noncompartmentalnoncompartmental vs. … · 1 noncompartmentalnoncompartmental vs. compartmental...

1

NoncompartmentalNoncompartmental vs. Compartmental vs. Compartmental Approaches to Approaches to

Pharmacokinetic Data AnalysisPharmacokinetic Data Analysis

Paolo Vicini, Ph.D.Paolo Vicini, Ph.D.Pfizer Global Research and Pfizer Global Research and

DevelopmentDevelopment

David M. Foster., Ph.D.David M. Foster., Ph.D.University of WashingtonUniversity of Washington

pp

2

Questions To Be AskedQuestions To Be Asked

PharmacokineticsPharmacokineticsWhat the body does to the drugWhat the body does to the drug

PharmacodynamicsPharmacodynamicsWh t th d d t th b dWh t th d d t th b dWhat the drug does to the body What the drug does to the body

Disease progressionDisease progressionMeasurable therapeutic effectMeasurable therapeutic effect

VariabilityVariabilityVariabilityVariabilitySources of error and biological variationSources of error and biological variation

3

Pharmacokinetics / Pharmacokinetics / PharmacodynamicsPharmacodynamics

7 3.5

1

2

3

4

5

6

Dru

g C

once

ntra

tion

0.5

1

1.5

2

2.5

3D

rug

Effe

ct

00 5 10 15 20 25

Time

PharmacokineticsPharmacokinetics“What the body does to “What the body does to

PharmacodynamicsPharmacodynamics“What the drug does to “What the drug does to

0

0.5

0 2 4 6 8 10

Drug Concentration

yythe drug”the drug”Fairly well knownFairly well knownUseful to get to the PDUseful to get to the PD

ggthe body”the body”Largely unknownLargely unknownHas clinical relevanceHas clinical relevance

4

PK/PD/Disease ProcessesPK/PD/Disease Processes6

7

on

3

3.5

0

1

2

3

4

5

Dru

g C

once

ntra

tio

0

0.5

1

1.5

2

2.5D

rug

Effe

ctPK PD

0 5 10 15 20 25

Time0 2 4 6 8 10

Drug Concentration

92

96

100

104

ase

Stat

us Disease

84

88

92

0 10 20 30 40

Dis

ea

Time

5

12.00

Hierarchical VariabilityHierarchical VariabilityNo VariationNo Variation

8.00

10.00

4.00

6.00

0.00

2.00

0.00 5.00 10.00 15.00 20.00 25.00

6

12.00

Hierarchical VariabilityHierarchical VariabilityResidual Unknown VariationResidual Unknown Variation

8.00

10.00

within-individual(what the model does not

explain – e g measurement

4.00

6.00explain – e.g. measurement

error)

0.00

2.00

0.00 5.00 10.00 15.00 20.00 25.00

7

12.00

Hierarchical VariabilityHierarchical VariabilityBetweenBetween--Subject VariationSubject Variation

8.00

10.00

between-individual(physiological variability)

4.00

6.00

0.00

2.00

0.00 5.00 10.00 15.00 20.00 25.00

8

12.00

Hierarchical VariabilityHierarchical VariabilitySimultaneously Present BetweenSimultaneously Present Between--Subject and Subject and

Residual Unknown VariationResidual Unknown Variation

8.00

10.00

Biological sources?

4.00

6.00

0.00

2.00

0.00 5.00 10.00 15.00 20.00 25.00

9

Pharmacokinetic ParametersPharmacokinetic Parameters

Definition of pharmacokinetic parametersDefinition of pharmacokinetic parametersDescriptive or observational Descriptive or observational Quantitative (requiring a formula and a means Quantitative (requiring a formula and a means to estimate using the formula)to estimate using the formula)to estimate using the formula)to estimate using the formula)

Formulas for the pharmacokinetic Formulas for the pharmacokinetic parametersparametersMethods to estimate the parameters fromMethods to estimate the parameters fromMethods to estimate the parameters from Methods to estimate the parameters from the formulas using measured datathe formulas using measured data

10

Models For EstimationModels For EstimationNoncompartmentalNoncompartmental

CompartmentalCompartmental

11

Goals Of This LectureGoals Of This Lecture

Description of the parameters of interestDescription of the parameters of interestUnderlying assumptions of Underlying assumptions of noncompartmental and compartmental noncompartmental and compartmental modelsmodelsmodelsmodelsParameter estimation methodsParameter estimation methodsWhat to expect from the analysisWhat to expect from the analysis

12

Goals Of This LectureGoals Of This Lecture

What this lecture is aboutWhat this lecture is aboutWhat are the assumptions, and how can What are the assumptions, and how can these affect the conclusionsthese affect the conclusionsMake an intelligent choice of methodsMake an intelligent choice of methodsMake an intelligent choice of methods Make an intelligent choice of methods depending upon what information is required depending upon what information is required from the datafrom the data

What this lecture is not aboutWhat this lecture is not aboutTo conclude that one method is “better” than To conclude that one method is “better” than anotheranother

13

A Drug In The Body:A Drug In The Body:Constantly Undergoing ChangeConstantly Undergoing Change

AbsorptionAbsorptionTransport in the circulationTransport in the circulationT t bT t bTransport across membranesTransport across membranesBiochemical transformationBiochemical transformationEliminationEliminationADMEADME→→ADMEADME

Absorption, Distribution, Absorption, Distribution, Metabolism, ExcretionMetabolism, Excretion

14


100

120

0

20

40

60

80

0 4 8 12 16 20 24

Dru

g

Time

12141618

0

2

4

6

8

10

12

0 4 8 12 16 20 24

Dru

g

Time

02468

1012

0 4 8 12 16 20 24

Dru

g

Time

5

10

15

20

25

Dru

g

6

0

0 4 8 12 16 20 24Time

0

1

2

3

4

5

0 4 8 12 16 20 24

Dru

g

Time

15

KineticsKineticsAnd PharmacokineticsAnd Pharmacokinetics

Kinetics Kinetics The temporal and spatial distribution of a The temporal and spatial distribution of a substance in a system.substance in a system.

