Pharmacokinetic–Pharmacodynamic Modeling of Opioids

S90 Journal of Pain and Symptom Management Vol. 29 No. 5S May 2005

Proceedings of the Symposium “Updates of the Clinical Pharmacologyof Opioids with Special Attention to Long-Acting Drugs”

Pharmacokinetic–PharmacodynamicModeling of OpioidsJorn Lotsch, MD, PhDpharmazentrum frankfurt, Institute of Clinical Pharmacology, Johann Wolfgang Goethe-UniversityHospital, Frankfurt, Germany

AbstractThe effects of opioids usually parallel the plasma concentrations but with a temporal shift.This temporal shift differs between opioids. It is small with alfentanil or remifentanil andvery long with the active metabolite of morphine, morphine-6-glucuronide (M6G). Themathematical and experimental techniques for modeling these pharmacokinetic-pharmacodynamic (PK/PD) relationships were developed in the late 1970s. The delaybetween plasma concentrations and effects is accounted for by the introduction of ahypothetic effect compartment, which is linked to the plasma compartment by a first-ordertransfer function with a rate constant ke0. The effects are then linked to the concentrationsat effects site by standard pharmacodynamic models such as sigmoid (“Emax”) models orpower models, depending on the actual effect measure. These principles were first applied tothe opioids fentanyl and alfentanil in 1985. Since then, PK/PD of opioids have beenrepeatedly assessed, using EEG derived parameters, pupil size, and experimental andclinical pain as effect measures. The opioids of the fentanyl group, methadone, morphine,and piritramid, are today well characterized with respect to their PK/PD properties.Alfentanil and remifentanil are very fast equilibrating opioids with equilibration half-livesbetween plasma and effect site of about 1 minute. They are followed by fentanyl andsufentanil, each with equilibration half-lives of about 6 min. Methadone equilibrates witha half-life of about 8 min. Morphine, in contrast, equilibrates with a half-life of 2–3 h.The slowest opioid with respect to plasma-effect site transfer is M6G, with an equilibrationhalf-life of about 7 h. PK/PD modeling has advanced the understanding of the time courseof the clinical effects of opioids after various dosing regimens. It may provide a rationalbasis for the selection of opioids in clinical circumstances. PK/PD modeling of opioids mayalso be employed for the design and the interpretation of experiments addressing clinicaleffects of opioids. J Pain Symptom Manage 2005;29:S90–S103. � 2005 U.S. CancerPain Relief Committee. Published by Elsevier Inc. All rights reserved.

Key WordsConcentration-response relationship, pain therapy, population modeling, effect compartment

Address reprint requests to: Jorn Lotsch, MD, PhD, Phar-mazentrum Frankfurt, Institute of Clinical Phar-macology, Johann Wolfgang Goethe-UniversityHospital, Theodor Stern Kai 7, D-60590 Frankfurt,Germany.Accepted for publication: January 5, 2005.

� 2005 U.S. Cancer Pain Relief CommitteePublished by Elsevier Inc. All rights reserved.

IntroductionDifferent dosing regimens of opioids may

result in different time courses of the opioidplasma concentrations, which in turn lead todifferent time courses of opioid effects. A bolus

0885-3924/05/$–see front matterdoi:10.1016/j.jpainsymman.2005.01.012

Vol. 29 No. 5S May 2005 S91Opioid PK/PD

injection is used when an immediate or a short-term effect is desired. Intravenous infusions,sustained release tablets, or transdermal deliv-ery systems are used when long-term effects areintended. The clinical observation of the effectsparalleling the plasma concentration is appar-ently supported by the reports of a good correla-tion between plasma concentrations ofmethadone and its clinical effects.1,2

Opioids differ with respect to the correlationbetween plasma concentrations and effects.The effects of fentanyl and alfentanil apparentlyparallel the plasma concentrations moreclosely than those of morphine and meperi-dine. When injecting a bolus of 10 mg mor-phine, the opioid effects increase during thefirst hour, although plasma concentrationshave their maximum immediately after the in-jection is finished. Thus, at least for some opi-oids, there appears to be a delay between thetime-course of the plasma concentrations andthe time course of the effects.

A clinical case will illustrate this delay betweenplasma concentration and effects. A 22-year-oldman with Goodpasture syndrome, end-stage renaldisease, and severe arterial hypertension under-went bilateral nephrectomy.3 He received 40 mgand 30 mg of morphine, respectively, as thesole analgesic at the beginning and at the end ofa 3.5-h surgery (intravenous bolus injections).Postoperative patient-controlled analgesia(PCA) using morphine was begun. The patientindicated mild pain at rest and severe pain whenmoving. He self-administered 36 mg of intrave-nous morphine during the first 18 h after sur-gery and another 4 mg during the following13 h. He became unconscious 31 h after surgeryand remained in that state for 45 h. This timecourse was also reflected by the results of vigi-lance tests administered in the postoperativeperiod (Galveston orientation and amnesiatest4), digital span test assessing how many ci-phers can be repeated correctly, and reactiontime to a visual stimulus). The patient under-went hemodialysis at 45 h, 88 h, and 162 hafter surgery. He was unconscious during thefirst hemodialysis and remained in that state34 h thereafter. Blood samples were drawn inthe pre- and postoperative period to assay forplasma concentrations of morphine and its glu-curonide metabolites using high performanceliquid chromatography.5 When the patientbecame unconscious 31 h after surgery, the

morphine plasma concentration had beenbelow the lower limit of quantification of25 ng/mL for more than 26 h.5 M6G concentra-tions had already been close to their maximumfor 26 h. The hemodialysis starting 45 h aftersurgery almost completely cleared M6G fromplasma. However, the patient did not regainconsciousness until 34 h after hemodialysis(Figure 1).

