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Meeting Abstracts  |   August 2005
A Physiologically Based, Recirculatory Model of the Kinetics and Dynamics of Propofol in Man
Author Affiliations & Notes
  • Richard N. Upton, B.Sc., Ph.D.
    *
  • Guy Ludbrook, M.B.B.S., Ph.D.
  • * Principal Medical Scientist/Senior Lecturer, †Professor and Head, Department of Anaesthesia and Intensive Care, Royal Adelaide Hospital, University of Adelaide.
Article Information
Meeting Abstracts   |   August 2005
A Physiologically Based, Recirculatory Model of the Kinetics and Dynamics of Propofol in Man
Anesthesiology 8 2005, Vol.103, 344-352. doi:
Anesthesiology 8 2005, Vol.103, 344-352. doi:
THERE is evidence that the kinetics of propofol are influenced by blood flow (e.g.  , cardiac output1–3 and cerebral blood flow4). It was hypothesized that there may be some benefit in developing a pharmacokinetic model for propofol that incorporated these and other blood flows as parameters. Such a model would provide a conceptual framework for understanding and predicting the consequence of physiologic perturbations (e.g.  , the coadministration of cardioactive drugs) on propofol disposition. We have previously developed a similar model for propofol based on detailed data sets collected in sheep.5–7 
The aim of this article is to extend this animal model to a physiologically based recirculatory model of propofol kinetics and dynamics in a “standard” man that is broadly consistent with published kinetic–dynamic data for propofol and with accepted physiologic values and mechanisms in the standard man. The hypothesis to be tested is that a range of clinical studies of propofol kinetics and dynamics can be accounted for by one physiologically based recirculatory model with blood flows as parameters. This model could be used to predict the quantitative effect of circulation changes on propofol disposition. It could therefore contribute to a physiologic understanding of some of the sources of variability in the kinetics and dynamics of propofol.
Materials and Methods
General Considerations
Data for model building were selected from the literature. Pharmacokinetic data were only used if they were based the measurement of plasma concentrations, to avoid uncertainty regarding the degree to which whole blood assays were affected by propofol sequestration into erythrocytes during sample storage.8 This assay difficulty aside, as the blood:plasma ratio of propofol can be calculated to be approximately 1, and the blood concentrations of propofol were assumed to be the same as the reported plasma concentrations.9 
The aim of the model was to define propofol kinetics in a “standard” patient. To facilitate subsequent simulations and scaling for body size, physiologic data (organ volumes and blood flows) for a standard man were derived from the National Health and Examination Survey III database via  the P3M program.10 Data were taken as the average of 1,000 men aged between 25 and 35 years of all races, excluding those with a body mass index greater than 25 kg/m2. This standard man was 30 yr old, weighed 69 kg, and was 174 cm tall.
Modeling was performed using Scientist version 2.01 (Micromath, Salt Lake City, UT) or MATLAB (The Mathworks, Natick, MA). The goodness of fit was assessed using the Model Selection Criterion (MSC).11 This is essentially the Akaike Information Criterion scaled for the size of the data set and has been discussed in detail previously4 —the higher the MSC is, the better the fit is. Model selection was based on the goodness of fit, the pattern of residuals, the SD and correlation of parameters, and the mechanistic plausibility of the model. Data points were given equal weighting. All distribution volumes are reported as apparent distribution volumes. The abbreviations used for the concentrations used in the model are Cpa(pulmonary artery), Cart(arterial), Cjug(jugular bulb), Cbrn(global brain), and Cmv(mixed venous).
Hybrid Modeling: General Principles
Hybrid modeling has been used in our laboratory to construct physiologically based models of a number of drugs.4–7,12,13 In brief, this involves modeling kinetics in key organs (such as the brain and lungs) by representing the measured input to the organ (afferent drug concentrations and organ blood flow) as empirical continuous functions. The form (e.g.  , exponential or polynomial) of these “forcing” functions is unimportant, provided they interpolate the observed data accurately. They are necessary because the numerical integration used to solve the model requires continuous functions rather than discrete data points. With appropriate forcing functions in place, the efferent concentrations from the organ can be curve-fitted to estimate the model parameters for the organ.
Hybrid Models of Cerebral Kinetics and Dynamics
This part of the model was based on the data of Ludbrook et al.  ,15 who reported the propofol concentrations in arterial blood, jugular bulb blood (efferent from the brain), and the brain (calculated via  mass balance principles14) after a two-stage infusion of propofol in man. Changes in cerebral blood flow (via  transcranial Doppler) and Bispectral Index (BIS® Aspect Medical Systems, Newton, MA) were also reported. Their mean data were characterized by relatively little uncertainty and were assumed to be representative of a standard man.
Cerebral Kinetics
Five different models of brain kinetics were examined: (1) a null model, where the jugular bulb concentration equaled the arterial concentration (this tested the hypothesis that the concentration difference across the brain was due to random fluctuations); (2) a single flow-limited compartment defined by a single distribution volume and organ blood flow; (3) a two-compartment “tank in series” model; (4) a single flow-limited compartment with an apparent first-order loss representing either deep distribution or metabolism; and (5) a two-compartment membrane-limited model with a permeability term describing distribution into a deep compartment. The equations for these models have been described elsewhere.4 
The measured arterial concentrations were represented as multiexponential forcing functions. The measured reduction in cerebral blood flow caused by propofol was represented as a polynomial forcing function, but with the baseline values based on those used by Ludbrook et al.  15 (brain volume of 1.4 l, brain blood flow of 0.770 l/min). The midazolam premedication of the patients was assumed not to change their cerebral blood flow.16 Both the jugular bulb alone and the jugular bulb together with the calculated brain concentrations were fitted to estimate model parameters.
Cerebral Dynamics
For the dynamic component of the model, both linear and maximum effect (Emax) functions were used to examine the relation between the jugular and calculated brain propofol concentrations and the changes in cerebral blood flow and the BIS. The best dynamic relations were then incorporated into a final cerebral kinetic–dynamic model. This final model was fitted to the observed data to estimate a final parameter set.
Validation
The final cerebral kinetic model was validated by testing its ability to predict independent data not used in constructing the model. The data chosen were those of Peacock et al.  ,17 who reported the jugular bulb concentrations of propofol during a short infusion at a rate of 6 mg · kg−1· h−1(their 12-mg · kg−1· h−1data were considered too noisy for modeling). The arterial concentrations reported by Peacock et al.  were used as the input forcing function for the model, and the model was used to predict the expected jugular bulb propofol concentrations.
