British Journal of Clinical Pharmacology

DOI:10.1111/bcp.12286

Pharmacokinetic– pharmacodynamic modelling in anaesthesia

Correspondence Dr Iñaki F. Trocóniz, Department of Pharmacy and Pharmaceutical Technology, School of Pharmacy, University of Navarra, Irunlarrea 1, Pamplona 31080, Spain. Tel.: +349 4842 5600 ext (806507) Fax: +349 4842 5740 E-mail: [email protected] -----------------------------------------------------------------------

Pedro L. Gambús 1

1,2,3

& Iñaki F. Trocóniz

Keywords

4

Systems Pharmacology Effect Control & Modeling (SPEC-M) Research Group, Anesthesiology

Department, Hospital CLINIC. Barcelona, 2Institut d’Investigacions Biomèdiques August Pi Sunyer 3

(IDIBAPS) Villarroel 170, Barcelona 08036, Spain Department of Anesthesia and Perioperative Care, University of California San Francisco (UCSF), San Francisco, CA, USA and 4Department of Pharmacy and Pharmaceutical Technology, School of Pharmacy, University of Navarra, Pamplona, Spain

anaesthesia, clinical applications, pharmacometrics, PKPD models -----------------------------------------------------------------------

Received 29 August 2013

Accepted 31 October 2013

Accepted Article Published Online 20 November 2013

Anaesthesiologists adjust drug dosing, administration system and kind of drug to the characteristics of the patient. They then observe the expected response and adjust dosing to the specific requirements according to the difference between observed response, expected response and the context of the surgery and the patient. The approach above can be achieved because on one hand quantification technology has made significant advances allowing the anaesthesiologist to measure almost any effect by using noninvasive, continuous measuring systems. On the other the knowledge on the relations between dosing, concentration, biophase dynamics and effect as well as detection of variability sources has been achieved as being the benchmark specialty for pharmacokinetic–pharmacodynamic (PKPD) modelling. The aim of the review is to revisit the most common PKPD models applied in the field of anaesthesia (i.e. effect compartmental, turnover, drug–receptor binding and drug interaction models) through representative examples. The effect compartmental model has been widely used in this field and there are multiple applications and examples. The use of turnover models has been limited mainly to describe respiratory effects. Similarly, cases in which the dissociation process of the drug–receptor complex is slow compared with other processes relevant to the time course of the anaesthetic effect are not frequent in anaesthesia, where in addition to a rapid onset, a fast offset of the response is required. With respect to the characterization of PD drug interactions different response surface models are discussed. Relevant applications that have changed the way modern anaesthesia is practiced are also provided.

Introduction The emergent sciences of pharmacometrics and systems pharmacology which currently are being applied in drug development, basic pharmacologic investigations and clinical use, integrate the knowledge gathered over the last three decades in pharmacokinetics (PK), pharmacodynamics (PD), population approach in data analysis, optimal control and design methods among others. The field of anaesthesia has contributed in a very significant way to almost all the aspects that have led to the birth of pharmacometrics and clinical practice in anaesthesia represents a paradigm of translational medicine. PK and PD models developed in controlled clinical trials drive the input rates of several anesthetics through computer controlled infusion pumps. The concept of context sensitive half-life or decrement time is another result of 72

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translational modelling with important practical implications for anaesthetic practice. Pharmacometrics and system pharmacology heavily rely on models describing and predicting in vivo drug action. The pioneer work establishing the relationship between PK and PD in cases of delayed response was performed using an agent within the class of drugs commonly used in anaesthetic practice, the neuromuscular blocking agents (NMBA) [1]. The framework of the indirect response models has also been applied in anaesthesia as a tool to describe respiratory depression due to opioid effects [2]. Drug combinations, a therapeutic practice needed not only in anaesthesia, have received considerable attention in this field in which models for competitive and noncompetitive drug interactions have been presented. The response surface model framework has also been investigated and widely used in anaesthesia [3]. © 2013 The British Pharmacological Society

