1

ONE COMPARTMENT

OPEN MODEL

S.Sangeetha., M.PHARM., (Ph.d)Department of PharmaceuticsSRM College of PharmacySRM University

2

ONE COMPARTMENT OPEN MODEL(Instantaneous distribution model)

‐The body is considered as a single, kineticallyhomogeneous unit.-This model applies only to those drugs thatdistributes rapidly throughout the body.-Drugs move dynamically in an out of thiscompartment-Elimination is first order(monoexponential)process with first order rate constant.-Rate of input(absorption)> rate of output(elimination)

Depending on rate of input, several one compartment open models are :1. one compartment open model, i.v. bolus

administration2. one compartment open model , continuous i.v.

infusion .3. one compartment open model, e.v.

administration, zero order absorption.4. one compartment open model, e.v.

administration, first order absorption

3

INTRAVENOUS  BOLUS  ADMINISTRATION When  drug is given in the form of rapid i.v. injection it takes about  one to three Minutes for complete circulation and therefore the rate of absorption is neglected

dX = Rate In – Rate OutdtdX = - Rate OutdtdX = - KEX dt

KE =first order elimination rate constantX= amt of drug in body at any time t remaining in the body

4

Estimation of pharmacokinetic parametersElimination rate constant:

dX = - KEX dt

Integrating above equation  yieldsln X = ln X0 – KEt

where Xo = amount of drug at time t=0

Equation can be written in exponential form as

X= Xo e‐Ket  

Transforming equation into logarithm form we get,Log X =  Log X0   ‐ KEt

2.3035

6

, KE   =   

Elimination half life:It is defined as time taken for the amount of drug in thebody as well as plasma concentration to decline by ½ or 50% its initial value.It is expressed in hrs or minst1/2 = 0.693  

KE

Half life is secondary parameter that depends upon theprimary parameters clearance and volume of distributionaccording to following equation,t1/2 = 0.693 Vd

ClT

7

Apparent volume of distribution:Vd= amt of drug in the body/ plasma drug conc.or Vd = X/CFor drugs given as i.v. bolus ,Vd  =    X0

KEAUC

For drugs administered extravascularly,Vd =    F   X0

KEAUC          where,X0 = dose administeredF = fraction of drug absorbed in

systemic circulation.  

8

9

ClearanceClearance is defined as the theoritical volume of bodyfluid containing drug from which the drug is completely

removed in a given period of time.

ClT =         Rate of eliminationPlasma drug concentration

Cl =          dX / dt (dx/dt = KE.X)

C          ClT =          KE Vd   ClT =           X0

AUC       

Total body clearance :It is estimated by dividing the rate of elimination byeach organ with the concentration of drug presentedto it.Renal clearanceClR = rate of elimination by kidney

CHepatic clearanceClH = rate of elimination by liver

CThus , ClT is also called as total systemic clearance ,is an additive property of individual organ clearances.It is represented as:

ClT = ClR + ClH+ Cl others

10

11

ORGAN CLEARANCE• Rate of Elimination = Rate of Presentation – Rate of

by an organ to organ (input) exit from organ

Rate of elimination      =      Q. C in  ‐Q. C out(Rate of extraction)

Extraction   Ratio     ER =         ( Cin – Cout)Cin

ER is an index of how efficiently the eliminating organ clearsthe blood flowing through it of drug.Based on ER values drugs can be classified as:• Drugs with high ER = above 0.7• Drugs with intermediate ER = between 0.7‐ 0.3                                       • Drugs with low ER = below 0.3 

Intravenous Infusion:Rapid i.v. injection is unsuitable when the drug haspotential to precipitate toxicity or when maintenance of astable concentration or amount of the drug in body isdesired. In such a situation , the drug is administered at aconstant rate (zero ordered) by i.v. infusion.Advantages of zero order infusion of drug include1. Ease of control of rate of infusion.2. Prevents fluctuating maxima and minima plasma level.

This is desired especially when the drug has a narrowtherapeutic index.

3. Other drugs , electrolytes and nutrients can beconveniently administered simultaneously by the sameinfusion line in critically ill patients.