PharmacokineticsPharmacokineticsPharmacokinetics Pharmacokinetics The temporal and spatial distribution of a drug The temporal and spatial distribution of a drug (or drugs) in a system.(or drugs) in a system.

16

Definition Of Kinetics: Definition Of Kinetics: ConsequencesConsequences

SpatialSpatial: : WhereWhere in the systemin the systemSpatial coordinatesSpatial coordinatesKey variables: (x, y, z)Key variables: (x, y, z)

y1

y3

Y-axis

Z-axis

TemporalTemporal: : WhenWhen in the systemin the systemTemporal coordinatesTemporal coordinatesKey variable: tKey variable: t

y1X-axis

t)t,z,y,x(c,

z)t,z,y,x(c,

y)t,z,y,x(c,

x)t,z,y,x(c

∂∂

∂∂

∂∂

∂∂

17


100

120

0

20

40

60

80

0 4 8 12 16 20 24

Dru

g

Time

12141618

0

2

4

6

8

10

12

0 4 8 12 16 20 24

Dru

g

Time

02468

1012

0 4 8 12 16 20 24

Dru

g

Time

5

10

15

20

25

Dru

g

6

0

0 4 8 12 16 20 24Time

0

1

2

3

4

5

0 4 8 12 16 20 24

Dru

g

Time

18


100

120

∂∂∂∂

0

20

40

60

80

0 4 8 12 16 20 24

Dru

g

Time

12141618

0

2

4

6

8

10

12

0 4 8 12 16 20 24

Dru

g

Time

zyxt ∂∂

∂∂

∂∂

∂∂ ,,, z

,y

,x

,t ∂

∂∂∂

∂∂

∂∂

02468

1012

0 4 8 12 16 20 24

Dru

g

Time

5

10

15

20

25

Dru

g

6

zyxt ∂∂

∂∂

∂∂

∂∂ ,,,

zyxt ∂∂

∂∂

∂∂

∂∂ ,,,

0

0 4 8 12 16 20 24Time

0

1

2

3

4

5

0 4 8 12 16 20 24

Dru

g

Time

zyxt ∂∂

∂∂

∂∂

∂∂ ,,,

zyxt ∂∂∂∂

19

Spatially Distributed ModelsSpatially Distributed Models

Spatially realistic models:Spatially realistic models:Spatially realistic models:Spatially realistic models:Require a knowledge of physical Require a knowledge of physical chemistry, irreversible thermodynamics chemistry, irreversible thermodynamics and circulatory dynamics.and circulatory dynamics.A diffi lt t lA diffi lt t lAre difficult to solve.Are difficult to solve.It is difficult to design an experiment to It is difficult to design an experiment to estimate their parameter values.estimate their parameter values.

While desirable, normally not practical.While desirable, normally not practical.e des ab e, o a y o p ac cae des ab e, o a y o p ac caQuestion: What can one do?Question: What can one do?

20

Resolving The ProblemResolving The Problem

Reducing the system to a finite number of Reducing the system to a finite number of componentscomponentsLumping processes together based upon Lumping processes together based upon time location or a combination of the twotime location or a combination of the twotime, location or a combination of the twotime, location or a combination of the twoSpace is not taken directly into account: Space is not taken directly into account: rather, spatial heterogeneity is modeled rather, spatial heterogeneity is modeled through changes that occur in timethrough changes that occur in timethrough changes that occur in timethrough changes that occur in time

21

Lumped Parameter ModelsLumped Parameter Models

M d l hi h k th t di tM d l hi h k th t di tModels which make the system discrete Models which make the system discrete through a lumping process thus through a lumping process thus eliminating the need to deal with partial eliminating the need to deal with partial differential equations.differential equations.Classes of such models:Classes of such models:

NoncompartmentalNoncompartmental modelsmodels•• Based on algebraic equationsBased on algebraic equations

Compartmental modelsCompartmental modelsCompartmental modelsCompartmental models•• Based on linear or nonlinear differential equationsBased on linear or nonlinear differential equations

22

Probing The SystemProbing The System

Accessible poolsAccessible pools: These : These are system spaces that are are system spaces that are available to the available to the experimentalist for test input experimentalist for test input and/or measurement.and/or measurement.Nonaccessible poolsNonaccessible pools: : These are spaces comprising These are spaces comprising the rest of the system which the rest of the system which are not available for test are not available for test input and/or measurement.input and/or measurement.

23

Focus On The Accessible PoolFocus On The Accessible PoolSOURCEINPUT MEASURESOURCE

SYSTEM

AP

ELIMINATION

24

Characteristics Of The Characteristics Of The Accessible PoolAccessible PoolKinetically HomogeneousKinetically Homogeneous

Instantaneously WellInstantaneously Well--mixedmixed

25

Accessible PoolAccessible PoolKinetically HomogeneousKinetically Homogeneous

(ref: see e.g. Cobelli et al.)

26

Accessible PoolAccessible PoolInstantaneously WellInstantaneously Well--MixedMixed

S1 S1

BA

S2 S2

A = not mixedA = not mixedB = well mixedB = well mixed

(ref: see e.g. Cobelli et al.)