This case illustrates that, in the setting ofrenal insufficiency, severe opioid side effectscan occur many hours after morphine plasmaconcentrations have peaked and M6G concen-trations have reached a plateau in plasma.These side effects are most likely mediated byM6G and not by morphine or morphine-3-glu-curonide (M3G). The main morphine metabo-lite M3G does not exert depressant effects in theCNS.6 Morphine itself equilibrates too quicklyfrom blood into the brain to account for ob-served delay of many hours between the plasmaconcentration versus time profile and the effectversus time profile, respectively. Despite twolarge intravenous doses of morphine duringsurgery, the patient experienced postoperativepain and continued to self-administer mor-phine by PCA. At this time, the high plasmaconcentrations of M6G did not seem to resultin clinically relevant analgesia. However, witha delay of many hours, similar plasma concen-trations of M6G resulted in toxic side effects,i.e., the patient became unconscious. As theslow transfer between plasma and effect com-partment is the reason for the delayed appear-ance of opioid side effects, it is also likely to bethe reason why the patient remained uncon-scious for a long period after M6G had disap-peared from plasma. Despite its eliminationfrom plasma, it was probably still present at theeffect site at high enough concentrations tomaintain clinical opioid effects.

Monitoring of morphine concentrationsplasma in the present clinical case would nothave given any indication for upcoming severeopioid side effects. Even monitoring of M6Gplasma concentrations would not necessarilyhave led to the conclusion of toxicity becauseM6G levels were high for a long time withoutclinical effects. The time course of the opioidsymptoms can only be explained when assum-ing that M6G very slowly equilibrates betweenplasma and the central nervous system. This

S92 Vol. 29 No. 5S May 2005Lotsch

Fig. 1. For bilateral nephrectomy, a total amount of 110 mg of intravenous morphine had been given to a youngman with chronic severe renal failure. Unconsciousness started at a time when morphine was no longer detectablein plasma, and M6G concentrations had been high for more than a day. Hemodialysis starting 45 h after surgeryalmost completely cleared M6G from plasma. However, the patient remained unconscious for another 34 h. Theslow equilibration of M6G between plasma and the brain seems to explain the long delay between reaching highplasma concentrations of M6G and the onset of unconsciousness, as well as the slow recovery from unconsciousnessafter clearing most M6G from plasma with hemodialysis. The time course of morphine dosage, the plasmaconcentrations of morphine, M6G, and M3G is displayed. From Angst et al.3

reasoning underscores the rationale for PK/PD modeling.

Principles of PK/PD ModelingThe principles of PK/PD modeling7–10 were

developed in the late 1970s.11,12 PK/PD model-ing links effects to the concentrations of anopioid. The effects depend on the concentra-tions at the site of effects, for example, the CNS,which the opioid may reach with some diffi-culty. An important application of PK/PD mod-eling is, therefore, to account for the delaybetween the time courses of the plasma concen-trations and that of the effects as were observedin the above-mentioned case.

The major points of so-called “indirectlink” PK/PD modeling start with the plasmaconcentration versus time profile of the opioid,to which the concentration versus time profileat the effect site can be linked via the conceptof an effect site. Dose response relationships ofanalgesics have been elsewhere reviewed,13 witha broader inclusion of non-modeling studiesin addition to PK/PD modeling approaches.

An opioid’s disposition can be regarded asmonotonically decreasing curve that can be de-scribed by a sum of exponentials

Civ(t) � �n

i�1Aie

�lit (1)


where Civ(t) denotes the concentration versustime profile after an intravenous dose Div. Thecorresponding unit impulse response of thesystem is given by

fD(t) �Civ(t)Div

� �n

i�1aie

�lit, (2)

where ai � Ai/Div.The concentration versus time profiles is gov-

erned by the unit impulse response (equation2) and by the specific dosing schedule by whichthe opioid is administered. For a bolus injec-tion, the input function, I(t), is Dose·Dirac(t),Dirac(t) being a function that has a value of 1when the term in the parentheses equals zero(i.e., when t � 0), and a value of zero otherwise.For an intravenous infusion, I(t) � k0·(Heavisi-de(t-Tstart)-Heaviside(t-Tend)), where Tstart is thetime of the start of the infusion, Tend the time ofthe end of the infusion, k0 the constant rateof the infusion, and Heaviside is a step functionthat has a value of 1 when the term in theparentheses is greater than zero, and a value ofzero if the term in the parentheses is smallerthan zero.

The plasma concentration versus time curve,Cp(t), of a drug can be regarded as a convolutionof the input or dosing function, I(t), and thedisposition function fD(t)

Cp(t) � I(t)*fD(t), (3)

where fD(t) is given by equation 2, and the as-terisk means “convolution.” For example, theplasma concentration versus time profile for adrug with three-compartment disposition afterbolus injection is

Cp(t) � Dose · (a1 · e�l1·t � a2 · e�l2·t (4)

� a3 · e�l3·t)

To account for the delay between the timecourses of the plasma concentrations and thetime course of the effects, an effect compart-ment with an equilibration time constant ke0between plasma concentrations and effect is in-troduced into the model.11,12 The concentra-tions at effect site, Ce(t), are obtained byconvolution of the plasma concentration versustime profile, Cp(t), with a transfer function, fT(t).A first-order function has been used in mostcases

fT(t) � ke0 · e�ke0·t (5)

The half-life, t½,ke0, of this first-order transfer isgiven by ln(2)/ke0. The concentrations at effectsite, Ce(t), are thus

Ce(t) � Cp(t)*fT(t) � ke0·e�ke0·t*Cp(t) (6)

where the asterisk denotes “convolution.”To the concentrations at the site of effect,

the effect E is linked by a pharmacodynamicmodel. This is often a sigmoid model of

E � E0 �Emax·C

ge

ECg50 � Cge(7)

where E0 is the baseline value of the effectsmeasure, Emax the maximum possible effect,EC50 the concentration at effect site that leadsto a half-maximum effect, and g determinesthe steepness of the concentration versus re-sponse relationship. The value of EC50 serves todefine the opioid’s potency. Other pharma-codynamic relationships can be used whenrequired. For example, a decreasing effect mea-sure such as pupil constriction after opioidadministration is described by

E � E0 � [(E0 � Emax) ·C ge

ECg50 � Cge] (8)

If the effects are the result of agonistic actionof two opioids, for example an active parentcompound and its active metabolite, pupil sizeas a function of the opioid concentration ateffect site would be described as

E � E0 � [(E0 � Emax) (9)

· ( Ce g,opioid1opioid1

EC g,opioid150,opioid1

�Ce g,opioid2

opioid2


1 �Ce g,opioid1

opioid1


�Ce g,opioid2

opioid2


)]Other effects such as pain tolerance are betterdescribedbya linearorpower model rather thana sigmoid model, because there is no ceiling ofthe effect:

E � Baseline � a · Cge (10)

where a is the slope of the effect versus concen-tration relationship, and g its shape factor.


