Introduction
Although we can be sure that a large bolus dose of an intravenous anaesthetic agent will cause all patients to lose consciousness and a large dose of muscle relaxant will paralyse all patients, the exact time of onset and offset of effect and side-effects is less definite. This is due to the large interindividual variability in the pharmacokinetics and pharmacodynamics among patients. Other specialties often monitor drug effects and adjust drug doses over days or even weeks, whereas anaesthesiologists sometimes find themselves waiting impatiently for a drug to act, even if the delay is only a matter of minutes or sometimes seconds. On the other hand, if it takes 15 min longer than expected for our patients to awaken and breathe following a long general anaesthetic, our surgical colleagues begin to wonder about our skills as anaesthesiologists. A thorough understanding of the patient factors and drug factors that determine the onset and offset of drug effect or side-effect is central to the practice of anaesthesia.

Anaesthesiologists develop an intuitive understanding of ‘onset’ and ‘offset’ of drug effect. Many of the drug effects we observe are binary — yes or no — parameters. For these binary response parameters, the onset and offset of drug effect is usually defined as the time to a predefined event, such as the time to loss or return of eyelash reflex or verbal response. In contrast, continuous measures of drug effect are available for some drugs. For example, we can measure the depression of the force of muscle contraction for muscle relaxants and the electroencephalographic changes for the hypnotic effects of intravenous and inhaled anaesthetic agents. Based on such continuous measures of drug effect, we can develop somewhat arbitrary endpoints to define precisely the onset and offset of effect. For muscle relaxants, the time to 95% twitch depression has been used as a descriptor of onset of drug effect [^{1} ]. Another descriptor of onset is the time to ‘maximum effect’. Note, however, that the time to 100% twitch depression after a high dose of a muscle relaxant is different from the time to the peak effect of a dose resulting in submaximum effect [^{1,2} ]. We regularly make use of this dose dependency when we give a larger dose of a muscle relaxant to achieve a more rapid onset of effect, e.g. a 100% block, and appreciate all too well that the larger dose results in a longer duration of effect.

The terms ‘onset time’ and ‘recovery time’ have been defined clearly in published guidelines for investigations with muscle relaxants [^{1} ]. However, in the context of this article we will use the general expressions ‘speed of onset’ and ‘speed of offset’ to describe the time course of a drug's effect. When drugs of the same group differ little in the balance of their desirable effects vs. side-effects profile, the rational selection of the most appropriate drug is, in large part, dependent on the clinical relevance of the available descriptors of onset and offset of drug effect. Fortunately, over the past 10 years several new descriptors have evolved. A brief description of pharmacokinetic and pharmacodynamic modelling is presented prior to comparison of the traditional predictors of onset and offset of drug effect with the new predictors derived from pharmacokinetic–pharmacodynamic models.

Pharmacokinetic and pharmacodynamic models
The relation between dose and effect can be broken down into the relation between dose and effect-site concentration (pharmacokinetics ) and the relation between concentration and effect (pharmacodynamics). Several different sets of parameters are commonly used to describe the relation between dose and concentration. A three-compartment model can be described by three volumes and three clearances (V _{1} , V _{2} , V _{3} , Cl _{1} , Cl _{2} , Cl _{3} ), or by the central volume and five first-order rate constants (V _{1} , k _{10} , k _{12} , k _{13} , k _{21} , k _{31} ), or by three coefficients and three hybrid rate constants of a tri-exponential equation (A _{1} , A _{2} , A _{3} , λ _{1} , λ _{2} , λ _{3} ), or by a variety of other combinations, e.g.MATH where the half-lives are equal to the natural logarithm of two divided by the respective hybrid rate constants. All the other parameters can be calculated with any set of six independent parameters and appropriate equations.

