12

Pharmacokinetic-Pharmacodynamic (PK-PD) Relationships of Opioid Analgesics

One of the goals of current pain research is improved analgesic dosing. The intensity of the pharmacodynamic (PD) effects of opioids such as methadone, morphine, fentanyl, alfentanil, and sufentanil is a function of their plasma concentration (1,3-6,9-11,13,14). The quantitative relationship between drug dose and biofluid opioid concentration can often be described by an equation (a pharmacokinetic [PK] model) shown schematically in Figure 1 . PK analysis can provide information that can be used to allow a more rational choice of drug dose, route, and frequency of administration (14). This information can be useful in adjusting dosage to compensate for changes in drug disposition that can result from alterations in the clinical status of the patient (8). The practical clinical utility of the PK approach depends, of course, on the extent to which the PK model yields concentrations that are capable of producing the desired intensity of effect (3,4,6). Therefore, implicit in PK studies aimed at optimizing drug dosage is a consideration of pharmacodynamics. Pharmacodynamics defines the relationship between drug concentrations and the intensity of the therapeutic and/or toxic pharmacological effects. This drug concentration-effect relationship can also be described by an equation (a PD model) as shown in Figure 1 .

During the past several years approaches have been developed that allow integration of PK and PD data (PK-PD modeling). Although the advantages of this approach for defining therapeutic concentrations are obvious, it may be of particular utility for analgesic drugs where patient factors, such as opioid tolerance and the intensity of the pain stimulus, can result in large interindividual variation and approaches are required to define individual PK-PD relationships (9).

A number of PD models have been described (2,7,16-18), and it is useful to test PK-PD data sets with more than one of these models. Figure 2 shows a representation of the concentration-effect relationship that is predicted for a receptor-mediated effect such as opioid analgesia. It is a sigmoid curve that defines the plasma opioid concentration-effect relationship from 0 (no effect) to 1.0 (the maximum effect). This type of concentration-effect relationship is described by a PD model that Holford and Sheiner (7) have called the Sigmoid E_max model. The model equation is:

where FR is the fraction of maximum effect, C_e is the concentration of drug at the effect site, C_ss50 is the steady-state concentration of drug at half maximum effect and a measure of individual sensitivity to the drug, and g is a slope factor that determines the steepness of the concentration-effect curve.

Both FR and C_e can be considered to be experimentally measurable quantities. FR is the observed effect (e.g., pain relief, Visual Analogue Scale [VAS]) transformed to a scale that extends from 0 to 1.0, and C_e can be obtained from the PK model that describes the concentration-time relationship (see below). The model equation above also contains two constants or PD parameters, g and C_ss50. These parameters are estimated from analysis of the experimental data by an iterative technique, as is usual for the PK parameters (11). Thus, if g and C_ss50 could be estimated from each patient's data set, a complete plasma concentration-effect curve could be generated using the model equation. These curves could be used to determine the plasma drug concentrations above which the majority of patients respond and those concentrations above which the majority develop limiting side effects (i.e., a therapeutic range). The onset and duration of effect can be described in terms of the time required to reach and leave the therapeutic range (18). The acquisition of this type of data for clinical populations is certainly one of the goals of PK-PD modeling of analgesics.

The objective of PK-PD modeling is to link the dose-concentration relationship described by a PK model to the concentration-effect relationship obtained by use of a PD model. This may be accomplished directly by equating the concentration predicted by a PK model to the drug concentration at the effect site (C_e in the equation). This direct approach requires that there is rapid equilibration between drug in the biofluid ``compartment'' being sampled (e.g., plasma) and the effect site concentration (e.g., CNS opioid receptors). This approach can be used to describe the PK-PD relationships for methadone in cancer patients receiving an intravenous infusion (11). However, a more common observation is that there is a lag between the time course of effect and plasma drug concentration, giving rise to a hysteresis loop in the concentration-effect plot. Figure 3 shows a plot of plasma morphine concentration-effect data for a cancer patient during and following an intravenous infusion of morphine. Note that when the infusion is discontinued and plasma morphine concentrations decrease, the intensity of the analgesic effect lags behind the change in plasma drug concentration. This time-dependent lag of effect behind a change in plasma drug concentration is called hysteresis and has been reported for many drugs (7,18). This type of hysteresis may result from an equilibrium delay between plasma drug concentration and effect-site concentration or as a result of the rapid formation and accumulation of an active metabolite. Both of these mechanisms may contribute to the hysteresis seen with morphine (12).

The practical consequence of the demonstration of hysteresis using a concentration-effect plot is that the measured plasma drug concentration (C_p) must be adjusted to reflect the temporal difference between a change in plasma drug concentration and a change in effect before it can be used in the PD model. This ``adjustment'' is accomplished by using the time course of the effect itself to define the rate of drug movement into a distinct ``effect compartment'' of negligible volume. The theoretical aspects of effect compartment models are discussed elsewhere (2,7,10,16-18). Figure 4 shows a representation of a PK-PD model with an effect compartment. The plasma drug concentrations are modeled by whatever PK model is appropriate to characterize the plasma drug concentration-time profile. The hypothetical effect compartment (E) is modeled as an additional exponential function. The rate constant for drug removal from the effect compartment is designated k_e ( Figure 4) and will precisely characterize the temporal aspects of equilibration between C_p and effect. k_e is a model parameter determined by nonlinear regression that will characterize the degree of hysteresis (2,7,16). Thus, by use of the observed C_p values and the estimates of k_e obtained from the model ( Figure 4 ), the effect-site concentrations (C_e) are predicted and C_e values are used by the PD model to estimate the desired PD parameters C_ss50 and g ( Figure 4).