This full parametric PK/PD modeling, thatis, mathematical modeling of both the PK andthe PD, has the disadvantage that a poor fit of thePK will carry over to the PK/PD and result ina poor PK/PD estimate. To minimize theimpact of poor PK estimates on the PK/PDestimate, semi-parametric PK/PD modeling14

reduces the mathematical description of thePK to a minimum model consisting of linearsplines, namely, of direct linear connection be-tween subsequent concentrations. The link tothe effect site compartment and the relation-ship between Ce and the effects are similar tothe full parametric PK/PD modeling.

The concepts of PK/PD modeling have directimplications for the time course of the opioideffects after various dosing regimens. De-pending on the speed of the transfer betweenplasma and effect site, the time course of theeffects can be close to the plasma concentra-tions or can follow them with a delay, whichwill be longer the slower the transfer is (Figure 2).This governs the onset and offset of the opioideffects. For an opioid with a fast transfer, thatis, a large value of ke0, the effect will start shortly

after administration is started, and the effectwill disappear together with the disappearanceof the opioid from plasma. Thus, the effectwill be controllable via the plasma concentra-tions. Immediate high plasma concentrationscan be easily achieved with a bolus injection,and the faster the opioid disappears fromplasma, the faster the effects will disappear andthe patient will recover. This time course ofopioid effects may be desirable, for example,in the setting of ambulatory surgery.

On the other hand, when the effects have tolast longer, such as after more painful surgicalprocedures or for long-term analgesic treat-ment, an opioid such as remifentanil with rapidelimination from plasma has to be continuouslyadministered in order to maintain the analgesiceffect. Alternatively, an opioid with slower elimi-nation from plasma can be administered. Here,a slower transfer between plasma and effectsite further prolongs the clinical effects of theopioid. The slower onset with building up ofsufficiently high concentrations at effect sites,possibly after several doses (Figure 2), mightnot be a problem for long-term therapy, or can

Fig. 2. Plasma and effect site concentrations after repeated intravenous bolus administration of opioids. Effectsof opioids with fast transfer between plasma and effect site such as alfentanil (half-life of the transfer of 6 min)follow the plasma concentrations closely. In contrast, effects of opioids with slow transfer between plasma andeffect site such as morphine (half-life of the transfer of 3 h) build up slowly, reaching a maximum only afterseveral repeated administrations, and then persist longer than the plasma concentrations would suggest. Theplasma concentrations of alfentanil and morphine have been set to be identical for demonstration purposes.


be compensated by administration of a loadingdose. For example, morphine has a muchslower transfer rate between plasma and effectsite than alfentanil or remifentanil. Therefore,it takes more time until the effect builds up butthe effect lasts longer, which is an additiveresult of its slower elimination from the site ofeffect and its slower elimination from plasmaas compared to remifentanil.

PK/PD Studies with OpioidsAn overview about the PK/PD studies on opi-

oids is given in Table 1. The first opioids ana-lyzed for their PK/PD properties were fentanyland alfentanil.15 Of twelve patients undergoingsurgery (lumbar laminectomy, femoral-popli-teal bypass, vasovasectomy, etc.), 6 receivedalfentanil at 1,500 µg/min and 6 receivedfentanyl at 150 µg/min. The EEG was continu-

ously recorded and arterial blood samples weredrawn frequently (0.5 to 1 min intervals duringthe infusion, and 2 to 4 min intervals thereafter,until the EEG had returned to baseline). TheEEG was submitted to Fast Fourier Analysis andthe 95% spectral edge was obtained as a mea-sure of the narcotic EEG effect. The total doserequired to produce delta waves was 600–825µg for fentanyl and 6000–9000 µg for alfentanil.Onset of respiratory depression occurred after3–5 min after starting the fentanyl infusion, andafter 1–2 min after starting the alfentanil infu-sion. A distinct time lag was observed betweenpeak fentanyl plasma concentration and peakchanges in the spectral edge. The spectral edgechanges paralleled the fentanyl concentrations,but with a temporal shift. This temporal shiftwas smaller with alfentanil than with fentanyl.After the end of the alfentanil infusion, the EEGreturned faster to baseline than with fentanyl. Asigmoid Emax model was used to describe the

Table 1Parameters Estimated By Means of PK/PD Modeling of Opioids

Opioid t½,ke0 EC50 [nmol/l] γ Effect measure Ref.

Fentanyl 6.4 � 1.3 min 20.5 � 4.5 4.9 � 1 EEG power spectrum analysis 15

6.6 � 1.7 min 24.1 � 6.5 6.2 � 1.8 EEG power spectrum analysis (spectral edge) 17

4.7 � 1.5 min 23.2 � 7.7 4.3 � 1.3 EEG power spectrum analysis (spectral edge) 16

5.4 � 2.1 min 29.1 � 24.7 4 � 3 EEG power spectrum analysis (spectral edge) 18

Alfentanil 1.1 � 0.3 min 1248 � 391 4.8 � 1.5 EEG power spectrum analysis 15

0.9 � 0.3 min 1150 � 651 4.8 � 2.4 EEG power spectrum analysis (spectral edge) 16

0.6 � 0.4 min 1385 � 655 6 � 2 EEG power spectrum analysis (spectral edge) 18

1 min 1522 7 � 3.3 EEG power spectrum analysis: bispectral index 19

1.3 min 1378 7 � 3.3 EEG power spectrum analysis (spectral edge) 19


Sufentanil 6.2 � 2.8 min 1.8 � 0.8 3.1 � 0.9 EEG power spectrum analysis (spectral edge) 17

Remifentanil 1.3 min 30 2.51 EEG power spectrum analysis (spectral edge) 23

1.6 � 0.9 min 53 � 14 4.3 � 2 EEG power spectrum analysis (spectral edge) 21

1.3 � 1.7 min 12 � 9 1.76 � 0.44a Experimental pain 25

0.79 min 31 4.27 EEG power spectrum analysis 27


Methadone 7.7 � 3.6 min 1016 � 1332 2.03 � 1.1 Analgesia (cancer pain) 28

9 � 14.7 minb 1258 � 554 4.4 � 3.8 Analgesia (cancer pain) 29

18.6 � 31.6 minb 1178 � 718 5.8 � 5.4 Sedation 29

Morphine 2.8 h 34 1.9 Pupil size 33

3.9 h 17 2.1 Pupil size 31

2.8 h 27 2.4 Pupil size 34

_ 1.7 h _ 250 2.6 Transcutaneous electrical stimuli: pain threshold 30

\ 3.9 h \ 146_ 1.6 h _ 268 2.8 Transcutaneous electrical stimuli: pain tolerance 30

