Intravenous Drug Delivery Systems

Key Points

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The pharmacokinetics of anesthetic drugs are described by multicompartment models. Accurate intravenous drug delivery requires adjusting the maintenance infusion rates to take into account the accumulation of the drug in the peripheral tissues.
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Biophase is the site of action of a drug. Initiation, maintenance, and titration of intravenous anesthetics must account for the delay in equilibration between plasma and the site of drug effect.
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Some drug effects directly reflect the concentration of the drug in the biophase (direct-effect models). Other drug effects reflect the alteration of feedback systems by anesthetics (indirect-effect models). The influence of opioids on ventilation reflects the dynamic influence of opioids on the feedback between ventilation and carbon dioxide and is thus an example of an indirect drug effect.
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The target concentration in the effect site is the same as the target concentration in plasma at steady state. Effect-site requirements are influenced by patient physiologic characteristics, surgical stimulation, and concurrent drug administration. Ideally, target concentrations should be set for the hypnotic (volatile anesthetic or propofol) and the analgesic (opioid) that properly accounts for the synergy between them.
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To achieve an effective target concentration, the conventional teaching of administering an initial dose as calculated by the product of target concentration and volume of distribution, followed by a maintenance rate as calculated as the product of target concentration and clearance, is inaccurate. The initial dose may be calculated as the product of target concentration and volume of distribution at peak effect. Maintenance rates must initially account for the distribution of drug in peripheral tissues and should only be reduced to the product of target concentration and clearance after equilibration of plasma and peripheral tissue concentrations.
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The terminal half-life does not reflect the clinical time course of drug plasma concentration. The context-sensitive decrement time is the time for a given decrement in drug concentration, as a function of the duration of infusion that maintains a steady plasma concentration. Context-sensitive decrement times properly incorporate the multicompartment behavior of intravenous anesthetics. The context-sensitive half-time is the time for a 50% decrement in concentration.
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Alfentanil, fentanyl, sufentanil, remifentanil, propofol, thiopental, methohexital, etomidate, ketamine, midazolam, and dexmedetomidine can all be administered as a continuous intravenous infusion. Specific caveats, infusion rates, and titration guidelines are presented in this text.
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Target-controlled infusions (TCIs) use pharmacokinetic models to determine intravenous anesthetic administration rates required to achieve specified plasma or effect-site drug concentrations. Various plasma and effect-site targeting TCI systems are commercially available worldwide (except within the United States) to administer hypnotics and opioids.
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Closed-loop drug delivery systems have used the median electroencephalographic frequency, bispectral index (BIS), or auditory-evoked potentials to control intravenous anesthetic delivery. Although these systems have generally performed well clinically, they remain under investigation.

Introduction

Drugs must reach their site of action to be effective. In 1628, William Harvey proved in Exercitatio Anatomica de Motu Cordis et Sanguinis in Animalibus that venous blood was transported to the arterial circulation and thus to body organs by the heart. That drugs injected into veins could be rapidly carried to the entire body was rapidly recognized. Consequently, for intravenous drug delivery to be successful, predictable intravenous access is essential.

The development of intravenous methods of anesthetic drug delivery has been made possible by technologic advances. In the middle of the 17th century, Christopher Wren and his Oxford contemporaries applied a feather quill and animal bladder to inject drugs into dogs and humans and rendered them unconscious. The hollow hypodermic needle and a functional syringe were developed by Frances Rynd (1801–1861) and Charles Pravaz (1791–1853), respectively. Contemporary needles, catheters, and syringes are descendants of these early devices. In the twentieth century, equipment began to be made of plastics, first polyvinyl chloride, then Teflon, and later, polyurethane. In 1950, Massa invented the Rochester needle ( Fig. 26.1 ), which led to the revolutionary concept of the “over-the-needle” catheter, which is still the gold standard for intravenous access for nearly all intravenous drug delivery today.

Fig. 26.1, Details of the assembly of the Massa plastic needle, sold by Rochester Products Company, Rochester, Minnesota.

Although fundamental principles of intravenously administering drugs were known in the 18th century, intravenous induction of anesthesia became common only in the 1930s after the discovery of barbiturates. Maintenance of anesthesia by intravenously administered anesthetics has become practical, safe, and popular in the past 2 decades. Intravenous drugs such as methohexitone and thiopental, although suitable for intravenous induction of anesthesia, are not suitable for use by infusion for the maintenance of anesthesia. In the case of thiopental, accumulation can lead to cardiovascular instability and delayed recovery, whereas methohexitone is associated with the excitatory phenomenon and epileptiform electroencephalographic (EEG) changes. Although the next generation of intravenous drugs, such as ketamine, althesin, and etomidate, possessed desirable pharmacokinetic characteristics, their use has been limited as a result of other side effects, including hallucinations, anaphylaxis, and adrenal suppression, respectively. The discovery of propofol in 1977 provided the anesthetic practice an intravenous drug suitable for both induction and maintenance of anesthesia; currently, propofol is still one of the most frequently used drugs for this purpose. Other drugs suitable for continuous infusion used today are some of the opioids such as alfentanil and sufentanil and certainly the short-acting opioid remifentanil. In addition, some of the nondepolarizing neuromuscular blocking agents are used as continuous infusions in specific situations.

Drugs are still predominantly injected as a bolus or continuous infusion using standard dosing guidelines, thereby ignoring the large interindividual variability in the dose-response relationship. In contrast to inhaled anesthetics, for which the inspired and end-tidal concentrations can be continuously measured in real time (“online”), the actual plasma or effect organ concentration of an intravenously administered drug is not immediately measurable in clinical practice. Therefore manually adjusting the intravenous drug injection regimens to maintain an online measured plasma concentration is impossible. It becomes even more complex if a specific effect-site concentration is the target. Optimal patient-individual dosing may be achieved by the application of pharmacokinetic-pharmacodynamic principles. Additionally, recent findings suggest that the pharmacokinetic and pharmacodynamic interactions during intravenous administration of various drugs are important and, as such, should be taken into account when optimizing drug administration. Computer technology can be used to assist the clinician in titrating intravenous drug administration by using the therapeutic end point as the feedback signal for dosing ( Fig. 26.2 ).

