characterizing carbon dioxide response and … · petco2 and the co2 sensitivity surface was a...

CHARACTERIZING CARBON DIOXIDE RESPONSE AND RESPIRATORY

DEPRESSION FOR REMIFENTANIL AND PROPOFOL COMBINATIONS

by

Benjamin Richard Randall

A thesis submitted to the faculty of

The University of Utah

in partial fulfillment of the requirements for the degree of

Master of Science

In

Bioengineering

Department of Bioengineering

The University of Utah

December 2006

ABSTRACT

Respiratory side effects are a significant problem when administering low doses of

propofol and reminfentanil to achieve mild sedation and analgesia for minimally invasive

procedures. Both drugs are respiratory depressants and the drug concentrations necessary

to achieve adequate anesthesia are definitely in a danger zone with respect to respiratory

depression. Reminfentanil and propofol concentrations as well as CO2 levels play a

factor in determining the respiratory response during mild anesthesia. It is anticipated

that certain drug combinations can provide necessary anesthesia while staying clear

dangerous respiratory side effects.

Seven volunteers were studied in order to characterize respiratory guard rails for

remifentanil and propofol combinations. Each subject was anesthetized with a random

series of drug combinations. At each steady state drug level, the response to surrogate

surgical stimuli and increased CO2 were measured. Ventilation was assessed in terms of

the presence of respiratory depression and airway obstruction. The CO2 response was fit

to a model relating end tidal CO2 to alveolar minute ventilation in order find the slope of

the ventilation response to CO2.

The response to CO2 could only be fit to the model only when the end tidal CO2 was

above a certain point which excluded the elbow in curve seen at lower levels. However,

this response could be measured accurately above the elbow up to CO2 levels just before

apnea occurred.

v

Remifentanil was the major player in causing respiratory depression and did so at very

low doses. The biggest respiratory side effect with propofol was not respiratory

depression, but airway obstruction. Because these side effects occurred at relatively low

doses of each drug, it is apparent that there is not a certain drug combination that will

provide adequate anesthesia without posing significant respiratory risks to the patient.

vi

TABLE OF CONTENTS

CHARACTERIZING CARBON DIOXIDE RESPONSE AND RESPIRATORY

DEPRESSION FOR REMIFENTANIL AND PROPOFOL COMBINATIONS i

ABSTRACT iv

TABLE OF CONTENTS vi

LIST OF TABLES viii

LIST OF FIGURES ix

CHAPTER 1 1

INTRODUCTION 1

Background 1

Recent studies to characterize effects of remifentanil and propofol 2

Hypotheses 6

CHAPTER 2 8

METHODS 8

Study Protocol 8

Data Analysis 19

CHAPTER 3 23

RESULTS 23

CO2 Response 23

CO2 Models 34

vii

Response Surfaces 41

Respiratory Guard Rails 42

CHAPTER 4 44

DISCUSSION 45

Most important 45

CO2 Response Curves 46

Response Surfaces 47

Limitations 48

Conclusion 48

REFERENCES 49

LIST OF TABLES

Table 1. Observer’s Assessment of Alertness/Sedation (OAA/S). 11

Table 2. Actual drug combinations given to each subject 16

Table 3. Drug combinations for CO2 curves with Linear parameters 24

LIST OF FIGURES

Figure 1. Representation of the study design.................................................................... 14

Figure 2. CrissCross design showing locations of drugs combinations used ................... 16

Figure 3. Example screen shot of the data collection and visualization software ............ 18

Figure 4. CO2 response curves for Subject 1, Period 1 .................................................... 25



Figure 7. CO2 response curves for Subject 2 Period 1 ..................................................... 28



Figure 10. CO2 response curves for Subject 4, Period 1 .................................................. 31



Figure 13. Fits for linear and power models for ventilation ............................................. 35

Figure 14. CO2 Model fits during a typical rebreathing phase......................................... 36

Figure 15. Effect Compartment Model Deficiency .......................................................... 37

Figure 16. Fits for linear and power models for ventilation using only the up step in

PetCO2.............................................................................................................................. 39

Figure 17. CO2 Sensitivity Surface .................................................................................. 41

Figure 18. CO2 Sensitivity Contour ................................................................................. 42

x

Figure 19. Respiratory depression surface........................................................................ 43

Figure 20. Respiratory depression contour ....................................................................... 43

CHAPTER 1

INTRODUCTION

Background

Respiratory depression is a major side effect of low doses of anesthetics given for

mild sedation. There has been much effort to effectively model and predict respiratory

depression for several anesthetic drugs.1,2,3,4,5,6,7,8,9 In particular, remifentanil has proven

to be a very potent respiratory depressant.4,9 Remifentanil is desirable because it provides

quick analgesia and quick recovery. Remifentanil and propofol combinations behave in a

synergistic manner and work together to provide analgesic and sedating effects making

them attractive for clinical use.

These low dose combinations of remifentanil and propofol can be useful in a

variety of surgical procedures including colonoscopies, catheterizations, and esophageal

ultrasound where an anesthesiologist is probably not present. It is important to define

dosing combinations that provide patients adequate analgesia and sedation without

causing negative respiratory side effects.

Response surface methodology (RSM) is a collection of statistical and

mathematical techniques useful for developing, improving, and optimizing processes.10

These methods are useful when several input variables potentially influence a particular

2

outcome. This outcome is referred to as the response or effect. RSM has been used

extensively in pharmacodynamic modeling of a variety of responses to different drug

inputs. Maximum effect sigmoid models have been found to give good estimations of

these surfaces.11,12 A nonlinear least squares method is used to estimate parameters of

these regression models. This is done iteratively by calculating residuals for each

parameter set and looking for minima. Isoeffect plots are also used to define all drug

combinations that can achieve a desired effect.

Recent studies to characterize effects of remifentanil and propofol

Characterizing analgesia and sedation

A volunteer response surface analysis study was performed at the University of

Utah recently to characterize the synergistic interaction of remifentanil and propofol in

blunting response to surrogate noxious stimuli.13 This study was done over a complete

range of clinically relevant concentrations. OAA/S (Observer Assessment of

Alertness/Sedation Score), tibial pressure algometry, electrical tetany, and response to

laryngoscopy were the surrogate measures used in this study. They found great synergy

between the two drugs and response surfaces for each of the stimuli. They did not study

respiratory effects.

