MULTI COMPARTMENT MODEL BIOPHARMACEUTICS..pdf

B.PHARMACY VI SEM

1. MULTI-COMPARTMENT
MODELS
M. BALASUNDARESAN,
ASSISTANT PROFESSOR,
ARUNAI COLLEGE OF PHARMACY,
TIRUVANNAMALAI.

2. •Ideallyatruepharmacokineticmodelshouldbetheonewitharateconstantfor
eachtissueundergoingequilibrium.
•Thereforebestapproachistopooltogethertissuesonthebasisofsimilarityin
theirdistributioncharacteristics.
•Thedrugdispositionoccursbyfirstorder.
•Multi-compartmentcharacteristicsarebestdescribedbyadministrationasi.v
bolusandobservingthemannerinwhichtheplasmaconcentrationdeclineswith
time.
Theno.Ofexponentialsrequiredtodescribesuchaplasmalevel-timeprofile
determinestheno.Ofkineticallyhomogeneouscompartmentsintowhicha
drugwilldistribute.
Thesimplestandcommonestisthetwocompartmentmodelwhichclassifiesthe
bodytissuesintwocategories:
1.Centralcompartmentorcompartment1
2.Peripheralortissuecompartmentorcompartment 2.

3. ADMINISTRATION:
Eliminationfromcentra
T
l
W
com
O
p
C
ar
O
tm
M
en
P
t
ARTMENTOPENMODEL-IVBOLUS
Fig:
•Aftertheivbolusofadrugthedeclineintheplasmaconc.Isbi-exponential.
•Twodispositionprocesses-distributionandelimination.
•ThesetwoprocessesareonlyevidentwhenasemilogplotofCvs.Tis
made.
•Initially,theconc.Ofdruginthecentralcompartmentdeclinesrapidly,due
tothedistributionofdrugfromthe centralcompartmenttotheperipheral
compartment.Thisiscalleddistributivephase.
1
Central
2
peripheral

4. ExtendingtherelationshipX=vdC
Dcc=K21xp–K12xc–KExc
Dtvp vc vc
X=Amt.Ofdruginthebodyatanytimetremainingtobeeliminated
C=drugconcinplasma
Vd =proportionalityconstapp.Volumeofdistribution
Xcandxp=amtofdruginC1andC2
Vcandvp=apparentvolumesofC1andC2
=K12xc–K21xp
Vc vp Onintegrationequationgivesconcofdrugincentraland
peripheralcompartmentsatanygiventimet
Cp=xo[(
Vc
K21–a)e
-at+(K12–b)e
-bt]
b–a a–b
Xo=ivbolusdose

5. •Therelationbetweenhybridandmicroconstantsisgivenas:
a+b= K12+K21+KE
Ab= K21KE
Cc=ae
-at+be
-bt
Cc=distributionexponent+elimination
exponent
AandBarehybridconstantsfortwoexponentsandcanberesolvedbygraph
bymethodofresiduals.
A=X0[K21-A]=CO[K21–A]
VC
B =X0
B –A
[K21-B]=
B –A
CO[K21–B]
VCA–B A–B
CO=Plasmadrugconcentrationimmediatelyafteri.v.Injection

6. •Methodofresiduals:thebiexponentialdispositioncurveobtainedafteri.V.
Bolusofadrugthatfitstwocompartmentmodelcanberesolvedintoits
individualexponentsbythemethodofresiduals.
C=ae
-at+be
-bt
Fromgraphtheinitialdeclineduetodistributionismorerapidthantheterminal
declineduetoeliminationi.E.Therateconstanta>>bandhencetheterme
-at
approacheszeromuchfasterthane
–bt
C=Be
-bt
LogC=logB–bt/2.303C=backextrapolatedpl.Conc.
•AsemilogplotofCvstyieldstheterminallinearphaseofthecurvehaving
slope–b/2.303andwhenbackextrapolatedtotimezero,yieldsy-interceptlog
B.Thet
1/2fortheeliminationphasecanbeobtainedfromequation
•t
1/2=0.693/b.
•Residualconcvaluescanbefoundas-
Cr=C–C=ae
-at
Logcr=logA–at
2.303
Asemilogplotcrvstgivesastraightline.

