Despite the ubiquitous use of propofol for anesthesia and conscious sedation and numerous publications about its effect, a pharmacodynamic model for propofol-induced ventilatory depression in the non-steady state has not been described. To investigate propofol-induced ventilatory depression in the clinically important range (at and below the metabolic hyperbola while carbon dioxide is accumulating because of drug-induced ventilatory depression), the authors applied indirect effect modeling to Paco2 data at a fraction of inspired carbon dioxide of 0 during and after administration of propofol.

Ten volunteers underwent determination of their carbon dioxide responsiveness by a rebreathing design. The parameters of a power function were fitted to the end-expiratory carbon dioxide and minute ventilation data. The volunteers then received propofol in a stepwise ascending pattern with use of a target-controlled infusion pump until significant ventilatory depression occurred (end-tidal pressure of carbon dioxide > 65 mmHg and/or imminent apnea). Thereafter, the concentration was reduced to 1 microg/ml. Propofol pharmacokinetics and the Paco2 were determined from frequent arterial blood samples. An indirect response model with Bayesian estimates of the pharmacokinetics and carbon dioxide responsiveness in the absence of drug was used to describe the Paco2 time course. Because propofol reduces oxygen requirements and carbon dioxide production, a correction factor for propofol-induced decreasing of carbon dioxide production was included.

The following pharmacodynamic parameters were found to describe the time course of hypercapnia after administration of propofol (population mean and interindividual variability expressed as coefficients of variation): F (gain of the carbon dioxide response), 4.37 +/- 36.7%; ke0, CO2, 0.95 min-1 +/- 59.8%; baseline Paco2, 40.9 mmHg +/- 12.8%; baseline minute ventilation, 6.45 l/min +/- 36.3%; kel, CO2, 0.11 min-1 +/- 34.2%; C50,propofol, 1.33 microg/ml +/- 49.6%; gamma, 1.68 +/- 21.3%.

Propofol at common clinical concentrations is a potent ventilatory depressant. An indirect response model accurately described the magnitude and time course of propofol-induced ventilatory depression. The indirect response model can be used to optimize propofol administration to reduce the risk of significant ventilatory depression.

## ArticlePlus

Click on the links below to access all the ArticlePlus for this article.

Please note that ArticlePlus files may launch a viewer application outside of your web browser.

ALTHOUGH propofol is frequently used for anesthesia and conscious sedation in spontaneously breathing patients, the ventilatory depressant action of propofol is not fully investigated. To date, no potency measure (C^{50}value) for non-steady state propofol-induced ventilatory depression has been reported. Because ventilatory depression in the clinical setting occurs at uncontrolled carbon dioxide concentrations and carbon dioxide displays its own kinetics and dynamics, these effects must be taken into account to describe the time course of the effects of propofol on ventilation. A non-steady state approach accounting for both drug and carbon dioxide kinetics and dynamics has been suggested and successfully applied to the ventilatory depressant effect of alfentanil in the non-steady state. 1The C^{50}for alfentanil calculated from non-steady state data was in good agreement with C^{50}estimates using steady state approaches. 2,3The purpose of this study was to determine the ventilatory depressant potency of propofol in the setting of target-controlled propofol administration, using an indirect response pharmacodynamic model to account for the non-steady state environment in which carbon dioxide accumulates during drug-induced ventilatory depression.

Additional material related to this article can be found on the Anesthesiology Web site. Go to http://www.anesthesiology.org, click on Enhancements Index, and then scroll down to find the appropriate article and link. Supplementary material can also be accessed on the Web by clicking on the “ArticlePlus” link either in the Table of Contents or at the top of the Abstract or HTML version of the article.

## Materials and Methods

The study was approved by the Stanford University Institutional Review Board (Stanford, California). Written informed consent was obtained from each subject. The reported data are a subset from a study of propofol and remifentanil aimed at identification of each drug’s ventilatory depressant effects, the pharmacokinetic interaction between both drugs, and the interaction of both drugs with regard to suppression of quantal responses to central nervous system stimulation and electroencephalographic effects.

### Subjects

We studied five male and five female healthy volunteers aged 33 yr (range, 20–41 yr) and weighing 70.3 kg (range, 52–120 kg). All volunteers received a physical examination, laboratory tests (complete blood cell count, blood chemistries [Sequential Multiple Analysis of 20 chemical constituents]), and an electrocardiogram.

