The interaction of sevoflurane and opioids can be described by response surface modeling using the hierarchical model. We expanded this for combined administration of sevoflurane, opioids, and 66 vol.% nitrous oxide (N2O), using historical data on the motor and hemodynamic responsiveness to incision, the minimal alveolar concentration, and minimal alveolar concentration to block autonomic reflexes to nociceptive stimuli, respectively.
Four potential actions of 66 vol.% N2O were postulated: (1) N2O is equivalent to A ng/ml of fentanyl (additive); (2) N2O reduces C50 of fentanyl by factor B; (3) N2O is equivalent to X vol.% of sevoflurane (additive); (4) N2O reduces C50 of sevoflurane by factor Y. These four actions, and all combinations, were fitted on the data using NONMEM (version VI, Icon Development Solutions, Ellicott City, MD), assuming identical interaction parameters (A, B, X, Y) for movement and sympathetic responses.
Sixty-six volume percentage nitrous oxide evokes an additive effect corresponding to 0.27 ng/ml fentanyl (A) with an additive effect corresponding to 0.54 vol.% sevoflurane (X). Parameters B and Y did not improve the fit.
The effect of nitrous oxide can be incorporated into the hierarchical interaction model with a simple extension. The model can be used to predict the probability of movement and sympathetic responses during sevoflurane anesthesia taking into account interactions with opioids and 66 vol.% N2O.
Opioids reduce potent volatile anesthetic minimal alveolarconcentration and minimal alveolar concentration-block autonomic reflexes
Response surface models describe the range of interactions between opioids and potent volatile anesthetics
These are mechanism-based models, the parameters of which represent clinically meaningful pharmacologic endpoints
The interactions of sevoflurane, fentanyl, and nitrous oxide for preventing movement and hemodynamic response to surgical incision were described with a response surface model of data from a previous study by using nitrous oxide as a model covariate
Nitrous oxide has an additive effect corresponding to 0.54 vol% sevoflurane and 0.27 ng/ml fentanyl
KATOH et al.1 described the reduction of minimal alveolar concentration (MAC) and MAC to block autonomic reflexes to nociceptive stimuli (MAC-BAR) of sevoflurane by fentanyl. MAC and MAC-BAR are the minimal alveolar anesthetic concentrations that evoke respectively “immobility” or “hemodynamic stability” after surgical incision in 50% of subjects. Nitrous oxide (N2O) is combined with inhaled anesthetics in anesthesia for its potentiating effects. For this reason, Katoh et al.1 also tested the MAC reduction evoked by nitrous oxide.
The classical MAC reduction studies use a logistic regression approach.2 The logistic regression approach as applied by Katoh et al. does not provide unique parameters describing the influence of nitrous oxide, independent of the type of response (movement or hemodynamic), thus not allowing to apply the results to other responses. Because the article of Katoh et al. does not provide the model parameters of the logistic regression analysis, it does not allow a flexible calculation of the probability of response at any drug level.
Currently, nonlinear mixed effect modeling is considered to be the definitive standard for modeling three-dimensional response surfaces on interaction datasets.3 One of the advantages of the response surface approach is that a single formula describes the full range of effect for any combination of drugs.2,3 Recently, Heyse et al.4 reviewed several equations that have been proposed as structural model for response surface modeling and found that the hierarchical model (with fixed C50 of opioids) performed best to predict the response to multiple noxious and non-noxious stimuli during a sevoflurane–remifentanil anesthesia. In contrast to the logistic regression approach, the hierarchical model is a mechanism-based model in which every parameter represents a pharmacologic endpoint, which has clinical meaning (e.g., potency of the opioid, potency of the hypnotic, steepness of the dose–response relationship). The parameters of the logistic regression are mathematical constants that have no physiological or pharmacologic correlate.
A response surface model that describes the effects of nitrous oxide has not yet been described. We hypothesized that the hierarchical model could be expanded further to allow flexible inclusion of the effects of 66 vol.% N2O in the interaction between opioids and sevoflurane.
The methodology for response surface modeling demands high numbers of observations in volunteers or patients, considerable costs, and manpower. Fortunately, we had the opportunity to reuse the historical dataset of Katoh et al.1 Contemporary analyzing methods can extract more information from this data compared with the classical MAC reduction approach. The data from Katoh et al.1 contain large numbers of observations with sufficient numbers of responders and nonresponders and a wide range of drug doses.
The purpose of the current study was to develop a response surface model for the combination of sevoflurane, nitrous oxide, and fentanyl using this dataset.
Materials and Methods
The study by Katoh et al.1 was performed after approval from the District Ethics Committee of Hamamatsu University Hospital (Hamamatsu, Japan), and individual informed consent had been obtained from all patients. We reanalyzed the raw data from Katoh et al.1 on the sevoflurane MAC and MAC-BAR reduction by fentanyl in the presence (n = 86 patients) or absence (n = 96 patients) of 66 vol.% N2O. Patients were randomly assigned to receive nitrous oxide or not. Patients of both sexes, between 20 and 50 years, were classified as American Society of Anesthesiologists physical status I and were scheduled for elective surgery of the abdomen, extremities, or body surface.
The anesthetic management and data collection were described in detail in the original article.1 In short, all patients fasted for at least 8 h before anesthesia and received no premedication. A target-controlled infusion of fentanyl (using the pharmacokinetic model published by Shafer et al.5 ) was initiated in all patients, according to a randomization list. Fentanyl plasma concentration targets ranged between 0 and 8 ng/ml. Inhaled induction was performed with sevoflurane 8% in 100% oxygen (control group) or sevoflurane 8% in oxygen with 66 vol.% N2O group. After loss of consciousness and precurarization with 0.02 mg/kg vecuronium, 1.5 mg/kg succinylcholine was administered and tracheal intubation was performed to secure the airway for the remainder of the study. Ventilation was adjusted to maintain normocapnia. An end-tidal concentration of sevoflurane was targeted according to a randomization list, between 0 and 4.5 vol.%. N2O was administered at an end-tidal concentration of 66 vol.%. End-tidal concentrations of sevoflurane, nitrous oxide, and carbon dioxide were measured continuously using an infrared multigas anesthetic analyzer (Capnomac Ultima, Datex, Helsinki, Finland), which was calibrated before anesthesia for each patient using a standard gas mixture. Gas samples were collected via a catheter placed at the tracheal end of the endotracheal tube.
After intubation and setting the drug targets for maintenance, a 20-min delay was respected to allow equilibration between the effect-site concentration and the end-tidal vapor pressure of sevoflurane and the plasma concentration of fentanyl, respectively. As such, all observations of the responses were performed during a pharmacologic pseudo steady state. This was confirmed by analysis of venous blood samples taken 5 min before and within 30 s after incision. Patients with a difference in measured plasma fentanyl concentration of more than 35% between samples were excluded from further analysis. The delay of 20 min also allowed recovery from the neuromuscular blocking agents, which was confirmed by monitoring the recovery.
Twenty minutes after intubation, the surgeon made an abdominal incision and somatic or hemodynamic responses were observed. Positive response was defined as a somatic or hemodynamic change within 60 s after incision. Coughing, chewing, or swallowing was not considered a purposeful movement. Hemodynamic response was defined as an increase in heart rate or systolic blood pressure of more than 15% over the preincision baseline value.
The probability of tolerance, P, to a certain stimulus (e.g., incision) can be expressed as
where U represents the normalized potency of a single drug or a combination of drugs. U is a function of the drug effect-site concentrations and model parameters, reflecting the relative drug concentration, and γ is the slope parameter reflecting the steepness of the concentration–effect relationship.4,6
The hierarchical model (equation 2) can be incorporated in equation 1 for calculating all levels of probability of response.6
UO and USEVO are the normalized opioid and sevoflurane effect-site concentrations, CO is the effect-site concentration of the opioid, CSEVO is the sevoflurane effect-site concentration, C50O is the effect-site concentration of the opioid that decreases C50SEVO by 50%, C50SEVO is the sevoflurane effect-site concentration that results in P = 0.5 in the absence of opioid, and γo is the slope parameter reflecting the steepness of the concentration-effect relationship of the opioid.
