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Response Surface Modeling of Remifentanil–Propofol Interaction on Cardiorespiratory Control and Bispectral Index
Diederik J.F. Nieuwenhuijs, M.D.; Erik Olofsen, M.Sc.; Raymonda R. Romberg, B.Sc.; Elise Sarton, M.D., Ph.D.; Denham Ward, M.D., Ph.D.; et al Frank Engbers, M.D.; Jaap Vuyk, M.D., Ph.D.; Rene Mooren, B.Sc.; Luc J Teppema, Ph.D.; Albert Dahan, M.D., Ph.D.
Author Affiliations & Notes
  • Diederik J.F. Nieuwenhuijs, M.D.
    *
  • Erik Olofsen, M.Sc.
    †
  • Raymonda R. Romberg, B.Sc.
    *
  • Elise Sarton, M.D., Ph.D.
    ‡
  • Denham Ward, M.D., Ph.D.
    §
  • Frank Engbers, M.D.
    ‡
  • Jaap Vuyk, M.D., Ph.D.
    ‡
  • Rene Mooren, B.Sc.
    ∥
  • Luc J Teppema, Ph.D.
    #
  • Albert Dahan, M.D., Ph.D.
    #
  • * Graduate Student, † Research Associate, ‡ Staff Anesthesiologist, § Visiting Professor, ∥ Research Analyst, # Associate Professor, Department of Anesthesiology, Leiden University Medical Center, Leiden, The Netherlands.
  • Received from the Department of Anesthesiology, Leiden University Medical Center, Leiden, The Netherlands.
Article Information
Education
Education   |   February 2003
Response Surface Modeling of Remifentanil–Propofol Interaction on Cardiorespiratory Control and Bispectral Index
Anesthesiology 2 2003, Vol.98, 312-322. doi:0000542-200302000-00008
Anesthesiology 2 2003, Vol.98, 312-322. doi:0000542-200302000-00008
THE COMBINED administration of opioids and anesthetics for induction and maintenance of anesthesia is common practice. The anesthetic is given to lose consciousness, prevent awareness, and reduce movement responses in the patient; the opioid is given to suppress somatic, stress, and adrenergic responses to surgical stimulation. An important advantage of combining an opioid and an anesthetic is the synergistic increase in these desired effects, with consequently the need for less drugs to attain the goal of adequate anesthesia relative to the amount of drug needed when only a single agent (i.e.  , an anesthetic) is given. 1 Since this is not only true for patients who are ventilated but also for patients who maintain their own breathing (for example, during minimal, moderate, and deep sedation), it is of interest to address the issue of the effect of drug combinations on respiration. While it is known that anesthesia induces many side effects, it is acknowledged that respiratory depression is potentially life-threatening. 2 Therefore, we studied the effect of the opioid remifentanil and intravenous anesthetic propofol on the cardiorespiratory control. This combination of drugs is frequently used in patients receiving monitored anesthesia care for minor (without additional regional anesthesia) and major (with additional regional anesthesia) surgery. Knowledge on the quantitative and qualitative (additive vs.  synergistic) nature of their interaction is clinically important and may lead to specific dosing regimens aimed at the titration of sedation/analgesia versus  respiratory effect.
To study the remifentanil–propofol interaction, we used the technique of response surface modeling. 3–7 This technique allows the observation of the concentration-effect relation among infinite combinations of remifentanil and propofol over the whole surface area in three-dimensional space. We previously made successful use of this technique to quantify the interactive effects of sevoflurane and alfentanil on cardiorespiratory control. 6 
Methods
Subjects and Apparatus
Twenty-two healthy male volunteers (aged 19–25 yr) participated in the protocol after approval was obtained from the local Human Ethics Committee (Commissie Medische Ethiek, Leiden University Medical Center, Leiden, The Netherlands). Oral and written consent was obtained from all volunteers.
An intravenous catheter was inserted in the left antecubital vein (for drug infusion) and an arterial line was placed in the right radial artery (for blood sampling) in each volunteer upon arrival at the laboratory. Subsequently, electrodes for electroencephalogram monitoring (BIS® Sensor, Aspect Medical Systems, Newton, MA) were placed on the head as specified by the manufacturer and the subjects rested for 20 to 30 min. Next a face mask was applied over the mouth and nose. Gas flow was measured with a pneumotachograph connected to a pressure transducer and electronically integrated to yield a volume signal. Corrections were made for the changes in gas viscosity due to changes in oxygen concentration of the inhaled gas mixtures. The pneumotachograph was connected to a T-piece. One arm of the T-piece received a gas mixture from a gas mixing system consisting of three mass-flow controllers (Bronkhorst High-Tec, Veenendaal, The Netherlands). A personal computer provided control signals to the mass-flow controllers so that the composition of the inspired gas mixtures could be adjusted to force end-tidal oxygen and carbon dioxide concentrations (PETO2and Petco2) to follow a specified pattern in time. The O2and CO2concentrations of inspired and expired gases and the arterial hemoglobin-O2saturation (Spo2) were measured with a Datex Multicap gas monitor and Datex Satlite Plus pulse oximeter, respectively (Datex-Engstrom, Helsinki, Finland).
The electroencephalogram was recorded using an A-2000 monitor with software version 3.3 (Aspect Medical Systems). The monitor computed the bispectral index over 2-s epochs. We averaged the bispectral index values during 1 min-intervals and used data points obtained at 3-min intervals.
Study Design
Resting ventilation and Petco2(i.e.  , without any inspired CO2), blood pressure, heart rate, bispectral index, and the ventilatory response to hypercapnia were measured before and during infusion of remifentanil, propofol, and the combined infusion of these agents. Initially control (i.e.  , without the administration of any agent) values were obtained. Next, the infusion of remifentanil was started and cardiorespiratory and bispectral index parameter values were obtained at steady state blood target concentrations. After this set of experiments, the infusion was terminated and the subject rested for 1 h. Next, the infusion of propofol was started and cardio-respiratory and bispectral index parameter values were obtained at steady state blood target concentrations. Subsequently, parameter values were obtained during the combined administration of remifentanil and propofol. In some subjects, two to three experiments were performed at different propofol–remifentanil combinations. The subjects were randomly assigned to a fixed scheme of target concentrations of remifentanil and propofol. The scheme was designed ensuring that, in the applied dose ranges, evenly spread data points were obtained.
