BJA Advance Access originally published online on June 22, 2007
British Journal of Anaesthesia 2007 99(2):226-236; doi:10.1093/bja/aem148
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Applying a physiological model to quantify the delay between changes in end-expired concentrations of sevoflurane and bispectral index
Radboud University Nijmegen Medical Centre, Department of Anaesthesia, Geert Grooteplein 10, 6500 HB Nijmegen, The Netherlands
* Corresponding author. E-mail: j.lerou{at}anes.umcn.nl
Accepted for publication March 25, 2007.
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Abstract |
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Background: The delay between changes in end-expired sevoflurane concentrations and bispectral index (BIS) may be characterized by a ‘rate constant’ (ke0). A smaller ke0 reflects a longer delay. Values for ke0 vary substantially among studies. The question arises how ke0 depends on experimental conditions, including ventilation and apparatus.
Methods: Increasing and decreasing sevoflurane concentrations were cyclically delivered to our validated model. First, we quantified theoretical ke0 values for distinct alveolar ventilations, estimating ke0 from sevoflurane tensions in alveolar space and grey matter. Secondly, we investigated the impact of experimental conditions. To predict BIS, the model was extended with a pharmacodynamic section, including ke0. Known values, matching theoretical values, were assigned to this ke0. These were recovered from end-expired concentrations and BIS. Possible determinants of error (difference between assigned and recovered ke0) were varied, that is fraction of dead space gas in end-expired gas (d), and time delays in measuring BIS (tBIS) and end-expired concentrations (tEE).
Results: Theoretical ke0s were 0.7, 0.53, 0.35, and 0.2 min–1 for an arterial PCO2 of 8, 6.67, 5.33 (normocapnia), and 4 kPa, respectively. For spontaneous ventilation, ke0 = 0.53 min–1. Recovered ke0s depended on d and 
t (= tBIS – tEE) and were smaller than assigned values (if t > 0). Errors increased with increasing d and t. For normocapnia, ke0 was between 0.32 and 0.23 min–1 (d = 0.1; any t = 0–60 s). For spontaneous ventilation, ke0 was between 0.51 and 0.40 min–1 (d = 0–0.1; t = 5–20 s).
Conclusions: Published ke0s (0.22–0.53 min–1), including our own for sevoflurane-depressed spontaneous ventilation (0.48 min–1), are in the ranges dictated by investigation-specific conditions.
Keywords: anaesthetics volatile, sevoflurane; model, pharmacodynamic; model, pharmacokinetic; monitoring, bispectral index
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Introduction |
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TopAbstractIntroductionMethodsResultsDiscussionSupplementary materialAppendixAcknowledgementsReferences |
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Anaesthetic effects on the electroencephalogram (EEG) may be described with measures derived from the EEG, such as the bispectral index (BIS). For inhaled anaesthetics, a delay or ‘hysteresis’ between changes in end-expired concentrations and EEG-derived measures has been found. This delay may reflect the time taken by the anaesthetic to move between alveoli and its effect-site in the brain. One may assume that the movement of drug obeys the mathematics of a wash-in/wash-out process characterized by one single rate constant. This ‘effect-site equilibration rate constant’ ke0 is the reciprocal of a time constant. A smaller ke0 reflects a longer delay between a change in anaesthetic dose and an effect.
Knowledge of such delays would be clinically relevant, but experimental values for ke0 differ substantially among clinical studies,1–3 including our own.4 Accurately predicting BIS may help to predict various end-points of anaesthesia, such as loss of wakefulness and return of consciousness. However, two different studies on sevoflurane and BIS gave estimates for ke0 which differ > two-fold.2 4 In our study, spontaneous ventilation was maintained,4 whereas in the other, manual support of ventilation was used.2 Thus, the question arises whether reported values for ke0 can be explained by experimental conditions, including alveolar ventilation and the apparatus used.
Our aim was to predict quantitatively the degree of hysteresis one might expect to find in a study on the dynamic relationship between end-expired sevoflurane and BIS. Using our physiologically based pharmacokinetic model, we systematically studied various determinants of ke0 under conditions that are not feasible in patients. First, we quantified the theoretical hysteresis originating solely from the physiological and pharmacological processes governing the transport of sevoflurane between alveoli and brain. Secondly, we quantified the impact of sources of error on actual measurements of ke0, such as the time needed to process the EEG and to calculate BIS. We therefore added a pharmacodynamic section to the model.
