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ANESTHETIC PHARMACOLOGY

Alveolar Recruitment and Arterial Desflurane Concentration During Bariatric Surgery

Juraj Sprung, MD, PhD^*, Francis X. Whalen, MD^*, Thomas Comfere, MD^*, Zeljko J. Bosnjak, PhD, Zeljko Bajzer, PhD, Ognjen Gajic, MD^||, Michael G. Sarr, MD, Darrell R. Schroeder, MS^¶, Lavonne M. Liedl, RT^*, Chetan P. Offord, RT, and David O. Warner, MD^*

From the *Department of Anesthesiology, Department of Surgery, Biomathematics Resource and Department of Biochemistry and Molecular Biology, ||Division of Pulmonary and Critical Care Medicine, ¶Department of Health Sciences Research, College of Medicine, Mayo Clinic, Rochester, Minnesota, and Anesthesia Research Medical College of Wisconsin, Milwaukee, Wisconsin.

Address correspondence and reprint requests to Juraj Sprung, MD, PhD, Department of Anesthesiology, College of Medicine, Mayo Clinic, 200 First St. SW, Rochester, MN 55905. Address e-mail to Sprung.juraj{at}mayo.edu.

	Abstract

Top Abstract Introduction METHODS RESULTS DISCUSSION APPENDIX REFERENCES

 
BACKGROUND: We investigated whether reversal of intraoperative atelectasis with the lung recruitment maneuver (RM) affects desflurane arterial concentrations during bariatric surgery.

METHODS: After anesthetic induction and maintenance with propofol, patients were randomized to receive alveolar RM at intervals (RM group) or not (controls). Desflurane 6% was initiated, and rate of increase of alveolar desflurane concentration (ratio of end-expiratory to inspiratory concentrations, F_A/F_I) and desflurane blood concentrations were measured in both groups. Blood and end-tidal desflurane concentrations were also measured after the discontinuation of anesthesia.

RESULTS: The RM group had higher intraoperative Pao₂/Fio₂ compared with the control group (both, P < 0.001). During induction, the rate of increase in blood desflurane concentrations was rapid in both groups. At comparable mechanical ventilation settings, median times to achieve 0.5 mM (approximately 3%) were 2.1 and 1.59 min (P = 0.09) in the control and RM group, respectively. The times to achieve 0.7 mM (approximately 4.2%) desflurane were 15.9 and 9.3 min in the control and RM group, respectively (P = 0.08). Desflurane blood concentrations tended to be higher during the first 30 min after induction in the RM group (P = 0.066). During maintenance or emergence, the blood desflurane concentrations were not different between control and RM groups. Consequently, the time to eye opening did not differ between groups.

CONCLUSION: Although the RM during bariatric surgery represents an effective method for improving intraoperative oxygenation, it does not significantly affect the desflurane blood concentrations during anesthesia or its elimination during emergence.

	Introduction

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Atelectasis develops within minutes after induction of anesthesia,1,2 especially in obese patients.1–6 Atelectasis increases intraoperative shunt and impairs arterial oxygenation.7 Lung expansion by alveolar recruitment maneuvers improves oxygenation in obese patients,8 presumably by decreasing atelectasis and shunting.

Pulmonary shunting decreases the proportion of cardiac output exposed to the alveolar anesthetic gas and thereby affects the rate of increase of inhaled anesthetic concentration in bloobd. Eger and Severinghaus9 modeled the effects of pulmonary shunt on the rate of increase of end-tidal and arterial concentrations and concluded that the magnitude of effect depends on the solubility of inhaled anesthetics (blood/gas partition coefficient, ). In their model, the effect of shunting was exaggerated with the use of cyclopropane, a poorly soluble anesthetic. Subsequently, this prediction was confirmed in a dog model with a pulmonary shunt created by occluding the ventilation to one lung; the rate of increase in arterial cyclopropane concentrations ( = 0.68) was significantly less during one-lung than during two-lung ventilation.10 Shunting caused by atelectasis could, in theory, affect the arterial concentration of inhaled anesthetics, but it is not known whether such an effect would be detectable or clinically relevant, given the relatively modest shunt present under most circumstances.