### Study Design

All volunteers were studied after fasting for at least 6 h. After arrival in the operating room, standard monitoring (noninvasive blood pressure monitoring, electrocardiography, and pulse oximetry) was established, one arterial cannula (radial artery of the nondominant hand) and two intravenous cannulas (both forearms) were inserted. The volunteers were supplied with a tight-fitting constant positive-airway pressure mask connected to a pressure differential spirometer/sidestream gas analyzer (D-lite flow sensor/gas sampler, AS/3; Datex-Ohmeda, Helsinki, Finland). The Bispectral Index was measured with an Aspect A1000 electroencephalographic monitor (Aspect Medical Systems, Newton, MA).

Before the study, the pressure differential spirometer underwent three-point calibration (500, 1,000, 1,500 ml) with a 3-l calibration syringe (Hans Rudolph, Kansas City, MO). The gas analyzer underwent two-point calibration with gas mixtures containing 4% and 8% CO^{2}. Blood pressure, heart rate, respiratory rate, tidal volume, minute ventilation (V̇^{E}), and inspiratory/expiratory oxygen and carbon dioxide were recorded every 5 s using the program Collect (Datex-Ohmeda).

#### Determination of Baseline Paco^{2}.

After a 5-min resting/equilibration period, an arterial blood sample for determination of Paco^{2}was drawn.

#### Determination of the Ventilatory Response to Carbon Dioxide.

Two anesthesia ventilation bags connected with a Y piece were filled with oxygen and connected to the flow sensor. The volunteers breathed from this reservoir and therefore rebreathed their exhaled carbon dioxide. Contrary to the classic Read design, 4the bags did not contain any carbon dioxide when rebreathing was initiated. At the volunteers’ request, the rebreathing bags but not the flow sensor were removed (on average after 5 min), enabling us to obtain blood gas and volume measurements during recovery to normal resting ventilation. Volunteers breathed room air after removal of the rebreathing bags. At the start of, during, and after the rebreathing part of the study, arterial blood samples were drawn every 1–2 min for the determination of Paco^{2}. After stabilization of V̇^{E}at baseline levels, this part of the study was terminated.

#### Determination of Propofol-induced Ventilatory Depression.

Twenty minutes after the end of rebreathing (measuring individual carbon dioxide responses), propofol was administered *via* target-controlled infusion with a Harvard infusion pump (Harvard Clinical Technology, Inc., South Natick, MA) driven by STANPUMP **running on a commercially available laptop computer. The pharmacokinetic parameters were those reported by Schnider *et al.* 5Propofol was administered in ascending steps targeting effect compartment concentrations until the end-tidal pressure of carbon dioxide (Petco^{2}) exceeded 65 mmHg and/or apnea periods of more than 60 s occurred. Thereafter, the respective drug concentration was allowed to decrease passively to 1 μg/ml. The steps used in the infusion are shown in table 1.

During and after the infusion, frequent arterial blood samples for determination of propofol concentrations and Paco^{2}were drawn. The fraction of inspired oxygen during the propofol-induced ventilatory depression experiments was always 1.0.

Figure 1displays a typical example of this administration schedule and the corresponding Paco^{2}values.

#### Sampling and Data Processing.

Blood sampling was timed according to pharmacokinetic, pharmacodynamic, and efficiency considerations. A blank sample was drawn after insertion of the arterial cannula. Sampling times were chosen on the basis of the following events: start of the infusion, step changes of the target concentrations, and decrease of the target concentrations at the end of the single drug administration.

#### Sampling Schedule.

Blood samples were drawn 2, 5, 10, and 15 min from the start of the infusion. For every additional step, 1 sample was drawn immediately before changing the target concentration. During the passive decrease down to 1 μg/ml, samples were drawn at 2, 5, 10, 15, 20, and 30 min after changing the target concentration. Propofol samples were centrifuged (3,000 rpm, 15 min) to obtain plasma and were stored at −20°C until assaying.

All arterial blood samples drawn for analysis of Paco^{2}were stored on ice immediately after being drawn and were analyzed within 20 min with a portable blood gas analyzer (i-stat Corporation, East Windsor, NJ). All volume measurements were converted to standard temperature and pressure (760 mmHg, 0° centigrade) before entering further calculations. Because Paco^{2}measurements were at best available every minute, V̇^{E}measurements every 5 s, and the further analysis required data pairs of partial pressure of carbon dioxide (Pco^{2}) and V̇^{E}matched by time, Petco^{2}values were used for calculation of the carbon dioxide response. To check for the validity of this approach, the average difference between Petco^{2}and Paco^{2}and the corresponding 95% CI was calculated. Paco^{2}measurements were used for calculation of drug-induced ventilatory depression.

