Noncompartmental vs. Compartmental Approaches to...
Transcript of 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 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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A Drug In The Body:A Drug In The Body:Constantly Undergoing ChangeConstantly Undergoing Change
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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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A Drug In The Body:A Drug In The Body:Constantly Undergoing ChangeConstantly Undergoing Change
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A Drug In The Body:A Drug In The Body:Constantly Undergoing ChangeConstantly Undergoing Change
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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
∫∫
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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
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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.
∫∫∫∫∞∞
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Estimating AUC And AUMC Using Estimating AUC And AUMC Using Sums Of ExponentialsSums Of Exponentials
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00dt)t(Cdt)t(Cdt)t(Cdt)t(CAUC
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tn
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Bolus IV InjectionBolus IV InjectionFormulas can be extended to other administration modesFormulas can be extended to other administration modes
∫∞
λ++
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∫∞
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λ=⋅=
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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.
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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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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 −−
−
− −+
⎟⎟⎠
⎞⎜⎜⎝
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)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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Extrapolating From Extrapolating From ttnn To InfinityTo Infinity
∫∞
− 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
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tzn
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Extrapolating From Extrapolating From ttnn To InfinityTo Infinity
Extrapolating function crucialExtrapolating function crucial
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Estimating The IntegralsEstimating The Integrals
To estimate the integrals, one sums up the individual components.
∫∫∫∫∞∞
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00dt)t(Cdt)t(Cdt)t(Cdt)t(CAUC
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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
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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
75
System Model with ExperimentSystem Model with Experiment
SAAM II software system, http://depts.washington.edu/saam2
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System Model with ExperimentSystem Model with Experiment
SAAM II software system, http://depts.washington.edu/saam2
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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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SAAM II software system, http://depts.washington.edu/saam2
91
SAAM II software system, http://depts.washington.edu/saam2
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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.