Pharmacokinetic–pharmacodynamic modeling in acute and chronic pain: an overview of the recent literature

Abstract

In acute and chronic pain, the objective of pharmacokinetic–pharmacodynamic (PKPD) modeling is the development and application of mathematical models to describe and/or predict the time course of the pharmacokinetics (PK) and pharmacodynamics (PD) of analgesic agents and link PK to PD. Performing population PKPD modeling using nonlinear mixed effects modeling allows, apart from the estimation of fixed effects (the PK and PD model estimates), the quantification of random effects as within- and between-subject variability. Effect-compartment models and mechanism-based biophase distribution models that incorporate drug-association and -dissociation kinetics are applied in PKPD modeling of pain treatment. Mechanism-based models enable the quantification of the rate-limiting factors in drug effect owing to drug distribution versus receptor kinetics (since receptor kinetics are nonlinear they are discernable from the linear effect-compartment kinetics). It is a helpful technique in understanding the complex behavior of specific analgesics, such as buprenorphine, but also morphine and its active metabolite morphine-6-glucuronide, especially with respect to the reversal of opioid-induced side effects, most importantly life-threatening respiratory depression. One approach in chronic pain studies is the application of mixture models. Mixture models do not necessarily need to take PK data into account and allow the objective differentiation of measured responses to analgesics into specific response subgroups, and as such, may play an important role in analyzing Phase I and II analgesia studies. Appropriate application of PKPD modeling leads to the improvement of current therapeutics with respect to dose design and outcome, understanding the interaction of analgesics within complex chronic pain disease processes and may play an important role in drug development. In the current article, novel observations using the aforementioned techniques on opioids, NSAIDs, epidural analgesia, ketamine and GABA-ergic drugs in acute and chronic pain are discussed.

Keywords::

• acute pain
• analgesics
• buprenorphine
• chronic pain
• effect compartment
• epidural analgesia
• ketamine
• modeling
• morphine
• opioids
• pain
• pharmacodynamics
• pharmacokinetics
• PKPD modeling

Figure 1. Effect of naloxone on opioid-induced respiratory depression.

(A)Mechanism-based pharmacokinetic–pharmacodynamic modeling of the ability of different doses of an opioid antagonist (naloxone, given at t = 55 min, low dose = 0.1 mg, high dose = 0.4 mg) to reverse the opioid-induced respiratory depression from an opioid agonist with relatively low k_off value (opioid agonist injected at t = 0 min). The speed of reversal is independent of the antagonist dose and is determined by the agonist opioid receptor k_off value. (B) Effect of variations in the k_off value of an opioid agonist (injected at t = 0 min over 90 s) on the effect of a fixed dose of an opioid receptor antagonist (naloxone, given at t = 55 min over 90 s) on reversal of opioid-induced respiratory depression.

K_off values are relative with 0.2 min^-1 = 0.2 × typical value, 1 = typical value, 5 = 5 × typical value. Here, the typical value = 0.03 min^-1.

K_off: Receptor dissociation constant.

Data taken from [17].

Figure 2. Population pharmacokinetic–pharmacodynamic modeling of epidural analgesia.

(A) Blockade probabilities after an epidural injection of ropivacaine 125 mg in a 50-year-old patient. The size of the dots is proportional to the probability of blockade with 90% (solid line), 75% (broken line) and 50% (dotted line) isoeffect lines. Time is expressed on the x-axis and the dermatome level on the y-axis. (B) Probability of blockade, P(block), for dermatomes S5 (broken line), Th11 (solid line) and Th7 (dotted line) versus time.

Data taken from [32].

Figure 3. Mixture analyses of the effect of a 1-week ketamine treatment (shaded bar) on pain scored in patients with chronic pain.

The analysis objectively divided the total population response into four categories, with (A) no response to treatment, (B) a response in the treatment week only, (C) a long-term response with a slow return towards pretreatment baseline and (D) full recovery.

Data taken from [33].

