2 – Pharmacokinetic and Pharmacodynamic Modelling in Anaesthesia


Pharmacokinetic/pharmacodynamic (PK/PD) modelling is a discipline currently under the umbrella of pharmacometrics, and aims to describe, understand and predict the time course of in vivo drug action.

In general PK/PD comprises three major elements: (1) pharmacokinetics (PK), (2) pharmacodynamics (PD), and (3) disease progression. However, given the fact that anaesthesia procedures take place in short periods of time, where the general state of the patient remains unaltered, in this chapter we will limit the focus to the interrelationship between PK and PD.

2 Pharmacokinetic and Pharmacodynamic Modelling in Anaesthesia

María J. Garrido and Iñaki F. Trocóniz


Pharmacokinetic/pharmacodynamic (PK/PD) modelling is a discipline currently under the umbrella of pharmacometrics, and aims to describe, understand and predict the time course of in vivo drug action.

In general PK/PD comprises three major elements: (1) pharmacokinetics (PK), (2) pharmacodynamics (PD), and (3) disease progression [1]. However, given the fact that anaesthesia procedures take place in short periods of time, where the general state of the patient remains unaltered, in this chapter we will limit the focus to the interrelationship between PK and PD.

Drug action in vivo is associated, in addition to the magnitude of drug effects, with the concept of dynamics. In fact, the onset and offset of drug action are important factors always considered when dosing a patient. If we aim to describe the entire course of a particular drug effect, we need to use mathematical models, but if we also want to understand and predict drug effect in new patients and/or under different administration paradigms, we require that those mathematical models resemble the main processes responsible for the clinical outcomes. Therefore PK/PD modelling does not represent just an exercise of applied mathematics. On the contrary, the main challenge is to extract from the available data the relevant underlying mechanisms.

One part of the present chapter is devoted to showing the most used PK/PD model in the specialty of anaesthesia, detailing the mathematical structure and interpreting the parameters that define the model.

One common characteristic of the in vivo response to any drug of any therapeutic area is variability across patients. The magnitude of such variability needs to be quantified accurately, and its causes identified so that the drug will be optimally used in the population it is intended for. In the discipline of pharmacometrics the population approach deals with variability in drug exposure and response and therefore a section in this chapter is devoted to population analysis.

Pharmacokinetic/Pharmacodynamic models

Figure 2.1 represents in a schematic form the processes taking place between drug administration and response, including PK, distribution to the biophase and the different steps between target engagement and measured response. The diagram might discourage anaesthesiologists from developing population PK/PD models because the data required seem overwhelming. We will show later in this section how the complex picture shown in Fig. 2.1 can be simplified on the basis of the rate limiting step concept, making the study of the time course of in vivo drug response tractable [2].

Fig. 2.1 Representation of the main processes involved in the in vivo time course of drug action. The terms Cp, Ce, R and Ce-R represent the plasma and effect-site concentrations, free target receptors, and the concentration–receptor complex, respectively; kon and koff correspond to the association and dissociation binding rate constants, respectively; ke0 is the first order rate constant governing the distribution equilibrium between the central and effect site compartment; kin and kout represent rate constants of turnover; and kTR is a first order rate constant controlling signal propagation.

PK principles will be discussed first, followed by PD, and finally how they are integrated [3, 4].


Prior to drug administration, and in the absence of circadian rhythms and during a short period of time, values of the measured response (i.e. baseline) remain constant, indicating that response, per se, does not vary over time. But as soon as a drug enters the body, the response variable will display a certain profile over time that is influenced by the PK properties of the anaesthetic drug. Therefore, one essential component in understanding the time profiles of a response is PK.

PK can be expressed in different ways: (1) characterization of the time course of the concentration of the active compound(s) [drug and/or metabolite(s)], generally in plasma, or (2) what the body does to the drug.

It has to be emphasized that PK is not just the measurement of plasma or tissue drug levels. Proper PK practice involves a modelling exercise with the aim to get precise PK parameters that accurately represent and quantify the main physiological process responsible for the fate of the drug in the body [3]. Before starting to describe the most commonly used models in anaesthesia, it is important to consider biopharmaceutics and data related aspects.

Usually during surgery, anaesthetics are administered intravenously, and in those cases exact information on the infusion rates and the corresponding amount delivered during each infusion have to be carefully recorded. In other areas of anaesthesia practice such as pain control, drugs are also given orally or subcutaneously, and therefore formulation type, mode and route of administration, as well as dose and dosing schedule are important factors to be taken into account.

