See discussions, stats, and author profiles for this publication at: http://www.researchgate.net/publication/13877791
`
`Basic concepts of
`pharmacokinetic/pharmacodynamic (PK/PD)
`modelling
`
`ARTICLE in INTERNATIONAL JOURNAL OF CLINICAL PHARMACOLOGY AND THERAPEUTICS · NOVEMBER 1997
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`Bernd Meibohm
`The University of Tennessee Health Science …
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`Hartmut Derendorf
`University of Florida
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`/11rernat/011af.lour'nal of Clinical Pharmacology and Therapeutics, Vol. 35, No. /0- 1997 (401-413)
`
`Basic concepts of pharmacokinetic/
`pharmacodynamic (PK/PD) modelling
`
`B. MEIBOHM and H. DERENDORF
`
`Department of Pharmaceutics, College of Pharmacy.University of Florida. Goinesvillr'. Fl., USA
`
`Abstract. Pharmacokinetic (PK) and pharmacodynamic (PD) information from the scientific basis of
`modern pharmacotherapy. Pharmacokinetics describes the drug concentration-time courses in body
`fluids resulting from administration of a certain drug dose, pharmacodynamics the observed effect
`resulting from a certain drug concentration. The rationale for PK/PD-modelling is to link pharmaco(cid:173)
`kinetics and pharmacodynamics in order to establish and evaluate dose-concentration-response
`relationships and subsequently describe and predict the effect-time courses resulting from a drug dose.
`Under phannacokinctic steady-state conditions, concentration-effect relationships can be described
`by several relatively simple pharmacodynamic models, which comprise the fixed effect model, the
`linear model, the log-linear model, the Emax-model and the sigmoid Emax-model. Under non steady(cid:173)
`state conditions, more complex integrated PK/PD-models are necessary to link and account for a
`possible temporal dissociation between the plasma concentration and the observed effect. Four basic
`attributes may be used to characterize PK/PD-models: First, the link between measured concentration
`and the pharmacologic response mechanism that mediates the observed effect, direct vs. indirect link;
`second, the respons.e mechanism that mediates the observed effect, direct vs. indirect response; third,
`the information used to establish the link between measured concentration and observed effect, hard
`vs. soft link; and fourth, the time dependency of the involved pharmacodynamic parameters, time(cid:173)
`variam vs. time-invariant. In general, PK/PD-modelling based on the underlying physiological process
`should be preferred whenever possible. The expanded use of PK/PD-modelling is assumed to be highly
`beneficial for drug development as well as applied phannacotherapy and will most likely improve the
`current state of applied therapeutics.
`
`Key words: pharmacology pharmacokinetics - phannacodynamics modelling
`
`Rationale for PK/PD-modelling
`
`The rational use of drugs and the design of effective
`dosage regimens is facilitated by the appreciation of the
`relatiunship5 between the administered dose of a drug, the
`resulting drug concentrations in body fluids accessible for
`measurements, and the intensity of pharmacologic effects
`caused by these concentrations l Gihaldi et al. 1971 J. These
`relationships and thus the dose of a drug required to achieve
`a certain effect is determined by its pharmacokinetic and
`pharmacodynamic properties.
`Pharmacokinetics describes the time course of the
`concentration of a drug in a body fluid, preferably plasma
`or blood, that results from the administration of a certain
`dose. In simple words, phannacokinetics is · wlwr rile hody
`does to the drug'. Pharmacodynamics describes the inten-
`
`Correspondence lo: Prof. Dr. H. Derendorf, 100494. College of
`Pharmacy. University of Florida. Gainesville. FL 32610, USA
`
`sity of a drug effect in relation to its concentration in the
`body tluid. It can be simplified to 'what the drug does to
`the body' [Holford and Sheiner 1982].
`Pharmacokinetic/pharmacodynamic (PK/PD)-mod(cid:173)
`elling combines both approaches and tries to estabiish
`models in order to describe the effect-time course directly
`resulting from the administration of a certain dose (Fig(cid:173)
`.ure I). Thus, a so-called integrated PK/PD-model consists
`of a pharmacokinetic model component that describes the
`time course of a drug in some body fluid and a pharma(cid:173)
`codynamic model component that relates the concentration
`in this fluid to the drug effect.