Ventilatory Response to Hypercapnia
The ventilatory response to CO2was obtained by using the “dynamic end-tidal forcing” technique. 8,9 After assessment of resting variables, three to eight increases in Petco2were applied to obtain data points for the steady-state ventilatory response. The increases varied from 3 to 19 mmHg. The increased Petco2readings lasted at least 8 min. When on-line analysis revealed that a ventilatory steady-state had not been reached, the duration of hypercapnia was extended. The order of increases was arbitrarily chosen. All hypercapnic studies were performed at a background of moderate hyperoxia (PETO2120 mmHg).
The increased Petco2and the corresponding V̇ibreath-to-breath data were averaged over 10-breaths. Data points were obtained at the end of the Petco2increase. This procedure yielded three to eight steady-state data points. We expressed V̇ias a linear function of Petco2:where S is the ventilatory CO2sensitivity and B the extrapolated Petco2at zero V̇i. Parameters S and B were determined by linear regression of V̇ion Petco2.
Remifentanil and Propofol Administration, Blood Sampling and Assays
Propofol and remifentanil were administered using target controlled infusion (TCI) systems. For propofol, we used a palm-top computer (Psion, London, UK) programmed with a three-compartment propofol pharmacokinetic data set to control an infusion pump (Becton Dickinson, St. Etienne, France). 10,11 For remifentanil, we used a custom built infusion pump that was programmed with a remifentanil pharmacokinetic data set (Remifusor, University of Glasgow, Glasgow). 12 These systems allow a specified target plasma concentration of remifentanil and propofol to be rapidly achieved and maintained. Hypercapnic studies were performed ∼10 min after blood remifentanil and propofol had reached their target concentrations. Since this equals more than five to 10 times the remifentanil and propofol blood-effect-site equilibration half-lifes, we assumed that brain and blood remifentanil and propofol concentrations were in equilibrium.
Before and after changes in target drug concentrations, arterial blood samples for determination of remifentanil and propofol concentrations were collected. Blood for propofol determination was collected in syringes containing potassium oxalate. Propofol concentrations were determined by reverse-phase high performance liquid chromatography. 13 Samples for the determination of blood remifentanil concentrations were collected into tubes containing sodium heparin and immediately transferred to tubes containing 50% citric acid (to inactivate esterases) before freezing at −20°C. The assay method is based on tandem mass spectrometry detection. 14 
Response Surface Modeling
Analysis was performed on the following variables: resting inspired minute ventilation (V̇i) and Petco2(i.e.  , without any inspired CO2), slope of the hypercapnic ventilatory response (S), ventilation at a fixed Petco2of 55 mmHg (V̇55, calculated from S and B), mean arterial pressure (MAP), heart rate (HR), and bispectral index. The basis of the pharmacodynamic model is similar to the previously published model. 6 The single-drug concentration-effect (C  -E  ) relationship is given by MATHwhere E  0is the baseline drug effect, C  50the value of C  that gives 50% depression, and γ a nonlinearity parameter; notice that the model is linear when γ= 1. A straightforward extension for two concomitantly administered drugs (C  r= remifentanil concentration, C  p= propofol concentration) is obtained by respecting Loewe additivity 15 :MATHNote that isoboles in the Cr-Cpplane are straight lines, irrespective of the value of γ. Deviations from additivity can be modeled as:with I  (Q  ) a smooth function (spline) with a parameter denoting maximum interaction I  maxat I  (Q  max) and Q  =U  r/(U  r+U  p), U  r=C  r/ C  50,r, U  p=C  p/ C  50,p. To limit the number of parameters γ(Q  ) was either a constant or a linear function going from γrat Q  = 1 to γpat Q  = 0. Since the concentration ranges used in the study for most variables lie below the C  50s, these parameters will be poorly estimated leading to wide asymmetric confidence intervals. A remedy would be to use C  10s or C  25s but one does not know the optimal parameters beforehand. In fact, it is better to use parameters that are centered according to the study design:where C  h,rand C  h,pthe values of C  rand C  pmidway in the measured concentrations range, and Q  redefined to be Q  =U  r/(U  r+U  p), U  r=C  r/ C  h,r, U  p=C  p/ C  h,p; λrand λpdenote the degree of depression from E  0when C  r=C  h,rand C  p= 0 and vice versa  , respectively. For variable Pco2, which increases from E  0, the model used was the same as equation 5, except the minus sign was replaced by a plus sign.
Parameter Estimation and Model Selection
The above model has the following parameters to be estimated:E  0, λr, λp, I  max, Q  max, γr, and γp. The following situations are of special interest:
  • I  max= 1, Q  max= 0.5 denoting additivity;
  • I  max≠ 1, Q  max= 0.5 denoting symmetric interaction;
  • I  max≠ 1, Q  max≠ 0.5 denoting asymmetric interaction.
Notice that when Q  max= 0.5 we could use the Minto parabolic function of Q  instead of the spline I  (Q  ). 5 Furthermore, when two drugs are pharmacodynamically equivalent apart from a difference in potency, we would expect a symmetric interaction (since Q  is based on normalized concentrations). For each of the above three cases, there are five situations that describe (non)linearity:
  • γr=γp= 1 denoting linearity;
  • γr=γp≠ 1 denoting nonlinearity described by one parameter;
  • γr≠ 1 and γp= 1 denoting nonlinearity for drug R  and linearity for P  ;
  • γr= 1 and γp≠ 1 denoting linearity for drug R  and nonlinearity for P  ;
  • γr≠ 1 and γp≠ 1 denoting nonlinearity described by two parameters.
This results in a total of fifteen models to be investigated (fig. 1). NONMEM was used to estimate the parameter values. 16 Since the models are nonnested, the likelihood ratio criterion is not applicable so the Akaike Information-theoretic Criterion (AIC) was used instead 16 : AIC =−2 LL  + 2 P  , where −2 LL  is the minimum value of the objective function calculated by NONMEM and P  denotes the number of parameters. The model with the lowest AIC  is considered “best.” The population analysis was done with the assumption of lognormally distributed model parameters and constant relative (except for Pco2where it was assumed to be additive) normally distributed intraindividual error.
Fig. 1. Schematic representation of the 15 different pharmacodynamic model (M) possibilities. Models 1 to 5: additive interaction between propofol (p) and remifentanil (r) with different values for γrand γpper model; models 5 to 10: nonadditive interactions at a value of Qmaxequal to 0.5 (i.e.  , symmetric interactions) with different values for γrand γpper model; models 11 to 15: nonadditive interactions at a value of Qmaxnot equal to 0.5 (nonsymmetrical interactions) with different values for γrand γpper model.
