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

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-
1. The body is considered as a single, kinetically homogeneous
unit that has no barriers to the movement of drug
2. 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
3. Drugs move dynamically, in (absorption) and out
(elimination) of this compartment
4. Elimination is a first order (monoexponential) process
with first order rate constant
6

7. 5. Rate of input (absorption)> rate of output(elimination)
6. 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
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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 tissuesDrug
Ka
Input
(absorption)
Ke
output
(Elimination )
Metabolism
Excretion
9

10.  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
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= -KE X (1.3)
Where KE= 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-
1. Elimination rate constant
2. Apparent volume of distribution
3. Elimination half life
4. Clearance
13

14. 14
Elimination rate constant (KE
)
 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= -KE X , now integrating this equation
lnX= ln X0 - KE t (1.4)
Where X0= amount of drug at time t = zero
Above equation can also be written in the following
monoexponential format as
X= X0 e-K
e
t
15

16.  Above equation we can write in the log to the base 10
form as
 logX = log X0 – KEt/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= Vd C
Where Vd = proportionality constant popularly known as the
apparent volume of distribution
16

17. 17
One compartment open modelDrugConc(C)
Time
log(C)
Time
logX = log X0 – KEt/2.303
X= X0 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 Vd 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
0
0
C
X
conc.initial
dose
Vd 

25. 25
The Extent of Distribution and Vd in a 70 kg
Normal Man
Vd, 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 Vd
 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 (t1/2)
 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:
VdKCl 

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.
Cl)hepaticandrenal,total,ClandCl,(Cl
ClClClCl
hrt
otherhrt



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:
a. Ease of control of rate of infusion to fit individual patient
needs
b. Prevents fluctuating maxima and minima plasma level
c. Other drugs, electrolytes and nutrients can be conveniently
administered simultaneously by the same infusion line in
critically ill patients
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40. One compartment open model:
Intravenous infusion-
 Model can be represent as : ( i.v infusion)
Drug
dX/dt=Ro-KEX …eq 23
X=Ro/KE(1-e-KEt) …eq 24
Since X=VdC
C=Ro/KEVd(1-e-KEt) …eq 25
=Ro/ClT(1-e-KEt) …eq 26
40
Blood & other
Body tissues
R0
Zero order
Infusion
rate
KE

41.  At steady state. The rate of change of amount of drug in
the body is zero ,eq 23 becomes
Zero=Ro-KEXSS …27
KEXSS=Ro …28
CSS=Ro/KEVd …29
=Ro/ClT i.e infusion rate ....30
clearance
Substituting eq. 30 in eq. 26
C=CSS(1-e-KEt) …31
Rearrangement yields:
 [CSS-C] =e-KEt
. ...32
CSS
log CSS-C = -KEt …33
CSS 2.303
41

42. 42
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.

43. 43

44. 44

45. 45
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 Cp is 50% of the
CSS, after 2 half-lives, Cp is 75% of the Css
…….

46.  If n is the no. of half lives passed since the start of
infusion(t/t1/2)
 Eq. can be written as
 C=CSS [1-(1/2)n] …34
46

47. Infusion plus loading dose-2,4
 Xo,L=CSSVd …35
 Substitution of CSS=Ro/KEVd
 Xo,L=Ro/KE …36
 C=Xo,L/Vd e-KEt+ Ro/KEVd(1-e-KEt) …37
47

48. 48

49. Assessment of pharmacokinetic parameter
 AUC=Ro T/KE Vd
=Ro T/ClT
=CSS 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.
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51. 51