\ 4.8 h \ 1152.8 h c c Transcutaneous electrical stimulation: pain tolerance 34

M6G 6.4 h 743 2.6 Pupil size 33

7.7 h 745 3.1 Pupil size 34

8.2 h c c Transcutaneous electrical stimuli: pain tolerance 34

6.2 h c c Transcutaneous electrical stimuli: pain tolerances 35

Piritramide 16.8 min 28 1.9 Postoperative analgesia 37

aThe use of the sigmoid model here is discussed in the text.bOnly in about half of the 15 subjects could, a ke0 parameter be proved to be part of the model. The given values are the average of the reportedindividual values of t½,ke 0, with values of zero when the data were fit without a ke 0.cA linear model was used to describe the effect versus concentration relationship; therefore, EC50 was not a parameter.


effect versus concentration relationship, andthe concentrations at effect site were ob-tained using an indirect-response model withan effect compartment, connected to theplasma compartment by a transfer function witha first-order rate constant ke0. The estimatedvalues of t½,ke0 were 6.4 min for fentanyl and1.1 min for alfentanil, indicating that both opi-oids equilibrate fast, yet alfentanil equilibratesfaster than fentanyl. The maximum effect wassimilar for both drugs, which is compatible withboth being full agonists at µ-opioid receptors.The EC50 of 6.9 and 520 ng/mL for fentanyl andalfentanil, respectively, indicated that fentanylwas 75 times more potent than alfentanil inshifting the EEG spectral edge to the left to-wards slower frequencies.

In a study again employing the EEG spectraledge as effect measure,16 alfentanil was adminis-tered to 17 patients and fentanyl was admin-istered to 20 patients aged 20–89 years. Thedose requirement of fentanyl or alfentanilwere found to decrease with increasing age (a50% decrease from age 20 to 89), whereas noage-related changes in the pharmacokinetic pa-rameters were found. This was interpreted asan indication for decreasing brain sensitivitywith age. The average PK/PD parameter valuesof t½,ke0 of 4.7 min, EC50 of 7.8 ng/mL, andγ of 4.3 corresponded to the values obtained inthe other studies. The same was true for theparameter values for alfentanil of t½,ke0 of 0.9min, EC50 of 479 ng/mL, and γ of 4.8.

The same authors compared the PK/PD offentanyl with that of sufentanil, employing asimilar methodology as in the previous study.17

Surgical patients received short infusions ofeither fentanyl (386–1042 µg/min, n � 9) orsufentanil (39–104 µg/min, n � 7), and the ef-fects were quantified by means of the EEG spec-tral edge. In a second study part, each of tenpatients received either 1,250 µg fentanyl or125 µg sufentanil as an intravenous bolus, andthe spectral edge of the EEG was monitoreduntil it came back to baseline. The study re-sulted in a similar PK/PD profile of fentanyl asthe previous study,15 the estimated t½,ke0 of 6.5minutes were almost identical for fentanyl andsufentanil, and identical maximum effects wereseen. Sufentanil had a 12-fold greater potencythan fentanyl in shifting the EEG spectral edgeto the left.

The effects of fentanyl (2.2 µg/min), alfen-tanil (22 µg/min), and the investigationalopioid trefentanil (22 µg/min) on the EEGspectral edge were compared in five healthyvolunteers in a crossover study.18 The estima-tions of the PK/PD parameters for fentanyl andalfentanil were similar to those previously ob-tained;15,17 trefentanil had a t½,ke0 of 1.2 �0.5 min, an EC50 of 429 � 313 ng/mL, and asteepness γ of the concentration–effect relation-ship of 5 � 3 for shifting the spectral edgetoward lower frequencies. The maximum ef-fects on the spectral edge were similar for allthree opioids.

The PK/PD properties of alfentanil were esti-mated from 31 volunteers who received an intra-venous infusion of 22 µg·min�1·kg�1 alfentaniluntil a plateau in the spectral edge was ob-served.19 The EEG was recorded and the opioideffects were quantified using the spectral edge,the delta power of the EEG and the bispectralindex. The obtained PK/PD parameters weresimilar to those previously obtained for the EEGeffects of alfentanil,15 with the exception thatthe EC50 obtained with the bispectral index wassignificantly greater than that obtained for theincrease in delta power (634 vs. 519 ng/mL);the PD parameters did not differ between thespectral edge and either delta power or bispec-tral index.

The PK/PD relationship for alfentanil-in-duced respiratory depression was investigatedin a study in 14 men who underwent majorurologic surgery.20 They received an intrave-nous infusion of 2.3 mg·min�1·kg�1 alfentaniluntil a cumulative dose of 70 µg/kg had beengiven, end-expiratory partial pressure ofcarbon dioxide exceeded 65 mmHg, or apneicperiods lasting more than 60 sec occurred. Theobtained PK/PD model treated CO2 as an en-dogenous metabolite that possesses its own ki-netic properties. The elimination of CO2 wasconsidered to be impeded by the opioid in aconcentration-related manner. Using an indi-rect response model with an effect site compart-ment as for the EEG effects of opioids (seeabove), the effect of the opioid on the elimina-tion of carbon dioxide rather than on thePaCO2 itself was modeled. Because carbon diox-ide is a strong respiratory stimulant, the modelaccounted for the fact that increasing carbondioxide concentrations stimulate carbon diox-ide elimination. The resulting PK/PD model


therefore substantially differed from the indi-rect-response models with sigmoid concentra-tion-effect relationship that described well theopioid effects on the EEG. The estimatedvalue of the EC50 for raising the PaCO2 was60.3 µg/L. The study did not estimate the ke0of alfentanil.