The objective of pharmacokinetic–pharmacodynamic (PKPD) modeling is the development and application of mathematical models to describe and/or predict the time course of dose-to-concentration (pharmacokinetics [PK]) and concentration-to-effect (pharmacodynamics [PD]) of pharmacological active agents in health and disease [1,2]. The PK model part of PKPD models describes the macrodistribution of the drug (drug distribution kinetics). Effect-compartment PKPD models add a hypothetical (infinitely small) effect compartment to describe the delay in effect (i.e., quantification of drug effect in terms of drug concentration at the target or effect site [C_E] vs measured effect). In compartmental models, the concentration–time profiles are described/explained by drug transfer between interconnected hypothetical compartments, mimicking drug absorption, distribution and elimination. Drug distribution towards the target site is described by the plasma effect-site equilibration constant (k_e0; or its half-life [t_½k_e0] = ln2/k_e0), whereas the concentration–effect relationship is often described by a sigmoid maximum effect (E_max) model of the form:

where A = C_E/C₅₀ (with C₅₀ as the measure of drug potency or the effect-site or steady-state concentration causing 50% of the effect, and γ as the Hill-factor). The first to propose a first-order rate constant between plasma and an effect-compartment was the Italian, Segre, in a pivotal article published in 1968 [3]. Segre described the transfer function of norepinephrine’s effect on the circulatory system and estimated a delay of 15 s between changes in plasma epinephrine concentration and blood pressure in a cat, showing that epinephrine’s target site (i.e., the ‘biophase’) is not in plasma. This concept was later further developed by Hull and Sheiner et al., in two separate studies, both in 1979 [4,5]. Effect-compartment models are empirical models that do not describe the physiological or mechanistic pathway of the drug from its concentration in plasma to effect. Recently, novel mechanism-based biophase distribution models were developed [1]. For example, mechanism-based PKPD models may incorporate concepts from receptor theory describing the drug–receptor interaction (receptor kinetics) in terms of drug-association and -dissociation kinetics and enable the quantification of the rate-limiting factor in drug effect owing to drug distribution versus receptor kinetics [6].

Rather than analyzing individual PKPD data, population PKPD analyses (i.e., performing a simultaneous analysis on the whole population dataset) has become the acceptable approach in current PKPD modeling studies. A population analysis will lead to an estimate of the population PK and PD estimates, identify the sources of variability that influence PK and PD, estimate the magnitude of between-subject variability, as well as intrasubject variability, and allow for the estimate of random residual variability (random effects) [7]. To separate random (i.e., variability parameters) from fixed effects (i.e., the PK and PD parameter estimates), nonlinear mixed-effects modeling is required [8]. Various statistical packages are available to perform nonlinear mixed-effects modeling, of which NONMEM^® is considered the most state-of-the-art [7–10]. NONMEM was developed by Lewis Sheiner and Stuart Beal at the University of California in San Francisco (CA, USA) and was initially developed to analyze PK data but is currently applied to model PK, PD and PKPD datasets [9,10]. Particularly in studies on the pharmacology of analgesics and anesthetics, drug–drug interactions, population PK, PD and PKPD analyses using NONMEM is currently the gold standard [8,10].

In this article, we will discuss population effect-compartment and mechanism-based PKPD and PD modeling studies published in the last 5 years on the effects of various analgesic agents in acute (in volunteers and postoperative patients) and chronic pain patients. End points covered are the relief of experimental and surgery- and disease-induced pain (i.e., antinociception and analgesia). In addition, we will discuss PKPD models of naloxone reversal of opioid-induced respiratory depression, the most life-threatening side effect of potent opioids.

PKPD modeling of acute antinociception & pain

Morphine & its metabolites

Morphine, first extracted from opium in 1806, is still considered the gold standard in the treatment of severe acute and chronic pain, although it has no clear superiority in efficacy or tolerability over other opioids [11]. In 2005, Lötsch listed six PKPD studies on morphine [12]. Studies were performed either on a surrogate measure of opioid effect, pupil size, or on electrical noxious cutaneous stimulation. One of the studies discussed demonstrated that sex was a significant covariate on pain relief responses, with greater morphine potency in women (C₅₀ = 250 nM in men vs 150 nM in women) (Table 1) but with a slower onset/offset time (blood effect-site equilibration t_½k_e0 for pain relief response [pain tolerance] was 1.6 h in men vs 4.8 h in women), without any sex differences in morphine’s PK [13]. These data were later confirmed in a systematic review and meta-analysis and account for the difference in opioid consumption between men and women in postoperative patient-controlled analgesia morphine studies [14]. Interestingly, while studies on pupil size had similar estimates for t_½k_e0 as observed in pain studies, the potency of morphine was much greater for miosis (C₅₀ ranging from 17 to 24 nM, without occurrence of any sex differences). Collectively, these data indicate that morphine has an onset time that is much smaller than observed in any of the other opioids currently in clinical use [12–14]. In comparison, t_½k_e0 (derived from yet another surrogate end point of opioid effect: changes in EEG power spectrum) is 5–6 min for fentanyl and sufentanil, 1–2 min for alfentanil and remifentanil, 8–19 min for methadone and 17 min for piritramide [12]. This makes morphine a relatively slow and consequently difficult to control opioid analgesic. Note, however, that currently no comparable data on morphine’s effect on EEG-related parameters are available.

