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The Effect of Using Different Values for the Effect-Site Equilibrium Half-Time on the Prediction of Effect-Site Sevoflurane Concentration: A Simulation Study

Kennedy, R Ross, MB, ChB, PhD FANZCA

doi: 10.1213/01.ane.0000168337.88469.9c
Anesthetic Pharmacology: Research Report

We have developed a predictive display that allows effect-site concentration (Ceff) to be used as a target for administration of inhaled anesthesia. Ceff is dependent on the half-time for plasma effect-site equilibrium [t½½(ke0)]. The t½(ke0) used in the predictive display is fixed and may differ from that in the patient. We wished to explore the effect of this difference on predictions of Ceff. In a computer simulation, fresh gas flow and vaporizer settings required to achieve a predefined time profile for Ceff were determined for t½(ke0) of 2.5, 3.5, and 5 min. The end-tidal values for each simulation were used to recalculate Ceff with each t½(ke0). The maximal deviation at predetermined points, measures of global fit, and the delay in “recovery” were calculated. With a predictive display t½(ke0) of 3.5 min, the maximal error in Ceff was 0.18 vol%, occurring during the wash-in phase and disappearing within 2–3 min. The difference in time for Ceff to decrease from 1.0 to 0.7 vol% was 1.3 min. Results with a display t½(ke0) of 2.5 min or 5 min and simulated patient t½(ke0) of 5 min or 2.5 min were approximately twice as large. These results suggest that Ceff is relatively insensitive to large (50%–100%) variations in t½(ke0).

IMPLICATIONS: A model-based predictive display to guide effect site targeting of volatile anesthesia is described. The effect of using different values for the rate of transfer of sevoflurane between central and effect site compartments is explored. The results suggest effect site concentration is relatively insensitive to 50-100% variations in half-time for plasma effect-site equilibrium.

Department of Anaesthesia, Christchurch Hospital & Christchurch School of Medicine and Health Sciences, Christchurch, New Zealand

Accepted for publication February 15, 2005.

This project was supported by a grant from the Canterbury Medical Research Foundation. April 15, 2005.

Address correspondence to R. Ross Kennedy, MB, ChB, PhD, FANZCA, Department of Anaesthesia, Christchurch Hospital & Christchurch School of Medicine and Health Sciences, Rolleston Ave., Christchurch, New Zealand. Address e-mail to

We previously described a model-based predictive display that permits manual delivery of target-controlled volatile anesthesia (1). The underlying model is a nine-compartment model of anesthetic uptake and distribution that, despite a number of theoretical limitations (2), predicts end-tidal (ET) isoflurane and sevoflurane concentrations at least as well as frequently used propofol models predict propofol blood levels (2). Target-controlled infusion for propofol and several other drugs including remifentanil and ketamine (3) is well described and commercial systems for propofol delivery are available in many parts of the world outside of the United States (4).

Our model-based predictive display for volatile anesthesia takes values of fresh gas flow (FGF) and vaporizer setting every 10 s to model the progress of the anesthetic and to predict future ET values, which are then corrected for the current ET values. By changing FGF and vaporizer dial setting, the anesthesiologist can rapidly see the effect of different “input” patterns and choose that which best suits the clinical needs. We have previously demonstrated that this display allows anesthesiologists to make step changes in ET sevoflurane levels at least twice as fast as when the display is unavailable (1).

Several studies have explored the value of using propofol effect-site concentration (Ceff) rather than plasma levels as the target. These studies have shown that Ceff are more closely related to certain clinical end-points than plasma levels (5,6) and that the addition of effect-site information may improve the stability of Ceff with both closed loop (7) and manual (8) propofol delivery systems. We consider that many of the benefits of effect-site targeting seen with propofol should also occur when Ceff is used as a control variable during administration of volatile anesthetics. As a first step in evaluating this assertion, we have extended our predictive display for target control of volatile anesthetics to allow targeting of Ceff.

Calculation of Ceff is dependent only on blood (or central compartment) concentration and on the value chosen for the rate of flux between the central and effect-site compartments (ke0). Whereas central-compartment concentrations can be approximated by ET values, defining an appropriate value for ke0 is less straightforward.

The equilibrium half-time (t½) for sevoflurane in the “brain” compartment in the multicompartment model underlying our target-control predictive display is 3.4 min. This compares well with the value of 3.5 ± 2 min (mean ± sd) reported by Olofsen and Dahan (9). Although the mean value is very close to that used in the model, these results do describe a range of values. We were interested in the effect on Ceff when the effect-site t½ in the patient differs from that being used by the predictive display system.

