ONE COMPARTMENT MODEL.pptx

ONE COMPARTMENT MODEL

1. Mrs. Arti Majumdar ONE COMPARTMENT MODEL

2. CONTENTS Introduction Types of one compartment open model: 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. ONE COMPARTMENT OPEN MODEL (Instantaneous Distribution Model) The time course of drug concentration determined after the administration can be satisfactorily explained by assuming the body as single well mixed compartment with first order disposition process. The term open indicates that the input and output are unidirectional. One – compartment open model is generally used to describe plasma levels following administration of a single dose of a drug.

4. ONE COMPARTMENT MODEL IS BASED UPON FOLLOWING ASSUMPTIONS The body is considered as a single kinetically homogenous unit that has no barriers to movement of drug. Applied for drugs that distributes rapidly throughout the body. Elimination is a first order process with first order rate constant. Drugs move dynamically in and out of the compartment. The rate of input is greater than rate of output. Any change in plasma drug concentration reflects proportional change in drug concentration throughout the body. However the model does not assume that the drug concentration in plasma is equal to that in other body tissues.

5. TYPES OF ONE COMPARTMENT OPEN MODEL Depending upon the rate of input, Following one compartment open models can be defined: 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

6. One-compartment open model Intravenous Bolus Administration When drug that distributes rapidly in the body is given in the form of a rapid intravenous injection, it takes about one to three minutes for complete circulation and therefore the rate of absorption is neglected in calculations. The model can be depicted as

7. The rate of drug presentation to the body is given by following expression dX / dt = Rate in (availability)- Rate out (elimination) (1) Since rate in or absorption is absent, the equation becomes dX / dt = -Rate out (2) If rate out or elimination follows first order kinetics then dX / dt = -K E X (3) Where K E = First order elimination rate constant and X= amount of drug in the body at any time t remaining to be eliminated Negative sign indicates that the drug is being lost from the body

8. Estimation of pharmacokinetic parameters –IV Bolus Administration For a drug that follows one compartment kinetics and administered as rapid IV injection, the decline in plasma drug concentration is only due to elimination of drug from the body and not due to distribution, the phase being called as elimination phase. Elimination phase can be characterized by 4 parameters- 1. Elimination rate constant 2. Apparent volume of distribution 3. Elimination half life 4. Clearance

9. Elimination rate constant (K E ) Elimination rate constant represents the fraction of drug removed per unit of time K E has a unit of reciprocal of time (e.g. minute-1, hour-1, and day-1) With first-order elimination, the rate of elimination is directly proportional to the serum drug concentration The equation for elimination rate dX / dt = -K E X …..3 now integrating this equation 3 lnX= ln X - K E t.........4 Where X0= amount of drug at time t = zero. Above equation can also be written in the following exponential format as X= X o e - KE t ………… 5 Above equation shows that disposition of drug in one compartment kinetics is monoexponential .

10. Elimination rate constant (K E ) Equation 4 can be written in common logarithm (log to the base 10 form) as logX = log X – K E t /2.303………..6 Since it is difficult to determine directly the amount of drug in the body X, advantage is taken of the fact that a constant relationship exists between drug concentration in plasma C and X thus X= Vd C………….7 Where Vd = proportionality constant popularly known as the apparent volume of distribution So equation 6 can be written as logC = log C – K E t /2.303………..8

11. One-compartment open model

12. Elimination half life (t 1/2 ) The elimination half life is sometimes called ‘‘biological half-life’’ of a drug The elimination half life is defined as the time taken for the amount of unchanged drug in the body as well as plasma concentration (h, min, day, etc.) to reduce to half (or 50%) of the initial amount of drug. The elimination half life is a secondary parameter that depends upon primary parameters clearance and volume of distribution. t 1/2 = 0.693 ………9 K E t 1/2 = 0.693V d ………10 Cl T

13. Apparent Volume of Distribution ( Vd ) Apparent volume of distribution may be defined as the hypothetical volume of body fluids into which a drug is distributed. The volume of distribution represents a volume that must be considered in estimating the amount of drug in the body from the concentration of drug found in the sampling compartment. In general, drug equilibrates rapidly in the body. When plasma or any other biologic compartment is sampled and analyzed for drug content, the results are usually reported in units of concentration instead of amount Each individual tissue in the body may contain a different concentration of drug due to differences in drug affinity for that tissue. Therefore, the amount of drug in a given location can be related to its concentration by a proportionality constant that reflects the volume of fluid in which the drug is dissolved.