PKPD models in anaesthesia

Quantifying and explaining inter-patient variability is essential for dose selection and personalized clinical practice. The population approach has been widely used in anaesthesia, and relationships between PK and PD parameters and patient characteristics have been established [4]. More recently genetic factors have been found to influence anaesthesia response and have been incorporated in the corresponding population PKPD models [5]. Clinical practice in anaesthesia represents the paradigm of translational medicine. Results obtained from the modelling approach in controlled clinical trials drive the input rates of several anesthetics through computer controlled infusion pump systems. Concepts such as context sensitive half-life with important practical implications represents another example of translational modellingbased results. It should be considered that in most fields of medicine, drugs are administered for days and weeks or even longer and plasma concentrations reach a steady-state. Conversely anaesthetic drugs are applied mostly in the nonsteady-state, the effect is needed within minutes or even seconds and also the decay of effect needs to be targeted precisely within minutes in order to maintain a satisfactory workflow in a busy operating environment. PD models enable the anaesthesiologist to improve timing, dosing and discontinuation of drug administration in order to achieve a sufficient effect at the required time, and to terminate the effect also in a timely manner. In general there is not a unique response elicited by an anaesthetic drug. Cardiovascular, analgesia and sedation are examples of drug effects associated with anaesthetics. Some of those responses are non-continuous response variables and therefore require proper treatment. Anaesthesiologists adjust drug dosing, administration system and kind of drug to the characteristics of the patient. Then they observe the expected response and adjust dosing to the specific requirements according to the difference between observed response, expected response and the context of the surgery and the patient. The approach above can be achieved because on one side quantification technology has made significant advances in the recent years allowing the anaesthesiologist to measure almost any effect by using non-invasive, continuous measuring systems. On the other because the knowledge on the relations between drug dosing, concentration, biophase dynamics, and drug effect as well as detection of variability sources is probably the highest amongst other medical specialties and it has been achieved as being the benchmark specialty for PKPD modelling. Overall, anaesthesia can be considered a paradigm for personalized medicine [6]. The aim of this review is to revisit the most common PKPD models applied in the field of anaesthesia through representative clinical and pre-clinical examples. Relevant applications that have changed the way modern anaesthesia is practiced are also provided. The following MeSH

terms were used during the literature search: ‘modelling in anaesthesia’, ‘pharmacokinetic–pharmacodynamic modelling of analgesics and anaesthetics’, ‘drug interactions in anaesthesia’.

PKPD models used in anaesthesia The purpose of PKPD modelling is to describe the time course of drug effect in relation to the dose over time. The PK part deals with the time course of the plasma concentration over time, while the PD part relates the plasma concentration with effect. However the impact of PK and PD on the time course of effect is sometimes not easily separated as it has been recently shown [7]. In combined PKPD studies the two components are investigated simultaneously which requires serial blood samples and continuous measurement of effect. Another approach is to use known PK models and relate predicted plasma concentrations with the measured effect. It is not uncommon to get access to response data and not to plasma concentrations. The concept of K-PD [8] modelling was first used in the field of anaesthesia for the drug vecuronium [9]. PK modelling is not the focus of this work but is also a field in which anaesthesia has provided significant advances. Physiological based models have been built for several anaesthetics for both animals and humans. Physiological based models consisted on a series of compartments resembling the different organs and tissues each of them characterized by the corresponding blood flow, real volume and partition coefficient. Building physiological based models requires considerable effort to gather and merge together data from different sources (i.e. in vitro data, in vivo pre-clinical experiments, human studies). In contrast compartmental models split the body into a number of compartments (usually three in the field of anaesthesia) which have no physiological meaning. Developing compartmental models requires just plasma drug concentration longitudinal data. It is generally accepted that physiological based models have better predictive performance to extrapolate PK profiles across different species and patient populations than the compartmental models. However, recently it has been shown that compartmental models can performed at least as well as physiologically based models [10]. Recent examples in anaesthesia are the works of (i) Levitt & Schnider [11] in which a human physiological based model for propofol is presented and (ii) Anderson & Larsson [12] on maturation of midazolam clearance. Most of PKPD models developed to date are data driven. Proper study designs are needed to ensure predictive models and accurate parameter estimates. There are very good examples in the field of anaesthesia showing how to perform a PKPD study taking into account critical aspects in the design. Schüttler and co-workers [13] Br J Clin Pharmacol

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characterized the concentration vs. effect of ketamine in healthy volunteers. In their study plasma concentrations of the two enantiomers of ketamine were measured and PD models were established for each of the enantiomers separately and in combination. Anaesthetics are characterized by a rapid response onset and therefore proper description of the disposition processes at early times after injection is essential. For example in the work of Schnider and co-workers [14] 14 blood samples were obtained during the first 120 min after drug administration. In addition the site of measurement (arterial vs. venous blood) can have also an impact for drugs with rapid onset of action. Maximal response can occur after the time of maximum drug concentration when measurements are taken from venous blood, suggesting apparent acute tolerance [15]. Several works have investigated the influence of the arterio-venous concentration differences on the PKPD parameter estimates [16, 17].