#### Model Building.

The pharmacokinetic models have been described previously. 6In brief, the concentration time courses of propofol were adequately predicted by a three-compartment model with the following parameters (typical value and interindividual coefficient of variation: V^{1}= 3.8 l ± 58.1%; V^{2}= 31.6 l ± 44.8%; V^{3}= 209.0 l ± 39.2%; Cl^{1}= 3.04 l/min ± 25.5%; Cl^{2}= 3.25 l/min ± 31.1%; Cl^{3}= 1.09 l/min ± 31.5%. The Bayesian predictions of this model were used for the pharmacodynamic calculations.

The program system NONMEM, version V, with the First Order Conditional Estimation method, was used for all model fits and empirical Bayesian estimation of the individual parameters. 7Proportional and exponential models were used to describe the interindividual variability of the parameters:

where P^{(i)}refers to the individual value of the respective parameter in the ith individual, θ is the typical value of the respective parameter in the population, and η varies randomly between individuals with mean zero and diagonal variance-covariance matrix Ω.

An additive (constant SD) error model was chosen for modeling residual variability of both V̇^{E}and Paco^{2}:

where DV^{obs}refers to the observed value of the dependent variable (V̇^{E}, V^{alv}), and DV^{exp}refers to the value predicted based on dose, time, and the individual pharmacokinetic and pharmacodynamic parameters. ε is a normally distributed random variable with mean zero and variance ς. 2

Model selection was performed using the log likelihood test. Model misspecification was checked for by plotting the predicted against the measured values of the dependent variable.

#### Modeling of the Carbon Dioxide Response Curves.

The dependent variable was Petco^{2}, which was modeled as a function of V̇^{E}. Hysteresis in the V̇^{E}-*versus* -Petco^{2}relation was modeled using an effect compartment for carbon dioxide:

where Pecco^{2}is the partial pressure of carbon dioxide at the effect site (biophase) (in mmHg), and k^{e0, co2}is the first-order equilibration constant between arterial and effect site Pco^{2}.

We simplified the steady state relation by assuming equal Petco^{2}(0) and Pecco^{2}(0), neglecting the carbon dioxide production of tissue adjacent to the central chemoreceptors, which leads to Petco^{2}(0) < Pecco^{2}(0).

Although the relation between V̇^{E}and Pecco^{2}above the resting metabolic hyperbola can be well described by a straight line, 8,9the shape of the curve changes near the hyperbola. We speculated that this change persists, and is even exaggerated, if ventilation is depressed below baseline. To account for our hypothesized shape of this relation, Pecco^{2}was used as an independent variable of a nonlinear expression 1:

where V̇^{E}(Pecco^{2}) is the V̇^{E}depending on Pecco^{2}(in l/min); V̇^{E}(0) is the baseline V̇^{E}(in l/min); Pecco^{2}(0,t) is the partial pressure of carbon dioxide at the effect site (compartment), at baseline and time t (in mmHg); and F is the gain determining the change of V̇^{E}for a given ratio of Pecco^{2}(t) and Pecco^{2}(0). For reasons of completeness, a linear carbon dioxide response was also calculated as

where SL is the slope of the carbon dioxide response curve. For each individual, baseline Pecco^{2}was fixed to the measured value of baseline Petco^{2}.

#### Modeling of Propofol-induced Ventilatory Depression.

Because changes in Paco^{2}during drug-induced ventilatory depression are not as fast as those during carbon dioxide rebreathing and Paco^{2}is less sensitive than Petco^{2}to artifacts such as poor mask fit or shallow breathing, Paco^{2}was used as dependent variable for modeling propofol-induced ventilatory depression. Although described previously, 1the essential steps of the modeling approach are repeated here.

