#### What We Already Know about This Topic

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

^{1},

^{2}Using the mathematically defined response surface, the corresponding drug effect for any two or more drug concentrations of the interacting drugs can be predicted.

^{1},

^{3}

*et al.*further developed a previously published response surface model by Greco

*et al.*to describe the interaction between dexmedetomidine and midazolam in rats.

^{4},

^{5}The Greco model can be considered as the basic approach to describe quantal response surface models, since it is the original and most simple model for drug interaction. As this model assumes identical slope factors and identical maximal effects for the single concentration effect courses of the interacting drugs, Minto

*et al.*extended the Greco model to make the response surface modeling more flexible. They defined a variable (originally called θ) as the proportion of one drug in the combination of two potentially interacting drugs.

^{1}More recently, Bouillon

*et al.*developed a novel mechanistic approach to the interaction between opioids and hypnotics. Using the knowledge that analgesia represents a drug action on ascending neuropathways and that hypnosis is a cortical response that balances the ascending noxious stimulus against drug-induced cortical suppression, they quantified opioid-hypnotic drug interaction in a sequential (also called hierarchical) model.

^{6}As some of the authors of the original paper thought that the initial form of their hierarchical model was overparameterized,

^{2}they designed a less complex form of the model, hereby called the Fixed C50

_{O}Hierarchical model,

^{7}which is now applied in one of the commercially available drug interaction displays (Smart Pilot View, Dräger, Lübeck, Germany). In contrast to the Greco and Minto models, the Scaled C50

_{O}and Fixed C50

_{O}Hierarchical models approach comes more close to the clinical pharmacological and physiologic reality.

*et al.*constructed a response surface for each pharmacodynamic response using a Logit model approach and found synergy between sevoflurane and remifentanil for all responses.

^{8}As this study suffered from nonsteady-state conditions at the moment of measurements, the authors reevaluated their data using effect-site sevoflurane concentrations and a Greco model instead of a Logit approach.

^{9}Accounting for the lag time between sevoflurane effect-site concentration and end-tidal concentration improved the predictions of responsiveness during anesthesia but had no effect on predicting a response to a noxious stimulus in the recovery room. They concluded that models may be useful in predicting events of clinical interest but large-scale evaluations with numerous patients are needed to better characterize model performance. Also, they did not test if other response surface models would describe the data more accurately.

#### Materials and Methods

^{10}

##### Subjects

##### Study Design

_{2}), M-entropy using a Datex S/5 Anesthesia Monitor (GE Healthcare, Helsinki, Finland) and bispectral index using a Aspect A-2000 monitor (Covidien, Norwood, MA) were connected, and a large forearm vein was cannulated. Thereafter, the patients were preoxygenated with 6 l/min O

_{2}at a F

_{I}= 1.0 for 5 min, using a tight-fitting facemask, which also served to sample exhaled air for end-tidal carbon dioxide measurement. All medical devices are approved for the purposes applied in the study. All drugs and the way of administration, either alone or in combination, are approved for clinical use under the studied conditions. No “off label” drug applications were used (European situation). Vital signs as well as end-tidal sevoflurane concentrations, respiratory data (tidal volume, minute volume, end-tidal carbon dioxide), and infusion related data (predicted concentrations, amounts infused) were continuously recorded on a computer hard disk using RUGLOOP II data recording software (Demed, Temse, Belgium).

##### Drug Administration

##### Technical Aspects.

*et al.*

^{11},

^{12}Remifentanil infusion was administered by using an Alaris Asena pump (Carefusion, Basingstoke, United Kingdom). RUGLOOP II TCI driver (Demed) controlled the pump at infusion rates between 0 and 1,200 ml/h

*via*an RS-232 interface. Sevoflurane was administered in 50% O

_{2}and 50% air by using a standard out of circle vaporizer and a standard breathing circuit of an ADU anesthesia workstation (Datex/Ohmeda, GE Healthcare). In all steps, the sevoflurane vaporizer was set to maximum until 80% of the target concentration was reached, then it was turned down to the target setting. A fresh gas flow above minute ventilation was used throughout the study.

