Model-based administration of inhalation anaesthesia. 3. Validating the system model

J. G. C. Lerou* and L. H. D. J. Booij

Institute for Anaesthesiology, University of Nijmegen, Geert Grooteplein 10, NL-6500 HB Nijmegen, The Netherlands*Corresponding author

Accepted for publication: August 31, 2001


    Abstract
 Top
 Abstract
 Introduction
 Patients and methods
 Results
 Discussion
 References
 
Background. We quantified the predictive performance of our computer model of the administration of inhalation anaesthesia from a Datex-Ohmeda Modulus CD circle-absorber system.

Methods. In 50 patients, desflurane anaesthesia was maintained with a fresh-gas flow (FGF) of 0.5 litres min–1 of both nitrous oxide and oxygen, preceded by fast (n=14) or slow (n=36) induction: FGF greater than total ventilation, Group F; FGF equal to 1.0 litres min–1, Group S. The two versions of the model studied differed in the size of their inter-tissue diffusion, as 0 (version 1) and 3% (version 2) of the cardiac output was shifted from the viscera to adipose tissue. Model performance was judged by comparing measured and predicted gas concentrations in terms of three variables for each gas concentration in each patient: root mean squared error (rmse=total error), bias (mean predicted – measured) (systematic error), and scatter (error around bias). These variables were then averaged over all patients. These measures were calculated overall, and separately for each group and each stage (1=induction or 2=maintenance).

Results. Model predictions were in reasonable to very good agreement with clinically obtained data. Version 2 performed better than version 1. Differences between groups were not demonstrated. The model performed better for stage 2, but only for desflurane. In group S, results (mean (SD); as percentages of the measured values for nitrous oxide, oxygen and desflurane) in the order rmse, bias, and scatter were for end-tidal concentrations of nitrous oxide: 8 (4), 8 (5), 2 (1)%; oxygen: 11 (4), –10 (6), 2 (1.1)%; nitrogen: 0.9 (0.6), –0.8 (0.6), 0.2 (0.1) vol%; carbon dioxide: 1.8 (0.6), 1.8 (0.6), 0.2 (0.1) vol%; desflurane, stage 2: 8 (4), 4 (7), 4 (2)%, vs 15 (6), –10 (8), 9 (4)% for stage 1.

Conclusion. Administration of inhalation anaesthesia can be based on version 2 of this model, but must be guided by active monitoring.

Br J Anaesth 2002; 88: 24–37

Keywords: anaesthetics volatile, desflurane; pharmacokinetics, models; equipment, breathing systems; anaesthetic techniques, inhalation


    Introduction
 Top
 Abstract
 Introduction
 Patients and methods
 Results
 Discussion
 References
 
We have developed a system model for the administration of inhalation anaesthesia from a particular circle-absorber breathing system with a standing bellows.1 The model includes almost all gases present in the system, and will handle a wide range of fresh gas flows (FGF). We have shown1 that its predictions agree reasonably well with some published clinical data and that it can reproduce well-known phenomena in the administration of inhalation anaesthesia.2

This paper presents a prospective comparison with clinical data and aims to quantify the predictive performance of the model. General anaesthesia with desflurane and nitrous oxide in oxygen was administered to 50 patients under low-flow conditions. We retrospectively compared the predicted inspired and end-tidal concentrations of desflurane, nitrous oxide, oxygen, carbon dioxide, and nitrogen with those measured in each patient. Two versions of the model were formulated. One version (1) is the basic model;1 another (2) accounts for large-scale diffusion between body compartments.3


    Patients and methods
 Top
 Abstract
 Introduction
 Patients and methods
 Results
 Discussion
 References
 
Anaesthetic management
Fifty consenting patients (ASA physical status I and II) were studied after approval of the Institutional Ethics and Research Committee. They were to undergo elective eye surgical procedures expected to last at least 30 min. Diazepam 10 mg and droperidol 2.5 mg were given orally 1 h before surgery. An i.v. catheter was inserted and basic monitoring (ECG, automatic non-invasive arterial pressure (NIBP), pulse oximetry) established. NIBP was measured with 1-min intervals for 20–30 min and a 3–5-min interval thereafter.

The anaesthetic breathing system (Fig. 1 in reference 1) was filled with air by flushing it with 9 litres min–1 medical air for 3 min. When the mask was applied to the face of the patient, fresh gas was switched from air to oxygen to perform denitrogenation and pre-oxygenation. Fentanyl 3 µg kg–1 i.v. was given. Two minutes later, anaesthesia was induced with an i.v. dose of propofol sufficient to obtund the eyelash reflex and rocuronium 0.5 mg kg–1 i.v.. The patient’s lungs were ventilated manually by mask for 2 min, using a FGF of 12 litres min–1 (nitrous oxide:oxygen, 1:1); an increment of propofol of 20–40 mg i.v. was then given, according to body mass, and changes in the NIBP as a result of inducing anaesthesia. After a further 30 s, topical anaesthesia of the larynx with 2–4 ml of lidocaine 4.3% spray was performed, the patient’s trachea was intubated with a cuffed tube, and the cuff was inflated.

