Pharmacokinetics and pharmacodynamics – is there anything new?

Abstract

The basic principles of pharmacokinetics (PK) and pharmacodynamics (PD) have been well understood for many years. An epic editorial in 1979 sets these out in a manner that remains relevant today [1], concluding: “the clinical anaesthetist may, if he has persisted this far, be forgiven for feeling the sources of variation are so legion that in the case of many drugs used in anaesthetic practice it is better to titrate dose against response than to attempt to predict the correct dose on theoretical groups. He is, of course, right . . .”. Why then bother with PK-PD and has our thinking moved forward? For PK to be useful we need intelligible information that clinicians use to adjust or understand their practice. Historically, drug disposition was described by half-lives (α, β ± γ depending on whether a two- or three-compartment model is preferred). Dissonance between these numbers and clinical experience of recovery from drug effects stimulated the creation of new descriptors, e.g. decrement time [2], context sensitive half time [3] and mean effect time [4] (Table 1). The latter integrates the influence of PD and PK on time to recovery. Similarly, at the start of anaesthesia, time to peak effect [5] (Fig. 1) is exactly what it says it is. Simulation based on a 20-year-old, 70-kg, 170-cm male, showing the time taken for concentrations of various synthetic opioids in the effect site to reach peak concentration (100%) following bolus injection. Effect site concentrations peak earlier with remifentanil (····) and alfentanil (- - -) than with fentanyl (—). Simulations are based on the Scott & Stanski [34] and Minto [35] models. Data kindly provided by Charles Minto. For the traditional two-stage approach, a PK model is constructed by sequential sample collection from an individual following bolus or infusion dosing. In the first stage, a model is developed by fitting (typically) a polyexponential curve to the drug concentration vs time data, and PK parameter values derived. In a second stage, the derived parameters from a group of subjects are averaged to create simple estimates of the mean parameter values in that population sample, and some indication of the variability associated with them. This approach is simple, computationally efficient and mathematically flawed. The fundamental problem is that it cannot differentiate between inter-individual variability and error due to other random effects. Hence, the imprecision resulting from estimation of group PK parameters is further increased by imprecision produced from estimating individual kinetics. The two-stage technique therefore overestimates PK variability within a population [6]. When the subjects are relatively homogenous and the dosing and sampling regimens well standardised (e.g. phase-1 clinical trial), a reasonable set of estimates may be computed. However, as soon as intra-subject heterogeneity is considered and different dosing/sampling regimens adopted, the system breaks down. Additionally, the frequent blood sampling required for successful execution of the two-stage approach renders it unsuitable for vulnerable groups e.g. neonates, the elderly and the critically ill, ironically those most likely to demonstrate altered kinetics. Mixed-effects population modelling, so called because it separately estimates both fixed (specific constants i.e. structural model parameters) and random (quantifying variability within the population) components, overcomes these limitations. Population models may be based on data where each subject contributes as few as 1-2 samples, the key feature of this type of modelling being that data, although analysed as a population, retain their individual identity. Hence, the population approach is able to separate and quantify the many possible sources of variation such as inter- and intra-patient variability, inter-occasion variability, and random residual error. Covariates, or factors influencing PK parameter values such as body weight and age, can be incorporated directly into the PK model structure if statistically justified, e.g. their inclusion reduces the magnitude of unexplained inter-individual error. Hence, statistically as well as practically, it is a superior methodology [6]. These analyses are commonly conducted using NONMEM® software (http://www.iconplc.com/technology/products/nonmem/), although other packages are available. Our analyses permit only cautious extrapolation from the patients studied to broader groups and different clinical scenarios. Administration of drugs to children is typically poorly supported by commercial drug development programmes and subsequent research, yet these patients are the most heterogeneous in age, size, metabolic capacity and body composition. Studying pharmacokinetics in children is particularly challenging, not least because ethical and safety considerations reduce the number and volume of samples obtained. Expanding the study ‘work space’ beyond a single patient allows the sharing of sample timings across individual study participants, thereby maximising the information yield from available subjects. Blood