Fig. 1. Schematic representation of the 15 different pharmacodynamic model (M) possibilities. Models 1 to 5: additive interaction between propofol (p) and remifentanil (r) with different values for γrand γpper model; models 5 to 10: nonadditive interactions at a value of Qmaxequal to 0.5 (i.e.  , symmetric interactions) with different values for γrand γpper model; models 11 to 15: nonadditive interactions at a value of Qmaxnot equal to 0.5 (nonsymmetrical interactions) with different values for γrand γpper model.
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Fig. 1. Schematic representation of the 15 different pharmacodynamic model (M) possibilities. Models 1 to 5: additive interaction between propofol (p) and remifentanil (r) with different values for γrand γpper model; models 5 to 10: nonadditive interactions at a value of Qmaxequal to 0.5 (i.e.  , symmetric interactions) with different values for γrand γpper model; models 11 to 15: nonadditive interactions at a value of Qmaxnot equal to 0.5 (nonsymmetrical interactions) with different values for γrand γpper model.
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Model Stability Assessment Using the Bootstrap
When, according to AIC  criterion, a model is chosen for a certain effect parameter, that choice is not associated with a measure of confidence in that model. One would like to be more certain that the choice is not an artifact of particular individuals in the current data set, and that when a new data set would be obtained, the same model would be chosen. A way to generate surrogate data sets is given by the method of the bootstrap. 17 Basically, a bootstrap data set is formed by selecting, with replacement, the data from individuals until a set is obtained with the same total number of individuals. This data set is then subject to the same fitting procedure, and by repeating the process N  times, N  parameter estimate sets are obtained with N  selections of one of the fifteen models. From the parameter estimates confidence intervals and histograms can be constructed. The impact of constraining certain parameters to fixed values, and therefore identifiability, can then be studied visually. The number of times a model is selected is a measure of our confidence in the model. In our analysis N  was set at 1000.
The bootstrap procedure was implemented in a C++ program that generates bootstrap data sets, NONMEM control files with appropriately fixed parameters, runs NONMEM and reads back the estimated parameter values and the minimum value of the objective function. When NONMEM returned an error status regarding parameter boundary problems (despite carefully chosen initial conditions and boundaries) or rounding errors, the model that was fitted was deemed to be not supported by the data. This, in principle, gives a bias towards the simpler models. Furthermore, to have a feasible procedure with respect to computer time, we opted not to investigate all possibilities for the statistical model. Initially, interindividual variability was assumed to be present only on parameters E  0, λr, and λp. When the number of times the corresponding variance was estimated to be negligible exceeded N  /2, this variability term was removed and the bootstrap redone. Confidence intervals were obtained in the traditional way (i.e.  , estimate ± 1.96 · SE) and the bootstrap BC  a(bias-corrected and accelerated) method. 17 
Results
All 22 subjects completed the protocol without major- nor-respiratory- side effects. The durations of the studies ranged from 3 to 4 h per subject. A total of 94 responses were obtained at different drug combinations. The range of the measured arterial remifentanil was 0–2 ng/ml. For propofol all measured concentrations were in the range of 0–2.0 μg/ml except one (2.6 μg/ml). Consequently C  h,rand C  h,pwere set to 1 ng/ml and 1 μg/ml, respectively, in the pharmacodynamic model.
A typical example of respiratory studies in one subject is given in figure 2. Its shows the control response (no drugs given) with a slope of 2.4 l · min−1· mmHg−1, the effects of 1.5 μg/ml propofol (a 66% reduction of the slope of the V̇i-CO2response to 0.8 l · min−1· mmHg−1) and 1 ng/ml remifentanil (a parallel shift of the response curve with a slope of 2.2 l · min−1· mmHg−1) alone, and the effect of that drug combination, which was greater than the sum of the effects of either drug alone (a > 90% depression of the slope to 0.2 l · min−1· mmHg−1).
Fig. 2. Four ventilatory carbon dioxide response curves of one subject. The control response had a slope of 2.4 l/min per mmHg. While propofol decreased the slope to 0.8 l/min per mmHg, remifentanil caused a parallel shift to higher Pco2values of about 12 mmHg (slope = 2.2 l/min per mmHg). The combined administration yielded both a reduction in slope of the response curve (slope = 0.2 l/min per mmHg) and a right-ward shift of about 20 mmHg. These observations suggest synergy on the slope of hypercapnic response and ventilation at a fixed Petco2.
Fig. 2. Four ventilatory carbon dioxide response curves of one subject. The control response had a slope of 2.4 l/min per mmHg. While propofol decreased the slope to 0.8 l/min per mmHg, remifentanil caused a parallel shift to higher Pco2values of about 12 mmHg (slope = 2.2 l/min per mmHg). The combined administration yielded both a reduction in slope of the response curve (slope = 0.2 l/min per mmHg) and a right-ward shift of about 20 mmHg. These observations suggest synergy on the slope of hypercapnic response and ventilation at a fixed Petco2.
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Fig. 2. Four ventilatory carbon dioxide response curves of one subject. The control response had a slope of 2.4 l/min per mmHg. While propofol decreased the slope to 0.8 l/min per mmHg, remifentanil caused a parallel shift to higher Pco2values of about 12 mmHg (slope = 2.2 l/min per mmHg). The combined administration yielded both a reduction in slope of the response curve (slope = 0.2 l/min per mmHg) and a right-ward shift of about 20 mmHg. These observations suggest synergy on the slope of hypercapnic response and ventilation at a fixed Petco2.
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In table 1the results of the bootstrap based model selection are given. For all respiratory variables, model 7 seemed best fitted to describe the data (i.e.  , nonlinear relationship between drugs and effect, synergistic interaction, Qmax= 0.5;i.e.  , indicating that interaction was symmetric) (fig. 1). The population estimates ± SE and 95% confidence intervals as derived from the NONMEM analysis of the response surfaces are given in table 2and for resting V̇i, resting Petco2, V̇55and S in figures 3–6. At 1 ng/ml and 1 μg/ml, remifentanil and propofol caused ∼28% and 13% depression of resting ventilation, respectively. Combining propofol and remifentanil at these same blood concentrations caused 58% depression (equation 4), indicating the synergistic nature of the interaction. Similar observations were made for resting Petco2, V̇55and S, although the synergistic interaction strength was less (Imaxresting V̇i= 1.9 vs.  Imaxresting Petco2, V̇55and S = 1.2–1.3). At the combined infusion of 1 μg/ml propofol and 1 ng/ml remifentanil the depression of V̇55was 82% (equation 4); the corresponding values for resting Petco2and S were 23% and 69%, respectively.