To perform a proper PK modelling exercise, in addition to the accurate dose- and administration-related aspects commented on above, longitudinal levels of the concentration of drug in plasma are required. Plasma drug concentrations can be obtained either from arterial or venous blood. The operating room setting with an unconscious patient offers the opportunity to gather several early blood samples which, as will be discussed later, has important implications in allowing proper characterization of the onset of drug action. For example, to characterize the PK profile of propofol, arterial blood samples were taken at 0, 1, 2, 4, 8, 16 minutes and beyond after the first bolus injection [5].

Empirical Pharmacokinetic Compartmental Models

The most common approach in PK is to consider the body of the patient to be made of several compartments. The three compartment model is one of the most frequently used models in anaesthesia [6]. Figure 2.2 represents schematically the structure of the three compartment model and its parameters. Chapter 1 discusses the compartment approach in more detail.

Fig. 2.2 Schematic representation of a three compartment PK model. V, V2 and V3, represent the apparent volume of distribution of the central, peripheral shallow and peripheral deep compartments, respectively; CLD1 and CLD2 are the inter-compartmental distribution clearances, and CL corresponds to the total plasma elimination clearance.

It is important to point out that besides using volumes and clearances, the primary PK parameters, PK models can be expressed in other parameters such as V, k12, k21, k13, k31 and k10. Both parameterizations are mathematically indistinguishable. In fact the above first order rate constants are equal to CLD1/V, CLD1/V2, CLD2/V, CLD2/V3 and CL/V, respectively. Primary PK parameters depend only on the physiopathology of the patient, whereas the first order rate constants are secondary PK parameters depending on the primary PK parameters. Therefore, if we want to investigate the influence of patient factors on the PK drug properties, use of volumes and clearances is the best parameterization approach.

It has to be realized that PK (and PD) parameters are not just numbers providing, within a specific mathematical structure, a good fit to the data. It is of great importance to recognize that PK parameters are a reflection of patient physiology.

Physiology-based Pharmacokinetic Models

A more mechanistic representation of the body is provided by the physiological PK models (PBPK), where all compartments do represent real body entities. Predictions of drug concentration over time are carried out using physiological variables such as tissue volumes and blood perfusion values, and a measure of the capacity of the tissue to be filled with drug, named the partition coefficient (KP), defined as the ratio between tissue and plasma concentrations once steady state is achieved.

A schematic representation of a PBPK model is shown in Fig. 2.3. Each of the tissues can be considered either (1) as a single compartment assuming perfusion limited distribution in which tissue drug kinetics can be predicted for a given value of arterial drug concentration, with the use of just KP, φ, the organ perfusion and tissue volume, or (2) as a number of compartments in cases of permeability limited distribution. From a data acquisition perspective, PBPK models are quite demanding because drug concentrations have to be measured in each tissue to compute KP (note that physiological variables such as organ perfusion and real volume values can be obtained from the literature because they are drug independent). Usually, PBPK are developed in laboratory animals and translated to humans assuming that KP remains equal across different species [13]. Once relevant drug properties are known (i.e. liposolubility, unbound fraction (fu), blood to plasma concentration ratio, elimination pathway, etc.) it would also be possible to use new technologies to provide concentrations. PBPK represents a good tool for hypothesis generation, to explore ‘in silico’ the impact of patient characteristics on drug concentration profiles using the systems pharmacology paradigm, and to predict PK dependent onset of drug action [14, 15].

Fig. 2.3 Representation of a physiology-based PK model. Compartments represent real physiological entities, and drug disposition is characterized by the tissue volume, blood flow and partition coefficient defined as the ratio between drug concentrations in tissue and plasma once the distribution equilibrium has been achieved. The inserted graphic provides a granular representation of the intestine. Clint, intrinsic clearance.

Adapted from: Jones H, Rowland-Yeo K: Basic concepts in physiologically based pharmacokinetic modeling in drug discovery and development. CPT Pharmacometrics Syst Pharmacol 2013; 2(8): e63.


PD can be seen as the relationship between drug concentration, generally measured in plasma (CP), and response. There are several aspects that can complicate the CP versus response relationship, which will be discussed in later sections. For now we assume that, first, drug concentrations at the site of action are in equilibrium with those in plasma and, second, that all processes taking place from target engagement to clinical response occur so fast that CP can be related directly to drug effects. Following the simpler description of PK analysis mentioned above, PD could be described as ‘what the drug does to the body’.