Hybrid Models of Lung Kinetics and Vascular Transport
The data used for defining a model of the lung kinetics of propofol in man were those reported by He et al.  18 In their first-pass dual-indicator uptake study, nine patients undergoing elective surgery and anesthetized with thiopental (250 mg) and then sevoflurane (0.5–1%) were given a simultaneous bolus of propofol (5 mg) and indocyanine green (15 mg) over 1 s into a central vein. Radial artery blood samples were collected at up to 1-s intervals for 1 min, and plasma was assayed for propofol. Indocyanine green was determined using a pulse-spectrophotometric monitoring system. Models were based on data interpolated electronically (using the Digitizer program1) from their figure 1(a representative patient) and from their reported mean values for cardiac output and mean transit times. Because their mean patient weight was 62.3 kg, volumes and cardiac outputs were scaled up to give representative values for a 70-kg standard man. Throughout, it was assumed that propofol itself had no effect on cardiac output, in keeping with previous work, which showed only a transient decrease in cardiac output after bolus administration of propofol.19,20 
Fig. 1. The submodel used to represent lung kinetics, based on the indocyanine green and propofol data of He  et al.  18 The venous mixing compartment is common to both drugs and accounts for the dispersion of the injected drug in transit from the injection site to the pulmonary artery (Cpa). For each drug, the lung itself is represented by a series of three compartments, each one third of the total apparent distribution volume for the drug (VICGand VPfor indocyanine green and propofol, respectively). All compartments are connected by a flow of blood given by cardiac output (QCO). Cmv= average mixed venous concentration downstream of the injection site; Cart= arterial concentration; Elng= extraction term for propofol across the lungs due to metabolism; Vmix= vascular mixing volume. 
Fig. 1. The submodel used to represent lung kinetics, based on the indocyanine green and propofol data of He  et al.  18The venous mixing compartment is common to both drugs and accounts for the dispersion of the injected drug in transit from the injection site to the pulmonary artery (Cpa). For each drug, the lung itself is represented by a series of three compartments, each one third of the total apparent distribution volume for the drug (VICGand VPfor indocyanine green and propofol, respectively). All compartments are connected by a flow of blood given by cardiac output (QCO). Cmv= average mixed venous concentration downstream of the injection site; Cart= arterial concentration; Elng= extraction term for propofol across the lungs due to metabolism; Vmix= vascular mixing volume. 
Fig. 1. The submodel used to represent lung kinetics, based on the indocyanine green and propofol data of He  et al.  18 The venous mixing compartment is common to both drugs and accounts for the dispersion of the injected drug in transit from the injection site to the pulmonary artery (Cpa). For each drug, the lung itself is represented by a series of three compartments, each one third of the total apparent distribution volume for the drug (VICGand VPfor indocyanine green and propofol, respectively). All compartments are connected by a flow of blood given by cardiac output (QCO). Cmv= average mixed venous concentration downstream of the injection site; Cart= arterial concentration; Elng= extraction term for propofol across the lungs due to metabolism; Vmix= vascular mixing volume. 
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Visual inspection of their data shows that they are characterized by a 13-s lag before any drug emerges from the lungs and first-pass peaks for indocyanine green and propofol that are asymmetric and delayed relative to the injection at time zero. This precludes several simple models for the lungs and indicates that models that introduce a dispersion or delay process would be preferred, such as the dispersion model of Roberts and Rowland21 or a tank in series model as used by Krejcie et al.  22 in this context.
Therefore, tank in series models (one to four tanks) were fitted concurrently to the indocyanine green and propofol data with an optional extraction term for propofol to allow for metabolism or deep distribution in the lungs (fig. 1). The model had a common vascular mixing compartment to allow for the dispersion of the drugs between the injection site and the pulmonary artery. This modeling device has been documented and validated previously.23 
Models of Systemic Kinetics
Models of systemic kinetics incorporated the best models of cerebral kinetics and dynamics and of lung kinetics. The task was therefore to find the simplest configuration for representing the remainder of the body (i.e.  , other than the lungs and brain) that was consistent with the known systemic kinetics of propofol. The arterial concentrations for two dose regimens were fitted simultaneously to determine the systemic kinetics. The first regimen was a rapid (5-min) and then slow (20-min) propofol infusion followed by a washout as used by Ludbrook et al.  15 The arterial concentrations expected for this dose regimen (0–300 min) for a standard man were simulated using the propofol population model reported by Schnider et al.  24 This model produced arterial concentrations that were almost superimposable with those directly observed by Ludbrook et al.  The second regimen was a constant rate infusion of 10 mg/min for 300 min, again simulated using the model of Schnider et al.  for a standard man.
The form of the systemic part of the model was determined semiempirically. The simplest tried was a single flow-limited compartment (receiving cardiac output less cerebral blood flow). Additional compartments were added, as dictated by the data, with organ blood flow as given by the standard man data derived via  the P3M program.10 Organs were nominally grouped into pooled compartments (e.g.  , well-perfused or poorly perfused) if necessary. Parameters for the systemic component of the model were estimated by simultaneous curve-fitting of the arterial concentration sets for both dose regimens.
In fitting the systemic component of the model, a range of fixed values for lung (0–30%) and fast compartment (0–60%) extraction ratios were tried to find the combination of hepatic, lung, and renal clearance that best fit the systemic data.
Sensitivity Analysis
Given the apparent role of cardiac output in the initial distribution of propofol, a sensitivity analysis of the model with respect to changes in cardiac output was performed. The time courses of the arterial blood concentrations and the BIS for 15 min after a propofol bolus dose of 140 mg given over 20 s in unpremedicated subjects were simulated for cardiac outputs of 2.5, 5.5, and 8.5 l/min. The cerebral blood flow was unchanged for each cardiac output; the blood flows to the other tissues were altered in proportion to the required cardiac output. The BIS for loss and recovery of consciousness were set at 80, and the times to these events were calculated.
Comparison with Clinical Data
The final model was used to simulate the outcome of the administration of propofol to provide some insight into the general fidelity of the model to clinical practice. The data simulated were those of Kazama et al.  , who infused propofol at rapid25 and slow rates26 into patients with a range of ages but similar body sizes. The simulation used the cardiac outputs reported by them for each age cohort (it declined progressively with age). Cerebral blood flow was also set to decrease progressively by a total of 20% between the ages of 20 and 80 years.27 Time to loss of consciousness was calculated assuming a threshold BIS of 80. No other changes were made, thereby testing the assumption that the differences in induction time reported by Kazama et al.  could be explained only by the differences in blood flow with age.
Results
Cerebral Kinetics and Dynamics
The decrease in the BIS was linked to the global brain concentration of propofol using a linear model (fig. 2A), but with constraints so that the BIS could not exceed 100 or be less than 0. The reduction of cerebral blood flow caused by propofol was linked to the global brain concentration of propofol using an Emaxmodel. This produced a curvilinear relation with minimal hysteresis (fig. 2B). The final cerebral kinetic–dynamic model was a membrane-limited model of the brain, with dynamic effects linked to the global brain concentration as described above (fig. 3). This gave an MSC of 4.71, and an excellent fit of all observed data (fig. 4). The MSC for the other models ranged from −2.14 for the null model to 3.11 for the flow with loss model.
Fig. 2. (  A  ) The data used for the initial estimates of the linear function linking the brain concentration (Cbrn) to the Bispectral Index (BIS). The data were fitted to the function BIS = BISslp× Cbrn+ BISint. The Model Selection Criterion was 3.62, and the parameter values were BISslp=−4.46 SD 0.16 units · mg−1· l and BISint= 77.7 SD 1.94 units. These were used as starting values for this component of the combined kinetic–dynamic model for the brain. (  B  ) The data used for the initial estimates of the Emaxfunction linking the brain concentration (Cbrn) to the percent reduction in cerebral blood flow from baseline (Qbrn,%). The data were fitted to the function Qbrn,%= Qbrn,baseline− (Emax,CBF× Cbrn)/(EC50,CBF+ Cbrn). The Model Selection Criterion was 3.95, and the parameter values were Baseline = 101.5 SD 1.8%, Emax= 79.5 SD 3.5%, and EC50= 6.17 SD 1.02 mg/l. These were used as starting values for this component of the combined kinetic–dynamic model for the brain. 
Fig. 2. (  A  ) The data used for the initial estimates of the linear function linking the brain concentration (Cbrn) to the Bispectral Index (BIS). The data were fitted to the function BIS = BISslp× Cbrn+ BISint. The Model Selection Criterion was 3.62, and the parameter values were BISslp=−4.46 SD 0.16 units · mg−1· l and BISint= 77.7 SD 1.94 units. These were used as starting values for this component of the combined kinetic–dynamic model for the brain. (  B  ) The data used for the initial estimates of the Emaxfunction linking the brain concentration (Cbrn) to the percent reduction in cerebral blood flow from baseline (Qbrn,%). The data were fitted to the function Qbrn,%= Qbrn,baseline− (Emax,CBF× Cbrn)/(EC50,CBF+ Cbrn). The Model Selection Criterion was 3.95, and the parameter values were Baseline = 101.5 SD 1.8%, Emax= 79.5 SD 3.5%, and EC50= 6.17 SD 1.02 mg/l. These were used as starting values for this component of the combined kinetic–dynamic model for the brain. 
Fig. 2. (  A  ) The data used for the initial estimates of the linear function linking the brain concentration (Cbrn) to the Bispectral Index (BIS). The data were fitted to the function BIS = BISslp× Cbrn+ BISint. The Model Selection Criterion was 3.62, and the parameter values were BISslp=−4.46 SD 0.16 units · mg−1· l and BISint= 77.7 SD 1.94 units. These were used as starting values for this component of the combined kinetic–dynamic model for the brain. (  B  ) The data used for the initial estimates of the Emaxfunction linking the brain concentration (Cbrn) to the percent reduction in cerebral blood flow from baseline (Qbrn,%). The data were fitted to the function Qbrn,%= Qbrn,baseline− (Emax,CBF× Cbrn)/(EC50,CBF+ Cbrn). The Model Selection Criterion was 3.95, and the parameter values were Baseline = 101.5 SD 1.8%, Emax= 79.5 SD 3.5%, and EC50= 6.17 SD 1.02 mg/l. These were used as starting values for this component of the combined kinetic–dynamic model for the brain. 
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Fig. 3. The final model of cerebral kinetics and dynamics based on the data of Ludbrook  et al.  15 The symbols used are described in  table 1.
Fig. 3. The final model of cerebral kinetics and dynamics based on the data of Ludbrook  et al.  15The symbols used are described in  table 1.
Fig. 3. The final model of cerebral kinetics and dynamics based on the data of Ludbrook  et al.  15 The symbols used are described in  table 1.
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Table 1. Parameter Values of the Final Model Shown with Their Symbols 
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Table 1. Parameter Values of the Final Model Shown with Their Symbols 
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Fig. 4. The lines of best fit (  solid lines  ) for the final cerebral kinetic–dynamic model shown together with the mean of the original data (  symbols  ) for percentage reduction in cerebral blood flow from baseline (Qbrn,%), the Bispectral Index (BIS), and the jugular bulb (Cjug) and calculated brain (Cbrn) propofol concentrations as reported by Ludbrook  et al.  .15 The model was a good representation of the data. 
Fig. 4. The lines of best fit (  solid lines  ) for the final cerebral kinetic–dynamic model shown together with the mean of the original data (  symbols  ) for percentage reduction in cerebral blood flow from baseline (Qbrn,%), the Bispectral Index (BIS), and the jugular bulb (Cjug) and calculated brain (Cbrn) propofol concentrations as reported by Ludbrook  et al.  .15The model was a good representation of the data. 
Fig. 4. The lines of best fit (  solid lines  ) for the final cerebral kinetic–dynamic model shown together with the mean of the original data (  symbols  ) for percentage reduction in cerebral blood flow from baseline (Qbrn,%), the Bispectral Index (BIS), and the jugular bulb (Cjug) and calculated brain (Cbrn) propofol concentrations as reported by Ludbrook  et al.  .15 The model was a good representation of the data. 
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When compared to the data of Peacock et al.  ,17 it was found that the predictions of the final model were in good agreement with the observed jugular bulb propofol concentrations (fig. 5). This suggests the general form of the cerebral kinetic model was appropriate.