Table 1. Results of the Bootstrap-based Model Selection
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Table 1. Results of the Bootstrap-based Model Selection
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Table 2. Population Pharmacodynamic Estimates
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Table 2. Population Pharmacodynamic Estimates
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Fig. 3. Response surface modeling of the interaction of remifentanil and propofol on resting ventilation. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.9 ± 0.2). Also, the dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.5 ± 0.1). (Right  ) Individual data points and 25, 50 and 75% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
Fig. 3. Response surface modeling of the interaction of remifentanil and propofol on resting ventilation. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.9 ± 0.2). Also, the dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.5 ± 0.1). (Right  ) Individual data points and 25, 50 and 75% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 3. Response surface modeling of the interaction of remifentanil and propofol on resting ventilation. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.9 ± 0.2). Also, the dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.5 ± 0.1). (Right  ) Individual data points and 25, 50 and 75% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 4. Response surface modeling of the interaction of remifentanil and propofol on resting end-tidal carbon dioxide concentration (Petco2). (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.3 ± 0.2). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.7 ± 0.1). Note that the x  and y  axes are different from the other response surface plots with the origin now facing the reader. (Right  ) Individual data points and 25% isobole. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
Fig. 4. Response surface modeling of the interaction of remifentanil and propofol on resting end-tidal carbon dioxide concentration (Petco2). (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.3 ± 0.2). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.7 ± 0.1). Note that the x  and y  axes are different from the other response surface plots with the origin now facing the reader. (Right  ) Individual data points and 25% isobole. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 4. Response surface modeling of the interaction of remifentanil and propofol on resting end-tidal carbon dioxide concentration (Petco2). (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.3 ± 0.2). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.7 ± 0.1). Note that the x  and y  axes are different from the other response surface plots with the origin now facing the reader. (Right  ) Individual data points and 25% isobole. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 5. Response surface modeling of the interaction of remifentanil and propofol on ventilation at a fixed Petco2of 55 mmHg. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.2 ± 0.1). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.4 ± 0.1). The model predicted apnea to occur at several combinations of propofol and remifentanil, for example, 1.6 ng/ml remifentanil and 2.0 μg/ml propofol or 2.0 ng/ml remifentanil and 1.6 μg/ml propofol. (Right  ) Individual data points and 25, 50, 75 and 100% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
Fig. 5. Response surface modeling of the interaction of remifentanil and propofol on ventilation at a fixed Petco2of 55 mmHg. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.2 ± 0.1). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.4 ± 0.1). The model predicted apnea to occur at several combinations of propofol and remifentanil, for example, 1.6 ng/ml remifentanil and 2.0 μg/ml propofol or 2.0 ng/ml remifentanil and 1.6 μg/ml propofol. (Right  ) Individual data points and 25, 50, 75 and 100% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 5. Response surface modeling of the interaction of remifentanil and propofol on ventilation at a fixed Petco2of 55 mmHg. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.2 ± 0.1). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.4 ± 0.1). The model predicted apnea to occur at several combinations of propofol and remifentanil, for example, 1.6 ng/ml remifentanil and 2.0 μg/ml propofol or 2.0 ng/ml remifentanil and 1.6 μg/ml propofol. (Right  ) Individual data points and 25, 50, 75 and 100% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 6. Response surface modeling of the interaction of remifentanil and propofol on the slope of the ventilatory response to carbon dioxide or CO2sensitivity. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.3 ± 0.1). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.4 ± 0.1). Note that the effect on slope was predominantly a propofol effect and to a lesser extend a remifentanil effect. (Right  ) Individual data points and 25, 50 and 75% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
Fig. 6. Response surface modeling of the interaction of remifentanil and propofol on the slope of the ventilatory response to carbon dioxide or CO2sensitivity. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.3 ± 0.1). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.4 ± 0.1). Note that the effect on slope was predominantly a propofol effect and to a lesser extend a remifentanil effect. (Right  ) Individual data points and 25, 50 and 75% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 6. Response surface modeling of the interaction of remifentanil and propofol on the slope of the ventilatory response to carbon dioxide or CO2sensitivity. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.3 ± 0.1). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.4 ± 0.1). Note that the effect on slope was predominantly a propofol effect and to a lesser extend a remifentanil effect. (Right  ) Individual data points and 25, 50 and 75% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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In order to get an indication of the spread of data points over the surface as well as on the goodness of fit, we included bubble plots that show the distance of individual measured data points from the population surface (i.e.  , residuals) (figs. 3–6). These plots show evenly spread data over the tested dose ranges and the absence of overt misfits.
The values of baseline mean arterial pressure and heart rate (values before any drug was given) indicate that the subjects were free of agitation or stress during the studies (table 2). The effects of remifentanil and propofol on MAP and HR rate were not as remarkable as their effects on the respiratory variables: depression at 1 ng/ml remifentanil and 1 μg/ml propofol ranged from 4 to 12% (table 2). The effect of their combination was expected from the concentration-response curve of the individual agents (i.e.  , additive interaction or Imax= 1, linear dose-effect relationship for MAP, nonlinear relationship for HR, table 1).
The BIS® monitor was unable to unearth any sedative effect of remifentanil in the dose range we studied (inert interaction) (fig. 7). Furthermore, the effect of propofol on the bispectral index was independent of the remifentanil concentration. The propofol–bispectral index relationship was linear with 19% depression of the bispectral index at 1 μg/ml plasma level.