Characterizing respiratory depression

Respiratory depression is more difficult to model than most drug effects, because

respiratory depression is not only a function of drug concentrations, but also the partial

pressure of carbon dioxide at the effect site in the body. Attempts have been done to

model respiratory depression of remifentanil alone and propofol alone while accounting

3

for carbon dioxide. One other attempt has been made to model respiratory depression as

a function of both drugs while clamping carbon dioxide.

Modeling ventilatory response to remifentanil and propofol separately

The recent approach to characterizing the response to remifentanil was attempted

by Bouillon et al.3 In his paper he argues that carbon dioxide should be an additional

input to any drug model used to describe respiratory ventilation and that CO2 has its own

kinetic properties.

To describe the partial pressure of CO2 (PCO2) at its site of action, an effect

compartment model can be used. A pharmacologic effect compartment model is a

theoretical compartment where drug concentration parallels the time course of drug

effect. This type of model can estimate the PCO2 at the theoretical respiratory effect

compartment by time correlating the readily available end tidal CO2 (PetCO2)

measurement with the ventilation signal. By determining this correlation, the hysteresis

lag between PetCO2 and its ventilatory effect can be collapsed so that ventilation as a

function of PecCO2 can be fit to a curve or line referred to as the CO2 response curve.

The effect compartment model allows for non-steady state approximation of the CO2

response which would be the most probable situation in a clinic. Bouillon found that the

CO2 response was better modeled as a power function than a straight line. He also

estimated that a straight line is adequate above PetCO2 of 45 mmHg.3

The stimulatory effect of CO2 multiplied by the inhibitory effect of increasing

drug concentrations gives an estimation model of minute ventilation.8 The CO2 response

and ventilation models used by Boullion are described in the methods section as they

were implemented in this study.

4

A similar study was done by Bouillon et al. to model ventilatory depressant

potency of propofol.1 The modeling techniques were the same as those used for

remifentanil.

Modeling ventilatory response to both drugs

One advance has been made to model remifentanil and propofol interaction on

respiratory depression.4 Data was collected in order to model CO2 response curves in the

presence of drug as well as response surfaces for minute ventilation at resting PetCO2,

minute ventilation at PetCO2 55 mmHg, PetCO2, and CO2 sensitivity.

Carbon dioxide response curves were modeled as a straight line and were done

using a steady state approach. Several CO2 levels were forced and allowed to reach

steady state for a minimum of eight minutes. End tidal carbon dioxide and minute

volume measurements were averages of the last ten breaths of the of the eight minute

periods so that PetCO2 and PecCO2 could be assumed to be equal.

It was not clear how minute ventilation measurements were taken for the surfaces

describing minute ventilation at resting CO2 and minute ventilation at a constant PetCO2

of 55 mmHg. It is assumed that the minute ventilation measurements were averaged over

some time period at steady state. The constant PetCO2 of 55 mmHg was achieved using

a mass flow controller.

The surface for PetCO2 also must have consisted of average steady state values of

PetCO2 and the CO2 sensitivity surface was a surface using slope values from the CO2

response curves.

5

Results indicated that remifentanil shifted the CO2 response curve to the right

without changing the slope. They also showed that propofol decreased the slope of the

curve without shifting. Together they were shown to shift the curve to the right and

decrease they slope. This synergism was also seen in the CO2 sensitivity surface.

Both minute ventilation surfaces indicated drug synergism in decreasing

ventilation.

Limitations

These recent studies have also had significant drawbacks. The one drug studies

did not attempt to look at the CO2 response at different drug levels. Response curves

were done at baseline without any drug present and were assumed to have the same

stimulatory effect with regard to ventilation at all drug levels studied. The two drug

experiment studied the CO2 response at several drug levels and indicated that there

should be a different CO2 response at every drug level.

The experiment involving two drugs provided some good information, but other

results were questionable. The minute ventilation surface at resting PetCO2 cannot be

considered valid because PetCO2 in unknown and unaccounted for. Minute ventilation

should not only be a function of drug concentrations, but also CO2 levels. The minute

ventilation surface at 55 mmHg eliminates this problem because the CO2 dimension is

eliminated. This, however, is not useful in most situations where PetCO2 cannot be held

constant. The minute ventilation at resting PetCO2 is skewed significantly. The isoeffect

plot showing 75%, 50%, and 25% isoboles for ventilation are not accurate. In particular,

the 50% isobole indicates that an infinite amount of propofol is needed to cut ventilation

in half without any remifentanil present. This is obviously not the case. The CO2

6

sensitivity isoeffect plot had a similar problem in that it indicates that an infinite amount

of remifentanil is needed to cut CO2 sensitivity in half, regardless of the concentration of

propofol. This is not true either. It is unclear whether this information is due the

selection of the drug interaction model or if insufficient data was collected. The

confounding information in these CO2 and minute ventilation surfaces make predicting

respiratory depression very difficult.

Another limitation to these respiratory depression studies is the neglect of airway

obstruction. None of these papers mention its effect on CO2 measurements or minute

ventilation measurements, but most likely airway obstruction was encountered during the

course of the studies.

Hypotheses

Respiratory guard rails

The overall objective of this research is to establish respiratory guard rails that

could lead to remifentanil and propofol dosing routines designed to reach adequate

sedation and analgesia while maintaining adequate ventilation.

Respiratory depression is a function of drug concentrations as well as effect

compartment CO2. It is expected that a model including these three input variables can

estimate ventilation and, therefore, respiratory depression.

CO2 response curves

In order to include CO2 in the respiratory model, it is anticipated that

measurements of the respiratory response to carbon dioxide can be made and will change

as a function of drug concentrations. These response curves should be able to be

7

measured using a steady state and a non-steady state approach. In the non-steady state,

one must compensate for the lag in minute ventilation as a response to PetCO2. It is

anticipated that an effect compartment model will provide adequate estimation for this

time correlation and allow for a non-steady state CO2 response curve. For the steady

state, average PetCO2 and minute ventilation measurements after reaching the steady

state should give good estimates of the CO2 response. It is anticipated that remifentanil

will be a potent respiratory depressant and will shift the CO2 response curve to the right.

It is also expected that propofol will change the CO2 response curve by decreasing the

slope.

Airway obstruction

It is also expected that airway obstruction will occur before respiratory depression

in some cases. Airway obstruction should be able to be detected using a combination of

measurements including airway flow and muscle excursion.

CHAPTER 2

METHODS

Study Protocol

A total of seven volunteers were included in this study. Each volunteer

participated in a screening visit and a study visit. During the screening visit, the subjects

first provided written informed consent to participate in the study. They were then

evaluated according to the following selection criteria to confirm eligibility.

Volunteers had to be class ASA I or II male or female (non-pregnant/non-

lactating) in order to be included in the study visit. In addition, they had to be at least 18

years of age, be healthy, have an uncomplicated airway anatomy, and have a body mass

index between 18 and 27.

Individuals were excluded if they had a neurological pathology, history of drug or

alcohol abuse, abnormal reaction to the drugs of interest, obstructive sleep apnea, or a

positive pregnancy test. Also, volunteers could not be prisoners, retarded, or

hypersensitive. Fasting for 8 hours prior to drug administration was also required.

If the selection criteria were met during the screening visit, volunteers returned on

a different day for administration of drugs, application of surrogate surgical stimuli, and

clinical measurements.

9

Setup

The subjects were monitored throughout the study using an array of instruments

including electrocardiogram, blood pressure cuff, arterial blood pressure monitor, pulse

oximeter, and bispectral index scale (Aspect Medical Systems, Inc). Breath

measurements were made using both a Datex gas analyzer and a Novametrix

pneumotachograph. Chest and abdominal wall excursion were measured using a

respitrace (Ambulatory Monitoring Inc., Ardsley, NY) which consists of two respiratory

inductance coils placed around the abdomen and chest. Pressure algometry, electrical

tetany, and esophageal probes were used to provide surrogate surgical stimuli. The

pressure algometer was placed on the leg so that pressure could be applied to the tibia.

Electrodes were positioned to stimulate the posterior tibial nerve during shock on the leg

opposite the algometer and esophageal probes were prepared for use.

Once the instrumentation setup was complete, a 20 gauge venous catheter was

placed in an antecubital vein under local anesthesia (0.2 mL of 0.5% lidocaine) for the

purpose of hydration and drug administration. A continuous IV infusion of 0.9% sodium

chloride was started at 1 mL/kg/hour. Continuous infusions of remifentanil and propofol

were infused into this same IV using Harvard Apparatus controlled infusion pumps.

A 20 gauge arterial catheter was placed in a radial artery under the same local

anesthesia for blood pressure monitoring and arterial blood draws.

Five minutes prior to drug administration, subjects were treated with .2 mg

glycopyrroloate IV to prevent bradycardia and 30 mL of sodium nitrate PO. Baseline

measurements were then recorded.

10

Procedures

After taking baseline measurements for a subject without drug, the first drug

combination was infused. The drug combinations were chosen randomly. The same

measurements made at baseline were made for every drug combination. These

measurements were done in order to assess the level of sedation, analgesia, airway

obstruction, CO2 sensitivity, and respiratory depression.

Sedation was evaluated using the OAA/S (Observer Assessment of

Alertness/Sedation scale). Subjects with an OAA/S of less than three were considered

sedated. The BIS monitor also provided a measure of sedation. The scale is described in

Table 1.

11

Table 1. Observer’s Assessment of Alertness/Sedation (OAA/S).

Responsiveness Speech Facial Expression Eyes Composite Score

□ 5 Responds readily to name spoken in normal tone

□ 5 Normal □ 5 Normal □ 5 Clear, no ptosis □ 5 (Awake)

□ 4 Lethargic response to name spoken in normal tone

□ 4 Mild slowing or thickening

□ 4 Mild relaxation□ 4 Glazed or mild ptosis (less than half the eye)

□ 4

□ 3 Responds only after name is called loudly and/or repeatedly

□ 3 Slurring or prominent slowing

□ 3 Marked relaxation (Slack Jaw)

□ 3 Glazed and marked ptosis (half the eye or more

□3

□ 2 Responds only after mild prodding or shaking

□ 2 Few recognizable words

□ 2

□ 1 Does not respond to mild prodding or shaking

□ 1 (Asleep)

Instructions: Within each of the four assessment categories check the statement which best describes the condition of the patient. Responsiveness should be evaluated first, assess speech by asking the patient to repeat a phrase (e.g., “The quick brown fox jumps over the lazy dog”). Assign the composite score corresponding to the lowest level at which any statement is checked.

Assessment Categories

12

Analgesia was measured with pressure algometry, electrical titanic stimulation,

and esophageal probing. During algometry measurements the volunteer started with a

stimulus of 0 PSI and the pressure was slowly increased until the volunteer indicated the

desire to stop or 50 PSI was reached. For electrical tetany the stimulus began at 0 mA

and was slowly increased as before until the subject indicated no more. A maximum of

50 mA of current was allowed. The esophageal probe was placed in the esophagus if the

subject allowed.

Airway obstruction was assessed by comparing the capnogram airway flow

waveform and the respitrace abdominal and chest waveforms. Zero airway flow in the

presence of muscle movement as indicated by the respitrace indicated obstruction.

Carbon dioxide sensitivity was measured by comparing Valv to end tidal carbon

dioxide (etCO2) with and without a CO2 rebreathing stimulus. Five minutes was allowed

before baseline and rebreathing measurements so that the etCO2 measurement could be

assumed to be the same as the partial pressure of CO2 at the effect site in the body. This

also provided time for a steady state value of Valv. The rebreathing stimulus was done

using a section of expandable hose. The volume in the hose was calibrated so that it

could be stretched to provide a volume that was equal to the volunteers’ tidal volume just

before rebreathing.

Respiratory depression was considered to be present if the alveolar minute

ventilation (Valv) dropped to 65% of the baseline measurement.

13

Experimental Design

The experimental design was similar to a previous drug study done at the

University of Utah.13 It consisted of three study periods for each subject. Figure 1 is a

representation of the study design.

