#### What We Already Know about This Topic

#### What This Article Tells Us That Is New

^{1–3}These variables were not related to predicted remifentanil effect-site concentrations. Conversely, the predicted remifentanil effect-site concentration combined with the bispectral index (BIS) was a significant predictor of a relevant hemodynamic response to tracheal intubation.

^{2}The prediction was not improved by adding the pulse wave response to electrical ulnar nerve stimulation.

^{2}Given the close correlation of the effect-site propofol concentration and the BIS,

^{4}we believe that the predicted effect-site propofol concentrations together with the predicted effect-site opioid concentrations and an appropriate interaction model provide sufficient information to predict the responsiveness of an anesthetized patient to noxious stimulation.

*et al.*

^{5}have described a response surface model for propofol and remifentanil in 2004. The model is the basis for a two-dimensional concentration domain interaction display in which predicted hypnotic and opioid concentrations are related to interaction isoboles such as the 50 and 90% tolerance of laryngoscopy isobole. To present the same information in a time-domain display, Schumacher

*et al.*

^{6}have defined the noxious stimulation response index (NSRI, see Methods section) based on the modified hierarchical interaction model by Bouillon.

^{7}Generally speaking, the NSRI is a univariate index calculated from the weighted propofol and remifentanil concentrations corrected for interaction and normalized to a range between 0 and 100, where 100 reflects 100% probability and values approaching 0 reflect close to 0% probability of responding to laryngoscopy.

_{K}) of the hypnotic state and the responsiveness to a noxious stimulus in anesthetized patients, using a previously published data set.

^{4}

#### Materials and Methods

##### Patients and Protocol of the Previous Study

*et al.*,

^{4}45 American Society of Anesthesiologists physical status 1 patients scheduled for ambulatory gynecologic surgery were enrolled and randomized to three treatment groups. Approval and written informed consent was granted for the original study by Institutional Ethics Committee of the Ghent University Hospital, Ghent, Belgium. The mean (SD) age in the three groups was 33 (5)–34 (4), and the mean weight and height were 63 (10)–66 (11) kg and 167 (6)–168 (6) cm, respectively. Propofol was infused in all groups according to a staircase protocol starting with effect-site target concentrations of 1.5 μg/ml in group 1 (no remifentanil) and 1.0 μg/ml in groups 2 and 3, in which remifentanil was added at effect-site target concentrations of 2.0 or 4.0 ng/ml, respectively. The infusion pumps were controlled by Rugloop II software (Demed, Temse, Belgium) using the pharmacokinetic parameter sets and effect-site equilibration constant (ke0) reported by Schnider

*et al.*

^{8,9}for propofol and Minto

*et al.*

^{10,11}for remifentanil.

##### The Hierarchical Propofol–Remifentanil Interaction Model

*et al.*

^{5}in 2004. The originally reported model was modified to increase parsimony while retaining its essential features (appendix).

^{7}On the basis of this modified model, the combination of predicted propofol and remifentanil concentrations can be expressed as probability to tolerate a certain reference stimulus, for example, tolerance of “shaking and shouting,” as indicator of deep hypnosis. The original and the modified model are illustrated in figure 1.

_{opioid}= effect-site opioid concentration, and Ce50

_{opioid}= effect-site opioid concentration associated with a 50% reduction of preopioid_intensity. Therefore, the Ce50

_{opioid}does not represent the opioid concentrations associated with half maximal effect on the probability of tolerating the stimulus but it is the ability to increase the effectiveness of the hypnotic by altering the respective Ce50 of the hypnotic (Ce50

_{hyp}, see Eq. 2). For a single stimulus, preopioid_intensity must be set to 1 to identify the C50 of the hypnotic (see Eq. 2). In this case, the postopioid_intensity is always a dimensionless number between 0 and 1, depending on the opioid concentration.

_{hyp}representing the hypnotic concentration that corresponds to a 50% probability of tolerance of a stimulus with preopioid_intensity in the absence of opioid.

*P*

_{no-response}= probability of nonresponse to a stimulus, Ce

_{hyp}= effect-site concentration of hypnotic, and φ = slope parameter.

_{opioid}) is identical for fractional suppression of stimuli of differing strength, only one parameter has to be added per additional stimulus, either “preopioid_intensity of stimulus

_{n}” (n = suffix for the nth stimulus) or, alternatively, the model can be parameterized with “C50

_{hyp n}” (n = suffix for the C50

_{hyp}related to the nth stimulus). If the second parameterization is chosen, the ratio of the respective C50s yields the relative strength of the stimuli. The second parameterization was chosen with “shake and shout” as reference stimulus with a preopioid_intensity of 1. The relative intensity of laryngoscopy then corresponds to the ratio of the propofol Ce50

_{TOSS}and the Ce50

_{TOL}(Eq. 3).