Fig. 7. Response surface modeling of the interaction of remifentanil and propofol on the bispectral index of the electroencephalogram (BIS). (Left  ) Population response surface showing that the propofol–remifentanil interaction is inert since remifentanil had no effect on bispectral index irrespective of the propofol concentrations. Over this dose range, propofol causes a linear decrease in bispectral index with a 25% decrease occurring at 1.4 μg/ml. (Right  ) Individual data points and 25,% isobole. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
Fig. 7. Response surface modeling of the interaction of remifentanil and propofol on the bispectral index of the electroencephalogram (BIS). (Left  ) Population response surface showing that the propofol–remifentanil interaction is inert since remifentanil had no effect on bispectral index irrespective of the propofol concentrations. Over this dose range, propofol causes a linear decrease in bispectral index with a 25% decrease occurring at 1.4 μg/ml. (Right  ) Individual data points and 25,% isobole. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 7. Response surface modeling of the interaction of remifentanil and propofol on the bispectral index of the electroencephalogram (BIS). (Left  ) Population response surface showing that the propofol–remifentanil interaction is inert since remifentanil had no effect on bispectral index irrespective of the propofol concentrations. Over this dose range, propofol causes a linear decrease in bispectral index with a 25% decrease occurring at 1.4 μg/ml. (Right  ) Individual data points and 25,% isobole. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Discussion
The main findings of our study are as follows. (1) In the dose range tested, remifentanil (0–2 ng/ml) and propofol (0–2.6 μg/ml) caused a dose-dependent depression of respiration, as observed by an increase in resting Petco2and decreases in resting V̇i, slope of the V̇i-CO2response and ventilation at a fixed Petco2of 55 mmHg. (2) While remifentanil shifts the V̇i-CO2response curve in a parallel fashion to higher Petco2levels, propofol reduces the slope of the response rather than shifting its position (pivot point at resting V̇i). (3) When combined, the depressant effect of propofol and remifentanil on resting V̇i, resting Petco2, S, and V̇55is synergistic, with the greatest synergy observed for resting V̇i. (4) The depressant effect of remifentanil and propofol on blood pressure and heart rate is modest, when given separately; when combined their depressant effect is additive. (5) The bispectral index is sensitive to propofol but not to remifentanil, even when these agents are combined.
Pharmacodynamic Modeling
The Model.
Similar to our previous study, 6 we used an asymmetric sigmoid function to describe the dose–effect relations. The function may be linear (γ= 1) or nonlinear (γ≠ 1). The advantages of this approach have been discussed previously. 6 In short, in contrast to classic sigmoid pharmacodynamic models, such as the inhibitory sigmoid Emaxmodel, 18 our model predicts apnea at and above certain finite drug concentrations; it can predict negative responses above certain drug concentrations (for example, negative responses may occur when testing the effect of opioids on the ventilatory response to hypoxia) 6,19; and finally, linear respiratory dose-responses may occur in limited dose ranges. 20 Interaction was modeled as suggested by Minto et al.  , 5 which is based on the following two ideas: (1) the combination of two drugs should be regarded as one new drug with its own properties, and (2) that these properties depend only on the concentration ratio Q  . As before, interaction was defined by the function I  (Q  ), for which we chose a spline (for details see ref. 6). Furthermore, the two drugs used in this study have dissimilar mechanisms of action so that we would not expect their γ to be equal at equipotent concentrations. Therefore, we also included the possibility of a linear γ(Q  ). To our surprise γr=γp=γ for all tested variables.
Parameterization
Frequently, pharmacodynamic models incorporate C50s to describe and compare potencies. Since, in our study, the applied concentration ranges are less than the C50s, these parameters are poorly estimated with wide and asymmetric confidence intervals. In order to overcome this problem, we introduced the parameter λ, which is the percentage depression at the concentration midway in the plasma concentration range (equation 4).
Bootstrap Model Selection
The method of the bootstrap was applied here to assess the stability of the model selection based on AIC  . Confidence in a model is then expressed as the number of times a model is chosen. Note that this confidence is not equivalent with the type I or type II error in traditional hypothesis testing. In the space of two nested models, however, the AIC  is closely related to the type I error and the model selection percentage closely related to the power of the test. 21 When NONMEM produced an error message concerning boundary errors, the model that was tested was most probably overparameterized and would not be selected by AIC  anyway.
Characteristics of Parameter Distributions
Parameter distributions can be estimated by constructing histograms of the estimated parameter values from the bootstrap runs. With the parameterization utilizing λs, their distributions were neither wide nor skewed so that the confidence intervals (obtained from the NONMEM population estimates ± 1.96 · SE, table 2) turned out to be equivalent with those obtained from the bootstrap parameter distributions. For example, for V̇55the corresponding bootstrap values are baseline value 29.0–34.0 l/min, λr51.0–67.0%, λp37.0–52.0%, Imax1.08–1.39 and γ 0.22–0.50.
Physiologic and Pharmacological Considerations
Opioids and anesthetics influence respiration by affecting chemical control of breathing, behavioral control of breathing, or, which happens most frequently, by affecting both. Chemical or metabolic control of breathing is coupled to the metabolism and depends on the chemical composition of arterial blood (pH, arterial Pco2, arterial Po2) and brainstem interstitial fluid (pH, brain tissue Pco2) via  actions at peripheral and central chemoreceptors. Behavioral control of breathing allows adjustment of breathing to speech, pain, sedation, arousal, et cetera  . We tested two sets of respiratory measures: resting variables (resting V̇iand resting Petco2) and variables obtained from the ventilatory response to inspired CO2(S and V̇55). While S and V̇55(note that V̇55determines the position of the ventilatory response slope to CO2, just like parameter B of equation 1) are predominantly chemical in nature, resting V̇iand resting Petco2have both behavioral and chemical components. Attempts have been made to combine all of these 4 variables into a single model. 18 We refrained from such an approach for the obvious reason that blind grouping of data obtained during CO2inhalation and resting variables has little physiologic meaning for the above mentioned reasons. Consequently, C50s and time constants may differ for data obtained without and with increased inspired CO2. We do believe, however, that grouping resting Petco2and resting V̇iinto a single model has obvious advantages. Such a model should be able to predict apnea at some finite opioid concentration and should possibly be independent of inspired CO2V̇i-response data. The proposed (sigmoid Emax) indirect response model lacks both characteristics. 18 Incorporation of our asymmetric sigmoid function will allow the modeling of apnea. Further -simulation and experimental- studies are needed to explore this matter.