The hierarchical model resulted in the best fit during sevoflurane–remifentanil interactions.4 We first used the same structural model to fit the interaction in the control group (sevoflurane, fentanyl, without N2O) of the study by Katoh et al.1
The data of the study by Katoh et al. did not allow a full response surface modeling of the three drugs (sevoflurane, fentanyl, N2O) because N2O was applied at a single concentration level (66 vol.%) only. Therefore, the influence of N2O was treated as a covariate in the hierarchical model of sevoflurane and opioids. For this purpose, we postulated that N2O interacts with sevoflurane and/or fentanyl as expressed below.
a. N2O is equivalent to a concentration of A ng/ml of fentanyl (additive interaction)
b. N2O reduces the C50 of fentanyl by a factor B (potentiation, nonadditive interaction)
a. N2O is equivalent to a concentration of X vol.% of sevoflurane (additive interaction)
b. N2O reduces the C50 of sevoflurane by a factor Y (potentiation, nonadditive interaction)
To incorporate these assumptions into the hierarchical model, parameter A, B, X, and Y were added to equations 3 and 4, as shown in equations 5 and 6:
Using equations 5 and 6 as a structural model, each of the postulated actions of N2O and any of the combinations of A, B, X, and Y were fitted simultaneously to all data for model parameter estimation.
The model parameters were estimated using NONMEM VI version 2.0 (Icon Development Solutions, Ellicott City, MD), using FOCE LAPLACE and LIKELIHOOD options. Platform was Windows XP (Microsoft, Redmond, WA) and compiler was G95. For all parameters, the interindividual variability was either assumed to be absent or to have a log-normal distribution. Model building was performed starting with the simplest form of each model and expanding the model with interaction terms and interindividual variability, until the decrease of the objective function value (OFV) was no longer statistically significant using the chi-square test. For each added parameter, a difference of 3.84 units in OFV was considered statistical significant at P < 0.05.
The NONMEM analysis was performed with various values for initial estimates and boundary values. The results were accepted as valid only if both minimization and covariance step were successful, unless stated otherwise.
To evaluate the uncertainty in the parameters of the final model, nonparametric 95% CIs of the model parameters were obtained from a bootstrap analysis, based on 500 sets of 177 patients each, randomly selected from the available 177 patients, using a custom program written in C. Results were analyzed in Microsoft Excel version 2010. In addition, log-likelihood profiles were calculated for each population parameter, and the nonparametric 95% CIs were obtained assuming a chi-square distribution with one degree of freedom and P = 0.025, resulting in a critical difference of 5.02 in the OFV.
Several performance measures were calculated from the prediction errors, i.e., the difference between the predicted probability of tolerance minus the observed response (0 for responsive, 1 for tolerant): Mean Prediction Error, Mean Absolute Prediction Error, and Root Mean Squared Error. In addition, the Prediction Error Score was calculated as the percentage of mispredicted responses, i.e., if tolerant, the probability of tolerance was less than 0.5, or if responsive, the probability of tolerance was more than 0.5.
All model parameters are reported as typical values with standard errors (expressed in % of the typical value) within parentheses, and clinical data are given as mean and SD.
Of the 96 patients not receiving nitrous oxide, two patients were excluded because of hypotension and the administration of ephedrine before the incision, and three patients were excluded because they were judged as being awake just before skin incision. Therefore, our analysis was performed on the remaining 177 patients of which 86 did receive nitrous oxide and 91 did not. Demographic data can be found in the original article.1 The measured drug concentration of fentanyl before incision ranged between 0 and 10 ng/ml. The measured end-tidal partial pressure of sevoflurane at the time of observation ranged between 0 and 4 vol.%.