Morphine is rapidly metabolized in the liver into the (presumably) inactive morphine-3-glucuronide (M3G) and the active morphine-6-glucuronide (M6G). Animal studies indicate that M6G is a µ-opioid receptor agonist with greater potency than morphine and rapidly crosses the blood–brain barrier (BBB) [15]. By contrast, PKPD studies in humans revealed a three- to five-fold lower potency than observed in animals after intravenous M6G administration coupled with a slow transfer of the drug across the BBB (C₅₀ = 750 nM and t_½k_e0 6–8 h) [12]. Using data obtained from healthy volunteers that were injected with either intravenous morphine or intravenous M6G, we were able to construct a population PKPD model of morphine’s metabolism into M6G and determine M6G’s contribution to morphine analgesia [16]. The fraction of morphine metabolized into M6G was 6.0 ± 0.2% (median value ± standard error); M6G formation was sex independent. Simulation studies demonstrated that repetitive morphine infusions (0.1 mg/kg at 8-h intervals) resulted in stable M6G effect-site (‘brain’) concentrations of 10–20 nM that contributed approximately 8 (women) and 15% (men) to the analgesic response. Under conditions of renal impairment, the effect-site M6G concentration rose by a factor of ten and its estimated contribution to effect increased by a factor of two. These findings derived from modeling studies and simulations are important as they tempered the initial aspirations of M6G as the replacement of morphine in the treatment of severe acute pain and exemplify the strength of population PKPD modeling studies in the development of new (analgesic) drugs [12,15,16]. The aspirations were based on early observations in animals and later in humans that M6G causes less respiratory depression and possibly less nausea and vomiting compared with the parent compound [15]. However, its low potency but especially its slow passage across the BBB and accumulation in renal impairment make it an analgesic that is even more difficult to control than morphine in perioperative patients and should be avoided in patients with reduced renal function.

Previously discussed studies used effect-site PKPD modeling and consequently remained uninformed on receptor kinetics. Using a mechanism-based PKPD approach, Olofsen et al. described naloxone reversal of morphine- and M6G-induced respiratory depression [17]. They estimated the following pharmacokinetic parameters: k_e0 or the biophase distribution constant, k_on or the receptor association constant, k_off or the receptor disociation constant of morphine and M6G, and k_e0 of naloxone. Naloxone is a µ-opioid receptor antagonist used both in clinical practice to reverse opioid-induced respiratory depression and in the treatment of opioid addiction. Since it was first synthesized in 1960 by Jack Fishman and further developed through the early 1970s by Harold Blumberg [18], little progress has been made on the policies required to administer appropriate doses of naloxone that cause rapid reversal with a limited chance of renarcotization. The study by Olofsen et al.[17] and especially the additional model simulations are insightful and enable the development of specific guidelines on the reversal of the toxic effects of morphine and its active metabolite M6G in clinical practice. The parameter estimates of the study are given in Table 1. Morphine and M6G both display slow receptor association/dissociation kinetics with an apparent potency (K_d = k_off/k_on) similar to C₅₀ values obtained in previous studies (see earlier). Naloxone had rapid receptor kinetics with t_½k_e0 of approximately 5 min on average, while its apparent estimated potency was greater for M6G reversal than for morphine reversal. The latter observation is hard to explain as naloxone is a competitive antagonist at the opioid receptors (i.e., its effect is described by a single equilibrium dissociation constant). Possibly, the differences in potency may be related to the ability of M6G to increase the affinity of naloxone for the µ-receptor [17]. Since the elimination t_½ of naloxone is between 15 and 30 min, naloxone is a limiting factor in the reversal of (long-acting) opioids. However, in addition, the slow receptor kinetics of morphine (and M6G) limits naloxone’s ability to disperse the opioid agonist from the receptor rapidly. This has clinical consequences as increasing the naloxone dose will not increase the speed of morphine reversal (i.e., the increase in ventilation over time), although the magnitude and duration of reversal will increase (Figure 1). Taken together, a sufficient high dose of naloxone is required, preferably administered as continuous infusion, to reverse morphine-induced respiratory depression with a reduced chance of renarcotization; the speed of reversal is not affected by the naloxone dose. Note that these observations regard only opioid agonists that have slow receptor association/dissociation kinetics (such as morphine, M6G and buprenorphine, see later) but are not valid for opioids with rapid kinetics such as fentanyl or any of the other fenylpiperidines (with the exception of remifentanil; if the remifentanil infusion is stopped, breathing resumes within minutes) [19]. A more rapid reversal of fentanyl just requires a greater dose of naloxone, as naloxone reversal is not dictated by receptor kinetics of the agonist (fentanyl).

