One compartment kinetics

1. P.N.M A LLIK A RJU N
A SSO C IA T E PRO FESSO R
V I G N A N I N S T I T U T E O F P H A R M A C E U T I C A L T E C H N O L O G Y
PHARMACOKINETICS

2.  The one- compartment open model is the simplest model. Owing
to its simplicity, it is based on following assumption
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 fluid (i.e. mixing) is attained instantaneously &
maintained at all times. This model is followed by only those
drugs that distribute rapidly throughout the body.
3) Drugs move dynamically, in (absorption) & out (elimination) of this
compartment.
4) Elimination is a first order (monoexponential) process with first
order rate constant.
5) Rate of input (absorption) > rate of output (elimination).
One Compartment Open Model
(Instantaneous Distribution Model)

3. One Compartment Open Model
Blood and
other body
tissues
Drug
One-compartment open model showing input
and output processes
Ka
Input
(Absorption)
Output
(Elimination)
Excretion
Metabolism
KE

4.  Depending 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

5. 1.One-compartment Open Model:
Intravenous Bolus Administration

6.  The drug is rapidly distributed in the body when
given in the form of intravenous injection (i.v.
bolus or slug). It takes about one to three minutes
for complete circulation & therefore the rate of
absorption is neglected in calculation.
Blood and
other body
tissues
KE

7.  The general expression for rate of drug presentation to the body is:
 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.
dX
 Ratein (absorption)- Rate out (elimination)
dt
 If the rate out or elimination follows first-order kinetics, then:
dt
 Since rate in or absorption is absent, the equation becomes:
dX
 - Rate out
E
dt
dX
 - K X

8.  Integration of equation yields:
In X = In X0 – KEt
 Where, X0 = amount of drug at time
zero i.e. Dose of the drug injected.
 Equation can also be written in the exponential form
as:
E
X =X0e-K t
 This equation shows one compartment kinetics is
monoexponential.

9.  Transforming equation into common logarithms (log base
10) we get:
 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 costant popularly known as the
apparent volume of distribution.
2.303
0
 KEt
log X  log X

10.  It is a pharmacokinetic parameter that permits the use
of plasma drug concentration in place of total amount
of drug in the body by equation therefore
becomes:
where C0 = plasma drug concentration immediately
after i.v. injection.
 Equation is that of a straight line and indicates that
semi logarithmic plot of log C vs t, will be linear with
Y-intercept as log C0
2.303
E
0
K t
log C  log C -

11. Estimation of pharmacokinetic
parameters – IV bolus administration
1) Elimination rate constant
2) Elimination half life
3) Apparent volume of Distribution
4) Total systemic Clearance

12. a) Cartesian plot of drug that follows one-compartment kinetics and given by rapid
injection
b) Semi logarithmic plot for the rate of elimination in a one-compartment model
Elimination Rate Constant

13. Elimination Half - Life
 Elimination half life :Also called as biological half life.
 The time taken for the amount of drug in the body as well as
plasma concentration to decline by 50% its initial value.
 It is expressed in hours or minutes.
 Half life expressed by following equation:
 The half – life is a secondary parameter that depends upon the
primary parameter clearance & apparent volume of distribution.
 According to following equation:
E
K
t1/ 2

0.693
T
Cl
t1/ 2

0.693Vd

14. The two separate & independent pharmacokinetic
characteristics of a drug distribution of a drug
.since, they are closely related with the
physiological mechanism of body, they are called
as primary parameters
1. Apparent volume of Distribution
2.Total systemic Clearance

15. Apparent Volume of Distribution
 The hypothetical volume of body fluids in which drug is completely
distributed or dissolved is called as Apparent volume of distribution.
 Apparent volume of distribution is denoted by Vd
 The units of Vd is litres
 Modification of equation defined apparent volume of distribution :
Vd 
plasma drug concentration

C
 The best and the simplest way of estimating Vd of a drug is
administering it by rapid i.v. injection, determining the resulting
plasma concentration immediately by using the following equation:
amount of drug in the body X
0 0
C C
Vd