Under non-steady state conditions, an apparent hysteresis is observed between the plasma drug concentration and the drug effect. This is because the plasma is not the site of drug effect. This hysteresis can be ‘removed’ by adding an effect-site to the pharmacokinetic model [^{3,4} ]. The rate of equilibration between the central compartment and the effect-site is characterized by a rate constant, commonly called the k _{e0} . For many drugs used by anaesthesiologists, the relation between effect-site concentration (C _{e} ) and drug effect is direct, and is often described by the sigmoidal E _{max} (Hill) model. Two important parameters of this model are C _{50} , which is the concentration at the half maximum effect, and γ , which describes the gradient of the concentration–response relation. There are, however, many examples where the relation between drug concentration and drug effect is indirect, i.e. the drug stimulates or inhibits the production or loss of endogenous substances that mediate the drug effect [^{5} ].

Pharmacokinetic–pharmacodynamic models can describe the predicted time course of plasma concentration, effect-site concentration and effect (for continuous measures of drug effect) or probability of effect (for binary measures of drug effect) after any dosing scheme. Computer simulations based on pharmacokinetic–pharmacodynamic models have been used to develop new descriptors of onset and offset of drug effect.

Predictors of onset of drug effect
The time after administration of a bolus dose of a drug to a predefined endpoint is always dose-dependent. This dose dependency of onset time was of particular interest with vecuronium because high doses were well-tolerated [^{6} ]. However, in order to compare the onset of effect for drugs of the same group we must find some way to normalize the doses to permit fair comparisons. The doses could be normalized so that they result in the same duration of effect, or we could consider normalizing the doses so that they cause the side-effects of a similar magnitude (e.g. haemodynamic depression). Based on these normalized doses, we can then compare the time to onset of effect, as determined by the time to achieve a predetermined endpoint. Although measurement of ‘onset times’ is helpful when determining optimal doses of a given drug for a certain purpose, it is not a particularly useful parameter for fair comparison of ‘speed of onset’ of different drugs.

It is often assumed that a drug with a higher k _{e0} (i.e. a shorter T _{1/2} k _{e0} ) will have a more rapid onset than a drug with a lower k _{e0} . This is not necessarily true, as illustrated in Figure 1 . The figure shows the predicted plasma and effect-site concentrations following an intravenous bolus dose for two drugs with the same k _{e0} , but with different plasma pharmacokinetic profiles. The initial high plasma concentrations decrease rapidly due to distribution and elimination processes. The changing concentration gradient between the plasma and the effect-site causes the effect-site concentrations to increase, then to decrease. The plasma and effect-site concentrations are equal at T _{peak} , the time of the peak effect-site concentration .

Figure 1.:
Time course of plasma concentration and effect-site concentration (at time 0 the concentration is 0) of two drugs. The ke0 of the two drugs are identical, but the plasma concentration of the one drug (solid line) decreases faster and intersects the curve of the effect-site concentration earlier. The faster onset is due to differences in the plasma pharmacokinetic profile.

The time to the peak effect-site concentration is a dose-independent pharmacokinetic parameter. It is, however, dependent on both the plasma pharmacokinetic parameter and the rate of equilibration between the plasma and the effect-site. Thus, T _{peak} is an important descriptor of onset of drug effect, which is particularly useful when comparing drugs of the same group. However, the T _{peak} coincides with only the time of peak effect when a submaximum dose is given. With ‘supramaximum’ doses, the maximum effect will occur prior to T _{peak} . However, even if a drug is administered by infusion and supramaximum doses have been administered, it is still possible to determine T _{peak} by simulations based on the pharmacokinetic–pharmacodynamic model [^{7–9} ].

Neither the central volume nor the volume of distribution at steady state is a useful parameter for calculating the bolus dose required to achieve a desired effect. In contrast, the T _{peak} concept has been used to calculate optimal initial bolus doses [^{10} ]. The volume of distribution (Vd _{pe} ) can be calculated at the time of the peak effect-site concentration following an intravenous bolus dose as:MATH where V _{1} is the central volume of distribution, C _{0} is the concentration at the T = 0 (as predicted by the pharmacokinetic model) and C _{peak} the predicted effect-site concentration at the time of the peak effect-site concentration . Using Vd _{pe} , we can calculate the loading dose to achieve the effect-site concentration (C _{peak} ) associated with a desired effect at T _{peak} as:MATH

In this way, we can compare drugs by calculating the time to a specified effect using the full pharmacokinetic–pharmacodynamic model.