`
`Models in general are simplified descriptions of true
`biological process and can be used for data reduction and
`interpolation, but their major value is derived from their
`ability to extrapolate relationships beyond the ~xisting
`data. In the case of PK/PD-models, that means to extrapo(cid:173)
`late dose-effect relationships for example from single to
`multiple dosing situations or from intravenous to oral or
`other administration routes.
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`. 402
`
`MPihnhm and DPrPndnrf
`
`Pharmacokinetics
`Dose¢ Cone.vs.time
`
`Time
`
`Pharmacodynamics
`Conc.¢Effect
`
`PK/PD
`Dose¢ Effect vs.time
`
`TI mo
`
`Fig. I
`Interrelationship
`between phannacokinetics,
`pharmacodynamic' and
`PK/PD-modelling.
`
`Evolution of PK/PD-modelling
`
`Early in the evolution of pharmacokinetics, relation(cid:173)
`ships between drug disposition and pharmacologic effect
`have been described [Gibaldi and Levy 1972, Gibaldi et al.
`1971, Levy 1964, Wagner 1968]. It could be shown that
`the intensity and time course of action of numerous directly
`and reversibly acting drugs are related to and largely deter(cid:173)
`mined by the time course of drug concentrations in the
`body. When the concentration-time courses and the con(cid:173)
`centration-effect relationships ot these drugs were known,
`it was often possible to predict the temporal pattern of their
`pharmacological effects, including maximum intensity and
`duration of action.
`However. for many other drugs, it was believed that
`there is no relationship between the drug concentration in
`plasma and the time course of action, predominantly hased
`on the observation that the pharmacologic effect of many
`drugs lags behind their concentration in plasma. Pharma(cid:173)
`cologic effects often increase in their intensity despite
`decreasing drug concentrations and may persist well be(cid:173)
`yond the time, when drug concentrations in plasma are no
`longer determinable. As a consequence plasma concentra(cid:173)
`tion vs. effect plots show a more or less pronounced hys(cid:173)
`teresis.
`The apparent dissociation between drug concentra(cid:173)
`tion and effect was first overcome by Sheiner and co-work(cid:173)
`ers [Holford and Sheiner 1982, Sheineret al. 1979] based
`on the concepts of Segre f 1968 J who proposed to use a
`hypothetical effect compartment to account for the lag
`between concentration and response, which is described in
`detail in one of the following sections of this paper. This
`approach led to a collapse of the hysteresis loop for drugs
`with a temporal delay between effect and plasma concen(cid:173)
`tration by plotting the effect intensity versus the concentra(cid:173)
`tion in the effect compartment. All of a sudden, the delay
`
`between the time courses of drug concentration and effect
`made sense [Levy 1994].
`Since then, PK/PD-modelling has exponentially
`evolved as a research area with increasing importance and
`dedication in academia, industry and regulatory authori(cid:173)
`ties. The present paper aims to give a short overview over
`the basic concepts in pharmacokinetic/pharmacodynamic
`modelling and provide some structural information for
`classification of the applied approaches.
`
`Pharmacokinetic models
`
`The plasma concentration-time course of a drug is
`determined by the pharmacokinetic processes of distribu(cid:173)
`tion, metabolism and excretion as well as absorption in case
`of nonsystemic administration. The currently used pharma(cid:173)
`cokinetic models can basically be distinguished into com(cid:173)
`partmental, physiological and statistical models.
`Although nonparametric and physiologically based
`pharmacokinetic models have been used as a basis for
`PK/PD-approaches [Holford et al. 1994, Unadkat et al.
`1986, Veng-Pedersen and Gillespie 1988], compartmental
`models are the most frequently preferred, probably due to
`the fact that they provide a continuous concentration-time
`profile in a body fluid that can be related to a continuous
`effect-time profile and that the popular effect compartment
`concept can easily be implemented. Thus, the present paper
`will focus on PK/PD-modelling based on compartmental
`pharmacokinetic models.