To identify the model for the interaction of sevoflurane and fentanyl, the data from patients not receiving nitrous oxide (control group) were analyzed using the hierarchical model with separate, unconstrained values for C50O for movement and sympathetic responses.4,6 Using the fixed C50O and the scaled C50O approach resulted in an increase of OFV by 29 and 37, respectively. Therefore, the model with separate values for C50O was considered the appropriate model for the interaction of sevoflurane and fentanyl for movement and sympathetic responses. Parameter estimations are listed in table 1.
For the simultaneous analysis of all data, the hierarchical model with different values of C50O and C50SEVO for movement and sympathetic responses was expanded by one of the factors X, Y, A, or B, resulting in a decrease of the OFV by 123, 54, 35, and 57, respectively, showing that factor X (called the sevoflurane-additivity model [X]) is the most influential factor in the model. Using different values of X for movement and for sympathetic responses in the sevoflurane-additivity model (X) did not result in a significant difference of OFV (–0.6).
In a second step, we expanded the sevoflurane-additivity model (X) further by including factors A or B in the equation. Inclusion of factor A in combination with factor X (sevoflurane/opioid-additivity model [XA]) resulted in a significant decrease of the OFV by 8.2 units. Using separate values of X and A for movement and sympathetic responses did not result in a significant drop of the OFV. Combining factor B with factor X did not result in a significant decrease of OFV (–1.5).
In a third step, we combined factor X with factor Y (sevoflurane-additivity/potentiation model [XY]), which resulted in a decrease of the OFV of –9.5 units compared with the sevoflurane-additivity model (X). This model mathematically combines an additive (X) and nonadditive (Y) interaction of nitrous oxidewith sevoflurane. This model does not support an independent interaction of nitrous oxide with fentanyl. The difference between the OFV of the sevoflurane/opioid-additivity (XA) and sevoflurane-additivity/potentiation (XY) models is not significant.
Finally, the combinations of three or four factors, including X (XYA, XYB, XAB, XYAB), did not result in lower OFV. In addition, the inclusion of intraindividual variability in any parameter did not result in a significant reduction of OFV for any model.
The final results of the parameter estimations for three adequately fitting models are listed in table 1. In all models, the values for X, Y, and A are identical for movement and sympathetic responses. Figures 1 and 2 show the observed somatic (fig. 1) and hemodynamic (fig. 2) responses to incision in the control group (fig. 2A) and in the nitrous oxide group (fig. 2B), and MAC and MAC95 (fig. 1) and MAC-BAR and MAC-BAR95 (fig. 2) for the sevoflurane/opioid-additivity model (XA). Figure 3 shows the three-dimensional response surface as predicted by this model for movement (fig. 3, A and C) and hemodynamic responses (fig. 3, B and D) after incision in the control group (fig. 3, A and B) and in the nitrous oxide group (fig. 3, C and D). MAC, MAC-BAR, MAC95, and MAC-BAR95 estimations for sevoflurane in the absence and presence of nitrous oxide and fentanyl for the sevoflurane/opioid-additivity model (XA) are also listed in table 2. The values are within comparable range of the results of Katoh et al.1 (table 3). The differences between the sevoflurane/opioid-additivity (XA) and sevoflurane-additivity/potentiation (XY) models with respect to the measures of “goodness-of-prediction” were rather small (table 1).
The results of the final sevoflurane/opioid-additivity model (XA) were checked by performing a bootstrap analysis. The median parameters of the bootstrap analysis were in reasonable agreement with the NONMEM results. The 95% CI for the parameters X (0.54 vol.% sevoflurane) and A (0.27 ng/ml fentanyl) were 0.41–0.67 vol.% and 0.07–0.62 ng/ml, respectively. The 95% CI obtained from the log-likelihood profiles was 0.42–0.68 vol.% for X and 0.05–0.58 ng/ml for A.
