The practice of closed-circuit anesthesia (CCA) has been described many decades ago [1-3]. Based on clinical observations of uptake of nitrous oxide, halothane, and methoxyflurane [2,4-8],1 Lowe and Ernst  developed the square root of time model (see Appendix), which has subsequently been used to predict uptake of any potent inhaled anesthetic during CCA [9-11]. The square root of time model indicates that anesthetic uptake gradually decreases according to the square root of time, and that the uptake is related to the patient's weight3/4.
Lowe HJ. Dose-regulated Penthrane[R] (methoxyflurane) anesthesia. Chicago: Abbott Laboratories, 1972.
However, recent clinical observations on uptake of inhaled anesthetics [12-15] conflict with the theory of Lowe and Ernst. Indeed, Lin and co-workers [16-18] suggested that the patient's uptake of nitrous oxide and potent inhaled anesthetics was fairly constant during the first 60-100 minutes of anesthesia. In addition, enflurane and isoflurane uptake could not be correlated with any patient characteristic [12,14].
It has also been theorized that uptake of potent inhaled anesthetics, defined as the amount of vapor that is taken up by the blood from the alveoli, is independent from minute ventilation whether high or low fresh gas flows are used [3,16]. On the other hand, isoflurane and enflurane uptake at constant end-expired concentration during high-flow anesthesia was as high or higher during spontaneous ventilation (SV) compared with controlled mechanical ventilation (CMV) .
The above considerations led us to investigate 1) desflurane and isoflurane uptake during CCA using liquid injection into the circuit; 2) whether patient characteristics can predict desflurane and isoflurane uptake; and 3) the influence of the mode of ventilation on uptake.
The study was approved by our institutional research review board, and informed consent was obtained from all patients. Forty-one ASA physical status I or II adults of both sexes were enrolled. The patients received general anesthesia for a variety of surgical procedures lasting at least 1 h. Patients were randomly assigned to one of four groups: desflurane anesthesia with SV (DS; n = 11), desflurane anesthesia with CMV (DC; n = 10), isoflurane anesthesia with SV (IS; n = 10), and isoflurane anesthesia with CMV (IC; n = 10).
Premedication consisted of intramuscular meperidine 1 mg/kg, midazolam 0.3 mg/kg, and atropine 0.5 mg. After breathing oxygen for a few minutes (end-expired nitrogen concentration <3%), anesthesia was induced with propofol 2-3 mg/kg. Endotracheal intubation was facilitated by succinylcholine 1 mg/kg in Groups IS and DS and by vecuronium 0.08 mg/kg in Groups IC and DC. The circle system was closed after endotracheal intubation, and liquid desflurane or isoflurane was injected into the circle system. The goal was to obtain 1.3 minimum alveolar anesthetic concentration (MAC) as soon as possible after the start of the infusion by adjusting the infusion rate based on the end-expired anesthetic concentration. The end-expired desflurane and isoflurane concentrations were maintained at 1.3 MAC throughout the remainder of the procedure (8.5% and 1.5%, respectively). An infusion pump with a volumetric accuracy of +/- 2% was used to inject the liquid anesthetic into the inspiratory limb. The syringe containing desflurane was continuously cooled with ice bags. Air and oxygen flows were titrated to keep the volume of the reservoir bag constant and to keep the fraction of inspired oxygen >0.30. Sufentanil and muscle relaxants were administered as clinically indicated.
Drager SA2 (Dragerwerk Akkiengesellschaft, Lubeck, Germany) anesthesia machines were used with soda lime as the CO2 absorber. The leak of each system was determined using constant pressure and during ventilation of a test lung. The fresh gas flow that was needed to continuously pressurize the circle system at 20 cm H2 O was <or=to30 mL/min in all systems. When ventilating a test lung at a peak inspiratory pressure of 20 cm H2 O, the mean leak was <or=to65 mL/min (the larger leak is due to the larger circuit, with more connections and tubing). All results have been corrected for the specific leak observed for each individual anesthesia machine. The volume of the circle system was approximately 5.7 L; the volume of the ventilator part of the circuit was 0.6 L.
The amount of liquid anesthetic needed to prime the circle system and an estimated functional residual capacity (FRC) of 2 L with 1.3 MAC of desflurane and isoflurane was 3.16 and 0.6 mL, respectively. During CMV, another 0.3 mL desflurane was used to prime the ventilator bellows (the extra amount of isoflurane needed was negligible: 0.05 mL).
Anesthetic gas concentrations were monitored with a multigas analyzer (Datex AS/3 Anesthesia System; Datex, Helsinki, Finland; or Rascal II Anesthetic Gas Monitor; Ohmeda, Salt Lake City, UT). The sampled gases (approximately 210 mL/min) were redirected into the circuit. End-expired carbon dioxide was maintained at 35 +/- 5 mm Hg during CMV (IC, DC). Hypotension, defined as a decrease in blood pressure >or=to25% from baseline, was treated with intravenous fluid administration and intravenous ephedrine (5 mg boluses).