14. Apparent Volume of Distribution ( Vd ) Drugs which binds selectively to plasma proteins, e.g. Warfarin have apparent volume of distribution smaller than their real volume of distribution Drugs which binds selectively to extravascular tissues, e.g. Chloroquines have apparent volume of distribution larger than their real volume of distribution. The Vd of such drugs is always greater than 42 L (Total body water) In order to determine the apparent volume of distribution of a drug, it is necessary to have plasma/serum concentration versus time data. ………11

15. Apparent Volume of Distribution ( Vd )

16. Clearance ( Cl ) Clearance is a measure of the removal of drug from the body. Plasma drug concentrations are affected by the rate at which drug is administered, the volume in which it distributes, and its clearance. A drug’s clearance and the volume of distribution determine its half life. It is the most important parameter in clinical drug applications and is useful in evaluating the mechanism by which a drug is eliminated by the whole organism or by a particular organ. For One compartment pharmacokinetics , clearance is calculated using: Cl = K/ Vd Cl = dX / dt C ………12 ………13

17. Clearance ( Cl ) Clearance (expressed as volume/time) describes the removal of drug from a volume of plasma in a given unit of time (drug loss from the body) Clearance does not indicate the amount of drug being removed. It indicates the volume of plasma (or blood) from which the drug is completely removed, or cleared, in a given time period. Figures in the following two slides represent two ways of thinking about drug clearance: In the first Figure, the amount of drug (the number of dots) decreases but fills the same volume, resulting in a lower concentration

18. Clearance ( Cl ) The amount of drug (the number of dots) decreases but fills the same volume, resulting in a lower concentration

19. Clearance ( Cl ) Another way of viewing the same decrease would be to calculate the volume that would be drug-free if the concentration were held constant as presented in the second Figure

20. Clearance ( Cl ) The most general definition of clearance is that it is ‘‘a proportionality constant describing the relationship between a substance’s rate of elimination (amount per unit time) at a given time and its corresponding concentration in an appropriate fluid at that time.’’ Clearance can also be defined as ‘‘the theoretical volume of blood (plasma or serum) or other biological fluids from which the drug is completely removed per unit time.’’ It is expressed as ml/min or ltr /hrs

21. Clearance ( Cl ) Drugs can be cleared from the body by different pathways, or organs, including hepatic biotransformation and renal and biliary excretion. Total body clearance of a drug is the sum of all the clearances by various mechanisms. Cl T = Cl R + Cl H + Cl others ………14 Cl T = K E X ………….15 C Cl T = 0.693Vd ………….16 T 1/2

22. ONE COMPARTMENT 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 drug in the body is desired. In such a situation, the drug (for example, several antibiotics, theophylline , procainamide , etc.) is administered at a constant rate (zero-order) by i.v . infusion. In contrast to the short duration of infusion of an i.v . bolus (few seconds), the duration of constant rate infusion is usually much longer than the half-life of the drug.

23. Advantages of zero-order infusion of drugs Ease of control of rate of infusion to fit individual patient needs. Prevents fluctuating maxima and minima (peak and valley) plasma level, desired especially when the drug has a narrow therapeutic index. Other drugs, electrolytes and nutrients can be conveniently administered simultaneously by the same infusion line in critically ill patients.

24. The model can be represented as: Ro= Zero order rate of drug infusion Ke = First order elimination rate constant

25. Estimation of pharmacokinetic parameters 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 / dt = R ‐ KeX .........1 (X=amount of drug in the body at any time t remaining to be eliminated.) Integration and rearrangement of above equation yields:- X=Ro/ Ke (1-e-Ket) …..2 As a constant relationship exists in between drug conc in plasma C & X. Thus: X= VdC Vd is a Proportionality constant(apparent volume of distribution)

26. Rearranging the eqn we get:- C= Ro/ KeVd (1-e-Ket)…..3 C = Ro/ Clt (1-e-Ket)…..4 The total body clearance, Clt , also called as total systemic clearance, is an additive property of individual organ clearances.