The patient was reconnected to the breathing system and the ventilator, and the lungs were artificially ventilated with a FGF of 9 litres min–1 (nitrous oxide:oxygen, 2:1) until the end-expiratory nitrogen concentration was less than 2 vol% (vol% is used for concentrations to avoid confusion with the per cent unit for the performance measures), or for a maximum of 10 min. Subsequent treatment depended on allocation to group S (‘slow scheme’) or group F (‘fast scheme’). Patients were not deliberately placed in one or the other group, but an increased proportion of patients (5:2, eventually) were assigned to group S (to achieve greater clinical experience with low-flow conditions).

In group S (n=36), the initial total FGF was set to 1 litre min–1 (i.e. 0.5 litres min–1 each of nitrous oxide and oxygen). Reducing the FGF from 9 to 1 litre min–1 necessitated adjustment of tidal volume because delivered tidal volume depended on the rate of FGF into the standing bellows ventilator.4 5 Once the total ventilation was stable, the vaporizer was set to deliver 12 vol% desflurane until its end-expired concentration was appropriate as judged by the attending anaesthetist (J.L.). Then, for the second stage, the vaporizer setting was reduced to 1 vol% above the target end-expired concentration, while the FGF was kept unchanged.

In group F (n=14), the initial total FGF exceeded at least the assumed alveolar ventilation, or was even greater than total ventilation, and the vaporizer was initially set to deliver 6 vol% desflurane until its end-expired concentration was appropriate as judged by the attending anaesthetist (J.L.). The second stage was the same as in group S. Each of the schemes reflected one out of two dosing strategies commonly used in the clinical environment.

During maintenance, we modified desflurane administration to provide adequate anaesthesia. We therefore carefully monitored the patient’s response to surgery by assessment of NIBP, heart rate, and heart rate variability judged by ear with the aid of pulse oximetry, and also end-tidal desflurane concentration. Extra fentanyl (0.05–0.1 mg) was given according to clinical needs. The end-tidal carbon dioxide concentration was maintained at 3.6–4.6 vol%. Desflurane was purchased from Pharmacia Nederland B.V. (Woerden).

Instrumentation
The anaesthetic equipment consisted of a Modulus CD anaesthesia system (Ohmeda, Madison, USA), which is routinely used in the operating theatres for ophthalmic surgery. The anaesthetic breathing system comprised a soda-lime canister (part of the Ohmeda GMS (Gas Management System) Absorber), two 1-m corrugated tubes in each limb, a water trap in each limb, and a Y-piece. The switch in the GMS allowed swift alternation between reservoir bag and ventilator: (i) a 2-litre bag at the end of 1-m length of corrugated tubing was used for spontaneous breathing and manual ventilation by mask; and (ii) a standing bellows ventilator (Ohmeda 7850) supported artificial ventilation of the lungs. A scheme of the breathing system with an internal volume of 6.6 litres, as used during artificial ventilation, was given in Figure 1 of reference 1. Leaks in the circuit were detected by plugging the Y-piece, pressurizing the breathing system to 4 kPa, and observing the volume and pressure gauge; a gas leak up to 60 ml min–1 was accepted. Desflurane was delivered in all patients by the same Ohmeda Tec 6 vaporizer. The accuracy of the Rotameters (oxygen, nitrous oxide, air) was checked against a bubble flow meter. The expiratory volumes were measured with an Ohmeda volume monitor; the turbine vane transducer sensor has an accuracy of ±5%.6

A respiratory mass spectrometer (QP 9000) was located in a nearby room. One of its two inlet ports was connected to a 2-m inlet probe, the other to a valve box which sampled via 30-m nylon probes,7 8 at 60 ml min–1, either at the Y-piece or from the FGF. Switching between the two inlet ports was done by pinch valves incorporated in the mass spectrometer, and between the long probes by a pair of electromagnetic valves. The hospital vacuum was used to draw continuously a matching 60 ml min–1 from whichever 30-m probe was not connected to the mass spectrometer. Normally the mass spectrometer continuously sampled gas from the Y-piece but, each time the setting of the vaporizer was changed, it was switched to sample fresh gas for 40 s.

The mass spectrometer was calibrated once or twice per day (morning and afternoon list) with just the 2-m inlet probe according to a procedure recommended by the manufacturer (CaSE, Gillingham, UK) using two gas mixtures of known composition. In addition, before starting each list we verified the mass spectrometer’s calibration with one of the long probes (both had same geometry) using room air and the same two calibration gas mixtures: (i) 1 vol% argon, 5 vol% carbon dioxide, 44 vol% oxygen in nitrous oxide; and (ii) 4 vol% desflurane, 44 vol% oxygen in nitrogen (AGA Gas, Amsterdam, The Netherlands). Six channels were tuned to the mass-charge ratios required to measure nitrogen (28), oxygen (32), carbon dioxide (12), nitrous oxide (30), argon (40), and desflurane (51). An eight-channel thermal array recorder (Nihon Kohden RTA 1300) running at 10 mm min–1 recorded the mass spectrometer signals.

A personal computer system located in the operating theatre processed the signals from the mass spectrometer (12-bit analogue-to-digital board (Keithley Metrabyte, USA)), was allowed to operate the switching mechanism for the valves, and recorded all variables acquired by the Modulus CD anaesthesia system (e.g. expiratory ventilation) from its RS232 output port. The data acquisition software was developed with the aid of ASYSTTM (Keithley Metrabyte). On-line analysis of the respiratory waveforms allowed continuous monitoring of the actual inspiratory and end-expired concentrations of nitrogen, oxygen, carbon dioxide, nitrous oxide, argon, and desflurane. Every 10-s the last inspiratory and end-expired concentrations of these gases were saved on hard disk for further data processing.

The model
Versions
Version 1 is the basic model that was quantified earlier.1 For desflurane it uses the tissue/gas partition coefficients reported by Yasuda and colleagues,9 and a blood–gas partition coefficient of 0.52 as reported by Lockwood and co-workers.10 Version 2 uses one of the amendments that Allott and colleagues11 made to their basic model as a simple means of mimicking inter-tissue diffusion. Part of the cardiac output was, therefore, redirected from the viscera, i.e. kidneys, heart, brain, and liver, to the adipose tissue. In successive simulation runs, the total fraction redirected cardiac output (fR) was increased, with steps of 0.01, until the cumulative uptakes of desflurane and isoflurane predicted by our model agreed closely with those reported by Hendrickx and co-workers.12 Their experimental conditions were mimicked in the way already described.1 As the sum of the fractions of cardiac output to the viscera in version 1 is 0.76, the fractions for version 2 were obtained by multiplying those in version 1 by (0.76–fR)/0.76.

Adapting the model to experimental conditions
Clinically important details were incorporated into the original model with the aim of performing this validation study. During the process of intubation there is a period of apnoea that may be short but also rather lengthy in case of a more difficult intubation. It is virtually impossible to fully describe the impact of removing the facemask and intubating the trachea on the composition of the gas mixture in the lungs and the anaesthetic breathing system. The model therefore assumed that, during apnoea, alveolar ventilation was zero and gas exchange continued across the alveolo-capillary membrane.

With a leak measured to be <=60 ml min–1 at a continuous breathing-system pressure of 4 kPa, plus a sample flow of 60 ml min–1, careful consideration of detail led us to conclude that, in use, a net loss as 75 ml min–1 of inspired mixture would be a good estimate of the effective loss. (Accuracy is minimally influenced by this legitimate assumption.)

The model also accounted for the sample flow drawn off continuously by the hospital vacuum at the common gas outlet (V·sf). If V·Fi was the FGF for each component i (nitrogen, oxygen, nitrous oxide, or desflurane), then its effective FGF entering the circle system was given as:


The ventilation in the model was matched to that measured for each patient as follows. A target value for the ‘ideal’ alveolar carbon dioxide tension (PAco2, assumed equal to PaCO2) was obtained from: (i) the ventilation measured (averaged over the period of administration of desflurane); (ii) a chosen value of 50% for wasted ventilation;1 and (iii) a specific, rearranged form of equation (24) in Appendix 2 of reference 1. The value obtained was used by the model to calculate alveolar ventilation (see below).

Simulated end-expired tensions were calculated according to Landon and co-workers13 from the ‘ideal’ alveolar (PA) and dead space (=inspired; PI) tensions predicted by our model: PE'=(1–d)PA+dPI. For each patient, the ‘dilution factor’ d (the fraction of dead space gas in end-expiratory gas)13 was deduced from the arterial and end-expired carbon dioxide tensions: d=(PaCO2PE'CO2)/PaCO2. Assuming that PaCO2PE'CO2=0.533 kPa (see discussion) and as PaCO2=PAco2, it follows that d=0.533/PAco2.

Comparing measurements with predicted values
This was a three-step process. During step one, the model input was generated. The model input consisted of the anthropometric data (age, sex, body weight, and height), the FGF rates (air, oxygen, and nitrous oxide), the desflurane concentration in the FGF (the measured signal averaged over 5 s), the target value for the ideal alveolar carbon dioxide tension, and the period of apnoea during intubation. Throughout step two, our model generated the predicted time courses of the inspired and end-expired concentrations of desflurane, nitrous oxide, oxygen, carbon dioxide, and nitrogen by running the model in a TutsimTM simulation program (Meerman Automation, Neede, The Netherlands). In the final step three, the differences between predicted and measured concentrations (the ‘predictive performance measures’, vide infra) were analysed in a Mathcad (version 7) worksheet (Mathsoft, Bagshot, Surrey, UK).