27

Probing The Accessible PoolProbing The Accessible PoolSOURCEINPUT MEASURESOURCE

SYSTEM

AP

ELIMINATION

28

The Pharmacokinetic The Pharmacokinetic ParametersParameters

Which pharmacokinetic parameters can Which pharmacokinetic parameters can we estimate based on measurements in we estimate based on measurements in the accessible pool?the accessible pool?Estimation requires a modelEstimation requires a modelEstimation requires a modelEstimation requires a model

Conceptualization of how the system worksConceptualization of how the system worksDepending on assumptions:Depending on assumptions:

Noncompartmental approachesNoncompartmental approachesNoncompartmental approachesNoncompartmental approachesCompartmental approachesCompartmental approaches

29

Accessible Pool & SystemAccessible Pool & SystemAssumptions Assumptions →→ InformationInformation

A ibl lA ibl lAccessible poolAccessible poolInitial volume of distributionInitial volume of distributionClearance rateClearance rateElimination rate constantElimination rate constantMean residence timeMean residence timeMean residence timeMean residence time

SystemSystemEquivalent volume of distributionEquivalent volume of distributionSystem mean residence timeSystem mean residence timeBioavailabilityBioavailabilityBioavailabilityBioavailabilityAbsorption rate constantAbsorption rate constant

30

Compartmental andCompartmental andCompartmental and Compartmental and Noncompartmental Noncompartmental

AnalysisAnalysis

The only difference between the two The only difference between the two methods is in how the nonaccessible methods is in how the nonaccessible

portion of the system is describedportion of the system is describedp yp y

31

The Noncompartmental ModelThe Noncompartmental ModelSOURCEINPUT MEASURESOURCE

SYSTEM

AP AP

ELIMINATION

32

RecirculationRecirculation--exchange exchange AssumptionsAssumptions

R i l ti / APRecirculation/Exchange

33

RecirculationRecirculation--exchange exchange AssumptionsAssumptions

R i l ti / APRecirculation/Exchange

34

Single Accessible Pool Single Accessible Pool Noncompartmental ModelNoncompartmental Model

Parameters (IV bolus and infusion)Parameters (IV bolus and infusion)Mean residence timeMean residence timeClearance rateClearance rateVolume of distributionVolume of distributionVolume of distributionVolume of distribution

Estimating the parameters from dataEstimating the parameters from dataAdditional assumption:Additional assumption:

Constancy of kinetic distribution parametersConstancy of kinetic distribution parametersConstancy of kinetic distribution parametersConstancy of kinetic distribution parameters

35

Mean Residence TimeMean Residence Time

The average time that a molecule of drug The average time that a molecule of drug spends in the systemspends in the system

100

120Concentration time-curve

20

40

60

80

100

Dru

g

Concentration time curve center of mass

∫∫

∞+

+∞

= 0

dt)t(C

dt)t(tCMRT

0

20

0 4 8 12 16 20 24

Time

∫0 )(

36

Areas Under The CurveAreas Under The CurveAUMCAUMC

Area Under the Moment CurveArea Under the Moment CurveAUCAUC

Area Under the CurveArea Under the CurveMRTMRT

“Normalized” AUMC (units = time)“Normalized” AUMC (units = time)

dt)t(tC∫+∞

AUCAUMC

dt)t(C

dt)t(tCMRT

0

0 ==

∫∫

∞+

37

What Is Needed For MRT?What Is Needed For MRT?

Estimates for AUC and AUMC.Estimates for AUC and AUMC.

100

120

20

40

60

80

Dru

g

00 4 8 12 16 20 24

Time

38

What Is Needed For MRT?What Is Needed For MRT?

Estimates for AUC and AUMC.Estimates for AUC and AUMC.

∫∫∫∫∞∞

++==n

n

1

1

t

t

t

t

00dt)t(Cdt)t(Cdt)t(Cdt)t(CAUC

∫∫∫∫∞∞ n1 tt

They require extrapolations beyond the time They require extrapolations beyond the time frame of the experimentframe of the experiment

∫∫∫∫∞∞⋅+⋅+⋅=⋅=

n

n

1

1

t

t

t

t

00dt)t(Ctdt)t(Ctdt)t(Ctdt)t(CtAUMC

Thus, this method is not model independent as Thus, this method is not model independent as often claimed.often claimed.

39

Estimating AUC And AUMC Using Estimating AUC And AUMC Using Sums Of ExponentialsSums Of Exponentials

∫∫∫∫∞∞

++==n

n

1

1

t

t

t

t


∫∫∫∫∞∞ n1 tt

∫∫∫∫∞∞⋅+⋅+⋅=⋅=

n

n

1

1

t

t

t

t


tn

t1

n1 eAeA)t(C λ−λ− ++= L

40

Bolus IV InjectionBolus IV InjectionFormulas can be extended to other administration modesFormulas can be extended to other administration modes

∫∞

λ++

λ==

0 n

n

1

1 AAdt)t(CAUC L

∫∞

1 AA∫ λ

++λ

=⋅=0 2

n

n21

1 AAdt)t(CtAUMC L

n1 AA)0(C ++= L

41

Estimating AUC And AUMC Estimating AUC And AUMC Using Other MethodsUsing Other Methodsgg

Trapezoidal Trapezoidal LogLog--trapezoidaltrapezoidalCombinationsCombinations

4

5

6

ugCombinationsCombinationsOtherOtherRole of extrapolationRole of extrapolation 0

1

2

3

0 4 8 12 16 20 24

Dru

Time

42

The IntegralsThe Integrals

These other methods provide formulas for the integrals between t1 and tn leaving it up to the researcher to extrapolate to time zero and time infinityzero and time infinity.

