Control of drugs such as propofol by target-controlled infusion (TCI) has gained widespread use throughout much of the world for >20 years and has been shown to perform reliably. TCI assumes that the pharmacokinetic model is known a priori. Different pharmacokinetic models have been used in TCI, but there is little question that no parameter set accurately describes all patients. Although pharmacokinetic modeling may reduce the impact of biological variability,^{1} considerable difference between the targeted concentration and the measured drug concentration may be seen with TCI.^{2},^{3} Any system that relies on an experimentally derived parameter set has this inherent limitation. Furthermore, even if the effect-site concentration is precisely known, the effect-site concentration that yields the intended clinical effect for an individual patient is variable. The variability in the error with which a specified target (be it drug concentration or clinical effect) is achieved across the population is a measure of the robustness of the system.

Although in many cases it is sufficient to choose a target significantly in excess of what is required for intubation and subsequently decrease the target, when using propofol for sedation, the consequences of target error can be more significant. Whereas TCI may adroitly maintain a constant depth of anesthesia in the postinduction period, control of the trajectory to the target during anesthetic induction is not a feature of current systems, and identification of a nonsteady-state concentration as a desired target during induction is not easily implemented.

This report describes an approach to drug delivery that uses the observation of a clinical end point by a skilled operator, during a slow increment in effect-site concentration, to identify patient sensitivity and determine an infusion that will maintain the desired clinical drug effect. We examined the impact of random errors in the parameter set of the pharmacokinetic model and random values for patient sensitivity on the ability of the system to achieve an effect-site concentration associated with the desired clinical effect in a population of 10,000 simulated patients. We compared this result with that obtained from a TCI targeting a fixed effect-site concentration associated with this clinical effect in the average patient. Our hypothesis was that this approach would reduce the variability of clinical response compared with TCI.

## METHODS

Two control systems were implemented using MATLAB 7.11 (MathWorks, Natick, MA), and are discussed in depth in the Appendix. Briefly, the first system, which we will refer to as the probability ramp control (PRC) system, generates 3 values for an infusion sequence: a bolus, an initial infusion rate, and a second infusion rate that is instituted after 175 seconds, as depicted in Appendix Figure 2. These values are determined through numerical minimization so that all patients experience a nearly identical trajectory of effect-site concentrations that pass through the effect concentration yielding 5% probability of loss of responsiveness (ED_{5}) at 90 seconds, the ED_{50} at 175 seconds, and the ED_{99.5} at 390 seconds. At some time before 390 seconds, almost all patients will become unresponsive. The effect-site concentration at that time is used as the target for a subsequent infusion sequence. The second system, a TCI system, used an open loop pole-zero cancellation method to generate an infusion sequence that tracks a fixed target.

The pharmacokinetic model was constructed with 8 parameters (*k*_{e0} and θ_{1}–θ_{7}) identified by Masui et al.^{4} using NONMEM analysis. These parameters were determined to vary independently and followed a log-normal distribution, and their nominal values and 95% confidence intervals (CIs) are listed in Table 1. The parameters θ_{1} to θ_{7} are used to calculate the volume and clearance of the compartments in a 2-compartment mammillary model using the covariates of patient age and weight and an infusion delay using pump rate, as shown in Appendix Table 1. The Masui et al. model predicted the effect-site concentration for an infusion sequence of propofol, which was converted to a probability of loss of responsiveness (LOR) using parameters determined by Johnson et al.^{5} The model equation and parameters are given in the Appendix (Equation A1 and Appendix Table 2). An individual patient either responds or does not, so this probability has a different interpretation for an individual, and can be seen as a measure of patient sensitivity. Thus, a patient who becomes unresponsive at 2.2 μg/mL (the ED_{50} for LOR) is said to have a sensitivity of 50%. The value for sensitivity is specified in the range of 0% to 100%, where 0% is extraordinary sensitivity, and 100% is extraordinary insensitivity.

