In a steady-state situation (meaning nonchanging drug concentrations), plasma or end-tidal concentration (C _{et} ) values and effect-site drug concentration (C _{eff} ) values are identical. However, in a nonsteady-state situation (e.g., during induction of anesthesia), C _{et} will differ from C _{eff} . Because C _{eff} usually cannot be measured directly, they are estimated by measuring the drug effect at the effect site (e.g., in the brain: electroencephalographic [EEG] suppression). When C _{et} and measured drug effect (corresponding to C _{eff} ) are both plotted over time, a time lag between peak C _{et} and peak drug effect at the effect site (maximal EEG suppression) is usually present. This time lag leads to a hysteresis loop (hysteresis originating from the Greek meaning “comes late”) when the effect (in our study EEG effect) is plotted over the plasma or C _{et} (Figs. 1–3A ). Pharmacokinetic pharmacodynamic (PKPD) models have been proposed that characterize the equilibration between C _{et} and C _{eff} . In these models, k _{e} _{0} describes the link between the time course of C _{et} and C _{eff} or, in other words, the equilibration time course of C _{et} and C _{eff} .

Figure 1: Figure 1.

Figure 2: Figure 2.

Figure 3: Figure 3.

Parametric and nonparametric approaches have been established to estimate k _{e} _{0} . Estimating k _{e} _{0} by simultaneous PKPD models^{1–4} and fitting the relationship between calculated C _{eff} and drug effect using a single sigmoid E_{max} model (Hill equation)^{5} gives an example for a parametric approach. Recently, a bisigmoidal curve was proposed to improve the relationship between EEG indices and anesthetic drug concentrations.^{6–9} This curve or model takes into account the previously described pharmacokinetic plateau^{10} that becomes evident before the occurrence of burst suppression. However, calculated k _{e} _{0} values differ depending on the underlying PKPD model used to fit the data.^{6,8,9} In addition, it has been questioned whether the plateau seen in the drug-response curve is physiological or may even be artificially obtained by falsely using sigmoidal or bisigmoidal models to fit the collected data.

Nonparametric approaches, on the other hand, do not depend on fitting the data with a previously chosen model (sigmoidal or bisigmoidal).^{11–13} Our aim was to apply a semiparametric approach of estimating k _{e} _{0} values to our data. Because estimating k _{e} _{0} is based on optimization of the correlation between drug concentration and effect and because the prediction probability (P _{K} ) can be used as a nonparametric measure of the goodness of the correlation between drug concentration and effect, we hypothesized that maximizing the prediction probability could be a promising new semiparametric method for estimating k _{e} _{0} . The prediction probability^{14} has become a standard measure for the performance of anesthetic depth monitors.^{4,6,15–17} Given 2 randomly selected data points with distinct anesthetic drug concentration, the P _{K} value describes the probability that the EEG parameter correctly predicts which of the data points is the one with the higher (or lower) anesthetic drug concentration. P _{K} has been developed from the need for a performance measure of 2 scales or databases that are polytomous (multivalued) ordinal with one scale (database) predicting the data of the other scale.^{18}

We investigated whether optimizing P _{K} to its maximum would allow the hysteresis loop to collapse yielding a single defined k _{e} _{0} value for each patient.

METHODS
We used the data of 3 published studies,^{7–9} wherein 45 male patients scheduled to undergo radical prostatectomy were studied using isoflurane, sevoflurane, and desflurane, respectively (15 patients each). All studies received IRB approval, and written informed consent was obtained from all patients. Exclusion criteria were a history of any disabling central nervous or cerebrovascular disease, substance abuse, or a treatment with opioids or any psychoactive medication.

All patients were premedicated with 7.5 mg midazolam orally on the morning before surgery. In the operating room, an IV catheter was inserted into a larger forearm vein, and standard monitors were applied. An epidural catheter was inserted in the lumbar space, and a test dose of 3 mL bupivacaine 0.5% was given. The electroencephalogram was recorded continuously using an Aspect A-2000 bispectral index (BIS) monitor (version XP, Aspect Medical Systems, Norwood, MA). After the skin of the forehead had been degreased with 70% isopropanol, the BIS electrodes (BIS-XP sensor, Aspect Medical Systems) were positioned as recommended by the manufacturer. Induction of anesthesia was started with a remifentanil infusion at 0.4 μg · kg^{−1} · min^{−1} . Five minutes later, the patients received 2.0 mg/kg propofol. After loss of consciousness occurred, oxygen was given through facemask ventilation, and patients received 0.5 mg/kg atracurium.