Fig. 26.2, Schematic representation of the dose-response relationship for both hypnotics and opioids. The pharmacokinetic (PK) (green area) and the pharmacodynamic (PD) (yellow area) parts of this relationship are shown. Closed-loop control of drug administration using the clinical measure is indicated as feedback control . PK and PD drug interactions are indicated.

The development of the first mechanical syringe pumps in the 1950s improved the quality of intravenous drug administration. A more recent technologic development in intravenous anesthesia is the introduction of computerized pharmacokinetic model–driven continuous-infusion devices, enabling the attainment of desired plasma levels of an intravenous anesthetic drug by using a computer-controlled infusion pump operated in accordance with the published pharmacokinetics of the drug. These efforts have resulted in the release of the first commercial target-controlled infusion (TCI) device in Europe, developed by Zeneca specifically for the administration of propofol. Since that time, several countries (with the exception of the United States) have approved the use of TCI devices for the delivery of anesthetic drugs.

The ultimate development in anesthetic delivery systems will be devices for closed-loop administration of intravenous drugs during anesthesia. Systems have been developed for closed-loop administration of various drugs such as neuromuscular blocking agents, hypnotics, and opioids. The control variables for these systems have included various pharmacodynamic measures derived from techniques such as acceleromyography, automated blood pressure measurement, and electroencephalography.

The dose-response relationship can be divided into three parts (see Fig. 26.2 ): (1) the time course of the relationship between the given dose and the plasma concentration is defined as pharmacokinetics, (2) the relationship between the plasma concentration and/or effect-organ concentration and the clinical effect is defined as pharmacodynamics, and (3) the coupling between pharmacokinetics and pharmacodynamics is required when the blood is not the site of drug effect.

Before reviewing delivery techniques and devices for intravenous anesthesia, this chapter presents some pharmacokinetic and pharmacodynamic principles as background for understanding how to administer intravenous drugs to their best advantage. Further discussion of the principles of pharmacokinetics and pharmacodynamics can be found in Chapter 23.

Pharmacokinetic Considerations

The aim of optimal intravenous drug dosing is to reach and maintain a desired time course of therapeutic drug effect as accurately as possible, thereby preventing dose-related adverse drug effects. To be useful in anesthesia, this time course should include a rapid onset of clinical effects, smooth maintenance, and fast recovery after the termination of drug administration. The pharmacokinetics of many intravenous drugs can be described using mammillary multicompartment pharmacokinetic models. These models assume that the drug is directly given and mixed in the plasma, resulting in an immediate peak in its plasma concentration.

The easiest clinical technique is to administer a single dose, calculated to keep the plasma concentration above the therapeutic target concentration for the required time ( Fig. 26.3 ). A constant concentration cannot be maintained, but it should not decrease to less than the therapeutic concentration. Unfortunately, when one single bolus is used, the initial dose must be large enough to maintain a concentration above the lowest therapeutic concentration even at the end of surgery. However, this sometimes very large dose of drug may cause numerous side effects attributable to the initially high concentrations in the body. It might be less harmful to keep the drug concentration above the lowest therapeutic level without very high initial concentrations by repeatedly injecting smaller doses; even with this technique, however, maintaining a stable plasma concentration is still impossible.

Fig. 26.3, Predicted propofol plasma concentration (Cp) and effect-site concentration (Ce) for a repeated bolus dose (1 mg/kg given at time 0 and after 5 and 10 minutes) (A), continuous infusion (10 mg/kg/h) (B), and a bolus (2 mg/kg) followed by a continuous infusion (10 mg/kg/h) (C). Simulated patient is a man, 45 years of age, 80 kg, 175 cm; Schnider model.

To produce a time course of drug effect that follows the time course of anesthetic requirement, a continuous infusion titrated to the perceived anesthetic requirement should be used. Typically, just enough amount of drug is given to achieve the therapeutic blood or plasma drug concentration. Drug administration thereafter should be continuously titrated throughout surgery. Although such a regimen does not overshoot the required concentration (and therefore avoids the risk of concentration-related side effects), yet another difficulty exists. Whereas, the large bolus approach produces an effective concentration (EC) from the onset, albeit with an excessive overshoot, a continuous infusion takes a long time to become effective because of the slow increase in concentration. Reaching steady state (see Fig. 26.3 ) takes a very long time during which the increase in concentration is rapid at first but then slows down as equilibrium is approached. For example, it will take longer than 1 hour for the propofol infusion to generate a plasma concentration that is at least 95% of the steady-state concentration. Consequently, although simple infusions are obviously very effective for maintaining constant blood concentrations once steady state is reached and for avoiding overshoot, infusions do not offer a clinically realistic approach. Therefore a combination of an initial bolus followed by a stepwise decreasing continuous infusion is more useful.

Pharmacokinetic models can be used to calculate the required drug-dosing regimen to reach and maintain a therapeutic drug concentration as fast as possible without overshooting or accumulation. In this chapter, an explanation of how pharmacokinetic models can be used to calculate accurate dosing schemes for use with intravenous drug delivery systems is offered.

