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Predictive Accuracy of a Model of Volatile Anesthetic Uptake

Kennedy, R. Ross MB, ChB, PhD, FANZCA; French, Richard A. MB, BS, FANZCA; Spencer, Christopher

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doi: 10.1097/00000539-200212000-00027
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A number of models of the uptake and distribution of volatile anesthetics have been described over the years (1–6). Those that have been subjected to validation have been shown to meet the various criteria set by these authors. Unfortunately, each group has used slightly different statistical methods to evaluate their model, making comparisons between models difficult.

A model of anesthetic uptake and distribution was developed in this department some years ago by Heffernan et al. (7,8). This model is based on the work of Mapleson (2,9) and was developed as an educational tool to illustrate factors affecting uptake and distribution. We have used this model to explore the relationship between cardiac output and volatile uptake (10), but the model has not been clinically validated.

Many new anesthesia machines, such as the Datex Anesthesia Delivery Unit (ADU), electronically export fresh gas flow rates and vaporizer dial settings, which allows the changes in these variables to be accurately documented. These fresh gas and vaporizer settings can then be used as inputs for uptake and distribution models, and the output of the model can be compared with actual values recorded from anesthesia monitors. This allows the pattern of volatile anesthetic administration from routine anesthesia to be used to study these models. The recent arrival of the ADU in our department allowed the opportunity to validate our model.

One of the several goals of modeling volatile anesthetics is to help develop systems for automated control, as is already available for IV drugs. A statistical method for analyzing the performance of computer-controlled infusion systems has been described by Varvel et al. (11). This method has been used in a number of studies of both propofol and opioid infusions. Propofol infusion systems such as the Diprifusor® are in common clinical use, and the performance of these systems may provide a useful standard against which to measure the performance of models of volatile anesthetic uptake.

The purpose of this study was to compare the expired volatile anesthetic concentrations predicted by our model with those actually occurring during a variety of anesthetics. We chose to use the statistical methods described by Varvel et al. (11) for the reasons outlined previously. Swinhoe et al. (12) have defined acceptable limits for a target-controlled infusion (TCI) system in terms of the method of Varvel et al. (11) as a median performance error of less than 10%–20% and a median absolute performance error of 0%–40%. These limits were used to define acceptable performance of the model.


The Canterbury Ethics Committee approved this study. Thirty patients were studied over a 2-mo period. All patients were undergoing elective surgery in one particular operating room at times when the required equipment and one of us (CS) were available. In all cases, the airway was secured with either an endotracheal tube or a laryngeal mask airway. There was no other selection of cases, and the anesthesiologist in charge of the case was given no instructions or guidelines as to how to administer the anesthetic or the choice of drugs. At the time this study was conducted, isoflurane and sevoflurane were the most common anesthetics in use, making up 40% and 46% of the primary anesthetics, according to audit data, with propofol-based anesthesia making up the balance. Desflurane was not available, and halothane use was minimal outside pediatric practice.

The data streams produced by the Datex ADU and Datex AS/3 monitor were collected by a Macintosh 8600 computer (Apple Computer, Inc., Cupertino, CA) by using separate but similar data logging applications we have developed. These data were later converted to text files, from which fresh gas flows, vaporizer settings, anesthetic selection, and measured expired volatile anesthetic concentrations were extracted. An observer (CS) was present throughout each case to ensure that the recordings were started and stopped at the appropriate times and that basic information about the case was recorded, but the observer took no part in the administration of the anesthetic. The age and weight of the patient were also recorded.

Anesthetic uptake and distribution was modeled with a multiple-compartment model based on that described by Heffernan et al. (7). The model is similar to those of Mapleson (2,9), with nine compartments—circuit, lung/blood, heart, brain, kidney, liver, muscle, fat, and poorly perfused tissues—and is described in more detail in Appendix 1. We have extended the model to include a number of volatile anesthetics and have also made a number of minor changes to the original model (10). The model does not include compartments to mimic blood transit time (2,4).

Other variables of the model, including the size of tissue compartments and relative blood flows, were as described by Heffernan et al. (7). Blood/gas and tissue/blood partition coefficients are those we have used previously (10,13).

Fresh gas flows and vapor dial settings collected from the ADU were run through a program containing the model. The model was initialized with the volatile anesthetic used and the patient’s weight. The volumes of the tissue compartments of the model were scaled linearly according to weight. Ventilation and cardiac output were adjusted by using Brody’s approach based on weight to the 3/4 power, as described by Lowe and Viljoen (14). The expired concentrations calculated by the model and those collected from the monitor were combined in a single spreadsheet and the timestamps aligned.

Data were analyzed with the methods described by Varvel et al. (11) as follows:

The performance error (PE) for each data point sampled is calculated as MATH where Cm is the measured concentration and Cp is the predicted concentration at each point in time.

Four variables are calculated from the PE. Median absolute PE (MDAPE) for each individual is a measure of the inaccuracy of the model and is the median of the absolute values of the PE in that individual. Divergence estimates the time-weighted deviation of the PE and is the slope of the linear regression line when |PE| is plotted against time. Units of divergence are percentage divergence per hour. Median predictive error (MDPE) is a measure of bias for the individual and is the median of the PE for the individual. Wobble is a measure of the variability of PE in the individual and is calculated as the median of the absolute value of the difference between MDPE and each PE.

