2.pharmacokinetics

One compartment kinetics

1. P.N.MALLIKARJUN
ASSOCIATE PROFESSOR
V I G N A NI N S T I T U T E O F P H A R M A C E U T I C A L T E C H N O L O G Y
PHARMACOKINETICS

2. Theone-compartmentopenmodelisthesimplestmodel.Owing
toitssimplicity,itisbasedonfollowingassumption
1)Thebodyisconsideredasasingle,kineticallyhomogeneous
unitthathasnobarrierstothemovementofdrug.
2)Finaldistributionequilibriumbetweenthedruginplasmaand
otherbodyfluid(i.e.mixing)isattainedinstantaneously&
maintainedatalltimes.Thismodelisfollowedbyonlythose
drugsthatdistributerapidlythroughoutthebody.
3)Drugsmovedynamically,in(absorption)&out(elimination)ofthis
compartment.
4)Eliminationisafirstorder(monoexponential)processwithfirst
orderrateconstant.
5)Rateofinput(absorption)>rateofoutput(elimination).
One Compartment Open Model
(Instantaneous DistributionModel)

3. One Compartment Open Model
Blood and
otherbody
tissues
Drug
One-compartmentopenmodelshowinginput
andoutputprocesses
K
a
Input
(Absorption)
Output
(Elimination)
Excretion
Metabolism
K
E

4. Dependingonrateofinput,severalone
compartment openmodelsare:
1.Onecompartmentopenmodel,i.v.
bolus administration
2.Onecompartmentopen model,
continuousi.v. infusion.
3.Onecompartmentopen model,e.v.
administration, zeroorder absorption.
4.Onecompartmentopen model,e.v.
administration, firstorder absorption

5. 1.One-compartment Open Model:
Intravenous Bolus Administration

6. Thedrugisrapidlydistributedinthebodywhen
givenintheformofintravenousinjection(i.v.
bolusorslug).Ittakesaboutonetothreeminutes
forcompletecirculation&thereforetherateof
absorptionisneglectedincalculation.
Blood and
otherbody
tissues
K
E

7. Thegeneralexpressionforrateofdrugpresentationtothebodyis:
whereK
E=first-ordereliminationrateconstant,and
X =amountof druginthebody atanytimetremainingto
beeliminated
Negativesignindicatesthatthedrugisbeinglostfromthe body.
dX
Ratein(absorption)-Rateout(elimination)
dt
If therateoutor eliminationfollowsfirst-orderkinetics,then:
dt
Sincerateinorabsorptionis absent,theequationbecomes:
dX
-Rateout
E
dt
dX
-KX

8. Integrationofequationyields:
In X=InX
0–K
Et
Where,X
0= amountofdrugattime
zero i.e. Dose ofthe druginjected.
Equation can also be written in the exponential form
as:
EX=X
0e
-Kt
This equation shows one compartment kinetics is
monoexponential.

9. Transforming equation into common logarithms (log base
10) weget:
Since it is difficult to determine directly the amount of drug in
the body X, advantage is taken of the fact that a constant
relationshipexistsbetweendrug concentrationinplasmaC and
X,thus
X=V
dC
where,V
d= proportionalitycostantpopularlyknownasthe
apparentvolumeofdistribution.
2.303
0
K
Et
logXlogX

10. Itisapharmacokineticparameterthatpermitstheuse
ofplasmadrugconcentrationinplaceoftotalamount
ofdruginthebodybyequationtherefore
becomes:
whereC
0=plasmadrugconcentrationimmediately
afteri.v.injection.
Equationisthatofastraightlineandindicatesthat
semilogarithmicplotoflogCvst,willbelinearwith
Y-interceptaslogC
0
2.303
E
0
Kt
logClogC-

11. Estimationofpharmacokinetic
parameters–IVbolusadministration
1)Elimination rateconstant
2)Eliminationhalflife
3)Apparent volume of Distribution
4)Total systemic Clearance

12. a)Cartesianplotofdrugthatfollowsone-compartmentkineticsandgivenbyrapid
injection
b)Semilogarithmicplotfortherateofeliminationinaone-compartmentmodel
EliminationRate Constant