`
`Pharmacodynamic models for steady-state situations
`
`Pharmacodynamic analysis involves quantifying
`drug concentration/effect relationships. Ideally, concentra(cid:173)
`tions should be measured at the effect site, the site-of-action
`or biophase, where the interaction with the respective bio-
`
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`Basic concepts ofpharmacokineric!pharmacodynamic (PK/PD) modelling
`
`403
`
`logical receptor system takes place, but this is in most cases
`not possible. Thus, concentrations in easily accessible body
`fluids like plasma or blood are frequently used to establish
`these relationships under the assumption, that the pharma(cid:173)
`cologically active, unbound concentration at the effect site
`is directly related to the one in the respective body fluid
`and that they are, under phannacokinetic steady-state con(cid:173)
`ditions, in equilibrium. Since the same proportionality
`holds for different steady-state plasma levels, steady-state
`plasma concentrations may serve as the only determinant
`of the observed effect.
`A drug effect can be defined as any drug-induced
`change in a physiological parameter when compared to the
`rc:spective prt::dost:: or bast::line value. The ba:seli11e value is
`the value of the same physiological parameter in the ab(cid:173)
`sence of drug dosing. Baseline values do not necessarily
`have to be constant but can change, e.g. as a function of
`time of day or of food intake. Furthermore, the term effect
`has to be clearly separated from the term efficacy. Efficacy
`is the sum of all therapeutically beneficial drug effects and
`is the most relevant target parameter in PK/ PD-modelling.
`However, in many PK/PD-studies, efficacy is difficult to
`quantify and thus, easily accessible surrogate markers as
`effect parameters are used instead. In these cases, it is
`necessary to present evidence that the pharrnacodynamic
`effect parameter used correlates with the desired efficacy
`to provide valid results.
`For steady-state conditions, the most commonly·used
`phannacodynamic models are the
`
`-
`
`fixed effect model,
`linear model,
`log-linear model,
`-
`- Emax-model, and
`sigmoid Emax-rnodel.
`-
`
`Fixed effect model
`
`A fixed effect model, also known as quanta! effect
`model, is a statistical approach based on a logistic regres(cid:173)
`sion analysis. It relates a certain drug concentration with
`the statistical likelihood of a predefined. fixed effect to be
`present or absent.
`The simplest case of a fixed effect model is a threshold
`model, where the effect Efoed occurs after reaching a
`certain threshold concentration Cthrcshold, as for example
`described for the ototoxicity occurring during gentamicin
`therapy with trough levels exceeding 4 ~Lg/ml for longer
`than l 0 days of therapy ! Mawer et al. l 97 4 j:
`
`E
`
`Et1xed
`
`if
`
`C 2: Critreshold
`
`(Eq. I)
`
`certain concentration will be a function of the threshold
`concentration distribution in the population. For example,
`at a digoxin plasma concentration of 2.0 ng/ml there is a
`50% probability to observe digox.in toxicity, whereas at a
`concentration of 4.1 ng/ml the probability is 90% [Beiler
`et al. 197 l, Holford and Sheiner 1982]. This approach may
`be useful in the clinical setting as an approximation of
`dose-response relationships but has major limitations for
`the prediction of complete effect-time profiles.
`
`Linear model
`
`The linear model assumes a direct proportionality
`between drug concentration and drug effect, as shown by
`Weaver et al. [ 1992] for the correlation of salivary flow rate
`and plasma concentration after pilocarpine infusions:
`
`E=m xC+ Eo
`where Eo is the baseline effect in the absence of drug
`and ma proportionality factor, that characterizes the slope
`of a plot of effect E versus concentration C. Although the
`linear model is the one that intuitively is the most popular,
`it rarely applies.
`
`(Eq. 2)
`
`log-linear model
`
`A much more common situation than the linear model
`is the log-linear model with
`
`E""' m x log C + b
`
`(Eq. 3)
`
`where m and bare slope and intercept in a plot of effect
`E versus the logarithm of the concentration C. Although b
`should have the unit of the effect, it is an empiric constant
`that has no real physiologic meaning, especially not that of.
`a baseline value. The log-linear model is applicable in
`many situations and can be considered a special case of the
`Emax-model, regarding the range between 20% to 80% of
`Emax, where effect E and logarithm of the concentration C
`follow a linear relationship. Nagashima et al. [ 1969] for
`example used it to relate the synthesis rate of prothrombin
`complex activity to the plasma concentration of warfarin.
