Journal of Pharmacokinetics and Biopharmaceutics, Vol. 24, No. 6, 1996
`
`Characterization of Four Basic Models of Indirect
`Pharmacodynamic Responses'
`
`Amarnath Sharma2,3 and William J. Jusko2,4
`
`Received May 17, 1995 -Final February 19, 1997
`
`Four basic models of indirect pharmacodynamic responses were characterized in terms of changing
`dose, 'max or Sma.,., and IC50 or SC50 to examine the effects of these fundamental drug properties
`on response profiles. Standard pharmacokinetic parameters were used for generating plasma con-
`centration, and response -time profiles using computer simulations. Comparisons to theoretical
`expectations were made. In all four models, the maximum response (Rmax) (inhibition or stimula-
`tion) and the time of its occurrence (TR .a) were dependent on the model, dose, 'max or Smax, and
`IC50 or SC50 values. An increase in dose or a decrease in IC50 or SC50 by the same factor produced,
`as theoretically expected, identical and superimposable pharmacodynamic response patterns in
`each of the models. Some parameters (TR,,ar, ABEC) were nearly proportional to log dose, while
`others (R,,,ax, CRmo,) were nonlinear. Assessment of expected response signature patterns as demon-
`strated in this report may be helpful in experimental designs and in assigning appropriate models
`to pharmacodynamie data.
`
`KEY WORDS: pharmacodynamics; indirect response models.
`
`INTRODUCTION
`In the field of pharmacodynamies, there are various approaches to
`correlate the time course of pharmacological effects with plasma drug con-
`centrations. However, the selection of the appropriate procedure for model-
`ing of pharmacokinetic -pharmacodynamic data should, if possible, be based
`on the mechanism by which a drug produces its response. Previously, four
`
`'Supported in part by Grant No. 24211 from the National Institute of General Medical Sciences,
`National Institutes of Health.
`2Department of Pharmaceutics, School of Pharmacy, State University of New York at Buffalo,
`Buffalo, New York 14260.
`3Department of Experimental Therapeutics, Roswell Park Cancer Institute, Buffalo, New York
`14263.
`4To whom correspondence should be addressed at 565 Hochstetter Hall, Department of Pharm-
`aceutics, School of Pharmacy, State University of New York at Buffalo, Buffalo, New York
`14260.
`
`611
`
`0090-466X/96/1200-0611509.50/0 © 1996 Plenum Publishing Corporation
`
`EXHIBIT
`
`3ag
`
`$
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`8a
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`612
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`Sharma and Jusko
`
`basic models were proposed for describing the pharmacodynamic responses
`of drugs produced by indirect mechanisms such as by inhibition or stimula-
`tion of the production or dissipation of factors controlling the measured
`response (1). The classic example of an indirect mechanism is the inhibition
`of prothrombin complex activity by the anticoagulant warfarin (2). The
`applicability of these models to a diverse array of drugs has recently been
`demonstrated (3).
`The pharmacokinetic /pharmacodynamic parameter(s) of a drug can
`be influenced by genetic, environmental, physiologic, or pathologic factors.
`Primary or secondary drugs given clinically can change pharmacokinetic
`and /or pharmacodynamic parameters or response profiles of the drug. For
`instance, gender affects both the kinetics (clearance) and dynamics (IC50) of
`methylprednisolone (4). The IC50 values for T- helper and T- suppressor cell
`trafficking effects increased significantly after multiple dosing of methyl -
`prednisolone in asthma patients (5). In the drug discovery process, it is
`commonplace to develop a congeneric series of compounds with differences
`in physicochemical, pharmacokinetic, and intrinsic potency properties, and
`thereby alter the pharmacodynamic profiles (6). At present, the availability
`of suitable experimental data is limited for full understanding of the effects
`of changes in intrinsic pharmacodynamic parameters on the overall response
`patterns. Such data include the drug concentrations and pharmacological
`effects simultaneously measured after administration of drugs at different
`rates or dose levels.
`In the present report, we have further examined response patterns (data
`signatures) expected from four basic indirect pharmacodynamic response
`models in terms of the dose, maximum inhibition or stimulation capacity
`(Imax or Smax), and drug concentration producing 50% inhibition or stimula-
`tion (IC50 or SC50). These are fundamental properties or variables of a drug
`and biological system. Full understanding of mechanism -based physiological
`models requires varied doses and /or administration rates to generate various
`pharmacodynamic response patterns. It was sought to determine whether it
`is possible to generalize the data signatures of the dynamics of drugs that
`have indirect response mechanisms and to provide simulations that comple-
`ment and extend theoretical relationships developed recently for these
`models (7,8).
`
`THEORETICAL
`The basic premise of this study is that the measured response (R) to a
`drug is produced by an indirect mechanism. The rate of change of the
`response over time with no drug present can be described as:
`dR
`dt
`
`_ kin -kout R
`
`(1)
`
`Page 2
`
`

