multicompartment models: intravenous bolus …

MULTICOMPARTMENTMODELS:INTRAVENOUSBOLUSADMINISTRATION

Dr.QutaibaAhmedAlAgaAssistantProfessorFacultyofPharmacy

PhiladelphiaUniversity-Jordan

LearningOutcomes

Attheendofthislessonstudentswillbeableto•  Understandthecompartmentalmodelingandit’ssignificance•  Understanddrugabsorption,distributionandelimination•  Understanddrugclearanceincluding(total,renalandhepatic

clearance)•  UnderstandpharmacokineticsandbiopharmaceuticsafterI.V

bolus,I.Vinfusion,and•  oraladministrationofdrugs.•  Understandproteinbindinganditseffects•  Understandbioavailabilityandbioequivalence•  UnderstandMultipledosageregimen•  Haveaknowledgeonbiopharmaceuticsconsiderationsin

dosageformdesign

TWO-COMPARTMENTOPENMODEL

•  Many drugs given in a single intravenous bolusdosedemonstrateaplasmalevel-timecurvethatdoes not decline as a single exponential (first-order)process.Theplasmalevel-timecurveforadrug that follows a two-compartment modelshows that the plasma drug concentrationdeclinesbiexponentially as the sumof two first-orderprocesses-distributionandelimination.

•  A drug that follows the pharmacokinetics of a two-compartment model does not equilibrate rapidlythroughout the body, as is assumed for a one-compartmentmodel.

•  The central compartment represents the blood,extracellularfluid,andhighlyperfusedtissues.

•  The drug distributes rapidly and uniformly in thecentralcompartment.

•  A second compartment, known as the tissue orperipheral compartment, contains tissues inwhichthedrugequilibratesmoreslowly.

•  The drug in the tissues that have the highest bloodperfusion equilibrates rapidly with the drug in theplasma.Thesehighlyperfusedtissuesandbloodmakeup the central compartment. While this initial drugdistribution is taking place, multicompartment drugsare delivered concurrently to one ormore peripheralcompartments composed of groups of tissues withlower blood perfusion and different affinity for thedrug.

•  Adrugwill concentrate ina tissue inaccordancewiththe affinity of the drug for that particular tissue. Forexample, lipid-solubledrugs tend toaccumulate in fattissues.

•  Drugs that bind plasma proteins may be moreconcentratedintheplasma,becauseprotein-bounddrugs do not diffuse easily into the tissues. Drugsmay also bind with tissue proteins and othermacromolecules,suchasDNAandmelanin.

•  Tissuesamplingisinvasive,andthedrugconcentrationin the tissue sample may not represent the drugconcentrationintheentireorgan.

•  GeneralGroupingofTissuesAccordingtoperfusion

Highlyperfused Slowlyperfused

Heart,brain,hepatic-portalsystem,kidney,andendocrineglands

Bone,ligaments,tendons,cartilage,teeth,andhair

•  Drug transfer between the two compartments isassumedtotakeplacebyfirst-orderprocesses.

Plasmalevel-timecurveforthetwo-compartmentopenmodel(singleIVdose)describedin(modelA).

Thereareseveralpossibletwo-compartmentmodels•  ModelA isusedmostoftenanddescribes theplasma level-time curve observed in the diagram below. By convention,compartment 1 is the central compartment andcompartment 2 is the tissue compartment. The rateconstantsk12andk21representthefirst-orderratetransferconstantsforthemovementofdrugfromcompartment1tocompartment 2 (k 12) and from compartment 2 tocompartment1 (k 21).Thetransferconstantsaresometimestermed microconstants , and their values cannot beestimateddirectly.

•  Model B assume that elimination occurs from theperipheralcompartmentmodel,asshownin(modelB)

•  Model C assume that elimination occurs from thecentral andperipheral compartmentmodel, as shownin(modelC)

•  The plasma level-time curve for a drug that follows atwo-compartment model may be divided into twoparts, (a) a distribution phase and (b) an eliminationphase.Thetwo-compartmentmodelassumesthat,att=0,nodrug is in the tissuecompartment.Afteran IVbolus injection,drugequilibratesrapidly inthecentralcompartment.

•  The distribution phase of the curve represents the initial,more rapid decline of drug from the central compartmentintothetissuecompartment(linea).

•  Although drug elimination and distribution occurconcurrently during the distribution phase, there is a nettransferofdrugfromthecentralcompartmenttothetissuecompartment.

•  The fraction of drug in the tissue compartment during thedistribution phase increases up to a maximum in a giventissue,whosevaluemaybegreateror less than theplasmadrugconcentration.