With respect to chemical control of breathing, we tested two agents with distinct respiratory properties and mechanisms of action. The opioid remifentanil caused a parallel shift of the V̇i-CO2response towards higher Pco2values with little effect on the slope (fig. 2). However, the anesthetic/sedative propofol caused a reduction of the slope of the V̇i-CO2response curve (S) with little to no effect on the position of the curve at resting Petco2values (fig. 2). We consider the parallel shift of the V̇i-CO2response curve a typical μ-opioid effect, and the reduction of the slope a typical effect of a hypnotic/sedative. Previously, we observed large differences in the effect of intravenous morphine on the slope of the V̇i-CO2response in men and women, 22,23 with no effect of morphine on the slope in men but a large reduction in women. Taken into account the above, it would be appropriate to suggest that in our previous studies morphine produced greater sedation in women than in men and consequently greater effects on S in women. Indeed, in a recent study in which we assessed the effect of morphine's active metabolite, morphine-6-glucuronide (M6G), on the level of sedation using a numerical rating score, we found greater sedation in women than men while plasma M6G concentrations were equal (R.R. Romberg, B.Sc., A. Dahan, M.D., Ph.D., unpublished observation, January 2002–January 2003). Note however, that our suggestions do not exclude more fundamental sex differences in CNS responses to opioids, such as sex differences in μ-opioid receptor density and affinity in regions involved in ventilatory control and pain response. 24 
In agreement with our previous study, 6 the magnitude of synergy was greatest for resting ventilation. The observed differences in synergy strength (table 2) may be related to the observation that (in)activation of behavioral control does not have a large effect on the chemoreflexes but does increase resting ventilation by a chemoreflex-independent tonic drive. 25 The effect of propofol and remifentanil on resting ventilation has a behavioral component (the patients falls “asleep”) and a chemical component (the direct effect of propofol and remifentanil on carotid bodies and respiratory neurons in the CNS) which leads consequently into a maximal synergistic interaction; the effect of both drugs on CO2-driven ventilation has much less of a behavioral component and is predominantly chemoreceptor-related, and consequently results in less synergistic interaction. The interactions described here are clinically important since they show marked synergistic interactions on resting ventilation and to a lesser extent on resting CO2at low drug concentrations.
Parameter Values
The effects of 1 ng/ml remifentanil and 1 μg/ml propofol on resting V̇iwere considerably less than their effect on V̇55(the ratio of λs is 0.5 for remifentanil and 0.3 for propofol). This is not surprising taking into account the fact that, while resting V̇iis measured in closed-loop conditions and part of the respiratory depression is offset by the gradual increase in resting Petco2, V̇55is measured in open-loop conditions and the pharmacokinetics and pharmacodynamic of CO2(and the effect the tested drugs have on CO2pharmacokinetics/pharmacodynamics) have been effectively removed.
The extrapolated C50values from this study correspond well with studies from the literature. For example, the remifentanil C50of V̇iat an increased and fixed Petco2obtained from a single bolus of 0.5 μg/kg was of the same order of magnitude as our observation (1.1 ng/ml vs.  0.7 ng/ml in this study, table 2). 26 Note that, in this latter study, remifentanil concentrations were not measured but obtained from the literature. These C50values are a factor of 10 smaller than those observed for changes in spectral edge frequency of the electroencephalogram, 12 and 4 to 5 times smaller than those observed for 50% probability of adequate anesthesia during abdominal surgery (in combination with 66% nitrous oxide). 27 These findings indicate the higher opioid sensitivity of CNS sites involved in ventilatory control compared to sites involved in behavioral state control and suppression of somatic and autonomic responses. Remifentanil is about 80–100 times more potent than alfentanil in depressing V̇55. 6,26 At present we are unaware of any previous respiratory pharmacokinetic or pharmacodynamic data for propofol.
Clinical Considerations
It is difficult to extrapolate the response surfaces to the clinical situation. Propofol and remifentanil are mostly given at a constant rate (resulting in constant plasma levels) with one of the drugs adjusted up as needed for additional analgesia/sedation or down if less respiratory depression is important. Therefore, we calculated the effect of changes in infusion rate (and hence changes in plasma concentration) on changes in resting Petco2(the more easily clinically monitored variable) for propofol at constant remifentanil concentration (fig. 8, top) and remifentanil at constant propofol concentration (fig. 8, bottom). The nonlinear shape of the resting Petco2response surface results in marked differences between these two figures: (1) increasing propofol has little effect on resting Petco2but adding remifentanil however has a marked -synergistic- effect (fig. 8, top); (2) increasing remifentanil increases Petco2regularly with only some potentiation by the addition of propofol (fig. 8, bottom). These graphs indicate that it is safer to titrate the propofol dose with a constant remifentanil background if more or less sedation is needed, but if less respiratory depression is required, then the remifentanil would need to be reduced.
Fig. 8. (Top  ) The influence of the steady-state or effect-site propofol concentration on resting Petco2at various constant remifentanil concentrations. (Bottom  ) The influence of the steady-state or effect-site remifentanil concentration on resting Petco2at various constant propofol concentrations. Changing remifentanil concentrations causes marked reductions in resting end-tidal Pco2, irrespective of the propofol concentration, while changes in propofol concentrations have less of an effect on resting Petco2, irrespective of the remifentanil concentrations.
Fig. 8. (Top  ) The influence of the steady-state or effect-site propofol concentration on resting Petco2at various constant remifentanil concentrations. (Bottom  ) The influence of the steady-state or effect-site remifentanil concentration on resting Petco2at various constant propofol concentrations. Changing remifentanil concentrations causes marked reductions in resting end-tidal Pco2, irrespective of the propofol concentration, while changes in propofol concentrations have less of an effect on resting Petco2, irrespective of the remifentanil concentrations.
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Fig. 8. (Top  ) The influence of the steady-state or effect-site propofol concentration on resting Petco2at various constant remifentanil concentrations. (Bottom  ) The influence of the steady-state or effect-site remifentanil concentration on resting Petco2at various constant propofol concentrations. Changing remifentanil concentrations causes marked reductions in resting end-tidal Pco2, irrespective of the propofol concentration, while changes in propofol concentrations have less of an effect on resting Petco2, irrespective of the remifentanil concentrations.
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The above applies best to patients who maintain their breathing during anesthesia. In order to extrapolate our findings to postoperative patients, in figure 9, we plotted the 10–60% isoboles of increasing resting Petco2with the isobole for 50% probability of regaining consciousness after general anesthesia for abdominal surgery (and the isobole for 50% probability of no somatic/autonomic response to surgical stimuli). (Data from Martijn Mertens, Ph.D. thesis, Leiden University, 2002.) The plot shows (1) the synergistic interaction between propofol and remifentanil on the 50% probability to “wake-up” after anesthesia (and thus shows in contrast to the bispectral index data (fig. 7) the sedative/hypnotic effect of remifentanil); (2) whether consciousness has been regained or not, ventilation improves best by reducing the remifentanil concentration (i.e.  , the return of the wakefulness drive is of limited importance at least when the subject is not stimulated or reminded to breathe); (3) without the addition of propofol, remifentanil concentrations up to 2 ng/ml cause only limited respiratory depression and may be applied for postoperative pain relief.