`

`Four Basic Models of Indirect Pharmacodynamie Responses
`
`613
`
`where kin represents the apparent zero -order rate constant for production
`of the response, ka t defines the first -order rate constant for loss of the
`response, and R is assumed to be stationary with an initial value of Ro . The
`response variable, R, can be a directly measured entity or it may be an
`observed response which is directly and immediately proportional to the
`concentration of a mediator. It is assumed that kin and ko t fully account
`for production and loss of the response..
`For the four models shown in Fig. 1, the rate of change of the response
`over time in the presence of drug can be described as :
`dR=
`dt
`Models I (n = 1) and II (n = 2) represent processes that inhibit the fac-
`tors controlling drug response (Fig. 1) where inhibition processes operate
`according to :
`
`kin {1 +H1(t) } -kot { I + H2(t)} R
`
`(2)
`
`Hn ( t) =
`
`'max- Cp
`IC50 + C
`
`(3)
`
`dR = kin . {1 +H1(t)} - kout {1 +H2(t)} . R
`dt
`
`Model
`
`H1(t)
`
`112(t)
`
`Condition
`
`0
`
`0 < Ima: < i
`
`Ima: C
`p
`IC + Cp
`
`-
`
`0
`
`<
`
`(Imaz CD)
`IC50 + Cp
`
`p
`
`S. C p
`SC50 +cp
`
`o
`
`Sma: Cp
`(SC50+ci;
`
`4 <Sma:
`
`I
`
`II
`
`III
`
`IV
`
`Key:
`
`IIC50 Inhibition
`
`SC50 Stimulation
`
`Fig. 1. Four basic indirect response models represent
`processes that inhibit or stimulate the factors controlling
`drug response.
`
`Page 3
`
`

`

`614
`
`Sharma and Jusko
`
`The value of Lax is always less than or equal to unity, i.e., 0 < Imax <1. The
`plasma concentration of drug (Cr) can be defined as a function of time and
`IC50 is the drug concentration which produces 50% of the maximum inhibi-
`tion achieved at the effect site.
`A more specific form of Model I is:
`
`while Model II is :
`
`dR
`dt
`
`{ 1 + H, (t) } - knut - R
`
`dR
`dt kin-kom-
`
`{1 +H2(t)} R
`
`(4)
`
`(5)
`
`Models III (n= 1) and IV (n = 2) represent processes that stimulate the
`factors controlling drug response (Fig. 1) where stimulation processes oper-
`ate according to :
`
`H,(t)
`
`Smax Cp
`SC50 + Cr
`
`(6)
`
`The SC50 represents drug concentration producing 50% of the maximum
`stimulation achieved at the effect side. The value of Smax can be any number
`greater than zero.
`The more specific form of Model III is:
`dR =k;,, {1 +H,(t)} -k.t R
`
`(7)
`
`and Model IV is:
`
`dt
`
`dR
`dt
`
`-knut { 1 +H2 (t)} R
`
`(8)
`
`A summary parameter used to characterize the overall effect of drug is
`the area between the baseline and the effect curve (ABEC) which is defined
`as
`
`AREC R0 tr _ AUEC tri
`
`(9)
`where R0 is the baseline value and AUEC is the area under or over the
`response vs. time curve over the time interval of 0 to tr . The value of tr is
`assumed -.4 oo .
`Some of the characteristics of the four basic indirect response models
`that have explicit solutions include the following (7,8) :
`
`Page 4
`
`

`

`615
`
`(10)
`
`(11)
`
`(12)
`
`(13)
`
`(14)
`
`(15)
`
`(16)
`
`(17)
`
`(18)
`
`ABEC = R0-2211 x In 1 +
`el
`
`C50
`
`(ABEC)
`
`Model I
`
`( 19)
`
`ABEC(D
`
`-> co)
`
`= Rolm"
`kei 1- 'max
`
`1
`
`In 1 +
`
`D/V
`IC5o
`
`if I. 0 1
`
`Model II
`
`(20)
`
`=
`
`RO
`
`21n2
`
`k°" `
`2(kel)
`
`1
`
`+
`
`D/V
`IC5o
`D/V
`
`ABEC = Ro Smax ln 1 +
`kel
`
`SC5o
`
`if Lax =1
`
`Model II
`
`Model III
`
`(21)
`
`Model IV
`
`(22)
`
`ABEC(D
`
`-- co)
`
`1
`
`1
`
`ln 1 + D/V
`= R0 Smax
`SCso
`kei
`Smax
`Equations (20) and (22) are solutions which can be obtained only at
`high doses of drug.
`Initial Slopes (SI) :
`The limiting values of the initial slope (S1) of the four models can be
`identified by setting Eqs. (4), (5), (7), and (8) equal to zero when Cr-* co. The
`limiting S1 value will also depend on the maximum inhibition or stimulation
`
`Four Basic Models of Indirect Pharmacodynamic Responses
`
`Maximum Response (Rmax) as Dose -> co or IC50 or SC50 --*0:
`Model I
`Rmax R0(1 -Imax)
`Model II
`Rmax Ro/(1 - Imax)
`Model II
`Model III
`Model IV
`Rmax Ro /(1 + Smax)
`Drug Concentrations occurring at Rmax (CRmax)
`
`Rmax
`
`oo
`
`Rmax T R0( + Smax)
`
`if Imax < 1
`
`if 'max =1
`
`I I
`
`I
`
`Model
`
`Model
`
`Model
`
`III
`
`CR
`
`CR
`
`CR
`
`IC50- (Ro - Rmax)
`max Rmax - (1 - Imax)R0
`IC50 (Rmax - R0)
`max R0 - (1 - Imax)Rmax
`- SC50 (Rmax - Ro)
`max Ro(1 + Smax) Rmax
`
`CRmax -
`
`Model
`
`IV
`
`SC50 (Ro - Rmax)
`Rmax(1 + Smax) - R0
`Area Between the Baseline and Effect Curve
`D/V
`
`Page 5
`
`