*R*

_{lar}= intensity ratio of laryngoscopy to the calibration stimulus shaking and shouting, Ce50

_{hypTOL}and Ce50

_{hypTOSS}= effect-site hypnotic concentrations associated with 50% probability of tolerating laryngoscopy and shake and shout, respectively. The parameter estimates (SE) for Ce50

_{hypTOL}and Ce50

_{hypTOSS}according to the modified model were 8.46 (1.98) and 2.99 (0.75) μg/ml

^{−1}, respectively. Intensity ratios compared with shake and shout can be computed for any other stimulus, provided the respective Ce50

_{hyp}is known.

##### Transformation of Probabilities of Tolerance into NSRI Units.

*P*

_{TOL}) can be computed according to equations 3 and 5.

*P*

_{TOL}of 0.5 corresponds to a NSRI of 50. The slope factor sl was calibrated to transform a

*P*

_{TOL}of 0.9 to an NSRI of 20, yielding sl = 2.18. The NSRI has the same underlying structural model but is not a direct mathematical transformation of

*P*

_{TOL}. The relationship between the NSRI and the probability of tolerance of laryngoscopy is depicted in figure 2.

##### Data Evaluation and Statistics

Equation 1 Image Tools |
Equation 4 Image Tools |
Equation 7 Image Tools |

^{4}were used to compute the related NSRI according to equations 1, 4, and 7.

*P*

_{TOL}was calculated according to equations 1 and 2. Primary independent variables (= predictors) were the NSRI and the predicted propofol and remifentanil effect-site concentrations. Primary dependent variables were the modified OAAS (full scale, table 1), the presence or absence of the eyelash reflex, and the presence or absence of a motor response to electrical tetanic stimulation of the forearm (dichotomous), BIS, and AAI values (continuous data). The BIS and the AAI were also used as predictors of OAAS and response to eyelash and tetanic stimulation. A similar analysis was performed for

*P*

_{TOL}.

_{K}s for all variables to be predicted were calculated. The prediction probability macro (PKMACRO; Excel spreadsheet) developed by Smith

*et al.*,

^{12}which was used for data evaluation in the previous article,

^{4}is designed for analysis of independent data. Because the data were not independent, we applied a bootstrap technique with 1,000 random samples of the 263 data points for each dependent variable for P

_{K}calculation using Matlab (The Mathworks Inc., Natick, MA). Each sample included one random data point per patient, that is, 44 data points. The P

_{K}value was then calculated for each sample using the PKMACRO functionality within Matlab. With this modification, the assumption of independence of the data was not violated. Because the P

_{K}values were not normally distributed, they are presented in box plots. To avoid assumptions on the distribution of the bootstrap samples, the 2.5–97.5 percentile range of the 1,000 P

_{K}was calculated to approximate the 95% CI of the resampled P

_{K}s. The differences between a median P

_{K}s of a given predictor (

*e.g.*, NSRI) and another predictor (

*e.g.*, BIS) in predicting the same variable (

*e.g.*, OAAS) were considered statistically significant if the median P

_{K}of the first was outside the 95% CI of the second predictor, corresponding to an [

*alpha*] of 0.05. Because statistical testing with calculation of

*P*values might be affected by the bootstrap distribution and the number of resamplings, we restrict our P

_{K}comparison to this rather crude and conservative method and do not present the calculated

*P*values.

#### Results

_{K}analysis are presented in figure 3.

_{K}values (95% CI) for prediction of OAAS by the effect-site propofol concentration, the BIS, the AAI, and the NSRI were 0.88 (0.81–0.93), 0.88 (0.82–0.93), 0.86 (0.80–0.92), and 0.77 (0.68–0.85), respectively.

_{K}values of NSRI, effect-site propofol concentration, BIS, and AAI for prediction of loss of response to tetanic stimulation were 0.87 (0.75–0.96), 0.68 (0.54–0.81), 0.73 (0.58–0.85), and 0.70(0.54–0.84), respectively, whereas the corresponding P

_{K}of the remifentanil effect-site concentration was 0.66 (0.50–0.80). The reason for the median propofol P

_{K}being slightly higher than the remifentanil P

_{K}might be explained by the study design including only two remifentanil concentrations.

_{K}s of the remifentanil effect-site concentration to predict OAAS, loss of eyelash reflex, BIS, and AAI were 0.43 (0.32–0.54), 0.41 (0.26–0.59), 0.36 (0.29–0.45), and 0.38 (0.30–0.46), respectively, which indicates a slight reverse prediction, most likely caused by study design (in groups with remifentanil, the propofol concentrations were lower).