12

At any time during infusion , the rate of change in amt.of drug in the body , dx/dt is the difference betweenthe zero order rate of drug infusion Ro and first orderrate elimination , ‐KE x:dX = R0‐ KEX , by integrating the equation  ,dt

x = Ro (1-e-kE t) , kE

Since, x=vdc the above equation can be transformed into concentration terms ,

C= Ro (1-e-kE t) = Ro (1-e-kE t)kEvd clT

13

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

14

At steady state , the rate of change in amt. of drug in body is zero , hence zero=RO-KE XSSKEXSS = RO , Transforming to concentration terms and rearranging the equation ,Css= Ro = Ro

KEvd ClTWhere , XSS and Css are the amt. of the drug in bodyand concentration of the drug in plasma at steady staterespectively.The value of KE can be obtain from the slope of straightline obtained after a semi logarithmic plot log c vs. time.By substituting the R0/ClT=CSS ,C=CSS(1 – e-KEt)

15

Rearrangement yields

Transforming into log form.

A semilog plot of (Css – C)/Css versus t results in a straight line with slope –KE /2.303

16

17

Infusion plus loading dose:It takes very long time for the drugs having longer halflives before the steady state concentration is reached.

An i.v. loading dose is given to yield the desired steady-state immediately upon injection prior to starting theinfusion.

It should then be followed immediately by i.v. infusion at arate enough to maintain this concentration.

The equation for the plasma concentration time profilefollowing i.v. loading dose and constant rate i.v. infusion,

C =X0,L e-kEt + R0 (1- e-Ket)Vd KEVd

18

One Compartment open modelEXTRAVASCULAR ADMINISTRATION

When drug is administered by extravascular route,absorption is prerequisite for its therapeutic activity. Therate of absorption may be described mathematically aszero-order or first-order process.After e.v. administration, the rate of change in the amountof drug in the body is given by

dx = Rate of absorption – Rate of eliminationdt

dX = dXev - dxedt dt dt

19

• During absorption phase, the rate of absorption isgreater than elimination phase.

dXev > dxedt dt

• At peak plasma concentration,dXev = dxedt dt

• During post absorption phase,dXev < dxedt dt

20

ZERO-ORDER ABSORPTION MODELR0 KE

Drug Blood Excretion

This model similar to that of constant rate infusion and all equation which applies to it are applicable to this model.

21

FIRST-ORDER ABSORPTION MODELKa KE

Drug Blood Excretionfirst order

From equ. dX = dXev - dxedt dt dt

Differentiating above equ. We get,dX = Ka Xa – KEX, Ka= absorption rate const.dt Xa= amt of drug remaining

to be absorbed.Integrating above equ.,

X = [ ]tKTK

Ea

oa aE eeKK

FXK −− −− )(

22

ABSORPTION RATE CONSTANTThis can be calculated by METHOD OF RESIDUALS.Method is also known as Feathering, stripping andpeeling.Drug that folllows one- compartment kinetics and

administered e.v. , the concentration of drug in plasmais expressed by biexponential equation:

Assuming A = Log Ka F X 0Vd (Ka – KE)

C = A e-kEt – A e-Kat

23

During the elimination phase, when absorption is mostover, Ka >>KE

C = A e-KetIn log form above equation is

Log C = Log A - Ket 2.303

Where, C = back extraplotted plasma conc. Values.Substracting true plasma conc. From extraploted one,

log(C – C ) =Cδ = Log A - Ket2.303

24

25

This method works best when difference between KaKE is large (Ka/KE >3)

If KE/Ka > 3 , the terminal slope estimates Ka and notKE whereas the slope of residuals line gives Ke and notKa.

This is called as flip-flop phenomenon since the slopesof the two lines have exchanged their meanings.

26

Wagner Nelson Method for Estimation of Ka The method involves determination of ka from percent un absorbed- timeplots and does not required assumption of zero or first- order absorption

After oral administration of single dose of drug at any given time ,the amountof drug in the body X and the amount of drug eliminated from the body XE.Thus:

X=VdC ,

The total amount of drug absorbed into systemic circulation from time zero toinfinite can be given as :

Since at t = ∞, ,the above equation reduce to :

27

The fraction of drug absorbed at any time t is given as:

Percent drug unabsorbed at any time is therefore:

28

29

References:1. D.M. Brahmankar, compartment model inBiopharmaceutics and Pharmacokinetics, Vallabhprakashan, second editon, 2009; p: