Inhalational anesthetics produce dose-dependent effects on electroencephalogram-derived parameters, such as 95% spectral edge frequency (SEF) and bispectral index (BIS). The authors analyzed the relationship between end-tidal sevoflurane and isoflurane concentrations (FET) and BIS and SEF and determined the speed of onset and offset of effect (t1/2k(e0)).
Twenty-four patients with American Society of Anesthesiologists physical status I or II were randomly assigned to receive anesthesia with sevoflurane or isoflurane. Several transitions between 0.5 and 1.5 minimum alveolar concentration were performed. BIS and SEF data were analyzed with a combination of an effect compartment and an inhibitory sigmoid Emax model, characterized by t1/2k(e0), the concentration at which 50% depression of the electroencephalogram parameters occurred (IC50), and shape parameters. Parameter values estimated are mean +/- SD.
The model adequately described the FET-BIS relationship. Values for t1/2k(e0), derived from the BIS data, were 3.5 +/- 2.0 and 3.2 +/- 0.7 min for sevoflurane and isoflurane, respectively (NS). Equivalent values derived from SEF were 3.1 +/- 2.4 min (sevoflurane) and 2.3 +/- 1.2 min (isoflurane; NS). Values of t1/2k(e0) derived from the SEF were smaller than those from BIS (P < 0.05). IC50 values derived from the BIS were 1.14 +/- 0.31% (sevoflurane) and 0.60 +/- 0.11% (isoflurane; P < 0.05).
The speed of onset and offset of anesthetic effect did not differ between isoflurane and sevoflurane; isoflurane was approximately twice as potent as sevoflurane. The greater values of t1/2k(e0) derived from the BIS data compared with those derived from the SEF data may be related to computational and physiologic delays.
GENERAL anesthetics produce dose-dependent effects on the electroencephalogram (EEG), causing an increase in power combined with a decrease in average EEG frequency. [1,2] Various EEG-derived parameters are used to describe the anesthetic related effects. Most parameters are based on spectral analysis of the EEG, such as the 95% spectral edge frequency (SEF), median frequency, and relative [Greek small letter delta]-power. Spectral analysis transforms a set of EEG measurements to a set of numbers in the frequency domain (i.e., the power spectrum). For example, the SEF is the highest frequency in the EEG, determined by the 95% percentile of the power spectral density.
A novel EEG-derived parameter is the bispectral index (BIS), which is (partly) based on the bispectral analysis of the EEG. In contrast to spectral analysis, bispectral analysis retains information on the interdependence of frequencies.  Such an analysis helps to disentangle the information about hypnotic or sedative components of general anesthesia contained in the EEG. The BIS is based on a combination of time domain, frequency domain, and second-order spectral subparameters.  It is optimized using a patient database to correlate with the level of hypnosis or sedation, defined by a sedation score and ranges from 100 (awake) to 0 (isoelectric EEG). [5–7] Recent studies indicate that the BIS correlates well with brain metabolism  and with hypnotic or sedative end-points such as sedation, loss of consciousness, and lack of awareness and memory (see  and references cited therein).
For intravenous anesthetics, a "hysteresis" or lag has been observed between blood concentration and anesthetic effects, in particular for EEG-derived measures such as SEF [2,9] and BIS. [7,10] For inhalational anesthetics, a lag between end-tidal and brain concentration has been recognized and physiologically modeled.  Quantitative analysis of the hysteresis provides information on the speed of onset and offset of anesthetic action. After having considered the hysteresis, the subsequent determination of the concentration-effect relationship enables the comparison of potencies of anesthetics.
To the best of our knowledge, there are, apart from a single abstract,  no studies that systematically investigated the dynamic relationship between inhalational anesthetics and the EEG or any of its derived indices. We therefore quantified the relationship between end-tidal concentrations (FET) of sevoflurane and isoflurane and bispectral index and spectral edge frequency in patients of American Society of Anesthesiologists physical status I or II before surgery.
Materials and Methods
After approval of the protocol by the local Medical Ethics Committee, 24 patients with American Society of Anesthesiologists physical status I or II, aged 18 to 60 yr, scheduled for elective surgery, were randomly allocated to one of two study groups, receiving either sevoflurane or isoflurane. Patient exclusion criteria were weight 25% or more above ideal body weight; patients undergoing cranial surgery; use of alcohol more than 3 U/day; illicit drug use; use of medication acting on the central nervous system; and a history of esophageal reflux, neurologic, cardiac, pulmonary, hepatic, or renal disease. Patients were included in the study after written informed consent was obtained. They were instructed to fast for at least 6 h before the study and received no premedication.