`

`616
`
`Sharma and Juskn
`
`capacity (Imax or Smax) of the drug. Since kin = kot Ro at steady- state, solu-
`tions are possible using either kin or k0.1. Thus :
`(Model I)
`(Model II)
`(Model III)
`(Model IV)
`
`Si _ --kin Imax = -kout . R0 Imax
`
`SI = kin Imax = kout Ro Imax
`SI = kin Smax = kont Ro . Smax
`SI = -kin Smax = -kaut Ro Smax
`
`(23)
`(24)
`(25)
`(26)
`
`METHODS
`
`Pharmacokinetics
`
`Methylprednisolone was selected as the model drug for simulation since
`its pharmacokinetics can be described using a linear, one -compartment
`model, and it has been found to produce several indirect pharmacodynamie
`responses. A volume of distribution (V) of 90 L and elimination rate con-
`stant (kei) of 0.3 hr -' were used to simulate monotonic plasma concentra-
`tion -time profiles at various doses (D) using
`
`(D ' 1000
`V
`where the factor 1000 converts the plasma concentrations to ng /ml for mg
`dose units.
`
`Cp
`
`e -ke' '
`
`(27)
`
`Pharmacodynamies
`
`The Imax or Sm, IC50 or SC50, and dose were varied individually to
`define their effects on the pharmacodynamic response. A wide range for the
`Imax (0.2 to 1.0), Smax (0.2 to 1.5), IC50 (10 to 500 ng /ml), SCso (10 to
`500 ng /ml), and dose (10 to 10,000 mg) were used for simulations. The
`differential equations for Models I to IV were used in the PCNONLIN
`program (SCI Software Inc., Apex, NC) to simulate the response versus
`time profiles. The initial condition (R0 = 30) and values of kin = 9 unit /hr
`and kot = 0.3 hr -' were chosen to produce reasonable response patterns.
`The ABEC was generated over 0 to t,. where t,. is the time taken by the
`response to return to baseline (Ro) after drug administration. The Initial
`Slope (S1) of the response versus time curve was calculated over 0 to 1 hr.
`
`RESULTS
`Model I
`
`Figure 2 shows the effects of changes in either Imax, ICso, or dose of a
`drug which produces its pharmacodynamie response by inhibition of the
`
`Page 6
`
`

`

`Four Basic Models of Indirect Pharmacodynamic Responses
`
`617
`
`MODEL I
`
`_
`
`--
`
`-. zoo
`,o
`
`.-
`
`I
`
`J
`
`I
`
`I
`
`DOSE:
`
`35
`
`30 -
`
`25
`
`20
`
`15
`
`10
`
`5 0
`
`30
`
`25
`
`20
`
`15
`
`10
`
`5 0
`
`30
`
`25
`
`20 -
`
`15
`
`10 -
`5-
`
`0
`
`0
`
`5
`
`10
`
`15
`
`z
`
`z z
`
`1000
`
`100
`
`10
`
`1
`
`1000
`
`100
`
`10
`
`i
`
`1000
`
`M+I
`
`.1 100
`
`s IO
`
`1
`
``
`
`30
`
`20
`
`25
`
`TIME
`Fig. 2. Model 1 simulations of the pharmacodynamic response variables (solid
`lines) with respect to time after a single iv bolus dose. Simulated pharmaco-
`kinetic profiles at the corresponding doses are shown by dashed lines. The
`indicated values of Ima, (0.2 to 1.0) (A); ICso (10 to 500) (8); and doses (10
`to 1000) (C); were used to study their effects on pharmacodynamic response
`and pharmacokinetic profiles. The heavy curves show the identical standard
`condition for all simulations (Imax =1.0, ICso =100 ng /ml, Dose =100). The
`dots in Panel B indicate the response and time when CP - ICso
`
`Page 7
`
`