The purpose of this investigation was to search for evidence that atelectasis caused by anesthesia could produce measurable changes in the rate of increase of arterial concentrations of the poorly soluble anesthetic desflurane ( = 0.42).11 We tested the hypothesis that the application of lung recruitment after the induction of anesthesia would hasten both the rate of increase of the arterial desflurane partial pressure (during induction) and its rate of decrease (during emergence) in the obese patient, presumably by reducing the extent of atelectasis. We chose specific experimental conditions to maximize the possibility of detecting such effects by studying obese patients known to develop more extensive atelectasis during anesthesia.

	METHODS

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The Mayo Clinic Institutional Review Board approved this investigation, and each subject gave written informed consent. Patients undergoing open bariatric surgery at Saint Mary’s Hospital, Rochester, MN, with a body mass index >40 kg/m² were eligible for enrollment. Exclusion criteria included substantive abnormalities in spirometry (forced expiratory volume in 1 s <50% of predicted, forced vital capacity <50% of predicted), active asthma (requiring bronchodilator therapy), previous lung surgery, or home oxygen therapy.

Patients were randomized into two groups based on intraoperative ventilatory management. Control patients received standard mechanical ventilation with 4 cm H₂O positive end-expiratory pressure (PEEP), and the lung recruitment group received a series of RM followed by 12 cm H₂O PEEP (see later).8,12 Randomization was accomplished through our Division of Biostatistics using a computerized random number generator.

Anesthetic Management
Before anesthetic induction, a thoracic epidural catheter was placed and a test dose was administered. In addition to the standard monitors recommended by the American Society of Anesthesiologists, arterial blood pressure was monitored with a radial arterial line. Anesthesia was induced with 2 mg/kg ideal body weight (IBW) of propofol supplemented by 2 µg/kg IBW of fentanyl and 2 mg of midazolam. Tracheal intubation with a size 8.0 mm internal diameter tracheal tube was facilitated with 1.5 mg/kg IBW of succinylcholine, and continuous infusions of fentanyl (2 µg/kg (IBW)/h) and propofol (150–200 µg/kg (IBW)/min) were initiated. Neuromuscular blockade was maintained with vecuronium. Esophageal temperature was monitored and maintained above 36.5°C using a forced air warmer (Bair Hugger®, Arizant Healtcare, Eden Prairie, MN).

After placement of an arterial catheter (after tracheal intubation) and obtaining a blood sample for determination of baseline blood gases and blood desflurane concentrations and performance of RM in the recruitment group (see later), the desflurane vaporizer was set to 6% (at 10 ± 3 min after induction), and the propofol infusion was tapered at the discretion of the attending anesthesiologist. The fentanyl infusion was maintained until 30 min before the end of operation, and the rate was adjusted according to hemodynamic variables at the discretion of the primary anesthesiologist. No long-acting IV opioids were given during the operation. Shortly before the surgical incision, hydromorphone hydrochloride (0.8 mg) was administered through the epidural catheter; no other medications were administered through the catheter during the operation.

Ventilatory Management
Mechanical ventilation (Datex-Ohmeda Aestiva/5 Smart Ventilator, Madison, WI) was initiated immediately after tracheal intubation using the volume-control mode at a rate of 8 breaths/min, tidal volume (V_T) of 8 mL/kg IBW, PEEP of 4 cm H₂O, inspiratory-to-expiratory time ratio of 1:2, inspired oxygen fraction of 0.5 (balance nitrogen), and a total gas flow to the anesthesia circle system of 10 L/min. If a rate of 8/min did not maintain the end-tidal CO₂ between 40 and 45 mm Hg, the respiratory rate was adjusted to achieve that target value. These settings were maintained in the control group throughout the operation.

In the alveolar RM group, the variables of initial mechanical ventilation were identical to those in the control group after tracheal intubation. After placement of an arterial catheter, a lung RM was performed as described previously.8 Briefly, RM was achieved by sequential increases in PEEP in three steps from 4 cm H₂O (baseline) to 10 cm H₂O (three breaths), 15 cm H₂O (three breaths), and 20 cm H₂O PEEP [or maximal peak airway pressure of 50 cm H₂O (10 breaths)]. After recruitment, the level of PEEP was set at 12 cm H₂O. All other ventilatory variables remained identical to the control group. This RM was repeated at 30 and 60 min after first recruitment and hourly afterward.

After skin closure, desflurane administration was discontinued. The times from desflurane discontinuation to eye opening were noted. Until eye opening, the patients’ lungs were ventilated with identical intraoperative settings. After eye opening, the patients were allowed to breathe spontaneously (without continuous positive airway pressure).