Monitoring and Data Acquisition
Standard monitoring (electrocardiogram, arterial oxygen saturation, and noninvasive blood pressure) was applied. The EEG was recorded using an Aspect A-1000 EEG monitor (software version 3.22; Aspect Medical Systems, Natick, MA). Electrodes (Zipprep, Aspect Medical Systems) were placed on the scalp according to the international 10/20 system for electrode placement at Fp1-A1 and Fp2-A2 for bipolar recordings of the EEG. Electrode impedances were found to be less than 2 k Omega before data acquisition started. The BIS and SEF were computed by the Aspect monitor. Bispectral index and spectral edge smoothing rates were 15 s and "off," respectively (these were the smallest possible values). Raw and processed EEG data and serial data (inspired and expired concentrations of the anesthetic, oxygen, and carbon dioxide) from a Datex Capnomac monitor (Datex, Helsinki, Finland) were collected by the Datalogger program (Aspect Medical Systems) using a four-channel communications adapter (QS-100D, Quatech, Akron, OH) and stored on disc for off-line data analysis.
Patients were randomly assigned to receive either sevoflurane or isoflurane. Patients were connected to a semiclosed anesthetic breathing circuit via a good-fitting face mask. Fresh gas flow was set at 6 l/min. The anesthetic agents were given using sevoflurane or isoflurane Vapor 19.3 vaporizers (Drager, Lubeck, Germany). Inspired and expired concentrations of the anesthetic agent, oxygen, and carbon dioxide were measured at the mouth. The Datex Capnomac monitor was calibrated using a gas mixture of known concentration (Quick Cal, Datex).
Each procedure started with a 2- to 5-min period of recording awake baseline values, during which the patients breathed 30% oxygen in nitrogen. Subsequently the anesthetic was added to the inspired gas mixture. To reduce the occurrence of artifacts the subjects were instructed to keep their eyes closed, and the number of blood-pressure measurements was minimized. The vaporizer setting was increased slowly until end-tidal concentrations reached 1% and 0.6% for sevoflurane and isoflurane, respectively (approximately 0.5 minimum alveolar concentration [MAC]). When consciousness was lost (tested by response to eyelash reflex and to verbal command), vecuronium bromide (0.07 mg/kg) was given, and the lungs of the patients were artificially ventilated (by machine) via the mask. Ventilatory settings were such that the end-tidal carbon dioxide concentration was between 35 and 45 mmHg. When BIS values stabilized, several transitions in end-tidal anesthetic concentrations were performed. For sevoflurane, the end-tidal concentration sequence was 3%(about 1.5 MAC) for 15–20 min, followed by 1.0–1.5% for another 15–20 min. For isoflurane the sequence was 1.6%(about 1.5 MAC) for 15–20 min, followed by 0.6–1% for another 15–20 min. When time permitted, additional transitions were performed. Subsequently the study period ended, the tracheas of the patients were intubated, and surgery was started.
Data averaged from right and left leads were taken as effect measurements. The EEG monitor determined BIS and SEF from at least 15 s and 2 s of EEG data, respectively (J. Sigl, Aspect Medical Systems, personal communication, November 1998) and transmitted the results every 5 s. End-tidal concentrations were collected with a maximum delay of 12 s (this includes sampling delay, data processing, and transmission). We assumed that the anesthetic end-tidal concentrations were constant from one EEG measurement to the next. For each EEG value (BIS, SEF), the corresponding anesthetic end-tidal concentration was computed using linear interpolation, which introduced negligible errors.
Pharmacokinetic-Pharmacodynamic Model. The purpose of pharmacokinetic-pharmacodynamic (PK/PD) modeling is the simultaneous description of the pharmacokinetics and pharmacodynamics of a drug. In some cases, it may not be necessary to measure the concentration of the drug in the blood.  In the present study, end-tidal concentrations of the inhalational anesthetic were measured. The pharmacokinetic part of the model need only describe the relationship between the end-tidal (CET) and brain or effect-site (Ce) concentration, which is given by:Equation 1where k (e0) is a rate constant determining the speed of equilibration; we estimated the effect-site equilibration half-life t1/2ke0= ln 2/k (e0). A similar Equation iswidely used in PK/PD modeling of intravenous drugs.