Predictive performance measures
Definitions
The summary measures that served to determine the predictive performance of the model are defined in Table 1. The prediction error (pe) is the difference between a predicted and a measured concentration; pe can be expressed as a vol% difference (nitrogen and carbon dioxide) or as a percentage of the measured value (N2O, O2, desflurane). Prediction error and squared prediction error (pe2) are calculated for each time period of 10 s. These two quantities are used to provide the following three predictive performance measures.


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Table 1 Definitions of predictive performance measures. Cp and Cm = pair of predicted and measured concentrations (one pair for each time period of 10 s). n = number of measurements per gas per patient in a defined period of time, pei is the i-th prediction error, and me = mean prediction error

 
1. Root mean squared error (rmse). The mean squared error (mse) is the average of the squared prediction errors. Rmse is defined as {surd}mse and is a measure of the total error budget for an individual patient in a defined period of time. Rmse is not influenced by the signs of the prediction errors. Rmse can be decomposed into two terms: rmse={surd}bias2 + scatter2 (Table 1).

2. Bias (me), that is the average of the prediction errors for an individual patient, is a measure of the systematic component of error. It can be either positive or negative, thus indicating over- or under-prediction, respectively. It does not provide information about the typical size of the prediction error if there are both under- and over-predictions in an individual patient.

3. Scatter is a measure of the variation of the prediction errors for an individual patient around the mean (or bias) for that patient.

These three measures were calculated for the inspired and end-tidal concentrations of desflurane, nitrous oxide, oxygen, and nitrogen, and for the end-tidal concentration of carbon dioxide—a total of 3x9=27 performance measures for each patient, or 54 when calculated separately for stages 1 and 2.

Overall performance measures
‘Overall’ measures evaluate the whole period including stages 1 and 2, which ends at the moment when artificial ventilation was stopped. Each of 27 performance measures per patient were averaged over all patients to yield the three group overall performance measures for each of the nine gas concentrations.

Performance measures per stage and group
Each of the performance measures was also assessed separately for each stage (1 or 2) and group (S or F). Calculating the numerical averages of the 36 (group S) or 14 (group F) performance measures—one per patient and per gas concentration—yielded the group performance measures for a gas concentration for each stage.

Performance measures per version
Performance measures were calculated once for version 1 and once for version 2 of the model. Figures 013F1 and 013F2 serve to illustrate the predictive performance measures. Figure 013F1 shows a selection of measured and predicted concentration– time profiles in a patient who was considered representative because she belonged to the largest group S, had average physical characteristics, and showed individual results in agreement with group results. Figure 013F2 illustrates the biases and scatters of end-tidal desflurane in the same patient.



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Fig 1 Measured and predicted (version 2) concentrations from a 44-yr-old female patient (weight 65 kg, height 1.58 m) for inspired oxygen, end-tidal nitrous oxide and nitrogen, and inspired and end-tidal desflurane. The following periods or events can be recognized: pre-oxygenation and denitrogenation; ventilation by face mask; intubation; period of 9 litres min–1 FGF, that is the period with 33% inspired oxygen. The ‘deviation spikes’ on the measured concentrations result from the three 40-s periods (encircled numbers 1, 2, and 3) during which fresh-gas is sampled by the respiratory mass spectrometer to measure the delivered desflurane concentrations. Note that the delivered desflurane concentration (11.81 vol%) goes off scale at the first sampling period. Stage 1 is the period of induction (vaporizer set to deliver 12 vol% desflurane in 0.5 litres min–1 of each of nitrous oxide and oxygen), and stage 2 is the maintenance period (vaporizer set to deliver 6 vol% desflurane in 0.5 litres min–1 of each of nitrous oxide and oxygen).

 


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Fig 2 (A) Measured and predicted end-tidal desflurane concentrations obtained with version 2 in the same patient as in Figure 1. (B) Time course of the prediction errors: the y-axis extends from –25 to +25% with zero in the middle. The overall rmse, bias, and scatter for end-tidal desflurane in this patient were 6.9, 3.1, and 6.1%, respectively. For stage 1, rmse, bias, and scatter were 15.2, –12.4, and 8.7%, respectively; for stage 2, they were 5.1, 4.8, and 1.8%, respectively.

 
Statistical analysis
Student’s t-test for unpaired data was used to test the difference between the means of the duration of stage 1 in group S vs F; P<0.05 was considered statistically significant.

Results for each of the 27 (or 54) performance indicators, per version, were expressed as mean of all patients and SD between patients, except where otherwise stated.

The sign test was used to compare the predictive performance of version 1 with that of version 2. We therefore tested the hypothesis of whether the rmse for version 2 was closer to zero. The same hypothesis was tested for bias and scatter.