Three minutes later, the trachea was intubated, and the lungs were ventilated to an end-tidal carbon dioxide concentration of 35 mm Hg. Immediately after intubation, the remifentanil infusion was stopped, and isoflurane, sevoflurane, or desflurane in 1.5 L/min fresh gas flow (0.5 L/min O_{2} and 1 L/min air) was given for hypnosis. End-tidal concentrations were measured using infrared absorption technology (PM 8050, Dräger, Lübeck, Germany).

After induction of anesthesia had been completed, patients received 12 mL bupivacaine 0.5% epidurally for intraoperative analgesia. Complete neuromuscular blockade during the investigation was ensured by repeated injections of 0.25 mg/kg atracurium and by neuromuscular monitoring, i.e., train-of-four and double burst stimulation monitoring.

Study Measurements
To exclude residual propofol or remifentanil effects and to ensure a condition of constant surgical stimulation, study measurements began with the dissection of the prostate, a minimum of 45 min after induction of anesthesia. At this time, a steady-state condition for the volatile anesthetics between gas C _{et} and gas C _{eff} was assumed. To obtain volatile drug concentration versus EEG response curves, C _{et} values were subsequently increased and decreased 2 times: starting at a C _{et} of 0.5, 1, or 3.0 vol% for isoflurane, sevoflurane, and desflurane, respectively, the vaporizer was opened to the maximum of 5, 8, and 14 vol% concentration until the C _{et} values of 2.3, 4, or 10 vol% for isoflurane, sevoflurane, and desflurane, respectively, were reached. Thereafter, the vaporizer was closed (0 vol% concentration) until the C _{et} values reached 0.5, 1, or 3.0 vol% or a BIS value of 60 was reached. Fifteen minutes later, this sequence was performed a second time. After the final suture was placed, the vaporizer was turned off, and patients were allowed to recover from anesthesia. The measurements ended when patients were tracheally extubated.

Study data were automatically recorded at intervals of 5 s using the software programs Proto 99 (version 1.0.2.0, Dräger) for the gas C _{et} and Hyperterminal (Microsoft, Redmond, WA) for BIS values.

Calculating k _{e} _{0} Values by Maximizing the Prediction Probability
In a first step, estimated C _{eff} values were calculated according to the following equation:

where C _{eff} (now) is the estimated effect-site drug concentration at the present time, C _{et} (now) is the measured end-tidal drug concentration at the present time (which is assumed to be constant over the time interval t _{now} − t _{prior} ), C _{eff} (prior) is the calculated effect-site concentration of the previous data point (in our study, one data point was obtained every 5 s), and k _{e} _{0} is the time constant (min^{−1} ) for the equilibration between C _{et} and C _{eff} . Because we used a time interval of 5 s, “t _{now} − t _{prior} ” was 5 s or 1/12 min throughout the calculation. Calculations were performed in Excel (Microsoft).

Hypothetical C _{eff} values were calculated for the following k _{e} _{0} values: 0.05; 0.06; 0.07 … 0.3; 0.4; and 0.5 … 1.2 min^{−1} leading to 35 different datasets of hypothetical drug C _{eff} values and their corresponding measured BIS values for each patient.

In a second step, the prediction probability (P _{K} ) was calculated as described by Smith et al.^{14} Prediction probability P _{K} has been defined as:

where P _{c} , P _{d} , and P _{tx} are the respective probabilities that 2 data points are a concordance, a discordance, or an x-only tie. The Excel software program PKMACRO^{14} was used to calculate P _{K} for each of the 35 datasets.

In a third step, the highest prediction probability for each patient was identified (Figs. 1–3E ) within the 35 datasets and the corresponding k _{e} _{0} value defined as optimal to describe the drug-response relationship between the investigated volatile anesthetic and BIS in this patient.