Pharmacokinetic models are mathematical descriptions of how the body disposes of drugs. The parameters describing this process are estimated by administering a known dose of the drug and measuring the resulting plasma concentrations. A mathematical model then relates the input over time, I(t), with the concentrations over time, C(t). These models can take many forms. Fig. 26.4 shows concentrations in plasma and effect site over time after a single intravenous bolus of drug at time 0. Drug concentrations continuously decrease after the bolus, and the rate of decrease is approximately proportional to the amount of drug in plasma. Typically, this behavior can be described with the use of exponential models. The curve might have a single exponent, in which case the plasma concentrations over time might be described by the function C(t) = Ae− ^kt , where A is the concentration at time 0 and k is a constant that describes the rate of decrease in concentration. The relationship appears to be a straight line when graphed as the log of concentration versus time. The pharmacokinetics of intravenous anesthetic drugs is more complex because after the bolus, a period of rapid decline is observed before the terminal log-linear portion (i.e., the part that is a straight line when described as log concentration vs. time). This process can be modeled by taking several monoexponential curves and adding them together. The result is a polyexponential curve. For example, the concentrations after an intravenous bolus might be described by an equation with two exponents, C(t) = Ae ⁻ α ^t + Be ⁻ β ^t , or an equation with three exponents, C(t) = Ae ⁻ α ^t + Be ⁻ β ^t + Ce ⁻ γ ^t .

The aforementioned is applied to single bolus dosing, which is, of course, only one way of administering intravenous anesthetic drugs. A more general way to think of pharmacokinetics is to decompose the input into a series of small bits (boluses) and consider each bit of drug separately. The general pharmacokinetic model of drug disposition commonly used in anesthesia independently considers each bit of drug and analyzes its contribution by means of polyexponential decay over time. The formal mathematic description of each bit of drug in terms of polyexponential decay over time is the relationship (Eq. 26.1)

C (t) = I (t) * \sum_{i = 1}^{n} A_{i} e^{- λ_{i} t}

$C (t) = I (t) * \sum_{i = 1}^{n} A_{i} e^{- λ_{i} t}$

where C(t) is the plasma concentration at time t and I(t) is drug input (i.e., a bolus or infusion). The summation after the asterisk (described later in this chapter) is the function describing how each bit of drug is disposed (hence the name, disposition function ). Note that this is again a sum of n exponentials, as described in the previous paragraph.

Pharmacokinetic modeling is the process of estimating the parameters within this function. The integer n is the number of exponentials (i.e., compartments) and is usually two or three. Each exponential term is associated with a coefficient A _i and an exponent λ _i . The λ values are inversely proportional to the half-lives (half-life = ln 2/λ = 0.693/λ), with the smallest λ representing the longest (terminal) half-life. The A values are the relative contribution of each half-life to overall drug disposition. If a drug has a very long terminal half-life but a coefficient that is significantly smaller than the other coefficients, then the long half-life is likely to be clinically meaningless. Conversely, if a drug has a very long half-life with a relatively large coefficient, then the drug will be long lasting even after brief administration. The asterisk (∗) operator is the mathematic process called convolution , which is simply the process of breaking the infusion into bits of drug and then adding up the results to observe the overall concentrations resulting from the disposition of the different bits up to a time point t.

The pharmacokinetic model shown has some useful characteristics that account for its enduring popularity in pharmacokinetic analysis. Most important, the model describes observations from studies reasonably well, the sine qua non for models. Second, these models have the useful characteristic of linearity . Simply stated, if the dose, I, is doubled (e.g., administering a bolus twice as large or an infusion twice as fast), then the resulting concentrations should be doubled.

More generally, linearity implies that the system (i.e., the body acting to produce a plasma drug concentration output from a drug dosage input) behaves in accordance with the principle of superposition. The superposition principle states that the response of a linear system with multiple inputs can be computed by determining the response to each individual input and then summing the individual responses. In other words, when the body treats each bit of drug by polyexponential decay over time, the disposing of each bit of drug does not influence the disposing of other bits of drug.

The third reason for the continuing popularity of these models is that they can be mathematically transformed from the admittedly nonintuitive exponential form shown earlier to a more intuitive compartment form ( Fig. 26.5 ). The fundamental parameters of the compartment model are the volumes of distribution (central, rapidly equilibrating, and slowly equilibrating peripheral volumes) and clearances (systemic, rapid, and slow intercompartment). The central compartment (V ₁ ) represents a distribution volume and includes the rapidly mixing portion of the blood and first-pass pulmonary uptake. The peripheral compartments are made up of tissues and organs that show a time course and extent of drug accumulation (or dissipation) different from that of the central compartment. In the three-compartment model, the two peripheral compartments may roughly correspond to splanchnic and muscle tissues (rapidly equilibrating) and fat stores (slowly equilibrating). The sum of the compartment volumes is the apparent volume of distribution at steady state (Vd _ss ) and is the proportionality constant relating the plasma drug concentration at steady state to the total amount of drug in the body. The intercompartment rate constants (k ₁₂ , k ₂₁ , and so on) describe the movement of drug between the central and peripheral compartments. The elimination rate constant (k ₁₀ ) encompasses processes acting through biotransformation or elimination that irreversibly removes drug from the central compartment.

Fig. 26.5, Three-compartment model (including the biophase) illustrating the basic pharmacokinetic processes that occur after intravenous drug administration. I, Dosing scheme as a function of time; k 10 , rate constant reflecting all processes acting to remove drug irreversibly from the central compartment; k, intercompartment rate constants; V 1 , central compartment volume, usually expressed in liters or liters per kilogram.

Despite their physiologic flavor, compartment models are simply mathematic transformations of the polyexponential disposition functions computed from observed plasma concentrations. Thus physiologic interpretation of volumes and clearances (with the possible exception of systemic clearance and Vd _ss [the algebraic sum of the volumes]) is entirely speculative.

