One compartment model intro

Introduction to One compartment model and details of One compartment open model IV infusion.

1. O ne compartment open model (Instantaneous Distribution Model)

2. One compartment 2

3. One compartment 3

4. More than one compartment 4

5. More than one compartment 5

6. Assumptions The one-compartment open model is the simplest model. Owing to its simplicity, it is based on following assumptions- The body is considered as a single, kinetically homogeneous unit that has no barriers to the movement of drug Final distribution equilibrium between the drug in plasma and other body fluids (i.e. mixing) is attained instantaneously and maintained at all times. This model thus applies only to those drug that distribute rapidly throughout the body Drugs move dynamically, in (absorption) and out (elimination) of this compartment Elimination is a first order (monoexponential) process with first order rate constant 6

7. Rate of input (absorption)> rate of output(elimination) The anatomical reference compartment is plasma and concentration of drug in plasma is representative of drug concentration in all body tissues ie . Any change in plasma drug concentration reflects a 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 7

8. One compartment: 8

9. The term open indicates that the input(availability) and output (elimination) are unidirectional and that the drug can be eliminated from the body. One – compartment open model is generally used to describe plasma levels following administration of a single dose of a drug. Blood and other Body tissues Drug Ka Input (absorption) Ke output (Elimination ) Metabolism Excretion 9

10. Depending upon the rate of input, Following one compartment open models can be defined: One –compartment open model, I. V. bolus administration One –compartment open model, continuous I.V. infusion One-compartment open model, E.V. Administration, zero order absorption One compartment open model E.V. Administration, first order absorption 10

11. 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 Blood and other Body tissues Ke 11

12. The general expression for rate of drug presentation to the body is dX / dt = Rate in (availability)- Rate out (elimination) (1.1) Since rate in or absorption is absent, the equation becomes dX / dt = -Rate out (1.2) If rate out or elimination follows first order kinetics then dX / dt = -K E X (1.3) 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 12

13. 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- Elimination rate constant Apparent volume of distribution Elimination half life Clearance 13

14. 14 Elimination rate constant ( K E ) Elimination rate constant represents the fraction of drug removed per unit of time K 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

15. Elimination rate constant The equation for elimination rate is dX / dt = -K E X , now integrating this equation lnX = ln X - K E t (1.4) Where X = amount of drug at time t = zero Above equation can also be written in the following monoexponential format as X= X e - K e t 15

16. Above equation we can write in the log to the base 10 form as logX = log X – K E t /2.303 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= V d C Where V d = proportionality constant popularly known as the apparent volume of distribution 16

17. 17 One compartment open model Drug Conc (C) Time log (C) Time logX = log X – K E t /2.303 X= X e - K e t

18. 18 Apparent Volume of Distribution ( Vd )

19. Apparent volume of distribution may be defined as the hypothetical volume of body fluids into which a drug is distributed. 19

20. 20 Apparent Volume of Distribution ( Vd ) 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 the drug is dissolved in

21. 21 The real Volume of Distribution has physiological meaning and is related to body water Plasma Interstitial fluid Total body water 42 L Intracellular fluid Plasma volume 4 L Interstitial fluid volume 10 L Intracellular fluid volume 28 L

22. 22 Apparent Volume of Distribution 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 V d of such drugs is always greater than 42 L (Total body water)

23. 23 Apparent Volume of Distribution Lipid solubility of drug Degree of plasma protein binding Affinity for different tissue proteins Fat : lean body mass Disease like Congestive Heart Failure (CHF), uremia, cirrhosis

24. 24 Apparent Volume of Distribution: Mathematics In order to determine the apparent volume of distribution of a drug, it is necessary to have plasma/serum concentration versus time data

25. 25 The Extent of Distribution and V d in a 70 kg Normal Man V d , L % Body Weight Extent of Distribution Examples with volume of distribution in litre 5, low 7 Only in plasma Warfarin-7, 5-20, medium 7-28 In extracellular fluids ibuprofen-10 20-40, High 28-56 In total body fluids. Theophylline -50 >40, very high >56 In deep tissues; bound to peripheral tissues Ranitidine-500, chloroquine-15000

26. Significance of V d It simply indicates how widely the drug is distributed in the tissues compared to plasma For example Vd of paracetamol is 0.950 l/kg body weight It means that 0.950 l of tissue is expected to contain the same concentration of paracetamol as that contained in the blood on the basis of average kg body weight. It does not mean that the remaining tissue contains zero drug concentration. It is conceptually assumed and expressed in this manner. 26