The first 5 min of the anesthetic were excluded from the analysis, primarily because models of this type are not able to track changes on a breath-by-breath basis (2). Other confounding factors during this period included frequent large changes in fresh gas flow and vaporizer settings, loss of gas and vapor during bag mask ventilation before intubation, and the effects of initial rapid uptake of nitrous oxide (if used). Many of these factors also apply near the end of a case, and, in addition, Cp approaches 0, making PE unduly sensitive to small changes in Cp. For this reason, we chose to also exclude the final 5 min of data from each case.

Results were pooled by calculating the means and 95% confidence intervals for MDAPE, divergence, MDPE, and wobble. Results with isoflurane and sevoflurane were compared by using a two-tailed Student’s t-test.


Data were collected from 30 elective general and vascular surgical cases. Patient ages ranged from 8 to 88 yr (mean, 56 yr), and weights were 30–90 kg (70 kg). Isoflurane was used in 14 patients and sevoflurane in 16 patients. The mean duration of anesthesia was 93 min (range, 19–260 min).

Figure 1 shows the PE for all cases up to 120 min. Except for occasional transients, PEs for all except one patient were between −50% and +30%.

Figure 1:
Performance error (PE) for all 30 cases plotted against time. This plot allows the performance of the model in each patient and in the group to be visualized, providing a graphical representation of the bias and time-weighted variability.

Table 1 shows the values for the various descriptors for all cases, separated by the volatile anesthetic used. Mean values are shown with 95% confidence intervals. The confidence intervals for MDPE and divergence both include 0, implying no overall bias or deviation with time.

Table 1:
Values for Median Performance Error (MDPE), Median Absolute Performance Error (MDAPE), Divergence, and Wobble for All 30 Cases, Separated by the Volatile Anesthetic Used

When the results for isoflurane and sevoflurane were compared by using a two-tailed Student’s t-test, there was no significant difference between the anesthetics in the predictive performance of the model (MDPE, P = 0.15; MADPE, P = 0.06; divergence, P = 0.62; wobble, P = 0.85).


The aim of this study was to evaluate the performance of a model of anesthetic uptake and distribution. The model used in this study was developed to demonstrate and teach various aspects of volatile kinetics and distribution and not for clinical use (7,8). The availability of anesthetic machines that allow direct capture of fresh gas flow rates and vapor settings to a computer allowed us to study the performance of the model we have used for several purposes over a number of years (10).

One reason for performing this assessment was that this model has a number of well recognized limitations. This model lumps all blood and lung volumes into a single compartment and does not include any transport delay from the lungs to and among the various tissue compartments. Mapleson (2) concluded that the effect of this simplification is apparent only briefly after major changes in concentration. The model also assumes ventilation and circulation to remain constant throughout the case and to be continuous rather than cyclical functions. The effect of this simplification is also apparent only briefly around transient changes. Other potential limitations inherent in this type of model have been recently summarized by Hendrickx et al. (15).

Despite these limitations, the model was able to predict expired concentrations of isoflurane and sevoflurane with reasonable accuracy and minimal bias or deviation. Acceptable limits for a TCI system have been defined by Swinhoe et al. (12) as MDPE <10%–20% and MDAPE 20%–40%. Our results with this model occur well within these values. The upper 95% confidence intervals for both the pooled results and for sevoflurane and isoflurane separately are all less than these limits.

A number of authors (12,16,17) have used the method of Varvel et al. (11) to analyze the propofol infusion model used in commonly available TCI systems. The range of mean values for MDAPE from these studies was 21% to 24.1%; for MDPE, −12.1% to 16.2%; for divergence, −17%/h to −2.9%/h; and for wobble, 7% to 11.6%. Our results for the model of volatile anesthetic uptake in this study (Table 1) are at or below the lower limit of these ranges, both for the overall results and also when isoflurane and sevoflurane are considered separately. Thus, our model was able to predict end-tidal volatile concentrations of both sevoflurane and isoflurane at least as well as the data sets in these studies describe the relationship between predicted and measured blood levels of propofol.

There are several parallels between this study and assessments of TCI systems. In both cases, a model is initialized with a small number of patient characteristics and details of the drug to be used. The only other information available to the modeling system is the various changes made in the rate of the administration of the drug, the infusion rate in the case of IV anesthetics, and total fresh gas flow and vaporizer settings for volatile anesthetics.

It is useful to consider the clinical implications of the various measures calculated. MDAPE is a measure of the size of typical error (miss). Assuming a target of 1 minimum alveolar anesthetic concentration (1.3 vol% for isoflurane and 2.0 vol% for sevoflurane), our results suggest a typical error of <0.2 vol% isoflurane and <0.3 vol% for sevoflurane. MDPE estimates the bias (0.003 vol% isoflurane and 0.005 vol% sevoflurane). Divergence represents any time-weighted trend away from or toward the targeted concentration (0.05 vol%/h isoflurane and 0.08 vol%/h sevoflurane). Wobble is the instability of the prediction (0.03 vol% isoflurane or 0.05 vol% sevoflurane).