13. EliminationHalf -Life
Eliminationhalflife:Alsocalledasbiologicalhalflife.
The time taken for the amount of drug in the body as well as
plasma concentrationtodeclineby50%itsinitial value.
Itisexpressedinhoursorminutes.
Half lifeexpressedbyfollowingequation:
The half –life is a secondary parameter that depends upon the
primaryparameterclearance&apparentvolumeofdistribution.
Accordingtofollowingequation:
E
K
t
1/2

0.693
T
Cl
t
1/2

0.693V
d

14. The two separate & independent pharmacokinetic
characteristics of a drugdistributionof adrug
.since,theyarecloselyrelatedwiththe
physiological mechanism of body, they are called
as primary parameters
1.Apparent volume of Distribution
2.Total systemic Clearance

15. ApparentVolumeofDistribution
The hypothetical volume of body fluids in which drug is completely
distributed or dissolved is called as Apparent volume of distribution.
Apparent volume of distribution is denoted by Vd
The units of Vdis litres
Modificationofequationdefinedapparentvolumeof distribution:
V
d
plasmadrugconcentration

C
Thebest andthe simplestwayofestimatingV
d
ofadrugis
administering it by rapid i.v. injection, determining the resulting
plasma concentrationimmediatelyby usingthe followingequation:
amountofdruginthebodyX
0 0
C C
V
d

X
0

i.v.bolusdose

16. Amoregeneral,amoreusefulnon-compartmentalmethod
that can be applied to many compartment models for
estimatingtheV
d
is:
Fordrugs givenas i.v.bolus,
X
0
=doseadministered
F = fractionof drugabsorbedintothe systemic circulation.
KAUC
E
Fordrugsadministeredextravascularly(e.v.),
X
0
V
d(area)
KAUC
E
FX
0
V
d(area)

17. Total systemic Clearance
Clearance:Clearanceisdefinedasthetheoreticalvolumeofbodyfluid
containing drug fromwhichthe drug iscompletelyremovedinagiven
periodof time.It is expressed in ml/minorlit/hr.
Clearanceisaparameterthatrelatesplasmadrugconcentrationwiththe
rateofdrugeliminationaccordingtofollowingequations-
plasmadrugconcentration
rateofelimination
clearance
Or
Cl
dx/dt
C
Clearanceisthemostimportantparameterinclinical drug
applications&isusefulinevaluatingthemechanismbywhichadrug
is eliminatedby thewhole organismsorbyaparticularorgan.

18. Thetotalbody clearance,Cl
T= Cl
R+Cl
H+Cl
other
Clearanceby allorgansotherthankidneyis sometimesknownas
nonrenalclearanceCl
NR
Itis thedifferencebetweentotalclearanceandrenalclearance
accordingtoearlierandefinition
SubstitutingdX/dt=K
EXinabove equ.weget
Since X/C =Vd,equation canbewrittenas
C

dx/dt
T
Cl
C

K
EX
T
Cl
Cl
TK
Ev
d

19. 1/2
t
T
Cl
0.693Vd
Parallelequationcanbewrittenforrenalandhepaticclearance
as:
Cl
R=KeV
d
Cl
H=KmV
d
Since, K
E=0.693/t1/2( from equa.11),clearancecanberelated
tohalf lifeby thefollowingequation

20. 2.One-compartment Open Model:
Intravenous Infusion
Rapidi.v.injectionisunsuitablewhenthedrughas
potentialtoprecipitatetoxicityorwhenmaintenance
ofastableconcentrationoramountofdruginthe
bodyisdesired.
Insuchasituation,thedrug(forexample,several
antibiotics,theophylline,procainamide,etc.)is
administeredataconstantrate(zero-order)byi.v.
infusion.
Incontrasttotheshortdurationofinfusionofani.v.
bolus(fewseconds),thedurationofconstantrate
infusionisusuallymuchlongerthanthehalf-lifeof
thedrug.

21. Advantagesofzero-orderinfusionofdrugsinclude—
•Easeofcontrolofrateofinfusiontofitindividual
patientneeds.
•Preventsfluctuatingmaximaandminima(peakand
valley)plasmalevel,desiredespeciallywhenthedrug
hasanarrowtherapeuticindex.
•Otherdrugs,electrolytesandnutrientscanbe
convenientlyadministeredsimultaneouslybythe
sameinfusionlineincriticallyillpatients.