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The work was not reviewed by an institutional review board because of its theoretical nature. Studies from which we used data, including our own,4 were approved by appropriate Ethical Committees. Data are expressed as mean (SD), unless stated otherwise.
The basic model
Structure
Our model5 6 describes changes in partial gas pressures with time in a subject attached to a circle breathing system with carbon dioxide absorption. Kinetics of nitrogen, oxygen, carbon dioxide, nitrous oxide, inhaled anaesthetic agents, and helium are calculated. We have modified this existing model. The brain is no longer one compartment supplied by a fixed fraction of cardiac output. The brain is now a two-compartment sub-model, providing a better description of sevoflurane kinetics in those parts of the brain where anaesthetic effects occur (Fig. 1). A time delay is present between alveolar and brain tensions as the model mimics circulation times in the body by including blood pools (Fig. 1).
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1 and A description, mathematical formulation, and supplementary validation of the new version of the model are given in the on-line supplement. There we have described 10 typical data sets quantifying the sub-model for the brain. However, the characteristics of the two brain compartments may be freely chosen. One can choose either to model grey matter and white matter in various ways or to treat the first brain compartment as a ‘sample brain’ with negligible volume. Here, we use only the two data sets from Table 1.
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In the new version, cerebral blood flow (CBF) changes by 30% from default within the range 3.33–10 kPa for each kPa change in arterial partial pressure of carbon dioxide (PaCO2).7 8 CBF decreases linearly with the arterial sevoflurane tension with a maximal reduction of 30% at 0.5 MAC.8 9 For each kPa change in PaCO2, cardiac output changes by 7% from baseline within the range 2.67–8 kPa (details in the on-line document).10
Body mass, height, age, and gender are inputs to the model and are used to calculate physiological variables. The model was set to simulate a standard human (40-yr-old man of 70 kg body mass and 1.80 m in height), unless stated otherwise.
Alveolar and end-expired partial pressures
A set point for PCO2 in the ‘ideal’ alveolar space (PACO2) is to be chosen and entered in the model. This set point is used to calculate the inspired alveolar ventilation [equation (24) in the work of Lerou and Booij5]. By definition, PACO2 equals the arterial CO2 tension.
End-expired partial pressures are needed for comparison with clinical measurements. Thus, an ‘ideal’ alveolar–end-expired partial pressure difference for CO2 (PCO2) is to be chosen. Default PCO2 = 0.533 kPa. A ‘dilution factor’, that is, the fraction of dead space gas in end-expired gas, can be deduced: d = PCO2/PACO2. The default dilution factor of 10% is similar to the average d reported [11.58% (range 4.44–23.81%)].11 This is an acceptable matching of our model to experimental data.12 13
Simulated end-expired sevoflurane tensions were calculated from the ‘ideal’ alveolar (PAsev) and dead space (=inspired; PIsev) tensions predicted by the model and the dilution factor: PE'sev = (1 – d)/PAsev + dPIsev.5
Clinical validation
Predicted sevoflurane tensions were tested against the clinical data of Nakamura and colleagues.14 They measured sevoflurane tensions in arterial and internal jugular venous blood (the latter reflecting brain tensions). Two scenarios were simulated for their average 56-yr-old male patient of 1.58 m height and 55 kg body mass.14 Our approach to recreate their experimental conditions is described in the on-line supplement.
The basic model extended with a pharmacodynamic section
The extended model separates the relationship between dose of anaesthetic and effect into two successive processes. The pharmacokinetic section (basic model) describes how, after the introduction of an anaesthetic gas at the upper respiratory tract, partial pressures in the tissues, including grey matter, vary with time. The pharmacodynamic section defines the relationship between the partial pressure of sevoflurane at its effect-site (Pe) and a measure for its anaesthetic effect (E), using four parameters:
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| (1) |
determines the shape and slope of the sigmoid curve.
In equation (1), Pe is yet to be defined. There are two distinct ways to calculate Pe. The first and obvious way in a physiologically based model is to assume that the calculated grey matter tension (Pgm) represents the tension Pe at the effect-site. Alternatively, one may leave physiological fidelity and derive the effect-site partial pressure directly from the calculated alveolar partial pressure PA using the relationship known from pharmacokinetic–pharmacodynamic models:
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Equation (2) implies that the larger the ke0, the more rapid the approach to equilibrium.