The last reason behind the popularity of these models is that they can be used to design infusion regimens. If the disposition function (Eq. 26.2)

\sum_{i = 1}^{n} A_{i} e^{- λ_{i} t}

$\sum_{i = 1}^{n} A_{i} e^{- λ_{i} t}$

is abbreviated as simply D(t), then the relationship among concentration, dose, and the pharmacokinetic model D(t) can be rewritten as (Eq. 26.3)

C (t) = I (t) * D (t)

$C (t) = I (t) * D (t)$

where ∗ is the convolution operator, as noted earlier. In the usual pharmacokinetic study, I(t) is known, the dose that is given the patient, and C(t) is measured, the concentrations over time. The goal is to find D(t), the pharmacokinetic disposition function. Pharmacokinetic analysis can be thought of as a simple rearrangement of Eq. 26.3 to solve for D(t) (Eq. 26.4)

D (t) = \begin{matrix} C (t) \\ \to\leftarrow \\ I (t) \end{matrix}

$D (t) = \begin{matrix} C (t) \\ \to\leftarrow \\ I (t) \end{matrix}$

where the symbol →← means deconvolution , the inverse operation of convolution. Deconvolution is similar to division, but of functions rather than simple numbers. When dosing regimens are designed from known pharmacokinetic models and a desired course for the plasma concentration over time, the known values are D(t) (the pharmacokinetics) and C _T (t) (the desired target concentrations), and the drug dosing scheme is (Eq. 26.5)

I (t) = \begin{matrix} C_{T} (t) \\ \to\leftarrow \\ I (t) \end{matrix}

$I (t) = \begin{matrix} C_{T} (t) \\ \to\leftarrow \\ I (t) \end{matrix}$

Thus the necessary infusion rates, I(t), can be calculated, given the desired target concentrations, C _T (t), and the pharmacokinetics, D(t), by applying the same tools used to calculate the original pharmacokinetics. Unfortunately, such a solution might require some negative infusion rates, which are obviously impossible. Because a drug cannot be retracted from the body (i.e., give inverse infusions), clinicians must restrict themselves to plasma concentrations over time that can be achieved with noninverse infusion rates.

The standard pharmacokinetic model has one glaring shortcoming. It assumes that after a bolus injection there is complete mixing within the central compartment such that the peak concentration occurs precisely at time 0. It actually takes approximately 30 to 45 seconds for the drug to make its transit from the venous injection site to the arterial circulation. This model misspecification over the first minute or so may not seem significant, but it can cause problems in attempts to relate the drug effect after a bolus to drug concentrations in the body, which becomes even more important when using effect-site TCI. The standard polyexponential pharmacokinetic models are being modified to provide more accurate models of plasma drug concentration in the first minute after bolus injection, also taking into account infusion rate. Recently, Masui and associates found that a pharmacokinetic model consisting of a two-compartment model with a LAG (the time shift of dosing as if the drug were, in fact, administered to the pharmacokinetic model at a later time) and presystemic compartments model accurately described the early pharmacologic phase of propofol during infusion rates between 10 and 160 mg/kg/h. The infusion rate has an influence on kinetics. Age was a covariate for LAG time ( Fig. 26.6 ). Besides compartment models, various physiologically based models have been developed to model the pharmacokinetic behavior of anesthetics. So far, these models are not superior at predicting the time course of drug concentration. None of these models have been used to control intravenous drug delivery devices.

Fig. 26.6, Scheme of the final two-compartment pharmacokinetic model with a LAG time and six TRANSIT compartments (C tr n ). The equilibration rate constants between the central and peripheral compartments were calculated using the following equations: k 12 = C l2 ÷ V 1 , k 21 = Cl 2 ÷ V 2 . The elimination rate constant was calculated using the following equation: k 10 = Cl 1 ÷ V 1 . Cl 1 , Clearance of central compartment; Cl 2 , clearance of peripheral compartment; V 1 , distribution volume of central compartment; V 2 , distribution volume of peripheral compartment. LAG time represents the time shift of dosing as if the drug was, in fact, administered to the pharmacokinetic model at a later time. TRANSIT compartments depict a multiple-step process represented by a chain of presystemic compartments.

Pharmacodynamic Considerations

The Biophase

The goal of drug titration during anesthesia is to reach and maintain a stable therapeutic drug concentration at the site of drug effect, also defined as the effect site or biophase . For most drugs used in anesthesia, the plasma is not the biophase and thus even after the drug has reached the arterial circulation, a further delay occurs before a therapeutic effect is observed. The reason is that additional time is required for the drug to be transported to the target organ, penetrate the tissue, bind to a receptor, and induce intercellular processes that ultimately lead to the onset of drug effect. This delay between peak plasma concentration and peak concentration at the effect site is called hysteresis . Fig. 26.7 illustrates an example of hysteresis revealed during an experiment published by Soehle and coworkers. Two periods of continuous propofol infusions were given. The time course of the plasma concentration and effect-site concentration are simulated using pharmacokinetic and pharmacodynamic models. The cerebral drug effect was measured using the EEG-derived bispectral index (BIS). A clear delay between the time course of the plasma concentration and that of the BIS can be observed. The plasma concentration versus effect curve forms a counterclockwise hysteresis loop. This loop represents the plasma concentration, which is not the site of drug effect. Using nonlinear mixed-effect modeling, the hysteresis is minimized to reveal the effect-site concentration versus clinical effect relationship. The typical sigmoidal population model is also depicted in Fig. 26.7 .

Fig. 26.7, (A) Time course of an experiment showing the hysteresis between plasma concentration (C p ) and hypnotic effect as measured by the electroencephalographic (EEG)-derived bispectral index (BIS). Propofol was given at a constant rate during the shaded periods, resulting in C p (orange line) and effect-site concentration (C e ) (blue line) . The corresponding BIS values are shown as a solid red line . (B) Relation between C p and BIS using data from the experiment reveals a hysteresis loop. (C) After modeling, this hysteresis is minimized as shown in the relation between the effect-site concentration and BIS.