Figures 1–3 show the effect of different k _{e} _{0} values on the calculated C _{eff} of the volatile anesthetics and the corresponding drug-response curves. When a chosen hypothetical k _{e} _{0} value is too high, an immediate equilibration between C _{et} and C _{eff} as shown in Figures 1B, 2B, and 3B for k _{e} _{0} = 1 min^{−1} will be assumed and gas C _{et} will equal C _{eff} values immediately, so that the hysteresis loop does not collapse. If the chosen k _{e} _{0} is too small, the equilibration between C _{et} and gas C _{eff} is assumed to be too slow and the resulting drug-response curve (Figs. 1D, 2D, and 3D ) will overlap. Both underestimation and overestimation of k _{e} _{0} have a direct effect on P _{K} (Figs. 1–3E ), which was therefore chosen to identify the optimal k _{e} _{0} value.

As a nonparametric measure, the P _{K} value is independent of scale units and does not require knowledge of underlying distributions or efforts to linearize or to otherwise transform scales. Furthermore, P _{K} can be computed for any degree of coarseness or fineness of the scales. Therefore, P _{K} fully uses the available data without imposing additional arbitrary constraints. These qualities are shared with other possible approaches.

Calculating k _{e} _{0} by Minimizing the Area Within the Hysteresis Loop
In parallel, an additional approach was undertaken to estimate k _{e} _{0} values for volatile anesthetics. A spreadsheet using the Excel 2000 software program was used to calculate the area within the hysteresis loop. In a first step, C _{eff} values were calculated according to Eq. 1 for the following k _{e} _{0} values: 0.05; 0.06; 0.07 … 0.3; 0.4; and 0.5 … 1.2 min^{−1} leading to 35 different datasets of hypothetical drug C _{eff} values and their corresponding measured BIS values for each patient.

The sum yielding the remaining area within the hysteresis loop for each of the 35 datasets was calculated according to

where n denominates the number of measured BIS values, BIS_{i} is the i th measured BIS value, and BIS_{Ceff (i)} is the mean BIS value of all BIS values measured at the same C _{eff} as BIS_{i} . Calculated C _{eff} values derived from Eq. 1 were rounded up to one decimal digit after the comma.

Statistical Analysis
Statistical analysis was performed using SigmaStat 2.03 and SigmaPlot 2000 computer software (SPSS, Erkrath, Germany). Statistical calculations were performed by a paired two-tailed Student’s t -test with a statistical significance defined as P < 0.05 to compare k _{e} _{0} values obtained by maximized P _{K} or minimized residual area. k _{e} _{0} values among isoflurane, sevoflurane, and desflurane were compared with between-groups analysis of variance, and Tukey honestly significant difference post hoc test as the global analysis of variance result was significant. Data are presented as mean and sd unless otherwise indicated.

RESULTS
A single k _{e} _{0} value leading to maximal prediction probability or a minimal residual area within the hysteresis loop could be identified in all patients for both semiparametric approaches for isoflurane, sevoflurane, and desflurane. The relationship between the range of chosen k _{e} _{0} values (0.05–1.2 min^{−1} ) and prediction probability and the area in the hysteresis loop are shown for 3 different patients anesthetized with isoflurane (Figs. 1E and F ), sevoflurane (Figs. 2E and F ), and desflurane (Figs. 3E and F ).

k _{e} _{0} values calculated for isoflurane yielded similar results when applying the maximized prediction probability (k _{e} _{0} = 0.18 ± 0.06 min^{−1} ) and when minimizing the area within the hysteresis loop (k _{e} _{0} = 0.15 ± 0.04 min^{−1} ) (P = 0.14). Also, no significant differences between both methods for estimating k _{e} _{0} were found for sevoflurane (k _{e} _{0} = 0.17 ± 0.08 min^{−1} when applying the maximized prediction probability and k _{e} _{0} = 0.16 ± 0.11 min^{−1} when minimizing the area within the hysteresis loop, P = 0.73) and desflurane (k _{e} _{0} = 0.30 ± 0.17 min^{−1} when applying the maximized prediction probability and k _{e} _{0} = 0.32 ± 0.25 min^{−1} when minimizing the area within the hysteresis loop, P = 0.46).

Although no significant differences were found between the k _{e} _{0} values for isoflurane and sevoflurane (P = 0.97), the k _{e} _{0} values for desflurane were significantly higher compared with isoflurane (P = 0.02) and sevoflurane (P = 0.003) (P values are given for the statistical analysis of k _{e} _{0} values obtained by the P _{K} maximizing approach).

k _{e} _{0} values that were estimated according to our semiparametric approach were lower for desflurane and sevoflurane compared with k _{e} _{0} values derived from the classical sigmoidal or bisigmoidal fits published in previous studies for the same data (Table 1 ). Our semiparametrically derived k _{e} _{0} values, however, were closer to the k _{e} _{0} values estimated by the bisigmoidal model compared with the classical sigmoidal model (Table 1 ).