Measurements
A noninvasive NICO₂^TM monitor (Novametrix Medical Systems, Wallingford, CT) was used to measure the following variables: (a) mean and peak airway pressures, (b) minute ventilation (MV), (c) expiratory tidal volume (V_TE), (d) respiratory system dynamic compliance (C_dyn, mL/cm H₂O, calculated as ratio of the maximum inspiratory volume over the difference between peak inspiratory and PEEP), (e) airway resistance (R_aw, cm H₂O · L^–1 · s^–1, computed by least square fittings of the raw waveform data of flow, volume, and pressure to a simple model), (f) physiologic dead space-to-V_T ratios (V_DPhysiol/V_T = Paco₂-end-tidal CO₂/Paco₂), (g) alveolar tidal volume (V_Talv in mL, calculated as difference between V_TE and total airway dead space), (h) alveolar MV (MV_alv, calculated as V_Talv times respiratory rate), and (i) noninvasive cardiac index by means of a differential form of the Fick equation. End-tidal (F_A) and inspired desflurane concentrations (F_I) were sampled from the port at the tracheal tube every 2 min (Anesthetic Gas Module, Philips Medical Systems; Model: M1026B.3000, Minuteman Road, Andover, MA).

Samples for arterial blood gases were drawn after tracheal intubation and the institution of mechanical ventilation for at least 2 min (before recruitment in the RM group), 2 min after initiation of desflurane anesthesia, 60, 120, 180 min after start of desflurane, at the end of the operation (immediately before desflurane discontinuation), and at arrival and discharge from the recovery room. Samples for determination of blood concentrations of desflurane (in millimoles, mM) were obtained after tracheal intubation and the institution of mechanical ventilation for at least 2 min (before recruitment in the RM group, time 0), at 2, 5, 10, 30, 60, 120, 180 min from initiation of desflurane, at the end of the operation (immediately before desflurane discontinuation), at 2, 5, 10 min after desflurane discontinuation, and at arrival to the recovery room. Arterial blood samples for determination of desflurane were collected and promptly placed in glass vials and sealed with specially designed crimp-caps with aluminum seals that have red rubber septa and an exact-fit aluminum foil lining disk and immediately refrigerated. Once stored on ice, the anesthetic concentration in the blood remains stable for at least 1 wk (data not shown). All standard and sample concentrations of desflurane were determined using a gas chromatograph (Shimadzu model GC 8-A, Shimadzu Corp., Kyoto, Japan).

Statistical Analyses
The ratio of arterial oxygen partial pressure to inspiratory oxygen concentration (Pao₂/Fio₂), ventilation variables (inspiratory and expiratory pressures, V_TE, alveolar ventilation), and respiratory mechanics (compliance and resistance) were summarized using mean ± sd. Baseline levels were compared between groups using the t-test. Data collected from 2 to 180 min after the initiation of desflurane were compared between groups using repeated-measures analysis of variance (ANOVA). The alveolar concentrations of desflurane were calculated as the ratio of end-tidal anesthetic concentrations (F_A) to inspired anesthetic concentrations (F_I). After discontinuation of desflurane, its elimination was calculated as the ratio of end-tidal anesthetic concentrations to the last end-tidal concentrations just before the vaporizer was turned off (F_A/F_{A-end surgery}). The rate of increase of alveolar concentration and the rate of elimination of desflurane were characterized by fitting a single exponential model to the alveolar desflurane concentrations obtained after initiation and discontinuation of its use. Models were fit separately for each individual, and the time constants were compared between groups using the t-test.

We characterized rate of increase of desflurane blood concentration by using a stretched exponential model, because a single exponential model did not result in satisfactory fit (Appendix). Data for each patient (from 0 time until the end of the administration of desflurane) were fitted by the least squares method. Using this model, we estimated (median and range) T_0.5 and T_0.7, i.e., times in minutes at which blood concentrations reached 0.5 mM (approximately 3%) and 0.7 mM (approximately 4.2%). We also estimated the slopes of concentration curves given by the model at times T_0.5 and T_0.7, i.e., the rates of change in blood desflurane concentration denoted as R_0.5 and R_0.7 in mM/min. To further analyze the blood desflurane concentrations, a repeated-measures ANOVA was performed. Since anticipated differences between groups should be most apparent after induction,9,10 this analysis was restricted to blood desflurane concentrations measured at 2, 5, 10, and 30 min after initiation of desflurane.