The cumulative dose (total amount of liquid anesthetic injected over time) and end-expired anesthetic vapor concentrations were recorded every minute. The individual uptake curves were fitted (linear and one-exponential) once 1.3 MAC was obtained. The cumulative doses after 4, 15, 30, 45, and 60 min as well as the variables of the fitted curves of each individual were correlated with patient characteristics. The desflurane and isoflurane uptake in each group was compared with the square root of time model (see Appendix). The uptake during SV and CMV were compared to determine the effects of mode of ventilation.
Patient demographics and variables of the fitted uptake curves were compared using analysis of variance. The variables of the fitted uptake curves and the cumulative dose after 4, 15, 30, 45, and 60 min were correlated with patient characteristics (age, height, weight, weight3/4, and body surface area [BSA]) using regression analysis. Values are given as mean +/- SD.
The four groups did not differ with regard to sex, age, height, weight, weight3/4, and BSA (Table 1). The end-expired CO2 concentration in each group was: DS 53.4 +/- 6.3 mm Hg; DC 35.1 +/- 2.8 mm Hg; IS 48.7 +/- 4.3 mm Hg; and IC 35.1 +/- 1.3 mm Hg. Patients tolerated the anesthetic technique well: five patients received 5 mg of ephedrine, and one patient received 20 mg of ephedrine.
In all our patients, an end-expired concentration of desflurane or isoflurane of 1.3 MAC was reached after 4 min, although there was a slight overshoot in the IS group (Figure 1). The rate of increase of the end-expired concentration was similar in all four groups. After 4 min, uptake was nearly constant over the first hour in all groups (Figure 2 and Figure 3; Table 2). Figure 2 and Figure 3 also include the uptake as predicted by the square root of time model of Lowe and Ernst. Within each group, there was interindividual variability in the uptake at all times, with a coefficient of variation of 8%-20%. A linear fit of the individual curves between 4 and 60 min, representing only patient uptake, gave excellent correlations (r2 ranged from 0.946 to 0.999). Because the cumulative dose over time curve during the first hour appeared to be almost linear, exponential curve fitting is meaningful only if uptake is studied over a much longer time period. In general, anesthetic uptake, represented by the slope of the linear fit (b1) of the cumulative dose versus time data, did not correlate with any of the patient characteristics (Table 3). In addition, the cumulative dose at 4, 15, 30, 45, and 60 min also did not correlate with any of the patient characteristics. Uptake was not different between spontaneously breathing patients (IS and DS) and ventilated patients (IC and DC) (P > 0.05).
We found that the rate of increase of the end-expired concentration was similar in all four groups. When closed circuit anesthesia with liquid injection of the anesthetic is used, the user will define the rate of increase of the end-expired concentration, which under these circumstances, is independent from the blood-gas solubility. After this initial three-to four-minute wash-in period, uptake of desflurane and isoflurane is almost constant during the first hour of CCA with end-expired anesthetic concentrations remaining constant at 1.3 MAC. The square root of time model, as described by Lowe and Ernst, overestimates uptake, especially during the first part of the procedure (Figure 2 and Figure 3).
The development of the square root of time model is based on uptake data in a limited number of patients, in a time period when several technical limitations were present. In 1954, Severinghaus  described the uptake of nitrous oxide in six patients; he observed that the uptake of nitrous oxide decreased over time, proportional to the square root of time (uptake = 1000 [center dot] t-0.5 mL/min). These findings were confirmed in two other studies [20,21]. Based on these data, on a theoretical analysis involving known organ uptake, if tissue volume, tissue/blood solubility of the anesthetic, and organ blood flow are known, and on experiments with halothane and methoxyflurane [5,8], Lowe and Ernst  developed the square root of time model. The data were generated in an era of more primitive measurement and analysis, using anesthetic gas monitors such as the Narko-Test, a device based on the solubility of anesthetics in silicon rubber . In addition, a large amount of methoxyflurane was dissolving into rubber circuit components, making it difficult to separate patient uptake from circuit uptake. Nevertheless, without much further experimental verification, the uptake of other potent inhaled anesthetics was assumed to behave according to the square root of time model [3,9-11].
Anesthetic uptake has traditionally been described using high-flow techniques during which the inspired concentration is held constant [11,23]. The relationship between the alveolar (FA) and inspired concentrations (FI) over time (FA/FI curve) is used to describe the pharmacokinetics, and computer simulations determine the effects of minute ventilation and cardiac output. However, several misconceptions exist regarding the meaning of the FA/FI curves. The initial portion of the FA/FI curves represents mostly wash-in of the anesthetic circuit and the FRC , and patient uptake is probably minimal during the first few minutes of an anesthetic . In addition, the FA/FI curves by themselves do not describe patient uptake: anesthetic uptake is represented by the area above the FA/FI curves and is the product of FI, fraction of uptake (1 - FA/FI), and alveolar ventilation. When, for example, alveolar ventilation increases, the FA/FI curve is shifted upward, which means that the fraction of uptake decreases and thus patient uptake is virtually unchanged . Nevertheless, increased minute ventilation will speed up induction by shortening the time-constant for wash-in of the FRC, but after FRC wash-in, anesthetic uptake is probably independent of minute ventilation.