∫∫∫∫∞∞

++==n

n

1

1

t

t

t

t


∫∫∫∫∞∞⋅+⋅+⋅=⋅=

n

n

1

1

t

t

t

t


43

Trapezoidal RuleTrapezoidal RuleFor every time tFor every time tii, i = 1, …, n, i = 1, …, n

)tt)](t(y)t(y[21AUC 1ii1iobsiobs

i1i −−− −+=

)tt)](t(yt)t(yt[21AUMC 1ii1iobs1iiobsi

i1i −−−− −⋅+⋅=

2

60

80

100

120

Dru

g

)t(y)t(y 1iobs −

0

20

40

0 2 4 6 8 10

Time

)t(y iobs

it1it −

44

LogLog--trapezoidal Ruletrapezoidal Rule

For every time tFor every time tii, i = 1, …, n, i = 1, …, n

)tt)](t(y)t(y[)t(y

1AUC 1ii1iobsiobsi

1i −−− −+⎟⎞

⎜⎛

=

)t(y)t(yln1iobs

iobs

−⎟⎟⎠

⎞⎜⎜⎝

⎛

)tt)](t(yt)t(yt[1AUMC 1ii1ib1iibii

1i −+⋅= )tt)](t(yt)t(yt[

)t(y)t(yln

AUMC 1ii1iobs1iiobsi

1iobs

iobs1i −−⋅−

−

− +

⎟⎟⎠

⎞⎜⎜⎝

⎛

45

Trapezoidal Rule Potential PitfallsTrapezoidal Rule Potential Pitfalls

20

40

60

80

100

120

Dru

g

20

40

60

80

100

120D

rug

? ?0

0 4 8 12 16 20 24

Time

00 4 8 12 16 20 24

Time

As the number of samples decreases, the As the number of samples decreases, the interpolation may not be accurate (depends oninterpolation may not be accurate (depends oninterpolation may not be accurate (depends on interpolation may not be accurate (depends on the shape of the curve)the shape of the curve)Extrapolation from last measurement necessaryExtrapolation from last measurement necessary

46

Extrapolating From tExtrapolating From tnn To InfinityTo Infinity

Terminal decay is assumed to be a Terminal decay is assumed to be a monoexponential monoexponential The corresponding exponent is often The corresponding exponent is often calledcalled λλcalled called λλzz..

HalfHalf--life of terminal decay can be life of terminal decay can be calculated:calculated:

tt l (2)/l (2)/ λλttz/1/2z/1/2 = ln(2)/ = ln(2)/ λλzz

47


∫∞

− λ==

nt z

nobsdatextrap

)t(ydt)t(CAUC

∫∞

+⋅

=⋅= nobsnobsn )t(y)t(ytdt)t(CtAUMC

From last data point:

∫− λ+

λ=⋅=

nt 2zz

datextrap dt)t(CtAUMC

From last calculated value:

∫∞ λ−

− λ==

nz

t

tz

calcextrapeAdt)t(CAUC ∫ λnt z

calcextrap )(

∫∞ λ−λ−

− λ+

λ⋅

=⋅=n

nznz

t 2z

tz

z

tzn

calcextrapeAeAtdt)t(CtAUMC

48


1000

Extrapolating function crucialExtrapolating function crucial

10

100

Dru

g

10 4 8 12 16 20 24

Time

49

Estimating The IntegralsEstimating The Integrals

To estimate the integrals, one sums up the individual components.

tt

∫∫∫∫∞∞

++==n

n

1

1

t

t

t

t


∫∫∫∫∞∞⋅+⋅+⋅=⋅=

n

n

1

1

t

t

t

t


50

Advantages Of Using Function Advantages Of Using Function Extrapolation (Exponentials)Extrapolation (Exponentials)

Extrapolation is automatically done as part Extrapolation is automatically done as part of the data fittingof the data fittingStatistical information for all parameters Statistical information for all parameters (e g their standard errors) calculated(e g their standard errors) calculated(e.g. their standard errors) calculated(e.g. their standard errors) calculatedThere is a natural connection with the There is a natural connection with the solution of linear, constant coefficient solution of linear, constant coefficient compartmental modelscompartmental modelscompartmental modelscompartmental modelsSoftware is availableSoftware is available

51

Clearance RateClearance Rate

The volume of blood cleared per unit time, The volume of blood cleared per unit time, relative to the drugrelative to the drug

ratenEliminatioCL

It can be shown thatIt can be shown thatblood in ionConcentrat

CL =

AUCDoseDrugCL =

52

Remember Our AssumptionsRemember Our Assumptions

If these are not verified If these are not verified the estimates will be the estimates will be incorrectincorrectIn addition, this approach In addition, this approach

AP

ppppcannot straightforwardly cannot straightforwardly handle nonlinearities in handle nonlinearities in the data (timethe data (time--varying varying rates, saturation rates, saturation processes, etc.)processes, etc.)

53

The Compartmental ModelThe Compartmental ModelThe Compartmental ModelThe Compartmental Model

54

Single Accessible PoolSingle Accessible PoolSOURCEINPUT MEASURESOURCE

SYSTEM

AP AP

ELIMINATION

55

Single Accessible Pool ModelsSingle Accessible Pool Models

Noncompartmental Noncompartmental CompartmentalCompartmental

APAP

56

A Model Of The System

Inaccessible Portion

Accessible Portion

57

Compartmental ModelCompartmental Model

CompartmentCompartmentInstantaneously wellInstantaneously well--mixedmixedKinetically homogeneousKinetically homogeneous

C t t l d lC t t l d lCompartmental modelCompartmental modelFinite number of compartmentsFinite number of compartmentsSpecifically connectedSpecifically connectedSpecific input and outputSpecific input and outputSpecific input and outputSpecific input and output

