ONE COMPARTMENT OPEN MODEL

S.Sangeetha., M.PHARM., (Ph.d) Department of Pharmaceutics SRM College of Pharmacy SRM University 1 ONE COMPARTMENT OPEN MODEL (Instantaneous distribution model) ‐The body is considered as a single, kinetically homogeneous unit. -This model applies only to those drugs that distributes rapidly throughout the body. -Drugs move dynamically in an out of this compartment -Elimination is first order(monoexponential) process with first order rate constant. -Rate of input(absorption)> rate of output (elimination)

2 ¾Depending on rate of input, several one compartment open models are : 1. one compartment open model, i.v. bolus administration 2. 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 Out dt dX = - Rate Out KE =first order elimination rate constant dt X= amt of drug in body at any time t dX = - KEX remaining in the body dt 4 ¾Estimation of pharmacokinetic parameters Elimination rate constant:

dX = - KEX dt

Integrating above equation yields

ln 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 2.303 5 , KE =

6 Elimination half life: It is defined as time taken for the amount of drug in the body as well as plasma concentration to decline by ½ or 50 % its initial value. It is expressed in hrs or mins t1/2 = 0.693 KE

Half life is secondary parameter that depends upon the primary parameters clearance and volume of distribution according 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/C For drugs given as i.v. bolus ,

Vd = X0 KE AUC

For drugs administered extravascularly,

Vd = F X0 KE AUC where,X0 = dose administered F = fraction of drug absorbed in systemic circulation.

8 Clearance Clearance is defined as the theoritical volume of body fluid containing drug from which the drug is completely removed in a given period of time.

ClT = Rate of elimination Plasma drug concentration Cl = dX / dt (dx/dt = KE.X) C ClT = KE Vd ClT = X0 AUC

9 Total body clearance : It is estimated by dividing the rate of elimination by each organ with the concentration of drug presented to it. Renal clearance ClR = rate of elimination by kidney C Hepatic clearance ClH = rate of elimination by liver C Thus , 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 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 clears the 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 11 Intravenous Infusion: Rapid i.v. injection is unsuitable when the drug has potential to precipitate toxicity or when maintenance of a stable concentration or amount of the drug in body is desired. In such a situation , the drug is administered at a constant rate (zero ordered) by i.v. infusion. Advantages of zero order infusion of drug include 1. Ease of control of rate of infusion. 2. Prevents fluctuating maxima and minima plasma level. This is desired especially when the drug has a narrow therapeutic index. 3. Other drugs , electrolytes and nutrients can be conveniently administered simultaneously by the same infusion 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 between the zero order rate of drug infusion Ro and first order

rate 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 in to 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 XSS KEXSS = RO , Transforming to concentration terms and rearranging the equation ,

Css= Ro = Ro KEvd ClT Where , XSS and Css are the amt. of the drug in body and concentration of the drug in plasma at steady state respectively. The value of KE can be obtain from the slope of straight line obtained after a semi logarithmic plot log c vs. time.

By substituting the R0/ClT=CSS , -KEt C=CSS(1 – e )

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 Infusion plus loading dose: It takes very long time for the drugs having longer half lives 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 the infusion.

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

17 The equation for the plasma concentration time profile following 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 model ™ EXTRAVASCULAR ADMINISTRATION When drug is administered by extravascular route, absorption is prerequisite for its therapeutic activity. The rate of absorption may be described mathematically as zero-order or first-order process. After e.v. administration, the rate of change in the amount of drug in the body is given by dx = Rate of absorption – Rate of elimination dt dX = dXev -dxe dt dt dt

19 • During absorption phase, the rate of absorption is greater than elimination phase. dXev > dxe dt dt

• At peak plasma concentration, dXev = dxe dt dt

• During post absorption phase, dXev < dxe dt dt

20 ZERO-ORDER ABSORPTION MODEL

R0 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 MODEL Ka KE Drug Blood Excretion first order From equ. dX =dXev-dxe dt 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.,

FXK X = oa [ − ETK − ee − atK ] − KK Ea )(

22 ABSORPTION RATE CONSTANT This can be calculated by METHOD OF RESIDUALS. Method is also known as Feathering, stripping and peeling. Drug that folllows one- compartment kinetics and administered e.v. , the concentration of drug in plasma is expressed by biexponential equation:

Assuming A = Log Ka F X 0 Vd (Ka – KE)

C = A e-kEt – A e-Kat

23 During the elimination phase, when absorption is most over, Ka >>KE

C = A e-Ket In 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 - Ket 2.303

24 25 This method works best when difference between Ka KE is large (Ka/KE >3)

If KE/Ka > 3 , the terminal slope estimates Ka and not KE whereas the slope of residuals line gives Ke and not Ka.

This is called as flip-flop phenomenon since the slopes of 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- time plots and does not required assumption of zero or first- order absorption

After oral administration of single dose of drug at any given time ,the amount of 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 to infinite 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 References: 1. D.M. Brahmankar, compartment model in Biopharmaceutics and Pharmacokinetics, Vallabh prakashan, second editon, 2009; p:

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