`

`618
`
`Sharma and Jusko
`
`Table I. Effect of Imax or Sm:,x on Properties of the Response Profiles
`Model II
`Model III
`Model IV
`
`Model 1
`
`Imax or Smax
`
`0.2
`0.4
`0.6
`0.8
`LO
`1.5
`
`0.2
`0.4
`0.6
`0.8
`1.0
`1.5
`
`0.2
`0.4
`0.6
`0.8
`1.0
`1.5
`
`0.2
`0.4
`0.6
`0.8
`1.0
`1.5
`
`0.2
`0.4
`0.6
`0.8
`1.0
`1.5
`
`26.2 (-13)
`22.4 (-25)
`18.6 (-38)
`14.8 (-51)
`11.0 (-63)
`NA'
`
`-1.4
`-2.8
`-4.2
`-5.6
`-7.0
`NA'
`
`50
`99
`149
`198
`248
`NA'
`
`6.2
`6.2
`6.2
`6.2
`6.2
`NA`'
`
`173
`173
`173
`173
`173
`NA'
`
`Rmx ('%Rm.,x)"
`34.4 (15)
`39.7 (33)
`45.9 (53)
`54.5 (82)
`65.1 (117)
`NA`'
`Initial Slopeh (Si)
`1.4
`3.0
`4.6
`6.3
`8.0
`NA`
`ABEC
`
`55
`121
`204
`309
`444
`NA`'
`
`TRma.
`
`6.2
`6.5
`7.0
`7.2
`7.5
`NA`'
`CR..
`
`173
`158
`136
`128
`117
`NA'
`
`%Rmax = [(Rmax- Ro) /Ro] 100
`h Initial slope= (AR /Ot)e,= i
`` NA: Not Applicable.
`
`33.8 (13)
`37.6 (25)
`41.4 (38)
`45.2 (51)
`49.0 (63)
`58.5 (95)
`
`26.6 (-11)
`23.8 (-21)
`21.4 (-29)
`19.4 (-35)
`17.7 (-41)
`14.5 (-52)
`
`1.4
`2.8
`4.2
`5.6
`7.0
`10.6
`
`50
`99
`149
`198
`248
`372
`
`6.2
`6.2
`6.2
`6.2
`6.2
`6.2
`
`173
`173
`173
`173
`173
`173
`
`-1.4
`-2.7
`-3.9
`-5.1
`-6.2
`-8.8
`
`45
`84
`116
`144
`169
`219
`
`6.2
`6.0
`5.7
`5.5
`5.2
`5.0
`
`173
`184
`201
`213
`234
`247
`
`factors controlling kin (Fig. 1: Model I). The numerical values of the proper-
`ties such as Rmax , Sr , ABEC, TRmax , and CRmax resulting from the simulations
`are provided in Tables I -III for the three parameters varied. The increase
`in Imax resulted in a proportional increase in the maximum inhibitory
`response (Rmax) up to the expected limit of O. The initial slope (S1) of
`response vs. time curves behaved similarly (Fig. 2A). A five -fold increase in
`Imax (from 0.2 to 1.0) produced an increase in the maximum percent Rmax
`and S1 by nearly the same magnitude (Rmax increased from 12.7 to 63.3 %,
`
`Page 8
`
`

`

`Four Basic Models of Indirect Pharmacodynamie Responses
`
`619
`
`Table II. Effect of IC50 or SC50 on Properties of the Response Profiles"
`Model I
`Model III
`
`IC50 or SC50
`
`Model II
`
`0
`10
`50
`100
`250
`500
`
`0
`10
`50
`100
`250
`500
`
`o
`10
`50
`100
`250
`500
`
`o
`10
`50
`100
`250
`500
`
`0
`10
`50
`100
`250
`500
`
`{Ro (1 -Imax)}
`4.0 (-87)
`8A (-73)
`11.0 (-63)
`15.5 (-48)
`18.4 (-39)
`
`-kin Imax
`-7.7
`-7.4
`-7.0
`-6.2
`-5.1
`
`09
`464
`312
`248
`169
`116
`
`09
`9.5
`7.0
`6.2
`5.2
`4.2
`
`0
`64
`136
`173
`234
`315
`
`Rmax (A Rmax)
`{Ro/(1 -Imax)}
`108.4 (261)
`74.8 (149)
`65.1 (] 17)
`52.0 (74)
`44.9 (50)
`Initial Slope (S1)
`kin
`Imax
`8.9
`8.5
`8.0
`6.9
`5.6
`AREC
`
`co
`1259
`638
`444
`254
`155
`
`TRn,ax
`
`cc
`12.5
`9.0
`7.5
`6.0
`5.0
`
`CRmux
`
`0
`26
`75
`117
`184
`248
`
`{Ro (1 +Smax)}
`56.0 (87)
`51.9 (73)
`49.0 (63)
`44.5 (48)
`41.6 (39)
`
`Smax
`
`kin
`7.7
`7.4
`7.0
`6.2
`5.1
`
`cc
`464
`312
`248
`169
`116
`
`09
`9.5
`7.0
`6.2
`5.2
`4.2
`
`0
`64
`136
`173
`234
`315
`
`Model IV
`
`{R(,/(1 +smax)}
`15.6 (-48)
`16.8 (-44)
`17.7 (-41)
`19.5 (-35)
`21.6 (-28)
`
`-kin Smax
`-6.7
`-6.5
`-6.2
`-5.5
`-4.6
`
`00
`279
`203
`169
`125
`92
`
`co
`7.5
`6.0
`5.2
`4.5
`4.2
`
`0
`117
`184
`234
`288
`315
`
`" Symbols are defined in Glossary.
`
`and Si increased from 1.4 to 7.0) (Table I). The area between the baseline
`and the effect curve (ABEC) was calculated to characterize the overall effect
`of the drug. The ABEC increased proportionally with the increase in Imax
`(Table I) as expected [Eq. (19)] . The time of occurrence of the maximum
`response TRm was independent of Imax and, therefore, plasma drug concen-
`trations at the time of maximal response Cam. remained constant with the
`change in Imax . This is a general expectation for these models (note Remark
`4: Ref. 7).
`The effect of ICS0 on the dynamic response profile is shown in Fig. 2B.
`Lower IC50 values yield more pronounced effects. The percent maximum
`
`Page 9
`
`

`

`620
`
`Sharma and Jusko
`
`Table III. Effect of Dose on Properties of the Response Profiles"
`Modelli
`Model IV
`Model III
`Model I
`Dose
`
`10
`100
`1000
`10000
`co
`
`10
`100
`1000
`10000
`00
`
`10
`100
`1000
`10000
`00
`
`10
`100
`1000
`10000
`co
`
`10
`100
`1000
`10000
`co
`
`22.5 (-25)
`11.0 (-63)
`4.0 (-87)
`1.3 (-96)
`{Rq(1-Imux)}
`
`-3.8
`-7.0
`-7.7
`-7.8
`-kin imax
`
`74
`248
`464
`659
`co
`
`4.0
`6.2
`9.5
`13.0
`x
`
`33
`173
`643
`2249
`00
`
`Rrnux (VoRmax)
`24.0 (-20)
`39.5 (32)
`37.2 (24)
`17.7 (-41)
`49.0 (63)
`65.1 (117)
`15.6 (-48)
`108.4 (261)
`56.0 (87)
`15.1 (-50)
`180.6 (502)
`58.7 (96)
`{Ro(1+Smax)f {Ro/(1+Smax)j
`{Ro/(1-Imax)}
`Initial Slope (S1)
`4.1
`8.0
`8.9
`9.0
`km - imax
`ABEC
`
`-3.5
`-6.2
`-6,7
`-6.8
`-kin Smax
`
`3.8
`7.0
`7.7
`7.8
`kin- Smax
`
`89
`444
`1259
`2493
`oo
`
`TRmax
`4.2
`7.5
`12.5
`18.0
`o0
`
`CRmax
`
`32
`117
`261
`502
`oo
`
`74
`248
`464
`659
`co
`
`4.0
`6.2
`9.5
`13.0
`0o
`
`33
`173
`643
`2249
`c0
`
`63
`169
`279
`369
`00
`
`4.0
`5.2
`7.0
`10.0
`00
`
`33
`234
`1360
`5532
`00
`
`"Symbols are defined in Glossary.
`
`inhibitory response ( %Rmax), Si , and ABEC increased with the decrease in
`IC50. The TRmax shifted to later times and, therefore, CRmax decreased with
`the decrease in IC50 (Table II). The limiting value of Rmax as IC50 -40 is
`R0 (1- Imax) [Table II and Eq. (10)].
`Figure 2C shows the effect of dose on the dynamic response and phar-
`macokinetic profiles. The curves show a typical declining response with a
`delayed nadir, later return to baseline, and greater effects with larger doses.
`This is expected to occur for any monotonic drug concentration profile (see
`Remark 1, Ref. 7). The %Rmax, S1 , ABEC, TRmax and.CRmax values increased
`with the increase in dose (Table III), but Rmax and S1 have limiting values
`of 0 [ =R0 (1- Imax)] and -9 (= - kin 'max) at larger doses (Dose -4 co). The
`ABEC and TRmax continue increasing in proportion to log dose. This was
`demonstrated previously for ABEC (9). Note the corresponding relationship
`
`Page 10
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`

`

`Four Basic Models of Indirect Pharmacodynamie Responses
`
`621
`
`for Rmax in Eq. (10) (7). However, both CRmax and TRmax increase to infinity
`as Dose -+ co [Eqs. (22) and (32), Ref. 7). This is also seen for CRmax in Eq.
`(15) when Rmax--0 and Imax =1.
`It is interesting to note that an increase in dose or decrease in IC50
`by the same factor (i.e., constant Dose /IC50 ratio) results in identical and
`superimposable pharmacodynamie response patterns for this and subsequent
`models (see Appendix). For instance, the dose of 1000 at IC50 =100 produced
`the same response as that produced at the dose of 100 at IC50 =10 (Fig.
`2B, C). This property allows use of the ratio (IC50 V) /D as a nondimen-
`sional parameter in seeking generalized solutions for these types of models
`(7)-
`
`Model II
`Figure 3 characterizes Model II with respect to changes in either Imax,
`IC50 , or dose. The drug described by Model II produces its pharmaco-
`dynamic response by inhibition of k0 (Fig. 1: Model II). The effect of
`Imax on the pharmacodynamic response variable is shown in Fig. 3A. The
`maximum stimulatory response (Rmax), initial slope, and ABEC values
`increased with the increase in Imax (Table I). The TRmax shifted to later times
`and, therefore, CRmax decreased with the increase in 'max (Table I). The Rmax
`will have a specific limiting value when 0 < Imax <1. However, if Imax =1,
`Rmax -9 co with large doses or low IC50 values [see Table III, Eqs. (11) and
`(12)]. This is a unique characteristic for Model II (7).
`The effect of IC50 on the dynamic response is shown in Fig. 3B. Lower
`IC50 values produce larger effect profiles. The Rmax, Si , and ABEC values
`increased with the decrease in IC50 (Table II). The limiting value of Rmax as
`IC50-+0 is R0 /(1 -Imax) [Table II, Eqs. (11) and (12)]. The TRmax shifted to
`later times and, therefore, CRmax decreased with the decrease in IC50 (Table
`II). The value of CRmax is proportional to IC50 [Eq. (16)].
`The effect of dose on the dynamic response and pharmacokinetic pro-
`files are shown in Fig. 3C. The curves show increasing observed effects with
`a delayed maximum and slow return to baseline; these effects increase in
`relation to dose. These type of patterns are expected to occur for any mono-
`tonic drug concentration profile (note Remark 1, Ref. 7). The Rmax , Si ,
`ABEC, TRmax and CRmax values increased with the increase in dose (Table
`III). The Si has a limiting value of 9 ( =k;,, 'max) at larger doses (Dose -,00),
`while ABEC, TRmax and CRmax values continue increasing in proportion to
`log dose. The value of Rmax increases in proportion to dose only when
`Imax =1, otherwise large doses produce a limiting value of R0 /C1- Imax)
`[Table III and Eq. (64), Ref. 71. Both TRmax and CRmax increase to infinity as
`Dose-k co [Eqs. (22) and (32) of Ref. 7].
`
`Page 11
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`