Demographics and postoperative outcomes were reported and compared across procedure groups using either t-tests or Wilcoxon’s ranked sum test when appropriate. For all calculations, P values <0.05 were considered statistically significant. Statistical analysis was performed with the SAS statistical software package (Statistical Analysis System, SAS Institute, Cary, NC).

	RESULTS

Top Abstract Introduction METHODS RESULTS DISCUSSION APPENDIX REFERENCES

 
We recruited 20 patients. In three patients, the study was aborted for the following reasons: 1) hypotension after anesthetic induction despite fluid and vasopressor administration (control group), 2) bronchospasm after tracheal intubation requiring repeated bronchodilator treatment (RM group), and 3) lost desflurane samples and malfunction of NICO₂^TM monitor (RM group). In one additional control patient, desflurane blood samples were lost, but the rest of the data on this patient were collected and used for the analysis. Therefore, 17 patients were analyzed (n = 9 in the control group and n = 8 in the RM group). The two study groups had comparable demographic characteristics: age 48 ± 9 vs 52 ± 9 yr (control vs RM, P = 0.336), weight 153 ± 34 vs 167 ± 41 kg (P = 0.447), body mass index 51 ± 5 vs 56 ± 11 kg/m², sex (four male patients in each group), and duration of anesthesia (433 ± 137 vs 458 ± 177 min, P = 0.757). This long duration of bariatric surgery reflects complex patients with previous abdominal operations, with many requiring simultaneous abdominal hernia repair. Seven patients in each group used continuous positive airway pressure preoperatively for obstructive sleep apnea. The RM was performed according to the planned protocol in all RM patients, and peak inspiratory pressure did not exceed 50 cm H₂O during recruitment in any patient. No patient experienced hemodynamic instability requiring treatment during the RM.

Cardiac index was comparable during the first 30 min of anesthesia (baseline, 3.3 ± 1.1 vs 3.5 ± 0.8 L · min^–1 · m^–2 in control and recruitment groups, respectively, P = 0.629). After 30 min, the average cardiac index increased in the control group to 4.4 ± 1.1 L · min^–1 · m^–2 (P = 0.0004, compared with baseline), and in the RM group remained unchanged (3.7 ± 1.0 L · min^–1 · m^–2, P = 0.40, compared with baseline). The difference between control and RM patients became significant (P = 0.027). The Pao₂/Fio₂ (Fig. 1, Table 1, P = 0.001) was greater in the RM group at all times after RM. Respiratory system dynamic compliance increased with RM and remained higher throughout the operation compared with the control group (by approximately 35%, P < 0.001, Fig. 1), whereas airway resistance tended to be lower (P = 0.051) (Fig. 1). V_DPhysiol/V_T was not different between the two groups (P = 0.335) (Table 1). The estimated rate of increase of the alveolar concentration (expressed as F_A/F_I) during induction did not differ between groups (Fig. 2, left panel) (P = 0.325 for comparison of time constants in a single exponential model [0.67 ± 0.28 and 0.83 ± 0.36 min^–1 for control and RM groups, respectively]).

View larger version (25K): [in this window] [in a new window]   Figure 1. Arterial oxygen partial pressure as a fraction of inspiratory oxygen concentration (Pao2/Fio2), dynamic respiratory system compliance (Cdyn), and airway resistance (Raw) in control and recruitment groups. Pao2/Fio2 and Cdyn increased after recruitment and remained greater in the recruitment group compared with control patients throughout the operation. Data are mean ± sd. Asterisk denotes P < 0.05 control versus recruitment group at same time intervals. P value is overall statistical significance from ANOVA. B = baseline.

View larger version (18K): [in this window] [in a new window]   Figure 2. The increase in alveolar desflurane concentrations (FA/FI) after initiation of anesthesia in control and recruitment groups. Data are mean ± sd.

Table 2 shows mean blood desflurane concentrations in both RM and control groups. Figure 3 shows the stretched exponential model fitted through mean (±sd) desflurane blood concentrations between 0 and 180 min of anesthesia (insert), including a more detailed presentation at the time of maximal uptake (first 30 min). A trend for a more rapid increase in blood desflurane concentration was present during first 30 min after induction (P = 0.066 from repeated-measures ANOVA using measurements at 2, 5, 10, and 30 min). The estimated times necessary to achieve blood desflurane concentrations (in min) of 0.5 mM and 0.7 mM were not significantly different between the two groups: T_0.5 (median, range) of 2.1 (0.56–5.85) and 1.59 (0.30–3.94) in control and RM group, respectively (P = 0.09), and T_0.7 of 15.9 (6.53–38.2) and 9.3 (2.39, 37.9), in control and RM group, respectively (P = 0.08). Similarly, the estimated rates of change of desflurane blood concentrations (in mM/min), R_0.5 [0.05 (0.016–0.105) vs 0.09 (0.028–0.219), P = 0.064)] and R_0.7 [0.005 (0.002–0.023) vs 0.01 (0.0002–0.077), P = 0.094)] also did not significantly differ between the control and RM groups, respectively.