Table 1: ke0 Values Obtained by Different Estimation Approaches

The relationship between prediction probability and k _{e} _{0} and the area within the hysteresis loop and k _{e} _{0} for all patients is displayed in Figure 4 .

Figure 4: Figure 4.

Data of all measurements displaying the relationship between semiparametrically derived drug C _{eff} values and BIS are shown in Figure 5 . Visual inspection is suggestive of a seemingly bisigmoidal (nonmonotonous) relationship of the drug-response curve in all 3 investigated volatile anesthetics.

Figure 5: Figure 5.

DISCUSSION
In this study, we have implemented maximizing prediction probability P _{K} as an alternative semiparametric approach to estimate the plasma effect-site equilibration rate constant k _{e} _{0} ; whereas k _{e} _{0} values of isoflurane and sevoflurane are comparable, the k _{e} _{0} value of desflurane is significantly higher. Visual inspection of the relationship between semiparametrically derived C _{eff} values of all 3 anesthetic gases and BIS (drug-response curve) is suggestive of a biphasic (bisigmoidal) correlation.

Prediction Probability
Prediction probability was introduced by Smith et al.^{14} as a measure for evaluating and comparing the performance of anesthetic depth indicators. We favored prediction probability to calculate the optimal k _{e} _{0} value because P _{K} is independent of scale units and does not require knowledge of underlying distributions or efforts to linearize or to otherwise transform scales. Furthermore, P _{K} fully uses the available data without imposing additional arbitrary constraints. In the same way, we believe that P _{K} is an alternative to any parametric estimation of k _{e} _{0} , with the given advantage that the data-fitting process is not restricted or the data transformed merely to suit the chosen model. Moreover, it is difficult to predict in advance whether the chosen model will meet the criteria to best describe the drug-response curve. One of the merits of this approach is the ability to actually “see” the effect versus C _{eff} relationship without having to first prespecify a model. Nevertheless, our approach of estimating k _{e} _{0} has its limitation. With our data, assuming that the effect versus C _{eff} relationship is decreasing (or technically speaking, and nonincreasing), this approach worked well. However, in the rare case of a “U”-shaped relationship of effect versus C _{eff} , maximizing P _{K} would not lead to reasonable k _{e} _{0} values. The flaws that might arise with model fitting are discussed in detail by Smith et al.^{14}

Prediction probability was initially developed to evaluate 2 scales or databases that are polytomous (multivalued) ordinal with one scale (database) predicting the data of the other scale. In our study, we applied the calculation of P _{K} to the BIS scale and to volatile anesthetic drug concentrations. It could be argued that the scale of the latter goes beyond ordinal to interval or ratio. However, if an anesthetic depth indicator (volatile anesthetic drug concentration) that has an interval or ratio output scale is to be compared with an indicator that has an output scale that is no more than ordinal (BIS), the same measure of performance must be used for both, namely, a performance measure applicable to ordinal scales.^{14} We therefore decided to treat the scale of measured volatile anesthetic drug concentration as no more than ordinal.

Maximizing P _{K} will identify a k _{e} _{0} value that transforms the hysteresis of the drug-response curve (where one single BIS value corresponds to at least 2 different C _{eff} values) into a drug-response curve that is ideally monotonously decreasing and where a single BIS value will correspond to only one single C _{eff} value.

It has been identified that when investigating drug effects of volatile anesthetics or propofol over the entire range of the drug-response curve including burst suppression electroencephalogram, a different PKPD model has to be used (bisigmoidal model),^{6–9} compared with fitting data where no burst suppression occurs (simple sigmoidal model). Until now, data in which burst suppression occurred were either cut off^{4} or investigations were limited to increasing drug concentrations,^{15} avoiding a hysteresis loop and thereby not being able to calculate a k _{e} _{0} value. Others simply refrained from applying a model to their data^{16} and evaluated only the performance of their anesthetic depth indicators by calculating P _{K} . This new approach that we are introducing using prediction probability can be applied to collapsing the hysteresis loop in the presence and in the absence of burst suppression yielding k _{e} _{0} values without an underlying model. Interestingly, our k _{e} _{0} values were very close to the k _{e} _{0} values previously estimated for the same data in a bisigmoidal model by Kreuer et al.^{7–9} (Table 1 ). Because visual inspection of the dose-response curves (Figs. 1–3C and 5A–C ) of C _{eff} and BIS also displayed a marked plateau for most patients before burst suppression occurred, although k _{e} _{0} values were estimated using the P _{K} approach (i.e., without predefining a specific model), we believe that this plateau is not artificial.

