Compartment modeling

Presented by,
Shilpi Biswas
M-
Pharm 2nd year
(Pharmaceutics)
GURUNANAK COLLEGE OF
Compartment Modeling

Introduction
 The time course of drug concentration determined
after its administration can be satisfactorily explained
by assuming the body as a single well-mixed
compartment with first-order disposition processes.
In case of other drug, two or more body compartment
may be postulated to describe mathematically the data
collected.
The one compartment open model treats the body as
one homogeneous volume in which mixing is
instantaneous input and output are from this one
volume.

One – Compartment
Open Model
(Instantaneous
Distribution Model)

One – Compartment Open Model
(Instantaneous Distribution Model)
 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).

Blood and
other body
tissues
Drug
FIG: One-compartment open model showing input
and output processes
Ka
Input
(Absorption)
Output
(Elimination)
Excretion
Metabolism
KE

 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

One-compartment
Open Model:
Intravenous Bolus
Administration

One-compartment Open Model:
Intravenous Bolus Administration
 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

One-compartment Open Model:
Intravenous Bolus Administration
 The general expression for rate of drug presentation to the body is:
 Since rate in or absorption is absent, the equation becomes:
 If the rate out or elimination follows first-order kinetics, then:
 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.
on)(eliminatioutRate-y)(availabitinRate
dt
dX
outRate-
dt
dX
XK- E
dt
dX
…(1)
…(2)
…(3)

Estimation of pharmacokinetic
parameters – IV bolus administration
 Elimination phase can be characterized by 3
parameters -
1) Elimination rate constant
2) Elimination half life
3) Clearance

Elimination Rate Constant:
 Integration of equation (3) yields:
In X = In X0 – KEt
 Where, X0 = amount of drug at time
t = zero i.e. the initial amount of drug injected.
 Equation (4) can also be written in the exponential form
as:
X =X0e-K
E
t
 This equation shows one compartment kinetics is
monoexponential.
…(4)
…(5)

Elimination Rate Constant
 Transforming equation (4) 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.
303.2
K
XlogXlog E
0
t
 …(6)
…(7)

 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 (6) therefore
becomes:
where C0 = plasma drug concentration immediately
after i.v. injection.
 Equation (8) 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
303.2
K
-ClogClog E
0
t
 …(8)

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

 The elimination or removal of drug from the body is the sum
of urinary excretion, metabolism, biliary excretion, pulmonary
excretion and other mechanisms involved therein.
 Thus, KE is an additive property of rate constant for each of
this processes and better called as overall elimination rate
constant.
 The fraction of drug excreted unchanged in urine Fe and
fraction of drug metabolized Fm can be given as
......K 1E  KKKK bme
E
e
e
K
K
F 
E
m
m
K
K
F 
…(9)
…(10)a
…(10)b

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 one- half or 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:
EK
t
693.0
2/1 
T
d
Cl
V
t
693.0
2/1 
…(11)
…(12)

Apparent Volume
of Distribution

Apparent Volume of Distribution
 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.
 Modification of equation (7) defined apparent volume of
distribution :
 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:
C
X
Vd 
ionconcentratdrugplasma
bodyin thedrugofamount
00
0
C
dosebolus..vi
C
X
Vd  …(14)
…(13)

Apparent Volume of Distribution
 A more 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,
 For drugs administered extravascularly (e.v.),
X0 = dose administered
F = fraction of drug absorbed into the systemic circulation.
AUCK
X
V
E
aread
0
)( 
AUCK
FX
V
E
aread
0
)( 
…(15)a
…(15)b

Clearance
 Clearance : 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.
 Clearance is a parameter that relates plasma drug concentration with the
rate of drug elimination according to following equations-
Or
 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.
ionconcentratdrugplasma
neliminatioofrate
clearance
c
/dd tx
Cl
…(16)

Total Body Clearance
 Clearance at an individual organ level is called as organ clearance.
 It can be estimated by dividing the rate of elimination by each organ
with the concentration of drug presented to it. Thus,
 Renal clearance
 Hepatic clearance
 Other organ clearance

C
kidneybyneliminatioofRate
ClR 
C
liverbyneliminatioofRate
Cl
C
organsotherbyneliminatioofRate
Cl
…(17)
…(18)
…(19)

 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 (equation 17)
Substituting dX/dt = KEX from equ.3 in above equ.we get
Since X/C = Vd ( from equation 13) the equ. (21) can be written as
C
dx/dt
TCl
C
XKE
TCl
dTCl vKE
…(20)
…(21)
…(22)

 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
1/2t
0.693v d
TCl 
…(23)

One-
Compartment
Open Model
Extravascular
Administration

One-Compartment Open Model
Extravascular Administration
 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.
 Difference between zero- order and first- order kinetics are given in
fig.

