The behavior of a drug that does not undergo redistribution can be described by a single compartment mathematical model. On injection, the drug distributes uniformly throughout a single volume, V, and the drug concentration in this compartment is the same as the plasma concentration. After a single bolus or an infusion, the drug concentration will decline because of metabolism or elimination, as described by the following equation:
where C _p(t) is the plasma concentration at time t, C _p(0) is the initial plasma concentration, k _el is the elimination rate constant, and t = 0 is the time of the bolus or the time at which the infusion ceased. Clearance (mL/h) can be calculated from k _el as follows:
If the relationship between drug concentration and time is plotted on linear axes then the exponential decline results in a curved graph (Fig. 31.2). If however, a semilogarithmic graph is used (i.e., the logarithm of the concentration is plotted) a straight line will result. Figure 31.3 shows the relationship between log_e C _p(t) and time.

As shown the elimination rate constant can be calculated from the slope of the line in Fig. 31.3. If the natural logarithm (log_e or “ln”) of the drug concentration is plotted against time, then the slope is simply equal to k _el. As there is only one rate constant influencing the rate of decline in drug concentration, the decline in plasma concentrations has a constant t _1/2 that can be calculated from k _el as shown previously.

The pharmacokinetics of most anesthetic drugs are best described by three compartment models. Each model describes the number of compartments, and their volumes, the rate of drug metabolism or elimination, and the rate of transfer of drug between the different compartments. The concept is summarized in Fig. 31.1.

By convention, the compartment into which the drug is injected is called the central compartment (V1 or Vc), which may be thought of as including the blood volume, although it can be larger than the blood volume. It is sometimes referred to as the initial volume of distribution. Elimination of active drug by metabolism usually occurs from within this compartment (as in the case of hepatic or renal metabolism). The rate of elimination is described interchangeably by a rate constant (k ₁₀) or a clearance (Clearance = k ₁₀ × V1). The second compartment, V2, is referred to as the “rapid redistribution” compartment since drug concentrations in V2 equilibrate rapidly with those in the central compartment. The rate constants k ₁₂ and k ₂₁ are used to describe the rate of drug transfer from V1 to V2 and from V2 to V1, respectively. Fast redistribution clearance, “Clearance 2,” can be calculated as:
The third compartment, V3, is often referred to as the “slow” compartment (because there is rather slower drug distribution between V1 and V3). Here the rate constants k₁₃ and k₃₁ are used to describe the rate of drug transfer from V1 to V3 and from V3 to V1, respectively. Slow redistribution clearance, “Clearance 3,” can be calculated as:
The second and third compartments are sometimes referred to as the “vessel rich” and “vessel poor” compartments, respectively, but these terms are best avoided since they encourage the false impression that these compartments represent distinct anatomical or physiological entities. The sum of V1, V2, and V3 gives the “volume of distribution at steady state,” Vd_ss.

The site of action of the anesthetic agents is, of course, not in the vascular system, but in the brain at a vaguely defined “effect site.” Thus, many models now also include the effect site as a fourth compartment, with the rate constant k _eo being used to describe the rate of equilibration between the central and effect-site compartments.

For a drug showing three compartment kinetics (such as propofol), the change in concentrations after a bolus or infusion cannot be described by a single rate constant or half-life. The decline in plasma concentration is more complex because it is influenced by several simultaneous exponential processes, each with a different rate constant, so that the time required for the concentration to fall by 50 % (or any other proportion) changes over time. Figure 31.4 shows a typical curve of the relationship between blood concentration and time after a single bolus dose of an anesthetic drug. The time course of changes in plasma concentration shown in Fig. 31.4 can be described mathematically as the sum of three exponential processes as follows:

where A, B, C, α(alpha), β(beta), and γ(gamma) are constants. As can be seen in Fig. 31.4, in the early phase after a bolus dose, the plasma concentration falls rapidly, being mostly influenced by rapid redistribution (described by a rate constant α[alpha]). Later on the rate of decline in plasma concentrations is influenced mostly by redistribution to less well-perfused tissues (described by a rate constant β[1]). Eventually the predominant factor is elimination (rate constant γ[gamma]). From these parameters the time-honored redistribution and elimination half-lives can be calculated.