sampling times can be block randomised so that subjects contribute one sample at a randomised time within a specified sampling window (Fig. 2). Hence, collectively the population dataset consists of multiple samples within the time frame of interest, but each individual has contributed only one sample per window. Ideally, to increase the likelihood of estimating PK parameter values with confidence, the sampling points within the window will have been selected based on their ability to inform the development of the PK model [7]. Study design can be enhanced in this manner when some PK information about the drug of interest in the target population is already available. Example of a simple randomised blood-sampling schedule in a PK study of remifentanil in critically ill children [14]. The black squares show typical blood-sampling time points. Three samples were collected during the first 25 min of remifentanil infusion, a further three samples were collected during the 20-min period after the first reduction in remifentanil infusion rate, and additional samples were obtained 20 min after the second and subsequent reductions in the remifentanil infusion rate and at the point of arousal, one sample per child per time window. A simple approach is to plot the partial derivatives of the predicted function, e.g. the mathematical function describing a two-compartment PK model, with respect to each of the model parameters against time, to reveal any turning points (local maxima or minima). Blood samples taken at the time of these turning points will contribute the maximum information on that particular parameter [8]. More sophisticated approaches entail the use of computer-aided sampling designs. These are constructed based on a chosen optimality criterion to produce a sampling scheme that is optimal for the specified model. The D-optimal approach is a commonly used criterion that maximises the determinant of the Fisher information matrix and hence, aims to minimise the variance-covariance matrix of the parameter estimates, based on the model under consideration. Whilst D-optimal designs can offer more robust estimates of PK parameter values [9], they must be interpreted pragmatically; e.g. a very late sampling time produced as part of a D-optimal scheme may be impractical for both patients and clinical staff. Typically, clinicians adjust drug doses against their patients’ weight, usually in a linear manner. Such confident scaling is seldom justified by theory. Firstly, clinical trial populations, especially young males in phase-1 studies, may be remarkably homogenous and hence weight may not be found to be a statistically justifiable covariate for describing drug disposition. This does not necessarily mean that concentration is independent of weight; rather, the study dataset does not include enough information to explore this properly. If narrow age ranges are studied in isolation, potential covariates may not vary widely within the study group and their relationship to one or more PK/PD parameters cannot be revealed. Population modelling permits the broad exploration of covariates from widely varying subjects receiving different doses of drug at different times. Appropriate examination of data from heterogenous groups, such as children, often reveals that the relationship between drug disposition and weight is best described non-linearly e.g. by the application of allometric scaling. Allometry describes changes that deviate from isometry, i.e. when a change in body size corresponds to a change in the proportion of a body function or component. In 1932, animal researcher Max Kleiber demonstrated that animals’ metabolic rate was proportional to their body mass raised to the power of 0.75 [10]. This relationship has been successfully applied to describe the relationship between body weight and drug clearance in paediatric populations [11-14]. Figure 3 demonstrates the non-linear relationship between the metabolic clearance of remifentanil and body weight in children ranging from neonates to 9 years. (a) Plot of metabolic clearance of remifentanil against body weight via an allometric relationship demonstrating a non-linear relationship in which clearance, on a per kg body weight basis, is higher in the smallest children. (b) Simulation of a 1-h infusion of remifentanil (0.8 μg.kg−1.min−1) using an allometric paediatric model [14], as above, and the adult simple weight-proportional model derived by Minto and colleagues [35]. Enhanced clearance and a larger central compartment volume (92 vs 67 ml.kg−1) results in reduced blood concentrations compared with adults. Smaller children require higher infusion rates (on a μg per kg basis) than larger children or adults, to achieve equivalent blood concentrations. Allometric models have also been usefully applied to describing body mass-related changes in drug disposition in adults. Extrapolation of popular target controlled infusion (TCI) models for propofol [15, 16] to obese and super-obese patients may produce paradoxical and possibly harmful induction recommendations [17]. Previous studies have determined that for obese patients, propofol dosing for induction is best