1 Introduction

Fentanyl (Fig. 1) was the first in a class of potent synthetic opioid analgesics developed in the 1960s. It is 100-fold more potent than morphine [1] and was designed by Paul Janssen [2] in a search for highly potent opioids with pronounced opioid receptor subtype specificity, which at the time was thought would improve drug safety [3]. Shortly afterwards in the 1960s, fentanyl was introduced into clinical practice as a narcotic analgesic [4]. Its use was restricted to anaesthesia until the 1990s when the development of non-injectable fentanyl formulations was intensively pursued. Elimination of the need for vessel puncturing subsequently allowed for the broader use of fentanyl as a prescriptive analgesic [5]. Nowadays, transdermal fentanyl patches are among the most frequently prescribed strong opioid analgesics [6, 7].

Fig. 1
figure 1

Fentanyl and its metabolites as found in an in vitro human liver microsomal system [26]. The major metabolite is norfentanyl, to which 99 % of the fentanyl is transformed. Norfentanyl is not known to produce clinically relevant pharmacodynamic effects [29]

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Fentanyl was the first opioid for which a delay between its concentration–time profile in the blood and that observed in the central nervous system (CNS) was experimentally established and mathematically modelled [8], using the relevant principles of pharmacokinetic/pharmacodynamic modelling established 7 years earlier with depolarizing neuromuscular-blocking agents [9, 10]. The development of non-injectable fentanyl formulations not only led to prolongation of the analgesic action of fentanyl, it also offered the possibility to generate a larger variety of plasma concentration–time profiles in patients in a formulation-dependent manner. This increased the therapeutic options because the clinical activity of fentanyl depends on its availability at μ-opioid receptors [11], predominantly in the CNS [12].

The different clinical indications for non-injectable fentanyl formulations, ranging from the treatment of breakthrough pain [13] with fast-acting formulations to the treatment of chronic pain with long-acting combinations, are based on the different pharmacokinetic properties of the pharmaceutical formulations. In particular, the different pharmacokinetic input characteristics of fentanyl into the central compartment, or, as in the case of nasal formulations, directly into the brain, are crucial. The key to a rational clinical use of fentanyl and all its available formulations, however, is based on the overall understanding of their pharmacokinetics, as discussed in this article.

2 Physicochemical and Pharmacological Properties of Fentanyl

Fentanyl represents a major success in the development of synthetic strong analgesic drugs. Subsequently developed members of the fentanyl class, such as sufentanil, carfentanil, lofentanil and alfentanil [14] or trefentanil differ from the prototype in their chemical structure mainly in the N-alkyl substituent of the piperidine ring (Fig. 1). In the majority of the pharmaceutical formulations currently available, fentanyl is present as the citrate salt. Besides intravenous formulations, which can also be injected intrathecally, epidurally or locally, into joints [15], fentanyl is available as transdermal patches; for nasal administration; or for oral transmucosal use as buccal lozenges, films, sublingual tablets or spray. Its physicochemical properties (Table 1), in particular its high lipophilicity (octanol:water partition coefficient >700) [16], mean that it quickly crosses between plasma and CNS effect sites with a transfer half-life (t½,ke0), of 4.7–6.6 min [8, 17, 18].

Table 1 Key physicochemical and pharmacological properties of fentanyl
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Fentanyl exerts its actions mainly by agonist binding at μ-opioid receptors [11] (Table 1). Its pharmacological effects consist of the inhibition of the accumulation of cyclic adenosine monophosphate, presynaptic Ca2+ influx and postsynaptic K+ efflux, leading to neuronal hyperpolarization. Its clinical effects are typical for opioids, including analgesia, respiratory depression, drowsiness, nausea, vomiting and decreased gastrointestinal motility. These effects seem to be altered individually by genetic polymorphisms known to modulate opioid effects, such as the OPRM1 variant 118A>G (rs1799971) coding for N40D μ-opioid receptors, which results in reduced expression [19] and signalling efficiency [20]. This seems to be associated occasionally [19] with a reduced analgesic response to fentanyl [21–23]. Moreover, downstream signalling components, such as potassium channels Kir3.2, appear to act as further pharmacogenetic modulators of the pharmacodynamics of fentanyl. For instance, the genotype KCNJ6 rs2070995 AA has been associated with increased opioid requirements, but without explicit reference to the effects of fentanyl as several other opioids were pooled in the analysis [24].