During and after administration of repeated bolus doses or infusions, the changes in drug concentrations vary in a complex matter since they are influenced by several simultaneous exponential processes, and the relative contributions of the different processes change for most anesthetic drugs as the duration of infusion increases. These factors make it difficult to predict drug concentrations without the assistance of computer programs.

The concept of “context-sensitive half-time” (CSHT) has been introduced as a simple metric that provides a summary of the interplay of time and the different half-lives after an infusion [31].

It describes the time taken for blood concentration of a drug to fall by 50 % after the end of an infusion of a specified duration—the context is thus the duration of infusion. The influence of duration of infusion on CSHT indicates the degree of drug accumulation and the balance between redistribution and metabolism/elimination. This metric only describes the time taken for the first decline of 50 %—the time taken for subsequent 50 % falls will be different. Also, it does not necessarily describe when clinical effects will cease, since these depend on the initial concentration, and pharmacodynamic factors such as the sensitivity of the patient to the drug. Nonetheless, it gives the physician a useful indicator of the rate at which drug concentrations will decline after an infusion and an indication of the influence of duration of infusion.

Although it is possible to maintain sedation or anesthesia with repeated boluses of an intravenous sedative agent, this is far from ideal. Firstly, stable levels of sedation are not possible since the blood and effect-site concentrations will be constantly either rising or falling. If the bolus size is too big, the patient state will oscillate from excessive sedation/anesthesia, with the attendant risks, to inadequate sedation. Secondly, it is difficult to judge the dose required to produce adequate, but not excessive blood concentrations. Finally, it is also difficult to judge the required interval between doses. Figure 31.6 shows the estimated blood concentrations arising from repeated 40 mg boluses of propofol administered to a 20 kg child. In these simulations, a bolus was administered each time the estimated concentration fell to 2 μ(mu)g/mL. As can be seen, as drug accumulates in peripheral tissues, the rate of decline in blood concentration after successive doses gradually decreases, resulting in an increase in the interval between doses.

Typically, blood concentrations of the order of 2–3 μ(mu)g/mL are required for sedation in children. Naturally the concentration required is influenced by multiple other factors such as co-administered drugs. Thus, it is not surprising that after cardiac surgery, Murray et al. found that the mean measured propofol concentration at recovery of consciousness was only 0.97 μ(mu)g/mL [42], whereas Rigouzzo et al. found that the EC50 (of measured blood propofol concentration at steady state) associated with loss of consciousness in healthy children was 4.0 μg/mL [43].

A commonly used deep sedation regimen for is an initial bolus of 2 mg/kg followed by an infusion at 10 mg/kg/h (in children <1 year of age, higher doses may be required, e.g., an initial bolus of 3 mg/kg and higher initial infusion rates). Figure 31.7 shows a simulation of the regimen, with the concentrations estimated by the Paedfusor model. At about 10 min after the initial bolus the blood concentrations reach a nadir of ~2.5 μ(mu)g/mL. If the infusion rate is kept constant at 10 mg/kg/h, the blood and effect-site concentrations, and clinical effect will gradually increase (reaching ~5 μ[mu]g/mL after several hours), which is why downward titration of the infusion rate is commonly required.

In a recent study, Koroglu and colleagues administered a 3 mg/kg bolus followed by infusions of 10–15 μ(mu)g/kg/min (i.e., 6–9 mg/kg/h) of propofol to 30 children between 1 and 7 years of age for sedation during MRI scans [22]. With this propofol regimen, sedation was adequate in 27 of the 30 children, cardiorespiratory stability was reasonable, and mean recovery and discharge times were 18 and 27 min, respectively.

Pharmacokinetic models for dexmedetomidine in children have recently been produced from studies involving single bolus administration [44], after short infusions [45], and after longer infusions [46] for postoperative sedation. Further studies are needed to compare the predictive accuracy of these models to determine which perform optimally in clinically relevant situations.

Despite the low α(alpha)1 affinity of dexmedetomidine, rapidly administered boluses cause bradycardia and hypertension. Typical infusion regimens thus usually comprise an initial bolus over 10 min, followed by a continuous infusion. Mason used an initial bolus of 2 μ(mu)g/kg over 10 min (repeated if Ramsay sedation score [47] of 4 not reached) followed by an infusion at 1 μ(mu)g/kg/min, in 62 patients with mean age 2.8 years and mean weight 15 kg, undergoing CT imaging [23]. Of these patients, 10 % were able to undergo their scan during the initial loading dose, 16 % required a second loading dose, and 90 % required the maintenance infusion. Two patients became agitated during the loading dose and were given alternative agents for sedation.