Fig. 5. Validation of the cerebral kinetics component of the model. The final cerebral kinetic model based on the data of Ludbrook  et al.  15 was used to predict the jugular bulb concentrations (Cjug) expected in the study by Peacock  et al.  17 in which propofol was infused at 6 mg · kg−1· h−1until loss of consciousness. There was good agreement between the observed (  open circle  ) and predicted (  solid line  ) concentrations. The forcing function used to represent the arterial concentrations is shown as a  dashed line  , and the observed arterial concentrations are the  closed circle  . 
Fig. 5. Validation of the cerebral kinetics component of the model. The final cerebral kinetic model based on the data of Ludbrook  et al.  15was used to predict the jugular bulb concentrations (Cjug) expected in the study by Peacock  et al.  17in which propofol was infused at 6 mg · kg−1· h−1until loss of consciousness. There was good agreement between the observed (  open circle  ) and predicted (  solid line  ) concentrations. The forcing function used to represent the arterial concentrations is shown as a  dashed line  , and the observed arterial concentrations are the  closed circle  . 
Fig. 5. Validation of the cerebral kinetics component of the model. The final cerebral kinetic model based on the data of Ludbrook  et al.  15 was used to predict the jugular bulb concentrations (Cjug) expected in the study by Peacock  et al.  17 in which propofol was infused at 6 mg · kg−1· h−1until loss of consciousness. There was good agreement between the observed (  open circle  ) and predicted (  solid line  ) concentrations. The forcing function used to represent the arterial concentrations is shown as a  dashed line  , and the observed arterial concentrations are the  closed circle  . 
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Lung Kinetics
A model of the lungs with three compartments in series (fig. 1) provided an excellent fit (MSC = 5.24) of the observed data for both indocyanine green and propofol (fig. 6). The MSC for the other models ranged from 0.09 to 3.79. A lag of approximately 13 s was needed to allow for the delayed emergence of the peaks in arterial blood after intravenous injection. Extraction of propofol by the lung (22%) was required to account for the single subject shown in figure 1of He et al.  18 Only 3% extraction was evident in their infusion data reported in the same article. In the final model, a lung extraction of 10% was used.
Fig. 6. The observed data of He  et al.  18 on the concurrent lung kinetics of indocyanine green (ICG;  open circles  ) and propofol (  open triangles  ). The  lines  show the best fit of the three-tank in series model (  fig. 6) that simultaneously described the lung kinetics of both drugs. 
Fig. 6. The observed data of He  et al.  18on the concurrent lung kinetics of indocyanine green (ICG;  open circles  ) and propofol (  open triangles  ). The  lines  show the best fit of the three-tank in series model (  fig. 6) that simultaneously described the lung kinetics of both drugs. 
Fig. 6. The observed data of He  et al.  18 on the concurrent lung kinetics of indocyanine green (ICG;  open circles  ) and propofol (  open triangles  ). The  lines  show the best fit of the three-tank in series model (  fig. 6) that simultaneously described the lung kinetics of both drugs. 
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Systemic Kinetics
The final systemic model is shown in figure 7. The liver and gut were represented as one flow-limited compartment with an extraction term. Two distribution compartments were used (nominally fast and slow). A single flow-limited compartment represented well-perfused tissues (perfusion > 0.5 l/min per l of tissue) receiving 22.8% of the cardiac output (nominally the heart, kidneys, and thyroid). This also had an extraction term representing renal clearance. A membrane-limited compartment (with a shallow and deep compartment separated by a permeability barrier) was used to represent the poorly perfused tissues (nominally muscle, fat, skin, and marrow). This received 38.3% of the cardiac output. Some difficulty was encountered finding a model that fit both the initial peak and the washout tail before adopting this configuration, because it was found that using flow-limited compartments alone caused the recirculated concentrations to increase too slowly. However, this concept of membrane limitation is consistent with measurements of the kinetics of propofol in the muscle of sheep.28 
Fig. 7. The model of the systemic kinetics of propofol incorporating detailed models of the brain and lungs based on separately derived hybrid models. QCOis the cardiac output, Qbrnis the cerebral blood flow. The remaining symbols are shown in  table 1.
Fig. 7. The model of the systemic kinetics of propofol incorporating detailed models of the brain and lungs based on separately derived hybrid models. QCOis the cardiac output, Qbrnis the cerebral blood flow. The remaining symbols are shown in  table 1.
Fig. 7. The model of the systemic kinetics of propofol incorporating detailed models of the brain and lungs based on separately derived hybrid models. QCOis the cardiac output, Qbrnis the cerebral blood flow. The remaining symbols are shown in  table 1.
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It was found that a fixed lung extraction of 10% and a fixed fast compartment extraction of 30% provided the best fit of the systemic data consistent with the criteria that the fitted hepatic extraction was between 70 and 100%. The final parameters of the model are shown in table 1, and the model is shown pictorially in figure 7. The best fits to the data are shown in figure 8. The MSC of the final model was 5.04. When used to simulate a bolus dose, the declining phase of the arterial concentrations predicted by the model could be described by four apparent exponentials, with half-lives of 1.1, 5.4, 23.4, and 167 min.
Fig. 8. The fit of the systemic model (  line  ) shown in  figure 7to the observed arterial concentrations (Cart,  symbols  ) for two dose regimens derived from the data of Ludbrook  et al.  15 and Schinder  et al.  24 
Fig. 8. The fit of the systemic model (  line  ) shown in  figure 7to the observed arterial concentrations (Cart,  symbols  ) for two dose regimens derived from the data of Ludbrook  et al.  15and Schinder  et al.  24
Fig. 8. The fit of the systemic model (  line  ) shown in  figure 7to the observed arterial concentrations (Cart,  symbols  ) for two dose regimens derived from the data of Ludbrook  et al.  15 and Schinder  et al.  24 
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Sensitivity Analysis
The time course of the arterial concentrations and BIS for a bolus of 2 mg/kg over 20 s is shown in figure 9. With respect to changes in cardiac output, the model behaved in a manner analogous to the previous models based on data collected in sheep.5–7 The height and area of the large first-pass arterial peak emerging from the lungs was inversely related to cardiac output, as previously measured in sheep.3 The brain concentrations and hence BIS were also affected by cardiac output, because this first-pass concentration peak also enters and passes through the brain. The loss of consciousness was predicted to occur at 35, 34, and 39 s for cardiac outputs of 8.5, 5.5, and 2.5 l/min, respectively. The recovery of consciousness was predicted to occur at 2.9, 8.6, and 18.7 min for the same respective cardiac outputs. Therefore, changes in cardiac output had only a minor effect on the time to loss of consciousness but greatly affected the time to recovery of consciousness and the overall depth of anesthesia (fig. 9).