Fentanyl is extensively metabolised and renal excretion accounts for only 10 % of the dose [25]. It undergoes pre-systemic metabolic elimination in the liver and intestinal wall, mainly by piperidine N-dealkylation to norfentanyl (Fig. 1) as the predominant degradative pathway in humans accounting for >99 % of the metabolism [26]. In addition, it is metabolised by amide hydrolysis to despropionylfentanyl and alkyl hydroxylation to hydroxyfentanyl, the latter being further N-dealkylated to hydroxynorfentanyl [26]. This presystemic elimination reduces the bioavailability of fentanyl to 32 % [26] when it is rapidly swallowed, thus largely bypassing oral transmucosal absorption.

The metabolism of fentanyl is mediated almost exclusively by cytochrome P450 (CYP) 3A4 [26, 27] together with CYP3A5 and 3A7 [28]. The metabolites seem to lack clinically relevant opioid agonist activity [29]. The involvement of CYP3A7 contributes to a pharmacokinetic change during early development as this enzyme is the main CYP3A isoenzyme in newborns [30], but the shift to CYP3A4 dominance occurs quickly after birth [31]. However, the major consequence of the almost exclusive CYP3A-dependent metabolism is the occurrence of drug interactions. CYP3A is the main drug-metabolising enzyme [32] and fentanyl metabolism is inhibited by several inhibitors of CYP3A [33] such as ritonavir [34] or diltiazem [35], but not by all inhibitors. Itraconazole apparently lacks influence on fentanyl metabolism [36], which can also be induced by CYP3A inducers such as phenobarbital [37]. Fentanyl may also act as an enzymatic inhibitor and reduce the clearance of co-administered drugs such as midazolam [38].

The major dependence of the fentanyl clearance on CYP3A makes it comparatively less vulnerable to genetic polymorphisms. The existence of the major CYP3A5*3 allele, associated with reduced CYP3A5 expression and thus reduced CYP3A function [39], has been reported to cause measurable differences in the metabolism of fentanyl [28]. While the known pharmacokinetically relevant pharmacogenetic factors play a minor role in the metabolic clearance of fentanyl, clinically measurable modulation of the distribution of fentanyl seems to be produced by genetic polymorphisms in the ABCB1 gene that encodes for P-glycoprotein (P-gp) for which fentanyl is a substrate [40]. Genotypes with lower net P-gp function, such as ABCB1 1236TT (rs1128503), 2677TT (rs2032582) and 3435TT (rs1045642), result in increased CNS retention of fentanyl. This is because P-gp is an outward transporter across the blood–brain barrier [41]. The compromised net P-gp function causes an increase in clinical CNS adverse effects such as respiratory depression [42, 43] and sedation [42] and reduces the need for analgesic rescue medications [44] and possibly the analgesic dosing requirements [45]. In contrast to its elimination from the CNS, the exact mechanism of the CNS uptake of fentanyl across the blood–brain barrier seems to be still unresolved and could involve passive diffusion as well as active transport by yet unspecified carriers.

3 Pharmaceutical Formulations for Non-Intravenous Administration of Fentanyl

For non-intravenous administration, preparations delivering fentanyl oral transmucosally, intranasally and transdermally are currently available. While transdermal patches release fentanyl in a constant sustained manner, thus being suitable for chronic pain management, the iontophoretic transdermal system and the oral transmucosal and intranasal routes of administration achieve a rapid onset of analgesia in acute pain conditions (Table 2).