Predictors of offset of drug effect
The most commonly quoted and traditional descriptor of drug offset is the drug's ‘terminal’ half-life [^{11} ], which, using the notation described earlier, can be calculated as log_{e} (2)/λ _{3} . Unfortunately, if we know only one (or even all three) of the half-lives, we cannot calculate the rate of decline of the drug concentrations; for this we need a computer and all six parameters. However, some insight into the relative contribution of each half-life can be obtained relatively easily by calculating A _{1} /λ _{1} , A _{2} /λ _{2} and A _{3} /λ _{3} as percentages of the total area under the curve (A _{1} /λ _{1} +A _{2} /λ _{2} +A _{3} /λ _{3} ) [^{12} ].

Sometimes it is difficult to determine whether the concentration vs. time data of a drug should be described by one-, two-, or three-exponential terms. Non-compartmental analysis may be helpful to overcome such problems, as well as the ‘terminal’ half-life not representing the half-life in the phase where the majority of the drug is eliminated. The clearance (Cl ), the volume of distribution at steady state (Vd _{ss} ) and mean residence time can be calculated based on methods that calculate the area under the concentration time curve and area under the first moment curve curve. However, mean residence time is rarely reported in the anaesthesia literature. Its utility (or lack of it) in comparing the offset of effect for drugs of the same group has not been reported.

Youngs and Shafer, using computer simulations, found that no single parameter of the pharmacokinetic model predicted fast recovery [^{13} ]. They showed that the implications of a change in a single parameter on the time course of onset and offset of drug effect could only be predicted if all the other parameters of the pharmacokinetic model remained unchanged. For example, if all the other parameters of the model were fixed, a small V _{1} and or large Cl _{1} predicted rapid recovery. However, a large Cl _{1} can be offset completely by other parameters of the pharmacokinetic model. Because no two drugs differ by only one parameter, comparison is not possible on a parameter-by-parameter basis.

Figure 2 shows the time course of the concentration of fentanyl after a bolus dose, after a 60-min infusion and after a 120-min infusion. Although the three half-lives are unchanged (they are ‘fixed’ parameters of the pharmacokinetic model), the time for the propofol concentration to decrease by one-half is longer when the duration of infusion is longer. Shafer and Varvel were the first to construct recovery curves based on computer simulations in order to enable rational opioid selection [^{14} ]. This is explained by the concept of the ‘context-sensitive half-time ’ introduced by Hughes and colleagues [^{15} ]. Figure 3 illustrates a family of ‘context-sensitive decrement time’ curves for fentanyl.

Figure 2.:
Time course of the plasma concentration of fentanyl after a bolus injection (solid line) after a 60-min, target-controlled infusion (dashed line) and a 120-min target-controlled infusion (dotted line). The time for a given decrease of the concentration increases with the duration of the infusion. A 50% decrease of the concentration (indicated by the horizontal line) after a bolus takes 1.2 min and, after a 60-min infusion, 27 min. After a 120-min infusion, it takes more than 60 min for the concentration to decrease by 50%.

Figure 3.:
Decrement time curves for fentanyl. The x-axis represents the duration of the target-controlled infusion and the y-axis the time required for a given decrement in concentration. The 50% decrement time is the same as the context-sensitive half-time . Although the context-sensitive half-time after a 120-min, target-controlled infusion is still reasonable, the 70% decrement is very long. Depending on the drug and clinical situation, a decrease of the concentration by more than 50% may be required. Because the gradient of the concentration–response relation may be different for different drugs, when comparing the offset of different drugs, comparisons of drugs based on the same percentage decrement may be misleading.