14

Was

ho

ut

#3

Was

ho

ut

#2

Was

ho

ut

#1

Dru

g C

on

cen

trat

ion

0

2

4

Dru

g C

on

cen

trat

ion

0

2

4

Time

Primary Agent

Secondary Agent

Primary Agent

Secondary Agent

1

3

2

5

4

1

3

2

5

4

11

13

12

15

14

6

8

7

10

9

Low

Medium

High

Period #1Period #1 Period #2Period #2 Period #3Period #3

Figure 1. Representation of the study design

15

Each study period included time for measurements at five drug combinations and was

followed by a washout phase. The primary agent for each subject was chosen randomly.

The concentration pairs represented by low, medium, high, and the numbers 1-15 were

also chosen randomly for each volunteer. These pairs were chosen from sets of

concentrations designed to best characterize the expected curves for sedation, analgesia,

and respiratory depression as estimated from previous studies.13,4 A crisscross trial

design was used because it has been shown to be the most efficient.14 Figure 2 illustrates

the placement of data points using this method. Table 2 shows the actual drug

combinations used.

16

CrissCross Design

0

1

2

3

4

5

0 1 2 3 4 5 6 7Remifentanil (ng/mL)

Prop

ofol

(ug/

mL)

OnceTwice

Figure 2. CrissCross design showing locations of drugs combinations used

Table 2. Actual drug combinations given to each subject

Remi Prop Remi Prop Remi Prop Remi Prop Remi Prop0.00 1.47 1.22 0 0.00 0.81 2.2 0 1.22 00.39 1.47 1.22 0.26 0.39 0.81 2.2 0.26 1.22 0.260.84 1.47 1.22 0.56 0.84 0.81 2.2 0.56 1.22 0.561.65 1.47 1.22 1.1 1.65 0.81 2.2 1.1 1.22 1.13.3 1.47 1.22 2.2 3.3 0.81 2.2 2.2 1.22 2.2

0.00 2.00 2.2 0 0.00 2 5 0 2.2 00.39 2.00 2.2 0.26 0.39 2 5 0.26 2.2 0.260.84 2.00 2.2 0.56 0.84 2 5 0.56 2.2 0.561.65 2.00 2.2 1.1 1.65 2 5 1.1 2.2 1.13.3 2.00 2.2 2.2 3.3 2 5 2.2 2.2 2.2

0.00 3.33 3 0 0.00 2.67 6.4 0 4 00.39 3.33 3 0.26 0.39 2.67 6.4 0.26 4 0.260.84 3.33 3 0.56 0.84 2.67 6.4 0.56 4 0.561.65 3.33 3 1.1 1.65 2.67 6.4 1.1 4 1.13.3 3.33 3 2.2 3.3 2.67 6.4 2.2 4 2.2

Subject 3 Subject 4 Subject 5

Drug Combinations GivenSubject 1 Subject 2

17

Data Collection

All computers used to collect data were time synchronized using an internet clock

before beginning each study. Datex monitor commercial software was used to collect

pulse oximetry, blood pressure, and gas data at 25 Hz.

Data from the Respitrace, BIS, and Novametrix pneumotachograph was sampled

using data collection and visualization software written using Borland C++ Builder 5.0.

The Respitrace signals were read from a Measurement Computing A/D converter and the

BIS and Novametrix were read from serial ports. The Respitrace data consisted of

abdominal and chest waveforms sampled at 100 Hz. The BIS data included the BIS

number and other parameters that can be used to evaluate the validity of the BIS number

sampled at .2 Hz. The Novametrix included waveforms sampled at 100 Hz and

calculated parameters sampled every breath. The raw data was written to four separate

text files at the rates described. Run time plots and discrete values were displayed on the

user interface so that results could be evaluated throughout the study. An example screen

shot of the graphical interface is shown in Figure 3.

18

Figure 3. Example screen shot of the data collection and visualization software

19

Another program was written by Noah Syroid to guide the researchers during

the study. It allowed for discrete data values to be entered after each study phase. This

software wrote an output file including the study phases with the same time stamp as the

data collection and visualization software so that data the data could be parsed more

easily. Comments were also entered with time stamps using this software.

Data Analysis

Alveolar minute ventilation and PetCO2 measurements were sampled each breath.

If the alveolar minute ventilation (Valv) values were less than twice the accompanying

dead space values, PetCO2 was considered not valid.

CO2 response model

The Valv versus PetCO2 (CO2 response curve) data was analyzed using two

approaches. In the first approach, data for ten breaths was averaged during the steady

state time periods corresponding to no rebreathing, 100% rebreathing, and after

rebreathing. This approach gives three PetCO2 and Valv to compare.

The second approach used the effect compartment model of Bouillon et al.8 to

correlate the PetCO2 and Valv measurements so that all the data pairs during the

rebreathing phase could be used to fit a CO2 response curve. The effect compartment

model is described by:

tPecCOtPetCOkdt

dPecCO22

2 (1)

20

where PecCO2 is the partial pressure of carbon dioxide at the effect compartment (in

mmHg), PetCO2 is the end tidal carbon dioxide measurement (in mmHg), and k is an

equilibrium constant (inmin-1). Solving the differential equation yields:

ktetPetCOPecCOtPetCOtPecCO 2222 0 (2)

and PecCO2(0) are assumed to be equal to PetCO2(0) at steady state.

The relationship between PecCO2 and Valv can be described using a linear

approximation8:

00 222 PecCOtPecCOSVPecCOV alvalv (3)

where Valv(PecCO2) is the alveolar minute ventilation as a function of PecCO2 (in L/min),

Valv(0) is the baseline Valv (in L/min), PecCO2(0 and t) are the effect compartment carbon

dioxide values at baseline and t (in mmHg), and S is the slope of CO2 response curve.

The CO2 response curve can also be described using a power function8:

F

alvalv PecCO

tPecCOVPecCOV

00

2

22 (4)

with F representing the gain parameter and other variables being the same as used in the

linear approximation.