Table 2 Summary of representative key descriptive pharmacokinetic parameters of non-intravenous pharmaceutical formulations of fentanyl. If available, onset and duration of analgesic effects are also given
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4 Concentration–Time Profiles of Fentanyl

The actions of fentanyl are related to its concentrations at opioid receptors expressed within its main effect site, the CNS. Except for intranasal administration, whereby fentanyl is also directly delivered to the CNS [46] (Fig. 2), the extent and time course of its effects are a function of the time course of its plasma concentrations, Cp(t). The profile of the latter is therefore clinically relevant and results from convolving the disposition function, fD(t), with an input function, fI(t), as in Eq. 1:

$$ {\text{C}}_{\text{p}} \left( {\text{t}} \right) = {\text{f}}_{\text{I}} \left( {\text{t}} \right) * {\text{f}}_{\text{D}} \left( {\text{t}} \right) $$
(1)

where asterisk denotes convolution of the input and disposition functions.

Fig. 2
figure 2

Schematic presentation of differences in the sites of fentanyl absorption in relation to different routes of non-intravenous fentanyl administration

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4.1 Input Functions of Non-Intravenously Administered Fentanyl

The input function, fI(t), represents the major difference in fentanyl pharmacokinetic properties following various non-intravenous routes of administration because, once fentanyl has reached the blood, its disposition function, fD(t), is similar for all non-intranasal routes of administration. Depending on the clinical indication for rapid versus sustained analgesia in the therapy of breakthrough or chronic pain, respectively, pharmaceutical formulations have been developed to provide either fast or slow input functions, respectively. Unfortunately, while descriptive pharmacokinetics of non-intravenous fentanyl formulations have often been reported, including comprehensive reviews of particular formulations [47], parametric characterizations of their input functions are rare. In fact, several input functions could be used to characterize oral or oral transmucosal routes, the simplest describing a first-order input with a rate constant, ka, as in Eq. 2:

$$ {\text{f}}_{\text{I,absorption}} ({\text{t}}) = {\text{Dose}} \cdot {\text{k}}_{\text{a}} \cdot {\text{e}}^{{ - {\text{k}}_{\text{a}} \cdot {\text{t}}}} $$
(2)

from which the absorption or input half-life can be calculated as ln(2)/ka.

Extravascular input of fentanyl also determines a bioavailability (F) of <1 (or <100 %), but for different reasons. Whereas incomplete absorption and pre-systemic elimination via CYP3A both reduce the bioavailability of formulations that deliver fentanyl into the gastrointestinal tract (Fig. 2), incomplete absorption of the dose from fentanyl patches, due to reduction of the concentration gradient, accounts for an F <1 with transdermal patches.

4.1.1 Fentanyl Formulations for Fast Analgesic Effects

Rapid analgesic effects are needed to cope with breakthrough pain [48], which is defined as a transitory exacerbated pain on the background of chronic pain managed with opioids [13]. It has a duration of around half an hour (range 1–240 min), is described as lancinating and has a high intensity (6–9 of an 11-point rating scale from 0 to 10) [49]. The necessary rapid analgesic onset can be achieved with oral transmucosal formulations or intranasal delivery of fentanyl. Transmucosal preparations attempt to mimic intravenous injections without vascular puncture. However, the fastest input is that given into the blood by a rapid bolus injection, which can be described by a Dirac delta function (Eq. 3) of:

$$ {\text{f}}_{\text{I,bolus}} ({\text{t}}) = {\text{Dose}} \cdot {\text{Dirac}}({\text{t}}) $$
(3)

which by definition takes a value of 1 when t = 0 and a value of zero otherwise, and models the input as if it were instantaneous. Fast-delivery non-intravenous fentanyl formulations attempt to draw close to this velocity of the fentanyl input.