For the time being, PD models will be constrained to the case of the analysis of continuous response (i.e. bispectral index, other EEG-derived measures, pupil diameter parameters, degree of neuromuscular blockade, heart rate or blood pressure, just to cite some commonly used and analysed clinical end-points in anaesthesia).

Models for Single Agent

Just like the three compartmental model is the most commonly used model in PK analysis, so is the sigmoidal EMAX model (Fig. 2.4a) the most widely used model in PD analysis [16]. This PD model is based on the receptor theory under the following assumptions: (1) more than one molecule of drug can bind to the receptor, (2) there is a single type of drug–receptor complex, (3) the effect is proportional to the drug–receptor complex and (4) the system is in equilibrium (i.e. fast association and dissociation receptor binding processes). A description of the EMAX model is available in Chapter 1.

Figure 2.4b shows some variants of the sigmoidal EMAX model. In the absence of sigmoidicity (γ = 1) Equation 7 collapses to the EMAX model. When CPC50CP≪C50, or CPγ≪C50γ, the EMAX and sigmoidal EMAX models simply regress to the linear and exponential models, respectively.

In cases where the drug diminishes the baseline response, the following expression (Equation 1) for the sigmoidal EMAX model is used (assuming that the response cannot take negative values) where EMAX is constrained between 0 and 1:


Figure 2.4c represents a characteristic profile corresponding to Equation 1. When EMAX equals 1, the previous expression allows the response value of 0 to be reached asymptotically. The additional PD profiles shown in Fig. 2.4c correspond to a model which allows a sharp decrease in response until it is abolished within the range of drug concentrations used in clinic [17, 18]. In that model, C50 represents CP eliciting a 50% decrease in effect with respect to E0.

Models for Drug Combinations

There are fields in medicine such as anaesthesia and analgesia where the administration of two or more drugs to achieve the pharmacological goals represents the rule rather than the exception. There are several types of PD interactions between drugs, which can be summarized as additivity (no interaction), antagonism, allosterism and synergism. The model in Equation 2 represents an empirical flexible model that can be used to describe additivity, antagonism or synergism. In fact, that model satisfactorily described the synergism between propofol (Prop) and remifentanil (Remi) on the BIS response in patients undergoing sedation for a gastrointestinal endoscopic procedure [19].


The parameter describing the type of interaction in Equation 2 is the parameter α, which can be equal to, greater or lower than 0, depending on the type of interaction occurring between the two drugs: absence of interaction (additivity), synergistic interaction or antagonistic interaction respectively. Usually the values of the steepness parameter γ differ from those of the individual drugs. In those cases the steepness of the concentration versus response curve resulting from the drug combination (γProp,Remi) can be described as a linear function of γProp and γRemi as follows [20]:

γProp,Remi =R • γProp + (1 − R) • γRemi
γProp,Remi= R • γProp+ (1 – R) • γRemi

where the variable R is calculated as rProp/(rProp+rRemi), rProp and rRemi being the ratio between drug concentration and potency (i.e. rProp = Ce,Prop/C50Prop) according to the parameterization originally proposed by Minto et al [21]. The R variable is therefore constrained between 0 and 1, and in the absence of either propofol or remifentanil γProp,Remi is equal to γRemi or γProp, respectively, and Equation 2 reduces to Equation 1.

In cases where one drug (B) cannot exert a measurable effect when administered alone (perhaps because of very low drug concentrations in plasma) but exacerbates the response elicited by a second compound, the following model resembling an allosteric modulation might be considered (Equation 4), where α is the interaction parameter with a value greater than 0.


The interaction between neuromuscular agents acting as antagonists of the postjunctional acetylcholine nicotinic receptor and the antagonist compounds that inhibit acetylcholinesterase, such as neostigmine or edrophonium, that subsequently increase the presence of the agonist acetylcholine has been modelled using the mechanistic model for competitive interaction [22].

All models mentioned above assume the same type of interaction, regardless of the magnitude of the concentration values and their drug interaction ratio. Surface response models [21] overcome that limitation. However, surface response models can prove cumbersome in the case of three or more interacting drugs. In fact, most published examples deal with only two drugs. Recently Wicha et al proposed a comprehensive framework applied to antibiotics that facilitates quantitative description of the response profiles that are the results of the interaction between two or more drugs [23]. Chapter 3 details the presence of synergy in anaesthetic drug interactions.

Pharmacokinetic/Pharmacodynamic Models

In Fig. 2.4 and Equations 14, time is not considered, as the assumption of time invariant PD is implicit. Therefore, to describe the time course of effect, both PK and PD models have to be integrated into a PKPD model, as represented by Fig. 2.5 [3, 4, 16].