Our purposes are to describe the extensions to the system to allow effect-site to be used as a control variable during administration of volatile anesthesia and to consider, in a computer simulation, the effect of using a range of values for t½(ke0) on predictions of Ceff with sevoflurane.

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Prediction and display of Ceff is an extension of our previously described system for targeting ET concentrations (1). We have previously shown that the model underlying this system predicts ET concentrations at least as accurately as many models for IV drugs with MDPE (median performance error, a measure of bias for the individual) of 3.4% and MADPE (median absolute performance error, a measure of the inaccuracy of the model) of 10.9% (2). One of the nine compartments of the underlying model represents brain. When initialized for a standard 70-kg person and sevoflurane, the characteristics of this compartment (volume 1430 mL, brain bloodflow 492 mL/min, solubility 1.7) give a central-compartment time constant (t) of 4.91 or t½ of 3.43 min. Further details of the model are given in Appendix 1.

Estimation and prediction of Ceff is in two phases. The current Ceff is calculated by using the current measured ET concentration as an estimate of plasma concentration, combined with a t½(ke0) of 3.43 min. This is achieved by implementing a separate “instance” of the brain compartment of the model which uses measured ET values in place of the modeled ET value at each iteration. The calculated Ceff at each point in time is displayed numerically and included in the system trend display along with the ET concentration.

The predictive component of the system takes the current state of the model, which is updated every 10 s, and then, using the current FGF and vaporizer dial settings, looks ahead 10 min to predict concentrations in each compartment of the model. The method for this prediction has been outlined recently (1). Predictions for the brain compartment are converted to vol% equivalents by multiplying by the brain/blood partition coefficient to allow direct comparison with ET values and then bought into line with the current Ceff by multiplying all predictions by the ratio (Ceff/current modeled brain concentration) in a similar manner to that used for ET values (1). Predicted effect-site values are displayed graphically and numerically along with ET predictions within an integrated trend display system (Fig. 1). As the user changes the inputs (FGF and vaporizer setting), the predictions for ET and Ceff are updated. This allows the anesthesiologist to select a combination of inputs that either maintains the current Ceff or to produce a change in Ceff with an optimum profile for the specific clinical requirements.

Figure 1.

Figure 1.

The physiological model of inhaled anesthetic uptake was implemented as a C program on an Apple Macintosh G4 system by adapting the program used clinically to generate the data for this simulation study. Values for the t½ for blood to brain transfer of sevoflurane in the model of 2.5, 3.5, or 5.0 min were produced by altering bloodflow to the brain compartment to simulate different values of t½(ke0). FGF and vaporizer settings were read from a text file and the model iterated at 10-s intervals. All other parameters of the model assumed a 70-kg subject.

For each t½(ke0), a pattern of “vaporizer” settings was determined by trial and error to produce a predetermined pattern in Ceff (within ±0.01 vol% of the target). FGF was maintained at 4 L/min in all simulations. The pattern for Ceff was:

  • Vaporizer on at t = 1 min, constant value until t = 20 min
  • Ceff to reach 1.5 vol% at 20 min
  • Ceff maintained at 1.5% until 60 min
  • Vaporizer off at t = 60 min
  • Vaporizer on when Ceff reached 1.0
  • Ceff maintained at 1.0 vol% until t = 100 min

Vaporizer off at t = 100 min, simulation continued until t = 160 min

Table 1 shows the input pattern for a t½(ke0) of 3.5 min.

Table 1

Table 1

The sequence of ET values produced was imported into an Excel spreadsheet, which calculated corresponding values for Ceff for any t½(ke0) using the same numerical method as in the main program. The performance of the spreadsheet implementation was validated by comparing results with those produced by the model and also by demonstrating that any step change in ET concentration produced an exponential wash-in with the expected characteristics.

For each set of ET values, Ceff was calculated in the spreadsheet with a t½(ke0) of 2.5, 3.5, or 5 min. The original “target” Ceff patterns can be considered the behavior of the simulated patient whereas results with other values for t½(ke0) represent the output of the predictive display when using different values for t½(ke0).

Data sets were compared in a number of ways. The difference at each point in time was calculated as vol% and as performance error (PE = [measured – predicted]/predicted * 100). The maximal difference and PE were determined.

The difference in Ceff in the simulated patient and the predictive display were determined at defined points in the simulation to see how much the system would lag or drift once the apparent target was reached. The points taken were the difference (vol%) at t = 20 and at the time the vaporizer was turned on after the decrease from 1.5% to 1% and the difference in time to reach 0.7 vol%. The maximal difference (vol%) occurring at any point in time after t = 2.5 min was also recorded.