The PRC system determines an infusion sequence that will bring the effect-site concentration to the value that corresponds to this sensitivity, referred to as the target. It is assumed that if the patient loses responsiveness at a given effect-site concentration, maintaining that concentration will meet the objectives of the anesthetic. Three outcome measures were defined. Because the time of transition cannot be anticipated, the drug that has entered the patient but not the effect site will cause the system to exceed the target. This is termed “overshoot.” To return to the target quickly, the infusion is paused briefly, which may lead the effect-site concentration to decrease below the target. This is termed “undershoot.” Finally, the infusion is restarted at a rate that will maintain the effect-site concentration target for a finite period. Because the propofol concentration will not achieve steady state by the end of this period, a difference between the effect-site concentration at the end of this finite period and the target is termed “terminal target error.” These terms are defined in units of sensitivity, and are graphically illustrated in Appendix Figure 2. Although the clinical significance of these 3 errors is unknown, it is assumed that lower is better. The TCI system did not generate undershoot errors in our testing paradigm, so only overshoot and terminal target error were determined for the TCI.

The analysis was conducted in 3 phases. In phase I, a covariance matrix was determined for each of the 8 pharmacokinetic parameters of Masui et al. at 3 values: the lower CI, the nominal value, and the upper CI. This produced a population of 6561 (3^{8}) patients at a fixed age (50 years), weight (70 kg), and sensitivity (50%; LOR at the ED_{50}). The outcome measures were determined for each of these patients for PRC and TCI. This permitted identification of the parameters that most affected the outcome measures and the behavior of the outcome measures with misparameterization.

In phase II, a population of 10,000 patients was generated using log-normal distributions for the 8 pharmacokinetic parameters. This analysis was performed 27 times for permutations of 3 sensitivities (25%, 50%, and 80%), ages (30, 50, and 70 years), and weights (50, 70, and 100 kg). The effect of age and weight on the outcome measures was not significant, and these were fixed at 50 years and 70 kg for subsequent analyses. Both phase I and phase II results indicated that maximal overshoot produced similar results to terminal target error, so only terminal target error was retained for phase III.

In phase III, the analysis of phase II was performed using a fixed age and weight and a random sensitivity. For each of the 10,000 simulated patients, the sensitivity was obtained from a uniformly distributed sequence in the range of 2.5% to 97.5%. The terminal target error was determined for both PRC and TCI targeting the ED_{50}. The resulting distributions were compared using the 2-sample Kolmogorov-Smirnov test using the Statistics Toolbox of MATLAB. Means and standard deviations for the 2 control systems were also calculated.

In generating the random pharmacokinetic parameter sets, log-normal distributions were used, because this was the distribution identified by Masui et al. using NONMEM analysis, and thus may approximate the distribution in the population. For patient sensitivity, a uniform distribution was used, because the transformation from effect-site concentration to response probability is the probability density function. The intent of the simulation was to determine the expected performance assuming that the population is accurately represented by the patients studied by Masui et al. and Johnson et al.

## RESULTS

The results of the phase I analysis are presented in Table 1. The covariances of the 8 parameters for the various outcome measures are listed with the nominal values and 95% CIs for the 8 parameters.

The phase II analysis demonstrated that the variation in the outcome measures was not appreciably affected by age and weight. Variation in sensitivity did affect these measures for PRC, as illustrated in Figure 1, but not TCI, as illustrated in Figure 2. When the sensitivity was known in advance, the effect of pharmacokinetic parameter variation was greater in PRC than for TCI, as depicted in Figure 3.

The result of the phase III analysis is depicted in Figure 4. The distributions of terminal target errors are visibly different in shape and width; this was confirmed using the 2-sample Kolmogorov-Smirnov test (*P* < 0.0001). The terminal target error of PRC was −0.76% ± 8.96% with a 95% CI of −20% to 18%, compared with 0% ± 27.6% and a 95% CI of −46% to 46% for TCI. Both standard deviation and 95% CI are reported because of the lack of normal distributions. PRC achieved the terminal target with a 3.1-fold–lower standard deviation compared with TCI.