`

`622
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`Sharma and Jusko
`
`A
`
`MODEL II
`
`ImIX:
`
`120
`
`100
`
`80
`
`60
`
`40
`
`100
`
`80
`
`60
`
`40
`
`100
`
`80
`
`60
`
`40
`
`1000
`
`100
`
`10
`
`1
`
`3
`
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`1000 [
`
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`&ug
`
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`
`1
`
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`1000 f
`
`100
`
`10
`
`1
`
`0,
`
`5
`
`10
`
`15
`
`20
`
`25
`
`30
`
`TIME
`
`Fig. 3. Model H simulations of the pharmacodynamie response variables (solid
`lines) with respect to time after a single iv bolus dose. Simulated pharmacokinetic
`profiles at the corresponding doses are shown by dashed lines. The indicated
`values of Imax (0.2 to 1.0) (A); IC50 (10 to 500) (B); and doses (10 to 1000) (C);
`were used to study their effects on pharmacodynamie response and pharmaco-
`kinetic profiles. The heavy curves show the identical standard condition for all
`simulations (Imax =1.0, IC50 = 100 ng /mI, Dose = 100). The dots in Panel B indi-
`cate the response and time when C = IC50.
`
`Page 12
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`

`

`Four Basic Models of Indirect Pharmacodynamic Responses
`
`623
`
`An increase in dose or decrease in IC50 by the same factor results
`in superimposable pharmacodynamic response patterns (Fig. 3B, C, and
`Appendix).
`
`Model III
`
`Figure 4 shows the effects of changes in either Smax, SC50, or dose for
`a drug which produces its dynamic response by stimulation of k;,, (Fig. 1:
`Model III). The effect of Smax on the dynamic response variable is shown in
`Fig. 4A. The maximum stimulatory response (Rmax) and initial slope
`increased proportionally with an increase in Sm.. The 7.5 -fold increase in
`Smax (from 0.2 to 1.5) resulted in an increase in the maximum percent Rmax
`and initial slope by nearly the same magnitude (Rmax increased from 12.7
`to 95.3% and initial slope increased from 1.41 to 10.55) (Table I). The former
`occurs because Rmax is proportional to R0(1 + Smax) at large doses [Eq. (13)].
`The ABEC also increased proportionally with an increase in Smax (Table I)
`as expected according to Eq. (21). TRmax remained constant with the increase
`in Smax and, therefore, so did the Climax . This is expected for any monotonic
`drug disposition profile (see Remark 4, Ref. 7).
`The effect of SC50 on the response variable is shown in Fig. 4B. Lower
`SC50 values produce pharmacologic effects with greater magnitudes and
`duration. The Rmax, SI , and ABEC values increased with the decrease in
`SC50 (Table II). The TRmax shifted to later times and, therefore, Cimax
`decreased with the decrease in SC50 (Table II).
`Figure 4C shows the effect of dose on the response variable and pharma-
`cokinetic profiles. The profiles show greater observed effects with a delayed
`maximum and slow return to baseline as dose is increased. These types of
`curves are expected to occur for any monotonic drug concentration pattern
`(note Remark 1, Ref. 7). The Rmax, Si, ABEC, TRmax and CRmax values
`increased with the increase in dose (Table III). The Rmax and SI have limiting
`values of 60 [ =R0 (1 + Smax)] and 9 ( =kin Smax) at larger doses (Dose -- co)
`[Eq. (13); also see Eq. (65) of Ref. 7]. ABEC increases nearly proportional
`to log dose as expected from Eq. (21). However, both CRmax and TRmax increase
`to infinity as Dose-- + oo [Eq. (22) and (32) of Ref. 7] .
`An increase in SC50 or a decrease in dose by the same factor produced
`identical responses (Fig. 4B, C, and Appendix).
`
`Model IV
`
`The effects of changes in either Smax, SC50, or dose for a drug whose
`pharmacodynamic response can be described by Model IV are shown in
`Fig. 5. This model represents a drug that stimulates lc... The effect of Smax
`on the response variable is shown in Fig. 5A. The maximum inhibitory
`
`Page 13
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`