Single agent Models for delayed response The delay between plasma concentration and effect is due to diffusion of the drug from plasma to the cell (mostly brain cells), the drug receptor interaction and the intracellular signal processing. The result is a hysteresis loop (as shown in Figure 1B and C, where the plasma concentration (CP) is related to effect in time order), which implies that a certain effect can occur at different plasma concentrations [a higher one during onset and a lower one during offset of effect (Figure 1A)]

A

and indicating that it is not possible to relate CP directly with response. There are several reasons (together with their corresponding models) that can be the cause of delayed responses. In the following we focus on those most used in the field of anaesthesia to describe and predict drug response. Effect compartment model The effect compartment model (ECm) described schematically in Figure 2A and first proposed by Segre [18], assumes that the absence of a direct relationship between CP and response is the result of a delay in the distribution process between the central and effect site compartment. The concept behind the (unobserved and many occasions unknown) effect compartment has had an enormous impact in clinical pharmacology, and opened the era of describing, understanding and predicting the time course of in vivo drug action. Although it has been applied in almost all therapeutic areas, the ECm was formalized and first applied using response measurements (percentage of blockade) obtained during and after administration of dtubocurarine [1], a neuromuscular blocking drug. The model assumes that (i) the effect site receives a small amount of drug, that is pharmacokinetic properties of the drug are not affected by postulating that additional compartment, (ii) drug transfer to and from the effect site follows first order kinetics, (iii) at equilibrium, concentrations of drug in plasma and biophase are the same and (iv)

B

Cp, Effect 1, Effect 2 vs. Time

C Effect 1 vs. Cp

Effect 2 vs. Cp

Cp

Effect 1

Effect 2

t1

t2

Counterclockwise hysteresis

Clockwise hysteresis

Figure 1 Time course of concentrations and effects. A) Time course of drug concentrations measured in plasma (Cp), and two response variables (Effect 1 and Effect 2). Sampling times, t1 and t2, show same values of Cp but correspond to very different levels of response. B and C) Effect vs Cp relationships. Arrows indicate time direction 74

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PKPD models in anaesthesia

C Effect compartment

A

Peripheral effect site

Biophase Cp

ke12

k1e

Ce

Response Cp

k1e

ke21

Central effect site

Response

ke0 ke0

B

Cp

k1p

Interstitial compartment

kpe

Effect compartment

ke0

Figure 2 Different interpretations of the relation between concentration and effect. Schematic representation of A) the effect compartment model proposed by Sheiner et al. [1] B) the interstitial ECm model (Schiere et al. [28]) and C) the two-compartments ECm (Björnsson et al. [29]). k1e, ke0, ke12, ke21, k1p and kpe, represent first order rate constants. Cp and Ce denote plasma and effect site drug concentrations, respectively

the concentration of drug bound in the biophase is negligible compared with free drug. The time course of the drug in the biophase (Ce) is predicted accordingly to equation 1, where ke0 (the fundamental parameter of the ECm) governs the rate of exit from the effect site, controlling therefore the time required to achieve the distribution equilibrium.

dC e = k e 0 × (C P − C e ) dt

equation 1

Ce (and not CP) is related directly to the response (E) as it is shown in equation 2, using as an example the sigmoidal Emax model, where the parameters E0, Emax, C50, and γ correspond to the baseline condition of the response, maximum change in response with respect to E0 that the drug can exert, level of Ce eliciting half of Emax, and the steepness parameter of the effect vs. Ce curve, respectively.

E = E 0 ± Emax ×

C eγ γ C + C50 γ e

equation 2

Examples of a perfectly designed studies with high resolution analysis are the PKPD studies on remifentanil, a short acting opioid, by Minto and co-workers [19, 20] and

the PKPD studies on propofol by Schnider and co-workers [14, 21]. In both studies a three compartment model for PK was combined with an effect compartment for PD and many model parameters are functions of patient characteristics, such age, gender, height and weight. These covariates improve the fit of the data in the study and allow a more precise titration of the drugs when the models are used in clinical practice. A similar approach has been used for inhalation anaesthetics, although the PK model included five compartments [22, 23]. There are many examples in the anaesthesia literature in which the ECm (equations 1 and 2) has been used using both continuous (i.e. percentage of neuromuscular blockade, processed electroencephalogram analysis) or non-continuous (i.e. pain or sedation scores) type of measurements [24–27]. Since its first applications, and due to its frequent use in many aspects and conditions in anaesthetic practice, the ECm has experienced further developments. Schiere et al. incorporated an intermediate (interstitial) compartment between the central and the effect compartments (Figure 2B) to describe the time course of the effect of mivacurium, a neuromuscular blocking agent, in humans. In the original manuscript values of k1p and ke0 of 0.374 and 0.151 min−1 were reported. When the standard effect compartmental model was fitted to the data the estimate obtained for ke0 was 0.041 min−1 [28]. Br J Clin Pharmacol