Fig. 9. A simulation of a bolus over 20 s of 2 mg/kg propofol in an unpremedicated 70-kg subject when cardiac output (Qco) was 2.5, 5.5, or 8.5 l/min. (  A  ) The time course of the predicted arterial propofol concentrations over a 15-min interval, showing the large cardiac output–dependent first-pass peak.  29 (Inset  ) The arterial concentrations shown on a log scale and for a longer time period. (  B  ) The time course of the predicted Bispectral Index (BIS). The threshold for loss and recovery of consciousness was a BIS of 80, giving the times for these events listed in the Results section. 
Fig. 9. A simulation of a bolus over 20 s of 2 mg/kg propofol in an unpremedicated 70-kg subject when cardiac output (Qco) was 2.5, 5.5, or 8.5 l/min. (  A  ) The time course of the predicted arterial propofol concentrations over a 15-min interval, showing the large cardiac output–dependent first-pass peak.  29(Inset  ) The arterial concentrations shown on a log scale and for a longer time period. (  B  ) The time course of the predicted Bispectral Index (BIS). The threshold for loss and recovery of consciousness was a BIS of 80, giving the times for these events listed in the Results section. 
Fig. 9. A simulation of a bolus over 20 s of 2 mg/kg propofol in an unpremedicated 70-kg subject when cardiac output (Qco) was 2.5, 5.5, or 8.5 l/min. (  A  ) The time course of the predicted arterial propofol concentrations over a 15-min interval, showing the large cardiac output–dependent first-pass peak.  29 (Inset  ) The arterial concentrations shown on a log scale and for a longer time period. (  B  ) The time course of the predicted Bispectral Index (BIS). The threshold for loss and recovery of consciousness was a BIS of 80, giving the times for these events listed in the Results section. 
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Comparison with Clinical Data
The simulation of the data of Kazama et al.  25,26 for a threshold BIS of 80 produced quantitative similarities but an obvious discrepancy between the model and the data. However, changing the BIS threshold (to 77) produced a match of the data (fig. 10). Therefore, induction time (particularly for their low dose rate) was found to be sensitive to the anesthetic threshold of the brain. In general, the model supported the hypothesis that cardiovascular changes with age can account for a significant component of the changing propofol dose requirements with age. It was considered that this comparison supported the utility of the model in simulating the clinical behavior of propofol.
Fig. 10. The predictions of the model (  lines  ) when adapted to simulate the data of Kazama  et al.  25,26 on the time to loss of consciousness (LOC) in cohorts of different age and cardiac output when propofol is infused at low and high dose rates. The observed data are shown as  symbols  . 
Fig. 10. The predictions of the model (  lines  ) when adapted to simulate the data of Kazama  et al.  25,26on the time to loss of consciousness (LOC) in cohorts of different age and cardiac output when propofol is infused at low and high dose rates. The observed data are shown as  symbols  . 
Fig. 10. The predictions of the model (  lines  ) when adapted to simulate the data of Kazama  et al.  25,26 on the time to loss of consciousness (LOC) in cohorts of different age and cardiac output when propofol is infused at low and high dose rates. The observed data are shown as  symbols  . 
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Discussion
In this model, particular attention was paid to representing those physiologic features that influence propofol bolus kinetics (vascular mixing with cardiac output and lung kinetics) and propofol dynamics (cerebral kinetics, cerebral blood flow, cerebral concentration–effect relations). The incorporation of blood flow terms into the model was considered important because surgery and anesthetic management affects the state of the circulation, and the state of the circulation affects the disposition of propofol.
The basic form of the cerebral kinetic model used here is essentially similar to that used previously to model the cerebral kinetics of propofol in sheep.6 This model has previously been shown to represent the changes in cerebral kinetics that occur with changes in cerebral blood flow.4 The fact that the cerebral kinetic model developed from the data of Ludbrook et al.  15 was able to predict the independent data of Peacock et al.  17 suggests that this component of this model is a reasonable account of the in vivo  situation.
In the previous sheep models, the distribution component of lung kinetics was described as a single compartment with a small volume (approximately 4.5 l),5 but this was based on data collected for a 2-min infusion. In the current model, a tank in series model was needed to represent the unique data of He et al.  ,18 which was based on a 1-s bolus and showed a significant vascular lag and skewed first-pass peak. It would be expected that this refined representation of the lungs would enhance the ability of the human model to describe the (often large) first-pass concentration peak that occurs after rapid intravenous bolus injections.29 The systemic component of the model (including the lungs) is most similar to the recirculatory compartmental models presented by Krejcie et al.  30,31 These models have been shown to successfully describe the front-end kinetics of a number of drugs after rapid bolus injection, particularly in data where both a first-pass and a second-pass recirculatory peak are evident.32 Recirculatory models by their nature divide the body into representations of the lungs and the rest of body.33,34 Cardiac output (and its distribution) is often an important kinetic parameter. The sensitivity analysis of the current model confirms this for propofol (fig. 9), in keeping with data collected in animals.3 Extensions of the current model to the basic recirculatory structure include the representation of hepatic clearance via  extraction across a hepatic compartment to better account for the influence of changes in hepatic blood flow. Furthermore, based on data on the muscle uptake of propofol in sheep,28 slow distribution was represented using a membrane-limited compartment, rather than a single compartment with a separate nondistributive pathway as used by Krejcie et al.  Both approaches function to increase the rate of recirculation of a drug compared with that predicted when using a single large compartment for deep distribution.
It should be emphasized that the approach used here was not directed toward developing a model that could be fitted to concentration–time course data collected in individual patients. With current technology, it is difficult to estimate the necessary regional distribution of blood flow in individual subjects (although this area is advancing rapidly). Rather, the intention was to use mean data collected from a number of patients in which the mean blood flows and organ volumes approach those expected for a standard man. Despite this limitation, the model in its current form is useful for answering questions that are directed toward defining the response of the average patient. This could include making quantitative predictions about how a drug that affects cardiac output would affect the kinetics and dynamics of propofol. There are a number of studies reporting possible interactions between cardioactive drugs and propofol35,36 for which the underlying mechanism is currently speculative and could be supported using modeling.