7. Ke=abc
Ab+ Ba
K12=ab(b-a)
2
C0(Ab +Ba)
K21=Ab+ Ba
C0
•Fortwocompartmentmodel,KEistherateconstantforeliminationofdrug
fromthecentralcompartmentandbistherateconstantforeliminationfrom
theentirebody.Overalleliminationt1/2canbecalculatedfromb.
Areaunder(auc)=a+b
Thecurve ab
X0=X0App.Volumeofcentral=
compartment C0KE(AUC)

8. App.Volumeof =VP=VCK12
PeripheralcompartmentK21
Apparentvolumeofdistributionatsteadystateorequilibrium
Vd,ss=VC+VP
Vd,area=X0
BAUC
Totalsystemicclearence=clt=bvd
Renalclearence=clr=dxu=KEVC
Dt
The rateof excretionof unchangeddruginurinecanberepresentedby:
dxu=KE Ae
-at+KEBe
-bt
Dt
Theaboveequationcanberesolvedintoindividualexponents bythemethodofresiduals.

9. TWO–COMPARTMENTOPENMODEL-I.V.
NFIUSION
Theplasmaorcentralcompartmentconcofadrugwhenadministeredasconstantrate(0order)i.V.Infusionis
givenas:
C =R0[1+(KE-b)e
-at+(KE-a)e
-bt]
VCKE b–a a-b
Atsteadystate(i.E.Attime infinity)thesecondandthethirdterminthebracketbecomeszeroandtheequation
reducesto:
Css=R0
Vcke
NowVCKE =vdb
Css= r0=r0
Vdbclt
TheloadingdoseX0,L=cssvc=R0
Ke
1
Central
2
Peripheral

10. TWO-COMPARTMENTOPENMODEL-
•
First-E
ordX
erT
absR
orpA
tioV
n:ASCULARADMINISTRATION
•Foradrugthatentersthebodybyafirst-orderabsorptionprocessand
distributedaccordingtotwo compartmentmodel,therateofchangeindrug
concinthecentralcompartmentisdescribedbythreeexponents:
•Anabsorptionexponent,andthetwousualexponentsthatdescribedrug
disposition.
Theplasmaconcatanytimetis
C=ne
-kat+le
-at+me
-bt
C=absorption+distribution+elimination
Exponentexponentexponent
•Besidesthemethodofresiduals,kacanalsobefoundbyloo-riegelmanmethod
fordrugthatfollowstwo-compartmentcharacteristics.
•Despiteitscomplexity,themethodcanbeappliedtodrugsthatdistributeinany
numberofcompartments.

11. CALCULATINGKausingWagner-
nelsonmethod(Bioavailability
parameters)

12. WAGNER-NELSONSMETHOD
THEORY:Theworkingequationscanbederivedfromthemassbalance
equation:Givesthefollowingeqautionwithtimeandmassbalance
•AboveequationIntegratinggives
• Totheequationamount
absorbedVERSUSTIME

13. WAGNER-NELSONSMETHOD
•Takingthistoinfinitywherecpequals0
•Finally(Amax-A),theamountremainingtobeabsorbedcanalsobe
expressedastheamountremainingintheGI,xg
•Wecanusethisequationtolookattheabsorptionprocess.If,andonlyif,
absorptionisasinglefirstorderprocess

14. WAGNER-NELSONSMETHOD
•Exampledataforthemethodofwagner-nelsonkel(fromIVdata)=0.2hr
-
Time
(hr)
Plasma
Concentratio
n
(mg/L)
Column
3
ΔAUC
Column
4
AUC
Column5
kel*AUC
A/V
[Col2+
Col5]
(Amax-A)/V
0.0 0.0 0.0 0.0 0.0 0.0 4.9
1.0 1.2 0.6 0.6 0.12 1.32 3.58
2.0 1.8 1.5 2.1 0.42 2.22 2.68
3.0 2.1 1.95 4.05 0.81 2.91 1.99
4.0 2.2 2.15 6.2 1.24 3.44 1.46
5.0 2.2 2.2 8.4 1.68 3.88 1.02
6.0 2.0 2.1 10.5 2.1 4.1 0.8
8.0 1.7 3.7 14.2 2.84 4.54 0.36
10.0 1.3 3.0 17.2 3.44 4.74 0.16
12.0 1.0 2.3 19.5 3.9 4.9 -
∞ 0.0 5.0 24.5 4.9 4.9 -