^{4}The P

_{K}for

*P*

_{TOL}was similar to NSRI (fig. 3B) because the NSRI is the transformed and rescaled

*P*

_{TOL}. The NSRI (SE of the estimate) associated with a 50% probability of loss of response to tetanic stimulation was 61 (SE, 3.8) (fig. 4).

#### Discussion

_{K}of the NSRI and

*P*

_{TOL}to predict the probability of response to a 2-s, 50-mA, 100-Hz tetanic stimulus is higher compared with all other investigated predictors. As expected, the P

_{K}of the mainly hypnotic endpoints (loss of eyelash reflex and OAAS) was intermediate, and the P

_{K}of electroencephalogram-based predictors (BIS and AAI) was low.

_{K}s of 0.89–0.94 were reported.

^{13}The performance in predicting a response to noxious stimulation with these variables was 0.82–0.87 with pure propofol anesthesia

^{13}and 0.72–0.75 with coadministration of remifentanil.

^{4}These findings reflect the poor sensitivity of electroencephalogram-based measurements to the effect of opioids.

*versus*light stimulation and low

*versus*moderate remifentanil effect-site concentrations.

^{14}The skin conductance variation induced by several noxious and nonnoxious stimuli is a sensitive measure of stress

^{15,16}but discriminates only between the presence or absence of low remifentanil effect-site concentrations (2 ng/ml).

^{16}Whether it discriminates different opioid concentration levels or predicts the response to clinical stimuli is not known. Our investigations of the pulse plethysmography response to a 5-s 60-mA tetanic stimulus of the ulnar nerve as a surrogate variable to measure the analgesic state or the hemodynamic responsiveness of anesthetized patients were disappointing.

^{1,2}One reason was the large and probably random interindividual variation

^{1}of the signal (tetanic stimulation-induced variation of the pulse plethysmography trace). Therefore, we assume that baseline variability may reduce the predictive performance of any analgesic state index that is derived from physiologic signals related to the sympathoadrenergic stress response. Because the NSRI takes into account predicted effect-site drug concentrations and their interaction only, these drawbacks do not apply. It seems that the prediction error of effect-site drug concentrations, which is greater or equal to 20%,

^{8,10}does not degrade the prediction performance of the NSRI. Because the NSRI accounts for the interaction of hypnotic and analgesic, it must be superior to single drug concentrations for prediction of any endpoint for which hypnotic/analgesic interactions have been demonstrated, that is, responsiveness to noxious stimuli during anesthesia.

_{K}does not imply that a given NSRI value correctly predicts the response in an individual patient, but it means that the probability of response is highly correlated with the NSRI. The calibration of NSRI and

*P*

_{TOL}as anesthetic depth indicators was beyond the scope of this study and needs to be prospectively evaluated.

_{K}of a single predictor to one predicted variable) and the predition probability difference macro (PKDMACRO) (comparing the P

_{K}s of different predictors) were used for validation of anesthetic depth indicators in the past, the assumption on independence of the data has been neglected. The reason for this is inherent in the study design with repeated measurements taken at several drug concentrations in the same subject. The resampling technique applied in this study is an attempt to solve this problem of the statistical analysis. Currently, it is not clear how far the resampling method affects the boundaries of our parameter estimates and to what extend a sampling bias could have been introduced. To clarify this, a formal evaluation of this technique under a range of circumstances in which the “true” bounds are known would be required, which is well beyond the scope of this study. Therefore, we have presented the 2.5–97.5 percentile ranges of the different P

_{K}s that approximate the 95% CIs and did not calculate any

*P*values. To reject the null hypothesis that two P

_{K}s are similar, the median P

_{K}of one predictor had to be outside the 95% CI of P

_{K}s of the other. Therefore, only large differences in the median P

_{K}s were accepted as significant, which are unlikely to be substantially affected by a potential sampling bias; for example, the difference between the P

_{K}s of NSRI and

*P*

_{TOL}and the P

_{K}s of all other predictors to predict response to a noxious stimulus (fig. 3). It is, therefore, unlikely that the main message of this study is affected by this yet unsolved statistical problem.

*post hoc*validation. Second, the selected propofol and remifentanil concentrations are not independent of each other. Third, the applied 2-s tetanic stimulus is substantially weaker than strong surgical stimuli such as skin incision,

^{17}which is illustrated by the high NSRI

_{50}for loss of response to tetanic stimulation. Fourth, the data used for this validation were recorded only in a female patient population. Therefore, this study only attests to the usefulness of the NSRI as predictor of the response to medium-intensity stimuli during coadministration of propofol and remifentanil. Future studies have to validate the NSRI in the clinical setting for both total intravenous and balanced (volatile plus opioid) anesthesia and in both sexes.