Fig. 9. Comparison of isoboles of respiratory depression (10–60% isoboles for increases in Petco2, data from this study), consciousness and adequate anesthesia (50% probability lines for consciousness and adequate anesthesia in patients undergoing abdominal surgery.
Fig. 9. Comparison of isoboles of respiratory depression (10–60% isoboles for increases in Petco2, data from this study), consciousness and adequate anesthesia (50% probability lines for consciousness and adequate anesthesia in patients undergoing abdominal surgery.
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Fig. 9. Comparison of isoboles of respiratory depression (10–60% isoboles for increases in Petco2, data from this study), consciousness and adequate anesthesia (50% probability lines for consciousness and adequate anesthesia in patients undergoing abdominal surgery.
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Since, in our study, ventilation and plasma drug levels were at steady state when data points were obtained, we did not get information about the time-course of respiratory effects. Furthermore, especially for rapidly acting drugs, such as remifentanil and propofol, the degree of nonsteady-state respiratory depression may be dependent on the rate of drug infusion. Further studies are needed to study ventilatory dynamics caused by different infusion schemes of opioids and anesthetics.
In conclusion, we observed dose-dependent respiratory depression from propofol and remifentanil. When combined, the respiratory effects were strikingly synergistic with clinically important respiratory depression at already low doses.
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Fig. 1. Schematic representation of the 15 different pharmacodynamic model (M) possibilities. Models 1 to 5: additive interaction between propofol (p) and remifentanil (r) with different values for γrand γpper model; models 5 to 10: nonadditive interactions at a value of Qmaxequal to 0.5 (i.e.  , symmetric interactions) with different values for γrand γpper model; models 11 to 15: nonadditive interactions at a value of Qmaxnot equal to 0.5 (nonsymmetrical interactions) with different values for γrand γpper model.
Fig. 1. Schematic representation of the 15 different pharmacodynamic model (M) possibilities. Models 1 to 5: additive interaction between propofol (p) and remifentanil (r) with different values for γrand γpper model; models 5 to 10: nonadditive interactions at a value of Qmaxequal to 0.5 (i.e.  , symmetric interactions) with different values for γrand γpper model; models 11 to 15: nonadditive interactions at a value of Qmaxnot equal to 0.5 (nonsymmetrical interactions) with different values for γrand γpper model.
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Fig. 1. Schematic representation of the 15 different pharmacodynamic model (M) possibilities. Models 1 to 5: additive interaction between propofol (p) and remifentanil (r) with different values for γrand γpper model; models 5 to 10: nonadditive interactions at a value of Qmaxequal to 0.5 (i.e.  , symmetric interactions) with different values for γrand γpper model; models 11 to 15: nonadditive interactions at a value of Qmaxnot equal to 0.5 (nonsymmetrical interactions) with different values for γrand γpper model.
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Fig. 2. Four ventilatory carbon dioxide response curves of one subject. The control response had a slope of 2.4 l/min per mmHg. While propofol decreased the slope to 0.8 l/min per mmHg, remifentanil caused a parallel shift to higher Pco2values of about 12 mmHg (slope = 2.2 l/min per mmHg). The combined administration yielded both a reduction in slope of the response curve (slope = 0.2 l/min per mmHg) and a right-ward shift of about 20 mmHg. These observations suggest synergy on the slope of hypercapnic response and ventilation at a fixed Petco2.
Fig. 2. Four ventilatory carbon dioxide response curves of one subject. The control response had a slope of 2.4 l/min per mmHg. While propofol decreased the slope to 0.8 l/min per mmHg, remifentanil caused a parallel shift to higher Pco2values of about 12 mmHg (slope = 2.2 l/min per mmHg). The combined administration yielded both a reduction in slope of the response curve (slope = 0.2 l/min per mmHg) and a right-ward shift of about 20 mmHg. These observations suggest synergy on the slope of hypercapnic response and ventilation at a fixed Petco2.
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Fig. 2. Four ventilatory carbon dioxide response curves of one subject. The control response had a slope of 2.4 l/min per mmHg. While propofol decreased the slope to 0.8 l/min per mmHg, remifentanil caused a parallel shift to higher Pco2values of about 12 mmHg (slope = 2.2 l/min per mmHg). The combined administration yielded both a reduction in slope of the response curve (slope = 0.2 l/min per mmHg) and a right-ward shift of about 20 mmHg. These observations suggest synergy on the slope of hypercapnic response and ventilation at a fixed Petco2.