We found low values for both wobble and divergence in this study. Low values for these variables allow the clinician to titrate the predicted values against patient response without knowing either the actual value or the bias in the individual patient (11). When wobble and divergence are low, the user can assume that the bias remains constant in an individual patient. This principle is part of what makes TCI systems useable in clinical practice.

A number of the studies of the predictive accuracy of models of volatile anesthetic uptake and distribution, such as those of Lerou et al. (4,5) and Beams et al. (3), have been developed for low-flow or closed-circuit anesthesia and, as such, are based on direct injection of the anesthetic into the circuit. While there are enthusiasts for very low flows and direct injection of liquid volatile anesthetic into the anesthetic circuit (18), this is not the way most anesthesiologists practice. Therefore, one aim of this study was to collect data covering a wide range of typical anesthetics administered by a number of unselected anesthesiologists.

Several models of volatile uptake and distribution based on physiological compartments have been described and tested (2,19–22). All reported results suggest that the model in question performed reasonably well, although it may be that negative results were not reported. Despite the theoretical problems identified with models of this type, as described previously, several studies have identified factors that are less critical in this type of model. Mapleson (2) demonstrated the value of allowing for the delay resulting from blood transit times in this type of model but also suggested that these are only necessary to follow changes on a breath-by-breath basis. Vermeulen et al. (23) suggested that incorporating factors such as age, sex, or various measures of body weight offers little or no benefit.

A number of compartmental models derived from measurements have also been described (6,24) and have also performed well. However, one major advantage of models based on physiological compartments is that the interrelationship between compartments can be demonstrated in a way that is easily understood. This is a reason why models designed to teach aspects of volatile kinetics based on physiological models have been successful (8,25).

It is difficult to compare our results with other studies of various models of volatile uptake and distribution because a wide variety of statistical methods have been used in these studies. Lerou et al. (4,21) and Lerou and Booij (5,22) have extensively evaluated a 14-compartment model that in more recent versions (5,22) includes all gases in the breathing system. This model administers volatile anesthetics as intermittent liquid boluses, and although it is designed to operate over a wide range of total fresh gas flows, it is primarily being developed as a tool to guide closed-circuit anesthesia. They have developed an alternative method to analyze the model performance, which they regard as particularly rigorous. We chose to use the method of Varvel et al. (11) for our statistical analysis, because we were interested in how well our model of volatile anesthetic uptake and prediction performs, and propofol models, which are in routine clinical use, provide a useful standard. We believe that by using what has become a standard method of analysis, it should be possible for future studies to make appropriate comparisons with our results.

In summary, we have shown that a relatively simple model of volatile anesthetic uptake and distribution is able to predict end-tidal values of isoflurane and sevoflurane under normal clinical conditions. This model predicts end-tidal anesthetic concentrations at least as well as propofol TCI models predict blood propofol levels.

Appendix 1: The Model

The model is based on that described by Heffernan et al. (7,8). Heffernan et al. developed the model to teach various principles of the uptake and distribution of halothane, including the effect of halothane concentration on cardiac output and ventilation. The model has 10 compartments, as listed in Table 2, which also lists the volume of each compartment along with the fractional blood flows and the partition coefficients used. The model assumes continuous rather than cyclical ventilation and circulation and does not allow for blood or gas transit delays.

Table 2:
Variables of the Compartments Used in the Model

The model assumes equilibrium between alveolar and arterial pressure and includes a lung tissue volume (Vlt) of 0.5 L and an arterial blood volume (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 a functional residual capacity of 2.5 L and half a tidal volume of 500 mL.

Thus, the effective volume of the lung compartment (Veff) is MATH where λb/g and λl/b are the blood/gas and lung tissue/blood partition coefficients, respectively.

By using the equations of Brody, as described by Lowe and Viljoen (14), cardiac output (𝑄̇) is calculated as 0.5 × kg3/4, where kg is the weight in kilograms as entered by the user. Similarly, minute ventilation (V̇) is calculated as 0.16 × kg3/4 to give an alveolar CO2 of 5%. 𝑄̇ was then empirically reduced by one fourth to allow for the effects of anesthesia. The tissue compartments of the model are scaled linearly as weight changes from 70 kg.

The Circuit

The mass of anesthetic (FF) entering the circuit is calculated as vapor % × total fresh gas flow (VF). Assuming that VF < circuit volume (i.e., some rebreathing), the change in circuit concentration (FB) is calculated as MATH where VB is the volume of the circuit, FA is the alveolar concentration, and V̇D is dead space ventilation. The expression FB · (V̇F − 0.3) represents loss from the circuit and allowing 300 mL representing oxygen uptake and losses to sampling equipment.


The change in alveolar concentration (FA) is calculated as MATH where 𝑄̇ is lung blood flow and FJOURNAL/asag/04.03/00000539-200212000-00027/ENTITY_OV0456/v/2021-09-14T033004Z/r/image-png 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.


From the Fick equation MATH assuming that 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 MATH

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


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