4.1.1.1 Oral Transmucosal Formulations

The oral route is the most accepted route of administration for drugs. However, like intravenous injections or infusions, transmucosal administration only provides rapid input into the circulation, while the time course of the effects also depends on the velocity of the transfer of fentanyl across the blood–brain barrier. As the latter is fast [8, 17, 18], oral transmucosal, either buccal or sublingual, delivery may provide the intended quick onset of analgesic effects. Hence, the pharmacokinetic profiles of fentanyl following transmucosal administration are characterized by rapid absorption. This is reflected in consistently reported short values for the time to reach the maximum (peak) plasma concentrations (tmax) after dosing, eventually supported by pharmacodynamic information on the onset and duration of analgesia (Table 2). For example, the pharmacokinetics of oral transmucosal fentanyl citrate, formulated as a sweetened lozenge on a stick and designed to dissolve slowly in the mouth, have been compared with those of a fentanyl buccal tablet formulation of transmucosal fentanyl citrate, which is manufactured to enhance the rate and extent of absorption of fentanyl [47]. Median values of tmax were 46.8 min (range 20–240 min) and 90.8 min (range 35.0–240.1 min) for the buccal tablet and oral stick, respectively, with mean values of the maximum plasma concentration (Cmax) of 1.02 ± 0.42 and 0.63 ± 0.21 ng/mL normalized to a dose of 400 μg. This is consistent with the analysis of pooled data from nine pharmacokinetic studies including a total of 365 healthy, non-opioid-tolerant adults, indicating rapid absorption from buccal tablets with tmax ranging from 20 min to 4 h post-dose and mean values of Cmax of 0.237 ng/mL normalized to doses of 100 μg [47]. The above assessment was also used for a parametric pharmacokinetic analysis [50], but was only published as an abstract which did not give the input variables [51]. Parametric modelling of another oral transmucosal fentanyl citrate formulation utilized a first-order input, estimating a ka of 0.018 min−1, which corresponds to an absorption half-life of 38 min [52], emphasizing that transmucosal input is slower than an intravenous injection.

The bioavailability of oral transmucosal fentanyl formulations depends on both transmucosal absorption and the absorption of the ingested dose fraction. The latter undergoes first-pass metabolism, which is bypassed by transmucosal absorption. A rapidly swallowed fentanyl solution achieved a bioavailability of 32 % [53]. A comparable value of 36 % was found in children following fentanyl lozenge application, suggesting that they swallow a considerable amount of the fentanyl [54]. Placing an oral transmucosal fentanyl unit of 15 mg/kg in the buccal pouch and sucking for 15 min raised the bioavailability to 50 % [53]. However, the input of contemporary oral transmucosal pharmaceutical preparations does not significantly depend on the dwell time (duration of presence in the oral cavity) [47]. Despite the fact that the sublingual mucosa is thinner than the buccal mucosa, bioequivalence has been shown between sublingual and buccal placement of fentanyl preparations [55], although sublingual formulations occasionally provided higher bioavailability (Table 2) of up to 75 % of a dose [56]. Finally, mucositis occurring during cancer treatment does not seem to affect transmucosal absorption of fentanyl, at least for grade I [57], although the product information, for instance for Subsys®, warns about a four- to sevenfold higher Cmax due to mucositis.

4.1.1.2 Intranasal Formulations

In contrast to intravenous, transdermal, subcutaneous, intramuscular, transbuccal and oral routes of administration, whereby the drug first enters the central blood compartment before crossing the blood–brain barrier to exert CNS effects, nasal routes additionally deliver the opioid directly to the CNS site of action (Fig. 2) and thus partly circumvent the blood–brain barrier [46]. Pathways that are considered likely to allow direct CNS access include the olfactory and trigeminal nerves, the vasculature, the cerebrospinal fluid and the lymphatic system [46]. The anatomy of the nose is well suited to the efficient transfer of exogenous agents into the brain [58]. This is accounted for [59] by the large surface of the nasal cavity, which is increased by the turbinates, and by the extremely high blood flow to the nasal mucosa which exceeds, relative to the tissue volume, that in muscle, brain and liver [60]. In adults, the 15–20 mL of the nasal cavity are enveloped by a surface area of 150–180 cm2, 5–10 cm2 of which is olfactory epithelium and the remaining 145–170 cm2 is respiratory epithelium [61, 62]. In particular, the olfactory nerve seems to provide a delivery path directly into the CNS, known as the olfactory vector hypothesis [63]. Following this path, even opioids that only slowly cross the blood–brain barrier, such as morphine [64] or morphine-6-glucuronide [65], can produce rapid central nervous analgesic effects [66]. Plasma concentrations measured after intranasal fentanyl administration provide an estimate of systemic exposure and compare well with concentrations measured after intravenous injection [67]. However, local fentanyl concentrations in the CNS are probably higher after intranasal than after systemic administration due to the additional direct delivery.