In this reanalysis of previously published data, we present new information on the interaction between opioids, sevoflurane, and nitrous oxide compared with the original work.1 We found that a simple extension of the hierarchical model (equations 2–6) integrates the additional effect of 66 vol.% N2O to the interaction of fentanyl and sevoflurane using a response surface modeling approach.
Although the opioid used was fentanyl, we often do refer to it using the more general term “opioid.” This is a deliberate choice because the model may be applicable for other opioids also, provided that equipotent doses of the other opioids are given.7
The major difference between our analysis (using the hierarchical model) and the original analysis by Katoh et al. (using logistic regression) is that our analysis was performed on all data simultaneously, i.e., a single analysis with parameters common for the groups (movement and sympathetic response, without and with nitrous oxide) where appropriate, rather than on all four groups of data separately. In principle, both approaches are equally valid, and we confirmed that the MAC and MAC-BAR values for both approaches are comparable. However, the separate logistic analysis does not provide unique parameters describing the influence of nitrous oxide, in contrast to the simultaneous response surface approach providing these parameters (A, B, X, Y). The unique parameters allow to apply the results to other responses to noxious stimuli, e.g., tolerance to laryngoscopy, taking into account the difference in intensity of the noxious stimuli. Once the interaction of sevoflurane and opioids has been adequately described by the hierarchical model, the interaction with nitrous oxide can be modeled by adding parameters A, B, X, and/or Y of the nitrous oxide interaction according to equations 5 and 6.
We used the hierarchical model in our analysis, because this model was found to describe the interaction between sevoflurane and remifentanil best.4 For comparison, we also tested the Greco model, reduced Greco model, and the logistic model using the simultaneous approach. For each of the tested models, and for each model for the nitrous oxide effect (X, XY, XA), the OFV values and model parameters were close to that of the hierarchical model. From a clinical point of view, the hierarchical model reflects the physiologic sequence of the opioid and hypnotic drug effect: Opioids reduce the afferent nociceptive transmission to medulla, thalamus, and cortex and thus the arousal response to the nociceptive stimulus and thus the hypnotic drug concentration to keep the patient asleep or unresponsive.
The process of minimizing the OFV is presented in separate stages because each stage represents a logical sequence where we tested a hypothesis on the interaction mechanisms that may explain the observed data.
In the first stage of modeling, the sevoflurane-additivity model (X) fitted the data best, supporting the hypothesis that an additive interaction of nitrous oxide with sevoflurane contributes strongly to the leftward shift of the isoboles. Model Y (potentiation of sevoflurane) was by far inferior to the sevoflurane-additivity model (X), which suggests that a reduction of C50SEVO by nitrous oxide is less likely than an additive effect. This finding is in concordance with the MAC additivity principle.
The second stage explored whether the fit on the data can be improved by modeling the analgesic potency of nitrous oxide. The flexibility of the structural model increases by assuming an additional interaction between nitrous oxide and fentanyl (in an additive (A) or nonadditive (B) way, respectively). We found that 66 vol.% N2O corresponds simultaneously to 0.27 ng/ml of fentanyl (A) and 0.54 vol.% of sevoflurane (X).
The parameter estimates of sevoflurane MAC and MAC-BAR presented in table 2 are consistent with the results of the original article for the sevoflurane/opioid-additivity model (table 3). The reduction of sevoflurane MAC by 66 vol.% N2O in the absence of fentanyl is 54 and 57% in the current and original study, respectively. The C50 of fentanyl is 2.07 ng/ml in the current study and 1.8 ng/ml in the original study. The small differences are presumably related to the differences in method of analysis. Stevens et al.8 reported a reduction of the isoflurane MAC in the presence of 70% N2O by 68%. Fragen and Dunn9 reported a reduction of the sevoflurane MAC in the presence of 65% N2O by 50%, whereas Rampil et al.10 reported a 50% reduction for desflurane in the presence of 60% N2O.