•  At maximum tissue concentrations, the rate of drug entryintothetissueequalstherateofdrugexitfromthetissue.

Thefractionofdruginthetissuecompartmentisnowinequilibrium(distributionequilibrium)withthefractionofdruginthecentralcompartment,andthedrugconcentrationsinboththecentralandtissuecompartmentsdeclineinparallelandmoreslowlycomparedtothedistributionphase.Thisdeclineisafirst-orderprocessandiscalledtheeliminationphaseorthebetaphase(lineb).

Sinceplasmaandtissueconcentrationsdeclineinparallel,plasmadrugconcentrationsprovidesomeindicationoftheconcentrationofdruginthetissue.Atthispoint,drugkineticsappearstofollowaone-compartmentmodelinwhichdrugeliminationisafirst-orderprocessdescribedbyb(alsoknownasbeta).

•  Themethodof residuals (also knownas featheringorpeeling)isausefulprocedureforfittingacurvetotheexperimental data of a drugwhen the drug does notclearlyfollowaone-compartmentmodel.Forexample,100 mg of a drug was administered by rapid IVinjectiontoa70-kg,healthyadultmale.Bloodsampleswere taken periodically after the administration ofdrug, and the plasma fraction of each sample wasassayedfordrug.Thefollowingdatawereobtained:

Time(hr) Plasma concentration (µg/ml)

0.25 43.00

0.5 32.00

1.0 20.00

1.5 14.00

2.0 11.00

4.0 6.50

8.0 2.80

12.0 1.20

16.0 0.52

•  Whenthesedataareplottedonsemilogarithmicgraphpaper, a curved line is observed. The curved-linerelationship between the logarithm of the plasmaconcentration and time indicates that the drug isdistributedinmorethanonecompartment.

•  Asshowninthebiexponentialcurve,thedeclineintheinitial distribution phase is more rapid than theelimination phase. The rapid distribution phase isconfirmed with the constant a being larger than therateconstantb.Therefore,atsomelatertimethetermA.e-atatwillapproachzero,whileB.e-btwillstillhaveavalue.AtthislatertimeEquationwillreduceto:

Which,incommonlogarithms,is

•  Therateconstantcanbeobtainedfromtheslope(-b/2.3)of a straight line representing the terminal exponentialphase. The t 1/2 for the elimination phase (beta half-life)canbederivedfromthefollowingrelationship:

•  Inthesamplecaseconsideredhere,bwasfoundtobe0.21hr-1. From this information the regression line fortheterminalexponentialorbphase isextrapolatedtothey-axis;theyinterceptisequaltoB,or15g/mL.

•  Values fromtheextrapolated lineare thensubtractedfrom the original experimental data points and astraightlineisobtained.Thislinerepresentstherapidlydistributedaphase.

ApplicationofthemethodofResidual

Time(hr) Cpobservedplasmalevel

Ć’pextrapolatedplasmaconcentration

Cp-Ć’presidualplasmaconcentration

0.25 43.0 14.5 28.5

0.5 32.0 13.5 18.5

1.0 20.0 12.3 7.7

1.5 14.0 11.0 3.0

2.0 11.0 10.0 1.0

4.0 6.5

8.0 2.8

12.0 1.2

16.0 0.52

•  Thenewlineobtainedbygraphingthelogarithmoftheresidual plasma concentration (C p-C' p) against timerepresentstheaphase.Thevaluefora is1.8hr-1,andthe y intercept is 45 g/mL. The elimination t 1/2b iscomputedfrombandhasthevalueof3.3hr.

•  Atanygiventime,theplasmaconcentrationcanbecalculatedby:

•  A number of pharmacokinetic parameters may bederivedbypropersubstitutionofrateconstantsaandbandyinterceptsAandBintothefollowingequations:

•  k 12 and k 21 are first-order rate constants thatgovern the rate of drug change in and out of thetissues.

ApparentVolumesofDistributionVOLUMEOFTHECENTRALCOMPARTMENT

Thevolumeof thecentralcompartment isuseful fordetermining thedrug concentrationdirectly after anIV injection into the body. In clinical pharmacy, thisvolumeisalsoreferredtoasViortheinitialvolumeofdistributionasthedrugdistributeswithintheplasmaand other accessible body fluids. This volume isgenerally smaller than the terminal volume ofdistribution after drug distribution to tissue iscompleted.