X 0

i.v.bolus dose

16.  Amore general, a more useful non-compartmental method
that can be applied to many compartment models for
estimating the Vd is:
 For drugs given as i.v. bolus,
X0 = dose administered
F = fraction of drug absorbed into the systemic circulation.
K AUC
E
 For drugs administered extravascularly (e.v.),
X0
Vd (area) 
K AUC
E
FX0
Vd (area) 

17. Total systemic Clearance
 Clearance : Clearance is defined as the theoretical volume of body fluid
containing drug from which the drug is completely removed in a given
period of time. It is expressed in ml/min or lit/hr.
 Clearance is a parameter that relates plasma drug concentration with the
rate of drug elimination according to following equations-
plasma drug concentration
rate of elimination
clearance 
Or
Cl 
dx/dt
C
 Clearance is the most important parameter in clinical drug
applications & is useful in evaluating the mechanism by which a drug
is eliminated by the whole organisms or by a particular organ.

18.  The total body clearance, ClT = ClR +ClH + Clother
 Clearance by all organs other than kidney is some times known as
nonrenal clearance ClNR
 It is the difference between total clearance and renal clearance
according to earlier an definition
Substituting dX/dt = KEX in above equ.we get
Since X/C = Vd ,equation can be written as
C
 d x / d t
T
C l
C

K E X
T
C l
C lT  K E v d

19. 1/2
t
T
Cl 
0.693Vd
 Parallel equation can be written for renal and hepatic clearance
as:
ClR = Ke Vd
ClH= Km Vd
Since, KE= 0.693/t1/2 ( from equa. 11), clearance can be related
to half life by the following equation

20. 2.One-compartment Open Model:
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.

21. 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 (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.

22. One-compartment Open Model:
Intravenous Infusion

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

24. The time to reach steady-state concentration is
dependent upon the elimination half-life and not
infusion rate. An increase in infusion rate will merely
increase the plasma concentration attained at steady-
state . If n is the number of half-lives passed since
the start of infusion (t/t½),
For therapeutic purpose, more than 90% of the
steady-state drug concentration in the blood is
desired which is reached in 3.3 half-lives. It takes
6.6 half-lives for the concentration to reach 99% of
the steady-state. Thus, the shorter the half-life (e.g.
penicillin G, 30 min), sooner is the steady-state
reached.

27. Infusion Plus Loading Dose
It takes a very long time for the drugs having longer
half- lives before the plateau concentration is reached
(e.g. phenobarbital, 5 days). Thus, initially, such
drugs have subtherapeutic concentrations. This can
be overcome by administering a
loading dose large enough 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

29. The equation describing the plasma concentration-time profile
following simultaneous i.v. loading dose (i.v. bolus) and constant rate
i.v. infusion
loading dose Xo,L
where, T = infusion time

30. One-compartment Open Model:
Extravascular administration

31.  When a drug is administered by extravascular route ,the rate of
absorption may be described by mathematically as a zero or first
order process.
 A large number of plasma concentration time profile can be
described by a one compartment model with first order absorption &
elimination.
 Zero order absorption is characterized by a constant rate of
absorption

32. Difference between zero- order and first- order kinetics

33.  After e.v. administration , the rate in the change of amount of drug in
the body dx/dt is difference between the rate of input
(absorption) dxev/dt and rate of output( elimination) dxE/dt .

34.  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
Elimination phase.

dt dt
dxev dxE
dt dt
 The plasma level time curve is characterized only by the
dx dx
ev
 E
dxev

dxE
dt dt

36. • Zero order
absorption
• First order
absorption
One compartment open-
Extravascular Administration

37. 3.Zero - OrderAbsorption Model
ExtravascularAdministration
 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.
Blood and
other body
tissues
R0
Zero order
absorption
KE
Drug at
e.v.site
Elimination

38. Rate of Presentation of Drug to the body=Rate in –Rate out
Rate of absorption – Rate of Elimination
Zero order absorption-First order Elimination

39. 4.First orderAbsorption Model
ExtravascularAdministration
 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
 The differential form of the drug the eque. (1)
 Ka = first order absorption rate constant
 Xa = amount of drug at the absorption site remaining to be
absorbed i.e.ARA.
a a E
dt
dx
 K x - K x
Blood and
other body
tissues
KE
Elimination
Drug at
e.v.site
Ka
First order
absorption