The model was required to be consistent with several studies. This was by necessity to estimate the required parameters but has the added advantage of increasing confidence in the model. This can be thought of as a meta-modeling approach that is analogous to the meta-analysis used in evidence based medicine. Components of the final model (lung and brain) were derived independently from appropriate studies in the literature by use of hybrid modeling. The underlying principle is that by pooling data, interpatient variability is reduced, and the data are representative of that for the average patient. A consequence of this approach is the degree of flexibility allowed in model structure—there are many aspects of the model that could be altered slightly without greatly affecting the behavior of the model (particularly for the systemic component of the model). Therefore, there is a subjective component to the model that reflects the experience and background of the model builder. A corollary of this is that there is substantial room for debate about final structure and parameter values of the model. However, we take the view that the model presented here is neither complete or a singular solution to the problem of propofol disposition. We expect the model will continue to evolve as the information contained in future (or past) data sets is compared to the predictions of the model. The approach used here does require careful consideration of which data sets are appropriate for model building—many in the literature are not suitable because of methodologic deficiencies or noisy mean data (high SE) or because inadequate information is provided.
In contrast to models where all parameters can be estimated from the data, the approach used here required some parameters to be estimated based on known physiology. The standard (or reference) man is a useful device in this regard and has a long history in the modeling of volatile anesthetics, toxicology, and radiation exposure. The current article made use of the P3M program10 for deriving organ volumes and blood flows. It is intended to make use of this database to adapt the model so that it can be scaled across body size. Ultimately, by adding variability to parameter values with appropriate correlation structures,37 the model may be suitable for clinical trial simulation; however, this is outside the scope of the current article.
The total body clearance of propofol exceeds liver blood flow, and the hepatic extraction of propofol is in the approximate range of 70–90%.38,39 In our model, this extrahepatic clearance of propofol must be attributed to specific organs. During the development of the systemic model, initially, no lung or kidney extraction was used. This invariably produced fitted hepatic extraction ratios that exceeded 1 (physiologically impossible). Incrementally adding either lung extraction or kidney extraction alone improved the fit of the model, but the best fit was found with a lung extraction of 10% and a fast compartment (via  kidney) extraction of 30%. The lung metabolism of propofol in man is controversial but has been demonstrated in man.40 However, other studies have shown no pulmonary extraction of propofol.18 These studies may be confounded by power problems or inhibition of lung extraction by concurrent volatile anesthetics, which we have observed in sheep.41 The inclusion of renal metabolism of propofol is supported by in vitro  42 and in vivo  data.43 The extraction of 30% used in the fast compartment of the model (which includes the kidneys, heart, and thyroid) is equivalent to a renal extraction of 40% if the other organs in the fast compartment are assumed not to metabolize propofol. In fact, the renal extraction of propofol has been reported to be nearly 60%.43 However, models in which the extrahepatic clearance of propofol was restricted only to the kidneys did not fit the systemic data as well as models in which clearance was in both the kidneys and the lungs. The sites of the extrahepatic clearance in the model will be reviewed as more data become available.
Another confounding issue is that of the delay between the BIS and the true anesthetic depth. The magnitude of this delay is not publicly documented but may be of the order of 30 s for the older ASPECT 1000 series monitors (Aspect Medical Systems). The Ludbrook study15 used an ASPECT 2000 monitor. Trials were conducted during the modeling allowing for up to a 30-s delay between brain concentration and BIS, but this was not supported by the data. In the final model, the effect of the delay was ignored. This may have implications when the brain concentrations are changing rapidly at the onset of loss of consciousness. The BIS reported by the model will reflect the underlying brain concentration of propofol, but the observed BIS may be higher as the monitor lags behind events in the brain.
There are several notable limitations to the current model. First, recirculatory models have the potential to simultaneously model drug concentrations in multiple sites in the body. The current model should in theory account for mixed venous, jugular bulb, pulmonary artery, and aortic propofol concentrations with acceptable accuracy. However, it was not developed using any peripheral venous data. It may require modification to predict these concentrations accurately. Second, the lumped description of the body (into fast and slow distribution compartments) is adequate for scaling body size and blood flow changes but may be less suitable for predicting changes in body composition (particularly percent body fat). This will await comparison with a suitable data set. Third, the model was based on data collected for relatively short administration regimens. Therefore, it may not adequately represent longer-term kinetic processes (such as uptake into fat), which dictate the behavior of multiday infusions (e.g.  , sedation in critical care) and the long-term washout of propofol. This is evident in the systematic discrepancy in the fit of the washout tail in figure 8.
In conclusion, it seems to be feasible to devise a recirculatory kinetic–dynamic model of propofol disposition in man that is consistent with a range of clinical data. It is hoped that the final model will represent the complex effect of the circulation on propofol disposition. When linked to a suitable model of the cardiovascular system, the basic recirculatory form of the model presented here may also be suitable for representing the converse effect of propofol on the circulation.44 We also note the growing call for integrative modeling of biologic systems.45 For example, the Physiome Project2is an ambitious attempt to model a human at the whole body system level. We see that model presented here is a small step in this direction for anesthesia and pharmacology. The current uses of the model include quantitatively predicting the outcome of circulatory changes on propofol kinetics and dynamics. The potential uses for this model may eventually include clinical trial (or practice) simulation, to assist in the development of target-controlled or closed-loop infusion devices (where it could act as reference model46), or more realistic anesthesia simulators. It is clear, however, that progress in this area will require more model development and new paradigms for both modeling and data sharing.
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Fig. 1. The submodel used to represent lung kinetics, based on the indocyanine green and propofol data of He  et al.  18 The venous mixing compartment is common to both drugs and accounts for the dispersion of the injected drug in transit from the injection site to the pulmonary artery (Cpa). For each drug, the lung itself is represented by a series of three compartments, each one third of the total apparent distribution volume for the drug (VICGand VPfor indocyanine green and propofol, respectively). All compartments are connected by a flow of blood given by cardiac output (QCO). Cmv= average mixed venous concentration downstream of the injection site; Cart= arterial concentration; Elng= extraction term for propofol across the lungs due to metabolism; Vmix= vascular mixing volume. 