Markers of the global fit of data, MDPE, MDAPE, and “wobble” were calculated. MDPE is the median PE for the individual, MADPE is the median of the absolute values of PE for the individual, and wobble is the median of the absolute value of the difference between MDPE and each PE. These measures were developed by Varvel et al. (10) to describe the fit of computer-controlled infusion pumps. They have become the standard for this type of measurement and have been used to compare other sets of time-based data (11).

The simulations and the underlying modeling were performed without consideration of inter- or intrasubject variability (12). The results are therefore purely descriptive and no further statistical analysis was performed.

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Figure 2 shows the values required for ET to produce the desired pattern of Ceff for a t½(ke0) of 3.5 min. The results are summarized in Table 2.

Figure 2.

Figure 2.



Table 2

Table 2

Of particular interest are the results representing a “display” t½(ke0) of 3.5 min because this represents the clinical situation that might occur with the predictive display using a t½(ke0) of 3.5 min but with the actual t½(ke0) in the patient of 2.5 or 5 min. The central two columns of Table 2 represent these situations.

When “adjacent” values for t½(ke0) (e.g., 2.5 and 3.5 or 3.5 and 5 min) are compared, the maximal deviation in Ceff of sevoflurane was up to 0.18 vol% whereas, for the larger differences (2.5–5 min), it was 0.33 vol%. These maximums occurred as the first target for Ceff of 1.0 vol% was approached. The maximal values for PE were 31% and 55%, respectively, occurring early in the simulation when the denominator for calculating PE was small.

Values for the measures of global performance are low, representing good performance. The maximal absolute value for MDPE was 2.72%, MADPE 6.8%, and wobble 2.7%. These are well within suggested guidelines for control systems and suggest that the overall differences between sets of values for Ceff are small.

The differences in “system” Ceff at t = 20 min and at the point the “patient” reaches a Ceff of 1.0 are ≤0.11 vol% for a single step and ≤0.23 vol% for the steps between t½(ke0) values of 2.5 min and 5 min. The difference in time to reach 0.7 vol% is <80 s for adjacent values of t½(ke0) and <140 s for the large difference.

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Target control of plasma levels offers a number of potential advantages over conventional methods of administration including better use of drugs, the ability to make changes faster and more accurately, and avoiding peaks and troughs in plasma levels. Although most of the work on target control relates to IV drugs, and in particular propofol, many of these advantages also apply to target-controlled delivery of volatile anesthetics. In addition, low flow techniques are facilitated and the predictive display system may help avoid inadvertent under- or overdosing (1). Targeting the effect-site of a drug has further advantages over targeting plasma levels, such as more accurately predicting loss of consciousness (5) and allowing onset of effect to be hastened without adverse hemodynamic effects (5,6). Making the Ceff of propofol available improves the stability of Ceff and bispectral index (BIS) in both manual (8,13,14) and automated (7) control systems. Combining Ceff and effect-site activity measures such as BIS may further enhance stability of both Ceff and BIS. Improved control of Ceff should also improve drug utilization.

During inhaled anesthesia, ET values are routinely available and provide more information than is available with nontarget-controlled propofol administration when no estimates of plasma or Ceff are available. The values for t½(ke0) of isoflurane and sevoflurane are similar to values frequently used for propofol. If effect-site information is useful in the administration of propofol, many of the same advantages should apply with inhaled anesthetics. Development of a system that displays an estimate of Ceff in real time and allows Ceff to be used as a control variable is the first step in establishing the value of effect-site control with inhaled anesthetics.

The results of this simulation study suggest that the error in predicted Ceff for an assumed t½(ke0) of 3.5 minutes but with actual values between 2.5 and 5 minutes would not be clinically significant with errors of <0.2 vol%, which disappear over a few minutes. In practice, many factors are used to determine the desired target for a particular patient at a particular point in a procedure. The actual value of the target is less important than the ability of the control system to maintain the desired values (10). Our results suggest that if these various factors indicated that the patient had reached a desired “state” and our system was used to maintain the Ceff, there may, at worst, be a small drift in Ceff over the few minutes after a large change.

Measures of global performance (MDPE, MADPE, and wobble) showed that the pairs of data were very similar. This is predictable in a situation in which the only differences between the data sets occur around the transients produced by the abrupt changes. Although others have used these parameters to compare other sets of data during anesthesia (11), a good fit as determined by these parameters may need to be treated with caution when used outside the setting for which these parameters were developed, which was to describe the performance of models, specifically in computer-controlled infusion pumps (10).