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

28. Plasma concentration-time profile for a drug given by constant rate i.v . infusion (the two curves indicate different infusion rates Ro and 2Ro for the same drug) 

29. AT STEADY STATE At steady state, the rate of change of amount of drug in the body is zero. So eqn 1 becomes:- Zero=Ro- KeXss KeXss =Ro …….5 Css =Ro/ Kevd ….6 Css =Ro/ Clt i.e infusion rate/clearance ….7 Xss = amount of drug in the body at steady state. Css = amount of drug in plasma at steady state.

30. Substituting the value ofCss =Ro/ Clt in eqn 4: C= Css (1-e-ket)…..8 Rearrangement yields: [ Css -c] =e- Ket Css log C SS -C = - Ket …….9 Css 2.303

31. The time to reach steady state concentration is dependent upon the elimination half life . If n is the number of half-lives passed since the start of infusion, then the eqn 7 can be written as:- C=C SS [1-(1/2)n]………10

32. It takes very long time for the drugs having longer half lives before the steady state concentration is reached.( Eg :- Phenobarbital) 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.

33. So the equation for computing the loading dose Xo,L can be given: As X = VdC , the loading dose Xo,L can be given: X 0,L = C ss V d …..11 Substitution of Css = Ro/ K E V d from equation 6 in above equation yields another expression for loading dose in terms of infusion rate: X 0,L = Ro/K E …….12 The equation describing the plasma concentration-time profile following simultaneous i.v . loading dose ( i.v . bolus) and constant rate i.v . infusion is the sum of two equations describing each process …….13

34. If we substitute CssVd for Xo,L and CssKEVd for Ro in above equation and simplify it, it reduces to C = Css …..14 indicating that the concentration of drug in plasma remains constant (steady) throughout the infusion time.

35. EXTRAVASCULAR ADMINISTRATION When a drug is administered by extravascular route (e.g. oral, i.m ., rectal, etc.), absorption is a prerequisite for its therapeutic activity. The rate of absorption may be described mathematically as a zero-order or first-order process. A large number of plasma concentration- time profiles can be described by a one- compartment model with first-order absorption and elimination. However, under certain conditions, the absorption of some drugs may be better described by assuming zero-order (constant rate) kinetics.

36. Distinction between zero-order and first-order absorption processes. Figure a is regular plot, and Figure b a semilog plot of amount of drug remaining to be absorbed (ARA) versus time t.

37. Zero-order absorption is characterized by a constant rate of absorption. It is independent of amount remaining to be absorbed (ARA), and its regular ARA versus t plot is linear with slope equal to rate of absorption while the semilog plot is described by an ever-increasing gradient with time. In contrast, the first-order absorption process is distinguished by a decline in the rate with ARA i.e. absorption rate is dependent upon ARA; its regular plot is curvilinear and semilog plot a straight line with absorption rate constant as its slope.

38. After e.v . administration, the rate of change in the amount of drug in the body dX / dt is the difference between the rate of input (absorption) dXev / dt and rate of output (elimination) dXE / dt . dX / dt = Rate of absorption – Rate of elimination

39. For a drug that follows one-compartment kinetics, the plasma concentration-time profile is characterized by absorption phase, post-absorption phase and elimination phase.

40. During the absorption phase, the rate absorption is greater than the rate of elimination At peak plasma concentration , the rate of absorption equals the rate of elimination and the change in amount of drug in the body is zero Post absorption is characterized by After completion of drug absorption, its rate becomes zero and the plasma level time curve is characterized only by the elimination phase.

41. Zero-Order Absorption Model This model is similar to that for constant rate infusion. All equation that explain the plasma concentration – time profile for constant rate i.v . infusion are also applicable to this model.

42. First order Absorption Model Extravascular Administration 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

43. Integration of eque . (5) gives Transforming in to concentration terms, the eque . becomes Where, F= fraction of drug absorbed systemically after e.v . administration .

44. REFERENCES Brahmankar D. M., Jaiswal S.B., Biopharmaceutics & Pharmacokinetics A Treatise, Second Edition 2009 Published by Vallabh Prakashan . Milo Glibaldi Biopharmaceutics and Clinical Pharmaceutics, Reprint 2006 , Fourth Edition published by Pharma book syndicate, Hyderabad