based on lean body weight [18] while maintenance dosing is more appropriately based on total body weight (TBW) [19]. Cortinez and colleagues comprehensively evaluated propofol pharmacokinetics in a pooled cohort comprising 51 individuals with BMIs of 16–52 kg.m−2 [20]. The model that best described the pooled data was one in which all structural model parameters (clearance and volume terms) were allometrically scaled to TBW. When only obese patients were included in a separate PK analysis, simple linear scaling to TBW was deemed sufficient, demonstrating the importance of studying a covariate effect such as body weight in a population sample in which it varies widely. The popularity of TCI has exposed the benefits and limitations of PK-PD modelling. The availability of generic propofol formulations heralded the introduction of ‘open TCI’ systems. With these, anaesthetists were no longer limited to the Diprifusor/Marsh [15] model but could select alternatives. The academic community has recognised the potential problems relating to the clinical use of multiple kinetic datasets for a single drug, and identified the need for a single ‘best’ PK-PD model for propofol and other drugs. The Open TCI Initiative [21] was proposed in 2008 to develop such a model. Interested academics are invited to upload raw data from (initially) propofol PK-PD studies, particularly those featuring patients at the extremes of age and weight. It is hoped that this extended dataset can be used to evaluate existing models further and ultimately, to develop a unified, robust model that might be incorporated in future TCI devices. Meanwhile, anaesthetists must remain aware that the most popular PK models for propofol, Marsh [15] and Schnider [16], calculate strikingly different bolus doses and initial infusion rates when used to induce and maintain general anaesthesia [17] (Fig. 4). Though things settle down as the infusion continues, this suggests that the early part of anaesthesia is not well characterised in these mathematical simplifications. Simulation of a target controlled infusion (target 4 μg.ml−1) in a 40-year-old male, height 170 cm, using the Marsh/Diprifusor () and Schnider () PK models [17, 18], if the patient weighs (a) 70 kg (BMI 24 kg.m−2) and (b) 118 kg (BMI 40 kg.m−2). The Schnider model has a fixed central compartment volume, i.e. does not vary with the patient’s weight, in part explaining some of the discrepancy between the two infusion profiles during induction. Solid lines, plasma site targeted; dashed lines, effect site targeted. Simulations produced using Stanpump software. If the reductionist simplification to a two- or three-compartmental model seems generous, then its application to induction seems absurd. The assumption that bolus doses of drug are instantly distributed throughout the volume of distribution and that the drug itself has no effect on its own distribution or elimination is hard to believe and biologically implausible. An empirical method of dealing with this limitation has been to incorporate dosing lag time into a model so that drug simply arrives in the central compartment a little later than the actual dosing time (Fig. 5). Transit (—) and lag time () models for a drug after intravenous injection. With a traditional lag time model, the arrival of the drug in the central compartment is simply offset by a defined time (fixed or estimated) after dosing. However, this discontinuous process is non-physiological and can also lead to numerical difficulties during the modelling process. In a transit compartment model, the drug moves through varying numbers of pre-systemic compartment before arrival in the central compartment. Movement of drug through the pre-systemic compartments is described by the rate constant, Ktr. Increasingly, modellers apply a more mechanistic approach to mathematical descriptions of early drug kinetics. This is particularly important when developing integrated PK-PD models, as the accurate derivation of Ke0, which describes the temporal relationship between drug concentration and effect, relies on a robust description of drug concentrations at onset of drug effect i.e. front-end kinetics [22]. The transit model approach utilises a varying number of pre-systemic compartments to characterise the initial delay in drug reaching the central compartment (Fig. 5). The number of transit compartments required to characterise the delay adequately may be established by stepwise addition and subsequent examination of the diagnostic plots and NONMEM objective function value, or for single-dose data, may be directly estimated as described by Savic et al. [23] Delaying the appearance of drug in the central compartment can partly account for the flawed assumption of instantaneous mixing. Recently, Masui et al. have used a combined transit and lag time approach to describe the early time course of arterial propofol concentrations after intravenous administration [24]. Mamillary models (standard two- or three-compartment models) lack heuristic validity in that they have limited predictive ability in patients significantly different from the population sample that contributed the original