The time course of the pharmacological response of neuromuscular blockers and intravenous anaesthetics cannot be understood without a model of the underlying pharmacokinetics which includes drug transport to the effect site. In traditional pharmacokinetics, drug distribution is assumed to take place between homogenous compartments which lack physiological reality. Thus, despite the fact that convection by blood flow is the primary transport process of drugs in the body, cardiac output does not play any role as a parameter in these models. A well-mixed sampling compartment, for example, is in contrast to noninstantaneous circulatory mixing and compartment models do not account for the effect of hemodynamics on the initial distribution volume of drugs. Much progress has been made in the understanding of pharmacokinetics in terms of physiologically based models, i.e., multi-organ models reflecting the structure of the circulatory system (PBPK). However, due to the fact that PBPK models are useful for simulation purposes but cannot be identified on the basis of plasma concentration time data, there is still some confusion regarding the selection of an appropriate pharmacokinetic model. On the other hand, one may ask whether there is any need for more detailed models in pharmacokinetic/pharmacodynamic modelling (PK/PD), especially when the model predictions are already in accordance with the observed data. Keeping in mind that the validity of a model is defined in terms of the modelling objectives, black box or behavioural models are sufficient if the objective is the quantitative prediction of drug effects, as in closed loop anaesthesia or empirical PK/PD modelling. However, structural models are necessary if we want to understand the behaviour of the system. Thus, the focus here is on the heuristic validity of PK/PD-models in view of the underlying transport processes.

The following modelling approaches are useful for a better understanding of the time course of drug effect: (i) multi-organ models based on the structure of the circulatory system and reflecting the convective transport of drugs by blood flow to the various organs and tissues, (ii) circulatory minimal models consisting of the pulmonary and systemic circulation as the essential subsystems of the body, (iii) "semi-physiological" models combining the system approach with structural information, and (iv) models of blood-tissue exchange, which describe the concentration at the effect site as a response to any arterial input function.

In PBPK models (i) distribution kinetics at the organ level is mostly simplified as transfer between well-stirred compartments. These models are useful to explain the influence of the physiologic state (changes in cardiac output and body composition) on pharmacokinetics based on blood-tissue partition coefficients obtained in animal experiments by destructive sampling, as demonstrated for fentanyl and alfentanil[1]. These models can be extended to include the effect of intravascular dispersion and "slow" diffusion within tissue parenchyma to overcome the limitation of the well-mixed approach at the organ level. The necessary input parameters can be obtained, in principle, in experiments with isolated perfused organs using blood-tissue exchange models (iv). Minimal recirculatory models (ii) have the advantage that a parameter estimation (model identification) is also possible on the basis of clinical data (arterial concentration-time curve) and that the effect of hemodynamic changes can be analysed since cardiac output acts as a model parameter. In contrast to the compartmental approach this model accounts for the effect of hemodynamics on the pharmacokinetics of a physiological marker in healthy volunteers[2]. A similar type of model (with the systemic circulation split into subsystems) has been successfully used to analyse the initial distribution kinetics of drugs and multiple indicators[3]. Emphasis was given on the role of the lungs as a first pass organ in early pharmacokinetics after bolus injection. If one is not particular interested in the effect of the early mixing process (as for short acting drugs) but in the effect of generated metabolites on the PK/PD of a drug, semi-physiological models (iii) may be sufficient[4]. These models explain, for example, differences of drug effects after oral administration due to hepatic first-pass generation of an active metabolite, as recently shown for morphine[5]. Detailed structural models of blood-tissue exchange (iv) are essential for an understanding of drug distribution at the organ level, which also determines the concentration at the site of action. The necessary information on the distribution process can be obtained in experiments with isolated perfused organs or under in vivo conditions, if the input and output concentration of an organ can be measured, as recently demonstrated for the pulmonary distribution of synthetic opioids[6]. Dynamic positron emission tomography and - if the distribution process is sufficiently slow - also microdialysis are promising methods. Models describing the transport from aorta to the effect site (iv) can then be combined with models (i) to (iii) to form hybrid models in order to explain the physiological determinants of the biophase response function (equilibration process) in classical PK/PD models. Physiological models have implications for dosage optimisation, as shown recently for the induction of anaesthesia with propofol[7].

From a theoretical point of view, the analysis of drug distribution and mixing in heterogeneous systems without the assumption of homogenous compartments (or linear differential equations as the mathematical counterpart) is the most challenging problem. Not only the existence of a well-mixed blood or plasma compartment is a fiction (with the consequence that the location of input and sampling points is crucial in PK/PD-modelling[8]), the instantaneous distribution assumption is also an approximation at the organ level, both for the vascular and extravascular space. The theory of residence or transit time distributions is an appropriate mathematical tool which is independent of a particular mechanistic or structural model of the transport process. While it is clear that the steady-state distribution volume can be obtained from the mean transit time of the system, less appears to be known on the dispersion of residence time as a determinant of the dynamics of the mixing (distribution) process. The action of drugs which influence their own distribution kinetics (e.g., via blood flow changes) and cellular pharmacokinetics are among the problems for future research.

References
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