The development of intranasal opioids [61] has been intensified in the last decade so that it is now possible to deliver the active compound almost instantaneously to the brain, partly bypassing the blood–brain barrier [46, 68] and reducing systemic exposure. Indeed, following intranasal administration of fentanyl 50–200 μg, the intensity difference in cancer pain at 10 min (11-point numerical rating scale from 0 to 10) was significantly better than following placebo administration, without serious adverse events, demonstrating that in opioid-tolerant patients, intranasal fentanyl is an effective treatment for breakthrough pain [69]. The bioavailability of intranasal fentanyl has been reported to be 89 % [59], perhaps due to more efficient avoidance of the oral/gastrointestinal mucosa than with transbuccal or sublingual formulations, resulting in a lower swallowed fraction of the dose. Inhaled fentanyl, by contrast, seems to be readily absorbed via the lung epithelium [70]. In order to avoid run-off through the pharynx and swallowing, the intended dose must be dissolved in a volume not more than 150 μL for intranasal administration [61], for which the high potency of fentanyl is an advantage. To enhance nasal penetration and lessen local irritation, additives such as pectin, which form a thin gel over the mucosa, have been added into newer spray formulations [71]. With this formulation, it was shown that intranasal administration provides a shorter tmax and a higher Cmax than with oral transmucosal administration [71]. However, the pharmacokinetic parameters seemed to be comparable to other intranasal products (Table 2). Moreover, the delivery could be further enhanced by using the novel breath-actuated devices that provide significantly larger deposition in clinically important regions, possibly enhancing the delivery of drugs from the nose into the brain [72], although this technique has not yet been made available for a fentanyl formulation.

4.1.2 Fentanyl Formulations for Slow but Sustained Analgesic Effects

4.1.2.1 Passive Transdermal Therapeutic Systems

Fentanyl can be used for chronic pain treatment with continuous input from transdermal patches. The first transdermal fentanyl formulations consisted of a drug reservoir separated from the adhesive layer by a rate-limiting membrane to provide controlled drug release [73]. However, reservoir systems bore the risk of dose-dumping as a consequence of incidental leakage releasing the entire dose within a short period of time [74]. Therefore, current passive transdermal fentanyl formulations exclusively use the matrix technique with the drug dissolved in an inert polymer matrix that controls drug release [73]. This method diminishes the risk of incidental drug leakage and furthermore complicates the extraction of the drug for abuse [75]. The two delivery systems were shown to be bioequivalent, providing similar systemic concentration–time profiles and comparable tolerability [75]. Delivery results from the concentration gradient between fentanyl in the patch and the skin. Transdermal fentanyl patches constantly deliver 12.5, 25, 50, 75 or 100 μg/h over 72 h, so that a nearly zero-order delivery with a rate constant k0 is technically provided. As described previously [76], the release of fentanyl is a rate-controlled process and the amount delivered is proportional to the surface of the resorption area [77, 78]. Nevertheless, because the skin and underlying tissue need to be crossed, in contrast to delivery from the patch, the input function, fI(t), into the blood is not zero-order. Specifically, fentanyl was detected in plasma after 1–2 h, but the onset of the full analgesic effect was obtained between 12 and 24 h after patch application and may vary with local conditions such as skin temperature [79]. The mean Cmax values are proportional to the delivery rate provided by the patch, i.e. 0.3, 0.6, 1.4, 1.7 and 2.5 ng/mL, respectively, and are maintained during steady state after repeated patch applications [79].

The input function from transdermal patches has not been characterized parametrically. A pharmacokinetic/pharmacodynamic modelling approach, which was taken to overcome the difficulty in describing the release rates [80], therefore modelled the input function by cubic spline functions, akin to an earlier proposal to use linear splines [81]. A comparable technique had been used earlier to identify the absorption characteristics of a fentanyl transdermal therapeutic system delivering nominally 100 μg/h, using the Loo-Riegelman method [82] consisting of a deconvolution of the absorption function from the plasma concentration–time profiles based on the profile following intravenous administration [83]. According to this analysis, the absorption increased during the initial 4–8 h, remained relatively constant at 90.9 ± 25.7 mg/h until 24 h and subsequently decreased with a terminal absorption half-life of 16.6 ± 3.7 h.