Changes of partial pressures of a gas in the body over time can be computed by mass balance equations. For a one-compartment model with constant input (carbon dioxide production) and constant output (carbon dioxide elimination) under baseline steady state conditions, the change of amount of carbon dioxide over time can be expressed as

where V^{d}co^{2}is the apparent volume of distribution of carbon dioxide (in liters), k^{in}is the production rate of carbon dioxide (in l/min), k^{out}is the elimination rate of carbon dioxide (in l/min), 760 is the atmospheric pressure at sea level (in mm Hg), and k^{out}(t) can also be expressed as the product of alveolar ventilation (in l/min) and the current Paco^{2}divided by the barometric pressure yielding:

Under the assumption that the production rate of carbon dioxide is always equal to the baseline elimination rate, the production rate can be substituted by the product of the baseline value of the normalized Paco^{2}and alveolar ventilation and becomes a constant:

Rearranging for the change of Paco^{2}, the dependent variable, over time yields:

At steady state, equation 9reduces to ventilation times carbon dioxide equals a constant, which is a restatement of the equation of the metabolic hyperbola.

A hypothetical effect compartment for propofol was introduced:

where Cp is the drug concentration in plasma calculated from the individual dosing histories and pharmacokinetic parameters; Ce is the drug concentration in the effect compartment; and k^{e0,prop}is the first-order rate constant governing the transfer of propofol out of the effect compartment.

The combined inhibitory effect of propofol (effect compartment concentration) and the stimulatory effect of Pecco^{2}on alveolar ventilation was then expressed as product of a fractional sigmoid Emax model and the nonlinear term for carbon dioxide response:

with V^{alv}(0) referring to baseline alveolar ventilation, Ce referring to the effect compartment concentration of propofol, and C^{50}referring to the concentration at which V^{alv}and therefore k^{out}will be decreased to 50% of the value in the absence of propofol, for unchanged Pecco^{2}. F was estimated from the individual carbon dioxide response curves. The equation is compatible with the notion that propofol decreases the carbon dioxide sensitivity of the respiratory controller. This equation also yields alveolar ventilation, normalized to baseline (divide both sides by V^{alv}(0)) and can therefore predict the time course of ventilatory depression after drug administration.

As an alternative approach, we used the power function advanced by Dahan *et al.* , 3with the restriction that predictions below 0 were fixed at 0:

After insertion into the mass balance equation, combining equations 9 and 11, the final equation to describe propofol induced hypercapnia can be obtained.

The volunteers received propofol in concentrations usually used for clinical anesthesia and were deeply unconscious as judged by the clinical impression and Bispectral Index. Because propofol decreases oxygen consumption/carbon dioxide production of up to 30% from baseline, 10a correction for the decreased carbon dioxide production was introduced using a negative sigmoid Emax model:

where CO^{2}prod^{min}is the minimal fractional carbon dioxide production under propofol anesthesia, fixed to 0.7. 10

We performed a sensitivity analysis by calculating the NONMEM objective function for values of the estimated parameters ranging from 0.5 to 2.0 times the typical value. For each calculation, the parameter being studied was fixed at the assigned value, and the other parameters were optimized by NONMEM. Based on the likelihood ratio test, we used a threshold of 3.84 (chi-square for 1 degree of freedom) for statistical significance, based on the notion that the value of the parameter being studied had deviated so far from the typical value that the model would benefit significantly from reintroduction of the original parameter.

## Results

### General

All volunteers completed the study. They were deeply anesthetized at propofol concentrations exceeding 3 μg/ml, as judged from both clinical observation and electroencephalographic data. However, spontaneous ventilation was maintained, a clearly different picture from that seen when administering opioids. The only adverse effects were mild-to-moderate hypotension during propofol administration.

### Carbon Dioxide Dynamics

The parameters of a nonlinear model characterizing the influence of carbon dioxide on V̇^{E}including effect compartment equilibration are summarized in table 2, and the respective values for a linear model are shown in table 3. The rate of Petco^{2}increase during rebreathing was 3.2–3.9 mmHg/min (95% CI). Calculations were performed with end-tidal carbon dioxide and not arterial carbon dioxide because of the improved time resolution. Bland-Altman analysis showed a bias between arterial and end-tidal Pco^{2}of 2.5–3.9 mmHg (95% CI). The estimated parameters are those characterized by the mixed-effects model. Baseline Petco^{2}of each individual was fixed at the measured value. Estimating this value rather than using the measured value significantly degraded the quality of the fit. As a measured value, there is no SE for baseline Petco^{2}, but the coefficient of variation is included.

The mean errors of the model predictions were 1.44 l/min for the nonlinear model and 1.67 l/min for the linear model. The nonlinear model yielded a highly significant improvement in the NONMEM objective function of 340 (*P* < 0.001). Therefore, the predictions of the linear model were not plotted, and its parameters were not used for further calculations.