`
`Einax-model
`
`In the maximum effect Ema"-model, concentration C
`and effect E are related as
`
`E- Emax xC
`Eso-1 C
`
`(Eq.4)
`
`where E is the measured effect and C is the measured
`concentration.
`Since the threshold concentration will vary among
`patients, the probability of the effect to be present at a
`
`where Emax is the maximum effect possible and E5o
`is the concentration that causes 50% of Emax. Emax refers
`to the intrinsic activity of a drug, E5o to its potency. Lalonde
`et al.! 1987) gave an ex.ample for applying the Emax-model
`
`II
`
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`
`404
`
`100
`
`80
`
`60
`
`40
`
`20
`
`u Q)
`~
`E
`:J
`E
`·;;:
`!1l
`E
`0
`~ 0
`
`slope: Ema/4
`
`100
`
`80
`
`60
`
`40
`
`20
`
`b
`
`intercept: In E50 - 2
`
`!
`//
`
`0-4-...... ~~.,.;._~~~~~~~~
`-4 -3 -2 -1 0 1 2 3 4 5 6 7 6 9 10
`
`Concentration (ng/ml)
`
`In C (ng/ml)
`
`Meibohm and Derendorf
`
`Fig. 2 Concentration-effect
`relationship resulting from a
`phurmacodynamic Emax-model,
`where the effect is either plotted
`against the concentration (a) or
`the logarithm of the concentra(cid:173)
`tion (b), the Inst resulting in the
`classical sigmoidal shape. [Em ax
`maximum effect, Eso
`con(cid:173)
`centration at 50% ofEmaxl.
`
`by describing the relationship between propranolol plasma
`concentrations and the resulting decrease in heart rate.
`The equation of the Emax-model (Eq. 4) is based on
`the receptor theory relationship [Ariens and Simonis 1964]
`that can be derived for the equilibrium interaction of a drug
`(D) with its site of action (R), e.g. a receptor, enzyme or
`ion channel, producing the effect E:
`
`_
`[D]+[R]<=:>[DR]Hc:ffect =>
`
`[DR]-
`
`[nto,]x[D]
`[ ]
`Kd+ D
`
`(E 5)
`q.
`
`where Kct is the equilibrium constant and Riot the total
`number of interaction sites. Under the assumption, that the
`observed effect E is directly proportional to the number of
`occupied interaction sites DR, Eq. 4 and Eq. 5 are equiva(cid:173)
`lent indicating that maximum effect would be observed if
`all interaction sites are occupied. Kct is the concentration at
`which half of the interaction sites are occupied and, hence,
`equivalent to Eso.
`The Emax-model describes the concentration-effect
`relationship over a wide range of concentrations from zero
`effect in the absence of a drug to the maximum effect at
`concentrations much higher than Eso (C >> Eso). In the
`presence of a baseline effect Eo, this term can simply be
`added to Eq. 4 as shown in Eq. 6:
`xc
`£ 50 +C
`The clear non-proportional concentration-effect rela(cid:173)
`tionship of the Emax-model is presented in Figure 2 as linear
`and semilogarithmic plot. Whereas small increases in con(cid:173)
`centration may result in significant increases of the effect
`for low concentrations, this is much less pronounced for
`higher concentrations where only small changes in effect
`
`(Eq. 6)
`
`E=E0
`
`will result from changes in concentration. From the
`semilogarithmic presentation, it is apparent that in the
`range from 20% to 80% of the maximum effect, the rela(cid:173)
`tionship between effect and the logarithm of the concentra(cid:173)
`tion is linear. This is consistent with the log-linear model
`(Eq. 3). The slope of the linear phase can be calculated as
`EmaJ4, the respective x-intercept as In Eso - 2, and the
`y-intercept b as Emax x (2
`In Eso)/4, [Hochhaus and
`lJerendort 1995 J. At concentrations below 20% and above
`80% of the maximum effect the Emax-model clearly devi(cid:173)
`ates from the log-linear model. For concentrations much
`smaller than Eso (C << Eso), it reduces to a linear model
`with a slope m of Emax/Eso. Hence, both, the log-linear as
`well as the linear model may be interpreted as special case
`of the Emax-model.