Two recent studies analyzed the effect of morphine on acute postoperative pain using a population PKPD analysis in NONMEM. Mazoit et al. modeled morphine’s effect taking the potential effects of M3G and M6G into account in patients in acute pain following a variety of surgical interventions with a mean age of 51 years [20]. They assumed additive agonistic effects from morphine and M6G and an antagonistic effect from M3G. Using a sigmoid E_max model with pain (as determined from a visual analogue scale [VAS] scoring system) as effect variable, they estimated a tenfold greater potency of M6G compared with morphine (C₅₀ = 124 [morphine] vs 13 nM [M6G]) and an inhibitory effect from M3G with C₅₀ of 880 nM. The analgesic effect of M6G was delayed significantly relative to morphine (t_½k_e0: 1.7 [morphine], 3 [M6G] and 3 h [M3G]). Age, sex and weight did not improve the PD model. The M6G potency is much greater than observed in earlier studies in volunteers [12,16,17] and, although one may argue that in the Mazoit study measurements were made for 24–48 h (compared with 6–10 h in previous studies), the M6G potency is an indirect estimate derived from a study where morphine, but not M6G, was administered. Consequently, the true potency of M6G remains unknown, in comparison with previous studies where only M6G was administered. Of further interest is the observation of an antagonistic or hyperalgesic effect of M3G with a relatively slow distribution into the biophase. This suggests that reduced morphine efficacy may coincide with M3G’s hyperalgesic effect, requiring the switch to other opioid analgesics in some patients.

Abbou Hammoud et al. modeled the effect of morphine titration in the first postoperative hours on pain relief (using VAS) in patients (mean age of 62 years) following orthopedic surgery [21]. They used an indirect response E_max model with variations in pain expressed by the equilibrium of K_in (a zero-order rate constant of the appearance of pain) and K_out (a first-order rate constant of the disappearance of pain) with morphine inhibiting K_in. They added a virtual kinetic compartment, as no PK data were available. They estimated a value for ED₅₀ (dose of morphine causing 50% pain relief) of 10.2 ± 0.8 mg with significant covariates including initial VAS (direct postoperative VAS just before dosing), and the use of additional pain medication (but not age, sex or weight). The model is of interest as it predicts that greater titration doses are associated with a significant reduction in time to achieve analgesia (especially in patients with a greater initial VAS), albeit at the expense of increased toxicity (sedation in their study).

Buprenorphine

There is a renewed and increasing interest in this 30-year-old semisynthetic opiate derived from the morphine precursor, thebaine. Currently, buprenorphine is applied in the treatment of chronic pain using a relatively novel patch formulation and in the treatment of addiction (Subutex™ or in combination with naloxone Subutex [Reckitt Benckiser, Slough, UK]). Buprenorphine is a partial agonist at the µ-opioid receptor and displays typical opioid behavior (analgesia, sedation, nausea, delayed gastric emptying and respiratory depression). Yassen et al. performed a series of experiments in men using a mechanism-based population PKPD approach [6,22–24]. The most important observations from these studies are:

• • In the clinically relevant intravenous dose range of 0.05–0.6 mg/kg, buprenorphine displayed a lack of ceiling effect for antinociception (related to the absence of saturation of receptor occupancy) [24];

• • Rate-limiting factors in the onset and offset of antinociceptive effect are biophase distribution and slow receptor association–dissociation (Table 1)[24];

• • In contrast to antinociception, ceiling was observed for respiratory depression at high buprenorphine concentrations (the intrinsic activity of buprenorphine is 56%) [23];

• • Reversal of buprenorphine-induced respiratory depression with naloxone is possible, however, owing to the slow receptor kinetics of buprenorphine, in combination with the fast elimination kinetics of naloxone, naloxone is best administered as a continuous infusion rather than a single bolus injection (optimal dosage is 2–4 mg naloxone per h) [6].

Naloxone reversal is slow, however, and cannot be accelerated by giving higher naloxone doses (see earlier). The findings from these modeling studies indicate that buprenorphine is an opioid analgesic that provides long-term analgesia without ceiling over the clinical dose-range and has a limited effect on the respiratory systems. In case life-threatening respiratory depression does occur, it can be antagonized but requires high doses and continuous infusions of naloxone. These data demystify some of the old beliefs around buprenorphine that can still be found in older textbooks, including the existence of a ceiling in analgesic effect and inability to reverse buprenorphine-induced respiratory depression.