An example of this approach is given in Figure 5 , which shows the simultaneous fit of the plasma morphine concentration-time profile and analgesic effect data (pain relief, VAS) to a PK-PD model with an effect compartment (10). Table 1 lists the mean PD parameters estimated from the curve-fitting procedure. Good agreement was found between parameters obtained by use of categorical and visual analogue measures of analgesia. The k_e parameter can be converted to a half-life value (Table 1) to provide a more direct illustration of the hysteresis lag. Thus, when plasma morphine concentrations are altered, the half-time for the analgesic effect to come into equilibrium with this change averages 15 to 20 min. The intrinsic sensitivity to morphine's analgesic effects in this group of patients (10) is given by the C_ss50 values, which averaged 0.052 and 0.063 ng/mL (Table 1). These values can be compared with C_ss50 values obtained in the same manner for other opioids (11). Estimating these PD parameters provides a quantitative method for determining whether patient variables such as age, sex, disease, or organ dysfunction alter drug effects by affecting kinetics, dynamics, or both. If an active metabolite (e.g., morphine-6-glucuronide) is found to participate in the analgesic effect observed in these patients, then the PK-PD model can be adjusted to estimate the contribution of both parent and metabolite.

Using electroencephalographic (EEG) stage as a measure of opioid effect, Scott and Stanski (13) found that the dose requirement for fentanyl or alfentanil decreased approximately 50% from age 20 to age 89. No age-related changes in PK parameters were found. However, brain opioid sensitivity as estimated by the C_ss50 required to produce EEG changes decreased significantly with age, as shown by regression analysis of the C_ss50 values for fentanyl or alfentanil, against age. This PK-PD approach led to the conclusion that the decreased dose requirement for these opioids in the elderly is due to a PD difference. Although this study was not designed to determine the mechanism of altered brain sensitivity to opioids with aging, it does demonstrate the power of PK-PD analysis to define the source of interindividual variation in response (17).

Shafer (14) has used PK-PD analysis to simulate plasma (C_p) and effect site (C_e) opioid concentrations after a bolus injection, brief infusion, or a prolonged infusion of fentanyl, alfentanil, or sufentanil used during anesthesia. The simulation compared the onset, duration, and rate of recovery of each of these opioids after termination of the infusion. The PK-PD analysis suggests that alfentanil is best used for operations longer than 6 to 8 hr when a rapid decrease in effect site opioid concentration is desired to permit rapid recovery after discontinuation of the infusion. This approach provides a rational basis for designing dosing regimens and for selecting among opioids for a particular application (14).

The preceding discussion has focused on the use of parametric PK-PD models, since this approach has been used successfully with analgesic data. One alternate approach utilizes a nonparametric method in which no a priori judgment is made about the form of the PK and or PD models and the value of k_e is evaluated until the area within the hysteresis loop is minimized (collapsed) (15,17,19).

Since hysteresis is a result of a dysequilibrium, it is most prominent when plasma drug concentrations are rapidly changing. At steady state, no dysequilibrium exists; rather, a constant proportionality must exist between plasma drug concentration and active site concentration. Therefore, if all the measurements of effect are made at steady state, the problem created by hysteresis is eliminated. Hill and colleagues (4,6) have used a method that employs a computer-controlled infusion pump and an algorithm derived from individual subject PK parameters to preselected target plasma opioid values. They have demonstrated concentration-effect relationships for analgesia (experimental pain) and ventilatory responses (4). Hill et al. (5) have extended their PK-tailoring models to pain patients. In bone marrow transplant patients, they have demonstrated that a system that allows patients to adjust their plasma morphine concentrations around a preselected target plasma concentration (pharmacokinetically based patient-controlled analgesia [PKPCA]) can yield average plasma morphine concentrations very close to the predicted target values and provide relief of pain due to oral mucositis.

Chay et al. (1) have examined the steady-state PK-PD relationship for sedation during morphine infusions in neonates. Their quantal data could be analyzed by use of a Sigmoid E_max model (see model equation) to yield an estimate of 125 ng/mL for the concentration of morphine required to produce adequate sedation in 50% of these patients, whereas concentrations above 300 ng/mL may be associated with adverse effects (1).

SUMMARY

Although the analysis of simultaneously collected PK and PD analgesic data is still in its infancy, it is clear that this approach provides a unique insight that, in addition to providing the most complete quantitative description of the clinical pharmacology of an analgesic, may also serve to streamline the drug development process. For example, PK-PD modeling permits predictions of concentration-effect relationships that extend beyond the observed data, allowing some extrapolation of the therapeutic dose range. PK-PD modeling can provide an efficient method for defining the relative contribution of parent and metabolites, or each stereoisomer, to the therapeutic and/or undesirable effects of an analgesic. This approach also has the potential to provide methods for quantitating the development of tolerance to the analgesic and other effects of opioids. Although the utility of any particular PK or PD analytical method discussed in this chapter needs to be validated by rigorous application to each analgesic and modified as required, the concept that analgesic study design must include concurrent collection and analysis of PK and PD data is not a hope of the future, but a necessity for the present.

ACKNOWLEDGMENTS

My research is supported in part by National Cancer Institute grant CA-32897, National Institutes on Drug Abuse grants DA-01457 and DA-05130, and a Bristol-Myers Squibb Pain Research grant (to Dr. K. Foley). This chapter is dedicated to a friend and colleague, the late Dr. Harlan Hill, who made important contributions to our understanding of the PK-PD relationships of opioids and to the improvement of the management of pain.

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