View larger version (20K): [in this window] [in a new window]   Figure 3. The rate of increase in desflurane blood concentration in control and recruitment groups characterized by fitting the stretch exponential model (Appendix) through all available data points from 0 to 180 min (Table 2). The main plot is an expansion of the first 30 min of anesthesia, and the inserted plot shows the entire stretch exponential curve. The means (±sd) of data are represented by open (recruitment) and closed circles (control). The best fit values for model parameters were kin = 0.144 mM/min, kout = 0.140 min–1, and b = 0.260 for control, and kin = 0.244 mM/min, kout = 0.284 min–1, b = 0.311 for recruitment.

After discontinuation of anesthesia, elimination of desflurane from the blood was similar in both groups over the first 10 min (P = 0.838, time constants 0.68 ± 0.14 and 0.66 ± 0.26 min^–1 for control and RM groups, respectively). Similarly, the rate of decrease in alveolar desflurane concentration (F_A/F_{A-end surgery}) also did not differ between groups during the elimination phase (P = 0.504, time constants 0.85 ± 0.48 and 1.10 ± 0.92 min^–1 for control and RM groups, respectively).

The times to eye opening after discontinuation of desflurane were not different between the two groups (8.9 ± 3.0 vs 6.43 ± 3.2 min, in control and RM groups, P = 0.362). Similarly, there were no differences in oxygenation in the recovery room (115 ± 50 mm Hg vs 106 ± 41 mm Hg, in control and RM groups, P = 0.758) or efficiency of ventilation (51 ± 6 mm Hg vs 51 ± 6 mm Hg, in control and RM groups, P = 0.930).

	DISCUSSION

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Although lung recruitment improved oxygenation in our morbidly obese patients, it did not significantly affect the rate of increase in desflurane blood concentrations during anesthetic induction or emergence.

Intraoperative atelectasis is associated with an increase in pulmonary shunting leading to impairment of gas exchange.4,7,13 Abnormalities in gas exchange are exaggerated in bariatric patients, presumably, in part, because they are more prone to developing atelectasis.2,6,14–17 Alveolar recruitment combined with higher levels of PEEP is an effective strategy to improve arterial oxygenation in these patients.8,18 In a previous study, we confirmed8 that the RM improves oxygenation in bariatric patients and that these effects were limited only to the intraoperative course; when PEEP was removed after tracheal extubation, gas exchange in the recovery was similar between groups. Although intraoperative atelectasis can clearly affect gas exchange, it is not known whether the degree of atelectasis produced by anesthesia is sufficient to significantly affect the blood anesthetic concentration, especially during induction and emergence. In their classic analysis, Eger and Severinghaus9 predicted that shunting would slow the rate of increase in blood concentration of the anesthetic during induction, an effect that was exaggerated with less soluble drugs, such as cyclopropane. This concept was confirmed experimentally in a dog model by comparing the rate of increase of blood cyclopropane concentration during one-lung ventilation by using the method that estimated, rather than directly measured, anesthetic blood levels.10 We reasoned that patients with morbid obesity, who develop substantial atelectasis after induction, represent a suitable model for studying the role of RM on the rate of increase of desflurane arterial concentration. Based on differences between groups with regard to oxygenation and mechanics of breathing (higher dynamic compliance, and lower resistance in the RM group), we infer that there were substantial differences in the amount of atelectasis between the two groups. Although we did not directly assess the degree of atelectasis in our morbidly obese patients, we used an improvement of oxygenation as a surrogate for reduction of atelectasis by the RM.