## DISCUSSION

The most significant finding of the study was the reduction in variability of target error when both pharmacokinetic parameters and response probability were randomly varied. This can be seen in the probability distributions shown in Figure 4. The distribution for PRC in Figure 4 is similar to that seen in Figure 3, because the controller identifies the sensitivity for every patient, and doing so for a patient with 50% sensitivity is subject to similar error because of pharmacokinetic uncertainty to that for one with 5% or 95% sensitivity. The distribution for TCI is the result of the error due to pharmacokinetic uncertainty (as depicted in Fig. 3) multiplied by the uniform distribution of patient sensitivity. Our results extend those of Hu et al.^{1} in demonstrating that biological variability can be reduced through model-based control. Using a goal of effect-site concentration reduces variability by a factor of 2 over simply choosing a drug dose, and using a goal of LOR reduces variability by a factor of 3.1 over using a goal of an effect-site concentration.

To further compare our results with those of Hu et al., consider the problem of administering propofol for testing of an implantable defibrillator, in which 2 cardioversions are performed at 10-minute intervals. For any patient, there is an effect-site concentration below which the patient will respond to the cardioversion, an undesirable outcome. As the concentration is increased above this threshold, the duration and severity of other undesirable outcomes such as hypotension and respiratory obstruction will increase. Thus, the ideal anesthetic would produce exactly this minimal effect-site concentration at these 2 points in time. Consider 3 scenarios: in the first scenario, a single dose of propofol intended to yield an effect-site concentration of 2.2 μg/mL is given before each cardioversion; in the second scenario, a TCI is given to a target of 2.2 μg/mL; and in the third scenario, PRC is used to determine the infusion rate. Comparing the first scenario with the second using the analysis described by Hu et al., the variability in achieved effect-site concentration for the second bolus will be 2-fold higher than the variability after the TCI. Comparing the second scenario with the third scenario, the variability in probability of response to either cardioversion will be >3-fold higher. This variability may influence the outcome in several ways: higher rates of recall, greater frequency of hypotension, greater requirement for airway support, and longer recovery times. By more consistently identifying the minimal effect-site concentration associated with LOR, PRC can be expected to yield fewer undesired outcomes than TCI or intermittent boluses.

Another result of this analysis is the relative importance of pharmacokinetic model parameters in the robustness of control systems. The parameters that most affect robustness of PRC are *k*_{e0} and θ_{6}, whereas TCI is more affected by θ_{3}. This is not surprising, because *k*_{e0} and θ_{6} (the time constant for the effect site and the TRANSIT model) alter the rise time of the model, which will strongly influence the value of the effect-site concentration inferred at the time of loss of consciousness. Conversely, θ_{3} (clearance from the central compartment) influences DC gain, which directly affects the terminal error of the open loop TCI. Other parameters have 3 to 5 orders of magnitude less effect on the robustness of either system. When determining model parameters in NONMEM analysis, the goal is typically the reduction of overall prediction error, but sacrificing precision in parameters that have little impact on the performance of a controller may improve precision in more critical parameters. This suggests that a reanalysis of pharmacokinetic data should be part of the controller design process, which is facilitated by open access to such data in repositories such as OpenTCI.

An advantage of PRC is the minimal number of adjustments in infusion rate required to implement the control strategy. An initial bolus and infusion rate are specified before induction. After 175 seconds (if the patient is still responsive), a second rate is entered. When the patient becomes unresponsive, the pump is stopped, and a new infusion is entered. This process can (and has^{a}) been implemented by a clinician observing the screen of a computer and manually entering infusion rates into a Baxter AS50 syringe pump.

Although there are technical means to assess depth of sedation with processed electroencephalogram, and Masui et al. include a model for Bispectral Index (BIS) response to propofol in their report, we have not used this technology in this effort. There are several reasons for this. First, BIS introduces a variable delay in the observation of transition to sedation, and this will increase the variability of the system. Second, BIS introduces the potential for artifact. In our study of patient-controlled sedation during colonoscopy,^{6} 7% of all BIS readings were artifacts, typically caused by sensor dislodgement. It is unlikely that in a clinical setting this rate would be lower than that obtained by a motivated clinical researcher, and loss of data at the time of transition to sedation would significantly degrade the performance of this system. Third, although we determined that anesthesiologist-administered sedation produced a median BIS of 72 and patient-controlled sedation yielded a median BIS of 88, it is unlikely that either of these targets could easily be maintained by closed loop infusion, and a target of 60 would clearly exceed the depth of anesthesia associated with a significant risk of apnea as demonstrated in this study. Finally, the design of this system emphasized simplicity in implementation. Incorporation of BIS would increase the complexity.