Ketamine

The third ‘senior’ analgesic that we discuss is the 50-year-old N-methyl-D-aspartate receptor (NMDAR) antagonist ketamine. Initially developed as an anesthetic, multiple studies show that at low (subanesthetic) doses, ketamine is a potent analgesic in the treatment of both acute and chronic pain [25]. In the last few years, a series of investigations on population PKPD modeling in children and adults has been published. The studies in children were aimed at the anesthetic properties of ketamine and will not be discussed here [26,27]. In healthy volunteers, Sigtermans et al. tested the effect of increasing doses of the S(⁺)-enantiomer of ketamine (S-ketamine) on antinociception using two different nociceptive assays: heat pain (a fixed-heat stimulus was applied to the arm giving a baseline [i.e., predrug] VAS of 6 cm or greater) and tolerance to electrical pain (a 10-Hz stimulus train increasing at 0.5 mA/s) [28]. Using an effect-compartment population PKPD analysis yielded a significant difference in ketamine’s potency in the two assays (Table 1) with a sixfold greater potency for pain relief of heat pain. Of further interest is the observation of hyperalgesic responses following S-ketamine withdrawal that was modeled with a linear trend term. This phenomenon has recently been described in experimental and clinical studies. A delay between concentration and effect (that is hysteresis) was not observed for any of the antinociceptive end points. This indicates an almost immediate passage of S-ketamine across the BBB and rapid receptor kinetics. Similar observations were made previously on the effect of ketamine racemic mixture on EEG slowing in adults and on estimates of arousal and recall memory during anesthesia in children [26].

COX inhibitors

Kowalski et al. constructed PKPD models to describe the effect of the experimental COX-2 inhibitor SC-75416 (Pfizer, MI, USA) and performed trial simulations to enhance the further development of this analgesic [29]. SC-75416 is a selective COX-2 inhibitor that has anti-inflammatory and analgesic activity. They studied postoral surgery patients and performed comparisons with placebo, ibuprofen, rofecoxib and valdecoxib. In an initial study, an oral solution of SC-75416 was followed by a capsule formulation. They constructed logistic effects models in NONMEM with three main components: placebo effect (modeled by an exponential function) and drug effect (modeled by a sigmoid E_max function); random effect; and investigating pain relief. The estimated EC₅₀ values for SC-7541, rofecoxib, valdecoxib and ibuprofen were 5.5, 0.3, 0.07 and 6.8 µg/ml, respectively. The most important results of the study were: the study and trial simulations prompted further study developments (pursuing a high-dose strategy) that otherwise might not have been considered; the approach resulted in time and cost savings by not having to repeat the postoral surgery study and formulation rework was carried out to optimize drug delivery.

Li et al. performed a PKPD analysis of dental pain relief by ibuprofen [30]. Their aim was to assess the efficacy of a novel effervescent formulation in comparison with the standard tablets of NSAIDs. In a group of patients following third-molar extraction surgery, the effect of the two formulations and placebo were modeled using an E_max pain relief model and hazard models to analyze time to pain relief and remedication. The data show differences in PK parameters between the two formulations with a greater oral absorption rate for the effervescent ibuprofen and greater effect-site concentrations between t = 0 and 2.5 h following dosing. This caused more rapid pain relief and less remedication for the effervescent formulation compared with standard ibuprofen. The authors constructed nomograms to correlate time to pain relief with PK profiles. Overall, these data indicate the usefulness of PKPD modeling in the development of new formulations of analgesic drugs (see later).

Acetaminophen

Another example where model-based analysis was used to aid the development of a new formulation is the study by Green et al.[31]. They performed PKPD analysis on pain relief data following administration of four different acetaminophen (paracetamol) formulations (Tylenol^®, Panadol^® Rapid and two that used new formulation technologies from Imaginot Pty, Ltd., Brisbane, Australia) to explore possible differences in PD outcomes. The authors used the PKPD data from Anderson et al. to do a simulation on the effects of the new formulation [32]. PK data were obtained from volunteers that received a single dose of 1 g of the four formulations; PD data were obtained from simulations on an earlier published PD model that quantified pain relief following tonsillectomy. Interestingly, their data indicate that the placebo effect was appreciable and there was a potential difference between acetaminophen and placebo at t = 0 and 30 min postdosing only. With respect to the comparison between formulations, the authors conclude that their analysis suggests a significant reduction in time to onset of effect with the newer formulations. The current approach of learning drug behavior from already published datasets was used ‘because running a confirmatory clinical trial was prohibitively expensive’ [31].