The population and Bayesian predictions and goodness of fit of the nonlinear model are summarized in figure 2. The *top* shows the carbon dioxide response curves of the volunteers (Bayesian predictions of V̇^{E}*versus* Pco^{2}in the effect compartment [Pecco^{2}]). The *bottom* of figure 2shows the Bayesian predictions of V̇^{E}*versus* the actually measured V̇^{E}. Because the data points are symmetrically distributed around the line of identity, the effect compartment model and nonlinear model for carbon dioxide response adequately capture the relation of V̇^{E}and carbon dioxide in the range of measurements. The Bayesian predictions of F, the gain of the nonlinear carbon dioxide response curves, and k^{e0, CO2}were used for the calculation of propofol-induced ventilatory depression.

### Indirect Response Model Describing Drug-induced Ventilatory Depression

The remaining parameters characterizing the indirect response model describing propofol-induced ventilatory depression are summarized in table 4. Calculations were performed with Bayesian estimates of the plasma concentrations and the known drug infusion regimen. The parameters k^{e0, co2}and F were estimated independently from the carbon dioxide response curves in drug-naive subjects (table 2) and treated as known variables for each subject in modeling propofol-induced ventilatory depression. k^{el, co2}is the ratio of alveolar ventilation at baseline and its volume of distribution and is therefore completely independent of drug action. The measured value of baseline Paco^{2}was used throughout the modeling.

We compared sigmoid Emax and power functions to relate propofol concentration to drug effect. Both models have identical numbers of parameters. The sigmoid Emax pharmacodynamic model resulted in a decrease in the NONMEM objective function of 79 points compared with the power model. Therefore, the power model was rejected.

A model not accounting for decreasing carbon dioxide production with increasing propofol concentration was statistically indistinguishable from the model chosen (difference in NONMEM objective function = 2.77). Because the SEs of the C^{50}estimates for depression of carbon dioxide production and elimination overlapped, one common C^{50}value was used. The mean error for the model shown in table 4was 2.98 mmHg.

Figure 3displays the population and Bayesian prediction of Paco^{2}for one volunteer (*top* ) and the diagnostic plot of both the population and Bayesian estimates for all volunteers (*bottom* ). Although the model is not misspecified, as can be seen from the excellent fit of the Bayesian predictions, the amount of scatter of the population predictions as well as the enormous interindividual variability of the parameters alerts us to the fact that the ventilatory response in individuals may be poorly predicted by population estimates.

Figure 4displays the results of the sensitivity analysis. The model is most sensitive to changes in γ and least sensitive to changes in the carbon dioxide elimination rate constant. Comparing the sensitivity analysis in figure 4with the SEs in table 4shows that the line crosses the boundary for statistical significance at approximately ± 2 SEs from the mean, as would be expected.

Figure 4also shows that the two parameters of the carbon dioxide response model that are used in the ventilatory depression model, k^{e0, co2}and F, are not optimal for the ventilatory depression model. Instead, the model would have performed better at larger values for k^{e0, co2}and lower values for F, although the improvement did not reach statistical significance.

## Discussion

### General

Pharmacokinetic-pharmacodynamic investigations contribute to clinical drug therapy by determination of potency and subsequent design of safe and efficient dosing regimens. Although the ventilatory depressant effect of propofol was noticed and studied as soon as the drug was approved in humans, 11,12a measure of potency for ventilatory depression in the non-steady state, typically the C^{50}value, has not been determined. Either the available studies failed to explore a high enough concentration range and/or the design and analysis procedure were not geared toward determination of potency. 11–14Determination of the potency of a ventilatory depressant drug can principally be achieved by two methods: opening the major feedback loop by keeping the expiratory partial pressure of carbon dioxide constant regardless of drug concentration or sampling data contaminated by the feedback response of the system and explicitly modeling the system during data analysis. In 1995, an approach for identification of the C^{50}of ventilatory depressants by opening the carbon dioxide feedback loop (isohypercapnic approach) was published. 15The approach, although not suitable for dose finding, relied on few assumptions and, using a fractional Emax model for data analysis, determined potency by standard methods. Unfortunately, this approach has never been applied to propofol. Therefore, the isohypercapnic C^{50}of propofol for ventilatory depression is not available for comparison with the value estimated using our indirect response model.