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The bispectral index (BIS) is a measure of hypnotic effect used in clinical anaesthesia and is based on complex mathematical analysis of the electroencephalographic waves. The effects of propofol on the BIS in healthy volunteers receiving propofol intravenously at different rates were described using a variant of the ECm consisting of two compartments, (central and a peripheral) within the effect site (Figure 2C). In their analysis, the authors reported values for ke0, ke12 and ke21 of 0.159, 0.114 and 0.0214 min−1, respectively. Drug effects were related to the predicted concentrations in the central effect site compartment [29]. The proposed model was presented as an alternative to the original ECm model in cases where, as the example of propofol, different values of ke0 were observed with different rates of administration [30]. One of the basic assumptions of the ECm model is that in the biophase, drug concentrations bound to the receptors are negligible in comparison with the unbound concentrations. There are situations such as the case of the neuromuscular blocking agents in which 85% of receptor occupancy is required to obtain 50% of maximum blocking effect, and therefore bound concentration in the biophase might not be negligible. Back in 1991 Donati & Meistelman questioned such an assumption and extended the ECm taking into account bound drug concentrations for the case of neuromuscular blocking agents [31]. The model resembles the one proposed by Wagner in the past accounting for non-linear tissue binding [32]. The model derived by Donati & Meistelman might provide a plausible explanation for the lack of change in the C50 parameter of rocuronium (836 ng ml−1), a low potency drug in patients undergoing chronic treatment with phenytoin [24] in contrast to the increase in C50 from 95 to 213 ng ml−1 for the case of vecuronium, a high potency drug, in the same type of patient population [33]. Phenytoin is known as an enzymatic inductor and its effects on drug clearance have been reported several times. Other types of proteins (i.e. acetylcholine receptors) can also be up-regulated in the presence of phenytoin. The assumption that drug response is related to the percentage of occupied receptors in the biophase, might explain why after the administration of equipotent doses of vecuronium and rocuronium, only the former experienced an apparent decrease in its potency questioning that drug bound in the biophase can be neglected. Within the ECm framework another important contribution that came from PKPD analysis in anaesthesia is related to alternative ways to establish the relationship between drug concentration in the biophase and response for which equation 2 represents the most popular expression. On several occasions the following expression (power model) has been used to relate Ce with drug effect [34–36]:

( )

C ⎡ E = E 0 × ⎢1± 0.5 × e C50 ⎣ 76

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γ

⎤ ⎥⎦

equation 3

Response

P. L. Gambús & I. F. Trocóniz

Ce

Figure 3 Pharmacodynamic models. Response vs. Ce profiles corresponding to the sigmoidal (red) and power (blue) models. Parameters used for the simulations: sigmoidal Emax model (E0 = 100; Imax = 1; C50 = 50 ng ml−1; γ = 2), power model (E0 = 100; C50 = 50 ng ml−1; γ = 2). In both simulations it is assumed that the drug induces an inhibitory effect

where C50 represents the effect site concentration eliciting a 50% increase or decrease in effect with respect to baseline. This response model is indicated when the drug concentrations are lower than those exerting the maximum effect. Equation 3 has been slightly modified estimating C25 and C100, parameters, which are defined similarly to C50. Figure 3 shows typical profiles corresponding to equations 2 and 3. Turnover models. These type of models [also called indirect response models in the PKPD literature [37] assume that the drug acts by modifying the process governing the turnover of the studied response. These types of models can also account easily for phenomena such as rebound and tolerance (up- and down-regulation) [38, 39]. In the area of anaesthetics these types of models have been applied to describe respiratory depressant effects of opioids in non-steady-state situations [2, 40, 41], although Romberg et al. in humans [42] and Yassen et al. in rats [43] used the ECm for the case of morphine-6-glucuronide and norbuprenorphine, respectively. Figure 4 represents the model for the respiratory effects of remifentanil in awake and propofol-sedated healthy volunteers developed by Olofsen et al. [41]. The general behaviour of the model is as follows: (i) remifentanil increases the apneic threshold through its effect site concentrations, which induces a delayed reduction on ventilation, (ii) reduction in ventilation produces an increase in the arterial carbon dioxide concentrations, which (iii) triggers a stimulatory effect on the respiratory controller. The model proposed by Olofsen et al. was built based on measurements of ventilation and end-tidal carbon dioxide concentrations, under the assumption that end-tidal PCO2 (PE) is an accurate indicator of PA [41].