Published experimental data for ke0
We searched the literature for studies on relationships between sevoflurane and measures of its effects on the EEG and noted published values for E0, Emax, P50, , and ke0. However, instead of ke0, some investigators reported ‘equilibration half-time’ given as:
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Equation (3) allows exact calculation of individual values for ke0 from t1/2 or vice versa. This is not true for a group mean value or for measures of dispersion around the mean (Appendix). Therefore, we sought to obtain individual values for t1/2. This was only possible from the study of Baars and colleagues15 using the Engauge Digitizer program V.2.12 (available at http://digitizer.sourceforge.net. accessed March 1, 2006). From their Figure 3, we first calculated individual values for ke0 and then the average (SD) ke0.
Study 1: quantifying theoretical ke0s
We used the basic model to estimate the unknown theoretical hysteresis which is solely determined by the transport of sevoflurane between alveoli and grey matter. Accordingly, we estimated ke0 from simulated time courses of sevoflurane tensions in the alveolar space and grey matter. We assumed that EEG changes caused by sevoflurane are directly related to its tension in grey matter. This is in fact a model reduction operation illustrated in Figure 2A. The following factors were varied to study their impact on ke0: alveolar ventilation, blood flows supplying the brain, and delivered sevoflurane concentrations. Blood flows were varied by choosing different data sets to quantify the brain sub-model.
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Estimating ke0 involved two steps. First, the basic model was run for 70 min to calculate sevoflurane tensions in response to a varying vaporizer setting (one data point per 5 s). Secondly, the model defined by equation (2), in which Pe was substituted with Pgm, was fitted to time courses of sevoflurane tensions in alveoli and grey matter. The value of ke0 was optimized with the Solver tool (Excel, Microsoft) by minimizing the sum of the squared differences between values for Pgm calculated by the basic model and those calculated from equation (2). The ke0 guess value was 0.5 min–1 (no constraints).
Constant ventilation
Thirty-two simulation runs were performed (4 x 2 x 4) using four constant levels for PaCO2, two data sets quantifying the brain sub-model, and four vaporizer settings.
PaCO2 was 4, 5.33, 6.67, or 8 kPa. The two data sets are given in Table 1.
Three vaporizer settings produced a sinusoidal pattern (period = 30 min) of delivered sevoflurane concentration varying between 0 and 0.005, 0.5, or 3 MAC, respectively. The value of 0.005 MAC represents a trace concentration that does not produce circulatory effects on CBF. Non-rebreathing conditions were used. The fourth vaporizer setting had a staircase pattern: the setting was increased by 1 vol% per 3 min from 0 to 5 vol%. Fifteen minutes after starting the administration, the vaporizer was turned off for 5 min. The sequence was repeated once, after which the vaporizer was turned off. The whole sequence, in conjunction with rebreathing conditions using a fresh-gas flow of 5 litre min–1, is similar to that used in our separate experimental study.4
Varying ventilation
Twelve additional simulation runs used varying alveolar ventilation to mimic ventilatory responses to varying sevoflurane concentrations. The staircase sevoflurane input and rebreathing conditions as described earlier were used. The 12 runs (2 x 3 x 2) used two data sets (Table 1), three dose–response (D/R) curves for PaCO2 vs sevoflurane tension, and two time lags between peak end-expired concentrations of sevoflurane and those of carbon dioxide.
The set point for PaCO2 varied in time according to data from Doi and Ikeda.16 They showed that after 15 min of stable end-expired levels of 1.1 and 1.4 MAC sevoflurane, PaCO2 was 6.51 [95% confidence interval (CI) = 6.28–6.74] kPa and 7.31 (95% CI = 7.02–7.60) kPa, respectively. Assuming that ventilation is further depressed at 2 MAC and nearly stops at 2.5 MAC sevoflurane,17 we chose set points for PaCO2 of 10 and 40 kPa. Set points for arterial PCO2 at 0, 1.1, 1.4, 2, and 2.5 MAC sevoflurane were: (i) 5.33, 6.28, 7.02, 10, and 40 kPa (low D/R); (ii) 5.33, 6.51, 7.31, 10, and 40 kPa (medium D/R); and (iii) 5.33, 6.74, 7.60, 10, and 40 kPa (high D/R), with linear interpolation.