Fig. 2.4 Different response versus concentration (PD) profiles together with their corresponding mathematical equations and parameters.

Fig. 2.5 Typical PK, PD and PKPD profiles.

Generally, the dynamics of the response are complex and not simply directly related to the time course of drug concentrations in plasma. This discrepancy between the time course of drug exposure and time to obtain a response can sometimes be solved by proper study design, e.g. in the case of drugs with a fast onset of action it is recommended to measure drug concentration in arterial blood rather than in venous blood. Most PK/PD studies in the anaesthesia setting are based on arterial blood sampling.

Still, even when high quality data are obtained, the lack of direct connection between PK and response persists. This phenomenon is called hysteresis. One of the main contributions of PKPD modelling to clinical pharmacology is to provide comprehensive models capable of describing the course of drug effect in relationship to plasma or blood concentrations in those situations.

Models for Delayed Response

In anaesthesia and pain treatment, the observation of a delayed response is very common. Fig. 2.6 shows a delayed response presented either with respect to time (Panel a), or with respect to drug concentrations in plasma (Panel b), where the hysteresis loop becomes apparent. Both panels show how the same drug concentration seems to be able to have more than one effect. There are several mechanisms that can explain the phenomenon of hysteresis. In anaesthesia the most commonly quoted one is a slow drug distribution between the central compartment (which includes plasma/blood) and the biophase (effect compartment).

Fig. 2.6 a) Plasma drug concentration (Cp; red) and effect (E, blue) versus time profiles. Points show the different magnitude of drug effects at different times for the same value of Cp; b) counter clockwise hysteresis plot; c) schematic representation of the effect site (link) model. Ce, predicted effect site drug concentrations; ke0, first order rate constant governing the drug distribution disequilibrium between the central and effect site compartments.

As described in Chapter 1 the ‘link’ model presented schematically in Fig. 2.6c was conceptualized by Sheiner et al in 1979 [25]. The assumptions of the model are: (1) the effect site receives a small amount of drug, (2) drug transfer to and from the effect site follows first order kinetics, (3) at equilibrium, concentrations of drug in the biophase (Ce) equal those in plasma and (4) total drug concentrations in the biophase are in free form (not bound to receptors). keo is the parameter that controls the time required to achieve distribution equilibrium between the central and effect compartments.


Drug effects are a function of Ce, instead of Cp in Equations 14.

Since its first use, the effect compartment model has experienced further developments, for example the need to have different keo values to describe the time course of BIS at different administration rates [26]. Another example is the discrepancy observed in the impact of chronic administration of phenytoin on the C50 of vecuronium (high potency drug) and rocuronium (low potency drug) where an increase and no effect was found, respectively, with respect to naïve patients [7, 27].

The quantitative characterization of spinal anaesthesia represents another interesting and practical application of the PKPD framework that couples compartmental PK models (under certain constraints/assumptions regarding drug disposition in the different segments) with the link model, an approach used by Olofsen et al to describe the effects of levobupivacaine and ropivacaine in humans [28].

The so-called turn-over or indirect response models [29] and those models that account for slow receptor deactivation are used to describe delayed response. The indirect response model has been used to describe the respiratory depressant effects of opioids, based on the main assumption that the studied response (i.e. pO2, pCO2) is the result of a balance between the rates of synthesis and degradation, each of them governed by the corresponding rate constant. A drug acts either by inhibiting or stimulating the synthesis or degradation rates, where the relationship between the synthesis or degradation rate constant and drug concentrations can be characterized by using the sigmoidal EMAX model or any of its variants. The indirect response models are flexible enough to incorporate complexities such as tolerance and rebound phenomena, as recently shown by Hannam et al [30].

A pharmacological response is the result of the interaction between the drug and its target receptor. Because in anaesthesia rapid onset and offset of drug effect are required, a rapid elimination and reversible drug–receptor binding characteristics are required. On the contrary, in the treatment of chronic pain, sustained analgesic effects are desirable, and in those cases a slower receptor deactivation might be indicated. In the analysis of the migraine effects of the first in class calcitonin gene-related peptide receptor antagonist, Trocóniz et al described the time course of blocked receptors (CPR) as represented by Equation 6, considering the response is related directly to CPR [31].


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Aug 31, 2020 | Posted by in ANESTHESIA | Comments Off on 2 – Pharmacokinetic and Pharmacodynamic Modelling in Anaesthesia

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