PKPD models in anaesthesia

Vco2 VAL ¥ PV

dPA = –V ¥ PA + l1 ¥ Q ¥ [PV – PA] dt

PA Q

V

VTS ¥

dPV = Q ¥ [PV – PA] + l2 ¥ Vco2 dt

+

–

1/t C B = B0 ¥ È1+ e_Rem C100 Î

È Î

[G¥(PE–B)]¥1/t

t ¥ dV = G ¥ EProp ¥ [PE – B] – V dt

Respiratory controller (V)

+

Effect site

ke0

V0 = G ¥ [PE_0 – B0]

Propofol

Remifentanil

dCe_Rem = ke0 ¥ [CRem – Ce_Rem] dt

Figure 4 Schematic adaptation of the model published by Olofsen et al. [41], to describe respiratory effects of remifentanil and propofol. PA and PV are the arterial and venous carbon dioxide pressure, respectively; VAL and VTS correspond to the alveolar and tissue volumes, respectively; Q is the cardiac output; V0 and V represent the inspired minute ventilation at baseline and during anaesthesia, respectively; VCO2 represents the carbon dioxide production; λ1 and λ2 are scaling parameters; G, gain of the ventilatory control system; PE_0, and PE, end-tidal PCO2 at baseline and during anaesthesia, respectively; B, apneic threshold; τ, time delayed constant, EProp parameter accounting for the reducing effects of propofol on G [=1, in absence of propofol]; CRem and Ce_Rem, plasma and predicted effect site concentrations of remifentanil, respectively; ke0, first order rate constant governing the elimination process from the effect site; C100, effect site concentrations of remifentanil that elicits a 100% increase in B

The previous model described has very interesting properties. It incorporates effect site dynamics, and turnover kinetics including rebound phenomena. Moreover it can be considered a nice application of the systems pharmacology approach since it keeps the physiological framework of the system using model parameters obtained from the literature. VAL, λ1, λ2 and τ are the parameter fixed during the analysis. The parameters to be estimated are V0, PE0, Q, C50, ke0 and G. Receptor binding models. In general is difficult, based just on in vivo response data, to characterize the drug– receptor binding characteristics in addition to the distribution to biophase. Yassen et al. described mechanistically the antinoceptive effects of buprenorphine in rats [35] and healthy volunteers [36], respectively. The following equation briefly describes the model:

dC eR = kon × C e × R − koff × C eR dt

equation 4

where R is the free receptor concentration, CeR represents the drug–receptor complex, kon, is the second order rate constant describing the association between drug and its receptor and koff, is the first order rate constant describing the dissociation process of the drug–receptor complex. Taking into account that Rtot, the total receptor concentration (arbitrarily set to 1) is equal to the sum of R and CeR,

and considering that R < < Ce, equation 4 can be rearranged as follows:

dC eR = kon × C e × [R tot − C eR] − koff × C eR dt

equation 5

Finally drug effect (E) was related directly to CeR and baseline response (E0) as:

E=

E0 1− C eR

equation 6

The values of kon, koff and ke0 reported for rats were 0.0228 ml ng−1·min−1, 0.0731 min−1 and 0.0242 min−1, respectively. The corresponding estimates for humans were 0.0631 ml ng−1 min−1, 0.0785 min−1 and 0.00447 min−1, respectively. Interestingly in the work of Yassen et al., under a similar experimental setting but applied to the opioid fentanyl, parameters kon and koff were not identifiable [35]. Models for spinal anaesthesia There are two different options in spinal anaesthesia techniques, epidural and subarachnoidal blockades. Describing the time course of epidural anaesthesia represents an interesting application of the ECm framework. Figure 5 summarizes the model Br J Clin Pharmacol

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T4

Effect site ke0_T4

T5 T6 T7

Effect site ke0_T7

T8 T9 T10

Effect site ke0_T10

T11 T12 L1

Effect site ke0_L1

L2 L3 L4

Effect site ke0_L4

L5 S1 S2

Effect site ke0_S2

S3 S4 S5

Effect site ke0_S5

Ce[t] < Cthr_T4 Ce[t] ≥ Cthr_T4

No block

Ce[t] < Cthr_T7 Ce[t] ≥ Cthr_T7

No block

Ce[t] < Cthr_T10 Ce[t] ≥ Cthr_T10

No block

Ce[t] < Cthr_L1 Ce[t] ≥ Cthr_L1

No block

Ce[t] < Cthr_L4 Ce[t] ≥ Cthr_L4

No block

Ce[t] < Cthr_S2 Ce[t] ≥ Cthr_S2

No block

Ce[t] < Cthr_S5 Ce[t] ≥ Cthr_S5

No block

Block

Block

Block

Block

Block

Block

Block

Figure 5 A model explaining the relations between concentration and effect after epidural administration of a local anaesthetic drug. Schematic adaptation of the model developed by Olofsen et al. [44] to describe the anaesthetic effects of levobupivacaine and ropivacaine after epidural administration

proposed by Olofsen et al. to deal with the effects of levobupivacaine and ropivacaine in humans [44]. The time course of both agents in the systemic circulation was first described using data after epidural and systemic administration. The selected PK model consisted on two depot absorption compartments and a three compartmental model describing drug disposition. It was assumed that (i) the longitudinal spread of the drug across the segments of the epidural space occurs instantaneously and (ii) the absorption parameters are equal between the different segments. The model allowed the estimation of two different parameters per segment, ke0 and Cth, the latter indicating drug sensitivity, using data of probability of blockade. Another interesting work proposed a model of two exponential terms to describe the extension of sensitive blockade, absence of any tactile and painful feeling in the patient, after subarachnoid injection of a bolus dose of 0.5% bupivacaine using the extension of anaesthetized dermatomes as a PD endpoint [45].