Performance indicators were tested for differences between the means of stage 1 vs 2 (for each group) with a t-test for paired data, and group S vs F (for each stage) with a t-test for unpaired data. A P<0.001 (correction for multiple comparisons) was considered statistically significant; n denotes number of patients.


    Results
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 Abstract
 Introduction
 Patients and methods
 Results
 Discussion
 References
 
Patients in group S and F had similar characteristics (Table 2). Figure 013F1 provides definitions for the stages and a visual impression of the quality of the predictions. Table 3 lists clinical details on settings of the FGF and vaporizer, as well as desflurane concentrations measured in the fresh gas. Slight variants of the schemes were applied in two patients of group S (in one patient 0.5 litres min–1 was used, and the vaporizer was set to 8 vol% in another) and in six patients of group F (4 vol% during stage 1 and 0.5 litres min–1 during stage 2; Table 3). Recovery from the anaesthetic was uneventful in all patients.


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Table 2 Patient characteristics in the total population and in the two groups. Values are mean (SD) [range] or number. Patients with a body mass index (BMI) less than 20 kg m–2 are considered slender; patients with a BMI greater than 25 kg m–2 are designated obese. Because of a BMI exceeding 34 kg m–2, which is greater than mean BMI plus twice SD, two males (36.8 and 37.1 kg m–2) and one female (34.6 kg m–2) may be designated as outliers
 

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Table 3 Clinical details on settings of FGF and vaporizer for the 50 patients, as well as desflurane concentrations (mean (SD)) measured in the FGF. In this table, patients are first categorized into the two groups (Slow and Fast; details see text), then in accordance with the settings of FGF and vaporizer during induction, and finally according to FGF settings and the first vaporizer setting during maintenance (in the order of increasing total FGF and vaporizer setting). n=number of patients. During maintenance, a second and third vaporizer setting was used in one*, two{dagger}, or three{ddagger} patients
 
The intraoperative samples used for data analysis were from a total of more than 3000 min of anaesthesia. Table 4 lists the duration of the various measuring periods (no account was taken of the 40-s sampling periods needed to analyse the delivered desflurane concentration). Stage 1 was approximately 4 min shorter in group F vs S. The target concentration for desflurane was attained twice as fast in group F: 3.5 vs 7.1 min on average. The duration of stage 2 was equal for both groups (study design).


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Table 4 Duration of measuring periods (min) (mean (SD) [range]). *P<0.001 group S vs group F
 
Model versions
Cumulative uptakes were computed for desflurane and isoflurane for the conditions used in the study by Hendrickx and co-workers;12 these computed uptakes agreed closely with the measured values if fR=0.03 (Fig. 013F3). Accordingly, version 2 uses the cardiac output partitioning listed in Table 5.



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Fig 3 Cumulative uptakes of desflurane or isoflurane computed with three versions of the system model using different fractions of cardiac output redirected from the viscera to the adipose tissue: fR=0.0 (version 1), 0.03 (version 2) and 0.06. The calculated uptakes are compared with those reported (a square with bars represents mean with 95% CI) by Hendrickx and co-workers.12 They measured uptake of each anaesthetic, expressed in millilitres liquid, in 10 patients who received closed-circuit anaesthesia at 1.3 MAC with mechanical ventilation of the lungs. The simulation is for their average 42-yr-old female patient of 1.65 m height, and 74 (desflurane) or 66 kg (isoflurane) body weight.

 

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Table 5 Cardiac output partitioning in the two versions of the system model. The blood flows through the compartments are expressed as fractions of cardiac output. Values for version 1 are data given by Lowe and Ernst.30 In version 2, a fraction fR of the cardiac output is redirected from the viscera to the adipose tissue (fR=0.03; see text for calculation of the fractions for the individual tissue compartments). The liver compartment includes the liver itself and the whole portal system
 
Overall predictive performance
Performance measures for desflurane, nitrous oxide, and oxygen have been expressed as percentages of the measured values, but for nitrogen and carbon dioxide vol% has been used for the following reasons. Calculating the scaled differences for nitrogen would have led to dividing by zero. For carbon dioxide the differences found between calculated and measured values were relatively large; thus scaling would be less meaningful.

Table 6 shows that total error budgets for inspired concentrations are similar to those for end-expiratory concentrations. For desflurane, nitrous oxide, and oxygen, values of rmse are less than 12%. For carbon dioxide, there is an important total error budget of approximately 1.75 vol%. Performance measures are further reported only for end-expiratory concentrations, as their group rmse values tend to be slightly higher for most gases.


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Table 6 Overall predictive performance of both versions of the system model. Group rmses, [total error budgets (mean (SD)], are given per version for inspiratory and end-expiratory gas concentrations
 
Version 1 vs 2
Each of the performance indicators was calculated four times, that is once for each version (1 or 2) and each stage (1 or 2). Results for biases for (end-expiratory) desflurane concentrations are summarized in Figure 013F4. For stage 1, all observations are nearly on the line of identity, indicating that the biases differ very little between versions. For stage 2, most observations are further from the line of identity and in the blank area, where the bias for version 2 is closer to zero. Regardless of version, the majority of observations suffer from under-prediction (negative bias) for stage 1 and over-prediction (positive bias) for stage 2 (Fig. 013F4).