`The Emax-model presented in Eq. 4 assumes an in(cid:173)
`crease of the effect with increasing concentrations, i. e. a
`stimulated effect. Opposite, inhibitory effects can be de(cid:173)
`scribed by Eq. 7:
`
`xC
`E10 +C
`where Eo is the baseline effect. If the maximum effect
`Emax is complete suppression of the baseline effect Eo, Eq.
`7 simplifies to Eq. 8:
`
`(Eq. 7)
`
`E=E0x(1--C-)
`
`(Eq. 8)
`
`E5o+C
`In accordance with the receptor theory the Emax(cid:173)
`model also allows to describe more complex concentra(cid:173)
`tion-effect relationships, e.g. competitive or noncompeti(cid:173)
`tive agonist and antagonist interactions at the response
`system, if appropriately modified Emax-equations are ap(cid:173)
`plied. This was shown by Braat et al. 11992] for the com-
`
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`Basic concepts of pharmacokineticlpharmacodynamic (PK/PD) modelling
`
`405
`
`100
`
`80
`
`60
`
`40
`
`20
`
`Q)
`
`u
`~
`E
`:::>
`E ·x
`ro
`E
`0
`~ 0
`
`0
`
`0
`
`Fig. 3 Concentration-effect re(cid:173)
`lationship resulting from a sig(cid:173)
`moid Emax-model, presented a~
`a function of various shape fac(cid:173)
`tors in a normal (a) and
`semilogarithmic plot (b).
`
`shape factor
`
`40
`
`a
`
`shape factor
`
`b
`
`2
`3
`4
`Concentrationti;0
`
`5
`
`101
`
`1 ()l
`
`1 ()1
`
`Concentrationll::so
`
`petitive effect of dexamethasone and hydrocortisone on
`lymphocytes.
`
`Sigmoid Emax-model
`
`The sigmoid Emax-model is an expansion of the Emax(cid:173)
`model. Effect and concentration are related as
`
`(Eq.9)
`
`xC"
`E
`£- ma,\
`E;o +C"
`Theoretically, this relationship can be derived to de(cid:173)
`scribe the interaction between n drug molecules and one
`interaction site similar to Eq. 5. However, in most cases n
`has no molecular basis and is merely used as an operational
`shape factor that allows a better data fit. This also explains
`why in many studies noninteger values for n are reported
`which could not be possible based on the mentioned deri(cid:173)
`vation. Figure 3 shows the effect of different values of non
`the concentration-effect curves. The larger n, the steeper
`the linear phase of the log-concentration-effect curve. It
`could be shown that the slope is n x Emax/4, hence directly
`proportional to n. For this reason, n is also referred to as
`the slope factor. The respective x-intercept is then In Eso
`2/n fHochhaus and Derendorf 1995]. The Emax-model can
`be considered as a special case of the sigmoid Emax-model
`with a slope factor n = l.
`The sigmoid Emax-model is the most versatile model
`of those presented. Meffin et al. f 1977} fur exam pit: used
`it to characterize the relationship between plasma concen(cid:173)
`tration and antiarrhythmic activity of tocainide. The re(cid:173)
`ported slope factor of n 2.3 - 20.6 for individuals indi(cid:173)
`cates a very steep concentration-effect curve.
`In summary, several relatively simple pharmacody(cid:173)
`namic models are available to describe the relationship
`
`between drug concentration and effect under steady-state
`conditions. The preferred model in any given situation
`depends on many factors, including the drug used, the
`response to be measured, the effect seen after administra(cid:173)
`tion of drug and of placebo, the degree of linearity in the
`effect-concentration-curve, and the potential for achieving
`the maximum possible response.
`
`Pharmacodynamic models for non steady-state
`situations
`
`Under non steady-state conditions, drug concentra(cid:173)
`tions in plasma (or respective other sampled body fluids
`like blood. or serum, which will be summarized in the
`expression plasma from hen: on) undergo time-dependent
`changes due to the involved pharmacokinetic process and
`an equilibrium between plasma and effect site concentra(cid:173)
`tion does not necessarily exist. Furthermore, another time
`consuming process might be involved in mediating the
`observed effect after interaction of the drug with its physi(cid:173)
`ological response structure. As a result, time courses of
`plasma concentration and effect might dissociate. Thus, to
`characterize the time course of drug action under non
`steady-state conditions, pharmacokinetics and pharma(cid:173)
`codynamics have to be adequately linked to predict firstly
`the dose/concentration-relationship, and secondly the con(cid:173)
`centration/effect-relationship. This link is provided by the
`multiple concepts of integrated PK/PD-models.