Subsequently, Mason reported the results of a study of the use of higher doses of dexmedetomidine in >700 patients undergoing MRI scanning, which is more stimulating, and in which movement causes significant image degradation [24]. With time their regimen evolved from an initial bolus of 2–3 μ(mu)g/kg and from an initial infusion rate of 1 μ(mu)g/kg/h to 1.5 and 2 μ(mu)g/kg/h. The highest doses were associated with successful sedation and image acquisition in 97.6 % of patients, but with reasonable cardiorespiratory safety.

Koroglu and colleagues used smaller doses for sedation during MRI scanning in 30 children with a mean age of 4 and mean weight of 14 kg; the bolus dose was 1.0 μ(mu)g/kg over 10 min, and this was followed by an infusion at 0.5 μ(mu)g/kg/h initially, but increased to 0.7 μ(mu)g/kg/h if a Ramsay score of 5 was not reached within 25 min [22, 24]. With this regimen, additional midazolam was required in 16 % of patients to facilitate successful scan completion.

A TCI is an infusion of a drug administered by an infusion pump controlled by a computer or microprocessor that is programmed to calculate and implement the drug infusion rates required to achieve in a patient the blood or effect-site concentrations required by the user. Simply put, with these systems, the user inputs a desired “target” concentration, and the system uses the parameters of a pharmacokinetic model for that drug and the patient parameters included as covariates in the pharmacokinetic model to calculate the infusion rates estimated to be necessary to achieve that concentration.

As previously explained, bolus doses of intravenous drugs for sedation are generally only suitable for short procedures. Although infusions do provide more stable conditions, they still do not provide stable blood concentrations. Even for propofol, a drug with rapid kinetics, blood concentrations continue rising for several hours when fixed rate infusions are used (see Fig. 31.7). There is thus a poor correlation between infusion rate and clinical effect. During the course of any procedure, the effect-site concentration required for adequate sedation will vary widely according to several other factors such as the influence of co-administered drugs (especially opioid analgesics), the onset of natural sleep, changes in the environment, and the severity of any noxious stimuli. The changing relationship between infusion rate and effect-site concentration, and the delay in blood–effect-site concentration equilibration, makes rational, precise, and rapid titration of the infusion very difficult. As can be seen in Fig. 31.7, stepwise changes in the infusion rate of 2 mg/kg/h result in very slow changes in blood and effect-site concentrations, so that it is difficult to assess the response to an infusion rate adjustment. These difficulties form an important part of the rationale for TCIs, where a computer or microprocessor is used to implement the infusion rates required to maintain steady-state blood concentrations. Since steady-state blood concentrations arise quite quickly, TCI systems allow the user to judge the clinical effect of a blood concentration and to then adjust the target blood concentration accordingly, rather than adjusting the infusion rate accordingly. An analogy is to compare the control a car driver has over the speed of his car, when he has a speedometer and cruise control system versus the control he would have with only a gas pedal and no cruise control system or speedometer.

When k _eo values for children have been validated and effect-site targeting is sufficiently developed for use in children, then a further refinement will be added since users will then be able to titrate the effect-site concentration titrate according to observed patient responses.

With blood and effect-site concentration targeting, absolute accuracy of the pharmacokinetic model is not important, since steady-state concentrations arise very quickly, and there remains wide variability in pharmacodynamic sensitivity among different patients to given blood and effect-site concentrations. Thus, even with the most accurate models and systems, titration according to pharmacodynamic responses will be required.

With TCI the user is able to set and alter a desired “target” drug concentration. The target is usually a blood concentration (although algorithms do exist for effect-site targeting [48] and have been implemented for propofol, remifentanil, and sufentanil use in adults). TCI systems use compartmental pharmacokinetic models with complex mathematical algorithms to calculate and implement the infusion rates required to achieve the target concentration. The system software calculates the drug amount in each of the compartments every 10 s, taking into account the amount of drug infused over the previous 10 s, the movement of drug into and out of the central compartment by redistribution, and the rate of removal of active drug from the central compartment by metabolism or elimination. It then calculates and implements the infusion rate required to maintain the target concentration over the subsequent 10 s.