58

Kinetics And The Kinetics And The Compartmental ModelCompartmental Model

Time and spaceTime and space

)t(Xt

,z

,y

,x ∂

∂∂∂

∂∂

∂∂

TimeTimet

)t,z,y,x(X,z

)t,z,y,x(X,y

)t,z,y,x(X,x

)t,z,y,x(X)t,z,y,x(X

∂∂

∂∂

∂∂

∂∂

→

→

dt)t(dX)t(X

dtd

→→

59

Demystifying Differential Demystifying Differential EquationsEquations

It is all about modelingIt is all about modeling rates of changerates of changeIt is all about modeling It is all about modeling rates of changerates of change, , i.e.i.e. slopesslopes, or, or derivativesderivatives::

80100120

tion

)t(dCd

020406080

Con

cent

rat

dt)t(dC)t(C

dtd

→→

Rates of change may be constant or notRates of change may be constant or not

00 4 8 12

Time

60

Ingredients Of Model BuildingIngredients Of Model Building

Model of the systemModel of the systemIndependent of experiment designIndependent of experiment designPrincipal components of the biological systemPrincipal components of the biological system

E i t l d iE i t l d iExperimental designExperimental designTwo parts:Two parts:

•• Input function (dose, shape, protocol)Input function (dose, shape, protocol)•• Measurement function (sampling, location)Measurement function (sampling, location)

61

Single Compartment ModelSingle Compartment ModelThTh t f ht f h ffThe The rate of changerate of change of of the amount in the the amount in the compartment, qcompartment, q11(t), is (t), is equal to what enters the equal to what enters the

t t (i tt t (i tq1(t)Plasma compartment (inputs or compartment (inputs or

initial conditions), minus initial conditions), minus what leaves the what leaves the compartment, a compartment, a q antit proportional toq antit proportional to

k(0,1)

quantity proportional to quantity proportional to qq11(t)(t)k(0,1) is a k(0,1) is a rate constantrate constant)t(q)1,0(k

dt)t(dq

11 −=

62

Experiment DesignExperiment DesignModeling Input SitesModeling Input Sites

TheThe rate of changerate of change ofofThe The rate of changerate of change of of the amount in the the amount in the compartment, qcompartment, q11(t), is (t), is equal to what enters equal to what enters the compartmentthe compartment

q1(t)Plasma the compartment the compartment

(Dose), minus what (Dose), minus what leaves the leaves the compartment, a compartment, a quantity proportionalquantity proportional

k(0,1)Dose(t)

quantity proportional quantity proportional to q(t)to q(t)Dose(t) can be any Dose(t) can be any function of timefunction of time

)t(Dose)t(q)1,0(kdt

)t(dq1

1 +−=

63

Experiment Design Experiment Design Modeling Measurement SitesModeling Measurement Sites

The measurement (sample) The measurement (sample) s1 does not subtract mass or s1 does not subtract mass or perturb the systemperturb the systemThe measurement equation The measurement equation s1 links qs1 links q11 with the with the experiment thus preservingexperiment thus preserving

q1(t)Plasma

s1(t)

experiment, thus preserving experiment, thus preserving the units of differential the units of differential equations and data (e.g. qequations and data (e.g. q11 is is massmass, the measurement is , the measurement is concentrationconcentration

k(0,1)Dose(t)

⇒⇒ s1 = qs1 = q11 /V/VV = volume of distribution of V = volume of distribution of compartment 1compartment 1

V)t(q)t(1s 1=

64

NotationNotation• The fluxes F describe material• The fluxes Fij (from j to i) describe material

transport in units of mass per unit time

Fi0

Fji

Fij

q iFij

F0i

65

The Compartmental Fluxes (FThe Compartmental Fluxes (Fijij))

Describe movement among, into or out of Describe movement among, into or out of a compartmenta compartmentA composite of metabolic activityA composite of metabolic activity

t tt ttransporttransportbiochemical transformationbiochemical transformationbothboth

Similar (compatible) time frameSimilar (compatible) time frameSimilar (compatible) time frameSimilar (compatible) time frame

66

A Proportional Model For The A Proportional Model For The Compartmental FluxesCompartmental Fluxes

q = compartmental massesq = compartmental massesp = (unknown) system parametersp = (unknown) system parameterskkjiji = a (nonlinear) function specific to the = a (nonlinear) function specific to the t f f i t jt f f i t jtransfer from i to jtransfer from i to j

)t(q)t,p,q(k)t,p,q(F ijiji ⋅=

(ref: see Jacquez and Simon)

67

Nonlinear Kinetics Nonlinear Kinetics ExampleExample

Remember the oneRemember the one--compartment compartment model:model:

)t(Dose)t(1q)10(k)t(1dq

+−=

V)t(1q

)t(1s

)t(Dose)t(1q)1,0(kdt

=

+=

What if we had a concentrationWhat if we had a concentration--dependent drug elimination rate?dependent drug elimination rate?

68

Nonlinear Kinetics: Nonlinear Kinetics: MichaelisMichaelis--MentenMenten

MichaelisMichaelis--MentenMenten kinetics:kinetics:

)t(Dose)t(1q)(

Vm)t(1dq+−=

VV i l t b li ti l t b li tV)t(1q

)t(1s

)()(q)t(cKmdt

=

+

VmVm = maximal metabolic rate= maximal metabolic rateKm = Km = MichaelisMichaelis--MentenMenten constantconstant

69

Linear vs. Nonlinear KineticsLinear vs. Nonlinear KineticsIf KIf K ((tt) th) thIf Km >> If Km >> cc((tt) then:) then:

)t(Dose)t(1qKmVm

dt)t(1dq

+−≅

The concentration declines at a rate The concentration declines at a rate ti l t it (ti l t it (fi tfi t dd ki tiki ti ))

V)t(1q

)t(1s =

proportional to it (proportional to it (firstfirst--order order kineticskinetics))This is true at This is true at lowlow concentrations (concentrations (w.r.tw.r.t. Km). Km)

70

Linear vs. Nonlinear KineticsLinear vs. Nonlinear Kinetics

If Km << If Km << cc((tt) then:) then:

)t(Dose1vVm)t(D)t(1q)t(c

Vmdt

)t(1dq+×−=δ+−≅

The concentration declines at a constant rate The concentration declines at a constant rate V)t(1q

)t(1s

)(

=

((zerozero--order kineticsorder kinetics))This is true at This is true at highhigh concentrations (concentrations (w.r.tw.r.t. Km). Km)

71

Tracking NonlinearitiesTracking Nonlinearities

How to find nonlinear behavior?How to find nonlinear behavior?