The two time lags between end-expired concentrations of sevoflurane and those of carbon dioxide were 1 and 2.5 min. The set point for PaCO2 depended on the sevoflurane tension in grey matter, not in end-expired gas, thus mimicking the physiological time delay between alveoli and respiratory centres. This is in accordance with data from Mapleson and colleagues18 who described that the same perfusion may apply to the site of action of inhaled anaesthetics or to medullary chemoreceptors. An extra delay was to be introduced to replicate the total time lag we observed in a separate experimental study.4 For 16 out of 18 patients, we could calculate an unequivocal time lag between the attainment of the first peak end-expired sevoflurane concentration and the ensuing peak end-expired carbon dioxide concentration. ‘Physiological’ time delay in the basic model was 1 min. The extra delay to be incorporated in the model was 1.5 min, as we had found that total delay was 2.5 (95% CI = 1.4–3.5) min, with a peak sevoflurane concentration of 2.6 (95% CI = 2.4–2.8) vol% and a peak PaCO2 of 6.9 (95% CI = 6.6–7.3) kPa (assuming PCO2 = 0.533 kPa).4
Study 2: sensitivity analysis
Study 1 does not reflect the conditions of a clinical study, as alveoli and grey matter are inaccessible. Experimental ke0s are derived from end-expired partial pressures and BIS. In study 2, we investigated how sensitive the estimates for ke0 are to dilution of alveolar gas by dead space gas and time delays in data acquisition and analysis. Our strategy, which recreated the methodology of a clinical study,4 15 is illustrated in Figure 2B.
Estimating ke0 involved three steps. First, we assigned known values to the five parameters in the extended model: ke0, E0, Emax, P50, and . Secondly, the model was run for 70 min to calculate time courses of end-expired partial pressures and BIS (one data point per 5 s). We used only data set #4 and one forcing function for sevoflurane, that is the staircase function described earlier. Thirdly, we estimated—in fact, here we recovered—values for the five parameters. The model defined by equations (1) and (2) was therefore fitted to time courses of end-expired partial pressures and BIS. Alveolar partial pressures in equation (2) were replaced with end-expired partial pressures. The values of the five parameters were optimized with Solver, minimizing the sum of the squared differences between the values for BIS calculated by the extended model and those calculated from equations (1) and (2).
We refer back to the known assigned values for ke0 as ‘true values’ and to those regained as ‘recovered values’. The dilution factor d and the time delays in measuring BIS and end-expired partial pressures, as possible determinants of a difference between a true and a recovered value, were varied to quantify their impact. If there was no dilution of alveolar gas by dead space gas and if there were no time delays, recovered values would have to be equal to true values.
Two hundred and fifty-six simulation runs were performed (4 x 4 x 4 x 4) using four dilution factors (d = 0, 0.1, 0.2, and 0.5), four delays in gas sampling and analysis (0, 5, 10, and 15 s), four delays in EEG processing (0, 15, 30, and 60 s), and four levels of ventilation (constant with PaCO2 of 4, 5.33, and 6.67 kPa or varying).
Assigned values for ke0 were not arbitrarily chosen but were the theoretical values found in study 1. Thus, they depended on ventilation: 0.2, 0.35, and 0.53 min–1 for PaCO2 of 4, 5.33, and 6.67 kPa, respectively and 0.53 min–1 for a varying PaCO2. Assigned values for E0, Emax, P50, and were: 95 (enables adding noise without exceeding 100), 20, 1.3 kPa, and 2, respectively. The assigned values were also the guess values for Solver.
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Clinical validation of the model
Figure 3 shows predicted and corresponding measured14 values for internal jugular venous blood tensions of sevoflurane, expressed as fractions of arterial tension. The dashed curve based on a uniform blood supply of the brain clearly overpredicts measured sevoflurane tension, thus predicting a more rapid brain wash-in than measured. Very close agreement was obtained for simulations based on the new version of the model as the values are all within the 95% CI.
