noncompartmental vs. compartmental approaches to...

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Noncompartmental vs. Compartmental Noncompartmental vs. Compartmental Approaches to Approaches to

Pharmacokinetic Data AnalysisPharmacokinetic Data AnalysisOctober 29, 2009October 29, 2009

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

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Questions To Be AskedQuestions To Be Asked

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

PharmacodynamicsPharmacodynamicsWhat the drug does to the body What the drug does to the body

Disease progressionDisease progressionMeasurable therapeutic effectMeasurable therapeutic effect

VariabilityVariabilitySources of error and biological variationSources of error and biological variation

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Pharmacokinetics / Pharmacokinetics / PharmacodynamicsPharmacodynamics

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PharmacokineticsPharmacokinetics““What the body does to What the body does to the drugthe drug””Fairly well knownFairly well knownUseful to get to the PDUseful to get to the PD

PharmacodynamicsPharmacodynamics““What the drug does to What the drug does to the bodythe body””Largely unknownLargely unknownHas clinical relevanceHas clinical relevance

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Hierarchical VariabilityHierarchical VariabilityNo VariationNo Variation

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Hierarchical VariabilityHierarchical VariabilityResidual Unknown VariationResidual Unknown Variation

within-individual(what the model does not

explain – e.g. measurement error)

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Hierarchical VariabilityHierarchical VariabilityBetweenBetween--Subject VariationSubject Variation

between-individual(physiological variability)

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

Residual Unknown VariationResidual Unknown Variation

Biological sources?

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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)

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

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Models For EstimationModels For EstimationNoncompartmentalNoncompartmental

CompartmentalCompartmental

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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 modelsmodelsParameter estimation methodsParameter estimation methodsWhat to expect from the analysisWhat to expect from the analysis

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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 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 To conclude that one method is ““betterbetter”” than than anotheranother

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

AbsorptionAbsorptionTransport in the circulationTransport in the circulationTransport across membranesTransport across membranesBiochemical transformationBiochemical transformationEliminationElimination

→→ADMEADMEAbsorption, Distribution, Absorption, Distribution, Metabolism, ExcretionMetabolism, Excretion

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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.

Pharmacokinetics 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.

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

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

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

x)t,z,y,x(c

∂∂

∂∂

∂∂

∂∂

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)

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

Z-axis

y1

y3

X-axis

Y-axis

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zyxt ∂∂

∂∂

∂∂

∂∂ ,,,

zyxt ∂∂

∂∂

∂∂

∂∂ ,,, zyxt ∂

∂∂∂

∂∂

∂∂ ,,,

zyxt ∂∂

∂∂

∂∂

∂∂ ,,,

zyxt ∂∂

∂∂

∂∂

∂∂ ,,,

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Spatially Distributed ModelsSpatially Distributed 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.Are 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.Question: What can one do?Question: What can one do?

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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 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 time

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Lumped Parameter ModelsLumped Parameter Models

Models 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:

Noncompartmental modelsNoncompartmental models•• Based on algebraic equationsBased on algebraic equations

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

equationsequations

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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.

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Focus On The Accessible PoolFocus On The Accessible Pool

SYSTEM

AP

INPUT MEASURESOURCE

ELIMINATION

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Characteristics Of The Characteristics Of The Accessible PoolAccessible PoolKinetically HomogeneousKinetically Homogeneous

Instantaneously WellInstantaneously Well--mixedmixed

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Accessible PoolAccessible PoolKinetically HomogeneousKinetically Homogeneous

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

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Accessible PoolAccessible PoolInstantaneously WellInstantaneously Well--MixedMixed

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

S1

S2

S1

S2

BA

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

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Probing The Accessible PoolProbing The Accessible Pool

SYSTEM

AP

INPUT MEASURESOURCE

ELIMINATION

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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 model

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

Noncompartmental approachesNoncompartmental approachesCompartmental approachesCompartmental approaches

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Accessible Pool & SystemAccessible Pool & SystemAssumptions Assumptions →→ InformationInformation

Accessible poolAccessible poolInitial volume of distributionInitial volume of distributionClearance rateClearance rateElimination rate constantElimination rate constantMean residence timeMean residence time

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

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Compartmental and Compartmental and Noncompartmental Noncompartmental

AnalysisAnalysis

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

portion of the system is describedportion of the system is described

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The Noncompartmental ModelThe Noncompartmental Model

SYSTEM

AP

INPUT MEASURESOURCE

ELIMINATION

AP

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RecirculationRecirculation--exchange exchange AssumptionsAssumptions

APRecirculation/Exchange

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RecirculationRecirculation--exchange exchange AssumptionsAssumptions