)t(Dose)t(1q)t(cKm

Vmdt

)t(1dq+

+−=

W t hW t h Si l t d t ti tiSi l t d t ti tiV)t(1q

)t(1s

)t(cKmdt

=

+

WatchWatch: Simulated concentration time : Simulated concentration time profile for D = 180 mg, Vm = 20 profile for D = 180 mg, Vm = 20 mg/L/hr, Km = 1 mg/L, v1 = 5 Lmg/L/hr, Km = 1 mg/L, v1 = 5 L

72

40Tracking NonlinearitiesTracking Nonlinearities

20

25

30

35

atio

n (m

g/L)

V)t(1q

)t(1s

)t(Dose)t(1q)t(cKm

Vmdt

)t(1dq

=

++

−=

5

10

15

20

Con

cent

ra

00 3 6 9 12

Time (days)

73

Tracking NonlinearitiesTracking Nonlinearities

0 1

1

10

ntra

tion

(mg/

L)

)t(Dose)t(1q)t(K

Vmdt

)t(1dq+−=

0.001

0.01

0.1

Log

Con

cen

V)t(1q

)t(1s

)t(cKmdt

=

+

0.00010 3 6 9 12

Time (days)

74

The Fractional Coefficients (kThe Fractional Coefficients (kijij))

• The fractional coefficients kij are called fractional transfer functions

• If kij does not depend on the compartmental masses then the kij iscompartmental masses, then the kij is called a fractional transfer (or rate) constant.

k)t(k ijij k)t,p,q(k =

75

Compartmental Models And Compartmental Models And Systems Of Ordinary Differential Systems Of Ordinary Differential

EquationsEquationsEquationsEquations

Good mixingGood mixingpermits writing qpermits writing qii(t) for the i(t) for the ithth compartment.compartment.

Kinetic homogeneityKinetic homogeneitypermits connecting compartments via the permits connecting compartments via the kkijij..

76

The iThe ithth CompartmentCompartment

nni F)t(q)tpq(k)t(q)tpq(kdq

⎟⎟⎞

⎜⎜⎛

∑∑

Rate ofchange of

qi

Fractionalinput from

qj

0i

ij1j

jiji

ij0j

jii F)t(q)t,p,q(k)t(q)t,p,q(k

dtq

++⎟⎟

⎠⎜⎜

⎝

−= ∑∑≠=

≠=

Fractionalloss of

Input from“outside”

(productionloss ofqi

rates)

77

Linear, Constant Coefficient Linear, Constant Coefficient Compartmental ModelsCompartmental Models

All transfer rates kAll transfer rates kijij are constant.are constant.This facilitates the required computations This facilitates the required computations greatlygreatlyg yg y

Assume “steady state” conditions.Assume “steady state” conditions.Changes in compartmental mass do not affect Changes in compartmental mass do not affect the values for the transfer ratesthe values for the transfer ratesthe values for the transfer ratesthe values for the transfer rates

78

The iThe ithth CompartmentCompartment

nni F)t(qk)t(qkdq

++⎟⎟⎞

⎜⎜⎛

∑∑

Rate ofchange of

Qi

Fractionalinput from

Qj

0i

ij1j

jiji

ij0j

jii F)t(qk)t(qk

dtq

++⎟⎟

⎠⎜⎜

⎝

−= ∑∑≠=

≠=

Fractionalloss of

Input from“outside”

(productionloss of

Qirates)

79

The Compartmental MatrixThe Compartmental Matrix

⎞⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−= ∑

≠=

n

ij0j

jiii kk

⎥⎥⎥⎤

⎢⎢⎢⎡

= n22221

n11211

kkkkkk

KMOMM

L

L

⎥⎥

⎦⎢⎢

⎣ nn2n1n kkk L

MOMM

80

Compartmental ModelCompartmental Model

A detailed postulation of how one believes A detailed postulation of how one believes a system functions.a system functions.

The need to perform the same experiment The need to perform the same experiment on the model as one did in the laboratory.on the model as one did in the laboratory.

81

Underlying System ModelUnderlying System Model

1 2 3

4EV-1

1Stomach

2Gut

3Plasma

5EV-2

82

System Model with ExperimentSystem Model with Experiment

1 2 3

4EV-1

1Stomach

2Gut

3Plasma

5EV-2

83

System Model with ExperimentSystem Model with Experiment

1 2 3

4EV-1

1Stomach

2Gut

3Plasma

5EV-2

84

ExperimentsExperiments

Need to recreate the laboratory Need to recreate the laboratory experiment on the model.experiment on the model.Need to specify input and measurementsNeed to specify input and measurementsK UNITSK UNITSKey: UNITSKey: UNITS

Input usually in mass, or mass/timeInput usually in mass, or mass/timeMeasurement usually concentrationMeasurement usually concentration

•• Mass per unit volumeMass per unit volumeMass per unit volumeMass per unit volume

85

ConceptualizationReality Data AnalysisModel Of The System?Model Of The System?