The effects of alfentanil on the EEG spectraledge were compared with those of remifen-tanil in ten healthy volunteers.21 In a crossoverdesign, they received intravenous infusions of3 µg·min�1·kg�1 remifentanil or of 1,500 µg·min�1

alfentanil. The study design was similar to the pre-vious studies on EEG effects of opioids.15 Theresults for alfentanil resembled those previouslyobtained. Although both drugs had identicalmaximum effects, remifentanil was 19 timesmore potent than alfentanil in slowing downthe EEG. Remifentanil had a transfer half-lifebetween plasma and effect site of 1.6 min. Com-pared to alfentanil, the high plasma clearanceof remifentanil by de-esterification via bloodesterases (Figure 3), combined with its small

steady-state distribution volume, resulted in arapid decline in blood concentration after ter-mination of an infusion. With the fast equilibra-tion in terms of ke0, the result of this rapiddisappearance from plasma is a very quickdecline of the effects of remifentanil. Thisdecline in effects is considerably faster than withalfentanil, although the equilibration of alfen-tanil is at least as fast as that for remifentanil.The two PK and PK/PD properties together,that is, rapid degradation in plasma and rapidequilibration with the effect site, makes remi-fentanil an opioid with very fast onset and espe-cially very fast offset of action.

The fast equilibration of remifentanil be-tween plasma and central nervous effect site wasalso seen in a study in 10 healthy women whowere administered an intravenous infusion at 3mg·kg�1·min�1 for 10 min.22 Again, the EEGspectral edge was employed as an effect mea-sure. Pharmacokinetic parameters estimatedfrom venous and arterial data differed signifi-cantly. When arterial concentrations were plot-ted against electroencephalographic effect, a

Fig. 3. Plasma and effect site concentrations after a two-hour infusion. Although both alfentanil16 and remifen-tanil23 have a very fast equilibration between plasma and effect site and the effects therefore follow the plasmaconcentrations closely, the effects of remifentanil disappear faster than those of alfentanil. This owes to the veryfast elimination of remifentanil from plasma by cleavage via blood-esterases (upper right, esteric group markedby a dotted frame), whereas alfentanil is eliminated by hepatic metabolism via CYP3A4. The dosing of the opioid hasbeen arbitrarily adjusted to produce similar maximum concentrations.


classic counterclockwise hysteresis loop was ob-served, indicating a time lag between changes inconcentration and changes in effect. However,concentrations from venous blood produced aclockwise hysteresis loop. This emphasized thatwith the fast degrading remifentanil, arterialblood samples need to be drawn. The studyresulted in a t½,ke0 of 0.8 min for remifentanil,an EC50 of 14.8 for moving the spectral edge tothe left, and a shape factor γ of the sigmoideffect versus concentration relationship of 2.8.

A further study on the EEG effects of remifen-tanil, with a focus on the effects of age andsex on the pharmacokinetics and the PK/PDrelationship, employed a similar design andestimated similar PK/PD parameters23 as theprevious study.21 Age was identified as a co-variate for EC50 and ke0. That is, the EC50 was13.1–0.148·(age – 40), and the ke0 was 0.595–0.007·(age – 40). This means that the EC50ranged from 16 ng/mL in a 20-year-old personto 7 ng/mL in an 80-year-old person; the trans-fer half-life for a 20-year-old person was ap-proximately 1 min, whereas it was 2 min for an80-year-old person. A prospective part of thestudy was applied to 11 participants. Themedian absolute prediction error of the EEGeffects was 23%. Age and lean body mass wereidentified as covariates for various volumes andclearances of the three-compartment pharma-cokinetic model that described the plasma con-centration versus time course of remifentanil.Simulations of the effect site concentrations ofremifentanil in populations ages 20, 50, and 80showed that elderly patients might have occa-sionally a slower recovery with remifentanilthan younger persons.24

The PK/PD parameters of effects of remifen-tanil on experimental pain were investigatedin 29 healthy volunteers and were comparedwith data obtained from 12 persons who had re-ceived alfentanil and 7 persons who hadreceived placebo.25 Analgesia was evaluated bypain tolerance to pressure exerted on the tibiaand on the sternum, with a maximum pressureof 10 kg/4.43 mm2. A sigmoid Emax model wasused to describe the analgesic effects of theopioids. The transfer half-life between plasmaand effect site was estimated to be 1.3 min, andthe concentration of remifentanil producinghalf-maximal analgesic response (i.e., the EC50)was estimated to be 4.66 �3.3 ng/mL. In addi-tion, a steepness factor γ of the Emax model

(equation 7) of 1.76 � 0.44 was estimated. Theuse of this model may be questioned becausepain tolerance did not reach a true maximum.In contrast, the maximum pressure was just anarbitrary cut-off. Therefore, a linear or powermodel might have been better suited to de-scribe the rising effects of remifentanil on thepainfulness of experimental stimuli.26 Theshort equilibration time found with the EEGparameters,21–23,25,27 however, was reproducedfor the pain parameter.

Estimates of pain relief were assessed after asingle intravenous dose (10–30 mg) of meth-adone to eight patients with chronic pain.28 Theeffects were related to the methadone concen-trations at effect site by means of an Emax model.No effects were observed in two patients. Forthe remaining 6 patients, analgesia and miosispeaked at 2–5 min after drug injection, pointingtoward a short delay between plasma and site.The t½,ke0 ranged from 1.3 to 31.1 min, with aharmonic mean at 3.6 min. The EC50 for painrelief with methadone was 0.29 � 0.38 µg/mL,and the steepness γ was 2.03 � 1.1.

The effects of methadone were again sub-jected to PK/PD analysis in a study in 15 cancerpatients who received continuous infusions ofmethadone for 180 to 270 minutes.29 An in-crease in plasma methadone concentration re-sulted in a rapid increase in pain relief orsedation. The values of t½,ke0 ranged from 3.3to 49.5 min for pain relief, and from 2.9 to 99min for sedation. In several patients, no ke0 wasobserved, that is, the delay was smaller thandetectable. Pain relief was related to theconcentrations at the site of effect by a sigmoidEmax model. The model had an EC50 of 0.359 �0.158 µg/mL and a steepness γ of 4.4 � 3.8.The parameters values for sedation were verysimilar.

In another study, no delay was observed be-tween the plasma concentrations and the pupilsize after oral administration of R-methadone.2

The parameters of the direct link sigmoid popu-lation effect model used to describe the effectsof R-methadone on pupil diameter were anEC50 of 2.3 � 1.4 ng/mL and a shape factor γof 9�6.2. This is 100 times less than it hadbeen observed for pain relief.28 However, thesampling frequency in that study was quite slowstarting at 0.5 h after methadone administra-tion and continuing at 1, 2, 3, 4, 6 h up to 96h. Therefore, the study might have missed the


previously observed28,29 short delay for metha-done equilibration.