other aspects of PKPD analysis, anaesthesia has contributed extensively to the description of the time course of the in vivo response of drug combinations. Often a drug is administered to antagonize a response as in the case of neostigmine or edrophonium, which have no intrinsic activity and act as competitive antagonists of the neuromuscular blocking agents. Verotta et al. modelled the reversal of neuromuscular blockade in humans by these two antagonist in humans [46]. The interaction between midazolam and flumazenil in rats provides another good example [47]. Equation 7 represents the competitive interaction between an agonist (drug A) and an antagonist (drug B), where Imax is the maximum attainable response limited between 0 and 1.

Models for drug interactions

Mandema et al., showed examples of modelling the interaction between active compounds (full and partial agonists, including active metabolites) for the case of benzodiacepines in rats and humans [48, 49]. The

It is a common practice in anaesthesia to achieve the desired response taking advantage of the interaction obtained by using combinations of different drugs. As in 78

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CA ⎡ ⎢ C50 ,A E A ,B = E 0 × ⎢1− Imax × CA C ⎢ 1+ + B C50 ,A C50 ,B ⎣

⎤ ⎥ ⎥ ⎥ ⎦

equation 7

PKPD models in anaesthesia

following expression can be used in the case of a competitive interaction between two active compounds. Equation 8 assumes that drug A is a full agonist and drug B is a partial agonist (0 < β < 1).

CA C ⎡ +β× B ⎢ C50 ,A C50 ,B E A ,B = E 0 × ⎢1− Imax × C C ⎢ 1+ A + B C50 ,A C50 ,B ⎣

⎤ ⎥ ⎥ ⎥ ⎦

equation 8

Despite the fact that sometimes an antagonistic effect is desirable, in most of the cases drugs are combined aiming at a synergistic effect. Equation 9 represents a general model [50] to describe antagonism (α < 0), additivity (α = 0) or synergism (α > 0). n

INT ⎡ ⎛ C A + CB + α C A × CB ⎞ ⎢ ⎝ C50 ,A C50 ,B C50 ,A C50 ,A ⎠ E = E 0 × ⎢1− Imax × nINT ⎢ ⎛ C A + CB + α C A × CB ⎞ 1 + ⎢⎣ ⎝ C50 ,A C50 ,B C50 ,A C50 ,B ⎠

⎤ ⎥ ⎥ ⎥ ⎥⎦

equation 9 In equation 9 the parameter nINT, which represents sigmoidicity has the form of

nINT = nA ⋅ θ + (1− θ) ⋅ nB where nA and nB are the sigmoidicity parameters of the E vs. CA, and E vs, CB curves respectively. θ is constrained between 0 and 1 and has the following expression (equation 10):

UA θ= UA + UB

equation 10

where UA and UB are CA/C50,A and CB/C50,B, respectively, using the notation proposed by Minto and co-workers [3]. The model represented by equations 9 and 10 reduces to the sigmoidal model (equation 2) when only a single drug is present. Recently Borrat et al., used the above model to describe the interaction between propofol and remifentanil during endoscopic anaesthesia [5]. The synergistic effects between propofol and remifentanil have also characterized using a simplified version of equation 9 in which the switch between nA, and nB as a function of θ was not considered [51]. So far the models describing drug interactions predict the same type of interaction between the active compounds regardless of the ratio between concentrations. Minto et al. introduced the concept of response surface models applied to anaesthetic drug interactions [3]. Response surface models have been applied repeatedly

to describe the interaction between propofol and remifentanil [52], propofol and sevoflurane [53] and remifentanil and sevoflurane [54]. In the response surface modelling framework Minto and co-authors proposed the following general expression [3]:

(

)

UA + UB n( θ ) ⎤ ⎡ ⎢ ⎥ U50 (θ) E = E 0 × ⎢1− Imax (θ) × n( θ ) ⎥ U + UB ⎢ ⎥ 1+ A ⎢⎣ ⎥⎦ U50 (θ)

(

)

equation 11

In their original work the authors chose fourth-order polynomials for Imax(θ), n(θ) and U50(θ). However the model can be simplified if drugs A and B can abolish completely E (i.e. Imax = 1), using equation 10 for n(θ), and the expression 1 − β x θ + β × θ2 for U50(θ) as suggested in [3]. Greco’s and Minto et al.’s models [3, 50] are empirical models for drug interactions. Bouillon et al. [55] proposed the hierarchical response model to characterize the interaction between propofol and remifentanil on pain intensity. This model represents basic pharmacological mechanisms as follows: First, opioid (remifentanil) concentrations lower the painful stimulus, which is then projected to the cortex where the hypnotic drug (propofol) acts decreasing the probability of response to a painful stimulus during surgery. Other representative works dealing with PD drug interactions in anaesthesia correspond with Schumacher and co-workers [56] and Heyse and co-workers [57], the latter comparing across different response surface models.

