The dosing scheme required to achieve and maintain a particular target plasma concentration of a drug can be determined using standard pharmacokinetic (PK) equations. However, these cannot be implemented easily during the course of anesthesia. PC-based PK simulation programs, e.g., STANPUMP ^{(1)}, IVASIM ^{(2)}, or RUGLOOP ^{(3)} enable PK simulations as well as the control of infusion pumps and thus the target-controlled infusion (TCI) of IV anesthetics. The only commercially available TCI device is the DIPRIFUSOR™ (AstraZeneca, London, UK). Disadvantages of the DIPRIFUSOR™ are the inventory costs as well as expensive operating costs through exclusive use of special prefilled syringes of propofol. Neither generic propofol nor propofol in smaller volumes than the 50-mL prefilled syringes can be used.

A simple pocket-sized tool that enables a sufficiently accurate estimation of the required loading dose/infusion rate for a required target concentration at each time point could offer anesthesiologists a solution to the above-mentioned dilemma. We describe the development and evaluation of a slide rule that enables the simple determination of the required infusion rate for a required target concentration at each time point for propofol.

According to the principle of linear PK, the bolus dose required to reach a given target plasma concentration is the product of the (weight-related) distribution volume and required concentration. Similarly, the infusion rate at a particular time point is the product of target concentration, body weight, and a correction factor that depends on the time elapsed from the start of the initial infusion. This factor can be determined for each time point using a PK simulation program.

The TCI slide rule we developed enables the above-mentioned multiplications, analogous to the principle of the classical slide rule, as addition of logarithms. The required infusion rate for propofol can be read for each required target concentration after adjusting for body weight and time elapsed from the start of the initial infusion.

#### Methods

All simulations were run on a 200-MHz PC using the PK simulation and infusion software STANPUMP. The program STANPUMP is freely available from the author (Steven L. Shafer, MD, Department of Anesthesia, Stanford University, Stanford, CA).

For the simulations, we used the propofol PK parameter set described by Marsh et al. ^{(4)}, which has been shown to have a low bias and a high accuracy ^{(5)}. This parameter set is also used in the DIPRIFUSOR™. For the target concentration, we used the plasma concentration.

The dose required to reach a target plasma concentration is the product of central distribution volume and target plasma concentration:MATH where D is the dose in milligrams, Vc is the weight-corrected central distribution volume in liters per kilograms, Wt is the body weight in kilograms, and C_{target} is the target plasma concentration in milligrams per liter.

According to the principle of dose proportionality, the dose to reach a target plasma concentration of 2 mg/L is twice as large as the dose to reach a target plasma concentration of 1 mg/L. From the superposition principle, the bolus dose required to increase the target plasma concentration from 0 to 2 mg/L is equal to the dose required to increase the target plasma concentration from 2 to 4 mg/mL. The corresponding bolus dose is calculated by multiplication of the weight-related unit bolus dose with body weight and the required size of the target concentration change.

According to the principle of linear PK, the infusion rate required at steady-state at each time point is proportional to the product of elimination clearance and the target concentration, equalling the net drug loss (DL) per time unit. To maintain a target concentration, the amount of drug lost from the central compartment DL must be replaced at each time point: where R(ss) is the steady-state infusion rate in milligrams per hour, C_{target}(ss) is the steady-state target concentration in milligrams per liter, and Cl is the elimination clearance in liters per hour. Normalizing to body weight yields:MATH where K is the elimination clearance normalized to Wt (L · h^{−1} · kg^{−1}).

In the nonsteady-state situation, the infusion rate for drugs with multicompartment kinetics can be calculated in a similar way provided that loss from the central compartment because of distribution is also taken into account. Distribution loss depends on the concentration difference between the central and peripheral compartment(s) and therefore the duration of the infusion.