Epidural anesthesia/analgesia

Epidural anesthesia and analgesia is obtained by an injection of a local anesthetic, an opioid or their combination into the epidural space. The injected drugs spread over the epidural space, diffuse towards the intrathecal space (where the drugs diffuse into and bind to neuronal tissue) and into blood vessels of the epidural space. Depending on the concentration used and type of drug injected, motor and sensory blockade develops and dissipates. Onset/offset times and spread of the effect across the spinal segments vary and depend on the kinetic and dynamic properties of the injected anesthetic/opioid. Olofsen et al. are the first to develop a population PKPD model of epidural anesthesia and analgesia [33]. Their analysis enables the development of predictive epidural anesthesia, in this case, analgesia models that may improve therapeutic outcome. An indirect assessment of the epidural drug concentration was made by estimation of the time course of systemic drug absorption from the epidural space (by measurement of the drug concentration after an epidural injection). Simultaneously, they assessed the level of sensory blockade over time. Tested drugs were the local anesthetics levobupivacaine and ropivacaine. The epidural segments were modeled by central and peripheral absorption compartment- and effect-sites. Differences were observed in the onset/offset half-lives between the two drugs (15 min for levobupivacaine and 25 min for ropivacaine). Age was a significant modifier with respect to onset/offset half-lives (increasing age reduced t_½ for levobupivacaine at all segments) and anesthetic potency (increasing age increased the potency at segments Th12 and higher for ropivicaine). The model enables individualized dosing depending on age, sex and drug, and gives a predictive indication of the time course of the sensory blockade (Figure 2).

PKPD modeling in chronic pain

Ketamine

In a reanalysis of a study on the long-term (12 week) effects of a 100-h S-ketamine infusion on pain relief in chronic pain patients owing to complex regional pain syndrome type 1, Dahan et al. used an inhibitory sigmoid effect-compartment model of the form: PAIN score = BASELINE PAIN/(1 + [C_E/C₅₀]γ), and onset/offset rate constant k [34]. This very simple model satisfactorily described the 12-week pain relief data with a value for C₅₀ of 10 ng/ml and t_½k of 10.9 days (95% CI: 5–21 days). The C₅₀ for relief of chronic (neuropathic) pain is approximately 50-times lower than that observed for the relief of acute nociceptive pain (in volunteers) and suggests a different mode of action of S-ketamine in these two distinct pain states. This an example derived from chronic pain data of the power of PKPD modeling in understanding the complex behavior of analgesic agents. The parameter t_½k is not synonymous to the blood effect-site equilibration parameter t_½k_e0. While the later parameter described the passage of the drug to the postulated receptor site, parameter k reflects the dynamics of disease modulation by S-ketamine. The plasma concentration of S-ketamine drops rapidly below detection values (within hours) following the withdrawal from ketamine, while its analgesic effect persists for weeks, probably owing to the initiation of a complex cascade of events of which NMDAR desensitization is the first step of this disease-modulatory process.

Assuming that PK of S-ketamine is irrelevant in describing its prolonged analgesic effect in chronic pain, in addition a (population-based) time-series analysis was performed using a mixture model and autoregressive (Kalman) filter constructed in NONMEM. The model assumes an exponential return of pain values towards baseline pain with autoregressive factor F, which is comparable to parameter k. Indeed, the observed value of t_½F corresponded well with the range observed for k: 15–41 days. The mixture model allows the estimation of the chance of an analgesic effect to ketamine (or placebo) by objectively subdividing the patient population into four subgroups: two types of nonresponders and two types of responders (Figure 3). This type of analysis is attractive as it rapidly allows a subdivision of analgesic responses into various subgroups that may be defined a priori without the need for PK data. Mixture models using time-series analysis may be especially advantageous when quantifying the analgesic efficacy of drugs in terms of onset/offset time, division into subgroups, magnitude of variability and placebo effect, or when quantifying the effect of drugs that act locally, such as local applications of lidocaine or capsaicine.