##### Dosing Regimen.

*et al.*

^{3}The choice of the sevoflurane/remifentanil concentrations pairs were based on the sevoflurane Ce50 (Ce50

_{sevo}) to suppress the response to skin incision (= minimal anesthetic concentration, MAC) of 1.85%

^{13}and a remifentanil Ce50 (= remifentanil concentration reducing the MAC

_{SEVO}by 50%) of 1.5 ng/ml

^{14}:

_{SEVO}, Ce

_{REMI}is the effect-site concentration of remifentanil, Ce

_{SEVO}is the effect-site concentration of sevoflurane, and Ce

_{SEVO(norm)}is the effect-site concentration of sevoflurane normalized to MAC, taking into account the opioid effect.

_{SEVO}was set at 3.5 vol.%, and maximum Ce

_{REMI}was set at 12 ng/ml. A maximum of four steps was used to explore a single slice of the response surface. No other drugs were given, except for a possible 0.1 mg bolus of phenylephrine if mean arterial blood pressure dropped below 50 mmHg.

##### Assessment of Clinical Response

^{15}(an OAA/S score less than 2 was considered as tolerant); (2) a tetanic stimulus of the ulnar nerve for 5 s by using the standard neurostimulator used in the clinical setting to test the level of muscle relaxation (100 Hz, 60 mA, Tristim NS3A peripheral nerve stimulator; Life Tech, Houston, TX); (3) insertion of a laryngeal mask airway (LMA size 3 for women and 4 for men, LMA Unique®, The Surgical Company, Amersfoort, The Netherlands); (4) laryngoscopy aiming at full visualization of the vocal chords by using a size-3 curved Macintosh-type blade (HEINE Optotechnik GmbH & Co KG, Herrsching, Germany). Verbal response, eye opening, grimacing, coughing, withdrawal, or any other purposeful or nonpurposeful movement, including jaw clenching and bucking after a stimulus, were defined as a response. Absence of a response implied tolerance of the stimulus and was labeled 0, and presence of a response implied no tolerance of the stimulus and was labeled 1 in the case report form. All assessments were performed by one investigator to minimize interobserver variability. If there was no response to the first stimulus, the next stimulus was applied 1 min after the response assessment of the first. The assessment at each drug concentration level was stopped as soon as a response was observed or the patient tolerated laryngoscopy. If there was no response to laryngoscopy at the highest predefined drug combination, data acquisition was stopped, and the patient's trachea was intubated after the administration of 0.9 mg/kg rocuronium.

##### Pharmacodynamic Analysis of Quantal Responses

^{4},

^{5}Reduced Greco model,

^{2},

^{16}Minto model,

^{1}Scaled C50

_{O}Hierarchical model,

^{6}and Fixed C50

_{O}Hierarchical model.

^{2},

^{7}Details of the models can be found in the appendix.

^{11},

^{12}age-dependent equilibration half-time of 0.94, 1.32, and 2.20 min for 20, 50, and 80 yr, respectively, and was used as the remifentanil effect-site concentration (Ce

_{REMI}) in the analysis. For sevoflurane, the alveolar concentration measured by the S5 Anesthesia Monitor (GE Healthcare)

*via*end-expiratory measurement after 12 min of equilibration was considered as the steady state concentration, and was used as sevoflurane effect-site concentration (Ce

_{SEVO}) in the analysis. To reduce data noise in Ce

_{REMI}and Ce

_{SEVO}, the median value of 11 measurements at 5 s intervals during 1 min preceding the assessment of the OAA/S score were used. The duration of equilibration of 12 min was five times the reported equilibration half-life for sevoflurane of 2.4 min.

^{17}

*a priori*considered less intense stimulus. Therefore the approach described by Bouillon

*et al.*

^{6}and by Schumacher

*et al.*,

^{10}combining the observed responses to the four stimuli into a single value, could not be applied. Instead the observed response to each stimulus was compared to the probability of that response according to the model, irrespective of the response to the other stimuli.

##### Parameter Estimation

*P*= 0.025, resulting in a critical difference of 5.02 in the OFV.

*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, P < 0.5, or if responsive, P > 0.5.

##### Statistical Analysis

#### Results

##### Data

##### Model Selection

_{REMI}and α became very large, whereas the OFV was higher than for the Reduced Greco model (data not shown). If the parameters were allowed to take very large values, the OFV approached that of the Reduced Greco model, and the ratio Ce50

_{REMI}/α approached Ce50

_{REMI}of the Reduced Greco model. Therefore the original Greco model was not further considered. The Akaike Information Criterion of the Minto model was markedly higher than for the other models, and the estimated Ce50

_{REMI}was above the highest remifentanil concentration applied. Compared with the other models, the Akaike Information Criterion of the Fixed C50

_{O}Hierarchical model was the lowest, indicating the best-fitting model.