The DL from the central compartment at time (t) can be described by the equation:MATH where DL(t) is the drug loss from the central compartment in milligrams per hour at time (t), Cp(t), C1(t), and C2(t) are the drug concentrations at time (t) in the corresponding compartment in milligrams per liter, and Q1 and Q2 are the distribution clearances in liters per minute.

The contribution of the distribution process decreases exponentially and must be compensated for by an exponentially decreasing infusion rate to maintain a constant target concentration (Fig. 1). Given a relatively constant concentration in the peripheral compartments during consecutive time epochs, the above equation with discrete, constant values for the corresponding concentrations within the respective epochs may be assumed.

Expressing the concentration in the peripheral compartment within the respective time interval as a fraction of the concentration in the central compartment (e.g., for t1, three-compartment model), yields:MATH MATH MATH

Equation U6A Image Tools |
Equation U6B Image Tools |
Equation U6C Image Tools |

Because of the changing balance between concentrations in the compartments during the nonsteady-state, parameters a and b differ for each time interval (t1…tn) after the start of the initial infusion. The term Cl+Q1 × (1−a(t1))+Q2 × (1−b(t1)) equals total clearance from the central compartment during time interval t1 and may be considered constant during an appropriately chosen interval. Note that for t = 0, both a and b are 0 (no drug in the peripheral compartments), and therefore,MATH

For t = ∞, the steady-state conditions are met (a = 1 and b = 1), and therefore,

Equation 8 is identical to equation 2.

Equation 2 Image Tools |
Equation 8 Image Tools |

Thus, the drug amount to be infused per time unit to maintain the Cp in the interval t1 is given by:MATH where DL(t1) is the drug loss per time unit during the time interval t1 in milligrams per hour, R(t1) is the corresponding infusion rate in milligrams per hour, Cp(t1) is the plasma concentration during t1 in milligrams per liter, and Cls(t1) is the summed clearance in liters per hour.

After adjusting the summed clearance for body weight and putting Cp = C_{target}, the corresponding infusion rate for each discrete time interval n is given by:MATH where R(t_{n}) is the infusion rate in the time interval t_{n} in milligrams per hour, Wt is body weight in kilograms, C_{target}(t_{n}) is the required target concentration in milligrams per liter, and K(t_{n}) is the time-dependent summed clearance normalized to body weight (L · h^{−1} · kg^{−1}).

Therefore, K(t_{n}) may be calculated as:MATH

Using STANPUMP simulations, R(t_{n}) and K(t_{n}) were determined for various time intervals (Table 1). Because the magnitude of the changes in infusion rate decreases over time (Fig. 1), we chose smaller time intervals (15 min) for the first hour from the start of the initial infusion and thereafter, larger intervals (60 min) for determining the time-dependent factor. The infusion rate in the middle of a time period was chosen as the infusion rate for the entire period. We chose a body weight of 80 kg and a target plasma concentration of 3 μg/mL for the mutual simulations. Subsequently, the simulations were validated using different weights and target concentrations.

The calculation of the infusion rates is done on the newly developed slide rule. Analogous to the principle of the classical slide rule, multiplication is performed as addition of logarithms. The problem of double multiplication was solved by starting from a point on the moveable center of the slide rule, plotting Wt increasing logarithmically to the left (starting with 1 kg), and, from the same point, the required target concentration (starting with 1 μg/mL) increasing logarithmically to the right. This was realized by having the 1 kg and the 1 μg/mL points lie exactly above one another.

The body weight scale is marked on the lower side of the moveable center (tongue) of the slide rule and the target concentration scale on the upper side. The time-dependent factors are entered on the lower side of the nonmoveable part and marked with the corresponding time measurements. The clinically relevant infusion rates (100–2000 mg/h) were entered on the upper side of the nonmoveable part (Fig. 2A).

When the body weight (lower scale on the tongue) is placed exactly above the corresponding time point (lower scale on the nonmoveable part), the required infusion rates to be administered during the corresponding time period can be read from the upper scale on the nonmoveable part above the required target concentration (upper side of the tongue) (Fig. 2b).

As control criteria, we calculated the percentage deviation of the predicted plasma concentration obtained from STANPUMP driven in the target-controlled mode from the predicted plasma concentrations that were obtained using the infusion rates determined from the TCI slide rule. This percentage deviation is independent of the absolute level of the target plasma concentration and also independent of Wt. We chose the following target plasma concentrations:

* constant target of 3 μg/mL

* initial target of 3 μg/mL, and after 15 min, changing to a target of 1, 2, 4, or 5 μg/mL of propofol

* 0- to 15-min target of 5 μg/mL (e.g., induction and intubation), 15- to 30-min target of 2 μg/mL (e.g., waiting for surgeon), 30- to 60-min target of 3 μg/mL (e.g., surgical preparation), 60- to 120-min target of 4 μg/mL (e.g., intense surgical stimulus), 120- to 150-min target of 3 μg/mL (e.g., control of bleeding and withdrawal), and 150- to 165-min target of 2 μg/mL (e.g., skin closure).

While changing to a higher target, we administered the unit bolus dose of 0.228 mg of propofol/kilograms/(microgram per milliliter propofol target plasma concentration change), i.e., for a body weight of 80 kg and a target change from 3 to 5 μg/mL of propofol—a bolus dose of 0.228 × 80 × 2 mg of propofol and continued immediately after with the new calculated infusion rate. While changing to a lower target, we set the infusion rate to zero for a time period previously determined by STANPUMP simulations (Table 2), e.g., for 2.1 min when changing the target from 3 to 2 μg/mL of propofol 15 min after the start of the initial infusion and continued immediately after with the new calculated infusion rate.

For targeting a constant target of 3 μg/mL, the predicted plasma concentration was compared with the target plasma concentration, and the mean and the maximum percentage deviation was determined for the time frame 0–15 min and 16–300 min.

We also evaluated scenarios in which the target concentration was changed after 15 min. The predicted plasma concentration was compared with the target plasma concentration, and the mean and the maximum percentage deviation was determined for the time frame 15–75 min (first hour after changing the target) and for the time frame 75–135 min (second hour after changing the target).

#### Results

The time-dependent factor at the relevant time frames (0–15 min, 15–30 min, and so on) determined using STANPUMP is given in Table 1. To reach a specific target plasma concentration, an infusion rate of K(t1..n) × Wt × C_{target} is given after a bolus of 0.228 mg/kg/(microgram per milliliter of propofol target plasma concentration change). To increase the target plasma concentration, one unit bolus dose of 0.228 mg/kg/(microgram per milliliter of propofol target plasma concentration change) is given and is followed by an infusion with K × Wt × C_{targetnew}. To decrease the target plasma concentration, the infusion rate is set to zero for a time period determined by the wanted percentage decrease of the propofol target plasma concentration (Table 2) and followed by an infusion with K × Wt × C_{targetnew}.

Aiming for a constant target plasma concentration of 3 μg/mL of propofol, the mean percentage deviation between the predicted plasma concentration and the targeted plasma concentration was 4.05% (maximum percentage deviation, 6.97%) for the time frame 0–15 min and 0.5% (mean, 2.03% maximum) for the time frame 16–300 min. The results when the target plasma concentration was changed after 15 min are given in Table 3.

Figure 3 shows the time course of the predicted plasma concentration reached by STANPUMP driven in the target-controlled mode and of the predicted plasma concentrations that were obtained using the infusion rates determined from the TCI slide rule with and without changing the target concentration.

Even with multiple changes of the propofol target plasma concentration between 2 and 5 μg/mL (0–15 min, 5 μg/mL; 15–30 min, 2 μg/mL; 30–60 min, 3 μg/mL; 60–120 min, 4 μg/mL; 120–150 min, 3 μg/mL; and 150–165 min, 2 μg/mL), the deviation between targeted concentration and predicted plasma concentration using the infusion rates obtained by the new TCI slide rule remained always less than 23% (mean percentage deviation, 5.77%).