Conceptualization(Model)

Reality (Data)

Data Analysisand Simulation

program optimizebegin model…

dend

86

Pharmacokinetic ExperimentPharmacokinetic ExperimentCollecting System KnowledgeCollecting System Knowledge

60000

20000

30000

40000

50000

60000

entra

tion

(ng/

dl)

1 2

0

10000

20000

0 1 2 3 4 5 6 7 8

Con

ce

Time (days)

The model starts as a qualitative construct, The model starts as a qualitative construct, based on known physiology and further based on known physiology and further assumptionsassumptions

87

Data AnalysisData AnalysisDistilling Parameters From DataDistilling Parameters From Data

60000 )t(dq1

20000

30000

40000

50000

60000

entra

tion

(ng/

dl)

)t(qK)t(qKdt

)t(dq

)t(IV)t(qK)t(q)KK(dt

)t(dq

2211122

221112101

−=

+++−=

0

10000

20000

0 1 2 3 4 5 6 7 8

Con

ce

Time (days)

• Qualitative model ⇒ quantitative differential equations with parameters of physiological interest

• Parameter estimation (nonlinear regression)

88

Parameter EstimatesParameter EstimatesPrinciples of model buildingPrinciples of model buildingPrinciples of model buildingPrinciples of model building

Model definition: structure, error modelModel definition: structure, error modelModel selection: parsimony criteriaModel selection: parsimony criteriaEstimation methods: maximum likelihoodEstimation methods: maximum likelihood

Model Model parameters: parameters: kkijij and volumesand volumesPharmacokinetic parameters: volumes, Pharmacokinetic parameters: volumes, clearance, residence times, etc.clearance, residence times, etc.ReparameterizationReparameterization -- changing the changing the parameters from parameters from kkijij to the PK parameters.to the PK parameters.

89

Recovering The PK Parameters Recovering The PK Parameters From Compartmental ModelsFrom Compartmental Models

Parameters can be based upon Parameters can be based upon The model primary parametersThe model primary parameters

Diff ti l ti tDiff ti l ti t•• Differential equation parametersDifferential equation parameters•• Measurement parametersMeasurement parameters

The compartmental matrixThe compartmental matrix•• Aggregates of model parametersAggregates of model parameters•• Aggregates of model parametersAggregates of model parameters

90

Compartmental Model Compartmental Model ⇒⇒ExponentialExponential)t(d

V)t(q)t(1s

)t(Dose)t(q)1,0(kdt

)t(dq

1

11

=

δ+−=

t)10(k1

t)1,0(k1

Dose)t(q)t(1

eDose)t(q

−

−⋅=

For a pulse input δ(t)

t)1,0(k1 eVV

)(q)t(1s ==

V)1,0(kCL ×=

91

Compartmental Residence TimesCompartmental Residence Times

1 2

R t t tR t t tRate constantsRate constantsResidence timesResidence timesIntercompartmental clearancesIntercompartmental clearances

92

Parameters Based Upon The Parameters Based Upon The Compartmental MatrixCompartmental MatrixCompartmental MatrixCompartmental Matrix

⎤⎡ kkk ⎟⎞

⎜⎛ ϑϑϑ n11211 L

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

nn2n1n

n22221

n11211

kkk

kkkkkk

K

L

MOMM

L

L

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

ϑϑϑ

ϑϑϑϑϑϑ

=−= −

nn2n1n

n22221

n11211

1KΘ

L

MOMM

L

Theta, the negative of the inverse of the compartmental matrix, is called the mean residence time matrix.

93

Parameters Based Upon The Parameters Based Upon The Compartmental MatrixCompartmental MatrixCompartmental MatrixCompartmental Matrix

Generalization of Mean Residence TimeGeneralization of Mean Residence Time

ijϑThe average time the drug entering compartment jfor the first time spends in compartment i beforeijϑ

jiij ≠ϑ

for the first time spends in compartment i beforeleaving the system.

The probability that a drug particle inj ill ll h hji,

ii≠

ϑcompartment j will eventually pass throughcompartment i before leaving the system.

94

Compartmental Models:Compartmental Models:AdvantagesAdvantages

Can handle nonlinearitiesCan handle nonlinearitiesProvide hypotheses about system Provide hypotheses about system structurestructureCan aid in experimental design, for Can aid in experimental design, for example to design dosing regimensexample to design dosing regimensC t t l ti l hC t t l ti l hCan support translational researchCan support translational research

95

Bias That Can Be Introduced By Bias That Can Be Introduced By Noncompartmental AnalysisNoncompartmental Analysis

N t i l i kN t i l i kNot a single sinkNot a single sink== Clearance rateClearance rate↓↓ Mean residence timeMean residence time↓↓ Volume of distributionVolume of distribution↑↑ Fractional clearanceFractional clearance↑↑ Fractional clearanceFractional clearance

Not a single sink / not a single sourceNot a single sink / not a single source↓↓ Clearance rateClearance rate↓↓ Mean residence timeMean residence time↓↓ Volume of distributionVolume of distribution↓↓ Volume of distributionVolume of distribution↑↑ Fractional clearanceFractional clearance

JJ JJ DiStefanoDiStefano III.III.NoncompartmentalNoncompartmental vsvs compartmental analysis: some bases for choice. compartmental analysis: some bases for choice. Am J. Am J. PhysiolPhysiol. 1982;243:R1. 1982;243:R1--R6R6