The concentration of drug in the biophase cannot be measured because it is usually inaccessible, at least in human subjects. The time course of drug effect can be calculated by using rapid measures of drug effect. Knowing the time course of drug effect, the rate of drug flow in and out of the biophase (or effect site) can be calculated with the use of mathematic models. As such, the time course of the plasma concentration and the measured effect can be linked using the concept of the effect compartment , developed by Hull and Sheiner. The effect-site concentration is not a real measurable concentration but rather a virtual concentration in a theoretic compartment without a volume and, as such, also without any significant amount of drug present. For any concentration in this virtual compartment, a corresponding assumed effect is observed. This relationship between the effect-site concentration and effect is usually nonlinear and static (i.e., does not explicitly depend on time). If the plasma concentration is maintained at a constant level, then the model assumes that, at equilibrium, the effect-compartment concentration equals the plasma concentration. The delay between the plasma and the effect compartment is mathematically described by a single parameter, defined as k _{e 0} , the effect-site equilibration rate constant (see Fig. 26.5 ).

Measures of drug effect used to characterize the time course of drug transfer between plasma and the biophase vary with the drug being evaluated. For some drugs, a direct measure of drug effect can be applied. For neuromuscular blocking agents, the response from peripheral nerve stimulation (i.e., the twitch) is an ideal measure of effect. Various authors have used the T1% (percentage change of the T1 response compared with baseline T1 response during supramaximal stimulus) derived from electromyogram to measure the drug effect of newer drugs such as rocuronium and cisatracurium. For other categories of drugs such as opioids and hypnotics, the real clinical effects (e.g., unconsciousness, amnesia, memory loss, antinociception) are not measurable. For these reasons, surrogate measures are used to quantify the time course of clinical effects. These surrogate measures can be categorical or continuous. For example, the Observer’s Assessment of Alertness/Sedation (OAA/S) scale was used to measure quantal changes in hypnotic drug effects during propofol administration. Egan and colleagues applied a noxious pain stimulus and used an algometer to measure the balance between nociception and antinociception during remifentanil infusion. Various spontaneous and evoked EEG-derived and processed measures were used to measure cerebral drug effects for opioids and hypnotics. Ludbrook and associates measured propofol concentrations in the carotid artery and jugular bulb to establish movement of propofol into and equilibration with the brain. They simultaneously measured the BIS and found a close correlation between brain concentration (calculated by mass balance) and changes in the BIS.

Direct-Effect Models

As with plasma pharmacokinetics, the biophase concentration is the convolution of an input function (in this case, the plasma drug concentration over time) and the disposition function of the biophase. This relationship can be expressed as (Eq. 26.6)

C_{b i o p h a s e} (t) = C_{p l a s m a} (t) * D_{b i o p h a s e} (t)

$C_{b i o p h a s e} (t) = C_{p l a s m a} (t) * D_{b i o p h a s e} (t)$

The disposition function of the biophase is typically modeled as a single exponential decay (Eq. 26.7)

D_{b i o p h a s e} (t) = K_{e 0} e^{- K_{e O} t}

$D_{b i o p h a s e} (t) = K_{e 0} e^{- K_{e O} t}$

The monoexponential disposition function implies that the effect site is simply an additional compartment in the standard compartment model that is connected to the plasma compartment (see Fig. 26.5 ). The effect site is the hypothetical compartment that relates the time course of plasma drug concentration to the time course of drug effect, and k _{e 0} is the rate constant of elimination of drug from the effect site. By convention, the effect compartment is assumed to receive such small amounts of drug from the central compartment that it has no influence on plasma pharmacokinetics.

Neither C _biophase (t) nor D _biophase (t) can be directly measured, but the drug effect can be measured. Knowing that the observed drug effect is a function of the drug concentration in the biophase, the drug effect can be predicted as (Eq. 26.8)

E f f e c t = f_{P D} [C_{p l a s m a} (t) * D_{b i o p h a s e} (t), P_{P D}, K_{e 0}]

$E f f e c t = f_{P D} [C_{p l a s m a} (t) * D_{b i o p h a s e} (t), P_{P D}, K_{e 0}]$

where f _PD is a pharmacodynamic model (typically sigmoidal in shape), P _PD represents the parameters of the pharmacodynamic model, and k _{e 0} is the rate constant for equilibration between plasma and the biophase. Nonlinear regression programs are used to find values of P _PD and k _{e 0} that best predict the time course of drug effect. This method is called loop-collapsing (see Fig. 26.7 ). Knowledge of these parameters can then be incorporated into dosing regimens that produce the desired time course of drug effect.

If a constant plasma concentration is maintained, then the time required for the biophase concentration to reach 50% of the plasma concentration (t _1/2 k _{e 0} ) can be calculated as 0.693/k _{e 0} . After a bolus dose, the time to peak biophase concentration is a function of both plasma pharmacokinetics and k _{e 0} . For drugs with a very rapid decline in plasma concentration after a bolus (e.g., adenosine with a half-life of several seconds), the effect-site concentration peaks within several seconds of the bolus, regardless of k _{e 0} . For drugs with a rapid k _{e 0} and a slow decrease in concentration after a bolus injection (e.g., pancuronium), the peak effect-site concentration is determined more by k _{e 0} than by plasma pharmacokinetics.