Hysteresis Area Fit
We additionally calculated a k _{e} _{0} value by minimizing the area within the hysteresis loop. Fuseau and Sheiner^{11} were the first to introduce a PKPD model in which the PD modeling (estimation of k _{e} _{0} ) was performed without postulating any particular parametric form of the investigated drug-response curve. Their model was further elaborated to also describe the PK data without any underlying model.^{12,13} A program that estimates k _{e} _{0} values on the basis of their model was written by Steven Shafer and was called k _{e} _{0} _{obj} . This program was used in several publications to calculate k _{e} _{0} semiparametrically.^{19–21} Although the model of Fuseau and Sheiner was developed for data with only one hysteresis loop, we extended this model for our data with more than one subsequent hysteresis loop. Instead of calculating the squared difference between the only 2 drug effects (2 BIS values) for the same drug concentration at its effect site in the model of Fuseau and Sheiner, we now calculated the squared differences between each (i.e., more than 2) BIS value and the mean BIS value at the same calculated gas C _{eff} ; whereas our approach is a generalization of the k _{e} _{0} _{obj} approach, it might be questioned whether it is appropriate to continue calling it an area.

A limitation of our approach of measuring the area within the hysteresis loop will arise in the absence of sufficient data. Our approach of minimizing the area within the hysteresis loop is dependent on more than one (at least 2 or more) data point (BIS) for every C _{eff} value. Otherwise, extrapolated BIS values have to be used. This limitation, however, did not affect our calculations because every C _{eff} value corresponded to more than one BIS value.

Time to Peak Effect
Minto et al.^{22} published an alternative approach of calculating k _{e} _{0} values for IV anesthetics by measuring the time of the maximal drug effect (t _{peak} ) at the effect-site (brain) after a submaximal drug bolus. The advantage of the t _{peak} method is that t _{peak} is dose independent as long as the dose is submaximal, and it is not necessary to investigate increasing and decreasing loops of IV or volatile anesthetics. In addition, t _{peak} is a useful PD parameter allowing linking to separate PK and PD parameters from different studies.^{22,23} However, this approach is limited to IV anesthetics in which a predefined bolus can be given. Cortinez et al.^{24} demonstrated that k _{e} _{0} values calculated by the time to peak effect for rocuronium were similar to k _{e} _{0} values calculated by the traditional sigmoidal E_{max} model approach.

k _{e} _{0} Values of Volatile Anesthetics
A significantly higher k _{e} _{0} value for desflurane, compared with isoflurane and sevoflurane, has been reported by Rehberg et al.,^{2} who investigated the effect of these volatile anesthetics on spectral edge frequency. Desflurane is known for its preferential pharmacological properties because of the lower fat/blood solubility coefficient leading to faster emergence from anesthesia, especially after long surgical procedures.^{25} However, the equilibration time constant k _{e} _{0} values obtained here are more likely to depend on the blood/gas and the brain/blood solubility coefficients, both of which are also lower for desflurane compared with isoflurane and sevoflurane. An increase of the desflurane C _{et} will therefore lead to a faster increase of anesthetic depth as expressed by BIS compared with isoflurane and sevoflurane. The nearly identical k _{e} _{0} values for isoflurane and sevoflurane can also be explained by the blood/gas and brain/blood solubility coefficients. Although the blood/gas coefficient of sevoflurane is lower compared with isoflurane, the brain/blood coefficient is higher, resulting in similar equilibration time constants between the gas C _{et} and C _{eff} .

In conclusion, we introduced an alternative approach to estimate k _{e} _{0} values for isoflurane, sevoflurane, and desflurane without an underlying model by calculating the maximal possible prediction probability. This approach seems to be a promising method researchers could use on an exploratory basis. The k _{e} _{0} value for desflurane was significantly higher compared with sevoflurane and isoflurane. The obtained drug-response curve between BIS and C _{eff} is suggestive of a seemingly bisigmoidal shape with a plateau for all 3 investigated volatile anesthetics.

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