The analgesic effects of morphine were sub-jected to PK/PD modelling in a study em-ploying transcutaneous electrical stimulation inhealthy 20 volunteers.30 Morphine was adminis-tered as an intravenous bolus of 100 µg/kg fol-lowed by a one-hour infusion of 30 µg·h�1·kg�1.Arterial blood was sampled and the attenuationof pain threshold and pain tolerance was re-lated to the concentrations at effect site by asigmoid Emax model for decreasing effectmeasures. The authors observed statisticallysignificant sex differences in the PK/PD of theanalgesic effects of morphine. Morphine wasonly half as potent in men as in women (EC50of 76 vs. 33 ng/mL for attenuation of pain toler-ance), and it equilibrated much slower inwomen than in men between plasma and effectsite (t½,ke0 of 4.8 versus 1.6 h for women andmen, respectively).

Pupil size as a measure of central nervousopioid was employed in a study in 15 healthyvolunteers.31 The time course of the pupil diam-eter after four inhalations of each 2.2 mg mor-phine (1 min interval) was similar to thatafter 8.8 mg morphine administered intrave-nously. Similarity of the effects between the twomorphine formulations was also observed forthe respiratory depressive actions of morphine,quantified by means of carbon dioxide re-breathing.32 The effects on pupil size wererelated to the concentrations at effect site by asigmoid model. The obtained values of EC50and γ were 4.8 ng/mL and 2.1, respectively.Morphine equilibrated slowly between plasmaand effect site, with a t½,ke0 of 3.85 hours.

Recent studies employed a PK/PD modelingapproach to the importance of the active metab-olite of morphine, M6G, for the effects of mor-phine.33 Employing pupil size as a measure ofcentral opioid effect, 8 healthy volunteers re-ceived morphine (0.5 mg as bolus, followed by10.7 mg as infusion over 4.7 h) or M6G (10.2mg as loading dose followed by 39.1 mg as infu-sion over 3.7 h) in a randomized two-way cross-over study. The duration of the infusion wastailored to achieve submaximum pupil constric-tion, namely, a visible constriction that did notreach a plateau throughout the experiment.Pupil diameter was assessed every 20 min forapproximately 18 h. The pupil size was linkedto the concentrations of morphine and M6G at

effect site by a sigmoid Emax model for decreas-ing effects. The estimated t½,ke0 of M6G was avery slow 6.4 h, and that of morphine was 2.8 h,which resembled the previously obtainedvalues.31 M6G was apparently 22 times lesspotent than morphine (EC50 � 740.5 nM forM6G vs. 36.2 nM for morphine). The steepnessof the sigmoid Emax model did not significantlydiffer between morphine and M6G (γ of 1.9and 2.6, respectively). To produce similar pupileffects, the M6G dose had to be 2.8 timesgreater than the morphine dose. When consid-ering that about 10% of a morphine dose ismetabolized into M6G, the metabolite appar-ently contributed very little to the effects ofmorphine.

In a subsequent study,34 the effects of mor-phine and M6G on pain were assessed in aplacebo-controlled investigation in 12 healthyvolunteers who received 63 to 112 mg of M6Gor 26 to 66 mg of morphine as an intravenousbolus plus infusion for 1.8 to 6.4 h. Analgesiawas assessed every 30 minutes for up to 16 hoursby means of transcutaneous electrical stimula-tion (sine wave, 5 Hz; intensity, 0–9.99 mA).Pupil diameter and side effects were recordedconcomitantly. The delay between the timecourse of the plasma concentrations and thetime course of the effects was longer for M6Gthan for morphine (t½,ke0 of 8.2 vs. 2.6 hoursfor pain tolerance, and of 7.7 hours vs. 2.8hours for pupil diameter). The slope of thelinear effect versus concentration relationshipfor pain tolerance was flatter for M6G than formorphine (0.05% vs. 0.6% increase in pain tol-erance per nmol/L of M6G and morphine ateffect site, respectively), pointing toward alower potency of M6G as compared with mor-phine. M6G was also less potent than morphinein producing pupil constriction (EC50 of 745nmol/L vs. 26.4 nmol/L for M6G and mor-phine, respectively). The steepness of the con-centration response relationship for the pupildata was 2.4 and 3.1 for morphine and M6G,respectively. Thus, the results of the previousstudy33 regarding pupil size were reproducedin that study. Again, M6G had apparently nocontribution to the effects of morphine in thehealthy volunteers.

The pharmacokinetic–pharmacodynamic in-terrelations of M6G also were addressed in astudy of 10 men and 10 women who received0.3 mg/kg intravenous M6G or placebo in a


randomized two-way crossover fashion.35 Paintolerance to electric stimuli consisting of 10 Hzsine waves at increasing intensity (cut-off 128mA) was used as effect measure. The delay be-tween the time course of the plasma concentra-tions of M6G and the time course of the effectswas long, with a transfer half-life t½,ke0 of 6.2 hand a large interindividual variability of 218%coefficient of variation. The effect site M6G con-centration causing a 25% increase in currentfor pain tolerance was 275 nM. For comparison,in the previous study34 that value was 500 nMfor the 5 Hz electric stimuli. The shape factorof the relationship between the increase in paintolerance and the concentrations of M6G ateffect site, described by a power model (equa-tion 10), was 0.71.

Whereas the above studies employed directadministration of M6G to assess its relative con-tribution to the effects of morphine, a studyused a pure PK/PD modeling approach to sepa-rate the relative effects of morphine and M6Gafter administration of morphine only.36 Afterintravenous administration of 10 mg morphinesulfate to 8 healthy volunteers, thresholds toexperimental heat pain evoked by means of athermode placed at the forearm were assessedas a measure of analgesic opioid effects. Theeffects were related to the concentrations usinga linear model. The estimated t½,ke0 of 0.16 hwas a hybrid of morphine and M6G. It was,however, substantially shorter than previouslyobtained.30,31,33,34 The contribution of M6G toanalgesia ranged from �0.1% to 66% and wasinversely related to the overall effect elicited bythe morphine dose. The study suggested thatwith increasing overall effect of morphine, thefractional contribution of M6G declines. How-ever, because M6G administration was not em-ployed, the results solely rely on the statisticsmade at the occasion of PK/PD modeling,whereas in previous studies33,34 they were sup-ported by direct experimental evidence.