APRecirculation/Exchange

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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 distribution

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

Constancy of kinetic distribution parametersConstancy of kinetic distribution parameters

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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

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Concentration time-curve center of mass

∫∫

∞+

+∞

=

0

0

dt)t(C

dt)t(tCMRT

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Areas Under The CurveAreas Under The CurveAUMCAUMC

Area Under the Moment CurveArea Under the Moment CurveAUCAUC

Area Under the CurveArea Under the CurveMRTMRT

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

AUCAUMC

dt)t(C

dt)t(tCMRT

0

0 ==

∫∫

∞+

+∞

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

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

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

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

They require extrapolations beyond the time They require extrapolations beyond the time frame of the experimentframe of the experimentThus, this method is not model independent as Thus, this method is not model independent as often claimed.often claimed.

∫∫∫∫∞∞

++==n

n

1

1

t

t

t

t

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

∫∫∫∫∞∞⋅+⋅+⋅=⋅=

n

n

1

1

t

t

t

t

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

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

∫∫∫∫∞∞

++==n

n

1

1

t

t

t

t


∫∫∫∫∞∞⋅+⋅+⋅=⋅=

n

n

1

1

t

t

t

t


tn

t1

n1 eAeA)t(C λ−λ− ++= L

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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

∫∞

λ++

λ=⋅=

0 2n

n21

1 AAdt)t(CtAUMC L

n1 AA)0(C ++= L

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Estimating AUC And AUMC Estimating AUC And AUMC Using Other MethodsUsing Other Methods

Trapezoidal Trapezoidal LogLog--trapezoidaltrapezoidalCombinationsCombinationsOtherOtherRole of extrapolationRole of extrapolation 0

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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 infinity.

∫∫∫∫∞∞

++==n

n

1

1

t

t

t

t


∫∫∫∫∞∞⋅+⋅+⋅=⋅=

n

n

1

1

t

t

t

t


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Trapezoidal RuleTrapezoidal RuleFor every time For every time ttii, i = 1, , i = 1, ……, n, n

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

i1i −−− −+=

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

i1i −−−− −⋅+⋅=

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Dru

g

)t(y iobs

it

)t(y 1iobs −

1it −

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LogLog--trapezoidal Ruletrapezoidal Rule

For every time For every time ttii, i = 1, , i = 1, ……, n, n

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

)t(y)t(yln

1AUC 1ii1iobsiobs

1iobs

iobs

i1i −−

−

− −+

⎟⎟⎠

⎞⎜⎜⎝

⎛=

)tt)](t(yt)t(yt[

)t(y)t(yln

1AUMC 1ii1iobs1iiobsi

1iobs

iobs

i1i −−⋅−

−

− −+⋅

⎟⎟⎠

⎞⎜⎜⎝

⎛=

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Trapezoidal Rule Potential PitfallsTrapezoidal Rule Potential Pitfalls

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As the number of samples decreases, the As the number of samples decreases, the interpolation 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

? ?

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Extrapolating From Extrapolating From ttnn 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 called called λλzz..

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

ttz/1/2z/1/2 = ln(2)/ = ln(2)/ λλzz

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∫∞

− λ==

nt z

nobsdatextrap

)t(ydt)t(CAUC

∫∞

− λ+

λ⋅

=⋅=nt 2

z

nobs

z

nobsndatextrap

)t(y)t(ytdt)t(CtAUMC

From last data point:

From last calculated value:

∫∞ λ−

− λ==

n

nz

t z

tz

calcextrapeAdt)t(CAUC

∫∞ λ−λ−

− λ+

λ⋅

=⋅=n

nznz

t 2z

tz

z

tzn

calcextrapeAeAtdt)t(CtAUMC

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Extrapolating function crucialExtrapolating function crucial

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Estimating The IntegralsEstimating The Integrals

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

∫∫∫∫∞∞

++==n

n

1

1

t

t

t

t


∫∫∫∫∞∞⋅+⋅+⋅=⋅=

n

n

1

1

t

t

t

t


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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) 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 modelsSoftware is availableSoftware is available

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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

It can be shown thatIt can be shown that

AUCDose DrugCL =

blood in ionConcentratrate nEliminatioCL =

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Remember Our AssumptionsRemember Our Assumptions

AP

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

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The Compartmental ModelThe Compartmental Model

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Single Accessible PoolSingle Accessible Pool

SYSTEM

AP

INPUT MEASURESOURCE

ELIMINATION

AP

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Single Accessible Pool ModelsSingle Accessible Pool Models