`
`Four basic attributes of PK/PD-models
`
`At present, usually applied integrated PK/PD-models
`for reversible drug effects can be classified by four major
`attributes (Figure 4), characterizing the link between the
`
`-
`
`••
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`
`406
`
`Meibohm and Derendorf
`
`Dose-concentration-effect relationship
`to be modeled
`
`Dhec.• Link
`
`""·
`
`Indirect Link
`
`Olrv4:,;l fi'npvnH'
`vs.
`lndlroct Response
`
`.•.
`
`Hiird Link
`
`Soft Link
`
`Time~lnv111d•11t
`vs .
`Time~variant
`
`Selected PK/PD-approach
`
`•
`
`•
`
`.,
`lso
`Q. «> w
`.5 30
`
`10
`
`I J ~
`
`0 ......... --.;:::::;;;.,.~....,....--.~--~...-....... ~..--.
`0
`2
`4
`IS
`I
`10 12 14
`tlS 18 ~
`Cp (µ.g/m~
`
`Fig. 4 Four basic attributes of PK/PD-models to be considered during
`the selection of an appropriate modelling approach.
`
`Fig. 5 Counter-clockwise hysteresis loop for the relationship between
`plasma concentration (Cp) of $-ibuprofen and its analgesic effect quanti(cid:173)
`fied as decrease in evoked potential (EP) amplitudes [from Suri et al.
`1997).
`
`plasma concentration and the response mechanism respon(cid:173)
`sible for the observed effect, the response mechanism by
`which the effect is mediated, the information used to estab(cid:173)
`lish the link between plasma concentration and observed
`effect, and the time-dependency of the involved parameters
`of the pharmacodynamic model component. The resulting
`alternatives are
`Direct link vs. indirect link models
`- Direct response vs. indirect response models
`- Soft link vs. hard link models
`Time-invariant vs. time-variant models
`
`Direct link versus indirect link
`
`Plasma and effect site concentrations can be linked
`either directly or indirectly, dependent on their temporal
`arrangement. Due to the constant change in drug plasma
`levels during non steady-state situations, the equilibrium
`between the concentrations in plasma and at the site of
`action 1s permanently readjusted. Since this re-equilibra(cid:173)
`tion is dependent on the involved distribution process and
`more or less time consuming, the concentration at the site
`of action may lag behind that in plasma. This is reflected
`in a more or less pronounced counter-clockwise hysteresis
`between plasma concentration and effect, as shown in
`Figure 5 for the analgesic effect of S-ibuprofen quantified
`by the amplitude decrease of evoked potentials [Suri et al.
`1997]. The extent of hysteresis is then dependent on the
`degree of delay between the concentrations in plasma and
`at the effect site.
`
`and at the effect site occurs rapidly and the concentrations
`are directly proportional to each other at any time despite
`the pharmacokinetic non steady-state conditions. Thus, the
`time of maximum measured concentration is also the time
`of maximum effect, if the pharmacodynamic response is
`directly mediated (see following chapter). In such situ(cid:173)
`ations effect and plasma concentration lack any hysteresis
`and may be linked by simple substitution of the pharma(cid:173)
`cokinetic model for concentration C in one of the pharma(cid:173)
`codynamic models for steady-state situations. Schaefer et
`al. [ 1997] for example used plasma concentrations directly
`in an Emax-model to describe the effect-time course, quan(cid:173)
`tified as blood pressure reduction, after different doses and
`dosage forms of nisoldipine (Figure 6).