Computer programs simulating anesthetic uptake are based on previously published values for bloodgas solubilities, tissue weight or volume, tissue-blood solubilities, and tissue-blood flows. However, partition coefficients vary by as much as 150%  and, because they are derived from tissue homogenates, may not completely represent what is happening in vivo. Other processes (known and unknown) along the partial pressure cascade of an inhaled anesthetic may also affect uptake models. Landon et al.  found that the end-expired to arterial gradient for isoflurane cannot be explained by shunting and dead space ventilation alone. They postulate that diffusion problems may be responsible for the remaining gradient. Also, the influence of cardiac output has not been quantified in vivo in humans. In an in vivo nuclear magnetic resonance study, Lockhart et al.  clearly showed that the brain time constants for isoflurane and desflurane were much longer than predicted by partition coefficients. Interestingly, Lowe and Ernst , while recognizing the limitations of computer-generated data, heavily relied on computer simulations themselves when describing the square root of time model.
Other recent investigations have also found the uptake pattern of potent inhaled anesthetics to be different from the square root of time model. Lin et al. [16-18] illustrated the near constant uptake of nitrous oxide and other potent inhaled agents during the first hour of CCA. In a high-flow system, Bengtson et al.  found that the square root of time model overestimated isoflurane and enflurane uptake during the first 35-45 minutes and underestimated uptake thereafter. Lockwood et al. [14,15] recently measured isoflurane and desflurane uptake during CCA: although they found that desflurane and isoflurane uptake decayed bi- and triexponentially, for practical purposes, the uptake curves appear to be almost linear after stable end-expired concentrations were achieved. Our findings on isoflurane and desflurane uptake are almost identical to the observations made by Lockwood et al.  and Walker et al. .
As expected, we observed interindividual variation in uptake in our patients. However, there was a poor correlation between uptake and patient characteristics. Therefore, interindividual variability in the uptake is likely not the result of patient characteristics. Other studies came to similar conclusions [12,14], although O'Callaghan et al.  found a weak correlation (r = 0.629) between uptake of isoflurane and BSA. Although the spontaneously breathing patients had a higher end-expired PCO2 level and therefore a lower minute ventilation, the uptake was similar to that in patients who were mechanically ventilated. This is in contrast to the results of Bengston et al. , who found a trend toward higher uptake during SV, possibly related to a higher cardiac output in SV patients.
Our findings have several clinical implications. Complex mathematical models such as the square root of time model have deterred many clinicians from the use of CCA. Because of our findings that uptake at a constant end-expired concentration is almost constant during the first hour of an anesthetic, there is limited need for dose (or vaporizer) adjustments once the desired end-expired concentration is obtained. This may greatly simplify CCA. In addition, pharmacoeconomical analyses should not be based on uptake models such as the square root of time model. Rather, real uptake data should be used to determine the potential cost-savings of future automated CCA machines.
In conclusion, desflurane and isoflurane uptake during the first hour was virtually constant after a four-minute wash-in period. Uptake varied from patient to patient, but patient characteristics did not predict the uptake. Mode of ventilation had no effect on anesthetic uptake. These findings greatly simplify CCA.
Lowe and Ernst's Square Root of Time Model of Anesthetic Uptake 
During CCA, the uptake of a potent inhaled anesthetic has been predicted by the square root of time model . A "unit dose" is taken up by the body during the first minute and during each subsequent time interval (3 min, 5 min, 7 min, 9 min, 11 min, 13 min, etc.). In addition, the "prime dose" is required to saturate the circuit, FRC, and arterial delivery system. The addition of the prime dose and unit doses over time results in the "cumulative dose."
1. Unit dose (L of vapor) = 2 [center dot] f [center dot] MAC [centered dot] lambdaB/G [center dot] Q
2. Prime dose (L of vapor) = f [center dot] MAC [center dot] lambda (B/G) [center dot] Q + vol [center dot] f [center dot] MAC
3. Cumulative dose (L of vapor) = prime dose + unit dose [center dot] ????t = f [center dot] MAC [center dot] lambdaB/G [center dot] Q + vol [center dot] f [center dot] MAC + 2 [center dot] f [center dot] MAC [centered dot] lambdaB/G [center dot] Q [center dot] ????t
4. Conversion of vapor into liquid: 1 L vapor = MW/(D [center dot] 24) L liquid
The abbreviations used are: f = fraction of MAC administered; MAC = minimum alveolar anesthetic concentration; lambdaB/G = blood/gas partition coefficient (for desflurane, 0.424; for isoflurane, 1.38); Q = cardiac output (CO) (L/min, based on DeBrodie: CO = 0.2 x weight (kg)3/4); t = time (min); vol = volume of circuit and FRC (L); MW = molecular weight (for desflurane, 168g; for isoflurane, 184.5g); and D = density (for desflurane, 1.45 [center dot] 103 g [center dot] L (-1) [center dot] for isoflurane, 1.496 [center dot] 103 g [center dot] L-1, at 25 degrees C).
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© 1997 International Anesthesia Research Society
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