21

Drug effect model

The previous study at the University of Utah that characterized analgesia and sedation13

used a maximum effect sigmoid model proposed by Greco et al.12:

150505050

50505050max

P

P

R

R

P

P

R

R

P

P

R

R

P

P

R

R

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

CE

E (5)

where E is the drug effect, Emax is the maximum drug effect, CR and CP are the

concentrations of remifentanil and propofol (in ng/mL and µg/mL), C50R and C50P are the

concentrations of remifentanil and propofol needed to achieve 50% of the maximum

effect (in ng/mL and µg/mL), γ is the slope of the sigmoid curve, and α is the drug

interaction term. If α = 0, the drug interaction is additive. If α < 0, the drug interaction is

antagonistic. If α > 0, the drug interaction is synergistic.13 This model is used to describe

a drug effect that increases as drug concentrations increase. For ventilation and CO2

sensitivity which are responses that decrease with increased drug concentrations the

model can be adapted as follows:

1

1

50505050

50505050max

P

P

R

R

P

P

R

R

P

P

R

R

P

P

R

R

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

CE

E (6)

where Emax now represents the baseline value before administering drug.

22

Ventilation model

Ventilation was modeled by multiplying the stimulatory effect of CO2 by the inhibitory

effect of the two drugs8 in a three input model:

0,, 2 alvPRalv VPecCOCCV

F

P

P

R

R

P

P

R

R

P

P

R

R

P

P

R

R

PecCO

tPecCO

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

CE

0

1

12

2

50505050

50505050max

(7)

which results in a four dimensional space.

Matlab

CHAPTER 3

RESULTS

CO2 Response

The sections below compartmentalize the CO2 response data into subject number

and period number. This is meant to illustrate the effect of increasing drug levels during

each period. For each drug period, the CO2 response curve is shown as calculated using

steady state averages, the non-steady state linear model, and the non-steady state power

model. The slope parameter is shown for the linear models and the gain parameter is

listed for the power model. Concentrations for remifentanil and propofol are also shown.

Only response curves which were measured above 45 mmHg were included. This is

because the CO2 response below this level did not fit the linear or power models, and

therefore, does not represent a valid response. In other words, the curves shown are those

that were measured with a baseline above 45 mmHg up until apnea was reached.

Between the seven subjects, a total of 17 response curves were measured in this region.

The drugs combinations for which response curves were fit are shown with their linear

model parameters in Table 3. The shading distinguishes between measurements taken

during the same trial period.

24

Table 3. Drug combinations for CO2 curves with Linear parameters

Remi Prop Slope PecCO2(0) Remi Prop Slope PecCO2(0)

0.39 1.47 0.00 50 1.22 0.57 0.13 250.84 1.47 0.04 34 1.22 1.10 0.19 58

0 2.00 0.03 277 0.00 0.00 3.28 300.39 2.00 0.74 14 2.20 0.26 0.40 420.39 2.00 0.22 45 2.20 0.26 2.20 280.84 2.00 0.15 58 2.20 0.56 0.68 621.65 2.00 0.00 740.2 2.67 0.24 9

0.39 2.67 0.11 110.84 3.33 0.00 950.84 4.27 0.15 41

Drug Combinations and Linear Model ParametersPrimary Remifentanil Primary Propofol

25

Subject 1

Period 1

Figure 4 shows the CO2 response curves for Period 1 for the first subject.

30 35 40 45 50 55 60 650

2

4

6

8

10Linear CO2 Response Curves

PecCO2 (mmHg)

Val

v (L)

R = 0.84, P = 3.33 S = 0.0038653

Subject 1, Period 1

30 35 40 45 50 55 60 650

2

4

6

8

10Linear CO2 Response Curves using averages

PecCO2 (mmHg)

Val

v (L)

R = 0.84, P = 3.33 S = -0.10391

Subject 1, Period 1

30 35 40 45 50 55 60 650

2

4

6

8

10Power CO2 Response Curves

PecCO2 (mmHg)

Val

v (L)

R = 0.84, P = 3.33 F = 0.1233

Subject 1, Period 1

Figure 4. CO2 response curves for Subject 1, Period 1

It is not suspected that these results are correct for this concentration. It is more likely

that partial airway obstruction had occurred without being detected.

26

Period 2

Figure 5 shows the CO2 response curves for Period 2 of the first subject.

30 35 40 45 50 55 60 650

2

4

6

8

10

Linear CO2 Response Curves

PecCO2 (mmHg)

Val

v (L)

R = 0, P = 2 S = 0.034013R = 0.39, P = 2 S = 0.74869

Subject 1, Period 2Subject 1, Period 2

30 35 40 45 50 55 60 650

2

4

6

8

10

Linear CO2 Response Curves using averages

PecCO2 (mmHg)

Val

v (L)

R = 0, P = 2 S = 0.26301R = 0.39, P = 2 S = 0.045229


30 35 40 45 50 55 60 650

2

4

6

8

10

Power CO2 Response Curves

PecCO2 (mmHg)

Val

v (L)

R = 0, P = 2 F = 2.2218e-014R = 0.39, P = 2 F = 62.3081



The curve at R=0 and P=2 did not converge correctly using the models. The ventilation

data was very noisy at this drug level which is the reason that the model did not fit

correctly, but the reason for the noise is unknown.

27

Period 3

Figure 6 shows the CO2 response curves for Period 3 for the first subject.

30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 0.39, P = 1.47 S = 5.758e-005R = 0.84, P = 1.47 S = 0.038504


30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 0.39, P = 1.47 S = 0.39595R = 0.84, P = 1.47 S = 0.099245


30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 0.39, P = 1.47 F = 19.94R = 0.84, P = 1.47 F = 0.41143



The ventilation data at R=.39 and P=1.47 was very noisy as before.

28

Subject 2

Period 1

Figure 7 shows the CO2 response curves for Period 1 for the second subject.

30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 1.22, P = 0.57 S = 0.13238R = 1.22, P = 1.1 S = 0.18725


30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 1.22, P = 0.57 S = 0.21779R = 1.22, P = 1.1 S = 0.067725


30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 1.22, P = 0.57 F = 2.0885R = 1.22, P = 1.1 F = 5.6035


Figure 7. CO2 response curves for Subject 2 Period 1

The curves show considerable inconsistency. The linear model suggests that

propofol shifts the curve to the right, whereas the averages line predicts a change in

slope.