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Published experimental data
Table 2 shows the variation in experimental conditions of published studies. They are arranged in chronological order of appearance and denoted A–H. Our study is denoted Z. There is large variation in devices and algorithms for EEG processing, time windows, and time delays, as well as in methods used to analyse raw data to obtain E0, Emax, P50,
, and ke0.
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A and C have same patientsValues reported for these five parameters can be compared in Table 3. ke0 found in Z is much larger than that in other studies. Values reported for t1/2 are also given, but not translated into ke0 because of inevitable errors (Appendix). However, individual values for t1/2 could be recovered from 23 digitized data points for BIS from study G.15 We found a t1/2 = 3.89 (2.57) min, which corresponds very well with the reported t1/2 = 3.9 (2.6) min. Calculating the 23 individual ke0s [equation (3)] and averaging yielded ke0 = 0.34 (0.38) min–1 (Appendix). The median value for ke0 from study D could not be translated into a correct t1/2, owing to an even number of subjects.3
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Studies A and C have the same patients. ¶Averages and variances could not be derived exactly from published data. 
Value that could be exactly calculated from published data (details see text) or from own data. ||Parameters derived from the data of individual patientsTable 3 shows variance in the values for ke0 not only between studies but also between subjects. The large variance among subjects in study G [coefficient of variation (CV)=0.67] is responsible for the large error it is possible to make while translating the average value for t1/2 into ke0 (Appendix). SD (two-stage approach) or CV (non-linear mixed effect modelling) reported for t1/2 cannot be translated into values for dispersion measures for ke0.
In comparison with ke0, there is much less variation in P50, despite the different experimental conditions (Table 3). In contrast, there is gross variation in with a range between 1.1 and 4.5 for BIS. E0 and Emax values were not always estimated.
Study 1: quantifying the theoretical ke0s
Constant ventilation
Table 4 shows that values for ke0 increase substantially with PaCO2. For a given condition, there is a 2.7- to 4.6-fold increase in ke0 when doubling PaCO2 from 4 to 8 kPa. Rate constants obtained with data set #4 are larger compared with those obtained with data set #1. Values for ke0 decrease with increasing sevoflurane maxima in the sinusoid forcing functions. Rate constants obtained with a 3 MAC sinusoid forcing function are virtually identical to corresponding rate constants estimated with a staircase vaporizer setting.
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Varying ventilation
Table 5 shows that equilibration rate constants estimated during spontaneous ventilation with varying PaCO2 do not differ much from those in Table 4 for a constant PaCO2 of 6.67 kPa. The impact of the extra time lag is small.
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Study 2: sensitivity analysis
Figure 4 displays recovered values for ke0 for a variety of conditions. Each of the four graphs in Figure 4 is devoted to one of the four alveolar ventilations. The impact of a time delay for BIS (tBIS) or that of a time delay for end-expired concentrations (tEE) is not shown separately. The results for ke0 could be summarized by plotting against the difference (
t) between both delays (t = tBIS–tEE). Errors induced by a time delay for BIS are reduced by a time delay for end-expired concentrations in the common situation that tBIS is greater than tEE.
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‘True’ values for ke0 could be regained without noticeable error if alveolar gas is not contaminated with dead space gas (d = 0) and if
t is zero (Fig. 4A–D). Absolute values for ke0 decrease with increasing t and d. For PaCO2 = 5.33 kPa, t = 30 s and d = 0.2, the apparent ke0
0.25 min–1, that is, a –30% deviation from the ‘true’ value (Fig. 4B).
Recovered values for E0, Emax, and P50 deviated by no more than 10% for all conditions of PaCO2, d, and t, except for d = 0.5 where deviations up to 18% were found. Percentage deviations for the parameter were less than 10% for t < 25 s, but increased up to 28% with increasing t.
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Discussion |
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TopAbstractIntroductionMethodsResultsDiscussionSupplementary materialAppendixAcknowledgementsReferences |
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The new validated version of our model describes sevoflurane kinetics in those parts of the brain where anaesthetic effects occur. Using this model, we quantified the delay one might expect to find, in a clinical study, between changes in end-expired sevoflurane concentrations and BIS. The magnitude of this delay may be traced to two main sources.
