### Part I. Model development

After obtaining IRB approval (OLV Hospital, Aalst, Belgium) and written informed consent, 50 ASA physical status I or II patients presenting for plastic, urologic, or gynecologic surgery were enrolled. All patients received oral alprazolam (0.5 or 1.0 mg) 1 h before the scheduled start of surgery. After preoxygenation (8 L. min^{-1} O_{2} FGF for 3 min), propofol (3 mg. kg^{-1}), rocuronium (0.7 mg. kg^{-1}), and sufentanil (0.1 μg. kg^{-1}) were administered intravenously. After tracheal intubation, ventilation was mechanically controlled by an ADU anesthesia machine. Tidal volume and respiratory rate were set fixed at 500 mL and 10 breaths. min^{-1}, respectively.

In a particular patient, one (fixed) O_{2} FGF and desflurane F_{D} combination was used, with the FGF ranging from 0.5 tot 5 L. min^{-1} and F_{D} from 6 to 18%; this FGF - F_{D} combination was chosen randomly (random function in excel). Preliminary trials indicated what combinations were likely to lead to a desflurane F_{A} less than 1% after 2 min or less than 2% after 5 min, and these were not considered.

Inspired and expired gases were analyzed by a multigas analyzer (Datex-Ohmeda Compact Airway Module M-CAiOV^{®}, Datex-Ohmeda, Helsinki, Finland) and downloaded into a spreadsheet every 10 seconds. Gases were sampled at the distal end of the endotracheal tube using a piece of sampling tubing placed through an Arndt Multi-Port Airway Adapter^{®} (Cook Medical Inc., Bloomington, IN). Gases sampled by the gas analyzer were redirected to the anesthesia circuit via the expiratory limb. The study was terminated after 5 minutes, or earlier when the end-expired desflurane concentration had reached 8%. All values above 8% were eliminated from further analysis. The 5 min period was somewhat arbitrarily defined as the wash-in period because it encompasses (1) anesthesia circuit wash-in; (2) FRC wash-in; (3) early uptake by the VRG; (4) and the waning effects of propofol after about 5 min.

All measurements were done with the same anesthesia machine and gas analyzer. The ADU^{®} circle system volume is 3.4 L. The fresh gas flow inlet is located distal to the inspiratory valve. Fresh gas flow compensation is used to compensate for the inspiratory fresh gas flow during inspiration. The vaporizer output was measured at the common gas outlet by the same gas analyzer and compared with the dial setting using linear and non-linear regression because (1) the vaporizer output may not match the dial setting; (2) we wanted to be able to generalize the results to other ADU^{®} units; and (3) we wanted to exclude that certain performance error patterns could be related to systematic vaporizer error. Based on 229 measurements in 48 patients in this study, the actual desflurane vaporizer output (%) could be described as -0.72 + 1.075*dial setting (r^{2} = 0.98); the vaporizer's output tended to increase with lower FGF (Figure 1).

A model was build to relate F_{A} to FGF, F_{D}, time, and the following patient covariates: age, height, and weight. A constant ventilation allowed us to at least standardize circuit and FRC wash-in; after 5 min, ventilation can easily be adjusted to the desired end-expired CO_{2} concentration. All values before 1 min were deleted because zero values are hard to work with mathematically, and because the model was only supposed to model the FGF - F_{D} - time relationship between 1 and 5 min. Initial model building and parameter exploration were done in Excel using Solver^{®} (Microsoft, Seattle, WA). The initial choice of mathematical functions was guided by three assumptions. First, because F_{A} rises exponentially, a one exponential function was used to describe the rise of F_{A}. Second, the effect of a higher F_{D} was modeled as curvilinear (assuming, for example, that doubling F_{D} would lead to a doubling of F_{A}). Third, the effect of FGF was modeled with a single exponential (increasing as FGF is lowered). By trial and error, these and additional functions were added, deleted, modified, etc. to minimize the sum of least squares (difference between measured and predicted F_{A}) using Solver (Excel). Residuals were plotted against height, weight, and age to search for covariate effects by visual inspection and by linear regression. Finally, the model was tested for parsimony using NONMEM's Minimum Objective Function (ICON Development Solutions, Dublin, Ireland).

### Part II. Prospective testing

The model equation derived in part I was solved for F_{D} using Mathematica (Mathematica for Windows, Version 4.0, Wolfram Research Inc, Champaign, IL). The resulting equation predicts the FGF - F_{D} combinations the anesthesiologist can use to attain a F_{At} within the same time interval used during model development (i.e., between 1 and 5 min) using a single F_{D} and FGF setting, and takes into account the covariate effects derived during model building (see results section for actual equation).

Management of the patients during prospective testing only differed in the manner in which FGF and F_{D} were selected. Patients received desflurane in O_{2} with the goal to reach a F_{At} of 3.5% after 3.5 min (n = 40), 5% after 5 min (n = 37), or 6% after 4.5 min (n = 37). The number of patients was chosen based on prior experience. After entering the time and F_{At} as well as significant patient covariates in the equation, the equation describes all possible FGF - F_{D} combinations that reach the F_{At} at the desired time for a patient with the particular characteristics entered into the equation. The fixed FGF that was going to be used in the individual patient was randomly selected (using Excel's random function) and entered in the equation, yielding the F_{D} to be used. Because the resolution of the desflurane vaporizer is 0.5%, the nearest value was chosen. FGF values requiring an F_{D} above the vaporizer limit (18%) obviously could not be tested.

To allow us to compare model performance between the three subgroups (3.5% after 3.5 min, 5% after 5 min, or 6% after 4.5 min), the performance error (PE) for each patient was calculated as 100*((F_{A} measured - F_{A} predicted)/F_{A} predicted), and the absolute performance error (APE) as the absolute value of PE. Next, for each subgroup, the following were examined: (1) bias and accuracy, using the median performance error (MDPE, median of all PE) and median absolute performance error (MDAPE, median of all APE) [8]; (2) the relationship between FGF and PE (and APE) using linear regression (linear correlation) and a third order polynomial (non-linear effects) to help assess whether the model systematically over- or underestimated the end-expired desflurane concentration with increasing FGF.