29

Subject 3

Period 2

Figure 8 shows the CO2 response curves for Period 2 for the third subject.

30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 0.39, P = 2 S = 0.2214R = 0.84, P = 2 S = 0.14997R = 1.65, P = 2 S = 5.5392e-009

Subject 3, Period 2Subject 3, Period 2Subject 3, Period 2

30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 0.39, P = 2 S = 0.21925R = 0.84, P = 2 S = 0.13646R = 1.65, P = 2 S = 0.020571


30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 0.39, P = 2 F = 4.6531R = 0.84, P = 2 F = 6.1592R = 1.65, P = 2 F = 2.3263e-014



This data set shows expected results. The response curve diminishes with increased

drugs. The table parameters above indicate a small decrease in slope and a shift to the

right with regard to the PecCO2 required to keep breathing.

30

Period 3

Figure 9 shows the CO2 response curves for Period 3 for the third subject.

30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 0.2, P = 2.67 S = 0.23572R = 0.39, P = 2.67 S = 0.11281


30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 0.2, P = 2.67 S = 0.54654R = 0.39, P = 2.67 S = 0.31232


30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 0.2, P = 2.67 F = 2.2851R = 0.39, P = 2.67 F = 1.4872



The airway was cleared using the head tilt and chin lift during these 2 measurements.

The difference in the curves behaved as expected. Increasing reminfentanil shifted the

curve and decreased the slope.

31

Subject 4

Period 1

Figure 10 shows the CO2 response curves for Period 1 for the fourth subject.

30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 2.2, P = 0.26 S = 0.40435R = 0, P = 0 S = 3.2825


30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 2.2, P = 0.26 S = 0.33856R = 0, P = 0 S = 0.39405


30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 2.2, P = 0.26 F = 3.9492R = 0, P = 0 F = 5.0651



The increasing drug caused a shift and slope decrease.

32

Subject 6

Period 1

Figure 11 shows the CO2 response curves for Period 1 for the sixth subject.

30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 2.2, P = 0.26 S = 2.1954R = 2.2, P = 0.56 S = 0.67964


30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 2.2, P = 0.26 S = 1.8043R = 2.2, P = 0.56 S = -0.25794


30 35 40 45 50 55 60 650

2

4

6

8

10


PecCO2 (mmHg)

Val

v (L)

R = 2.2, P = 0.26 F = 98.9351R = 2.2, P = 0.56 F = 32.5573



Increasing propofol caused a shift and a decrease in slope of the curve.

33

Subject 7

Period 3

Figure 12 shows the CO2 response curves for Period 3 for the seventh subject.

30 35 40 45 50 55 60 650

2

4

6

8

10Linear CO2 Response Curves

PecCO2 (mmHg)

Val

v (L)

R = 0.84, P = 4.27 S = 0.14546

Subject 7, Period 3

30 35 40 45 50 55 60 650

2

4

6

8

10Linear CO2 Response Curves using averages

PecCO2 (mmHg)

Val

v (L)

R = 0.84, P = 4.27 S = 0.12477

Subject 7, Period 3

30 35 40 45 50 55 60 650

2

4

6

8

10Power CO2 Response Curves

PecCO2 (mmHg)

Val

v (L)

R = 0.84, P = 4.27 F = 2.5918

Subject 7, Period 3


This was the only data point available for Subject 7, but it does seem to be a good slope

measure at this concentration.

34

CO2 Models

The results for the linear and power models used to predict ventilation at each

drug level as a function of PetCO2 are summarized by Figure 13.

35

Fit

0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

12Linear Ventilation model fit

Predicted Valv (L)

Mea

sure

d V a

lv (L

)

Predicted

Measured

0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

12Power Ventilation model fit

Predicted Valv (L)

Mea

sure

d V a

lv (L

)

Predicted

Measured

-1 0 1 2 3 4 5 6 7 8 9-6

-4

-2

0

2

4

6

8

Predicted Valv (L)

Residuals for Linear Model

min=-4.2

max=6.5

std=1.16

Residuals

min

maxmean

std

-1 0 1 2 3 4 5 6 7 8 9-6

-4

-2

0

2

4

6

8

Predicted Valv (L)

Residuals for Power Model

std=1.15

max=6.4

min=-4.4

Residuals

min

maxmean

std

Figure 13. Fits for linear and power models for ventilation

The top two plots show the measured ventilation as a function of the predicted

ventilation for the linear and power models. The bottom two plots are the residuals as a

function of predicted ventilation. Maximum and minimum residuals are shown as well as

the standard deviation for the residuals. There was almost negligible difference in the

accuracy between the two models. The 95% confidence interval would be approximately

±2.25 (L) for both models. The results show great variability between subjects.

Because of the great variability in predicted ventilation, the models were

examined more closely. Figure 14 shows a particular baseline trial from Subject 5.

36

0 20 40 60 80 100 120 140 160 180 200

-5

0

5

10

Valv compared to linear model

Breaths

Val

v (L)

stdRes = 1.35

Valv

Valvmodel

Residuals

0 20 40 60 80 100 120 140 160 180 200

-5

0

5

10

Valv compared to power model

Breaths

Val

v (L)

stdRes = 1.42

Valv

Valvmodel

Residuals

Subject 5, R = 0, P = 0

Figure 14. CO2 Model fits during a typical rebreathing phase

The plots show similar results for both models. In both cases, the model follows

the measured Valv very well during the increase in effect compartment CO2. However,

the residuals become very significant during the CO2 down step. In order to understand

the lack of fit during the down slope, the effect compartment kinetic model was examined

as in Figure 15.