•  Atzerotime(t=0),allofthedruginthebodyisinthecentralcompartment.C0pcanbeshowntobeequaltoA+Bbythefollowingequation

•  Att=0,e0=1.Therefore

APPARENTVOLUMEOFDISTRIBUTIONATSTEADYSTATEAtsteady-stateconditions,therateofdrugentryintothetissue compartment from the central compartment isequal to the rate of drug exit from the tissuecompartmentintothecentralcompartment.Thistermisusedforthecalculationoftheloadingdose

VOLUMEOFDISTRIBUTIONBYAREAThistermisusedforthecalculationoftheclearance

Questions

•  1.Doyouagreewiththefollowingstatementsforadrugthatisdescribedbyatwo-compartmentpharmacokineticmodel?AtpeakCt,thedrugiswellequilibratedbetweentheplasmaandthetissuecompartment,Cp=Ct,andtheratesofdrugdiffusionintoandfromtheplasmacompartmentareequal.

•  2.WhathappensafterpeakCt?•  3.Whyisaloadingdoseused?•  4.WhatisVi?HowisthisvolumerelatedtoVp?•  5.Whatpopulationfactorscouldaffecttheconcentrationofazithromycin?

FREQUENTLYASKEDQUESTIONS•  Whatistheapparentvolumeofdistribution,andwhyaretheresomanydifferentvolumesofdistribution?

Apparentvolumesofdistributionarenotrealtissuevolumes,butratherreflectthevolumeinwhichthedrugiscontained.Forexample,Vp=initialorplasmavolumeVt=tissuevolume(VD)SS=steady-statevolumeofdistribution(mostoftenlistedintheliterature)thesteadystatedrugconcentrationmultipliedby(VD).SSyieldstheamountofdruginthebody.(VD)isavolumeusuallydeterminedfromareaunderthecurve(AUC),anddiffersfrom(VD)SSsomewhatinmagnitude.(VD)multipliedbybgivesclearanceofthedrug.

•  Ifphysiologicmodelsarebetterthancompartmentmodels,whynotjustusephysiologicmodels?

Aphysiologicmodelisadetailedrepresentationofdrugdispositioninthebody.Themodelrequiresbloodflow,extractionratio,andspecifictissueandorgansize.Thisinformationisnotoftenavailablefortheindividual.Thus,thelesssophisticatedcompartmentmodelsareusedmoreoften.

•  CanIjustlearnclearanceandforgetabouttheotherpharmacokineticparameters,becauseclearanceisthetermmostoftenusedinclinicalpharmacy?

Clearanceisusedtocalculatethesteady-statedrugconcentrationandtocalculatethemaintenancedose.However,clearancealoneisnotusefulindeterminingthemaximumandminimumdrugconcentrationsinamultiple-dosingregimen.

•  WhatistheerrorifIassumeaone-compartmentmodelinsteadofatwo-compartmentormulticompartmentmodel?

Ifthetwo-compartmentmodelisignoredandthedataaretreatedasaone-compartmentmodel,theestimatedvaluesforthepharmacokineticparametersaredistorted.Forexample,duringthedistributivephase,thedrugdeclinesrapidlyaccordingtodistributionhalf-life,whileintheelimination(terminal)partofthecurve,thedrugdeclinesaccordingtobeliminationhalf-life.

•  Whatkindofimprovementintermsofpatientcareordrugtherapywasmadeusingthecompartmentmodel?

Compartmentmodelshavebeenusedtodevelopdosageregimensandforthedevelopmentofpharmacodynamicmodels.Compartmentmodelshaveimprovedthedosingofdrugssuchasdigoxin,gentamicin,lidocaine,andmanyothers.Theprincipaluseofcompartmentmodelsindosingistosimulateaplasmadrugconcentrationprofilebasedonpharmacokinetic(PK)parameters.ThisinformationallowscomparisonofPKparametersinpatientswithonlytwoorthreepointstoapatientwithfullprofilesusinggeneratedPKparameters.

•  H.W.•  AfterasingleIVbolusdoseof1000mgofanantiarrhythmicdrug,thefollowingconcentrationswereobtained:

a.Usingthemethodofresiduals,calculatethefollowingparameters:t1/2α,t1/2β,k,k12,k21,C0pandVp.b.Whatwillbetheamountofdrugremaininginthebodyafter15h?Assumingthe(VD)ss=10liter

RecommendedBooks

ü  Applied Biopharmaceutics and Pharmacokinetics., Shargel and A.B.C. Yu., Appleton & Lange/MacGraw-Hill, New York., 4th edition 1999. ISBN 0-8385-0129-X

ü  Applies clinical pharmacokinetics, Bauer, Larry A. Appleton & Lange/MacGraw-Hill, New York., 2nd edition 2008. 10.1036/0071476288

ü  ClinicalPharmacokineticsConceptsandApplications.MALCOlMROWlANDandTHOMASN.TOZER.,1994,3rdedition.LIpPINCOTTWILLIAMS&WILKINS