The pharmacodynamic part of the model describes the relationship between the brain concentration and the measured effect, in our case BIS and SEF. We assume that this relationship is nonlinear and monotonically decreasing and can be described by the inhibitory sigmoid-Emaxmodel, also called the Hill equation (cf. Figure 1in ):Equation 2where E is the effect measure (BIS or SEF), Emaxand Eminare maximal and minimal effect values, Ceis the effect-site concentration, IC50the concentration that results in 50% inhibition, and [Greek small letter gamma] a steepness parameter.
An analog combination of the above effect compartment and Hill Equation wasproposed in 1979 by Sheiner et al.  to describe the relation between the arterial concentration of the muscle relaxant d-tubocurarine and the resulting muscle relaxation. This model has been applied to describe many effects of anesthetics (including BIS and SEF), e.g., opioids, [9,10,15,16] thiopental,  and propofol. 
Parameters of the model were estimated using nonlinear regression.  Because artifact rejection by the EEG monitor is not available for the SEF, we discarded corrupted initial SEF data.
Cross-validation. For the BIS-data, a cross-validation method using the "leave-one-out" procedure, as described by Fiset et al.,  was used to determine the predictive power of the model. In short, a population model is constructed from n - 1 patients by leaving patient i out and used to predict the BIS-time data of the ith patient. This is repeated for all n patients. This procedure can be shown to provide approximately unbiased estimates of the performance of the population model.  The performance is quantified by calculating the median (absolute) cross-validation errors. The cross-validation errors can be depicted in a histogram, for a visual inspection of the model's performance. For the population model the arithmetic means of the individual parameter estimates were used.
Two-sample, unpaired t tests were performed to test whether estimated mean model parameters of the sevoflurane and isoflurane study groups were different. Paired t tests were performed to compare parameters derived from the BIS and SEF data. P values < 0.05 were considered significant.
Anthropometric data of the study groups, given in Table 1, show that the sevoflurane and isoflurane groups were not different with respect to sex ratio, age, weight, and height. All patients completed the protocol without the occurrence of side effects.
Upon the introduction and withdrawal of the anesthetics we observed changes in raw EEG (Figure 1) and in BIS with an evident time lag between the BIS and the end-tidal anesthetic concentrations. Figure 2(left) shows the end-tidal sevoflurane versus BIS relationship with a hysteresis loop in one subject. Notice that there are four consecutive transitions in target end-tidal concentrations. When the time lag (t1/2ke0[approximate] 1.8 min) was taken into account, the loop collapsed (Figure 2, right).
(Figure 3 and Figure 4) show the best, median, and worst data fits for the sevoflurane and isoflurane BIS data and corresponding data fits for SEF. Overall, inspection of the individual data fits showed that the inhibitory sigmoid Emaxmodel adequately described the FET-BIS data. Mean values of the estimated parameters are given in Table 2. Values of t (1/2) ke0, Emax, Emin, and [Greek small letter gamma] did not differ between sevoflurane and isoflurane. Compared with isoflurane, the mean value of sevoflurane's IC50was greater by a factor of about 2. The values of t1/2ke0derived from the BIS data were greater than those derived from the SEF data (P < 0.05).
Individual and mean BIS versus effect-site sevoflurane and isoflurane concentration relationships are plotted in Figure 5. Apart from the potency difference between the two anesthetics, the Figure showsthat at sevoflurane concentrations > 1.5%(until 3%) and at isoflurane concentrations > 0.75%(until 1.5%) the BIS reached a plateau (BIS values [approximate] 40).
In Figure 6, histograms of the cross-validation errors of the sevoflurane and isoflurane groups are shown. Median and median absolute values for sevoflurane were 1 and 7 (range, -58 to 57) and for isoflurane 2 and 7 (range, -47 to 63), respectively. From the histograms and the median absolute cross-validation errors, it appears that the predictive power of the model for BIS is high, with large deviations having a small probability of occurring.
We analyzed the relationship between end-tidal sevoflurane and isoflurane concentrations and the BIS and SEF, using a combination of an effect compartment and an inhibitory sigmoid Emaxmodel. Our approach is similar to those of studies on the influence of intravenous anesthetics and opioids on EEG indices. In summary, our findings indicate that the model described the FET-BIS relationship adequately and has a high predictive power; the speed of onset and offset of anesthetic action, as defined by parameter t1/2ke0, did not differ between sevoflurane and isoflurane; the values of t1/2ke0derived from the BIS were greater than those obtained from the SEF; and isoflurane was twice as potent than sevoflurane (potency defined by parameter IC50).