37

Effect Compartment Model Deficiency

44

47

50

53

56

0 50 100 150 200 250 300 350

Time (s)

PC

O2

(mm

Hg

)

PetCO2 PecCO2 Incorrect

Figure 15. Effect Compartment Model Deficiency

The PetCO2 step was created using excel in order to resemble a typical CO2

rebreathe. The PecCO2 trace is the result of the effect compartment model using the

published value k=.96 l/min.3 The PecCO2 curve seems to resemble what would happen

physiologically, but the portion of the curve from 300 seconds and beyond is obviously

incorrect because the PecCO2 follows the PetCO2 exactly. The data shows that

ventilation has a similar lag behind PetCO2 during the CO2 drop. Bouillon mentions in

his paper that the effect compartment model assumes that PecCO2(0) = PetCO2(0).3 This

means that the drift is observed when the PetCO2 is steadily moving in one direction. As

soon as the PetCO2 changes directions, the PecCO2 does not work properly until steady

state is achieved. Despite the deficiency in the effect compartment CO2 model, it was

38

still used because of its simplicity, but data was only fit up till the beginning of the down

slope. By fitting the data only to the up step of PetCO2, the fit became much better. A

measure of the fit using the up step portion of the curve only is shown in Figure 16.

39

0 2 4 6 80

2

4

6

8

10Linear Ventilation model fit

Predicted Valv (L)

Mea

sure

d V

alv (

L)PredictedMeasured

0 2 4 6 80

2

4

6

8

10Power Ventilation model fit

Predicted Valv (L)

Mea

sure

d V

alv (

L)

PredictedMeasured

0 2 4 6 8 10-4

-3

-2

-1

0

1

2

3

4Residuals for Linear Model

Predicted Valv (L)

min=-3.13

std=.77

max=3.09 Resminmaxmeanstd

0 2 4 6 8-4

-3

-2

-1

0

1

2

3

4Residuals for Power Model

Predicted Valv (L)

min=-3.18

std=.80

max=3.06 Resminmaxmeanstd

Figure 16. Fits for linear and power models for ventilation using only the up step in PetCO2

40

The standard deviation of the residuals decreased from 1.16 to .77.

Response Surfaces

CO2 sensitivity

Figure 17 is the response surface for CO2 sensitivity. Sensitivity is expressed in L/min

per mmHg. The data plotted is the slope parameter from the linear CO2 response model

for each drug combination.

02

46

02

46

0

1

2

3

4

5

Remifentanil (ng/ml)

CO2 Sensitivity Surface

Propofol (g/ml)

CO

2 S

ensi

tivity

(L/m

in p

er m

mH

g)

Figure 17. CO2 Sensitivity Surface

The surface indicates that remifentanil causes the loss of CO2 sensitivity. Figure 18 is a

isoeffect plot showing the 75%, 50%, and 25% isoboles.

42

0 0.02 0.04 0.06 0.08 0.1 0.120

1

2

3

4

5CO2 Sensitivity Contour


Pro

pofo

l ( g

/ml)

7550

25

Figure 18. CO2 Sensitivity Contour

The isoboles cross at very low concentrations of remifentanil. These are much lower

than expected.

Respiratory Guard Rails

Three Dimensional Model

The three dimensional model was implemented, but the data was not sufficient for

convergence and therefore did not yield useful results.

Probability of respiratory depression

Because respiratory depression could not be predicted using the three dimensional

model, respiratory depression was plotted as binary data. Respiratory depression was

considered to be present if intervention to stimulate breathing was given. This was done

if a volunteer went 30 seconds without 2 breaths. Figure 19 is the binary respiratory

depression surface.

43

02

46

02

46

0

20

40

60

80

100


Respiratory Depression

Propofol (g/ml)

Res

pira

tory

Dep

ress

ion

Figure 19. Respiratory depression surface

Spontaneous breathing is represented by 100 and respiratory depression is 0. The 25, 50,

75, and 95 isoboles are shown in Figure 20.

0 0.5 1 1.5 20

1

2

3

4

5

6

7Respiratory Depression


Pro

pofo

l ( g

/ml)

95755025

Figure 20. Respiratory depression contour

According to the contour plot, remifentanil causes respiratory depression very quickly.

Propofol does not.

Table 4 lists the model parameters for the two surfaces described.

44

γ α C50P C50R EmaxCO2 Sensitivity 0.18 11.01 920.17 0.00 4.15

Respiratory Depression 3.07 -0.88 35.64 1.14 100.00

Model Parameters for CO2 Sensitivity and Respiratory Depression

The negative value for α in respiratory depression suggests slight antagonism

between drugs. However, the value for α in the CO2 sensitivity model indicates a large

amount of synergism in blocking respiratory drive.

CHAPTER 4

DISCUSSION

Characterizing respiratory depression for drug combinations is process that must

be broken down into pieces. In order to establish guard rails describing minute

ventilation as a function of drug combinations and CO2, the stimulatory effect of CO2

must be well defined for the drug surface first. Because of the hysteresis in relationship

between PetCO2 and Valv, this correlation becomes difficult to identify in itself. If the

minute ventilation could be accurately described in the non-steady state as a function of

PetCO2 for different drug combinations, a three input model could possibly be developed

to predict ventilation as a function of remifentanil, propofol, and PetCO2.

Most important

The most valid and important respiratory results from this study occur when the

PetCO2 was high enough that the CO2 response curve could be accurately predicted

using a linear model. This point was considered to be around 45 mmHg. Not only are

the measurements below this CO2 level not as accurate, but they are less relevant in

characterizing respiratory depression because the drug level is small enough that the

volunteer is still breathing well. It is important to characterize the respiratory response

when the drug starts to take effect up until respiratory depression is induced. This is

especially the region of interest in this study because clinical applications are being

considered in which only mild sedation and analgesia is required.

46

CO2 Response Curves

The CO2 response curves gave an indication of the effect of the drugs on CO2 sensitivity.

Both drugs seemed to cause the curves to shift to the right and decrease in slope. The

region drug region between 45 mmHg and apnea ended up being a very small region. It

was difficult to take many measurements in a single run to observe the trend more

closely. This is largely because the study protocol was set up to measure many different

drug effects. Many times several drug levels were increased above apnea in order to

blunt the effects of the drugs to the stimuli. Only one or two data points were inside the

region of interest for any one trial. It would have been a much more telling protocol if

the drug increments were smaller inside the region of interest so that more data points

could be gathered. It would also have been helpful to have the ability to clamp the CO2

at levels between 45 and 60 mmHg to make many measurements.

A few criteria were established for aborting the rebreathing phase during a trial.