Piritramide is a µ-opioid agonist that is widelyused in Germany but of minor importance inother countries. In 24 patients who underwentabdominal surgery, its effects on postoperativepain were assessed. Piritramide was infused at3 µg·min�1·kg�1 until analgesia was consideredsufficient or up to a maximum dose of 0.2 mg/kg.37 An inhibitory sigmoid Emax model was usedto describe the relation between effectsite concentration and perceived pain. The

resulting t1/2,ke0 was 16.8 min, the EC50 was12.1 ng/mL, and the slope factor γ was 1.9.Because piritramide equilibrates somewhatslower than as compared to the substances ofthe fentanyl group, the authors advised an ini-tial intravenous bolus of at least 5 mg when afast onset of the analgesic effects is desired.

Therapeutic Consequences Derivedfrom PK/PD Modeling

An important clinical application of PK/PDmodeling is for rational opioid selection. Usingpublished PK/PD models and parameters, thedecrease in plasma fentanyl, alfentanil, andsufentanil concentration after intravenous ad-ministration by bolus injection, brief infusion,or prolonged infusion was simulated.38 Thesecomputer simulations quantified the relation-ship between infusion duration and the timerequired for recovery after termination of theinfusion. The analysis suggested that alfentanil isbest used for operations longer than 6–8 h,when a rapid decrease in effect site opioid con-centration is desired after discontinuation ofthe infusion. In contrast, when fentanyl wasused with infusions longer than one hour, therecovery time rapidly increased to more than2 h (Figure 4). The time required for the effectsite concentration of fentanyl and sufentanil todecrease by a given percentage increases overat least the first 10 h of the infusion, whereasit does not increase after 4 h for alfentanil.This makes alfentanil the preferable drug forinfusions longer than 10 h when quick recoveryof the patient is desired. Remifentanil was notyet clinically available in 1991 when these simu-lations were performed. Its transfer half-life be-tween plasma and effect site, similar to that ofalfentanil, together with its very fast eliminationfrom plasma by blood esterases rather than byliver metabolism, makes it an opioid with evenfaster recovery than alfentanil (Figure 3).

Simulations conducted to determine thetime required for a 50% reduction in effectsite concentration after an infusion designedto maintain a constant effect site concentrationssimilarly showed that the time required for a50% reduction in the effect site concentrationof remifentanil was with 3.65 min. This wasconsiderably less than that for sufentanil (33.9


Fig. 4. Plasma and effect site concentrations after intravenous infusions of different durations τ. With a one-hour infusion, recovery from fentanyl and alfentanil it not very different. With longer infusions, the recovery offentanyl becomes more and more slowly as compared to the recovery from alfentanil, as demonstrated with 6-hour infusions. Simulations were made similar as Shafer and Varvel.38

min), alfentanil (58.5 min), and fentanyl(262 min).39

Morphine has a considerably slower transferbetween plasma and effect site. It appears there-fore to be quite unsuitable for short-term inter-ventions. The time to build up the effects islonger and, more importantly, the effects per-sist for a longer time than those of alfentanilor fentanyl. On the other hand, when longereffects are desired, morphine will require lessattention to short-interval or continuousdosing, and for long-term therapy, the slowerequilibration-time is no problem.

A further application of PK/PD modeling isits use for target infusion regimens. By means ofcomputerized infusions,40 it is possible to main-tain target opioid concentrations at effect siterather than in plasma, which provides a highdegree of accuracy and controllability of theopioid therapy.

ConclusionsPK/PD modeling has advanced the under-

standing of the time course of the clinical ef-fects of opioids after various dosing regimens.

It provides a rational basis for the selection ofopioids in clinical circumstances. PK/PD mod-eling of opioids may also be employed for thedesign and the interpretation of experimentsaddressing clinical effects of opioids.

References1. Hiltunen AJ, Lafolie P, Martel J, et al. Subjective

and objective symptoms in relation to plasma metha-done concentration in methadone patients. Psycho-pharmacology (Berl) 1995;118:122–126.

2. Boulton DW, Arnaud P, DeVane CL. Pharmaco-kinetics and pharmacodynamics of methadone en-antiomers after a single oral dose of racemate. ClinPharmacol Ther 2001;70:48–57.

3. Angst MS, Buhrer M, Lotsch J. Insidious intoxica-tion after morphine treatment in renal failure: De-layed onset of morphine-6-glucuronide action.Anesthesiology 2000;92:1473–1476.

4. Levin HS, O’Donnell VM, Grossman RG. TheGalveston Orientation and Amnesia Test. A practicalscale to assess cognition after head injury. J NervMent Dis 1979;167:675–684.

5. Bourquin D, Lehmann T, Hammig R, et al. High-performance liquid chromatographic monitoring of


intravenously administered diacetylmorphine andmorphine and their metabolites in human plasma.J Chromatogr B 1997;694:233–238.

6. Gardmark M, Karlsson MO, Jonsson F, Hammar-lund-Udenaes M. Morphine-3-glucuronide has aminor effect on morphine antinociception. Phar-macodynamic modeling. J Pharm Sci 1998;87:813–820.

7. Holford NH, Sheiner LB. Understanding thedose-effect relationship: clinical application of phar-macokinetic-pharmacodynamic models. Clin Pha-rmacokinet 1981;6:429–453.

8. Holford NH, Sheiner LB. Pharmacokinetic andpharmacodynamic modeling in vivo. CRC Crit RevBioeng 1981;5:273–322.

9. Colburn WA. Pharmacokinetic/pharmacody-namic modeling: what it is! J Pharmacokinet Bio-pharm 1987;15:545–555.

10. Derendorf H, Hochhaus G. Handbook of phar-macokinetic/pharmacodynamic correlation. BocaRaton, London, Tokyo: CRC Press, 1995.

11. Hull CJ, Van Beem HB, McLeod K, et al. A phar-macodynamic model for pancuronium. Br J Anaesth1978;50:1113–1123.

12. Sheiner LB, Stanski DR, Vozeh S, et al. Simulta-neous modeling of pharmacokinetics and pharmaco-dynamics: application to d-tubocurarine. ClinPharmacol Ther 1979;25:358–371.