_{REMI}and the slope parameters among the different stimuli in the modeling process did not significantly reduce the OFV. Even estimating a different Ce50

_{REMI}for the nonnoxious shaking and shouting compared with the Ce50

_{REMI}for tetanic stimulation, LMA insertion, and laryngoscopy did not result in a significant improvement of the OFV. In the final model a common Ce50

_{REMI}and common slope parameters for all stimuli were obtained, whereas the Ce50

_{SEVO}was stimulus specific (table 2).

_{SEVO}significantly improved the OFV (P < 0.01); but not interindividual variability of other parameters. We performed a covariate analysis on patient weight, height, age, gender, and order of administration of remifentanil and sevoflurane. None of these covariates did improve the fit significantly.

_{O}Hierarchical model if γ

_{O}is fixed to 1, and therefore both models may be compared using the likelihood ratio. Given the reduction of 8.5 in OFV it can be concluded that the Fixed C50

_{O}Hierarchical model fits significantly better to the data than the Reduced Greco model. The Scaled C50

_{O}and Fixed C50

_{O}Hierarchical models cannot be compared using the likelihood ratio, because both models have the same number of parameters. However, the reduction of 4.1 in the OFV indicates that the Fixed C50

_{O}Hierarchical model fits better to the data.

Table 3 Image Tools |
Fig. 1 Image Tools |

*i.e.*, Fixed C50

_{O}Hierarchical model with interindividual variability in Ce50

_{SEVO}, were checked by performing a bootstrap analysis, based on 1,000 sets; 994 sets resulted in a successful minimization, and 983 sets gave a successful covariance step. The results of the bootstrap analysis were in good agreement with the NONMEM results (table 3). Also, the CIs estimated from the bootstrap analysis and from the log-likelihood profiles were comparable (table 3). The log-likelihood profiles for the parameters of the Fixed C50

_{O}Hierarchical model are depicted in figure 1.

##### Response Surface and Isoboles

Fig. 2 Image Tools |
Fig. 3 Image Tools |
Fig. 4 Image Tools |

Fig. 5 Image Tools |
Fig. 6 Image Tools |
Fig. 7 Image Tools |

_{O}Hierarchical model and the observed responses. Figure 5 compares the isoboles for 50% probability of tolerance to the four stimuli for the four models. In figure 6, the isoboles for 5%, 50%, and 95% probability of tolerance to the four stimuli for the Fixed C50

_{O}Hierarchical model are shown. Figure 7 shows the isoboles for 95% probability of tolerance to laryngoscopy for the four models, illustrating the clinical significant difference between the Minto model and the three other models.

#### Discussion

^{2},

^{6},

^{7}The main finding of this study is the validity of the Fixed C50

_{O}Hierarchical model

^{2},

^{7}assuming an identical Ce50 and slope parameter for the opioid and an identical slope parameter of the hypnotic for different stimuli, but keeping different Ce50

_{hypnotic}for different stimuli. The model is thus validated not only for the propofol-remifentanil but also for the sevoflurane-remifentanil combination. The flexibility of the Fixed C50

_{O}Hierarchical model where only the Ce50

_{opioid}, the Ce50

_{hypnotic}and slope parameters for the opioid and hypnotic are needed as input parameters, is of importance for the parsimonious description of the interaction and therefore very useful in the context of anesthesia drug displays.

_{REMI}of 14.3 ng/ml was statistically inferior, whereas the original Greco model did not even support a reliable estimation of the Ce50

_{REMI}(estimated values above 50 ng/ml). This is in agreement with the clinical experience that in the absence of a hypnotic drug opioids do not suppress the response to stimulation, at least at clinically reasonable opioid concentrations. The Hierarchical models are semimechanistic models that have been developed to detect synergism for the combination of an analgesic and a hypnotic drug using a simple reconstruction of neuropathic pathways, as opposed to other more generalistic models. These Hierarchical models, as well as the Reduced Greco model, assume no relevant opioid effect if given alone, and therefore these models fitted better to the data than the Greco and Minto models. The differences between the Reduced Greco model, Scaled C50

_{O}Hierarchical model, and Fixed C50

_{O}Hierarchical model were rather small, and each of these three models fitted reasonably well to the data. However, the OFV and Akaike Information Criterion unequivocally showed that the Fixed C50

_{O}Hierarchical model fit best to our data.