Pregabalin

Pregabalin is an analog of GABA and acts at the α2δ protein associated with voltage-gated calcium channels. It has anxiolytic, antiepileptic and analgesic properties. Byon et al. performed a pregabalin exposure-response analysis in fibromyalgia chronic pain patients [35]. They modeled the probability of an observed specific daily pain score (on an 11-point categorical scale) using a proportional odds logistic regression model in NONMEM with additive components: baseline pain, placebo response (exponential function) and drug effect (E_max and linear models were tested). The logistic regression approach is used when pain or pain relief is scored with an ordered categorical scale and was first introduced by Sheiner in 1994 [36]. Byon et al. incorporated data from four separate randomized, placebo-controlled studies including 2758 patients with daily dosing for 8–14 weeks in the dose range of 150–600 mg two- to three-times per day [35]. The average drug concentration was estimated from the pregabalin dose and renal function. The E_max drug effect model gave the best results and they estimated an EC₅₀ of 1.4 µg/ml (equivalent to 174 mg/day in a patient with creatinine clearance of 100 ml/min), the delay in drug effect had a t_½ of 11 h. Sex and age were significant covariates, with increasing values for E_max with increasing age and in men compared with women.

Expert commentary & five-year view

In the last 5–6 years, only a limited number of studies have been published on population PKPD modeling of analgesic agents used in acute and/or chronic pain. The majority of studies focused on older analgesics, including morphine, buprenorphine and ketamine, while studies on newer agents were scarce and limited to morphine’s metabolite M6G, the experimental NSAID SC-75416 and pregabalin. We believe that population PKPD modeling is an important tool, not only in the description of the time course of drug effect (which enables the appropriate use of the drug), but equally important to understand the interaction of a specific pharmacological agent within the often complex disease processes and possibly even understand the complexities of the disease itself. Furthermore, model-based analysis using a population PKPD approach is useful in the development of new formulations of older drugs such as was shown for ibuprofen [29,30] and acetaminophen [31]. Simulation-based PKPD modeling may be a cheap but valid alternative to expensive large clinical trials [31].

In the upcoming 5 years, we foresee various advances in the use of population PKPD models in acute and chronic pain, most importantly in the development of new analgesic agents, in the development of new formulations or the switch towards new indications of already registered analgesics. Particularly in the development of new analgesic agents, population PKPD modeling enables simultaneous assessment of the various properties of analgesics by calculation of utility or safety functions. There are various possible calculations of the therapeutic utility functions possible such as the probability of analgesic effect minus the probability of side effect. In an animal study, Yassen et al. performed a logistic regression analysis to characterize the relationship between drug exposure and two effects of buprenorphine and fentanyl: antinociception and respiratory depression [37]. Odds ratios for both end points were calculated and were, for buprenorphine, 29 and 2 for antinociception and respiratory depression, respectively, while for fentanyl these values were 3 and 2.5. The safety index, calculated as the odds ratio for analgesia divided by the odds ratio for a side effect of 1.2 for fentanyl and 14 for buprenorphine, suggests a distinct margin of safety for these two opioids, with greater safety for buprenorphine compared with fentanyl. Respiratory depression, as a complication of potent opioid analgesics, is seldom studied, while respiratory events are potentially life-threatening side effects of these agents [38]. Hence, we encourage the industry to determine the safety index in the development of new analgesic agents. Particularly when developing potent analgesic agents or new formulations (such as for fentanyl and sufentanil), PKPD studies on opioid side effects (including respiratory depression, nausea/vomiting, delayed gastric emptying and sedation) should be part of the preregistration study process.

Another advancement would be the use of PD mixture models in NONMEM to predefine response groups into responders and nonresponders. Both responders and nonresponders require further a priori descriptions with patients that experience at least 50% pain relief throughout the treatment period as most clinically relevant. Apart from drug effect per se, the variability within the different groups is an important marker of drug efficacy. Moreover, patients in the different response groups may be further examined to link specific patient characteristics (such as parameters derived from quantitative sensory testing, duration of disease, severity of pain symptoms and variability in pain symptoms) to drug efficacy, eventually allowing prediction of drug response based entirely on these pretreatment assessments [39]. We predict an important place for mixture modeling in NONMEM in the development of new drugs or new indications of already registered drugs.

The chronification of pain is complex and poorly understood. Both peripheral and central processes play important roles, although the specifics vary per disease. To fully understand the effect of analgesics on these processes, disease-modulatory PKPD models are required that describe the effects of specific analgesics within the various pain pathways [31]. These models need to be dynamic as processes vary over time. For example, it may well be that chronic pain in its initial phase has predominant peripheral inflammatory components, while over time structural plastic changes within the spinal cord or at supraspinal sites develop. Abandanes et al. use PET to predict brain target occupancy of duloxetine, a serotonin-reuptake inhibitor with analgesic properties, with various dosing regimens, using a mechanism-based PK brain target occupancy models [40]. They define 50% receptor occupation (OC₅₀) as the drug plasma concentration that achieves 50% receptor occupation with OC₅₀ = k_off/k_on. Applying these techniques to specific pharmacological targets of chronic pain within the different timeframes of the lifecycle of the disease and linking receptor occupation to pain relief is an example of the building of disease-modulatory PKPD models. Such models will increase our insight into the disease process (such as the process of chronifcation of pain) and will improve treatment development.