In study 1, we assessed the contribution of the transport of sevoflurane between alveoli and grey matter. As a result of an increased CBF, a higher arterial carbon dioxide tension leads to more rapid approach to equilibrium between alveolar and brain tensions, characterized by a larger ke0. Theoretical ke0s range from 0.2 to 0.7 min–1 for an arterial carbon dioxide tension from 4 to 8 kPa. For sevoflurane-depressed spontaneous ventilation, ke0 is 0.53 min–1.
In study 2, we assessed the contributions of end-expired–alveolar differences and time delays occurring in devices for EEG processing and gas analysis. The difference in these time delays (t) proved to affect the experimental estimation of the theoretical values defined in study 1. As t is greater than zero in most circumstances (t = tBIS–tEE), measured ke0 is smaller than theoretical values, thus suggesting an erroneously slow brain wash-in. The deviation further increases with an increasing dilution factor. For normocapnia, apparent ke0s are approximately 10–35% smaller than the theoretical values for any t between 0 and 60 s with a dilution factor of 0.1. For sevoflurane-depressed spontaneous ventilation, one may expect to find a ke0 between 0.51 and 0.40 min–1 for any t between 5 and 20 s with any dilution factor between 0 and 0.1.
Factors influencing ke0 have been categorized previously.2 19 The list includes: (i) time needed for gas sampling and computing end-expired concentrations; (ii) end-expired to arterial tension differences; (iii) cardiac output-dependent transit time to the brain; (iv) anaesthetic wash-in and wash-out of the effect-site (depending on volume, perfusion, and partition coefficients); (v) neuronal dynamics; and (vi) time needed for parameter computing by the EEG device. The fifth factor was not incorporated in our model, on the assumption that its role is minimal in comparison with other factors.
This comprehensive list lacks the quantitative evaluation of the present study. The list does not recognize that the difference in time delays between monitoring devices for EEG processing and gas analysis plays a crucial role. Measurement error can even be minimized by similar delays. Inspired sevoflurane tensions are not explicitly mentioned, although they may grossly influence ke0 by determining CBF (Table 4).
The ke0 of 0.48 min–1 we report for spontaneously breathing patients4 agrees well with the expected value. Pilge and colleagues20 reported that the average delay is 19 s for BIS values changing with small steps and BIS > 20, which are conditions similar to those in our study Z. We defined empirically an average delay for end-expired concentrations of 8 s. Thus, with t = 11 s, the range of expected ke0s is 0.49 – 0.45 min–1 for any d = 0–0.05 (Fig. 4C). This range encompasses our experimental ke0 of 0.48 min–1. A small dilution factor d is consistent with the condition that gas was sampled from a nasal catheter and not from the facemask via an elbow adapter.4
The ke0 values obtained in four studies which aimed at normocapnia are also within the expected ranges. The ke0 for BIS was 0.31 and 0.34 min–1 in studies F and G, respectively. The theoretical ke0 is 0.35 min–1, but Figure 4B shows that it is likely to obtain estimates between 0.31 and 0.34 min–1, if d is small and t is between 20 and 0 s. Exact data on t are not available, but a t of 10 s would be compatible with data from study G. In study D, ke0 is 0.23 min–1. This value may be expected as Figure 4B shows that for t = 30 s (not an unlikely situation with the older A-1000 device) and d = 0.2, the theoretical value of 0.35 min–1 turns into an expected 0.25 min–1. In study E, ke0 is 0.41 min–1 for SEF95. The discrepancy with BIS may be explained by a larger delay for BIS, which is consistent with the longer computation time needed for BIS.
We did not discuss data from studies that reported only average equilibration half-times (Table 3) because of the possible error associated with converting a group mean t1/2 into a ke0. The exact error is unclear when investigators directly convert average values for ke0 in t1/2 or vice versa, in an attempt to compare their data with those of others.1 21 As there is evidence that the error could be more than trivial (Appendix), we propose that published results should always include values for ke0.
The model does not include carbon dioxide transport and storage. This has no consequences for the vast majority of simulations where we assumed that a steady state with a chosen PaCO2 was reached before administering sevoflurane. For simulations with sevoflurane-depressed spontaneous ventilation, we matched the time lag in the model (between peak end-expired concentration of sevoflurane and that of carbon dioxide) to that in our measurements.4 This is a well-known solution in the absence of sufficient data to build a perfect physiological model.