96

Nonlinear PharmacokineticsNonlinear PharmacokineticsE l tib d h ki tiE l tib d h ki tiExample: antibody pharmacokineticsExample: antibody pharmacokineticsOften, antibodies exhibit targetOften, antibodies exhibit target--mediated mediated disposition, and thus their elimination may disposition, and thus their elimination may occur at sites remote from plasma due tooccur at sites remote from plasma due tooccur at sites remote from plasma due to occur at sites remote from plasma due to binding and internalization processesbinding and internalization processesThis is one of many possible biological This is one of many possible biological processes causing nonlinear (capacityprocesses causing nonlinear (capacity--p g ( p yp g ( p ylimited) pharmacokinetic behaviorslimited) pharmacokinetic behaviors

97

Impact of Impact of NoncompartmentalNoncompartmentalAnalysis AssumptionsAnalysis Assumptions

When drug elimination is influenced by When drug elimination is influenced by binding to its pharmacological target, the binding to its pharmacological target, the assumptions of assumptions of noncompartmentalnoncompartmentalanalysis may not be met to a varyinganalysis may not be met to a varyinganalysis may not be met to a varying analysis may not be met to a varying degree and parameter estimates may be degree and parameter estimates may be misleadingmisleadingNoncompartmentalNoncompartmental analysis always analysis always pp y yy yrequires linearity and time invariance, but requires linearity and time invariance, but it can be useful to explore nonlinearitiesit can be useful to explore nonlinearities

98

Example of Dose NonlinearitiesExample of Dose Nonlinearities100

0.1

10

1000

0 1 2 3

AU

C

D ( /k )

0.11

10100

0 1 2 3

Cle

aran

ce

D ( /k )Dose (mg/kg) Dose (mg/kg)

0100200300

Vss

050

100150

Hal

f-Life

0 1 2 3Dose (mg/kg)

0 1 2 3Dose (mg/kg)

PK example from PK example from SheremataSheremata et al. (1999)as reported in et al. (1999)as reported in MagerMager (2006)(2006)TargetTarget--mediated drug disposition and dynamicsmediated drug disposition and dynamicsBiochemical Pharmacology 72(2006) 1Biochemical Pharmacology 72(2006) 1--1010

99

TargetTarget--Mediated Drug DispositionMediated Drug Disposition

33Tissue

k

ksynktp kpt

1Dosing

2Plasma

4Free

Receptor

5Receptor Complex

+kon

koff

ka

kdeg kmkelMagerMagerTargetTarget--mediated drug disposition and dynamicsmediated drug disposition and dynamicsBiochemical Pharmacology 72(2006) 1Biochemical Pharmacology 72(2006) 1--1010

100

Take Home MessageTake Home Message

To estimate traditional pharmacokinetic To estimate traditional pharmacokinetic parameters, either model is probably adequate parameters, either model is probably adequate when the sampling schedule is dense, provided when the sampling schedule is dense, provided all assumptions required for noncompartmental all assumptions required for noncompartmental analysis are metanalysis are metSparse sampling schedule and nonlinearities Sparse sampling schedule and nonlinearities may be an issue for noncompartmental analysismay be an issue for noncompartmental analysisNoncompartmental models are not predictiveNoncompartmental models are not predictiveNoncompartmental models are not predictiveNoncompartmental models are not predictiveBest strategy is probably a blend: but, careful Best strategy is probably a blend: but, careful about assumptions!about assumptions!

101

Selected ReferencesSelected ReferencesGeneral references (compartmental models)General references (compartmental models)

JJ JAJA C t t l A l i i Bi l d M di iC t t l A l i i Bi l d M di iJacquezJacquez, JA. , JA. Compartmental Analysis in Biology and MedicineCompartmental Analysis in Biology and Medicine. . BioMedwareBioMedware 1996. Ann Arbor, MI.1996. Ann Arbor, MI.Cobelli, C, D Foster and G Cobelli, C, D Foster and G ToffoloToffolo. . Tracer Kinetics in Biomedical Tracer Kinetics in Biomedical ResearchResearch. . KluwerKluwer Academic/Plenum Publishers. 2000, New York.Academic/Plenum Publishers. 2000, New York.

Theory of Theory of noncompartmentalnoncompartmental and compartmental modelsand compartmental modelsJJ JJ DiStefanoDiStefano III. III. NoncompartmentalNoncompartmental vsvs compartmental analysis: some compartmental analysis: some bases for choice Am Jbases for choice Am J PhysiolPhysiol 1982;243:R11982;243:R1 R6R6bases for choice. Am J. bases for choice. Am J. PhysiolPhysiol. 1982;243:R1. 1982;243:R1--R6R6DG DG CovellCovell et. al. Mean Residence Time. Math. et. al. Mean Residence Time. Math. BiosciBiosci. 1984;72:213. 1984;72:213--24442444JacquezJacquez, JA and SP Simon. Qualitative theory of compartmental analysis. , JA and SP Simon. Qualitative theory of compartmental analysis. SIAM Review SIAM Review 1993;35:431993;35:43--7979

Selected applications (nonlinear pharmacokinetics)Selected applications (nonlinear pharmacokinetics)MagerMager DE. TargetDE. Target--mediated drug disposition and dynamics. mediated drug disposition and dynamics. BiochemBiochemPh lPh l 2006 J 28 72(1) 12006 J 28 72(1) 1 1010PharmacolPharmacol. 2006 Jun 28;72(1):1. 2006 Jun 28;72(1):1--10.10.Lobo ED, Hansen RJ, Lobo ED, Hansen RJ, BalthasarBalthasar JP. Antibody pharmacokinetics and JP. Antibody pharmacokinetics and pharmacodynamics. J pharmacodynamics. J PharmPharm SciSci. 2004 Nov;93(11):2645. 2004 Nov;93(11):2645--68.68.

Thanks: Kenneth Luu (PGRD)Thanks: Kenneth Luu (PGRD)

noncompartmentalnoncompartmental vs. … · 1 noncompartmentalnoncompartmental vs. compartmental...

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