 Zero order absorption is characterized by a constant rate of
absorption .
 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 .
 dx /dt = rate of absorption – rate of elimination
dt
dx
-
dt
dx Eev

dt
dx …(1)

 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
 The plasma level time curve is characterized only by the
Elimination phase.
dt
dxE

dt
dxev
dt
dxE

dt
dxev …(3)
…(2)
dt
dxE

dt
dxev …(4)

Zero- Order
Absorption Model
Extravascular
Administration

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.
Blood and
other body
tissuesZero order
absorption
R0 KEDrug at
e.v.site
Elimination

First order
Absorption Model
Extravascular
Administration

First order Absorption Model
 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.
xk-xk Eaa
dt
dx
Blood and
other body
tissuesFirst order
absorption
KE EliminationDrug at
e.v.site
Ka
…(5)

First order Absorption Model
 Integration of eque. (5)
Transforming in to concentration terms, the eque. Becomes
Where, F= fraction of drug absorbed systemically after e.v.
administration .
][
K-K
FXK
Ea
0a tatE
kk
eeC


][
)K-Vd(K
FXK
Ea
0a tatE
kk
eeC

 …(7)
…(6)

Assessment of
Pharmacokinetic
Parameters
Extravascular
administration

Assessment 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(7)
 On simplifying ,the above eque. Becomes:
 Converting to logarithmic form,
Zero][
)K-Vd(K
FXK
Ea
0a
  ta
k
e
tE
k
e
aE
KK
dt
dc
ta
k-
e
a
K tE
k
e
E
K
2.303
k
-klog
2.303
k
-log
tt
a
a
E
Ek
…(8)
…(9)
…(10)

Assessment of Pharmacokinetic Parameters
Extravascular administration
 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 .
Ea
Ea
max
K-K
)/KK(2.303log
T
d
01
d
0
max
V
FX0.37
V
FX
 
eC
max
d
0
max
V
FX tE
k
eC


…(13)
…(12)
…(11)

 The parameter can be computed from the elimination phase of the
plasma level time profile.
 For most drugs administered e.v., absorption rate is significantly
greater than the elimination rate.
 At such a stage, when absorption is complete, the change in
plasma conc. Is depend only on elimination rate and eque. (7)
reduces to
 Transforming into log form, the eque. Becomes ,
tE
k
eC


)K-(KV
FXK
Ead
0a
2.303
K
)K-(KV
FXK
loglog
t
E
Ead
0a
C …(15)
…(14)

Absorption Rate Constant
 It can be calculated by the method of residuals.
 For a drug that follow one compartment kinetics & administered e.v., the
concentration of drug in plasma is expressed by a biexposnential equation (7)

 During the elimination phase, when absorption is almost over, Ka>KE value of
second exponential e-Kat is zero. Whereas the exponential e-K
E
t retaines some finite
value. at this time the eque.(15) reduced to:
 In log form, the above equation is:
]e[
)K-(KV
FXK ta
K-
Ead
0a

 tEK
eC
:henconstant thybridaA,)K-(K/VFXK Ead0a if
tatE
-k-k
Ae-AeC
t
C EK-
Ae

2.303
K
-Aloglog Et
C 

…(17)
…(16)
…(15)
…(14)

 Where, C represents the back extrapolation plasma
concentration value.
 Substraction of true plasma conc. Values i.e. equa.(15) from
the extrapolated plasma concentration values Cr
t
CC aK-
r AeC)( 

…(18)

 In log form, the eque.is
 A plot of log Cr vs t yields a straight line with –Ka/2.303 &
y intercept log A
 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,
303.2
K
-logAlog at
Cr  …(19)

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

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