_{O}Hierarchical, and Fixed C50

_{O}Hierarchical model is 1.59, 1.27, 1.08, and 1.19 vol.%, respectively. This illustrates the deviating characteristics of the Minto model, and the relatively small differences between the Reduced Greco and both Hierarchical models. Clinicians aim at titrating their drugs during anesthesia at least at a level of 95% probability of tolerance, so at a specific remifentanil concentration, applying the Minto model would result in the use of a clinically relevant higher sevoflurane concentration than when using the other models.

_{O}Hierarchical model, the Ce50s of the hypnotic are used to rank different stimuli according to their intensity. The Ce50

_{SEVO}for tolerance of shaking and shouting (nonnoxious) and tetanic stimulation (noxious) were similar. The Ce50

_{SEVO}for tolerance of LMA insertion and for laryngoscopy were also similar but substantially higher. In the previous study on the interaction of sevoflurane and propofol performed with the same stimuli by the same investigators,

^{10}the Ce50 values for sevoflurane for tolerance to shake and shout, tetanic stimulation, LMA insertion, and laryngoscopy were 1.03, 2.11, 2.55, and 2.83 vol.% respectively, which is markedly different from that found in the present study (1.47, 1.48, 2.09, and 2.00 vol.%, respectively). Furthermore, the slope reported by Schumacher

*et al.*was 17.6, whereas in the present study it was 7.41. To elucidate the cause of these differences, the data points of the previous and the current study where sevoflurane was given alone were reanalyzed (table 4). The parameter estimates obtained from the “sevoflurane alone” data of the two studies still differ, although the difference is smaller and the order of the Ce50s was similar in both studies. We can only speculate why the Ce50

_{SEVO}for tolerance to shake and shout was lower and the Ce50

_{SEVO}for tolerance to laryngoscopy was higher in the previous compared to the current study. Age, weight, and height were similar in the two studies. Classification of the subjects in responders and nonresponders was similar (a response was assumed if there was an observed response to a given stimulus, and if there was a response to a lower intensity stimulus and when the subsequent higher intensity stimulus was not applied). The pattern and current intensity of the electrical stimulus was also the same. The individual airway anatomy of the patients may affect the force to be applied during LMA insertion and the pressure applied with the laryngoscope to visualize the vocal cords. This may explain in part the difference between the Ce50s for tolerance to LMA insertion and laryngoscopy but not the difference between Ce50s for tolerance to shake and shout and tetanic stimulation.

^{8},

^{9}However, their parameter estimates differ markedly from those of the current study. Several reasons may explain this discrepancy: Whereas Manyam

*et al.*used a logistic regression model, Johnson used the Greco model in his reanalysis of the same data. In the study of Manyam

*et al.*the stimuli were given 5 min after achieving a stable end-tidal sevoflurane concentration, whereas in our study the equilibration was allowed for 12 min, which is five times the reported equilibration half-life for sevoflurane of 2.4 min.

^{17}To compensate for this disequilibrium, Johnson

*et al.*

^{9}used an estimated effect-site concentration to describe the hysteresis with the end-tidal concentration. The Ce50

_{REMI}for OAA/S≤1 during emergence (no return of consciousness) reported by Johnson

*et al.*was 50.9 ng/ml, which is far above the investigated concentration range applied and may thus not be reliable, although it reflects the weak hypnotic potency of opioids. The Ce50

_{REMI}for tolerance of tibial pressure was 1.3 ng/ml, which is in the range of the common Ce50

_{REMI}in the current study as well as in the previous studies.

^{2},

^{6},

^{7}In our study the Ce50

_{REMI}estimated with the Reduced Greco model was 2.28 ng/ml for OAAS≤1 (table 2), which is lower than the value calculated from the ratio Ce50/α reported by Johnson

*et al.*(50.9/9.4 = 5.4 ng/ml).

*et al.*reported a different slope for OAAS≤1 (5.2) and tolerance of tibial pressure (2.7), we did not find a significant difference between the slopes for the different stimuli, and in our final model the common slope was 7.4. It seems that the data from Manyam, reanalyzed by Johnson and our data are difficult to compare because of the different methodology and the different endpoints used.