Transdermally delivered fentanyl is not subject to first-pass metabolism (Fig. 2) and therefore provides a greater bioavailability than oral administration. However, its bioavailability from passive formulations is <1 because towards the end of the delivery interval, the concentration gradient between the transdermal therapeutic system and the skin decreases until the delivery ceases, which reduces the bioavailability of fentanyl from the patches to 92 % [84]. The absorption rate is also subject to changes in local conditions such as raised skin temperature, for instance during fever, with estimated increases in fentanyl plasma concentrations by approximately 33 % at body temperatures of 40 °C [85], with the risk of acute opioid overdose [79, 86]. In addition, in cancer patients the absorption was reported to be highly variable [87] and to depend on the patient’s age, with lower absorption with higher age, and surprisingly also varied with the type of cancer, with breast or digestive cancer associated with higher absorption than lung cancer [88].

4.1.2.2 Iontophoretic Transdermal Systems

Iontophoresis is a method for transdermal administration of ionisable drugs in which the electrically charged components are propelled through the skin by an external electric field [89]. In the fentanyl formulations, the transfer is mediated by a small electric current rather than by passive diffusion. The patch comprises a drug-containing hydrogel sandwiched between two electrodes that are arranged parallel to the skin surface, with the lower electrode attached closely to the skin via an adhesive layer [90]. No fentanyl was detected after passive (0.0 mA) fentanyl delivery, whereas with an electric current of 1–2 mA, mean times to detectable concentrations of plasma fentanyl were 33 and 19 min, respectively, and the tmax was observed at 122 and 119 min, respectively [91]. A voltage application of 2 V for 60 s released approximately 315–340 μg of fentanyl and a single-voltage application at 16 h produced a Cmax of approximately 200 pg/mL. Consecutive voltage applications at 16 and 40 h produced a Cmax of approximately 730 pg/mL [89, 90]. Iontophoretic systems may provide both rapid onset, for example within 15 min [92], and a long duration of analgesia. In this respect, 31.9 doses (40 μg per dose) were shown to provide analgesia for 24 h, which corresponds to a mean duration of analgesia per dose of approximately 45 min [93].

As shown for the growth hormone-releasing factor and R-apomorphine, the drug delivery from iontophoretic systems can be described as a zero-order input from the patch into the skin based on a constant iontophoretic driving force with a negligible lag time for the drug to enter the skin. The release from the skin into the plasma, however, is a passive process determined by a first-order skin-release rate constant [94]. The bioavailability depends on the duration of use, achieving only 41 % in the first hour but almost 100 % after 10 h, suggesting that the increased absorption over time may be due to alterations in the electrical conductance of the skin that occur during exposure to electric current [95].

4.2 Plasma Disposition of Fentanyl

The disposition function can best be assessed following intravascular administration and has been parametrically modelled using differential equation systems or, more simply, as a sum of exponentials, such as in Eq. 4:

$$ {\text{f}}_{\text{D}} ({\text{t}}) = \sum\limits_{{{\text{i}} = 1}}^{\text{n}} {{\text{A}}_{\text{i}} } \cdot {\text{e}}^{{ - \uplambda_{\text{i}} }} $$
(4)

where i denotes the number of the compartment (for fentanyl n = 2 or 3), A the amount of drug in the respective compartment and λ the first-order elimination rate constant from the respective compartment. Usually, arterial sampling has been employed in pharmacokinetic/pharmacodynamic assessments of fentanyl because of arteriovenous concentration differences [67]. As with remifentanil [96], venous samples may lead to false pharmacokinetic/pharmacodynamic conclusions for fentanyl. Mean reported [8] parameter values of a three-compartment pharmacokinetic model were 0.069, 0.0059 and 0.00190 L−1 for α1, α2 and α3, respectively, and those of λ1, λ2 and λ3 were 0.673, 0.0370 and 0.00146 h−1, respectively, where α and λ are the coefficients and exponents of the disposition function, respectively, described as a sum of exponentials. Alternatively, the disposition of fentanyl has been described by a two-compartment model [97] parameterized as a differential equations system as follows (Eq. 5):