Another approach to which our system can be compared is computer-assisted personalized sedation, as implemented in SedaSys®.^{7} Under this regimen, no pharmacokinetic model is used (personal communication, J. F. Martin, PhD, Endo Ethicon, 2009), and increments in propofol administration are determined by a schedule, subject to reduction by failure of the patient to respond to an automated response monitor, desaturation, or apnea. This approach is similar to ours in that a clinical observation is incorporated into the control loop, but differs in several respects. First, the device is limited to a target of moderate sedation by the dependence on the automated response monitor and requires continuous input from its pulse oximeter and capnograph. Second, the device does not adjust the dosing schedule to produce similar trajectory to sedation across a range of ages and weights. These differences are attributed to the intended operator; SedaSys is claimed to “permit nonanesthesia professionals with an on-label means by which to safely and effectively administer propofol for minimal to moderate sedation during endoscopic procedures,”^{8} whereas our system is intended for use by an anesthesia provider who determines the clinical end point. Again, the design of this system emphasizes simplicity: an anesthesiologist, a computer, a pump, and no cables.

### Limitations

Our system as described does not use a model for opioid coadministration, although this can easily be added, as the pharmacodynamic model presented in Equation A1 includes the interaction of propofol and remifentanil. Our previous study^{6} used this approach. The system as described uses the clinical end point of LOR to verbal stimulus and is intended for short, minimally invasive procedures such as endoscopy. Although this might seem a severe limitation, colonoscopy may be the single most frequent procedure for which propofol is administered. The principal utility of the system as described is to permit an anesthesia provider with no prior information to guide dosing, but who has the skill to recognize deep sedation, to produce and maintain this for a brief period using a pump and a syringe of propofol. Although this has been done in clinical research, claims of generalizability are beyond the scope of this analysis, and clinical validation will be required.

In conclusion, a system that produces a slow transition in the probability of LOR, and uses observation of the time of transition to determine an infusion sequence that maintains the effect-site concentration at the value associated with LOR has been described. The robustness of this control system to parameter error was compared with that of a TCI system with numerical simulation, and found to decrease the variability of target error (the difference between the achieved probability of LOR and the intended probability) by a factor of 3.1. The system uses a small number of adjustments in infusion rate, making operation by an anesthesiologist entering rates into an infusion pump practical. The utility of this system will require clinical validation.

## DISCLOSURES

**Name:** Jeff E. Mandel, MD, MS.

**Contribution:** This author helped design the study, conduct the study, analyze the data, and write the manuscript.

**Attestation:** Jeff E. Mandel has seen the original study data, reviewed the analysis of the data, approved the final manuscript, and is the author responsible for archiving the study files.

**Name:** Elie Sarraf, MDCM.

**Contribution:** This author helped design the study, conduct the study, analyze the data, and write the manuscript.

**Attestation:** Elie Sarraf has seen the original study data, reviewed the analysis of the data, and approved the final manuscript.

**This manuscript was handled by:** Dwayne R. Westenskow, PhD.

a A Pharmacokinetic Approach to Rapid Titration of Propofol to Moderate Obstruction for Diagnosis of Sleep Apnea. Available at: http://www.isaponline.org/files/2812/9900/8585/ISAP_2010_Abstract_14.pdf. Accessed September 9, 2011.

Cited Here...

b MATLAB Implementation of Early Phase Propofol Model. Available at: http://www.opentci.org/doku.php?id=code:matlab:masui. Accessed September 14, 2011.

Cited Here...