After initiation of desflurane, the times necessary to achieve blood concentrations of 0.5 and 0.7 mM tended to be less in patients receiving RM. Also, the observed mean arterial desflurane concentrations in patients with recruited lungs were 22.5%, 15.5%, 10.9%, and 5.2% larger at 2, 5, 10, and 30 min after induction when compared with control patients (repeated-measures ANOVA P = 0.066 restricted to the first 30 min of anesthesia). Thus, although a tendency was present, we could not confirm our hypothesis. Given the rapid increases in desflurane concentration during the first few minutes of induction in both groups, the precise timing of blood sampling is important. Slight variability in the timing of the samples likely accounts for some of the observed variability in blood concentrations. It is possible that if more patients had been studied, the tendency toward more rapid increases in desflurane blood concentrations in the RM group may have reached statistical significance. However, the magnitude of the potential effect observed is consistent with the results of Stoelting and Longnecker,10 who reported that the estimated blood concentrations of cyclopropane during two-lung ventilation in dogs were 30%, 26%, and 22% higher at 3, 5, and 10 min after induction than during one-lung ventilation. The degree of shunt during one-lung ventilation is likely higher than that in bariatric patients without recruitment.

Increased intrathoracic pressure by higher PEEP in the RM group can decrease cardiac output. Indeed, during maintenance of anesthesia, cardiac index was lower in recruitment patients. Lower cardiac output would decrease desflurane uptake and should increase, thereby, the alveolar concentration of anesthetic (not apparent in Fig. 2). If anything, a decreased uptake should increase the rise of desflurane blood concentration, especially the early rise, and, thus, cannot explain the trend for lower plateau values of blood desflurane observed in the RM group by the end of anesthesia (Fig. 3). It also cannot explain why we did not find a difference in the rates of increase of desflurane partial pressures in the control versus the RM groups.

Desflurane has gained popularity for bariatric surgery due to favorable profiles of emergence and recovery. Sturm et al.19 reported faster emergence, recovery, and greater oxyhemoglobin saturations on entry to the recovery room in bariatric patients undergoing anesthesia with desflurane compared with sevoflurane. In the present study, the rate of decrease in arterial desflurane concentration during emergence was fast and comparable in the two groups. Consistent with this finding, there were no differences between study groups in time to eye opening. Furthermore, we found that immediate postoperative oxygenation and ventilation, while in the recovery room, were not different between the two groups. Our findings regarding these end points should be considered with the limitations related to the small number of patients studied.

In conclusion, we confirmed that the RM is an effective method to improve intraoperative oxygenation in patients undergoing bariatric surgery. However, an assumed reduction in atelectasis, indicated by improved oxygenation, was not sufficient to change the arterial desflurane concentrations during either induction or emergence from anesthesia in a manner that was statistically or clinically significant.

	ACKNOWLEDGMENTS

 
We are thank Dr. Edmond I. Eger II for constructive suggestions during preparation for this project, Mrs. Mary Ziebell for help in measuring blood desflurane concentrations and Andrew Hanson (data analyst).

	APPENDIX

Top Abstract Introduction METHODS RESULTS DISCUSSION APPENDIX REFERENCES

 
In this appendix, we present some details of the stretched exponential model which was used to describe desflurane concentrations in blood.

The change of concentration C(t) of desflurane in blood per unit of time (net rate dC/dt) can be modeled according to the mass balance equation:

where a(t) is the rate of desflurane absorption in blood, and r(t)C(t) is the rate of desflurane disappearance from blood which depends on the current concentration of desflurane in blood and on the rate coefficient r(t). If the rate of absorption can be assumed constant, a(t) = k_in and r(t) = k_out can also be assumed constant, then the earlier equation can be solved to yield:

which corresponds to the one-compartment model with the constant input. By using this model and the least-squares method, we fitted individual data sets for 16 considered subjects. It turned out that the fits for most of the subjects were not satisfactory. If the exponential function is generalized to the stretched exponential function, one obtains:

where b is an additional parameter. The fits of this model to data were satisfactory. By using the modified Akaike information criterion for model selection,20–22 we found that for 14 of 16 data sets, model (3) was preferred over the model (2).

Equation (3) corresponds to the solution of differential Equation (1) when a(t) and r(t) are allometric with time:

These expressions resemble the rate coefficient for fractal kinetics.23 One may speculate that they are related to the fractal structures of the lung airways and vascular trees.

	Footnotes

 
Accepted for publication July 12, 2008.

Supported by Department of Anesthesiology, Mayo Clinic, Rochester, MN and Anesthesia Research Medical College of Wisconsin, Milwaukee, WI.

	REFERENCES

Top Abstract Introduction METHODS RESULTS DISCUSSION APPENDIX REFERENCES

 

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