Physiologically based pharmacokinetic modelling: a sound mechanistic basis is needed

Pharmacokinetics is the science of drug absorption, distribution, and elimination, or more specifically the quantification of those processes, leading to the understanding, interpretation, and prediction of blood concentration-time profiles. Occasionally, concentration-time data from other physiological fluids or tissues are available, but it is the lack of data in relevant tissues and organs that limits one's ability to get at the underlying mechanisms determining the blood profile. For example, blood concentration data are used to evaluate drug absorption for orally administered drugs, even though absorption is a multistep erratic process under the control of many factors, and measurements within the gut are not usually available.

Nevertheless, blood concentration can be a useful biomarker, and sometimes a surrogate [1], to guide therapy. The idea is that pharmacokinetics accounts for some of the variability in the dose-response relationship. However, in order to transform dose into concentration, a pharmacokinetic model is required, based on an analysis of concentration-time data and preferably incorporating relevant patient characteristics, allowing it to be used for individualized therapy [2]. The complexity of a pharmacokinetic model depends on the level and quality of information available and on its purpose.

A hierarchy of pharmacokinetic models

Pharmacokinetic models can be classified in order of increasing complexity. At the lowest level is the empirical model most often described by a sum of exponential terms, as in the following equation:

This model adequately describes typical concentration-time profiles and can be used to derive primary pharmacokinetic parameters, such as clearance and half-life. It can also be used to devise a dosage regimen for a subject who has the same set of parameters [C_i, λ_i]. Although useful for data description and interpolation, empirical models are very poor at extrapolation. This is because the parameters do not have a physiological interpretation, and it is difficult to predict how they change when the underlying physiology changes. For example, with a biexponential model after an intravenous bolus dose it is not obvious how the coefficients C₁ and C₂ change with age. This problem can be partly addressed by adopting a compartmental approach with a so-called physiological parameterization. A classical two-compartment model is shown in Figure 1. The concentration-time profiles that result from this model and a biexponential model are the same.

Figure 1.

Two-compartment model. V₁ and V₂ are the volumes of compartments one and two, respectively; CL is the clearance from compartment 1; and Q is the intercompartmental clearance

However, with this model it is easier to relate the parameters to physiological processes. For example, clearance can be related to renal function, perhaps through creatinine clearance measurements, and the effect of renal impairment on the shape of the concentration-time profile can be predicted. Nevertheless, the compartments in these classical compartment models do not represent real physical spaces, and their meaning has been misrepresented by many authors. Consequently, these models should be viewed as semi-mechanistic models.

The most comprehensive pharmacokinetic model is based on physiological considerations. An example is shown in Figure 2. The physiologically based pharmacokinetic (PBPK) model is a compartmental model, but differs from classical pharmacokinetic models in that the compartments represent actual tissue and organ spaces and their volumes are the physical volumes of those organs and tissues.

Figure 2.

An example of a physiologically based pharmacokinetic model. The compartments represent tissues and organs; connecting arrows represent blood supplies; ST is stomach; SPL are splanchnic organs; and CL_int is intrinsic hepatic clearance

In the simplest model, the compartments (that is, the tissues) are assumed to be homogeneous. This assumption can be relaxed if data at a lower level, for example the cellular level, are available. The drug enters the compartments in arterial blood and returns to the heart in the venous blood. Elimination occurs in specific organs, such as the kidney and liver. It is commonly assumed, at least for lipophilic drugs, that the uptake of drug by the tissue is blood-flow limited. If that is not the case, uptake is described by a permeability-limited model. Mass balance equations are written for each compartment, and hence the model is described by a series of differential equations.

One of the earliest descriptions of a PBPK model was that of Teorell [3], one of the pioneers of pharmacokinetics [4]. During the 1960s and 1970s, Bischoff and Dedrick developed a number of PBPK models, particularly for anticancer drugs [5]. For a review of the early work on PBPK models see Himmelstein and Lutz [6]. In a parallel development, researchers in the environmental protection field applied PBPK modelling to the risk assessment process of a number of environmental pollutants [7]. Likewise, Mapleson and colleagues in Cardiff developed PBPK models for anaesthetic agents [8]. An important application of PBPK modelling is the prediction of the pharmacokinetics of a drug in man from animal data [9].

The major limitation of PBPK models, and a deterrent to their wider use, is the lack of human data, making them rely heavily on animal studies and intensive sampling of tissues. The latter problem is being addressed by the use of in vitro predictions for both distribution [10] and elimination [11]. Consequently, very few clinical applications have appeared. A notable exception is the full PBPK model of Björkman, used to predict theophylline and midazolam disposition in infants and children [12].

Given the complexity of PBPK modelling, more use has been made of partial PBPK models or semi-mechanistic models. An example appears in the current issue of the Journal, in which Ito et al.[13] describe a recirculating hepatic clearance model that they used to describe the CO₂ exhalation rate after administration of uracil [14]. However, the physiological component of the model in this application is minimal, and it is doubtful whether it is necessary. A more interesting example appeared in a paper in a previous issue of the Journal, in which a semi-mechanistic model was used to model the inhibition of dextromethorphan by quinidine [15]. The model that was used is shown in Figure 3. The key to this analysis is the ability of the model to differentiate between first-pass metabolism and systemic metabolism, which are linked by a recirculating hepatic clearance model.

Figure 3.

Semi-physiologically based drug-metabolite model of dextromethorphan and dextrorphan. D stands for dextromethorphan and M for dextrorphan; G is the gut; S is the systemic circulation; P is a peripheral space; F is the bioavailability; k_a is the absorption rate constant; CL and CL_M are the clearances of the drug and metabolite, respectively; V and V_M are the volumes of distribution of the drug and metabolite, respectively; and k₁₂ and k₂₁ are the distribution rate constants for the drug between compartments one and two (S and P)

Systems biology

Although pharmacokinetics is sometimes a surrogate for the clinical outcome, in most cases it is necessary to link pharmacokinetics to response. In clinical practice, most commonly the simplest empirical pharmacodynamic models are used, and in many cases quite effectively. Just as with PBPK, at the other extreme to the simple pharmacodynamic models that are used in clinical practice, considerable effort is being applied to developing models from the bottom up, the so-called systems biology approach [16]. It is now time that these two methods came together, so that PBPKPD models can be put on a firm mechanistic basis. Again, some very nice semi-mechanistic work has appeared, particularly from Jusko and colleagues in Buffalo [17], who in this issue of the Journal describe a semi-mechanistic pharmacokinetic–pharmacodynamic model of the antimalarial effect of artemisinin [18]. The pharmacokinetic model incorporates autoinduction and saturable metabolism and is linked to a pharmacodynamic model reflecting different stages of the parasite's life-cycle.

Empirical models will always be the mainstay of dosage regimen calculations. However, it is important that they have a sound mechanistic basis.