This included a PetCO2 of greater than 60 mmHg, SPO2 of less than 95%, or less than

two spontaneous breaths during a 30 second period. On any of these occasions the

anesthesiologist intervened by prompting the subject to breathe or with mechanical

ventilation if necessary. A volunteer was considered apneic if intervention was required

and rebreathing measurements were no longer taken. However, these criteria were not

established during the first study and several rebreathing phases were done even while

the subject was periodically prompted to breathe. It was noted qualitatively that during

these phases, the subject had a greater respiratory drive and needed less prompting when

47

the rebreathing hose was present. This would suggest that even after drug induced

respiratory depression occurs; additional CO2 may initiate breathing again.

Experimenting with very high CO2 levels is difficult to justify in a volunteer study.

Response Surfaces

Data did not fit response surfaces well because of the lack of ability to study the

edges of the surface. The CO2 sensitivity surface does not give useful results with regard

to the isoboles for 25%, 50%, and 75% sensitivity. This is because the region from no

drug present up to 45 mmHg is not defined. In addition, the C50 for propofol is

abnormally high because sensitivity was not measured for very high levels of propofol so

that the fitting process did not know how the effect behaved as propofol approached

infinity.

In order to better describe a CO2 sensitivity surface, the protocol would need to

be changed to focus more on respiratory effects rather than response to surgical stimuli.

High levels of propofol would need to be studied and the CO2 level would need to be

driven up above 45 mmHg for data points with low drug doses. With this type of

protocol, a much better surface could be described.

Because of the false CO2 sensitivity fit caused by the lack of information at the

tails of the surface, the three dimensional model would not converge at sensible results.

For this reason, a binary respiratory depression surface was fit in order to establish some

type of respiratory guard rail. Perhaps respiratory guard rails would be easier to establish

48

by analyzing the probability of respiratory of respiratory depression as a function of

drugs rather than trying to fit a three input model to predict ventilation.

Limitations

The task of characterizing the respiratory side effects of remifentanil and propofol

is a challenging process. Many factors contribute to the respiratory response and must be

well controlled in order to isolate and analyze the effects of the drugs and CO2 levels on

respiratory drive. Talking and moving by the volunteers made respiratory measurements

impossible at times throughout the study. Airway obstruction was a significant

confounding factor on the CO2 response. Partial airway obstruction is suspected to have

gone undetected several times during CO2 rebreathing especially when larger amounts of

propofol were infused.

clclaConclusion

The respiratory effects of remifentanil and propofol are very distinct and add together to

give a dangerous combination. Remifentanil causes a person to stop breathing with very

low doses even though they are very conscious of the situation. Airway obstruction does

not seem to be a factor with remifentanil because the person is very awake. On the other

hand, propofol does not seem to cause complete respiratory depression, but rather

decreases the CO2 response slope and decreases ventilation. A person will continue to

breathe spontaneously at any of the propofol concentrations studied with no remifentanil

present. The problem encountered with propofol is airway obstruction. Obstruction was

49

observed in several cases at relatively low doses of propofol and suspected to have

occurred undetected in some cases. If propofol is used, measures must be taken to clear

the airway constantly. By designing a protocol using more drug combinations that are

below respiratory depression, the CO2 response could be better defined and fit to a three

input model to describe ventilation. It may be better to approach the problem by looking

at the probability of airway obstruction and respiratory depression as caused by propofol

and remifentanil.

REFERENCES

1. Bouillon T, Bruhn J, Radu-Radulescu L, Andresen C, Cohane C, Shafer SL: Mixed-effects modeling of the intrinsic ventilatory depressant potency of propofol in the non-steady state. Anesthesiology 2004; 100: 240-50

2. Magosso E, Ursino M, van Oostrom JH: Opioid-induced respiratory depression: a mathematical model for fentanyl. IEEE Trans Biomed Eng 2004; 51: 1115-28

3. Bouillon T, Bruhn J, Radu-Radulescu L, Andresen C, Cohane C, Shafer SL: A model of the ventilatory depressant potency of remifentanil in the non-steady state. Anesthesiology 2003; 99: 779-87

4. Nieuwenhuijs DJ, Olofsen E, Romberg RR, Sarton E, Ward D, Engbers F, Vuyk J, Mooren R, Teppema LJ, Dahan A: Response surface modeling of remifentanil-propofol interaction on cardiorespiratory control and bispectral index. Anesthesiology 2003; 98: 312-22

5. Dahan A, Nieuwenhuijs D, Olofsen E, Sarton E, Romberg R, Teppema L: Response surface modeling of alfentanil-sevoflurane interaction on cardiorespiratory control and bispectral index. Anesthesiology 2001; 94: 982-91

6. Olofsen E, Nieuwenhuijs DJ, Sarton EY, Teppema LJ, Dahan A: Response Surface modeling of drug interactions on cardiorespiratory control. Adv Exp Med Biol 2001; 499: 303-8

7. Babenco HD, Conard PF, Gross JB: The pharmacodynamic effect of a remifentanil bolus on ventilatory control. Anesthesiology 2000; 92: 393-8

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8. Bouillon T, Schmidt C, Garstka G, Heimbach D, Stafforst D, Schwilden H, Hoeft A: Pharmacokinetic-pharmacodynamic modeling of the respiratory depressant effect of alfentanil. Anesthesiology 1999; 91: 144-55

9. Glass PS, Iselin-Chaves IA, Goodman D, Delong E, Hermann DJ: Determination of the potency of remifentanil compared with alfentanil using ventilatory depression as the measure of opioid effect. Anesthesiology 1999; 90: 1556-63

10. Myers R, Montgomery D: Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 2nd Edition, John Wiley & Sons, Inc., 2002

11. Minto CF, Schnider TW, Short TG, Gregg KM, Gentilini A, Shafer SL: Response surface model for anesthetic drug interactions. Anesthesiology 2000; 92: 1603-16

12. Greco WR, Bravo G, Parsons JC: The search for synergy: a critical review from a response surface perspective. Pharmacol Rev 1995; 47: 331-85

13. Kern SE, Xie G, White JL, Egan TD: A response surface analysis of propofol-remifentanil pharmacodynamic interaction in volunteers. Anesthesiology 2004; 100: 1373-81

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