22. One-compartment Open Model:
Intravenous Infusion

23. At the start of constant rate infusion, the amount of
drug in the body is zero and hence, there is no
elimination. As time passes, the amount of drug in the
body rises gradually (elimination rate less than the
rate of infusion) until a point after which the rate of
elimination equals the rate of infusion i.e. the
concentration of drug in plasma approaches a
constant value called as steady-state, plateau or
infusion equilibrium.

24. The time to reach steady-state concentration is
dependent upon the elimination half-life and not
infusion rate. An increase in infusion rate will merely
increase the plasma concentration attained at steady-
state . If n is the number of half-lives passed since
the start of infusion (t/t½),
For therapeutic purpose, more than 90%of the
steady-state drug concentration in the blood is
desired which is reached in 3.3 half-lives. It takes
6.6 half-lives for the concentration to reach 99%of
the steady-state. Thus, the shorter the half-life (e.g.
penicillin G, 30 min), sooner is the steady-state
reached.

27. InfusionPlusLoadingDose
Ittakesaverylongtimeforthedrugshavinglonger
half-livesbeforetheplateauconcentrationisreached
(e.g.phenobarbital,5days).Thus,initially,such
drugshavesubtherapeuticconcentrations.Thiscan
beovercomebyadministeringa
loadingdoselargeenoughtoyieldthedesired
steady-stateimmediatelyuponinjectionpriorto
startingtheinfusion.Itshouldthenbefollowed
immediatelybyi.v.infusionatarateenoughto
maintainthisconcentration

29. The equation describing the plasma concentration-time profile
following simultaneous i.v. loading dose (i.v. bolus) and constant rate
i.v. infusion
loading dose Xo,L
where, T = infusion time

30. One-compartment Open Model:
Extravascularadministration

31. When a drug is administered by extravascular route ,the rate of
absorptionmaybedescribedbymathematicallyasazeroorfirst
orderprocess.
A large number of plasma concentration time profile can be
describedbyaonecompartment modelwithfirst orderabsorption&
elimination.
Zeroorderabsorptionis characterizedbyaconstantrateof
absorption

32. Differencebetweenzero-orderand first-orderkinetics

33. Aftere.v.administration, therateinthe changeof amountof drug in
thebodydx/dtis differencebetweentherateofinput
(absorption) dx
ev/dtand rateofoutput(elimination)dx
E/dt.

34. Duringtheabsorptionphase,therateabsorptionis greaterthanthe
rateof elimination
Atpeak plasma concentration , the rate of absorption equals the
rateof eliminationand thechange inamountof druginthebodyis
zero
Eliminationphase.

dtdt
dx
evdx
E
dtdt
Theplasma leveltimecurveischaracterizedonly by the
dxdx
ev

E
dx
ev

dx
E
dtdt

36. •Zero order
absorption
•First order
absorption
One compartment open -
ExtravascularAdministration

37. 3.Zero-OrderAbsorptionModel
ExtravascularAdministration
Thismodelissimilartothatfor constantrateinfusion.
Allequation thatexplaintheplasmaconcentration–time
profileforconstantratei.v.infusionarealsoapplicable
to thismodel.
Blood and
otherbody
tissues
R
0
Zeroorder
absorption
K
E
Drugat
e.v.site
Elimination

38. Rate of Presentation of Drug to the body=Rate in –Rate out
Rate of absorption –Rate of Elimination
Zero order absorption-First order Elimination

39. 4.First orderAbsorptionModel
ExtravascularAdministration
A drug that enters the body by a first order absorption process gets
distributed in the body according to one -compartment kinetics and
is eliminated by a first -order process, the model can be depicted as
follows
Thedifferentialformofthedrugtheeque.(1)
Ka=firstorderabsorptionrateconstant
Xa=amountofdrugattheabsorptionsiteremainingtobe
absorbedi.e.ARA.
aa E
dt
dx
Kx-Kx
Blood and
otherbody
tissues
K
E
EliminationDrugat
e.v.site
K
a
Firstorder
absorption