An accurate estimation of k _{e 0} demands an integrated pharmacokinetic-pharmacodynamic study combining rapid blood sampling with frequent measurements of drug effect, yielding an overall model for the dose-response behavior of the drug. Historically, the time constants of pharmacokinetic models and the k _{e 0} of pharmacodynamic studies were sometimes naively merged, possibly leading to inaccurate predictions of the clinical drug effect. Coppens and colleagues proved that pharmacodynamic models of BIS in children developed by using estimates of plasma propofol concentrations from published pharmacokinetic models and estimating the pharmacodynamic model do not ensure good pharmacokinetic accuracy or provide informative estimates for pharmacodynamic parameters. If no integrated pharmacokinetic-pharmacodynamic model exists, then the time to peak effect (t _peak ) after a bolus injection can be used to recalculate k _{e 0} using the pharmacokinetic model of interest to yield the correct time to peak effect. Under these circumstances, this alternative approach might lead to a more accurate prediction of the dose-response time course. However, the correct covariates for t _peak should be estimated in a specific population. A second caveat is that the time course of drug effect is specific for a given effect (e.g., cerebral drug effect as measured by a specific processed EEG). The time course of other side effects (e.g., hemodynamic effect for hypnotics) most frequently follows a different trajectory. The time to peak effect and the t _1/2 k _{e 0} for several intravenous anesthetics are listed in Table 26.1 .

TABLE 26.1

Time to Peak Effect and t _1/2 k _{e 0} After a Bolus Dose

Drug	Time to Peak Drug Effect (min) ∗	t _1/2 k _e0 (min)
Morphine	19	264
Fentanyl	3.6	4.7
Alfentanil	1.4	0.9
Sufentanil	5.6	3.0
Remifentanil	1.8	1.3
Ketamine	—	3.5
Propofol	1.6	1.7
Thiopental	1.6	1.5
Midazolam	2.8	4.0
Etomidate	2.0	1.5

k _{e 0} is the rate constant for transfer of drug from the site of drug effect to the environment.

t _1/2 k _{e 0} = 0.693/k _{e 0}

∗ Measured by electroencephalography.

All methods discussed so far incorporate k _{e 0} values calculated on the assumption that hysteresis between plasma concentration and clinical effect is explained by a delay in drug transfer between plasma and biophase and thus that anesthesia is a smooth, path- and state-independent, symmetric process. Although still commonly used, this assumption might be suboptimal. Data from animal experiments suggest that neural processes and pathways involved in anesthesia induction and recovery are different. In an animal study, measured brain drug concentrations at loss of consciousness and at return of consciousness were significantly different. If these data are confirmed, then a more complex model (e.g., one incorporating a second, serial effect-site model) might be required to depict the time course of drug effect. Several groups have investigated this hypothesis in humans, and so far the published findings are not consistent. One clinical study specifically designed to address this topic showed evidence supportive of the concept of neural inertia. Two other studies involving secondary analyses of existing data, of which one indicated supportive evidence and the other showed that neural inertia was not present in all subjects, and appeared to occur only with propofol (and not with sevoflurane), and to be present only with certain pharmacodynamic endpoints.

Indirect-Effect Models

Thus far, clinical effects that are an instantaneous function of drug concentration at the site of drug effect have been discussed, as implied by Eq. 26.8 . For example, once a hypnotic drug reaches the brain or a neuromuscular-blocking drug reaches the muscles, drug action is almost immediately observed. On the other hand, some effects are significantly more complex. For example, consider the effect of opioids on ventilation. Initially, opioids depress ventilation, and arterial tension of carbon dioxide (CO ₂ ) gradually accumulates. Yet, the accumulation of CO ₂ at normal conditions is a strong stimulant for ventilation, thereby partly counteracting the ventilatory depressant effects of opioids. Ventilatory depression is an example in which direct and indirect drug effects are incorporated. The direct effect of the opioid is to depress ventilation, and the indirect effect is to increase arterial tension of CO ₂ . Modeling the time course of opioid-induced ventilatory depression requires consideration of both components. Bouillon and colleagues developed a model of ventilatory depression that incorporates both direct and indirect effects. As is generally the case with indirect-effect models, characterizing drug-induced ventilatory depression requires a consideration of the entire time course of drug therapy, which is embodied by the following differential equation (Eq. 26.9) :

\begin{matrix} \begin{matrix} \frac{d}{d t} P a C O_{2} = k_{e 1} • {[1 - \frac{C p (t) γ}{C_{50} γ + C_{p} (t) γ}]}_{F} \\ • [\frac{P_{b i o p h a s e} C O_{2} (t)}{P_{b i o p h a s e} C O_{2} (0)}] • P a C O_{2} (t) \end{matrix} \end{matrix}

$\begin{matrix} \begin{matrix} \frac{d}{d t} P a C O_{2} = k_{e 1} • {[1 - \frac{C p (t) γ}{C_{50} γ + C_{p} (t) γ}]}_{F} \\ • [\frac{P_{b i o p h a s e} C O_{2} (t)}{P_{b i o p h a s e} C O_{2} (0)}] • P a C O_{2} (t) \end{matrix} \end{matrix}$

where partial pressure of arterial carbon dioxide (Pa co ₂ ) is arterial CO ₂ , P _biophase co ₂ is CO ₂ in the biophase (i.e., brainstem respiratory control circuits), k _el is the rate constant for the elimination of CO ₂ , C ₅₀ is the effect-site opioid concentration associated with a 50% reduction in ventilatory drive, and F is the steepness or gain of the effect of CO ₂ on ventilatory drive.

Dose Implications of the Biophase

The delay in onset of clinical effects has important clinical implications. After a bolus, the plasma concentration peaks nearly instantly and then steadily declines. The effect-site concentration starts at zero and increases over time until it equals the descending plasma concentration. The plasma concentration continues to decline. After the moment of identical concentrations, the gradient between plasma and the biophase favors removal of drug from the biophase, and the effect-site concentration decreases. The rate at which the effect-site concentration rises toward the peak after a bolus dictates how much drug must be injected into plasma to produce a given effect. For alfentanil, its rapid plasma effect-site equilibration (large k _{e 0} ) causes the effect-site concentration to rise rapidly, with a peak produced in approximately 90 seconds. At the time of the peak, approximately 60% of the alfentanil bolus has been distributed into peripheral tissues or eliminated from the body. For fentanyl, the effect-site concentration rises significantly more slowly and peaks 3 to 4 minutes after the bolus. At the time of the peak, more than 80% of the initial bolus of fentanyl has been distributed into tissues or eliminated. As a result of slower equilibration with the biophase, relatively more fentanyl than alfentanil must be injected into plasma, which makes the rate of offset of drug effect after a fentanyl bolus slower than after an alfentanil bolus.