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2. Frolich MA, Dennis DM, Shuster JA, Melker RJ. Precision and bias of target controlled propofol infusion for sedation. Br J Anaesth 2005;94:434–7

3. Masui K, Upton RN, Doufas AG, Coetzee JF, Kazama T, Mortier EP, Struys MM. The performance of compartmental and physiologically based recirculatory pharmacokinetic models for propofol: a comparison using bolus, continuous, and target-controlled infusion data. Anesth Analg 2010;111:368–79

4. Masui K, Kira M, Kazama T, Hagihira S, Mortier EP, Struys MM. Early phase pharmacokinetics but not pharmacodynamics are influenced by propofol infusion rate. Anesthesiology 2009;111:805–17

5. Johnson KB, Syroid ND, Gupta DK, Manyam SC, Egan TD, Huntington J, White JL, Tyler D, Westenskow DR. An evaluation of remifentanil propofol response surfaces for loss of responsiveness, loss of response to surrogates of painful stimuli and laryngoscopy in patients undergoing elective surgery. Anesth Analg 2008;106:471–9

6. Mandel JE, Lichtenstein GR, Metz DC, Ginsberg GG, Kochman ML. A prospective, randomized, comparative trial evaluating respiratory depression during patient-controlled versus anesthesiologist-administered propofol-remifentanil sedation for elective colonoscopy. Gastrointest Endosc 2010;72:112–7

7. Pambianco DJ, Whitten CJ, Moerman A, Struys MM, Martin JF. An assessment of computer-assisted personalized sedation: a sedation delivery system to administer propofol for gastrointestinal endoscopy. Gastrointest Endosc 2008;68:542–7

8. Pambianco DJ, Vargo JJ, Pruitt RE, Hardi R, Martin JF. Computer-assisted personalized sedation for upper endoscopy and colonoscopy: a comparative, multicenter randomized study. Gastrointest Endosc 2011;73:765–72

9. Goodwin GC, Graebe SF, Salgado ME. Control System Design. Upper Saddle River, NJ: Prentice-Hall, 2001: 184–7

### Probability Ramp Control Implementation

The probability ramp control (PRC) system is composed of the following elements:

- A pharmacokinetic model
- A pharmacodynamic model
- A solution for the probability ramp
- A solution for the maintenance infusion

### Pharmacokinetic Model

The purpose of the pharmacokinetic model is to simulate the effect-site concentration for a specified infusion sequence. Pharmacokinetic models of propofol have traditionally been geared toward long-term administration. A recent advance is the model of Masui et al., which is concerned with the early phase pharmacokinetics of the drug. The Masui et al. model uses a central compartment whose volume is a function of patient weight, a peripheral compartment, a pure delay that is a function of patient age, and a delivery process (referred to as the TRANSIT model) that is dependent on pump infusion rate. The values for these parameters were identified by NONMEM analysis, yielding 7 parameters, as indicated in Appendix Table 1. The Masui et al. model was implemented in state space form using MATLAB 7.11 (MathWorks), and is available at OpenTCI.^{b} Although the Masui et al. model has been used in this effort, other models could easily be substituted. A state space implementation is used to facilitate numerical solution to the minimization processes used in 3 and 4.

### Pharmacodynamic Model

The purpose of the pharmacodynamic model is to determine the probability of a clinical event given an estimated effect-site concentration. The model of Johnson et al.^{5} is used, as indicated in Equation A1.

This model considers both propofol and remifentanil and provides response probability predictions for 4 different levels of stimulation. The model uses a sigmoid relationship between effect-site concentration and response probability, with an interaction term for the 2 drugs. Although the current effort is directed at propofol as a single agent and loss of responsiveness, the system could be used for propofol and remifentanil, either separately controlled, or as a mixture in a single syringe, and could use 1 of the other end points. The parameters of the Johnson et al. model are listed in Appendix Table 2.

### Solution for the Probability Ramp

The probability ramp is formed by specifying 3 targets. The infusion sequence consists of a bolus (B) delivered at the maximal infusion rate of the pump followed by an infusion (I1) until 175 seconds, and a second constant infusion (I2) for the remaining time. The elements of the infusion sequence are indicated in Appendix Figure 2. The probability of loss of responsiveness (*P*_{LOR}) is calculated once per second for the infusion sequence using the pharmacokinetic and pharmacodynamic models. The infusion sequence is determined by minimization of a loss function (L), given in Equation A2:

where P_{LOR}(90) is the predicted probability of loss of responsiveness at 90 seconds, P_{LOR}(175) is the prediction at 175 seconds, and P_{LOR}(390) is the prediction at 390 seconds. At some time *t*_{LOR} before 390 seconds, loss of responsiveness should be observed in 99.5% of the population. The above times are somewhat arbitrary, but permit the average patient to lose responsiveness in approximately 3 minutes and very few patients to remain responsive after 6 minutes.