15. WAGNER-NELSONSMETHOD
•Thedata(Amax-A)/Vversustimecanbeplottedonsemi-logandlinear
graphpaper

16. WAGNER-NELSONSMETHOD
•Plotting(Amax-A)/Vversustimeproducesastraightlineonsemi-loggraphpaperanda
curvedlineonlineargraphpaper.Thiswouldsupporttheassumptionthatabsorptioncanbe
describedasasinglefirstprocess.Thefirst-orderabsorptionrateconstant,ka,canbe
calculatedtobe0.306hr
-1fromtheslopeofthelineonthesemi-loggraphpaper.
ADVANTAGES:
•Theabsorptionandeliminationprocessescanbequitesimilarandaccuratedeterminationsof
kacanstillbemade.
•Theabsorptionprocessdoesn'thavetobefirstorder.Thismethodcanbeusedtoinvestigate
theabsorptionprocess.
DISADVANTAGES:
•Themajordisadvantageofthismethodisthatyouneedtoknowtheeliminationrateconstant,
fromdatacollectedfollowingintravenousadministration.
•Therequiredcalculationsaremorecomplex.

17. RESIDUALMETHODOR
FEATHERINGTECHNIQUE
•Absowhenadrugisadministeredbyextravascularroute,absorptionisa
prerequisiteforitstherapeuticactivity.
•Theabsorptionrateconstantcanbecalculatedbythemethodof
residuals.
•Thetechniqueisalsoknownasfeathering,peelingandstripping.

18. φItiscommonlyusedinpharmacokineticstoresolvea
multiexponentialcurveintoitsindividualcomponents.
φForadrugthatfollowsone-compartmentkineticsand
administeredextravascularly,theconcentrationofdrug
inplasmaisexpressedbyabiexponentialequation.
C=
????????????�??????0
??????�(????????????−??????�)
[e
-K
E
t–e
-K
a
t] (1)
IfKaFX0/Vd(Ka-KE)=A,ahybridconstant,then:
C=Ae
-K
E
t–Ae
-K
a
t
(2)

19. φDuring
almost
theeliminationphase,whenabsorptionis
over,Ka<<KEandthevalueofsecond
exponentiale
-Katapproacheszerowhereasthefirst
exponentiale
-KEtretainssomefinitevalue.
φAtthistime,theequation(2)reducesto:
??????
−
=??????&#3627408466;
−
??????&#3627408440;??????(3)
φInlogform,theaboveequationis:
LogC
−
=logA-
??????&#3627408440;??????
2.303
(4)

20. Where,
C
−=backextrapolatedplasmaconcentrationvalues
φAplotoflogCversust yieldabiexponentialcurvewitha
terminallinearphasehavingslope–KE/2.303
φBackextrapolationofthisstraightlinetotimezeroyieldsy-
interceptequaltologA.
70

21. Plasmaconc.-Timeprofileafteroraladministrationofasingledoseofadrug

22. φSubtractionoftrueplasmaconcentrationvaluesi.e.
equation(2)fromtheextrapolatedplasma
valuesi.e.equation(3)yieldsaseries
concentration
ofresidual
concentrationvalueCτ.
(C
−-C)=Cτ=A e
-K
a
t
(5)
φIn logform,the equationis:
τ
2.303
log C =logA-
??????????????????
(6)

23. φAplotoflogCτversustyieldsastraightlinewithslope-
Ka/2.303andy-interceptlogA.
φThus,themethodofresidualenablesresolutionofthe
biexponentialplasmalevel-timecurveintoitstwo
exponentialcomponents.
φThetechniqueworksbestwhenthedifferencebetween
KaandKEislarge(Ka/KE≥3).