data. This is because standard compartmental models are unable to account for differences due to physiology (e.g. cardiac output [25]) or to reflect differences caused by a change in drug administration rate [16]. Recirculatory models, like transit models, are an extension of a traditional mamillary model (Fig. 6). Recirculatory models are able to overcome some of the limitations of a mamillary model without requiring the complex (and often unavailable) tissue:blood partition coefficient data needed for full physiologically based PK modelling [26]. Instead, recirculatory models incorporate cardiac output, lung kinetics and injection rate. Drug movement between compartments is described as drug concentrations in venous and arterial blood, rather than the theoretical rate constants used in mamillary models. Hence, recirculatory models do not make the flawed assumption that drug is instantaneously and uniformly distributed within arterial and venous blood after administration. Upton provides a useful and accessible introduction to recirculatory pharmacokinetics for the interested reader [26]. (a) Classic 2-compartment mamillary model, comprising a central compartment, typically representing the circulating blood volume and the well perfused tissues, and a peripheral compartment representing less perfused body tissues. Drug transfer between the two compartments is described by the rate constants, k12 and k21, but drug is only eliminated (clearance, CL) from the central compartment. (b) The recirculatory model also has two compartments but these represent the lungs (VL) and the rest of the body (VB). In this model, drug is administered into the lung compartment but cleared from the body compartment (QCL). Drug transfer between compartments is described as arterial (Ca) and mixed venous (Cv) drug concentrations. Cardiac output (QCO) provides the rate of blood circulation between the two model compartments. Adapted from Upton [26]. Almost all PK modelling is underpinned by blood samples, ideally arterial but frequently venous. From a mathematical standpoint, anaesthesia can be simplified to drug administration and drug effect with everything in between forming a ‘black box’. Blood sampling allows us to deconstruct the black box and to describe the intra-/inter-patient variability associated with each process. However, where a robust measure of drug effect is available it may be possible to derive PD parameters without blood sampling [27]. Modelling using predicted drug concentrations, such as those from PK models driving TCI systems, has been popular recently, especially in children [28-30]. Using predicted concentrations overcomes ethical and practical issues relating to blood sampling and storage, and negates the need for expensive drug assays. However, it is well known that predicted drug concentrations may differ substantially from true concentrations, particularly when the latter are changing rapidly during the onset or offset of anaesthesia [31]. So how might discrepancies between predicted and measured drug concentrations impact on the subsequent estimation of PD model parameters? A recent study by Coppens et al. [32] examined such a scenario. Blood samples and Bispectral Index (BIS) monitoring output from 28 children undergoing propofol anaesthesia during dental treatment were collected. The measured propofol concentration in the collected samples and the BIS data were then used to derive a classic PK-PD model. Subsequently, the group used various literature models to generate predicted propofol concentrations for the samples collected in their study, and then used these predicted values to re-estimate the pharmacodynamics. This analysis revealed that, despite accurate predictions of BIS, the use of predicted drug concentrations may lead to incorrect estimation of PD model parameters. So while avoiding blood sampling is possible, it is preferable to collect samples for assay. When blood sampling is not achievable, researchers should be aware of the limitations and potential pitfalls of modelling based on effect data alone. Pharmacokinetics and pharmacodynamics have some a long way since 1979, and enhancements in model construction with clinically relevant descriptors of onset and offset of drug concentration and effect have brought the information they yield closer to clinicians. It is hoped that further refinement can help predict more complex situations such as the interactions of multiple drugs and time. For example, the reversal of rocuronium by intravenous sugammadex requires a complex PK-PD model accounting for both drugs and their interaction [33]. Once a working model has been developed from sugammadex administration at various intervals after rocuronium, the investigator can then explore alternate theoretical scenarios including immediate reversal (as might occur in a clinical emergency) and a broad range of intermediate states whereby varying degrees of blockade and time elapsed since rocuronium administration could be tackled with different doses of sugammadex. Clearly, not all of these can be explored with a finite set of clinical trials. No external funding and no competing interests declared.