Noncompartmental Noncompartmental CompartmentalCompartmental

APAP

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Accessible Portion

Inaccessible Portion

A Model Of The System

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Compartmental ModelCompartmental Model

CompartmentCompartmentInstantaneously wellInstantaneously well--mixedmixedKinetically homogeneousKinetically homogeneous

Compartmental modelCompartmental modelFinite number of compartmentsFinite number of compartmentsSpecifically connectedSpecifically connectedSpecific input and outputSpecific input and output

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Kinetics And The Kinetics And The Compartmental ModelCompartmental Model

Time and spaceTime and space

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

t,

z,

y,

x

∂∂

∂∂

∂∂

∂∂

→

→∂∂

∂∂

∂∂

∂∂

dt)t(dX)t(X

dtd

→→

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Demystifying Differential Demystifying Differential EquationsEquations

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

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

020406080

100120

0 4 8 12Time

Con

cent

ratio

n

dt)t(dC)t(C

dtd

→→

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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

Experimental designExperimental designTwo parts:Two parts:

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

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Single Compartment ModelSingle Compartment ModelThe 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 compartment (inputs or compartment (inputs or initial conditions), minus initial conditions), minus what leaves the what leaves the compartment, a compartment, a 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 −=

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Experiment DesignExperiment DesignModeling Input SitesModeling Input Sites

The 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 compartment the compartment (Dose), minus what (Dose), minus what leaves the leaves the compartment, a compartment, a quantity proportional quantity proportional to to q(tq(t))Dose(tDose(t) can be any ) can be any function of timefunction of time

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

)t(dq1

1 +−=

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

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

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

transport in units of mass per unit time

Fji

Fij

F0i

Fi0

q i

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The Compartmental Fluxes (The Compartmental Fluxes (FFijij))

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

transporttransportbiochemical transformationbiochemical transformationbothboth

Similar (compatible) time frameSimilar (compatible) time frame

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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 transfer from i to jtransfer from i to j

)t(q)t,p,q(k)t,p,q(F ijiji ⋅=

(ref: see Jacquez and Simon)

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The Fractional Coefficients (The Fractional Coefficients (kkijij))

• The fractional coefficients kij are called fractional transfer functions

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

ijij k)t,p,q(k =

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

EquationsEquations

Good mixingGood mixingpermits writing permits writing qqii(t(t) for the i) for the ithth compartment.compartment.

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

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The iThe ithth CompartmentCompartment

0i

n

ij1j

jiji

n

ij0j

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

dtdq

++⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−= ∑∑

≠=

≠=

Rate ofchange of

qi

Fractionalloss of

qi

Fractionalinput from

qj

Input from“outside”

(productionrates)

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Linear, Constant Coefficient Linear, Constant Coefficient Compartmental ModelsCompartmental Models

All transfer rates All transfer rates kkijij are constant.are constant.This facilitates the required computations This facilitates the required computations greatlygreatly

Assume Assume ““steady statesteady state”” conditions.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 rates

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The iThe ithth CompartmentCompartment

0i

n

ij1j

jiji

n

ij0j

jii F)t(qk)t(qk

dtdq

++⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−= ∑∑

≠=

≠=

Rate ofchange of

Qi

Fractionalloss of

Qi

Fractionalinput from

Qj

Input from“outside”

(productionrates)

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The Compartmental MatrixThe Compartmental Matrix

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−= ∑

≠=

n

ij0j

jiii kk

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

nn2n1n

n22221

n11211

kkk

kkkkkk

K

L

MOMM

L

L

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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.

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Underlying System ModelUnderlying System Model

SAAM II software system, http://depts.washington.edu/saam2

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System Model with ExperimentSystem Model with Experiment


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System Model with ExperimentSystem Model with Experiment


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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 measurementsKey: 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 volume

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Conceptualization(Model)

Reality (Data)

Data Analysisand Simulation

program optimizebegin model…end

Model Of The System?Model Of The System?

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Pharmacokinetic ExperimentPharmacokinetic ExperimentCollecting System KnowledgeCollecting System Knowledge

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

0

10000

20000

30000

40000

50000

60000

0 1 2 3 4 5 6 7 8

Time (days)

Con

cent

ratio

n (n

g/dl

)

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Data AnalysisData AnalysisDistilling Parameters From DataDistilling Parameters From Data

0

10000

20000

30000

40000

50000

60000

0 1 2 3 4 5 6 7 8

Time (days)

Con

cent

ratio

n (n

g/dl

)

)t(qK)t(qKdt

)t(dq

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

)t(dq

2211122

221112101

−=

+++−=

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

• Parameter estimation (nonlinear regression)

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Parameter EstimatesParameter Estimates