40. Transforming intoconcentrationterms, theeque.Becomes
Where, F=fractionofdrugabsorbedsystemicallyaftere.v.
administration.
K
a-K
E
[e
k
E
t
e
k
a
t
]
Integrationofequation
X
K
aFX
0
K
aFX
0
[e
k
E
t
Vd(K
a-K
E)
e
k
a
t
]C

41. Estimationof PharmacokineticParameters
Extravascularadministration
C
maxandt
max:Atpeakplasmaconcentration,therateofabsorption
equalsrateofeliminationi.e.K
aX
a=K
EXandtherateofchangein
plasmadrugconcentrationdc/dt=zero.Differntiatingequation
On simplifying,theaboveeque.Becomes:
Convertingtologarithmicform,
kt
ea
k t
eE
E a
[K K]Zero
dc

K
aFX
0
dtVd(K
a-K
E)
-k
a
t
a
e
k
E
t
e
E
K K
KK
tt
a
2.303
a
E
2.303
logk-
E
logk-

42. Where t ist
max.Rearrangementofaboveeque. yield
:
C
maxcanbeobtainedbysubstitutingeque.(11)ineque(7),simplereque
forthesameis:
It hasbeenshown that atC
max,whenK
a= K
E,t
max=1/K
E.
Hencethe aboveeque. Further reduced to:
Since,FX
0/V
drepresent C
0.
a E
max
K-K

2.303log(K
a/K
E)
T
d d
max
V V

FX
0
e
1

0.37FX
0
C
d
max
V
C
FX
0
e
k
E
tmax

43. EliminationRate Constant
Theparametercanbe computedfromthe
eliminationphaseof the plasma leveltimeprofile.
KE = -2.303 X Slope of the Elimination phase
EliminationHalf Life

0.693
t1/2
KE

44. Absorption Constant Can Be Determined By Two Methods:
1.METHOD OF RESIDUALS
2.WAGNER NELSON METHOD
AbsorptionRate Constant

45. Method of Residuals
Themethodofresidualsis also called as feathering, peeling and stripping
For a drug that follow one compartment kinetics & administered e.v., the
concentrationofdruginplasma isexpressedbyabiexposnentialequation.

Duringtheeliminationphase,whenabsorptionisalmostover, Ka>KEvalueof
E
secondexponentiale-K
atiszero.Whereastheexponentiale-
K t
retainessomefinite
value.atthistimetheequationreducedto:
V
d(K
a-K
E)
K
aFX
0
e
-K
a
t
][e
K
Et
C
ifK
aFX
0/V
d(K
a-K
E)A,ahybridconstantthen:
E a
-kt -kt
C Ae-Ae
t
E
-K
CAe
Inlogform, theaboveequationis:

2.303
E
Kt
logClogA-


46. Where, C represents the back extrapolation plasma
concentrationvalue.
The slope of the terminal linear portion of the true plasma
curve gives elimination rate constant
a
-Kt
r
(CC)CAe
Substraction of true plasma conc. Values i.e. equationfrom
theextrapolatedplasmaconcentrationvaluesC
r


47. Inlogform,theequationis
AplotoflogCrvstyieldsastraightlinewith–Ka/2.303&y
interceptlogA
Insomeinstance,theKEobtainedafteri.v.bolusofsomeof
thedrugisverylargerthantheKaobtainedbyrecidualmethod
andKE/Ka>3,
2.303
K
at
r
logClogA-

48. flip-flop phenomenon since the
slopes of the two lines have
exchanged their meanings.
When KE/Ka ≥3.
No flip flop When Ka/KE ≥3
Lag time: is defined as the
time difference between
drug administration and
start of absorption.

49. Wagner Nelson Method

51. References
Biopharmaceutics&pharmacokineticsbyD.M.
Brahmankar,SunilB.Jaiswal
Aplliedbiopharmaceuticsandpharmacokineticsby
& pharmacokineticsby
&pharmacokineticsbyMilo
LeonSargel
Biopharmaceutics
Venkateswarlu
Biopharmaceutics
Gibaldi
Biopharmaceutics&clinicalpharmacokineticsan
introductionbyRobertE.Notaril