40. Transforming in to concentration terms, the eque. Becomes
Where, F= fraction of drug absorbed systemically after e.v.
administration .
Ka - KE
[e
kEt
 e
kat
]
 Integration of equation
X
Ka FX0
KaFX0
[e
kEt
Vd(K a -KE )
e
kat
]
C 

41. Estimation of Pharmacokinetic Parameters
Extravascular administration
 Cmax and tmax : At peak plasma concentration , the rate of absorption
equals rate of elimination i.e. KaXa =KEX and the rate of change in
plasma drug concentration dc/dt = zero . Differntiating equation
 On simplifying ,the above eque. Becomes:
 Converting to logarithmic form,
k t
e a
k t
e E
E a
[K  K ]  Zero
dc

Ka FX0
dt Vd(K a - KE )
- k
a t
a e
k
E t
e
E
K  K
K
K t
t
a
2.303
a
E
2.303
 log k -
E
log k -

42.  Where t is tmax . Rearrangement of above eque. yield:
 Cmax can be obtained by substituting eque. (11) in eque (7), simpler eque
for the same is:
 It has been shown that at Cmax, when Ka = KE, tmax = 1/KE.
Hence the above eque. Further reduced to:
Since, FX0/Vd represent C0 .
a E
m ax
K - K

2.303log ( Ka /KE )
T
d d
max
V V

FX0
e1

0.37 FX0
C
d
m ax
V
C 
FX 0
e
kEtm a x

43. Elimination Rate Constant
 The parameter can be computed from the
elimination phase of the plasma level time profile.
 KE = -2.303 X Slope of the Elimination phase
Elimination Half Life

0.693
t1/2
KE

44. A bso r pt i o n Co nst ant Can B e Det er mi ned B y T w o M et h o ds:
1.METHOD OF RESIDUALS
2.WAGNER NELSON METHOD
Absorption Rate Constant

45. Method of Residuals
 The method of residuals is also called as feathering, peeling and stripping
 For a drug that follow one compartment kinetics & administered e.v., the
concentration of drug in plasma is expressed by a biexposnential equation .

 During the elimination phase, when absorption is almost over, Ka>KE value of
E
second exponential e-Kat is zero. Whereas the exponential e-K t retaines some finite
value. at this time the equation reduced to:
Vd (K a - K E )
K a FX 0
 e
-K a t
]
[e  K E t
C 
if Ka FX0 /Vd (Ka - KE )  A, a hybrid constant then :
E a
-k t -k t
CAe -Ae
t
E
-K
C  Ae
 In log form, the above equation is:

2.303
E
K t
log C  log A-


46.  Where, C represents the back extrapolation plasma
concentration value.
 The slope of the terminal linear portion of the true plasma
curve gives elimination rate constant
a
-K t
r
(CC)C Ae
 Substraction of true plasma conc. Values i.e. equation from
the extrapolated plasma concentration values Cr


47.  In log form, the equation is
 A plot of log Cr vs t yields a straight line with –Ka/2.303 & y
intercept logA
 In some instance , the KE obtained after i.v. bolus of some of
the drug is very larger than the Ka obtained by recidual method
and KE/Ka> 3,
2.303
Kat
r
log C  logA -

48. flip-flop phenomenon since the
slopes of the two lines have
exchanged their meanings.
When KE/Ka ≥3.
No flip flop When Ka/KE ≥3
Lag time: is defined as the
time difference between
drug administration and
start of absorption.

49. Wagner Nelson Method

51. References
 Biopharmaceutics & pharmacokinetics by D.M.
Brahmankar, Sunil B.Jaiswal
 Apllied biopharmaceutics and pharmacokinetics by
& pharmacokinetics by
& pharmacokinetics by Milo
Leon Sargel
 Biopharmaceutics
Venkateswarlu
 Biopharmaceutics
Gibaldi
 Biopharmaceutics & clinical pharmacokinetics an
introduction by Robert E. Notaril