The only previous study on the dynamic relationship between end-tidal anesthetic concentrations and BIS was performed by Billard et al.  They studied the influence of sevoflurane on the BIS in five female patients. They observed a t1/2ke0of about 4.6 min, which is 30% greater than our estimate. There are several differences between their study and ours: The FET-BIS relationship was analyzed using data obtained during surgery (from induction until recovery), with opioids given during the study; only three end-tidal sevoflurane transitions were applied; and the FET-BIS lag was quantified using a nonparametric approach, i.e., the mathematical function that governs the concentration-effect relationship was left unspecified.
The identical values of t1/2ke0for isoflurane and sevoflurane indicate similar on- and offset times for both agents when the end-tidal concentrations are used as reference point. The lag between measured end-tidal anesthetic concentrations and EEG effect depends on several factors. These are (1) the time necessary for end-tidal concentrations sampling, processing, and transmission;(2) the measured end-tidal-to-alveolar concentration gradient caused by physiologic dead space and the use of a mask (elimination of this factor would increase the value of t1/2ke0);(3) the alveolar-to-arterial concentration gradient caused by pulmonary shunting (elimination of this factor would decrease t1/2ke0);(4) the cardiac output-dependent delivery of the anesthetic to the brain (i.e., transit time);(5) anesthetic wash-in and wash-out into and out of the brain compartment (this factor depends on brain volume, cerebral blood flow, blood-brain partition coefficient; see Equation 3);(6) cortical and subcortical neuronal dynamics; and (7) EEG-parameter calculation by the EEG monitor. The equilibrium half-life (t1/2) for anesthetic wash-in and wash-out into and out of the brain (factor 5) is given by:[11,20]Equation 3where Vbris the brain volume, Qbrcerebral blood flow, [Greek small letter lambda]T/Gthe brain-gas partition coefficient, and [Greek small letter lambda]B/Gthe blood-gas partition coefficient. Using data from the literature (see Katoh et al.  and references cited therein), we calculated a t1/2of 1.7 min for isoflurane and 2.1 min for sevoflurane. Because factors 1–4 and 7 affect the value of t1/2ke0in opposite directions and the overall effect of these factors on t1/2ke0is relatively minor, the smaller values of t1/2compared with the values of t1/2ke0measured in our study indicate the importance of factor 6 for the pharmacodynamic FET-BIS and FET-SEF relationship and further factor 7 for the FET-BIS relationship.
This suggests that for both SEF and BIS, apart from the lag due to the anesthetic wash-in and wash-out into and out of the brain, there is an additional lag between brain anesthetic concentration and EEG effect. Neuronal or receptor-related dynamics may be responsible for this extra lag. The differences of t1/2ke0derived from the BIS and SEF data are partly related to the difference in the time necessary to determine these indices. Furthermore, Rosow and Manberg  speculate that the BIS takes into account, apart from EEG slowing or speeding, interactions between cortical and subcortical neuronal generators. Brain blood flow differs greatly among and within central nervous system areas, and also the blood-flow response to anesthetics is not uniform within the brain.  These differences may result in a slower anesthetic uptake at subcortical areas (compared with the cortex), and consequently cortical-subcortical interactions may change relatively slowly in response to changing arterial concentrations of anesthetics. This may cause a further physiologic delay in the response time of the BIS to changes in anesthetic concentrations. In contrast, the SEF may respond immediately to anesthetic brain concentrations because this EEG parameter is less or not dependent on cortical-subcortical interactions.
It is of interest to note that the values of t1/2ke0of isoflurane and sevoflurane are in the same range as that of propofol (3.5 min) but greater than that of alfentanil (1.1 min; all values derived from BIS). 
