to obtain a "rise-time-corrected" signal, c'(t), from a real, continuous (i.e., not a step response) signal as follows

(8)

Note how this algorithm is a first-order differential equation; because the data only require differentiation once, the system is potentially less sensitive to signal noise than the second-order model of Arieli and Van Liew (3) and Wong et al. (21).

Characterization of the Step Response

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 70-ms 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 four-parameter 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 of b 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 rise-time-corrected step-response 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 APPENDIX.

The second-order exponential model. The step-response data were also used to determine the optimum values of ₁ and ₂ of the second-order model by the method of Arieli and Van Liew (see APPENDIX). Values obtained by this means were called the Arieli-Van Liew parameters. With the use of the correction described in Eq. 2, the "rise-time-enhanced" step response signal was constructed.

Evaluation of Correction Methods

A simulated measured signal was produced by convolving an artificially created "true" signal (shown in Fig. 3 ) with the impulse response (see Fig. 1B), using the method of Bates et al. (4) and summarized in the APPENDIX of this paper. The efficiency of the algorithms was checked in a number of ways.

View larger version (16K): [in this window] [in a new window]   Fig. 1.   A: step response for oxygen as measured by the Datex Ultima. Plot shows real experimental data (dotted line) and the theoretical Gompertz function (solid line), where b = 0.175. Lines are superimposed and appear as one. See text for details. B: impulse response for oxygen as measured by the Datex Ultima (derivative of data displayed in A). Plot shows real data (dotted line), theoretical Gompertz impulse for b = 0.175 (solid line, smaller peak), and theoretical twin exponential impulse with Arieli-Van Liew parameters, 1 = 0.134 and 2 = 0.157 (solid line labeled E). Curves are aligned such that their peaks are coincident. C: impulse response for oxygen as measured by the Datex Ultima. Lines are same as in B. Theoretical twin exponential impulse (E) is shown with "optimum" parameters, 1 = 0.16 and 2 = 0.185.

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 response-time 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 single-breath 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 min¹. Measurements were made for three cycles as carbon dioxide was washed out and O₂ 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 in-practice 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 mirror-image oxygram is not completely equivalent to a capnogram (due to nonunity of respiratory quotient and within-breath non-steady-state 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, rise-time 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 end-tidal to inspired concentration begins. Further details of this alignment algorithm are given in the APPENDIX. 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 mixed-expired concentration would be largely, if not solely, due to the finite response time.

	RESULTS

TOP ABSTRACT INTRODUCTION BACKGROUND AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Figure 1A 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 (R² = 0.999976). The plot of this function (Eq. 4) and the real data are superimposed and together form the (inseparable) solid curve.

Figure 1B shows the impulse functions (dc/dt) of the data in Fig. 1A. 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 of ₁ and ₂ are the Arieli-Van Liew parameters (see APPENDIX), which were found to be 0.134 and 0.157 respectively. The plots are drawn such that their peaks are aligned. Figure 1C shows the effect of selection of values of ₁ and ₂ (0.16 and 0.18, respectively), which correspond to a least-squares fit of the twin exponential model to the peak of the impulse response. These are what Bates et al. (4) termed the "best-fit parameters."

Figure 2 shows the results of rise-time correction of the actual measured step-response for oxygen, using both the second-order 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 second-order exponential model is also shown for values of ₁ and ₂ corresponding to the Arieli-Van Liew (dotted line) and best-fit parameters. The best-fit parameters appear to produce a slightly faster response than the Arieli-Van Liew parameters but at the expense of greater undershoot and overshoot.

View larger version (14K): [in this window] [in a new window]   Fig. 2.   Effect of correction of real measured oxygen step-response using Gompertz and twin exponential models. Step response was created by thrusting a needle attached to the sample catheter, through a membrane, from room air into a mixture containing 86.5% O2 in N2. T, assumed true "step" signal; R, raw data; E, corrections based on twin exponential model. Arieli-Van Liew parameters (1 = 0.134 and 2 = 0.157) are shown as solid line. Optimum parameters (1 = 0.16, 2 = 0.185) shown as a dotted line. G, correction based on Gompertz model (b = 0.175), shown as bold line.

The t_0.1-0.9 values for the raw and corrected signal are shown in Table 1 along 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

View larger version (17K): [in this window] [in a new window]   Fig. 3.   Signal simulation. A simulated true signal (a) is contrived to have a front edge that is a step response, from 0 to 1 arbitrary units. The rear edge begins as a step response from 1 to 0.5 units and thereafter decays exponentially. This protocol is similar to that designed by Bates et al. (4). The simulated measured response (d) is created by convolving the simulated true response with the measured impulse response of the Datex Ultima (see Fig. 1B). The simulated measured response is then subjected to enhancement according to the Gompertz model (b) and twin exponential model (Arieli-Van Liew parameters; c).

Analysis of Real Respiratory O₂ Signals

View larger version (15K): [in this window] [in a new window]   Fig. 4.   Comparison of corrected mirror image oxygram with fast mainstream capnogram from a breath generated by sine pump and fixed volume deadspace. Raw O2 mirror signal (dotted lines) created by subtraction of raw O2 signal from 20.7%. Line with ×, correction based on twin exponential model with Arieli-Van Liew parameters. Line with , correction based on Gompertz model. CO2 concentration (solid line), measured with fast mainstream (Novametrix) capnometer, is shown as a reference.

View larger version (15K): [in this window] [in a new window]   Fig. 5.   Comparison of volumetric capnograms (Fowler plots) using data from Fig. 4. Solid line, fast mainstream Novametrix CO2; line with ×, correction based on twin exponential model; line with , correction based on Gompertz model; dotted line, uncorrected "raw" oxygen signal in mirror image.

	DISCUSSION

TOP ABSTRACT INTRODUCTION BACKGROUND AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

The utility of measuring breath-by-breath 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, volume-ventilation, and ventilation-perfusion relationships (18, 19) in the clinical environment, using nonspecialized monitors. This perturbation technique cannot be applied to a tidal ventilation gas-exchange model without breath-by-breath 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

Step Responses

Real Expiratory Signals

It can be shown that the relationship between the relative (fractional) deadspace volume error (VD) and the relative (fractional) mixed-expired gas concentration error (F) is as follows

(9)

Therefore, the error in VD as a result of an error in F is bounded. Rearrangement of Eq. 9 reveals that the relative error in mixed expired concentration in our bench study is about one-fifth of the relative error for deadspace. For the Gompertz correction, in which the absolute deadspace error was 22 ml, this represents a relative error of 22/165 = 0.15. The relative error in mixed expired concentration is therefore 0.03.

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.