### Experimental and Model-building Considerations

#### Pharmacokinetics.

Ideally, a dosing regimen designed for maximal disturbance of the system should be applied in non-steady state studies. Propofol concentrations were increased in a stepwise fashion with a target-controlled infusion device up to an individually determined concentration associated with severe ventilatory depression. Duration of the concentration steps was 15 min. Thereafter, the concentration was allowed to decrease passively to 1 μg/ml. This design provides both non-steady state (transition between concentration steps) and pseudo-steady state conditions and is therefore well suited for model identification.

#### Carbon Dioxide Dynamics.

Because it would be nearly impossible to simultaneously determine carbon dioxide dynamics and propofol pharmacodynamics from data with constantly changing propofol and carbon dioxide concentrations, the experimental design included a drug-naive non-steady state carbon dioxide response curve in all subjects. We preferred Petco^{2}to Paco^{2}values for determination of the carbon dioxide response because of the higher resolution of Petco^{2}values and the small bias encountered (2.5–3.9 mmHg [95% CI]). Besides adequately describing the non-steady state carbon dioxide response in our study (fig. 2, *bottom* ), the model adequately described carbon dioxide response curves simulated with a more complex physiologic model, which can be downloaded. ††

Despite fulfilling the pragmatic demand of describing the measurements, our simplistic approach neglects two well-documented peculiarities of the carbon dioxide response: (1) The controller dynamics were lumped together, and no effort was made to differentiate between contributions of a central (slower) and peripheral (faster) part of the carbon dioxide response 16with different gains. (2) Because the central chemoreceptors are embedded in carbon dioxide-producing tissue, the assumption that steady state effect compartment concentrations equal arterial concentrations is physiologically wrong.

#### Carbon Dioxide Kinetics and Pharmacodynamics.

The core of the model is a modified indirect response model, which has been described previously. 1The carbon dioxide kinetic model and its parameters can be checked for plausibility by predicting the rate of increase of the Paco^{2}during apnea. The model predicts that Paco^{2}increases 4.5 mmHg/min during apnea, which is well in agreement with the rate of increase of carbon dioxide during rebreathing (3.7–4.2 mmHg/min [95% CI]) and a standard text on respiratory physiology (3–6 mmHg/min). 17

The underlying paradigm for our model choice was that propofol causes hypercapnia by decreasing ventilation and thereby, transiently, carbon dioxide elimination (carbon dioxide mass balance approach). Originally, our modified indirect response model was built on the notion that carbon dioxide production remains constant throughout the study regardless of drug concentrations. Although this simplification can be applied when modeling opioid-induced ventilatory depression, 18it may introduce error when modeling ventilatory depression induced by hypnotics/sedatives. Moderate sedation (sluggish response to tapping on the forearm and loud voice) with propofol reduced oxygen consumption and therefore carbon dioxide production by 15%. 18This was confirmed by Pavlin *et al.* , 19who determined a 16% reduction of carbon dioxide production at a propofol concentration corresponding to moderate sedation. Clinical anesthesia with propofol and alfentanil reduces carbon dioxide production/oxygen uptake by 30%. 10Based on this evidence, we fixed the maximal effect of propofol-induced reduction of carbon dioxide production to 30% of the baseline value. As shown in the steady state equations below, omission of the effect of a respiratory depressant on carbon dioxide production leads to an overestimation of steady state ventilation for a given degree of hypercapnia (unclamped carbon dioxide).

The practical implications of our model have been summarized in a simulation (fig. 5). Equal *transient* concentrations of propofol lead to different degrees of ventilatory depression, depending on the administration schedule. A 100-mg intravenous bolus of propofol causes a pronounced decrease of ventilation. The onset of propofol drug effect is too fast (half-life equilibration between arterial and propofol effect site concentrations = 2.6 min) and the increase in Paco^{2}occurs too slowly to effectively counteract the drug effect. In fact, Paco^{2}continues to increase after the nadir of V̇^{E}has passed, demonstrating the inertia of the system. For an infusion designed to achieve the identical peak concentration (fig. 5, *top* ) the maximum depression of ventilation is much less because of the ventilatory stimulation from the increased Paco^{2}. Eventually, the Paco^{2}effect on stimulating ventilation collapses at high carbon dioxide concentrations, yielding carbon dioxide narcosis. This has not been investigated in humans for obvious reasons and must be viewed as speculation (we hereby caution the readers against extrapolations of our model into concentration/Paco^{2}ranges not explored in the original study). These data support the propofol dosing guidelines in the package insert that recommend avoiding large boluses in patients when spontaneous respiration is desired (*e.g.* , for conscious sedation). A continuous infusion reaching concentrations below 3 μg/ml, as recommended in the package insert, should not severely impair ventilation.