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Fig 4 A scatterplot of the bias for version 2 vs the bias for version 1 obtained for end-expired desflurane concentrations. Each patient (n=50) is represented by one symbol per stage. One diagonal of the square is the line of identity, the other is the line where the biases have identical magnitudes but opposite signs. Observations in shaded areas are closer to zero bias for version 1, whereas observations in blank areas are closer to zero bias for version 2.

 
Figure 013F5 displays the results for the biases for nitrous oxide and oxygen. All observations are not far from the line of identity. Differences between versions are less than 2.5% for most observations. Over-prediction predominates for nitrous oxide, but under-prediction for oxygen. Observations for nitrous oxide and oxygen are lying predominantly in blank areas, indicating that their biases for version 2 are closer to zero (Fig. 013F5).



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Fig 5 A scatterplot of the bias for version 2 vs the bias for version 1 obtained for end-expired nitrous oxide and oxygen concentrations. Each patient (n=50) is represented by four symbols: one per stage and per gas. One diagonal of the square is the line of identity, the other is the line where the biases have identical magnitudes but opposite signs. Observations in shaded areas are closer to zero bias for version 1, whereas observations in blank areas are closer to zero bias for version 2.

 
Table 7 outlines complete results, including statistical significances, for desflurane, nitrous oxide, and oxygen. For desflurane, results depend on stage. For stage 2, version 2 performs better than version 1 because rmse and bias are closer to zero. For nitrous oxide and oxygen, all performance variables are closer to zero for version 2, except for the scatters in stage 2.


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Table 7 Predictive performance of version 1 vs version 2. Performance measures (mean (SD)) for end-expiratory gas concentrations are calculated per stage for both versions. The direction (sign) of the bias follows from its definition given in Table 1, that is predicted minus measured concentration. *P<0.001 version 2 vs version 1, for each stage
 
For end-expiratory concentrations of carbon dioxide and nitrogen, rmse values were similar to those in Table 6, with biases of –1.1 to –0.7 vol% for nitrogen and 1.6 to 1.7 vol% for carbon dioxide.

Version 2: performance per stage and group
Each of the performance measures was calculated four times, that is once for each group (S or F) and stage (1 or 2). In Figure 013F6, each patient is represented by two symbols on each of the xy plots, that is one symbol per stage. The iso-rmse semicircles allow a visual impression of the predictive performance of version 2.



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Fig 6 Predictive performance measures obtained with version 2 for end-expired desflurane, nitrous oxide and oxygen concentrations. The individual scatters of the 50 patients (36 in group S and 14 in group F) are plotted against their individual biases. Ideally, all observations for one gas would coincide with the black dot at (0,0) representing the point of ideal performance (rmse=0%). The semicircles are the iso-rmse lines from 10 to 25%. The distance between the black dot and a symbol is the rmse representing the total error budget for an individual. The mathematical relationship between rmse, bias, and scatter (Table 1) is visualized in the graph for nitrous oxide. This relationship is the same as that of the sides of a right-angled triangle. If bias is the base of the triangle and scatter the perpendicular, then the rmse is the length of the hypotenuse.

 
The total error budget was generally greater for desflurane (Fig. 013F6A) than for nitrous oxide (Fig. 013F6B) and oxygen (Fig. 013F6C). Most observations for nitrous oxide are in the over-prediction zone, whereas most observations for oxygen suffer from under-prediction. The observations in Figure 013F6B and C stick to the x-axis, lying predominantly beneath the 5% scatter line. Thus, scatter made a much smaller contribution than bias to the error budget for nitrous oxide and oxygen.

Version 2: differences in performance between stages
Figure 013F6 and Table 8 show that notable differences between stages only exist for desflurane (plus signs vs circles; crosses vs squares). Figure 013F6A shows that only a minority of the observations (24%) for stage 1, but a majority (74%) for stage 2 is found in the semicircular area where rmse is less than 10%. Most observations for stage 1 are in the under-prediction zone, whereas observations for stage 2 appear to be evenly dispersed either side of the +2% bias line (Fig. 013F6A).


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Table 8 Predictive performance of version 2. Performance measures (mean (SD)) for end-expiratory gas concentrations are calculated per stage, in the S(low) group (n=36) and F(ast) group (n=14). The direction (sign) of the bias follows from its definition given in Table 1, predicted minus measured concentration. *P<0.001 stage 1 vs stage 2, for each group. This study did not demonstrate differences between groups
 
The differences for desflurane are substantial in terms of bias (Table 8). In group S (n=36), mean bias in stage 1 is –10.1% vs +3.6% in stage 2. The magnitude of the difference between these sample mean biases is 13.7%, with a 99.9% confidence interval (CI) from 8.0 to 19.4%. In group F (n=14), this difference is 9.1%, with a 99.9% CI from 0.8 to 17.5 %.