For SEF and BIS, the ratio of sevoflurane IC50to isoflurane IC50is approximately 2 (see Table 2). This is in agreement with the finding that the ratio of sevoflurane MAC-awake to isoflurane MAC-awake equals 2.  MAC-awake is the concentration threshold preventing or allowing the response to verbal command during recovery from anesthesia. These observations indicate a potency difference of a factor of 2 between these two anesthetics with respect to their sedative or hypnotic properties. It is of interest to note that the ratio of sevoflurane MAC to isoflurane MAC is 2.  MAC is a measure of the motor response to a noxious stimulus, which is predominantly mediated by spinal pathways.  Whether our observation of a similar anesthetic potency ratio for effects at spinal and supraspinal sites is related to similar molecular modes of anesthetic action at these distinct sites needs further study.
Model Limitations and Extensions
The PK/PD model used by us should reflect and predict, to a fair approximation, the EEG changes that occur in response to the inhalation of volatile anesthetics. The cross-validation analysis shows that the predictive power of the model used to analyze the BIS data was high (see Figure 6). Large deviations from the model did occur, but at a low probability. We tried to avoid disturbing noxious and auditory stimuli as much as possible. Large deviations may be further related to anesthesia-induced excitation (noticeable in the EEG as high-frequency activation) or burst suppression of the EEG (described subsequently).
Several physiologic properties and phenomena were not incorporated in the model:
1. High concentrations of inhalational anesthetics increase brain blood flow.  The value of t1/2ke0is smaller at high brain blood-flow levels and vice versa (see Equation 3). Therefore, an anesthetic concentration-dependent parameter t1/2ke0seems more appropriate than a parameter that is concentration-independent.
2. We observed in 15 of 24 patients that after the administration of anesthetic agents there was an initial increase in the power of fast waves in the EEG ([Greek small letter beta]-band), resulting in a rise in both BIS and SEF, which was followed within 2–3 min by a decrease in power and reduction in BIS and SEF. Concomitantly, there was an increase in high-frequency activity of the surface electromyogram (SEMG), as determined by the EEG monitor. This phenomenon, which we designate excitation, may be equivalent to observations in humans by Stanski et al.  for thiopental and Seifert et al.  for propofol, and in rats by Dutta et al.  for propofol. It seems to be a general property of hypnotics and general anesthetics to cause excitement and disinhibition before depression. Following Stanski et al.,  we performed a separate data analysis in which we discarded the periods of high-frequency EEG activation. For both BIS and SEF we observed only a 10% decrease in the value of t1/2ke0, without affecting IC50.
3. At end-tidal concentrations higher than applied in our protocol, burst suppression of the EEG is a common phenomenon. Our model does not permit predictions at these concentrations. At the highest concentrations studied (1.5 MAC), we observed 10–30% burst suppression in 6 of 24 patients. This caused short-lived deviations from the model (30- to 60-s drops in BIS from 30 to 0–10).
Further studies are warranted to investigate whether improvements to the model, incorporating the previously mentioned phenomena, will improve data fits and predictive power. A first attempt to extend the model was of limited success. To reduce the variability in BIS residuals that occurred because of excitation (item 2), we incorporated the SEMG output in the model in the episode before muscle relaxation. [double dagger] A significant improvement of the model fit occurred in only 1 of 15 data sets.
For a complete PK/PD model for inhalational anesthetics, our PD part could be supplemented by a multicompartment model such as described by Tanner  and Yasuda et al.  Such a PK/PD model is completely analogous to the models used for intravenous anesthetics, considering the fact that ventilation times inspired concentration is analogous to infusion (except for the lag caused by the gas-delivery system of the ventilator). In a first attempt, we analyzed the inspired end-tidal data using a multicompartment kinetic model. We obtained adequate model fits using two or three compartments. Sevoflurane and isoflurane differed in the rate constants of the second compartment (sevoflurane equilibration half-life for lung uptake is 1.0 +/- 0.6 min versus 4.7 +/- 3.3 min for isoflurane, P < 0.05). Evidently, this is related to differences in blood-gas partition coefficients  and explains the greater speed of onset and offset for sevoflurane compared with isoflurane when the inspired concentrations are used as reference point.
In conclusion, we modeled the isoflurane and sevoflurane concentration-BIS relationship using an inhibitory sigmoid Emaxmodel. The model, which adequately described the data, is most suitable to predict BIS values during monoanesthesia with sevoflurane and isoflurane and hence may predict various endpoints of anesthesia, such as loss of wakefulness and return of consciousness.
The authors thank Drs. J. Swen and A. Schipper for their help in the recruitment of patients and performance of experiments. The authors also thank Prof. Dr. J.G. Bovill for his critically reading of the manuscript and valuable comments.