New concepts derived from the application of PKPD modelling in anaesthesia The specific idiosyncrasy of anaesthesia practice has originated several new pharmacologic concepts that are now applied to the understanding of the relationship between drug dose, concentration and effect.

tpeak: defining the time of onset of drug effect and its applications The time of peak effect, tpeak, can be defined as the time of maximal intensity of effect after the injection of a submaximal dose of any drug. tpeak is a parameter independent of the amount of the bolus of drug administered and independent of model of drug behaviour. It is obviously dependent on the effect that is being measured. For instance for a hypnotic like propofol tpeak, for changes in BIS of the EEG, will not be necessarily the same as maximal depression of arterial blood pressure. When combined with a PK model tpeak can be used to estimate the value of Br J Clin Pharmacol

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ke0 yielding the effect site concentration of the drug at the time of tpeak [58]. Then a prediction can be made on the time course of effect site concentration for any given administration method. tpeak has been estimated for propofol, remifentanil, thiopental [58] and rocuronium [24]. tpeak has a significant value from a pharmacologic point of view since it allows comparison of onset times between drugs regardless of the kind of drug. From a clinician’s perspective the choice of anaesthetic drug can be made based on onset time. Rational dosing with regard to onset and offset can be planned in advance based on objective concepts besides clinical knowledge and experience.

Estimation of decrement of drug effect Continuous administration is used most of the time to maintain the anaesthetic state by means of stable drug concentration and effect without the peaks and troughs associated with the administration of an intravenous bolus. Most of the estimations for duration of effect of anaesthetics are based on the value of the terminal half-life of the drug, a parameter that is useless per se for this purpose, in drugs whose behaviour is described by multicompartment models and this is the case for most anaesthetic drugs. Based initially on PK models and later on full PKPD models Shafer et al. proposed a new look for the decay on plasma concentration, or biophase concentration, after continuous infusion of varying duration. The concept of ‘context sensitive decrement time’ reflects this view [59, 60]. It allows a graphical description of the time it takes a given drug concentration, in plasma or effect site, to fall by a given percentage after stopping the infusion. It is not a single number but a graphical representation and it will always depend on the ‘context’ which is the duration of infusion of drug (Figure 6). The practical application of this concept is that the anaesthesiologist now can have an estimate of duration of effect as a function of the PKPD of the drug, duration of administration and intensity of dosing.

Application of PKPD modelling to drug administration: target controlled infusion (TCI) systems A PKPD model allows the prediction of an effect site concentration for a given dose. It can also estimate the inverse problem, the required dose to be given to achieve a certain, predefined target plasma or effect site concentration and maintain it for as long as it is required by adjusting the administration rate every time unit. This is what a TCI system does. Although the principles were established more than 40 years ago [61] its development in the experimental and clinical setting arrived with the widely spread use of com80

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Figure 6 Context sensitive decrement times for propofol according to the PKPD model of Schnider et al. [14]. The graph shows the time it would take for effect site propofol concentration to fall by an 80% as a function of infusion duration in a 80-year-old subject as compared with a 40-year-old. Also represented is the time it would take for a decrease of 50% in an 80-year-old vs. a 40-year-old

puters in the operating theatre. Most of the research in this area was done in the USA during the 1980s and 1990s [62–64]. Due to regulatory issues, the USA is now one of the few countries where TCI systems are not approved for use in anaesthesia. The goal of a TCI system is to achieve rapidly and maintain as long as necessary a target concentration in the plasma or effect site of the patient. It uses the covariate model to estimate the dose that must be administered to achieve the target concentration model based on the characteristics of the patient (age, gender, weight, height). In commercially available systems this information is recalculated every 10 s so that target concentrations can be kept stable by compensating for the loss of drug by elimination or distribution to peripheral compartments. If a new higher target is required, a new dose is calculated and given to reach the new level. If a lower target is chosen the system stops infusing but keeps estimating predicted concentration and when the new target is reached it starts infusing again. The time course on a hypothetical case where the opioid remifentanil is administered by a TCI system can be seen in Figure 7. The amount of work that proves the clinical utility of this technology is growing fast. During the early 1990s STANPUMP (Steven Shafer, Anesthesia Department, Stanford University, USA) and CACI, the acronym for computer assisted continuous infusion (James R Jacobs, Anesthesia Department, Duke University, USA) were the more prevalent systems used in clinical pharmacology research. Nowadays different companies have already commercialized TCI systems and its use is becoming a standard in intravenous anaesthesia techniques. The PKPD models used by the TCI systems usually integrate information from covariate factors: age, weight,