Fig. 1. The submodel used to represent lung kinetics, based on the indocyanine green and propofol data of He  et al.  18The venous mixing compartment is common to both drugs and accounts for the dispersion of the injected drug in transit from the injection site to the pulmonary artery (Cpa). For each drug, the lung itself is represented by a series of three compartments, each one third of the total apparent distribution volume for the drug (VICGand VPfor indocyanine green and propofol, respectively). All compartments are connected by a flow of blood given by cardiac output (QCO). Cmv= average mixed venous concentration downstream of the injection site; Cart= arterial concentration; Elng= extraction term for propofol across the lungs due to metabolism; Vmix= vascular mixing volume. 
Fig. 1. The submodel used to represent lung kinetics, based on the indocyanine green and propofol data of He  et al.  18 The venous mixing compartment is common to both drugs and accounts for the dispersion of the injected drug in transit from the injection site to the pulmonary artery (Cpa). For each drug, the lung itself is represented by a series of three compartments, each one third of the total apparent distribution volume for the drug (VICGand VPfor indocyanine green and propofol, respectively). All compartments are connected by a flow of blood given by cardiac output (QCO). Cmv= average mixed venous concentration downstream of the injection site; Cart= arterial concentration; Elng= extraction term for propofol across the lungs due to metabolism; Vmix= vascular mixing volume. 
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Fig. 2. (  A  ) The data used for the initial estimates of the linear function linking the brain concentration (Cbrn) to the Bispectral Index (BIS). The data were fitted to the function BIS = BISslp× Cbrn+ BISint. The Model Selection Criterion was 3.62, and the parameter values were BISslp=−4.46 SD 0.16 units · mg−1· l and BISint= 77.7 SD 1.94 units. These were used as starting values for this component of the combined kinetic–dynamic model for the brain. (  B  ) The data used for the initial estimates of the Emaxfunction linking the brain concentration (Cbrn) to the percent reduction in cerebral blood flow from baseline (Qbrn,%). The data were fitted to the function Qbrn,%= Qbrn,baseline− (Emax,CBF× Cbrn)/(EC50,CBF+ Cbrn). The Model Selection Criterion was 3.95, and the parameter values were Baseline = 101.5 SD 1.8%, Emax= 79.5 SD 3.5%, and EC50= 6.17 SD 1.02 mg/l. These were used as starting values for this component of the combined kinetic–dynamic model for the brain. 
Fig. 2. (  A  ) The data used for the initial estimates of the linear function linking the brain concentration (Cbrn) to the Bispectral Index (BIS). The data were fitted to the function BIS = BISslp× Cbrn+ BISint. The Model Selection Criterion was 3.62, and the parameter values were BISslp=−4.46 SD 0.16 units · mg−1· l and BISint= 77.7 SD 1.94 units. These were used as starting values for this component of the combined kinetic–dynamic model for the brain. (  B  ) The data used for the initial estimates of the Emaxfunction linking the brain concentration (Cbrn) to the percent reduction in cerebral blood flow from baseline (Qbrn,%). The data were fitted to the function Qbrn,%= Qbrn,baseline− (Emax,CBF× Cbrn)/(EC50,CBF+ Cbrn). The Model Selection Criterion was 3.95, and the parameter values were Baseline = 101.5 SD 1.8%, Emax= 79.5 SD 3.5%, and EC50= 6.17 SD 1.02 mg/l. These were used as starting values for this component of the combined kinetic–dynamic model for the brain. 
Fig. 2. (  A  ) The data used for the initial estimates of the linear function linking the brain concentration (Cbrn) to the Bispectral Index (BIS). The data were fitted to the function BIS = BISslp× Cbrn+ BISint. The Model Selection Criterion was 3.62, and the parameter values were BISslp=−4.46 SD 0.16 units · mg−1· l and BISint= 77.7 SD 1.94 units. These were used as starting values for this component of the combined kinetic–dynamic model for the brain. (  B  ) The data used for the initial estimates of the Emaxfunction linking the brain concentration (Cbrn) to the percent reduction in cerebral blood flow from baseline (Qbrn,%). The data were fitted to the function Qbrn,%= Qbrn,baseline− (Emax,CBF× Cbrn)/(EC50,CBF+ Cbrn). The Model Selection Criterion was 3.95, and the parameter values were Baseline = 101.5 SD 1.8%, Emax= 79.5 SD 3.5%, and EC50= 6.17 SD 1.02 mg/l. These were used as starting values for this component of the combined kinetic–dynamic model for the brain. 
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Fig. 3. The final model of cerebral kinetics and dynamics based on the data of Ludbrook  et al.  15 The symbols used are described in  table 1.
Fig. 3. The final model of cerebral kinetics and dynamics based on the data of Ludbrook  et al.  15The symbols used are described in  table 1.
Fig. 3. The final model of cerebral kinetics and dynamics based on the data of Ludbrook  et al.  15 The symbols used are described in  table 1.
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Fig. 4. The lines of best fit (  solid lines  ) for the final cerebral kinetic–dynamic model shown together with the mean of the original data (  symbols  ) for percentage reduction in cerebral blood flow from baseline (Qbrn,%), the Bispectral Index (BIS), and the jugular bulb (Cjug) and calculated brain (Cbrn) propofol concentrations as reported by Ludbrook  et al.  .15 The model was a good representation of the data. 
Fig. 4. The lines of best fit (  solid lines  ) for the final cerebral kinetic–dynamic model shown together with the mean of the original data (  symbols  ) for percentage reduction in cerebral blood flow from baseline (Qbrn,%), the Bispectral Index (BIS), and the jugular bulb (Cjug) and calculated brain (Cbrn) propofol concentrations as reported by Ludbrook  et al.  .15The model was a good representation of the data. 
Fig. 4. The lines of best fit (  solid lines  ) for the final cerebral kinetic–dynamic model shown together with the mean of the original data (  symbols  ) for percentage reduction in cerebral blood flow from baseline (Qbrn,%), the Bispectral Index (BIS), and the jugular bulb (Cjug) and calculated brain (Cbrn) propofol concentrations as reported by Ludbrook  et al.  .15 The model was a good representation of the data. 