Finally, we would like to explain why patients experience only minimal impairment of the steady state V̇^{E}at the C^{50}for ventilatory depression and demonstrate that, even under the assumption of drug-dependent changes of carbon dioxide production, steady state ventilation in the absence of inspired carbon dioxide can directly be determined from the isohypercapnic/indirect response C^{50}value and the carbon dioxide sensitivity.

In the spontaneously breathing subject at steady state without inspired carbon dioxide, there is a hyperbolic relation between Pecco^{2}and (alveolar) ventilation, defined by the equation (alveolar) ventilation times Pco^{2}equals a constant. More specifically, alveolar ventilation equals the ratio of a constant, a, incorporating carbon dioxide production and Pecco^{2}at the respective steady state and *vice versa* :

Because a includes carbon dioxide production, we can explicitly express this aspect to account for drug-dependent changes in carbon dioxide production.

Combined with the equation for V̇^{E}accounting for both carbon dioxide and drug effects (substituting Pecco^{2}), we obtain:

with a(0) equaling the product of alveolar ventilation in the absence of drug and Paco^{2}in the absence of drug. We can rearrange equation 17to solve for the fractional change in V̇^{E}:

This equation can be used to determine the fractional steady state V̇^{E}at uncontrolled carbon dioxide from the isohypercapnic C^{50}and the gain of the carbon dioxide response curve. What becomes evident is the huge difference between the expected effect at the isohypercapnic C^{50}in the presence and absence of free-floating carbon dioxide. At constant Pecco^{2}, V̇^{E}decreases, by definition, 50% at the (isohypercapnic) C^{50}. Incorporating the carbon dioxide effect (*e.g.* , F = 4.37) and neglecting the effect on carbon dioxide production, V̇^{E}decreases only by 12.1%. Including the effect on carbon dioxide production, V̇^{E}decreases by 23% at the (isohypercapnic) C^{50}. This difference illustrates the error margin when applying a combined model neglecting effects of a respiratory depressant on carbon dioxide production.

This explains why concentrations equal to the C^{50}obtained after appropriate treatment of carbon dioxide effects (either by modeling them or canceling them out with an isohypercapnic approach) and leading to 50% reduction of V̇^{E}at clamped carbon dioxide are perfectly safe in spontaneously breathing subjects in the absence of inspired carbon dioxide, provided the levels are attained slowly. The most striking support for this prediction comes from our own dosing regimen, which was designed to avoid apnea. The dosing regimen allowed for pronounced increases of the Paco^{2}and led to the administration of concentrations exceeding the isohypercapnic (= indirect response) C^{50}value by a factor of 5.

### Limitations

The carbon dioxide response curves in this study were generated from a rebreathing study methodology. 4This technique has been criticized as yielding biased results. 20Nevertheless, some investigators believe the rebreathing technique is best suited for non-steady state experimental designs, such as drug-induced ventilatory depression. 21We believe this would be best settled by repeating the experimental design using both the rebreathing design and steady state determinations of the carbon dioxide response curve, and seeing which approach yielded the best model of drug-induced ventilatory depression.

Our rebreathing study was compromised by our failure to administer 100% oxygen after removal of the rebreathing apparatus, resulting in abrupt transition from hyperoxia to normoxia. We have simulated the proper conduct of the study (administration of 100% oxygen throughout) and the manner in which we conducted the study (room air on conclusion of the rebreathing portion) using a respiratory simulator developed by Duffin *et al.* ††As seen from these simulations, the transition from hyperoxia to normoxia had no appreciable influence on the rebreathing curves generated.

We do not explore the effect of hypoxia in our experimental paradigm. Because the peripheral chemoreceptors are relatively insensitive at normoxic and mildly hypoxic levels and shut off at hyperoxic levels, 22it is impossible to separate these effects with our design. Because propofol profoundly depresses the peripheral hypoxic drive at sedative concentrations under isohypercapnic conditions, 14limiting the model to the influence of propofol on the hypercapnic ventilatory response seems justified.