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Fig. 3. Response surface modeling of the interaction of remifentanil and propofol on resting ventilation. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.9 ± 0.2). Also, the dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.5 ± 0.1). (Right  ) Individual data points and 25, 50 and 75% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
Fig. 3. Response surface modeling of the interaction of remifentanil and propofol on resting ventilation. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.9 ± 0.2). Also, the dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.5 ± 0.1). (Right  ) Individual data points and 25, 50 and 75% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 3. Response surface modeling of the interaction of remifentanil and propofol on resting ventilation. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.9 ± 0.2). Also, the dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.5 ± 0.1). (Right  ) Individual data points and 25, 50 and 75% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 4. Response surface modeling of the interaction of remifentanil and propofol on resting end-tidal carbon dioxide concentration (Petco2). (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.3 ± 0.2). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.7 ± 0.1). Note that the x  and y  axes are different from the other response surface plots with the origin now facing the reader. (Right  ) Individual data points and 25% isobole. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
Fig. 4. Response surface modeling of the interaction of remifentanil and propofol on resting end-tidal carbon dioxide concentration (Petco2). (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.3 ± 0.2). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.7 ± 0.1). Note that the x  and y  axes are different from the other response surface plots with the origin now facing the reader. (Right  ) Individual data points and 25% isobole. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 4. Response surface modeling of the interaction of remifentanil and propofol on resting end-tidal carbon dioxide concentration (Petco2). (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.3 ± 0.2). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.7 ± 0.1). Note that the x  and y  axes are different from the other response surface plots with the origin now facing the reader. (Right  ) Individual data points and 25% isobole. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 5. Response surface modeling of the interaction of remifentanil and propofol on ventilation at a fixed Petco2of 55 mmHg. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.2 ± 0.1). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.4 ± 0.1). The model predicted apnea to occur at several combinations of propofol and remifentanil, for example, 1.6 ng/ml remifentanil and 2.0 μg/ml propofol or 2.0 ng/ml remifentanil and 1.6 μg/ml propofol. (Right  ) Individual data points and 25, 50, 75 and 100% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
Fig. 5. Response surface modeling of the interaction of remifentanil and propofol on ventilation at a fixed Petco2of 55 mmHg. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.2 ± 0.1). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.4 ± 0.1). The model predicted apnea to occur at several combinations of propofol and remifentanil, for example, 1.6 ng/ml remifentanil and 2.0 μg/ml propofol or 2.0 ng/ml remifentanil and 1.6 μg/ml propofol. (Right  ) Individual data points and 25, 50, 75 and 100% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 5. Response surface modeling of the interaction of remifentanil and propofol on ventilation at a fixed Petco2of 55 mmHg. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.2 ± 0.1). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.4 ± 0.1). The model predicted apnea to occur at several combinations of propofol and remifentanil, for example, 1.6 ng/ml remifentanil and 2.0 μg/ml propofol or 2.0 ng/ml remifentanil and 1.6 μg/ml propofol. (Right  ) Individual data points and 25, 50, 75 and 100% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 6. Response surface modeling of the interaction of remifentanil and propofol on the slope of the ventilatory response to carbon dioxide or CO2sensitivity. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.3 ± 0.1). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.4 ± 0.1). Note that the effect on slope was predominantly a propofol effect and to a lesser extend a remifentanil effect. (Right  ) Individual data points and 25, 50 and 75% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
Fig. 6. Response surface modeling of the interaction of remifentanil and propofol on the slope of the ventilatory response to carbon dioxide or CO2sensitivity. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.3 ± 0.1). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.4 ± 0.1). Note that the effect on slope was predominantly a propofol effect and to a lesser extend a remifentanil effect. (Right  ) Individual data points and 25, 50 and 75% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 6. Response surface modeling of the interaction of remifentanil and propofol on the slope of the ventilatory response to carbon dioxide or CO2sensitivity. (Left  ) Population response surface showing that the propofol–remifentanil interaction is synergistic (I(max) = 1.3 ± 0.1). The dose-response relationships between drugs and effect was not linear (for both drugs γ= 0.4 ± 0.1). Note that the effect on slope was predominantly a propofol effect and to a lesser extend a remifentanil effect. (Right  ) Individual data points and 25, 50 and 75% isoboles. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 7. Response surface modeling of the interaction of remifentanil and propofol on the bispectral index of the electroencephalogram (BIS). (Left  ) Population response surface showing that the propofol–remifentanil interaction is inert since remifentanil had no effect on bispectral index irrespective of the propofol concentrations. Over this dose range, propofol causes a linear decrease in bispectral index with a 25% decrease occurring at 1.4 μg/ml. (Right  ) Individual data points and 25,% isobole. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
Fig. 7. Response surface modeling of the interaction of remifentanil and propofol on the bispectral index of the electroencephalogram (BIS). (Left  ) Population response surface showing that the propofol–remifentanil interaction is inert since remifentanil had no effect on bispectral index irrespective of the propofol concentrations. Over this dose range, propofol causes a linear decrease in bispectral index with a 25% decrease occurring at 1.4 μg/ml. (Right  ) Individual data points and 25,% isobole. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 7. Response surface modeling of the interaction of remifentanil and propofol on the bispectral index of the electroencephalogram (BIS). (Left  ) Population response surface showing that the propofol–remifentanil interaction is inert since remifentanil had no effect on bispectral index irrespective of the propofol concentrations. Over this dose range, propofol causes a linear decrease in bispectral index with a 25% decrease occurring at 1.4 μg/ml. (Right  ) Individual data points and 25,% isobole. Open circles denote data point above the surface, closed circles denote data points below the surface. The area of the circles is proportional to the distance from that data point to the surface.
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Fig. 8. (Top  ) The influence of the steady-state or effect-site propofol concentration on resting Petco2at various constant remifentanil concentrations. (Bottom  ) The influence of the steady-state or effect-site remifentanil concentration on resting Petco2at various constant propofol concentrations. Changing remifentanil concentrations causes marked reductions in resting end-tidal Pco2, irrespective of the propofol concentration, while changes in propofol concentrations have less of an effect on resting Petco2, irrespective of the remifentanil concentrations.
Fig. 8. (Top  ) The influence of the steady-state or effect-site propofol concentration on resting Petco2at various constant remifentanil concentrations. (Bottom  ) The influence of the steady-state or effect-site remifentanil concentration on resting Petco2at various constant propofol concentrations. Changing remifentanil concentrations causes marked reductions in resting end-tidal Pco2, irrespective of the propofol concentration, while changes in propofol concentrations have less of an effect on resting Petco2, irrespective of the remifentanil concentrations.
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Fig. 8. (Top  ) The influence of the steady-state or effect-site propofol concentration on resting Petco2at various constant remifentanil concentrations. (Bottom  ) The influence of the steady-state or effect-site remifentanil concentration on resting Petco2at various constant propofol concentrations. Changing remifentanil concentrations causes marked reductions in resting end-tidal Pco2, irrespective of the propofol concentration, while changes in propofol concentrations have less of an effect on resting Petco2, irrespective of the remifentanil concentrations.
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Fig. 9. Comparison of isoboles of respiratory depression (10–60% isoboles for increases in Petco2, data from this study), consciousness and adequate anesthesia (50% probability lines for consciousness and adequate anesthesia in patients undergoing abdominal surgery.
Fig. 9. Comparison of isoboles of respiratory depression (10–60% isoboles for increases in Petco2, data from this study), consciousness and adequate anesthesia (50% probability lines for consciousness and adequate anesthesia in patients undergoing abdominal surgery.
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Fig. 9. Comparison of isoboles of respiratory depression (10–60% isoboles for increases in Petco2, data from this study), consciousness and adequate anesthesia (50% probability lines for consciousness and adequate anesthesia in patients undergoing abdominal surgery.
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Table 1. Results of the Bootstrap-based Model Selection
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Table 1. Results of the Bootstrap-based Model Selection
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Table 2. Population Pharmacodynamic Estimates
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Table 2. Population Pharmacodynamic Estimates
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