Version 2: differences in performance between groups
The study failed to demonstrate differences in model performance between groups (Table 8).


    Discussion
 Top
 Abstract
 Introduction
 Patients and methods
 Results
 Discussion
 References
 
We quantified the predictive performance of our model1 2 during desflurane anaesthesia using various regimens in settings of vaporizer and FGF. The principal findings were: (i) predictions largely agreed with clinical data; (ii) version 2 performed better than 1; (iii) clinically important differences between groups or between stages were not demonstrated, except for desflurane for which the model performed better during maintenance. Apart from the large discrepancy for carbon dioxide, most systematic errors were small, 2–10%.

Structure of the model
Despite these findings we cannot conclude that the complete structure of the model is correct. This is a general problem of complex physiological models.14 Validating each and every variable in such a model may present insurmountable difficulties.15

The value for fR in version 2 was found on the basis of data from an other institution.12 The results in Figure 013F3 lend credence to the broad validity of the model, but do not prove that version 2 is entirely correct. Our results even do not indicate that direct diffusion of desflurane from well-perfused organs into adjacent fat is an indispensable part of a physiological model. Nonetheless, large-scale diffusion between body compartments is a well-known physical process.3 11 1618 The over-prediction we reported earlier, especially during longer anaesthetics,19 might have been minimized by adopting inter-tissue diffusion.

Diverting 3% of the cardiac output from viscera to fat (Table 5) represents inter-compartment diffusion only indirectly. Furthermore, an fR in the range from 0.01 to 0.05 would also have been a legitimate choice as the uptake curves obtained with these values were within the 95% CIs shown in Figure 013F3.

Our results also do not prove that we are right in using a blood–gas partition coefficient for desflurane of 0.52.10 This value is different from the most cited value2 (0.42), but lies within the range of measured values. Even a value of 0.58 (SD 0.06) has been reported.20 Illustrations of the impact of various values on desflurane wash-in and reasons for using 0.52 were given elsewhere.2 It is noteworthy that our model uses the tissue–gas partition coefficients reported by Yasuda and co-workers.9 Tissue–blood partition coefficients were obtained by dividing the latter coefficients by 0.52 (Table 5 in reference 1).

Accuracy for different gas species
There are some basic criteria to judge the accuracy of the model. First, valid models should not under-predict or over-predict reality in a systematic way. It is reasonable to expect a certain degree of bias for each patient. Nonetheless, the group bias should approximate zero. A value within ±10% can be defended as an acceptable approximation on the grounds that this is within the limits that can be achieved for control of concentrations clinically. Second, if the group bias approximates zero, the total error budget (rmse) should be acceptable for a majority (>=90%) of the patients. A 0–10% group bias can indeed coexist with a significant dispersion of the individual results for bias and scatter. The accuracy of the model becomes unacceptable when the individual rmses are greater than can be reasonably expected on the basis of the biological variability found in a general human population. Although we arrived at defining objective criteria to determine whether a physiological model has ‘acceptable’ accuracy for the special case of closed-circuit anaesthesia,19 further research is needed to define such rules for low-flow conditions, if at all possible.

For desflurane, nitrous oxide and oxygen, group biases are <=10% (Tables 7 and 8). For desflurane, rmse values are less than 25% (stage 1) and less than 17% (stage 2) for >=93% of the patients (Fig. 013F6). For nitrous oxide and oxygen, rmse values are less than 15% for 92–100% of the individuals.

The accurate performance of the model for nitrogen (rmse{approx}1 vol%) is not totally surprising. The largest stores of nitrogen are in the lungs and are washed out almost completely during the first minutes of an anaesthetic. The discrepancies between theory and experiment might have been greater during these very early stages, but they were omitted for reasons discussed further.

The mass balance model of Beams and co-workers21—based on the same specific breathing system—seems to represent clinical low-flow anaesthesia better than ours. In their study, however, actual patient exchanges for isoflurane, oxygen, nitrous oxide, and nitrogen were measured and used as input for the model. This points out that the predictive performance of a model heavily depends on variability in exchange rates of gases and vapours.

Can the cause of the systematic errors be elucidated?
In this study we could exclude errors induced by departures from the settings of the vaporizer as we measured the actual delivered concentration at each setting. Although deviations between checks of the Rotameters cannot be excluded, it is very unlikely that temporary deviations would always occur in the same direction.

The under-prediction for oxygen could be simply explained by assuming that actual oxygen uptake was lower than predicted. This would conflict with prior notions that reported values were greater than those predicted by our model.1 In addition, this single factor governed explanation might be too simplistic. There are many factors influencing the kinetics of the various gases, and their kinetics are inter-dependent.2 22 This leads to propagation of errors and thus presents nearly insuperable difficulties in tracing the exact sources of error. As oxygen plus nitrous oxide make up about 90% of the gas mixture, it is almost certain that a negative bias in one will be associated with a positive bias in the other.