Although more of our own data could have been introduced into the model, depression of alveolar ventilation was based on other published data.16
Predicted peak end-expired concentrations of carbon dioxide agreed with those measured. The impact of the extra time lag was small (Table 5).
Previous publications have suggested that physiologically based models incorrectly represent the uptake of inhaled anaesthetics by the brain as the predicted half-times were shorter than those measured. Nakamura and colleagues14 proposed to eliminate discrepancies between their clinical data and their own four-compartment model by assuming a brain partition coefficient twice as great as the accepted one. Others22 have postulated controversial23 24 ‘barriers’ between alveolar gas and blood and between blood and brain. Investigators calculated half-times using a chosen fixed CBF. However, experimental brain uptake depends on a CBF that may differ from that assumed and may vary dynamically (on-line supplement). Our results suggest that a model can explain clinical data without postulating concepts that are not consistent with previous evidence. Others have suggested alternative explanations, such as the combination of diffusion limitation and local shunt in the brain.25
Confidence is justified when the modified model is used for an analytical separation of the factors contributing to erroneous estimations of ke0. The model's credibility is related to the fact that surrogate measures of anaesthetic effects are indirect estimations of perfusion and sevoflurane tension in inaccessible parts of the central nervous system.18 The model describes sevoflurane tensions in grey matter where anaesthetic effects occur. For both the wash-in into the brain (Fig. 3) and the effects of sevoflurane on BIS, agreement was observed between theory and experiment. Thus, each gives further confidence in the validity of the other.
In conclusion, published data for ke0 are similar to theoretical values after correction for investigation-specific experimental conditions. Uncorrected ke0s for BIS suggest an erroneously slow wash-in of grey matter. Experimental conditions and interpatient variability, caused, for example, by variation in the dilution factor d, grossly influence the experimental results. As a corollary, there may be no unique ke0, but there may only be a ke0 for a unique patient under unique conditions.
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Supplementary material |
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A comprehensive description of the development of the new version of the model and more simulations can be found as supplementary material in British Journal of Anaesthesia online.
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Appendix |
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A value for an average ke0 that is calculated from a published group mean t1/2, or vice versa, may be more or less erroneous. Table A1 lists the 23 individual values for t1/2 we could recover from study G and three synthetic data sets for ke0.
For study G, we found an average t1/2 of 3.89 (2.57) min, which is equal to the reported t1/2 = 3.9 (2.6) min.15 Calculating the 23 individual ke0s [equation (3)] and averaging yields ke0 = 0.343 (0.382) min–1 (Table A1). Using equation (3) to calculate ke0 directly from the reported average t1/2 = 3.9 min would yield the erroneous ke0 = 0.18 min–1 which is nearly half the correct value.
Three synthetic data sets were obtained by simulation. Each data set of 4 x 106 random numbers was generated with a log normal distribution from which very unlikely values (ke0 < 0.069 or ke0 > 2.082) were excluded. Varying the SD yielded three data sets with different CV. From each of these three data sets, 100 subsets of 23 values were randomly selected, thus simulating 100 studies in 23 patients. The three subsets in Table A1 were those with a CV nearest to 1/3, 2/3, and 1.
Calculating first t1/2 for each individual [equation (3)] and then averaging yields the correct values of 1.546, 1.692, and 2.959 min for sets S1, S2, and S3, respectively. These values are to be compared with the erroneous values 1.394, 1.179, and 1.282 min for sets S1, S2, and S3, respectively. For S3, error factor=2.959/1.282=2.31.
The problem offered can be generalized. Suppose mean and variance of a random variable x are known. Can we calculate mean and variance of any function of x such as 2x2 or, as in our case, ln(2)/x? According to Armitage and Berry,26 there is no simple exact formula and an approximation is only available when the CV is small. Our examples illustrate that the error increases with the variance in the data. The error factors seem to be roughly 1 + CV2 in the three examples, but they were mostly larger in other subsets.
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We wish to thank our colleagues Drs J. Bruhn and D. Snijdelaar for their critical remarks.
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1 Ellerkmann RK, Liermann VM, Alves TM, et al. Spectral entropy and bispectral index as measures of the electroencephalographic effects of sevoflurane. Anesthesiology (2004) 101:1275–82.[CrossRef][Web of Science][Medline]
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