A drug described by one compartment model has only one ‘half-life.’ It can be called the ‘terminal’ half-life. It can also be called the ‘elimination’ half-life, because it is the time to eliminate one-half of the drug from the body (this is not true for a drug that has pharmacokinetics that exhibit more than one half-life). It is numerically the same as the ‘context-sensitive half-time ’, which will not change with the duration of infusion. For a drug described by a one-compartment model, the plasma concentration will decrease by 50% every half-life. Thus, for a one-compartment model the time for the concentration to decrease by 75% will be exactly twice as long as the time for the concentration to decrease by 50% (two half-lives). This is not true for a drug described by a multicompartment model.

Bailey generalized the concept of context-sensitive half-time to ‘relevant decrement times’, which incorporate the pharmacodynamic model into the simulations [^{16} ]. Bailey considered the relevant decrement time as the time required for the drug concentration to decrease from the C _{90} (considered the concentration to maintain anaesthesia) to the C _{50} (considered the concentration required for recovery). The magnitude of the difference between these two concentrations is influenced by the gradient of the concentration-effect relation. Bailey has also evaluated the recovery characteristics of volatile anaesthetics using these concepts [^{17} ].

Figure 4 shows effect vs. effect-site concentration for two different drugs, which differ with regard to the gradient of this relation. If we dose intraoperatively at the 90% level and clinical recovery from the drug effect is to be expected at a 10% level, the required concentration decrement for recovery is different. Therefore, in order to compare the offset of effect of the two drugs, we would generate curves showing the relevant decrement times for one drug based on its 90% decrement curve and for the other drug based on its 50% decrement curve.

Figure 4.:
Two drugs with different gradient of the concentration vs. effect relation. For a decrease of the effect from the 0.9 to the 0.1 level (i.e. 90% to 10% of maximum effect), the concentration must decrease by 90% for the drug with the shallower gradient (top panel) and by only 50% for the drug with the steeper gradient.

Bailey extended these concepts further by evaluating the duration of drug effect when drug effect is assessed in a binary, response/no response fashion [^{16,18} ]. The mean effect time is the area under the probability of drug effect curve as a function of time after drug administration is discontinued, assuming that the probability of responsiveness to surgical stimulation was reduced to 10% (C _{90} ) during the infusion. Bailey calculated mean effect time using the population average pharmacokinetic values and the C _{90} and γ values. Strictly, it should be calculated as an average over the distribution of individual pharmacokinetic, C _{90} , and γ values. Although these are usually not reported, one approach that has not yet been evaluated would be to perform computer simulations based on estimates of interindividual variability obtained by population modelling methods. However, as noted by Bailey, the pharmacodynamic analysis of binary data is assessed mostly by logistic regression of data pooled from multiple patients. With this methodology, the gradient of the pharmacodynamic model is often relatively low, reflecting high interindividual variability. If the slope of the individual concentration–response relation is steep, and if the anaesthesiologist has titrated the dose to effect, the time to offset of drug effect may be much shorter than that predicted by the mean effect time.

Conclusion
Traditional predictors of onset, such as the time to a predetermined endpoint, lack general applicability. Although they may be helpful in comparing different doses of the same drug, they are not particularly useful when comparing different drugs of the same group, even if the doses are normalized to some clinical endpoint. The implications of a change in a single parameter (e.g. V _{1} , Cl _{1} , k _{e0} ) on the time course of onset and offset of drug effect can be predicted only if all the other parameters of the pharmacokinetic model remain unchanged. Because no two drugs differ by only one parameter, comparison is not possible on a parameter-by-parameter basis. The time to peak effect-site concentration is an informative dose-independent descriptor of the onset of drug effect following an intravenous bolus dose. The relevant decrement time (for continuous measures of drug effect) and mean effect time (for binary measures of drug effect) build upon the context-sensitive half-time concept, by considering the time required for the concentrations to decrease from one clinically relevant level of drug effect to another. These newer descriptors facilitate the process of rational drug selection [^{17,19} ].

Acknowledgments
Supported in part by the Swiss National Science Foundation (grant no. 32-51028.97).

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