#### Discussion

We developed a TCI slide rule for propofol that enables us to obtain and maintain a required target plasma concentration, to increase the target plasma concentration quickly and accurately, and to maintain the new level or to decrease the target plasma concentration quickly and accurately and to maintain the new level over the complete duration of anesthesia.

The evaluation showed only a small mean and maximum percentage deviation between the predicted plasma concentration reached by STANPUMP driven in the target-controlled mode and the predicted plasma concentrations that were obtained using the infusion rates determined from the TCI slide rule. Therefore, the TCI slide rule enables the user to maintain a constant target plasma concentration with clinically acceptable deviations.

A rational form of anesthesia administration consists of holding the hypnotic components (e.g., propofol) at a sufficient level constant over the duration of anesthesia to avoid intraoperative consciousness, whereas the analgesic components are adjusted according to the relevant operative-induced pain stimulus. However, when an increase in the propofol target during anesthesia is required, the change in the infusion rate can be read from the TCI slide rule in combination with a calculated bolus dose of 0.228 mg/kg for each 1-μg/mL change in the target concentration. For a decrease in the target concentration, the infusion must be stopped until the new, smaller concentration is reached. The stop time, which is dependent on the duration of the infusion and the required percentage decrease in the target concentration, is given in Table 2. However, it must be borne in mind that the predicted plasma concentrations from the infusion rates calculated by the TCI slide rule might become more inaccurate with multiple changes in target concentrations at short time intervals.

The newly developed, simple method to determine the TCI infusion rates is based on a three-compartment PK model and the PK parameter set of Marsh et al. ^{(4)}. This parameter set assumes that the central distribution volume is proportional to the weight of the patient, and its application assumes that the PKs of propofol are fully linear. This set is used in the commercial TCI system DIPTIFUSOR™ for use in adults. Marsh et al. ^{(4)} investigated the performance of this PK parameter set in a population of children. A systematical but acceptable (bias, 18,5%) over-prediction of the measured blood concentration of propofol was observed. This should be kept in mind when using our slide rule for children.

PK parameter sets are determined from groups of patients or subjects. The individual PK parameter set of an individual patient can vary from these. In addition, anesthesia- and surgery-related changes in cardiac output, perfusion of liver and kidney, or blood loss ^{(6)}, which influence the distribution and elimination of the anesthetics, are not considered in this simple PK model. However, even though the prediction error (difference between predicted and measured concentration) amounts to 30%^{(7)}, a TCI-oriented administration of anesthesia is still useful. In the study of Swinhoe et al. ^{(8)}, the control of depth of anesthesia with a TCI system for propofol was considered as good in all patients undergoing major surgery, and the predictive performance of the TCI system was considered acceptable for clinical purpose.

Roberts et al. ^{(9)} described a manual, TCI-oriented infusion regimen for propofol. For a target plasma concentration of 3 μg/mL, the following propofol infusions were recommended after a propofol bolus of 1 mg/kg: 10 mg · kg^{−1} · h^{−1} for the first 10 minutes, 8 mg · kg^{−1} · h^{−1} for the next 10 minutes, and thereafter 6 mg · kg^{−1} · h^{−1}. This pattern is simple and of acceptable accuracy for obtaining a 3-μg/mL target concentration of propofol, leading to its use as a standard infusion rate with propofol. However, it is inflexible in comparison with the TCI simulations using STANPUMP, IVASIM, or RUGLOOP. The PK software in these systems or that integrated in the DIPRIFUSOR™ is more exact but associated with a certain technical, material, and financial cost.

In contrast, the TCI slide rule for propofol that we developed can be constructed with minimal theoretical or technical knowledge from a copy of the diagram. This TCI slide rule combines the advantages of minimal financial and technical cost with good accuracy. A TCI slide rule can further be developed for each drug/each patient group for which the corresponding PK parameters are available.