This difference in pharmacokinetics indicates that k _{e 0} must be incorporated into dosing strategies on which rational drug selection is dependent. For rapid onset of effect, a drug with a large k _{e 0} (short t _1/2 k _{e 0} ) should be chosen. For example, for rapid sequence induction of anesthesia, alfentanil or remifentanil may be the optimal opioid because its peak effect-site concentration coincides with the likely time of endotracheal intubation. However, for a slower induction of anesthesia in which a nondepolarizing neuromuscular blocking agent is used, an opioid with a slower onset of drug effect should be selected to coincide with the peak effect of the neuromuscular blocking agent. In this case, a bolus of fentanyl or sufentanil at the time of induction may be more appropriate. The time to peak effect for the commonly used opioids is shown in Fig. 26.8 . Knowing k _{e 0} (or time to peak effect) also improves titration of the drug by identifying the time at which the clinician should make an assessment of drug effect. For example, midazolam has a slow time to peak effect, and repeat bolus doses should be spaced at least 3 to 5 minutes apart to avoid inadvertent overdosing.

Fig. 26.8, Simulated onset and time to peak effect of commonly used opioids based on their k e 0 and pharmacokinetic parameters. k e 0 , Rate constant for the transfer of drug from the site of drug effect to the environment.

An accurate k _{e 0} is also crucial during TCI titrating to a specific effect-site concentration because the initial bolus given to reach the targeted effect-site concentration depends on both the pharmacokinetics and the k _{e 0} .

Drug Potency

Single Drugs

Knowledge about adequate therapeutic drug concentration is crucial to achieve the aim of providing optimal anesthetic conditions. Therefore information on drug potency is essential. Analogous to the concept of minimum alveolar concentration (MAC), the concentration associated with a 50% likelihood of movement in response to skin incision for inhaled anesthetics is the C ₅₀ for intravenous drugs, which provides a measure of relative potency between intravenous anesthetics.

There are many ways to look upon C ₅₀ , taking into consideration whether the clinical effect is a binary or continuous effect. When considering binary effects, the C ₅₀ might be the drug concentration that prevents response (e.g., movement, hypertension, release of catecholamines) to a particular stimulus (e.g., surgical incision, endotracheal intubation, spreading of the sternum) in 50% of patients. In this case, each combination of stimulus and response may have a unique C ₅₀ . When C ₅₀ is defined as the drug concentration that produces a given response in 50% of patients, a 50% probability of response is also likely in a given patient. Defining C ₅₀ as the concentration that produces a given drug effect in 50% of individuals implicitly assumes that the effect can be achieved in all individuals. Some drugs exhibit a ceiling effect. For example, a ceiling effect may exist on the ability of opioids to suppress response to noxious stimulation. When a ceiling in drug effect exists, some patients may not exhibit the drug effect even at infinitely large doses. In this case, C ₅₀ is not the concentration that causes the drug effect in 50% of patients but is the concentration associated with the drug effect in one half of whatever fraction of patients is able to respond.

Several studies have been performed to establish appropriate concentrations of intravenous anesthetics and opioids for various clinical endpoints and the effect of drug interactions ( Table 26.2 ).

TABLE 26.2

Steady-State Concentrations for Predefined Effects

Drug	C ₅₀ for EEG Depression ∗	C ₅₀ for Incision or Painful Stimulus ^†	C ₅₀ for Loss of Consciousness ^‡	C ₅₀ for Spontaneous Ventilation ^§	C ₅₀ for Isoflurane MAC Reduction	MEAC
Alfentanil (ng/mL)	500-600	200-300	—	170-230	50	10-30
Fentanyl (ng/mL)	6-10	4-6	—	2-3	1.7	0.5-1
Sufentanil (ng/mL)	0.5-0.75	(0.3-0.4)	—	(0.15-0.2)	0.15	0.025-0.05
Remifentanil (ng/mL)	10-15	4-6	—	2-3	1.2	0.5-1
Propofol (μg/mL)	3-4	4-8	2-3	1.33	__	—
Thiopental (μg/mL)	15-20	35-40	8-16	—	—	—
Etomidate (μg/mL)	0.53	—	0.55	__	__	—
Midazolam (ng/mL)	250-350	—	125-250	—	—	—

Values in parentheses are estimated by scaling to the alfentanil C ₅₀ (see text for details).

EEG, Electroencephalography; MAC, minimum alveolar concentration; MEAC, minimum effective plasma concentration providing postoperative analgesia.

∗ C ₅₀ for depression of the EEG is the steady-state serum concentration that causes a 50% slowing of the maximal EEG, except for midazolam, in which the C ₅₀ is associated with 50% activation of the EEG.

† C ₅₀ for skin incision is the steady-state plasma concentration that prevents a somatic or autonomic response in 50% of patients.

‡ C ₅₀ for loss of consciousness is the steady-state plasma concentration for absence of a response to a verbal command in 50% of patients.

§ C ₅₀ for spontaneous ventilation is the steady-state plasma concentration associated with adequate spontaneous ventilation in 50% of patients.

Another interpretation of C ₅₀ is the concentration that produces 50% of the maximum possible physiologic response. For example, the C ₅₀ for an EEG response is the drug concentration that provides 50% depression of the maximal EEG effect. The C ₅₀ for EEG response has been measured for the opioids alfentanil, fentanyl, sufentanil, and remifentanil. It has also been determined for thiopental, etomidate, propofol, and benzodiazepines (see Table 26.2 ). Other measures such as pupillary dilation in response to a noxious stimulus and pressure algometry were used to measure opioid potency and revealed slightly different values for C ₅₀ , which indicates that observation of drug potency also depends on the applied measure of drug effect.