Assuming that the actual uptake of nitrous oxide was greater than that predicted would not only explain the over-prediction found for nitrous oxide, but also the under-prediction found for oxygen. Figure 013F7 confirms this by re-plotting data from Figure 013F6B and C to show a strong link between the individual biases for nitrous oxide and oxygen: r2=0.58 and r2=0.70 for stages 1 and 2, respectively. Others23 also observed an actual uptake of nitrous oxide greater than predicted.



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Fig 7 Scattergram showing relationship between the individual biases obtained with version 2 for the end-expired nitrous oxide and oxygen concentrations. There is one symbol per stage for each of the 50 patients. Note that one unit of bias has the same length on both axes. The regression lines have intercepts not far from zero and slopes not far from minus one.

 
The complex array of sources for error is also not in favour of a simple explanation of the errors being less in stage 2 than in stage 1, but only for desflurane (Table 8). Under-prediction in stage 1 is no surprise as erroneous prediction of uptake of nitrous oxide may lead to erroneously predicted concentrations of oxygen (see above) and desflurane alike.2 However, we also showed that certain combinations of cardiac output and ventilation present in a patient did not necessarily lead to grossly erroneous predictions for desflurane.2 The actual value of each of the factors governing gaseous homeostasis in circle systems very likely differed from that assumed, but as a result of their complex interplay, systematic modelled-to-reality differentials in desflurane concentrations were absent during stage 2.

The systematic over-prediction for carbon dioxide (bias{approx}1.8 vol%) may be partially related to systematic measurement errors, despite the use of mass-charge ratio 12. Error-free assessment of relatively low carbon dioxide levels in gas mixtures containing high concentrations of nitrous oxide is a well-known problem in respiratory mass spectrometry. Carbon dioxide readings lower than expected have been reported.24 We found that carbon dioxide levels, as measured by the mass spectrometer, were invariably lower than those assessed by the infrared analyser incorporated in the anaesthesia system, at least during controlled ventilation. It is puzzling, however, why calibration mixtures showed no error. Notably, two infrared analysers may differ up to 0.4 vol%.25

Research design
Early stages of the anaesthetic
A reservoir bag at the end of a 1-m length of corrugated tubing was used for spontaneous breathing and manual ventilation by mask with high flows of fresh gas. This was not mimicked in the simulation as the volume of tubing and bag (2 litres when full, but much <2 litres when squeezed) averaged over a respiratory cycle closely matches the volume of the standing bellows (1.5 litres when full). Future research might be improved by using the standing bellows for both spontaneous breathing and artificial ventilation of the lungs by mask—as advocated for two-handed mask ventilation of the difficult airway by a single individual.26

Early stages, such as pre-oxygenation and manual ventilation of the lungs, were omitted from calculations of performance measures because the values obtained for inspired and end-tidal concentrations were too inaccurate before intubation.

Nitrous oxide
As the present study may suffer from some error as a result of the use of nitrous oxide (as described above), further research would preferably also include a group of subjects breathing oxygen in air. This would allow studying the impact of nitrous oxide and (better) assessing the behaviour of nitrogen and carbon dioxide.

Alveolar ventilation
The value for arterial–end-expired PCO2 difference, 0.533 kPa, was chosen to yield a 10% dilution factor for all patients. Others found a similar difference.25 27 Future research will need to include the invasive sampling of arterial blood for blood–gas analysis if one would want to assess individual alveolar ventilation. This would help to investigate the sources for a bias for carbon dioxide.

Clinical implications
The target end-expired desflurane concentration was rapidly attained, that is within 6.1 min on average, with both FGF regimens. The average difference between the fast and slow scheme, 3.6 min, would probably be clinically unimportant in many cases (Table 4). The low solubility of desflurane thus allows the use of a low FGF even during induction.

Although our results are based on a limited array of FGF and anaesthetics lasting 1 h, we are confident that the model can be safely used to develop drug-dosing regimens for various purposes.28 It is a unique feature of a physiological model to conceive such regimens for maintaining steady concentrations under different physiological conditions.29 Obviously, model-based predictions must be validated and are no substitute for actual monitoring of gases and vapours.

Our results suggest that version 2 of the new system model is an adequate representation of desflurane anaesthesia in the clinical setting. It allows, with a known uncertainty, prediction of the behaviour of most gases present in the anaesthetic breathing system. Nonetheless, administration of inhalation anaesthesia based on this model must be guided by active monitoring.


    Acknowledgement
 
We thank M.C.J. De Ruiter, BSc, for technical assistance and data management.



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    References
 Top
 Abstract
 Introduction
 Patients and methods
 Results
 Discussion
 References
 
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