##### Appendix: Modification of the Hierarchical Interaction Model

^{7}are described. The original model

^{5}was modified to avoid overparameterization. The resulting modified model was found to be mathematically equivalent to a reduced Greco model, implying strong synergism. Its C50 for the opioid can be interpreted in analogy to a C50 for reduction of the minimal alveolar concentration of a volatile anesthetic.

##### Model Modifications

_{hypnotic}, the slope factor, and the concentration of the hypnotic. As is evident from equation A,

^{5}only the product of Ce50

_{hypnotic}and postopioid_intensity, but not its individual components, is identifiable (Eq. A).

*P*

_{responsiveness}= probability that the patient responds to the incoming stimulus, Ce

_{hypnotic}= effect-site hypnotic drug concentration, Ce50

_{hypnotic}= effect-site hypnotic drug concentration associated with a 50% probability of nonresponsiveness, and φ = slope parameter.

_{opioid}= effect-site opioid concentration, Ce50

_{opioid}= the common opioid concentration reducing the intensity of an incoming stimulus by 50%, and gamma (γ) = slope parameter.

*per se*, if only one stimulus is investigated and preopioid_intensity must be fixed to 1 to obtain the C50 of the hypnotic. For n stimulus strengths, the number of parameters describing stimulus strength equals n − 1. These parameters describe relative strength of stimulus compared with the reference stimulus with the intensity of 1. Alternatively, the model can be parameterized in terms of one C50 for the hypnotic per stimulus applied.

_{opioid}= 0, and

*P*

_{nonresponsiveness}= 0.5. The Ce

_{hyp}equals the C50 of the hypnotic.

_{hyp}for

*P*

_{nonresponsiveness}= 0.5 by 50%. The Ce

_{opioid}that lowers the preopioid_intensity from 1 to a postopioid_intensity of 0.5 equals the Ce50

_{opioid}(Eqs. B and C).

_{hyp}for

*P*

_{nonresponsiveness}= 0.5 by 50%, for another stimulus with preopioid_intensity of 2. According to equation B (original model), the Ce

_{opioid}= 6 × Ce50

_{opioid}, whereas according to equation C (modified model), the Ce

_{opioid}= 3 × Ce50

_{opioid}.

^{5}: Ce50

_{propofol, TOSS}= 2.99 (0.75) μg/ml, Ce50

_{propofol, TOL}= 8.46 (1.98) μg/ml, and Ce50

_{remifentanil, TOSS}= 1.16 (0.48) ng/ml, whereas the Ce50

_{remifentanil, TOL}is implicitly modeled and not estimated from the data. The non-linear mixed effects modeling objective function was 80.2.

##### Discussion

^{7}the non-linear mixed effects modeling objective function value of the modified hierarchical model and the Greco model was equal, whereas it was 69 in the original model.

^{5}However, the small SEs of the parameter estimates in the original model are indicators of overparameterization.

^{18,19}and clinical experience. In contrast, the C

_{50, propofol}for tolerance of shaking and shouting (corresponding to the C

_{50, propofol}for loss of consciousness) estimated with the modified model was 2.99 μg/ml, which is well within the range of published data.

^{13,19,20}

^{7}simplifying comparisons with existing studies. The C

_{50, opioid}in our model equals the reciprocal ε′ of that model according to equation D.

_{50, opioid}= opioid concentration associated with half maximal attenuation of a stimulus in our model and ε′ = the modified Greco interaction parameter for constellations in which the opioid effect in the absence of hypnotic is too weak to be identified but profoundly changes the potency of a coadministered hypnotic. This situation was encountered in the interaction study by Mertens

*et al.*

^{19}The proof of interconvertability of the two models has been described elsewhere.

^{7}Interestingly, the Ce

_{50, remifentanil}estimated with the simplified Greco model from the propofol–remifentanil interaction data is 1.39 and 1.45 ng/ml for return of consciousness and for tolerating laryngoscopy, respectively, which is almost identical despite completely different stimulation strength and approximates the C

_{50, remifentanil}estimated with our modified hierarchical model (1.16 ng/ml).

*P*

_{nonresponse}= probability of tolerance of a given stimulus, Ce

_{hypnotic}= effect-site hypnotic drug concentration, Ce50

_{hypnotic}= effect-site hypnotic drug concentration associated with a 50% probability of nonresponse, preopioid_intensity = intensity of the stimulus without opioid attenuation, Ce50

_{opioid}= effect-site opioid drug concentration reducing the preopioid_intensity by 50%, and Ce

_{opioid}= effect-site opioid concentration.