If P_{LOR} were an algebraic function of B, I1, and I2, an analytical solution would be possible. Because this is not the case, computational numerical analysis is used. The loss function is continuous and monotonic with respect to B, I1, and I2, making it amenable to simplex minimization. For appropriately chosen targets, the global minimum for the loss function is zero.

The responses of the model for the calculated infusion sequences for 50-year-old patients with weights ranging from 50 to 100 kg (in 10-kg increments) and for 75-kg patients with ages ranging from 20 to 70 years (in 10-year increments) are depicted in Appendix Figure 1. Although the shape of the curves differs slightly, they all pass very close to 5% at 90 seconds, 50% at 175 seconds, and 99.5% at 390 seconds, and are monotonically increasing. The infusion sequence drives the effect site through a monotonic increase over 390 seconds with a relatively constant rate of change of response probability. This permits a unique mapping of the time to loss of responsiveness (*t*_{LOR}) to the desired effect-site concentration. The determination of loss of responsiveness is the clinical observation made by the operator, and it is assumed that this is a precise and unbiased measure.

### Solution for the Maintenance Infusion

Having identified the time of loss of responsiveness, the effect-site concentration associated with this end point is known (within the accuracy of the model). This will immediately be exceeded because of the quantity of propofol that has entered the patient but not yet entered the effect site. To return to this effect-site concentration in the minimal time without undershoot, a second infusion sequence must be determined comprising a period P over which the pump rate is zero, followed by a constant pump rate M for the remainder of the epoch. The solution for the terminal infusion rate M is derived from the DC gain (K) of the state space model, as shown in Equation A3:

where A, B, C, and D are the continuous state space matrices. M is given by Equation A4:

Where C_{e} is the effect-site concentration of propofol at a given time. Note that this is equivalent to the terminal infusion rate calculated from compartmental clearance.

Having specified the terminal infusion rate (M), the period for which the pump is paused after identification of the target (P) is solved for. This is found by minimizing a second loss function given in Equation A5:

Where T is the probability of loss of responsiveness (LOR) corresponding to C_{e}(*t*_{LOR}), and ‖ is the H_{2} norm of the target error over the period.

The functioning of the system is depicted in Appendix Figure 2. In this example, the patient loses responsiveness at 240 seconds, corresponding to a sensitivity of 69.7%. The peak effect-site concentration occurs 48 seconds later, corresponding to a sensitivity of 91.8%. For purposes of illustration, the system was detuned to produce a visible undershoot and terminal target error, because these are typically too small to observe.

The system is designed to produce a relatively constant rate of increase in the probability of loss of responsiveness over time. Thus, if 30,000 patients of varying weights and ages are subjected to this titration that takes 300 seconds, every second, another cohort of approximately 100 patients will become unresponsive. The 150th cohort will lose responsiveness after 150 seconds at the ED_{50}. By knowing that a single patient undergoing the probability ramp becomes unresponsive at 150 seconds, the infusion sequence that maintains the effect site at the ED_{50} can be chosen to maintain this clinical effect. The utility of maintaining propofol as a single agent at the effect-site concentration associated with loss of responsiveness varies with procedure, and it is unknown whether a patient who exhibits loss of responsiveness at the 50th percentile will achieve other end points at the same sensitivity. Although it is unknown whether the sensitivity for one clinical end point is predictive of another, most anesthesiologists check for a lash reflex during induction to determine when to proceed with intubation, and if there was no predictive value in time to loss of lash reflex, this practice would probably not have evolved.

### TCI Implementation

The TCI system was implemented using open-loop pole-zero cancellation and pole assignment using algebraic manipulation^{9} to obtain an infusion sequence that was constrained to the limits of a Baxter AS50 infusion pump with a 60-mL syringe (0–366 mL/h). A significant limitation in the Masui et al. model is the nonlinearity of the Transit model; this was dealt with by using the maximal infusion rate.