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Fig. 5. Validation of the cerebral kinetics component of the model. The final cerebral kinetic model based on the data of Ludbrook  et al.  15 was used to predict the jugular bulb concentrations (Cjug) expected in the study by Peacock  et al.  17 in which propofol was infused at 6 mg · kg−1· h−1until loss of consciousness. There was good agreement between the observed (  open circle  ) and predicted (  solid line  ) concentrations. The forcing function used to represent the arterial concentrations is shown as a  dashed line  , and the observed arterial concentrations are the  closed circle  . 
Fig. 5. Validation of the cerebral kinetics component of the model. The final cerebral kinetic model based on the data of Ludbrook  et al.  15was used to predict the jugular bulb concentrations (Cjug) expected in the study by Peacock  et al.  17in which propofol was infused at 6 mg · kg−1· h−1until loss of consciousness. There was good agreement between the observed (  open circle  ) and predicted (  solid line  ) concentrations. The forcing function used to represent the arterial concentrations is shown as a  dashed line  , and the observed arterial concentrations are the  closed circle  . 
Fig. 5. Validation of the cerebral kinetics component of the model. The final cerebral kinetic model based on the data of Ludbrook  et al.  15 was used to predict the jugular bulb concentrations (Cjug) expected in the study by Peacock  et al.  17 in which propofol was infused at 6 mg · kg−1· h−1until loss of consciousness. There was good agreement between the observed (  open circle  ) and predicted (  solid line  ) concentrations. The forcing function used to represent the arterial concentrations is shown as a  dashed line  , and the observed arterial concentrations are the  closed circle  . 
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Fig. 6. The observed data of He  et al.  18 on the concurrent lung kinetics of indocyanine green (ICG;  open circles  ) and propofol (  open triangles  ). The  lines  show the best fit of the three-tank in series model (  fig. 6) that simultaneously described the lung kinetics of both drugs. 
Fig. 6. The observed data of He  et al.  18on the concurrent lung kinetics of indocyanine green (ICG;  open circles  ) and propofol (  open triangles  ). The  lines  show the best fit of the three-tank in series model (  fig. 6) that simultaneously described the lung kinetics of both drugs. 
Fig. 6. The observed data of He  et al.  18 on the concurrent lung kinetics of indocyanine green (ICG;  open circles  ) and propofol (  open triangles  ). The  lines  show the best fit of the three-tank in series model (  fig. 6) that simultaneously described the lung kinetics of both drugs. 
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Fig. 7. The model of the systemic kinetics of propofol incorporating detailed models of the brain and lungs based on separately derived hybrid models. QCOis the cardiac output, Qbrnis the cerebral blood flow. The remaining symbols are shown in  table 1.
Fig. 7. The model of the systemic kinetics of propofol incorporating detailed models of the brain and lungs based on separately derived hybrid models. QCOis the cardiac output, Qbrnis the cerebral blood flow. The remaining symbols are shown in  table 1.
Fig. 7. The model of the systemic kinetics of propofol incorporating detailed models of the brain and lungs based on separately derived hybrid models. QCOis the cardiac output, Qbrnis the cerebral blood flow. The remaining symbols are shown in  table 1.
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Fig. 8. The fit of the systemic model (  line  ) shown in  figure 7to the observed arterial concentrations (Cart,  symbols  ) for two dose regimens derived from the data of Ludbrook  et al.  15 and Schinder  et al.  24 
Fig. 8. The fit of the systemic model (  line  ) shown in  figure 7to the observed arterial concentrations (Cart,  symbols  ) for two dose regimens derived from the data of Ludbrook  et al.  15and Schinder  et al.  24
Fig. 8. The fit of the systemic model (  line  ) shown in  figure 7to the observed arterial concentrations (Cart,  symbols  ) for two dose regimens derived from the data of Ludbrook  et al.  15 and Schinder  et al.  24 
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Fig. 9. A simulation of a bolus over 20 s of 2 mg/kg propofol in an unpremedicated 70-kg subject when cardiac output (Qco) was 2.5, 5.5, or 8.5 l/min. (  A  ) The time course of the predicted arterial propofol concentrations over a 15-min interval, showing the large cardiac output–dependent first-pass peak.  29 (Inset  ) The arterial concentrations shown on a log scale and for a longer time period. (  B  ) The time course of the predicted Bispectral Index (BIS). The threshold for loss and recovery of consciousness was a BIS of 80, giving the times for these events listed in the Results section. 
Fig. 9. A simulation of a bolus over 20 s of 2 mg/kg propofol in an unpremedicated 70-kg subject when cardiac output (Qco) was 2.5, 5.5, or 8.5 l/min. (  A  ) The time course of the predicted arterial propofol concentrations over a 15-min interval, showing the large cardiac output–dependent first-pass peak.  29(Inset  ) The arterial concentrations shown on a log scale and for a longer time period. (  B  ) The time course of the predicted Bispectral Index (BIS). The threshold for loss and recovery of consciousness was a BIS of 80, giving the times for these events listed in the Results section. 
Fig. 9. A simulation of a bolus over 20 s of 2 mg/kg propofol in an unpremedicated 70-kg subject when cardiac output (Qco) was 2.5, 5.5, or 8.5 l/min. (  A  ) The time course of the predicted arterial propofol concentrations over a 15-min interval, showing the large cardiac output–dependent first-pass peak.  29 (Inset  ) The arterial concentrations shown on a log scale and for a longer time period. (  B  ) The time course of the predicted Bispectral Index (BIS). The threshold for loss and recovery of consciousness was a BIS of 80, giving the times for these events listed in the Results section. 
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Fig. 10. The predictions of the model (  lines  ) when adapted to simulate the data of Kazama  et al.  25,26 on the time to loss of consciousness (LOC) in cohorts of different age and cardiac output when propofol is infused at low and high dose rates. The observed data are shown as  symbols  . 
Fig. 10. The predictions of the model (  lines  ) when adapted to simulate the data of Kazama  et al.  25,26on the time to loss of consciousness (LOC) in cohorts of different age and cardiac output when propofol is infused at low and high dose rates. The observed data are shown as  symbols  . 
Fig. 10. The predictions of the model (  lines  ) when adapted to simulate the data of Kazama  et al.  25,26 on the time to loss of consciousness (LOC) in cohorts of different age and cardiac output when propofol is infused at low and high dose rates. The observed data are shown as  symbols  . 
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Table 1. Parameter Values of the Final Model Shown with Their Symbols 
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Table 1. Parameter Values of the Final Model Shown with Their Symbols 
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