^{13},

^{14},

^{18}

^{–}

^{21}Whereas studies using multiple stimuli and several combinations of a hypnotic and an opioid in a criss-cross design

^{6},

^{10}only one stimulus (skin incision) at one randomly assigned combination of the two drugs in one patient was applied in the traditional MAC depression studies. The advantage of the former is a reduction of the number of subjects while maintaining a sufficient number of data points for parameter estimation.

_{O}Hierarchical model appears to be the most appropriate to define the reference lines or numbers to guide the clinician in rational dosing. The two stimuli used in interaction displays as reference must be clearly different in intensity,

*i.e.*, significantly differ in their Ce50

_{hypnotic}. According to the present and previous data, “shaking and shouting” and “laryngoscopy” with their clearly distinct Ce50s, are therefore reasonable reference stimuli representing a superficial (near loss of consciousness) and a deeper state of anesthesia needed for surgery.

_{O}Hierarchical model best fits our data on sevoflurane remifentanil interaction and it appears to be an appropriate model for use in hypnotic-opioid drug interaction displays. However, the prediction performance was not essentially different between the Reduced Greco, Scaled C50

_{O}Hierarchical, and Fixed C50

_{O}Hierarchical models.

##### Appendix: Binary Response Models

*e.g.*, in Luginbuhl,

^{7}N and φ, respectively).

_{O}and U

_{H}are the normalized opioid and hypnotic effect-site concentrations, C

_{O}is the effect-site concentration of the opioid, C

_{H}is the effect-site concentration of the hypnotic, C50

_{O}is the effect-site concentration of the opioid that results in

*P*= 0.5 in the absence of the hypnotic, and C50

_{H}is the effect-site concentration of the hypnotic that results in

*P*= 0.5 in the absence of opioid.

##### Greco Model

^{2},

^{4},

^{5}:

_{H}and U

_{O}are the normalized concentrations of the hypnotic and opioid respectively.

_{O}, C50

_{H}, γ, and α. In the case of multiple (N) stimuli, there are 4.N model parameters; assuming equal values for γ and α for each stimulus, there are 2.N + 2 parameters (C50

_{O}and C50

_{H}for each additional stimulus). The model can be further reduced by assuming a common value for C50

_{O}for each stimulus; in this case there are N + 3 parameters (one additional parameter for each stimulus).

##### Reduced Greco Model without Effect of the Opioid Alone

*P*may be too small to accurately assess the C50

_{O}(i.e., the actual value of C50

_{O}is very high). The Greco model can then be easily modified by leaving out the term U

_{O}from Eq. A4, creating

Equation 7 Image Tools |
Equation 8 Image Tools |

_{O}and α cannot be estimated independently, since only their ratio α/C50

_{O}appears in Eq. A6. Therefore Bouillon

^{2}replaced the term α/C50

_{O}by a single parameter α′, resulting in A7:

_{O}may now be interpreted as the concentration of the opioid that decreases C50

_{H}by 50%: If C

_{O}= C50

_{O}(U

_{O}= 1), U = 2×U

_{H},

*i.e.*, the concentration of the hypnotic required to achieve a certain potency U, and thus a certain probability of tolerance

*P*, is reduced by a factor 2, compared to the concentration in the absence of the opioid.

_{O}can be directly interpreted as the concentration that decreases C50

_{H}by 50%, whereas the meaning of the term α′ in the Bouillon method is more difficult to explain.

_{H}, γ, and α′ in the Bouillon method, and C50

_{O}, C50

_{H}, and γ in the second method. In the case of multiple (N) stimuli, there are 3.N model parameters; assuming an equal value for γ for each stimulus, there are 2.N + 1 parameters. The model can be further reduced by assuming a common value for C50

_{O}for each stimulus; in this case there are N + 2 parameters.

##### Minto Model

_{H}plus U

_{O}. Equation A11,

_{50}is the potency of two drugs in the combination θ yielding half maximal effect, and β

_{U50}is a dimensionless interaction coefficient relating θ (fraction of hypnotic) and 1-θ (fraction of opioid) to U

_{50}(higher-order functions of θ may be used to accommodate more complex shapes of interaction). Equation A12,

_{50}.