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

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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

•• Differential equation parametersDifferential equation parameters•• Measurement parametersMeasurement parameters

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

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Compartmental Model Compartmental Model ⇒⇒ExponentialExponential

V)t(q)t(1s

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

)t(dq

1

11

=

δ+−=

t)1,0(k1

t)1,0(k1

eV

DoseV

)t(q)t(1s

eDose)t(q

−

−

==

⋅=

For a pulse input δ(t)

V)1,0(kCL ×=

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Compartmental Residence TimesCompartmental Residence Times

Rate constantsRate constantsResidence timesResidence timesIntercompartmentalIntercompartmental clearancesclearances

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Parameters Based Upon The Parameters Based Upon The Compartmental MatrixCompartmental Matrix

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

nn2n1n

n22221

n11211

kkk

kkkkkk

K

L

MOMM

L

L

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

ϑϑϑ

ϑϑϑϑϑϑ

=−= −

nn2n1n

n22221

n11211

1KΘ

L

MOMM

L

L

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

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Parameters Based Upon The Parameters Based Upon The Compartmental MatrixCompartmental Matrix

Generalization of Mean Residence TimeGeneralization of Mean Residence Time

ijϑ

ji,ii

ij ≠ϑ

ϑ

The average time the drug entering compartment jfor the first time spends in compartment i beforeleaving the system.

The probability that a drug particle incompartment j will eventually pass throughcompartment i before leaving the system.

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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 regimensCan support translational researchCan support translational research

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Bias That Can Be Introduced By Bias That Can Be Introduced By NoncompartmentalNoncompartmental AnalysisAnalysis

Not a single sinkNot a single sink== Clearance rateClearance rate↓↓ Mean residence timeMean residence time↓↓ Volume of distributionVolume of distribution↑↑ 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↑↑ Fractional clearanceFractional clearance

JJ JJ DiStefanoDiStefano III.III.NoncompartmentalNoncompartmental vsvs compartmental analysis: some bases for choice. compartmental analysis: some bases for choice. Am J. Physiol. 1982;243:R1Am J. Physiol. 1982;243:R1--R6R6

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Noncompartmental Versus Noncompartmental Versus Compartmental Approaches To PK Compartmental Approaches To PK

Analysis: An ExampleAnalysis: An Example

Experiment designExperiment designBolus injection of 100 mg of a drug into Bolus injection of 100 mg of a drug into plasmaplasmaSerial plasma samples taken for 60 hoursSerial plasma samples taken for 60 hours

Analysis using:Analysis using:Trapezoidal integrationTrapezoidal integrationSums of exponentialsSums of exponentialsLinear compartmental modelLinear compartmental model

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91


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TrapezoidalAnalysis

Sum of Exponentials

Compartmental Model

Volume 10.2 (9%) 10.2 (3%)Clearance 1.02 1.02 (2%) 1.02 (1%)MRT 19.5 20.1 (2%) 20.1 (1%)λ z 0.0504 0.0458 (3%) 0.0458 (1%)AUC 97.8 97.9 (2%) 97.9 (1%)AUMC 1908 1964 (3%) 1964 (1%)

ResultsResults

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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 all assumptions required for noncompartmentalnoncompartmentalanalysis 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 predictiveBest strategy is probably a blend: but, careful Best strategy is probably a blend: but, careful about assumptions!about assumptions!

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Some ReferencesSome ReferencesJJ DiStefano III. Noncompartmental JJ DiStefano III. Noncompartmental vsvscompartmental analysis: some bases for choice. Am compartmental analysis: some bases for choice. Am J. Physiol. 1982;243:R1J. Physiol. 1982;243:R1--R6R6DG Covell et. al. Mean Residence Time. Math. DG Covell et. al. Mean Residence Time. Math. BiosciBiosci. 1984;72:213. 1984;72:213--24442444Jacquez, JA and SP Simon. Qualitative theory of Jacquez, JA and SP Simon. Qualitative theory of compartmental analysis. SIAM Review 1993;35:43compartmental analysis. SIAM Review 1993;35:43--7979Jacquez, JA. Jacquez, JA. Compartmental Analysis in Biology and Compartmental Analysis in Biology and MedicineMedicine. . BioMedwareBioMedware 1996. Ann Arbor, MI.1996. Ann Arbor, MI.Cobelli, C, D Foster and G Toffolo. Cobelli, C, D Foster and G Toffolo. Tracer Kinetics in Tracer Kinetics in Biomedical ResearchBiomedical Research. . KluwerKluwer Academic/Plenum Academic/Plenum Publishers. 2000, New York.Publishers. 2000, New York.

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