`

`624
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`Sharma and Jusko
`
`60
`
`55
`
`50
`
`45
`
`40
`
`35
`
`30
`
`55
`
`50
`
`45
`
`40
`
`35
`
`30
`
`55
`
`50
`
`45
`
`40
`
`35
`
`30
`
`1000
`
`100
`
`10
`
`1
`
`1000
`
`100
`
`10
`
`1
`
`1000
`
`100
`
`10
`
`i
`
`20
`
`25
`
`30
`
`5
`
`15
`
`10
`TIME
`Fig. 4. Model III simulations of the pharmacodynamic response variables (solid
`lines) with respect to time after a single iv bolus dose. Simulated pharmaco-
`kinetic profiles at the corresponding doses are shown by dashed lines. The
`indicated values of Smax (0.2 to 1.0) (A); SC51, (10 to 500) (B); and doses (10
`to 1000) (C); were used to study their effects on pharmacodynamic response
`and pharmacokinetic profiles. The heavy curves show the identical standard
`condition for all simulations (Smax =1.0, SCs,, =100 ng /ml, Dose =100). The
`dots in Panel B indicate the response and time when Cp = SCsu
`
`Page 14
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`

`

`Four Basic Models of Indirect Pharmacodynamic Responses
`
`625
`
`MODEL IV
`
`zo
`
`Z zo
`
`1000
`
`100
`
`10
`
`1
`
`1000
`
`100
`
`10
`
`1
`
`1000
`
`100
`
`10
`
`1
`
`30 -
`
`25 -
`
`20 -
`
`15-
`
`W ;
`
`To
`
`30L
`ó 25
`
`O 20 -
`et
`
`U'
`
`15
`
`C
`
`30 -061
`
`25 -
`
`20 -
`
`15 -
`
`0
`
`5
`
`10
`
`15
`
`20
`
`25
`
`30
`
`TIME
`Fig. 5. Model IV simulations of the pharmacodynamic response variables (solid
`lines) with respect to time after a single iv bolus dose. Simulated pharmaco-
`kinetic profiles at the corresponding doses are shown by dashed lines. The
`indicated values of Sp,,, (0.2 to 1.0) (A); SC50 (10 to 500) (B); and doses (10
`to 1000); (C) were used to study their effects on pharmacodynamic response
`and pharmacokinetic profiles. The heavy curves show the identical standard
`condition for all simulations (S,= 1.0, SC50=100 ng /ml, Dose =100). The
`dots in Panel B indicate the response and time when Cp = SCSI,.
`
`Page 15
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`

`626
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`Sharma and Jusko
`
`response (Rmax), initial slope and ABEC values increased with the increase
`in Smax (Table I). The limiting value of Rmax is Ro /(l + Smax) ; see Eq. (14).
`The TRmax shifted to earlier times and, therefore, CRmax increased with the
`increase in Smax (Table I). Proportionality between CRmax and Smax is expected
`at higher doses [Eq. (18)]. ABEC is proportional to Smax /(1 +Smax); see
`Eq. (22).
`Figure 5B shows the effect of SC50 on the dynamic responses. Lower
`SC50 values produce response patterns with greater nadirs and duration. The
`percent maximum inhibitory response ( %Rmax), Si , and ABEC increased
`with the decrease in SC50 (Table II). TRmax shifted to later times and, there-
`fore, CRmax decreased with the decrease in SC50 (Table II). These behaviors
`of ABEC and CRmax are in accordance with Eqs. (18) and (22).
`The effect of an increase in dose on the dynamic response is shown in
`Fig. 5C. The curves show increasing observed effects with a delayed maxi-
`mum and slow return to baseline ; such effects increase with dose. The Rmax
`Sy, ABEC, TRmax, and CRmax values increased with the increase in dose (Table
`III). The Rmax and S, have limiting values of 15 [ =R0 /(1 + Smax)] and -9
`( = -kin Smax) at larger doses (Dose -4 oo) [Eq. (26), Table III, and Eq. (27)
`of Ref. 7]. ABEC and TRmax continue increasing in proportion to log dose.
`However, both CRmax and TRmax increase to infinity as Dose-+ oo (Eqs. (22)
`and (32) of Ref. 7). Again, these patterns are predicted by Eqs. (18) and
`(22).
`
`An increase in dose or decrease in SC50 by the same factor results in the
`identical pharmacodynamic response patterns (Fig. 513, C, and Appendix).
`
`Effects of Dose
`Since dose is the most readily manipulated variable in a pharmaco-
`dynamic study, it is of interest to assess how selected parameters relate to
`a wide range of doses of drug for each of the models. Figure 6 shows that
`TRmax is nearly linear with log dose over a wide range of doses for all models
`(see Eq. (62) of Ref. 7). Similar behavior occurs with ABEC. This is the
`theoretical expectation for all models except Model II when Imax =1, ABEC
`will be proportional to 1n2 D (8). Thus, curvature is seen in Fig. 6 (Model II).
`As shown in Fig. 7, Rmax shows a lower limit with log dose for Models
`I and IV (also see Eqs. (63) and (66) of Ref. 7), and an upper limit with
`log dose for Model III (also see Eq. (65) of Ref. 7). Rmax continues increasing
`nonlinearly with log dose for Model II when Imax =1. However, if Imax < 1,
`'max) with dose for Model II (see Eq.
`Rmax has an upper limit of R0/(1
`(64) of Ref. 7). These relationships are- in accordance with Eqs. (10) -(14).
`It can also be seen in Fig. 7 that CRmax shows hyperbolic behavior with log
`dose for all four models. These patterns may be inferred from Eqs.
`(15) -(18).
`
`Page 16
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`