In the final stage of modeling, we tested several—more complex—combinations of interactions. Only the combination of parameters X and Y resulted in a comparably low OFV (but without significant difference with the sevoflurane/opioid-additivity model [XA]). The sevoflurane-additivity/potentiation model (XY) is less consistent with current pharmacologic concepts because it describes the interaction between nitrous oxide and sevoflurane as a combined additive and nonadditive interaction with sevoflurane. In addition, no independent analgesic effect of nitrous oxide is included in this model. In addition, the C50 values of the sevoflurane/opioid-additivity model (XA) are closer to the C50 values in conditions without nitrous oxide (control group) compared with that of the sevoflurane-additivity/potentiation (XY) model (table 1). For these reasons, we consider the sevoflurane/opioid-additivity model (XA) to be a better suited structural model to estimate responsiveness during opioid, sevoflurane, and nitrous oxide interaction.
A potential limitation of this study is based on the fact that data were derived from only one particular population (Asian). Even though MAC has limited variability within a population, interpopulation (ethnic) differences exist.11 Therefore, the validity of the proposed response surface model must be tested prospectively in a population with wider ethnic variation.
Our response surface model only applies to the clinical endpoint of a somatic or hemodynamic response to a noxious stimulus, MAC and MAC-BAR, respectively. For the response to stimuli that test the hypnotic state of the patient, such as “shake and shout,” “MACawake,” “name calling,” “the observer’s assessment of alertness and sedation score,” and the “isolated forearm technique,” we were not able to model a response surface because these endpoints were not included in our dataset. Moreover, the available literature suggests that the nature of the interaction for hypnotic endpoints of anesthesia might be different compared with somatic responsiveness to noxious stimuli. Data from Katoh et al. suggest the interaction between sevoflurane and nitrous oxide for MACawake to be infra-additive and so is the effect on learning.12–14 This has also been confirmed with thiopental and ethanol.15,16 This differential characteristic of the interaction for responses on noxious and non-noxious stimuli is probably evoked through differential balance of the N-methyl-d-aspartate and γ-aminobutyric acid receptor type A receptor inhibition in the neural networks, especially in the unstimulated patient.12 In addition, several electroencephalographic-derived measures of cerebral hypnotic drug effect do not detect the addition of nitrous oxide both during intravenous and inhalational anesthesia in unstimulated patients.17 In conclusion, our final model currently is only applicable for responses to a noxious stimulus and does not allow extrapolation to hypnotic endpoints of anesthesia.
Another important limitation of the dataset used in this analysis is that only one concentration of nitrous oxide has been tested and that data for other concentrations of nitrous oxide are currently lacking. However, if we assume that lower concentrations of nitrous oxide (< 66 vol.%) reduce the two interaction parameters X and A in a proportional way, the proposed the sevoflurane/opioid-additivity model (XA) can be generalized for different concentrations of nitrous oxide (CN2O) according to equations 7 and 8:
Because equations 7 and 8 cannot be verified from our data, but are based on one assumption of linearity in the interaction between 0 and 66 vol.% N2O, the clinical performance of this proposed solution needs to be prospectively validated.
The final response surface equation presented in this study can easily be applied in advisory systems that provide bedside pharmacokinetic–dynamic information based on the demographics of the patient and the administered drug doses. Such devices are currently being commercialized for clinical practice; however, prospective validation is still mandatory to evaluate the population-based reference as a tool for the clinician to target desired levels of responsiveness in an individual case.11
The influence of 66 vol.% N2O was best described by a combination of an additive effect corresponding to 0.27 ng/ml fentanyl (A) and an additive effect corresponding to 0.54 vol.% sevoflurane (X). With a simple extension, the effect of 66 vol.% N2O can be incorporated in the hierarchical interaction model of sevoflurane and opioids and allows to model the triple interaction in a response surface.