Table 1. Population pharmacodynamic model estimates for various analgesic agents and naloxone using pain relief or respiration as the effect parameter: studies performed in healthy volunteers or patients.

Drug	t½ke0	kon (ml/ng/min)	koff^† (min^-1)	Kd (koff/kon)	C50	Effect parameter	Population	Ref.
Morphine (men)	1.6 h				250 nM	Antinociception	V	[13]
Morphine (women)	4.8 h				125 nM	Antinociception	V	[13]
Morphine-6-glucuronide	6–8 h				750 nM	Antinociception	V	[12,16]
Morphine	1.2 h	0.85	0.14	160 nM		Respiration	V	[17]
Morphine-6-glucuronide	2.7 h	0.04	0.03	880 nM		Respiration	V	[17]
Naloxone	5–8 min				0.5–2 nM	Respiration	V	[17]
Morphine	1.7 h				124 nM	Postoperative analgesia	P	[20]
Morphine-6-glucuronide	3 h				13 nM	Postoperative analgesia	P	[20]
Buprenorphine	75 min	0.25	0.01	0.09 nM		Respiration	V	[6]
Fentanyl	16.4 min	>100	>100		1140 ng/ml	Respiration	V	[6]
Buprenorphine	155 min	0.06	0.08	2.66 nM		Antinociception	V	[24]
S-ketamine	^‡				373–2200 ng/ml^§	Antinociception	V	[28]
S-ketamine	11 days^¶				10.5 ng/ml	Chronic CRPS pain relief	P	[34]
Pregabalin	11 h				1.54 µg/ml	Chronic fibromyalgia pain relief	P	[35]
SC-75416					5500 ng/ml	Postoral surgery pain relief	P	[29]
Rofecoxib					284 ng/ml	Postoral surgery pain relief	P	[29]
Valdecoxib					68 ng/ml	Postoral surgery pain relief	P	[29]
Ibuprofen					6840 ng/ml	Postoral surgery pain relief	P	[29]
Ibuprofen	28 min				10,200 ng/ml	Postoral surgery pain relief	P	[30]
Acetaminophen	53 min				10 mg/l	Post-tonsillectomy pain relief	P	[32]

^†t_½k_off is calculated as ln2/k_off.

^‡No hysteresis observed between plasma concentration and effect.

^§C₅₀ varied depending on the nociceptive assay employed (see text).

^¶The value of 11 days represent a rate-constant form disease modulation (t_½k) unrelated to the blood effect-site equilibration t_½.

C₅₀: Steady-state concentration causing 50% of the effect; CRPS: Complex regional pain syndrome; K_d: Equilibrium dissociation constant; k_e0: Effect-site equilibration constant; K_off: Receptor dissociation constant; K_on: Receptor-association constant; P: Patients; t_½: Half-life; V: Volunteers.

Key issues

Population pharmacokinetic–pharmacodynamic modeling of analgesics enables:

• • Estimation of the population pharmacokinetic and pharmacodynamic parameters, including potency and onset/offset parameters.

• • Identification of the sources of variability that influence pharmacokinetics and dynamics.

• • Estimation of the magnitude of between-subject variability as well as intrasubject variability.

• • Individualized dosing.

• • Predictive assessment of the time course of drug effect.

• • Provision of aid in the development of new formulations of analgesic compounds.

Unmet needs include:

• • Development of disease-modulation-oriented pharmacokinetic–pharmacodynamic modeling models in acute and chronic pain.

• • Systematic application of mixture models in the development of novel analgesic agents to identify specific response subgroups, allowing optimization of therapy and outcome.

• • Extended use of utility functions in the development of novel analgesics by performing simultaneous pharmacokinetic–pharmacodynamic modeling of effect (analgesia) and side effects (toxicity).

Financial & competing interests disclosure

Ashraf Yassen is an employee of Astellas Pharma Europe BV, Leiderdorp, The Netherlands. Astellas Pharma Europe BV has active research programs in clinical and preclinical discovery for the treatment of chronic pain. The authors have no other relevant affiliations or financial involvement with any organization or entity with a financial interest in or financial conflict with the subject matter or materials discussed in the manuscript apart from those disclosed.

No writing assistance was utilized in the production of this manuscript.