An approach taking into account different steady state conditions of the central (arterial) and effect compartment (brain) at the beginning of rebreathing requires simultaneous modeling of the kinetics and dynamics of carbon dioxide. One essential parameter of this model must be total carbon dioxide production, the *dose* in pharmacokinetic terms. It is extremely unlikely that carbon dioxide production remains constant during rebreathing (up to sixfold increase of V̇^{E}), and it is impossible to measure it under pronounced non-steady state conditions. More specifically, what we call measurement of carbon dioxide production is truly measurement of carbon dioxide excretion, which equals production in the steady state only.

In addition, fractional carbon dioxide production in the brain and the ratio of cerebral blood flow and effective cerebral volume of distribution of carbon dioxide (*microconstant* for kineticists) must be estimated. Because this cannot be estimated from our data, we settled for the physiologically wrong assumption of equal Paco^{2}and Pecco^{2}, which enables us to collapse the hysteresis between changes in Petco^{2}and V̇^{E}with a standard first-order equilibration process.

The propofol-induced model of ventilatory depression is sensitive to the accuracy of the propofol pharmacokinetic model and the carbon dioxide response model. The predictions of these models enter the propofol-induced ventilatory depression model as known quantities. Any misspecification or inaccuracy in the parameter estimates of these pharmacokinetic and carbon dioxide response models influences the prediction of the model of propofol-induced ventilatory depression.

The sensitivity analysis (fig. 4) showed improvements in model fit at larger values of k^{e0, co2}and lower values of F. It is expected that the values of these parameters are not optimal for the drug-induced ventilatory depression model, as these two parameters were not derived in the optimization of the parameters of the ventilatory depression model. The sensitivity analysis indicates that the ventilatory depression model would perform better if we used a larger value of k^{e0, co2}, producing even faster plasma effect site equilibration. Given that the nominal value of k^{e0, co2}has a half time of equilibration of only 0.73 min in the current model, this suggests that a more parsimonious model that eliminated the carbon dioxide equilibration delay altogether would, in fact, have likely performed marginally better than the final model in the analysis. We chose to leave carbon dioxide equilibration in the model for completeness, even though it does not seem to have contributed to the quality of the fit.

The model would have performed marginally better if the values of F estimated from the carbon dioxide response data had been lower. Examination of the fits shows that the improvement comes by offsetting changes in F with changes in C^{50}of propofol. This validates our previous observation that it is not possible to concurrently estimate C^{50}and F from ventilatory depression data alone. 1Nevertheless, because both k^{e0, co2}and F can change by 30–40% with no significant change in −2 log likelihood, the model of ventilatory depression is not unduly sensitive to the parameters of the carbon dioxide response model.

Use of our model necessitates estimation of the influence of carbon dioxide on ventilatory drive. Because we cannot concurrently estimate the effect carbon dioxide and drugs on ventilatory drive, we gathered our information on carbon dioxide response while the volunteers were unmedicated and hyperventilating as a result of inspired carbon dioxide. We then applied this model of carbon dioxide drive to the volunteers when they were hypoventilating from drug-induced ventilatory depression. Typically, one wants to avoid extrapolating predictions beyond where the data were gathered, and no one has demonstrated that carbon dioxide response in the hyperventilating, hypercapnic patient accurately describes the carbon dioxide response in the hypoventilating, hypercapnic patient with drug-induced ventilatory depression. The fact that the model worked well provides evidence the carbon dioxide response curve can be extrapolated from hyperventilating patients to hypoventilating patients, but this must be explored further.

We could have modeled ventilatory depression using V̇^{E}, Paco^{2}, or both. Although we gathered both V̇^{E}and Paco^{2}, we chose to model Paco^{2}because (1) Paco^{2}is the definitive standard for measuring ventilation; (2) the Paco^{2}data were of higher quality than the V̇^{E}data, owing to the difficulty of maintaining a good mask fit during 75 min of drug-induced ventilatory depression; and (3) we implicitly model fractional alveolar ventilation and not V̇^{E}, so inclusion of the V̇^{E}data would necessitate calculation of dead space, which changes with changing tidal volumes. Therefore, we focused on Paco^{2}, which proved to be sufficient to characterize the C^{50}of propofol-induced ventilatory depression.

## Conclusion

In conclusion, we extended the applicability of the modified indirect response model to propofol and showed its validity even when using an application scheme consisting of multiple concentration steps. This model can be used to design dosing regimens to minimize propofol-induced ventilatory depression. Our raw data, NONMEM control streams, and an implementation of the model in the STELLA simulation are available on the Anesthesiology Web site at http://www.anesthesiology.org.