27. Continued…… Higher the Vd of a drug, more extensive is its distribution in the tissue If the plasma drug concentration is low, it can be inferred that the Vd is higher for a given dose If Vd is small then the drug concentration is more in plasma and less distributed in tissue. If Vd is 100% of body weight, then it may be assumed that the drug is concentration in certain tissue compartments If a drug is restricted to the vascular spaces and can freely penetrate erythrocytes, the drug has a volume of distribution of 6 litre. If the drug cannot permeate the RBC’s the available space is reduced to about 3 litre 27

28. 28 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 (h, min, day, etc.) at which the mass (or amount) of unchanged drug becomes half ( or 50%) of the initial mass of drug

29. Increased physiological understanding of pharmacokinetics shows that half life is a parameter that depends upon the primary parameters clearance and apparent volume of distribution, according to following equation 29

30. 30 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

31. 31 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 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 resented in the second Figure

32. 32 Clearance ( Cl ) the amount of drug (the number of dots) decreases but fills the same volume, resulting in a lower concentration

33. 33 Clearance (Cl)

34. 34 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 hypothetical volume of blood (plasma or serum) or other biological fluids from which the drug is totally and irreversibly removed per unit time.’’

35. 35 Clearance (Cl) estimation For One compartment pharmacokinetics , clearance is calculated using:

36. 36 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.

37. 37 Elimination rate The elimination rate at any time can be calculated using: Elimination rate = K*X(t) OR Elimination rate = Cl*C(t) where X(t) is the amount of drug in the body at time t, C(t) is the concntration of drug at time t

38. One –compartment open model, continuous I.V. Infusion 38

39. IV infusion is administered 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 eg . Theophylline, procainamide, antibiotics etc is administered at a constant rate(zero order) by IV infusion. Advantages of zero order infusion of drugs include: Ease of control of rate of infusion to fit individual patient needs Prevents fluctuating maxima and minima plasma level Other drugs, electrolytes and nutrients can be conveniently administered simultaneously by the same infusion line in critically ill patients 39

40. One compartment open model: Intravenous infusion- Model can be represent as : ( i.v infusion) Drug dX / dt =R o -K E X … eq 23 X=R o /K E (1-e -KEt ) … eq 24 Since X= V d C C=R o / K E Vd (1-e -KEt ) … eq 25 =R o / Cl T (1-e -KEt ) … eq 26 40 Blood & other Body tissues R Zero order Infusion rate K E

41. At steady state. The rate of change of amount of drug in the body is zero , eq 23 becomes Zero=R o -K E X SS …27 K E X SS =R o …28 C SS =R o / K E V d …29 =R o / Cl T i.e infusion rate ....30 clearance Substituting eq. 30 in eq. 26 C=C SS (1-e -K E t ) …31 Rearrangement yields: [C SS -C] =e - K E t . ...32 C SS log C SS -C = - K E t …33 C SS 2.303 41

42. 42 Increasing the Infusion Rate If a drug is given at a more rapid infusion rate, a higher SS drug concentration is obtained but the time to reach SS is the same.

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45. 45 Steady State Concentration (C ss ) Rate of Infusion = Rate of Elimination  The infusion rate (R) is fixed while the rate of elimination steadily increases  The time to reach SS is directly proportional to the half-life  After one half-life, the C p is 50% of the C SS , after 2 half-lives, C p is 75% of the C ss …….

46. If n is the no. of half lives passed since the start of infusion(t/t 1/2 ) Eq. can be written as C=C SS [1-(1/2) n ] …34 46

47. Infusion plus loading dose- 2,4 Xo,L = C SS V d …35 Substitution of C SS =R o / K E V d Xo,L =R o /K E …36 C= Xo,L / V d e - K E t + R o / K E V d (1-e -K E t ) …37 47

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49. Assessment of pharmacokinetic parameter AUC=R o T/K E V d =R o T/ Cl T =C SS T Where T=infusion time 49

50. Conclusion- In contrast to short duration of infusion of an i.v bolus (few second) ,the duration of constant rate infusion is usually much longer than half life of drug. The time course of drug conc determined after its administration by assuming the body as single well mixed compartment. 50

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