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Figure 7 An example of the time course of plasma and effect site predicted concentrations when using a TCI system to administer remifentanil. The device is targeting the effect site, hence the relative overshooting in plasma to achieve faster the pseudoequilibrium at the biophase. Targeted concentrations are 1, 4, 3 5 and 0 ng ml–1

height, gender, so that the size of the parameters of the model adapt to the characteristics of the patient decreasing variability in the expected response. Using a TCI system by definition will always be more on target (less variable) than a bolus or continuous infusion [65]. A potential application of TCI systems is automatic closed loop administration of anaesthesia. Several authors have published comparisons between manual and automatic control of the administration of NMBA [66] hypnotics [67, 68] or analgesics [69]. In general it can be said that automatic control is able to react faster to change and is able to stay on target for more time and more accurately than manual administration. Even though there are promising results in patients it has not been introduced commercially for clinical use. It should be expected to integrate of models of synergy in TCI systems to allow a better prediction of drug effect especially when, as usual, hypnotic and analgesic drugs are given in combination during clinical anaesthesia.

Other applications of PKPD models in anaesthesia

might be painful or uncomfortable otherwise. The therapeutic range for sedation usually is above the unconsciousness level, trying to keep him/her responsive to verbal or tactile stimuli, and it must be also above the respiratory depression level. The patient must be comfortable, without pain, arousable to stimulation and breathing on his/her own. TCI systems provide the ability of slow, stepwise, titration of drug effect according to the characteristics of the patient and the response of the patient to the dosing scheme with respect to the therapeutic (sedation, analgesia) [70] and side effects (airway obstruction, respiratory depression, haemodynamic unstability) [71]

Monitoring systems Technical evolution in anaesthesia has improved the way drug effects are controlled. Hypnotic effects can be measured by means of different indicators derived from processing the electroencephalographic signal. Haemodynamics and ventilator function control is possible through cardiovascular monitoring and ventilator systems as well as pulse oximetry or capnography. More sophisticated methods are also available in the care of critical patients like the possibility of measuring continuously cardiac output, respiratory rate, response to fluid administration, probability of response to noxious stimulation by a variety of systems. Information from dosing can be immediately captured by monitoring systems and become the input for PKPD models including interaction between different drugs given. These models establish predictions of expected effect according to the characteristics of each individual patient. By comparing the expectations of effect to the actual observation of effects the anaesthesiologist can have access to updated personalized information on how the patient is doing through the process of anaesthesia and surgery and detect and adapt in a timely manner to any change occurring. This approach of integration of information is already available in some anaesthesia workstations. The main systems are the Anaesthesia Navigator Applications Suite (GE Healthcare, USA) and the Anaesthesia Smart Pilot View (Dräger, Germany). Both seems to be very promising tools in clinical anaesthesia although there is not enough available information yet.

Modelling sedation analgesia One particular area of anaesthesia where PKPD modelling and its further developments (TCI systems, EEG measures of effect, modeling side effects) might have a wider appplication is sedation analgesia in patients undergoing diagnostic or therapeutic procedures where spontaneous ventilation and some degree of cooperation of the patient is required so that the patient should not be unconscious. Sedation means the administration of anaesthetic drugs with the goal of providing comfort to a patient undergoing diagnostic or therapeutic procedures that

Conclusions Anaesthesia means controlling the effect of very powerful drugs given to a patient, who is undergoing surgery or critical care, under highly dynamic and potentially dangerous conditions. Quantification, titration and adaptation are essential in anaesthesia. PKPD models have provided anaesthesiologists with the ability to individualize drug administration under such dynamic conditions by means Br J Clin Pharmacol

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of sophisticated mathematical approaches and new concepts to understand drug behaviour better. Anaesthesia has been a benchmark where new visions in PKPD modelling have been developed and tested. A wide introduction of genetic principles or physiologic changes due to individual patient responses to surgery will provide new knowledge and, from there, new applications must be expected for the highly synergistic relation between anaesthesia and PKPD modelling.

Competing Interests All authors have completed the Unified Competing Interest form at http://www.icmje.org/coi_disclosure.pdf (available on request from the corresponding author) and declare no support from any organization for the submitted work, no financial relationships with any organizations that might have an interest in the submitted work in the previous 3 years and no other relationships or activities that could appear to have influenced the submitted work. Dr Gambús is supported by FIS (Fondo de Investigaciones Sanitarias, Health Department, Government of Spain)/FEDER (European Regional Development Fund. (ERDF)) grants n° FIS PI/050072, FIS PS09/01209 and BAE 2012/00069.

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