As mentioned, C ₅₀ can be used to compare potency among drugs. For example, Glass and colleagues determined the potency of remifentanil compared with alfentanil using ventilatory depression as the measure of opioid effect. In their study, the C ₅₀ for depression of minute ventilation was 1.17 ng/mL and 49.4 ng/mL for remifentanil and alfentanil, respectively. Using this difference in C ₅₀ , they concluded that remifentanil is approximately 40 times more potent than alfentanil.

To be entirely independent of dosing history, C ₅₀ must be determined at steady state, which is rarely possible because most anesthetic drugs do not reach steady state during a continuous infusion until many hours have passed. However, if the drug exhibits rapid equilibration between plasma and the effect site and the investigator waits long enough after starting the infusion, then this choice can be reasonably satisfactory. For example, Ausems and colleagues used a continuous infusion of alfentanil in their experiments, which quickly equilibrated. They also recorded their measurements after the effect-site concentration had equilibrated with plasma.

A second alternative to performing a true steady-state experiment is to use mathematic modeling to calculate the effect-site concentrations of drug at the time of measurement, as proposed by Hull and colleagues and Sheiner and colleagues. The relationship between effect-site and plasma concentrations is graphically represented in Fig. 26.5 and mathematically in Eq. 26.6 . Calculating effect-site concentrations is the same as attempting to determine the steady-state plasma concentrations that produce the observed drug effect. When the C ₅₀ reflects effect-site concentrations, it is represented as Ce ₅₀ to distinguish it from values of C ₅₀ that are based on plasma concentrations, which are then termed Cp ₅₀ . However, the distinction is artificial. In both cases, C ₅₀ is intended to represent the steady-state plasma drug concentration associated with a given drug effect.

A third alternative to performing a steady-state experiment is to establish a pseudo–steady state with the use of computer-controlled drug delivery. By this term we mean that a steady state is assumed to exist, because a varying rate drug infusion is being administered with the intent of achieving stable plasma concentrations. This method has become the state of the art for determining the C ₅₀ for anesthetic drugs, and many of the C ₅₀ values referenced earlier were determined at pseudo–steady state with the use of computer-controlled drug delivery. Commonly, two or more measurements of the plasma concentration during pseudo–steady state conditions are performed to verify whether this is indeed the case. Typically, maintaining a constant plasma steady-state concentration for four to five plasma effect-site equilibration half-lives (e.g., 10-15 minutes for fentanyl) is required. Such a long delay is not necessarily needed when computer-controlled drug delivery is used.

Effect-compartment TCIs can be used to target the concentration at the effect site rather than the plasma concentration and thereby rapidly establishing plasma–effect site equilibration. For example, Kodaka and associates reported predicted values of propofol effect-site concentration C ₅₀ between 3.1 μg/mL and 4.3 μg/mL for insertion of various laryngeal airway masks, depending on the type of laryngeal mask. Cortinez and associates used TCI to determine the C ₅₀ of remifentanil and fentanyl for accurate pain relief during extracorporeal shock wave lithotripsy in relation with possible side effects and found that remifentanil and fentanyl C ₅₀ were 2.8 ng/mL and 2.9 ng/mL, respectively. At C ₅₀ , the probability of having a respiratory rate less than 10 breaths per minute was 4% for remifentanil and 56% for fentanyl.

Likewise, TCI has been used to estimate the following C ₅₀ values for dexmedetomidine: the C ₅₀ for half-maximal effect on the BIS (BIS ∼48) was 2.6 ng/mL, for half-maximal effect on the Modified Observer’s Assessment of Alertness/Sedation (MOAA/S) scale was 0.438 ng/mL, for half-maximal hypotensive effect was 0.36 ng/mL, and the C ₅₀ for half-maximal hypertensive effects was 1.6 ng/mL.

Thus there are several ways to establish C ₅₀ in terms of steady-state concentrations. C ₅₀ can be estimated through mathematic effect-site modeling or can be experimentally measured with the use of computer-controlled drug delivery to establish a pseudo–steady state quickly. Either way, when performing studies to define the concentration-effect relationship, equilibrium must exist or be modeled between the biophase (the site of effect) and plasma or blood (where the concentration is actually measured).

When C ₅₀ is defined in terms of the concentration associated with a response in one half of a population, that same C ₅₀ is the concentration associated with a 50% probability of response in a typical individual. However, individual patients are not typical individuals but rather will have their own value for C ₅₀ . Expressed in clinical terms, different patients have different anesthetic requirements for the same stimulus. For example, the minimal effective analgesic concentration of fentanyl is 0.6 ng/mL, but it varies among patients from 0.2 to 2.0 ng/mL. The minimal effective analgesic concentrations of alfentanil and sufentanil similarly vary among patients by a factor of 5 to 10. This range encompasses both variability in the intensity of the stimulus and variability of the individual patient.

One factor known to be responsible for this interindividual pharmacodynamic variability is the patient age. Thus with the pharmacodynamic for propofol developed by Eleveld and colleagues, and based on pharmacokinetic-pharmacodynamic data from a large number of patients and volunteers, the estimated effect-site propofol concentrations required for a BIS of 47 for patients whose ages are 20, 40, and 70 years, are 3.5, 3.1, and 2.6 ug/mL, respectively.

Although age has a strong effect on the C ₅₀ , it does not explain all the variability. This wide range reflects the clinical reality that must be accounted for when dosing regimens are designed. Because of this variability, intravenous anesthetics should be titrated to each patient’s unique anesthetic requirement for the given stimulus.

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