`If the pharmacokinetic properties of a drug are de(cid:173)
`scribed by a multicompartmental phannacokinetic model,
`the effect might not only be related to the central compart(cid:173)
`ment, but also to the concentration predicted for one of the
`peripheral compartments. This approach was used hy
`Hochhaus et al. [ l 992b] to characterize the pulmonary and
`cardiac effects of fenoterol (Figure 7), where the plasma
`concentration-time profile was described with a three com(cid:173)
`partment mammillary model and the concentration pre(cid:173)
`dicted for the shallow tissue compartment was used as an
`input function for a sigmoid Emax-model. Derendorf et al.
`[ 1991] used a similar approach with a two companmem
`model to relate fasting blood sugar levels to methylpred(cid:173)
`nisolone concentrations in the peripheral compartment. For
`most drugs, however, a specific compartment does not
`reflect the drug concentration at the site of action.
`
`Direct link models
`
`lndin:cl link rrwdels
`
`A direct link has to be applied in a situation where
`equilibration between the drug concentrations in plasma
`
`An indirect link is needed if the time courses of
`concentration and effect are dissociated and the observed
`
`Page 7 of 14
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`MYLAN PHARM. v YEDA
`IPR2015-00644
`
`

`
`Basic l'Oncepts of pharmacokineticlpharmacodynamic (PK/PD) modelling
`
`407
`
`a
`
`b
`
`c
`
`Dose
`Plasma concentration
`First-order rate constant
`
`a
`
`D
`
`D
`Dose
`C
`Plasma concentration
`c01 , cp2 Peripheral compartment
`concentration
`k10, k~ 1 , k,,, k.,, k 13
`First-order rate constants
`ko
`Zero-order rate constant
`
`I Effect I
`
`b 1600
`
`::::;
`e
`
`' Cl s-
`ti z
`0
`u
`
`1200
`
`800
`
`400
`
`tw-+1----1}
`t+h
`
`U
`
`H
`
`JO
`
`l4
`
`0
`
`0
`
`40
`
`80
`
`120
`TIME (mlnl
`
`160
`
`200
`
`240
`
`c
`
`•
`
`Time !hi
`
`•
`
`~r.::============;~~~~~~~~~-,
`
`_...... pndlacd ~· t_ mw.:1 I
`I -0- metl' \-al1,1e ~f ,wdy OIQ.4 \
`
`l 5
`
`'i
`;:: 2S
`~
`.! JO
`=
`= J5
`i 10
`
`•
`...
`-.-:r-;-AA-------"--------~
`....
`-~~.!
`~AA--..A..--x------~----­
`----0
`--'2..------
`,1-1{ 0
`10
`0 I
`........... ________________ /
`o/'
`I
`
`• . .,,,(cid:173)
`
`.,...
`./
`
`0
`
`10•& !'( N11t-<W'I'
`u .....
`
`5-s N .._...
`
`10.1 N MhtiM
`
`lO•t N tOhltiN
`
`•o+-~-+-~-+~~'--~f--~+-~~~-+~--1
`1.00
`D.00
`2.00
`3.00
`Time (hr)
`
`Fig. 6 Direct link model for the effect of oral nisoldipine (Nl on diastolic
`blood pressure (DRP): (a) PK/PD-model scheme: (b) plasma concentra(cid:173)
`tion vs. time and (c) effect vs. time profiles for different doses and dosing
`forms (study means) ffrom Schaefer et al.1997].
`
`counter-clockwise hysteresis in a plot of effect versus
`plasma concentration is most likely related to a distribu(cid:173)
`tional delay. The important conceptual advance in indi(cid:173)
`rectly linking pharmacokinetic and pharmacodynamic
`models was the realization that the time course of the effect
`itself can be used to define the rate of drug movement to
`the effect site as expressed in the effect compartment model
`of Holford and Sheiner [ 1981 a, 1982]
`
`Fig. 7 Direct link model for the effect of fenoterol on heart rate and
`airway resistance'. (a) PK/PO-model scheme; (b) plasma concentration vs.
`time profile after i. v. bolus (25 µg; solid line) or constant i. v. infusion
`incl. loading dose (200 µg/180 min+ 25 µg loading dose; dashed line;
`• ti) and airway resistance(•. 0)
`mean± SD); (c) effect on heart rate (
`vs. time pmfilf' ~ftl"r boil" (0. i\) resp. constant infusion(•.
`) (study
`means. lines are modeled) [from Hochhaus ct al. 1992a].