13. Suri A, Estes KS, Geisslinger G, Derendorf H.Pharmacokinetic-pharmacodynamic relationshipsfor analgesics. Int J Clin Pharmacol Ther1997;35:307–323.

14. Unadkat JD, Bartha F, Sheiner LB. Simultaneousmodeling of pharmacokinetics and pharmacody-namics with nonparametric kinetic and dynamicmodels. Clin Pharmacol Ther 1986;40:86–93.

15. Scott JC, Ponganis KV, Stanski DR. EEG quantita-tion of narcotic effect: the comparative pharmacody-namics of fentanyl and alfentanil. Anesthesiology1985;62:234–241.

16. Scott JC, Stanski DR. Decreased fentanyl andalfentanil dose requirements with age. A simultane-ous pharmacokinetic and pharmacodynamic evalua-tion. J Pharmacol Exp Ther 1987;240:159–166.

17. Scott JC, Cooke JE, Stanski DR. Electroencepha-lographic quantitation of opioid effect: comparativepharmacodynamics of fentanyl and sufentanil. Anes-thesiology 1991;74:34–42.

18. Lemmens HJ, Dyck JB, Shafer SL, Stanski DR.Pharmacokinetic-pharmacodynamic modeling indrug development: application to the investigationalopioid trefentanil. Clin Pharmacol Ther 1994;56:261–271.

19. Billard V, Gambus PL, Chamoun N, et al. A com-parison of spectral edge, delta power, and bispectralindex as EEG measures of alfentanil, propofol, andmidazolam drug effect. Clin Pharmacol Ther1997;61:45–58.

20. Bouillon T, Schmidt C, Garstka G, et al. Pharma-cokinetic-pharmacodynamic modeling of the respi-ratory depressant effect of alfentanil. Anesthesiology1999;91:144–155.

21. Egan TD, Minto CF, Hermann DJ, et al. Remifen-tanil versus alfentanil: comparative pharmacokinet-ics and pharmacodynamics in healthy adult malevolunteers [published erratum appears in Anesthesi-ology 1996 Sep;85(3):695]. Anesthesiology 1996;84:821–833.

22. Hermann DJ, Egan TD, Muir KT. Influence ofarteriovenous sampling on remifentanil pharmacoki-netics and pharmacodynamics. Clin Pharmacol Ther1999;65:511–518.

23. Minto CF, Schnider TW, Egan TD, et al. Influ-ence of age and gender on the pharmacokineticsand pharmacodynamics of remifentanil. I. Model de-velopment. Anesthesiology 1997;86:10–23.

24. Minto CF, Schnider TW, Shafer SL. Pharmacoki-netics and pharmacodynamics of remifentanil. II.Model application. Anesthesiology 1997;86:24–33.

25. Glass PS, Hardman D, Kamiyama Y, et al. Prelimi-nary pharmacokinetics and pharmacodynamics of anultra- short-acting opioid: remifentanil (GI87084B).Anesth Analg 1993;77:1031–1040.

26. Lotsch J, Angst MS. The mu-opioid agonist remi-fentanil attenuates hyperalgesia evoked by blunt andpunctuated stimuli with different potency: a pharma-cological evaluation of the freeze lesion in humans.Pain 2003;102(1–2):151–161.

27. Gambus PL, Gregg KM, Shafer SL. Validation ofthe alfentanil canonical univariate parameter as ameasure of opioid effect on the electroencephalo-gram. Anesthesiology 1995;83:747–756.

28. Inturrisi CE, Colburn WA, Kaiko RF, et al. Phar-macokinetics and pharmacodynamics of methadonein patients with chronic pain. Clin Pharmacol Ther1987;41:392–401.

29. Inturrisi CE, Portenoy RK, Max MB, et al. Phar-macokinetic-pharmacodynamic relationships ofmethadone infusions in patients with cancer pain.Clin Pharmacol Ther 1990;47:565–577.

30. Sarton E, Olofsen E, Romberg R, et al. Sex differ-ences in morphine analgesia: an experimental studyin healthy volunteers. Anesthesiology 2000;93:1245–1254. discussion 1246A.

31. Dershwitz M, Walsh JL, Morishige RJ, et al. Phar-macokinetics and pharmacodynamics of inhaledversus intravenous morphine in healthy volunteers.Anesthesiology 2000;93:619–628.

32. Read DJ, Simon H, Brandi G, Campbell EJ. Regu-lation of ventilation during rebreathing at imposedrespiratory frequencies. Respir Physiol 1966;2:88–98.

33. Lotsch J, Skarke C, Schmidt H, et al. The transferhalf-life of morphine-6-beta-glucuronide fromplasma to effect site assessed by pupil size measure-ment in healthy volunteers. Anesthesiology 2001;95:1329–1338.


34. Skarke C, Darimont J, Schmidt H, et al. Analgesiceffects of morphine and morphine-6-glucuronide ina transcutaneous electrical pain model in healthyvolunteers. Clin Pharmacol Ther 2003;73:107–121.35. Romberg R, Olofsen E, Sarton E, et al. Pharmaco-kinetic-pharmacodynamic modeling of morphine-6-glucuronide-induced analgesia in healthy volunteers:absence of sex differences. Anesthesiology 2004;100:120–133.36. Murthy BR, Pollack GM, Brouwer KL. Contribu-tion of morphine-6-glucuronide to antinociceptionfollowing intravenous administration of morphineto healthy volunteers. J Clin Pharmacol 2002;42:569–576.37. Kietzmann D, Bouillon T, Hamm C, et al. Phar-macodynamic modelling of the analgesic effects of

piritramide in postoperative patients. Acta Anaesthe-siol Scand 1997;41:888–894.

38. Shafer SL, Varvel JR. Pharmacokinetics, pharma-codynamics, and rational opioid selection. Anesthesi-ology 1991;74:53–63.

39. Westmoreland CL, Hoke JF, Sebel PS, et al. Phar-macokinetics of remifentanil (GI87084B) and itsmajor metabolite (GI90291) in patients undergoingelective inpatient surgery. Anesthesiology 1993;79:893–903.

40. Shafer SL, Gregg KM. Algorithms to rapidlyachieve and maintain stable drug concentrations atthe site of drug effect with a computer- controlledinfusion pump. J Pharmacokinet Biopharm 1992;20:147–169.

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Pharmacokinetic–Pharmacodynamic Modeling of Opioids