_{O}, C50

_{H}, γ

_{O}, γ

_{H}, β

_{γ}), and may be written as a linear interpolation between γ

_{H}, and γ

_{O}, and an interaction term analogous to Eq. A11 (higher-order functions of θ may be used to accommodate more complex shapes of interaction): A13,

^{1}to clarify the interaction.

_{O}, C50

_{H}, γ, and β

_{U50}, or six model parameters: C50

_{O}, C50

_{H}, γ

_{O}, γ

_{H}, β

_{U50}, and β

_{γ}. In the case of multiple (N) stimuli, there are 4.N (or 6.N) model parameters; assuming an equal value for γ's and β's for each stimulus, there are 2.N + 2 (or 2.N + 4) parameters. The model can be further reduced by assuming a common value for C50

_{O}for each stimulus; in this case there are N + 3 (or N + 5) parameters. In the current implementation using Eq. A1 the Minto model implies that both drugs on their own may yield the maximal effect.

##### Hierarchical Model

^{6},

^{7}may be written as A14,

Equation 16 Image Tools |
Equation 17 Image Tools |

^{6}was considered overparameterized.

^{2}The parameters preopioid_intensity, C50

_{H}, and C50

_{O}cannot be estimated uniquely, since the values of C50

_{H}, and C50

_{O}can always be adjusted to offset any value of preopioid_intensity.

Equation 18 Image Tools |
Equation 19 Image Tools |

*i.e.*, by adding an exponent γ

_{O}to U

_{O}in Eq. A9, yielding Eq. A18.

_{O}and the Fixed C50

_{O}Hierarchical model.

##### Scaled C50_{O} Hierarchical Model

*et al.*,

^{6}where the C50

_{o}is multiplied by preopioid intensity to reflect the decreasing potency of opioids in attenuating pain as the intensity of the pain increases.

_{O}Hierarchical model constrains C50

_{Oi}and C50

_{Hi}for i > 1 to:

_{Oi}= C50

_{O1 *}preopioid_intensity

_{i}

_{Hi}= C50

_{H1 *}preopioid_intensity

_{i}

_{O}Hierarchical model is that the stimulus intensity is a factor by which the C50s of both drugs are multiplied.

_{O}, C50

_{H}, γ, and γ

_{O}. In the case of multiple (N) stimuli, there are 4.N model parameters; the constraints on C50

_{Oi}and C50

_{Hi}reduce the number of free parameters to 3.N + 1; assuming that γ and γ

_{O}are not affected by the type and intensity of the stimulus, there are N + 3 parameters (C50

_{O1}, γ, γ

_{O}, and N values of C50

_{Hi}; values of C50

_{Oi}for i more than 1 follow from the constraints).

##### Fixed C50_{O} Hierarchical Model

^{2}introduced a different constraint on C50

_{Oi}that is also reasonable and testable: C50

_{O}is the same for all stimuli. Therefore this model is referred to as 'Fixed C50

_{O}Hierarchical model', constraining C50

_{Oi}and C50

_{Hi}for i > 1 to:

_{Oi}= C50

_{O1}

_{Hi}= C50

_{H1 *}preopioid_intensity

_{i}

_{O}Hierarchical model is an extension of the Reduced Greco model,

*i.e.*, by adding an exponent γ

_{O}to U

_{O}in Eq. A9 and with a common parameter C50

_{O}for all stimuli. Note that in the Fixed C50

_{O}Hierarchical model proposed by Bouillon

^{2}γ

_{O}was assumed to be 1 (equation on page 481 of that paper), making the model identical to the Reduced Greco model.

_{O}Hierarchical model (C50

_{O}, γ, γ

_{O}, and N values for C50

_{Hi}).

##### Relationships between Models

_{O}and α of the Greco model are infinitely large, and their ratio C50

_{O}/α) is equal to C50

_{O}of the Reduced Greco model;

_{O}for all stimuli and the Fixed C50

_{O}Hierarchical model are identical if the parameter γ

_{O}is fixed to 1;

_{O}Hierarchical model assumes that the C50 of opioid (C50

_{O}) and hypnotic (C50

_{H}) are multiplied by a common factor representing the intensity of the stimulus,

*i.e.*, the ratio C50

_{O}/C50

_{H}are the same for each stimulus;

_{O}Hierarchical model assumes a common C50

_{O}for each stimulus;

_{O}Hierarchical model is identical to the Scaled C50

_{O}Hierarchical model.