The predictive display uses a value of 3.4 minutes for t½(ke0) of sevoflurane. This is close to the mean value of 3.5 minutes determined by Olofsen and Dahan (9). The 95% confidence intervals for the results of Olofsen and Dahan are 2.4 and 4.6 minutes. The values of 2.5 and 5 minutes used in this simulation study, therefore, approximate the range of values for patient t½(ke0).

The pattern of Ceff chosen for this simulation was purely arbitrary and was designed to explore the effect of large changes in Ceff on the ability of the system to track Ceff rather than to represent the conduct of an idealized anesthetic, although the pattern displays many features of an anesthetic. The initial increase occurs over 20 minutes, rather than more rapidly, to represent the usual situation in which inhaled anesthetic is introduced after an IV induction and the time delay between induction of anesthesia and skin incision. The target value for Ceff was kept at each of the two steps for a reasonable length of time to bring the total system nearer to equilibrium before imposing a change. Both these factors will exaggerate the delay in changes in Ceff, the different effects of t½(ke0), and the difference between Ceff and ET values (Fig. 2). Although an arbitrary pattern, values for Ceff of 1.5 and 1.0 vol% sevoflurane are frequently seen in clinical practice, especially in association with adjunct drugs and when effect-site activity monitoring, such as BIS, is used. In clinical practice, we often find required changes in Ceff are 0.1–0.2 vol% sevoflurane rather than the 0.5 vol% change simulated. The decrease in Ceff to 0.7 vol% (0.3 minimum alveolar anesthetic concentration) simulates time to wake. Differences between simulations in the time for the decrease to 0.7 vol% are approximately 1 minute and are likely to be masked by the effect of other drugs such as opioids.

In conclusion, we have described a system that displays predictions of Ceff for sevoflurane and, with suitable modifications, other inhaled anesthetics. This predictive display can be used to guide effect-site target control of sevoflurane anesthesia. The effect of using a set value for t½(ke0) of 3.5 minutes for the predictive display does not seem to greatly affect predicted Ceff over a wide range of simulated patient values for t½(ke0). Further data on t½(ke0) for sevoflurane under a variety of circumstances and in combination with other drugs would help refine this system and formal evaluation of the clinical value of effect-site targeting of volatile anesthesia is required.

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Appendix 1: The Model

The model is based on that described by Heffernan et al. (15,16) developed to teach various principles of the uptake and distribution of halothane including the effect of halothane concentration on cardiac output and ventilation. These feedback effects are not included in the present implementation of the model. The model has 10 compartments (arterial, lung, heart, brain, liver, kidney, muscle, fat, vessel poor group, circuit, and shunt) each of which has a volume, fractional bloodflow, and tissue/blood partition coefficient (2). The model assumes continuous rather than cyclical ventilation and circulation and does not allow for blood or gas transit delays.

The model assumes equilibrium between alveolar and arterial pressure and includes lung tissue volume Vlt of 0.5 L and arterial blood (Vart) of 1.03 L in the lung compartment. The remainder of the blood volume is considered as part of the tissue compartments. Because ventilation is assumed to be continuous, the lung gas volume (VA) is taken as 2.75 L representing function residual capacity of 2.5 l and half a tidal volume of 500 mL. Thus, the effective volume of the lung compartment (Veff) is

where λb/g and λl/b are the blood/gas and lung tissue/blood partition coefficients, respectively.

Using the equations of Brody as described by Lowe et al. (17), cardiac output (???) is calculated as 0.5. kg3/4 where kg is the weight in kilograms as entered by the user. Similarly, minute ventilation (???) is calculated as 0.16. kg3/4 to give an alveolar CO2 of 5%. Cardiac output was then empirically reduced by 3/4 to allow for the effects of anesthesia. The tissue compartments of the model are scaled linearly as weight changes from 70 kg.

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The Circuit

The mass of drug (FF) entering the circuit is calculated as vapor % times total fresh gas flow (VF).

Assuming VF < circuit volume (i.e., some rebreathing), the change in circuit concentration (FB) is calculated as

where VB is the volume of the circuit, FA is the alveolar concentration, VD and is dead space ventilation. The expression represents loss from the circuit and allowing 300 mL representing O2 uptake and losses to sampling equipment.

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The change in alveolar concentration (FA) is calculated as

where ??? is lung bloodflow and Fv is mixed venous content. The first part of this equation represents movement in and out of the lungs whereas the final term allows for the concentration effect.

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From the Fick equation

assuming venous blood in each tissue is at the same partial pressure as that in the venous blood leaving it. The total venous fraction of blood is

The 7 tissue compartments are numbered 2–8. The bloodflow through the peripheral shunt (compartment 10) has concentration of arterial blood.

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