`
`This concept uses a hypothetical effect compartment
`that is modeled as an additional compartment of a pharma(cid:173)
`cokinetic compartment model and represents the active
`drug concentration at the effect site. It is linked to the
`kinetic model by a first-order process but receives negli(cid:173)
`gible mass of drug. Therefore, the first-order rate constant
`
`-
`
`...
`
`Page 8 of 14
`
`YEDA EXHIBIT NO. 2038
`MYLAN PHARM. v YEDA
`IPR2015-00644
`
`

`
`. 408
`
`Meibohm and Derendorf
`
`for the transfer from the respective pharmacokinetic com(cid:173)
`partment into the effect compartment is negligible. Thus,
`the time-dependent aspects of the equilibrium between
`plasma concentration and effect are only characterized by
`the first-order rate constant ke0, which describes the disap(cid:173)
`pearance of the drug from the effect compartment and is
`not directed to any of the pharmacokinetic compartments.
`The time course of the effect site concentration (Ce)
`for a one compartment body model with bolus input is
`
`a
`
`0
`
`D Dose
`C Plasma concentration
`c. Effect compartment
`concentration
`k10• k1 • k10, koa
`First-order rate constants
`
`ka0 ~::~-.~~) ~ I Effect I
`
`Effect compartment
`
`(Eq. 10)
`
`where Dis the dose, Vd the volume of distribution in
`the central compartment, and k the first order elimination
`rate constant of the drug. The respective equations for other
`compartment models and input functions are described
`elsewhere in detail [Colburn 1981, Holford and Sheiner
`198lb].
`Suri et al. [ 1997] used the effect compartment concept
`to model the analgesic effect of S-ibuprofen, Sheiner et al.
`[ 1979]
`to describe
`the muscle relaxant effect of
`d-tubocurarine. In the first example, the effect compart(cid:173)
`ment was attached to a pharmacokinetic one-compartment
`body model (Figure 8), in the last to the central compart(cid:173)
`ment of a three-compartment body model (Figure 9). For
`other drugs, it might theoretically also be possible to link
`the effect compartment to peripheral pharmacokinetic
`compartments as
`theoretically evaluated by Colburn
`[ 1981 ]. However, this seems only adequate, if the drug
`concentration is also measured in this pharmacokinetic
`compartment, for example by invasive techniques or by
`imaging technology. Otherwise, basing a hypothetical ef(cid:173)
`fect compartment concentration on a predicted peripheral
`compaitment concentration seems vague and obscures the
`elaborated PK/PD-relationship.
`The effect compartment concept achieved wide popu(cid:173)
`larity in PK/PD-modelling of various drugs. However, the
`link between measured plasma concentration and observed
`effect is based on an unk.11ow11 1m::cha11is111, a "black. bux".
`Therefore, the validity of the relationship has to be estab(cid:173)
`lished thoroughly by evaluating concentrations and effects
`under a variety of doses and input functions, before predic(cid:173)
`tions and extrapolations to other therapeutic situations may
`be performed.
`
`1~-1-. ....... ..,...,.. .................. .,.......,...,...,..,..,~ ................ .....-. ....... ..,...,...-j
`0
`
`Tme(hra)
`
`..
`
`c
`
`¢
`
`i•
`~
`i5., ID
`
`... Q c10
`i;.ll "' ·~.
`" £ ID
`
`40
`
`•
`s
`Time (hrs)
`
`Fig. 8
`Indirect link model for the analgesic effect of 400 mg oral
`ibuprofen quantified by subjective pain intensity rating: (a) PK/PD-model
`scheme; (b) plasma concentration vs. time courses of the enantiomcrs
`R-ibuprofen (OJ and S-ibuprofen (•), (c) effect vs. time profile modeled
`with the concentration of the active S-enantiomer (mean± SD. lines arc
`modeled) [from Suri et al. 1997].
`
`Direct response versus indirect response
`
`Direct response models
`
`Two general pathways can be distinguished for medi(cid:173)
`ating the effect of a drug, a direct response pathway, where
`the interaction of the drug with a response structure at its
`effect site directly results in the observed effect, and an
`indirect response pathway, where a physiological factor
`that governs the observed effect, is modulated.
`
`Direct response models