`Four Basic Models of Indirect Pharmacodynamie Responses
`
`627
`
`30
`
`20
`
`10
`
`0
`
`400 -
`
`Model IV
`
`200
`
`10
`
`100
`
`1000
`
`10,000 100,000
`
`o
`
`t
`10
`
`{
`
`100
`
`I
`
`1000
`
`I
`
`10,000
`
`DOSE
`Fig. 6. The time to reach maximum effect (TRmax) and the area between the baseline and effect
`curve (ABEC) vs. the log of drug dose for Models I to IV. Values of Imax =1.0, Smax =1.0,
`IC50= 100 ng /ml, and SC50 =100 ng /ml were used.
`
`Page 17
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`628
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`Sharma and Jusko
`
`Model I
`
`L
`
`I
`
`Model II
`
`Model III
`
`Model IV
`
`100
`
`1000
`
`10,000
`
`2400
`
`1600
`
`800
`
`-
`
`0
`
`I
`
`800
`
`400
`
`0
`
`E 3000
`1./
`
`2000
`
`1000
`
`-
`
`0
`
`6000
`
`-
`
`3000
`
`- 1
`
`0
`
`0
`
`1
`
`f
`
`1000
`
`I
`10,000
`
`Model I
`
`Model II
`
`Model III
`
`Model IV
`
`- -
`
`30
`
`20
`
`10 -
`
`0
`
`200 -
`
`lao
`
`o
`
`ea
`
`Eg so
`
`45
`
`30
`
`30
`
`20
`
`10
`
`I
`
`10
`
`I
`
`100
`
`DOSE
`Fig. 7. The maximum effect (Rmax) and the plasma concentration of drug at the time of
`maximum effect (CRO,.) vs. the log of drug dose for Models I to IV. Values of Ima, =1.0, Smax=
`1.0, IC5o= 100 ng /ml, and SC50= 100 ng /ml were used.
`
`Page 18
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`

`

`Four Basic Models of Indirect Pharmacodynamic Responses
`
`629
`
`Model Identification
`
`For pharmacodynamic modeling with a mechanistic basis, it is essential
`to assign appropriate models to pharmacodynamic data based on the funda-
`mental actions of the drug. It is also helpful to anticipate the nature of
`expected model behavior. Pharmacodynamic response patterns occur down-
`ward for Model I and IV, and upward for Model II and III. For instance,
`if the drug causes a decrease in the pharmacodynamic response from its
`baseline value, either Model I or IV may be able to characterize the general
`pattern of response for one dose level. Similarly, if response increases from
`its baseline value in presence of drug, either Model II or III may appear to
`be applicable.
`While an understanding of the mechanism of action of the drug is
`the best approach to construction of the model, the following two methods
`can be used fo- complete experimental identification of an appropriate
`indirect response model : (i) a single iv dose study at more than one dose
`level; and (ii) a steady -state iv infusion study at more than one administra-
`tion rate.
`In a single iv dose study, it is important that one of the dose levels be
`sufficiently high to produce either full inhibition or stimulation of the system.
`Pharmacodynamic parameters such as Imax or Smax and ICso or SCso can be
`obtained by fitting the experimental data to two of the four models. These
`parameters, in turn, can be used to estimate the maximum responses (Rmax)
`at large doses (Dose -400) according to Eqs. (10) -(14). Thus in the absence
`of knowledge about the mechanism of action of the drug, one can determine
`which model is more suitable by comparing experimental Rmax values
`obtained at larger doses with estimated Rmax values for models which
`describe responses produced in the same direction.
`In an infusion study, it is critical that the length of infusion be
`sufficiently long not only to produce steady -state pharmacokinetics but also
`steady -state conditions in the pharmacodynamic system. In other words, the
`time of infusion should be based on the km, value. For instance, if kout is
`small, a longer infusion time is required, and vice- versa.
`Figure 8 shows the effects of change in the infusion rate on the time to
`reach the maximum response ( TRmax) for the four models. In Models I and
`III, the TRmax remained constant with the change in the infusion rate (Table
`IV; Fig. 8) because the drug affects kit, for these two models (inhibits for
`Model I and stimulates for Model III), and kin has no influence on the
`time required by the pharmacodynamic system to reach steady -state under
`continuous drug infusion. However, in Models II and IV, the TRmax changed
`with the infusion rate (Table IV; Fig. 8) because the drug affects kout for
`these models (inhibits for Model II and stimulates for Model IV). The knot
`
`Page 19
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`630
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`Sharma and Jusko
`
`Model I
`
`0
`
`10
`
`20
`
`30
`
`40
`
`50
`
`60
`
`TIME
`Fig. 8. Simulations of the pharmacodynamic response variables (solid lines) with respect to
`time during and after a 24 hr iv infusion for the four models. Pharmacokinetic profiles at the
`corresponding doses are shown by dashed lines. Values of Ima =1.0, S..= 1.0, IC50 =
`100 ng /ml, SC50 =100 ng /ml, and the indicated values of infusion rate (1 to 100 mg /hr) were
`used to produce pharmacodynamic response and pharmacokinetic profiles.
`
`value influences the time required by the pharmacodynamic system to reach
`steady -state under continuous drug infusion. For instance, in Model II where
`a drug produces its pharmacodynamic response by inhibition of kout, the
`TRme shifted to late