$$ \begin{aligned} {\text{dA}}_{1} /{\text{dt}} & = {\text{CL}} \cdot {\text{C}}_{1} - {\text{Q}} \cdot {\text{C}}_{1} + {\text{Q}} \cdot {\text{C}}_{2} \\ {\text{dA}}_{2} /{\text{dt}} & = {\text{Q}} \cdot {\text{C}}_{1} - {\text{Q}} \cdot {\text{C}}_{2} \\ \end{aligned} $$
(5)

where the amounts of drug in a compartment are denoted by A followed by the number of the compartment (i.e. compartment 1 is the central compartment and compartment 2 is the peripheral distribution compartment); the total body clearance and the intercompartmental clearance are denoted by CL and Q, respectively, and the volumes come into play as scaling factors between concentrations (C) and amounts, i.e. A = C · V. The numerical values from Yassen et al. [97] were CL = 0.98 L/min, Q = 3.51 L/min, V1 = 19.5 L and V2 = 150 L. The different volumes may also be responsible for the apparently highly variable description of the distribution volumes of fentanyl between 60 and 300 L [25]. These may depend on the pharmacokinetic models used and become inaccurate with simpler approaches such as descriptive or one-compartmental analysis and short observation times. Furthermore, the pharmacokinetics of fentanyl seem to be complicated by its possible sequestration in the lungs [98, 99], with consequences for the temporal course of its brain delivery, which has trigged the development of re-circulatory physiological pharmacokinetic models [100].

4.3 Concentrations at Effect Site

The effects of fentanyl can be related to its concentrations at effect site (Ce) using a sigmoid pharmacodynamic model [101] as in Eq. 6:

$$ {\text{PDmeasure}} = {\text{E}}_{0} + \frac{{{\text{E}}_{ \hbox{max} } \cdot {\text{C}}_{\text{e}}^{\gamma } }}{{{\text{EC}}_{50}^{\gamma } + {\text{C}}_{\text{e}}^{\gamma } }} $$
(6)

where E0 denotes the baseline value of the pharmacodynamic measure, Emax its possible maximum and EC50 the concentration of fentanyl at half-maximal effect. As described previously [76], plasma concentration ranges are available to which clinical effects can be associated. Thus, 50 % pain relief was achieved at concentrations of 1.35 and 1.9 ng/mL and quantified in healthy volunteers by either pain-related cortical potentials or pain intensity ratings after dental electrical stimulation, respectively [102]. To achieve clinical analgesia, fentanyl plasma concentrations between 0.3 and 1.5 ng/mL are recommended, whereas adverse effects significantly increase at concentrations >2 ng/mL [103]. The most dangerous opioid effects on respiration are reported with EC50 values between 3.5 ± 1.4 ng/mL for respiratory rate and 6.1 ± 1.4 ng/mL for ventilatory volume [104]. The EC50 for maximal slowing of EEG was 6.9 ± 1.5 ng/mL [18]. Although the calculations were performed on plasma concentrations, they involve steady-state considerations [105] and therefore reflect the required local CNS concentrations. However, to produce these effects, fentanyl molecules in blood must cross the blood–brain barrier. As a lipophilic compound (Table 1), fentanyl is transferred quickly with a t½,ke0 of 6.6 min for its effects on EEG [17] and of 16.4 min for its effects on respiration [97]. The half-lives were obtained as ln(2)/ke0, which is the rate constant of the first-order transfer function [fT(t); Eq. 7] between plasma and a virtual effect compartment [10, 105], which for fentanyl can be assumed to represent the CNS.

$$ {\text{f}}{}_{\text{T}}({\text{t}}) = {\text{k}}_{{{\text{e}}0}} \cdot {\text{e}}^{{ - {\text{k}}_{{{\text{e}}0}} \cdot {\text{t}}}} $$
(7)

The resulting short delay between the time course of fentanyl concentrations in plasma and that at the effect site is of major relevance in settings where plasma concentrations change quickly, such as following administration in fast-delivery formulations.

5 Future Directions

Despite the many pharmacokinetic reports on non-intravenous fentanyl preparations, mainly driven by the need for plasma concentration data for the approval of generic formulations, most are limited to descriptive pharmacokinetics such as tmax, Cmax and area under the plasma concentration–time curve. Only a few parametric assessments have been published and this incomplete knowledge on the characteristics of particular formulations impedes pharmacokinetic predictions of plasma and effect-site concentration profiles. As demonstrated 20 years ago, such information may be a valuable basis for rational opioid selection [106]. With respect to non-intravenous fentanyl, input models for sustained-release formulations are particularly lacking, and this impedes covariate associations such as the dependency of the input on age, sex, skin properties or body temperature. The occasionally used [80, 83] splines can only partially close this gap. More complex input functions provide meaningful parameters for sustained-release administration, such as an inverse Gaussian distribution (Eq. 8) [107] of:

$$ {\text{f}}{}_{\text{I}}({\text{t}}) = {\text{Dose}} \cdot {\text{F}} \cdot \sqrt {\frac{\text{MIT}}{{2 \cdot \pi \cdot {\text{CV}}_{\text{I}}^{2} \cdot {\text{t}}^{3} }}} \cdot {\text{e}}^{{ - \frac{{({\text{t}} - {\text{MIT}})^{2} }}{{2 \cdot {\text{CV}}_{\text{I}}^{2} \cdot {\text{MIT}} \cdot {\text{t}}}}}} $$
(8)

where MIT denotes the mean input time and \( {\text{CV}}_{\text{I}}^{2} \) the normalized variance of the distribution, which provides a flexible input function [108]. This might provide a basis for a parametric input function for transdermal fentanyl administration, as exemplified in Fig. 3. Moreover, the pharmacokinetics and pharmacokinetics/pharmacodynamics of intranasal fentanyl administration have also only been addressed parametrically in an incomplete way. The model would need to take into account that the fentanyl CNS concentrations originate from direct input, bypassing the blood–brain barrier, and from input via the blood compartment.

Fig. 3
figure 3

Exploratory assessment of an inverse Gaussian distribution [107] as a possible input function for transdermally delivered fentanyl. The absorption rate over time was read from a published graph (figure 5 of Varvel et al. [83]) of the input characteristics described as splines, here shown as dots connected by a solid line. Subsequently, the values of mean input time and coefficient of variation of the input function (oral bioavailability being set to a value of 1) were roughly adapted using weighted (1/y) least squares regression (Gnumeric spreadsheet solver, http://projects.gnome.org/gnumeric). This resulted in mean input time = 25.5 h and coefficient of variation = 0.76. A more exact analysis was not done as data were reconstructed from a figure without exact numerical information. It demonstrates, nevertheless, that the inverse Gaussian distribution (dashed line) may serve as a basis for the development of a suitable input function for transcutaneous fentanyl

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6 Conclusions

From a pharmacokinetic perspective, it is the input function, fI(t), into the body by which non-intravenous formulations of fentanyl mainly differ from each other. Designed half a century ago [2], fentanyl is nowadays one of the most successful opioid analgesics [6, 7]. This success has been made possible by the pharmaceutical development of a variety of formulations, which allow non-invasive administration of fentanyl in several clinical settings of pain. These include rapid delivery from intranasal, transmucosal or transdermal iontophoretic formulations to treat breakthrough pain, in the paediatric setting for acute post-operative pain, as well as slow and partly controllable release from passive formulations with transdermal delivery for treating chronic pain. Current approaches to new drug development in the pharmaceutical industry include reformulation of previously known active substances, the development of new indications and specific patient populations, and the development of novel mechanisms of action. Among these, the development of non-intravenous formulations of fentanyl may be regarded as successful reformulation based on pharmacokinetic research, in concert with pharmaceutical technology, which expanded its use beyond the restriction to the operating [4] or recovery room. To further facilitate the clinical utility of fentanyl, parametric pharmacokinetics, in particular the mathematical characterization of input functions from the various formulations and generics, are required for the rational selection of non-intravenous fentanyl formulations for individualized pain therapy. Clinical pharmacokinetics has the necessary knowledge and tools to achieve this.