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EMPIRICAL PHARMACOKINETICS EXPONENTIAL MODELS

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EMPIRICAL PHARMACOKINETICS EXPONENTIAL MODELS
Lecture 6
MECHANISTIC
PHARMACOKINETICS:
COMPARTMENTAL MODELS
Adv PK/PD 2006
Lecture 6
1
One Compartment Model
Drug Administration
I
V
Volume of
Distribution
Model =V1, k10
k
Adv PK/PD 2006
Lecture 6
2
Two compartment model
Drug Administration
I
V2
k 12
V1
Peripheral
Compartment
k 21
Central
Compartmen
Model =V1, k10, k12, k21
V2 = V1 k12 / k21
CL1 = V1 k10
CL2 = V1 k12
Adv PK/PD 2006
Lecture 6
k 10
3
Three compartment model
Drug Administration
I
V2
k 12
V1
k 13
V3
Rapidly
Equilibrating
k 21
Central
Compartmen
k 31
Slowly
Equilibrating
Model =V1, k10, k12, k21, k13, k31
V2 = V1 k12 / k21
V3 = V1 k13 / k31
CL1 = V1 k10
CL2 = V1 k12
CL3 = V1 k13
Adv PK/PD 2006
k 10
Lecture 6
4
Two Compartment Model
Drug Administration
I
A2
V2
k 12
Peripheral
Compartment
k 21
A1
V1
Central
Compartmen
k 10
dA1
 I  A2 k21  A1 (k10  k12 )
dt
dA2
 A1k12  A2 k21
dt
Adv PK/PD 2006
Lecture 6
5
Two Compartment Model
dA1
 I  A2 k21  A1 (k10  k12 )
dt
dA2
 A1k12  A2 k21
dt

Cp (t )  C1e
Adv PK/PD 2006
 1t
 C2 e
Lecture 6
 2t
 * I (t )
6
If you know volumes (V1, V2) and clearances (CL1, C
then the micro-rate constants are:
k10 = CL1 / V1
k12 = CL2 / V1
k21 = CL2 / V2
Two Compartment Model
dA1
 I  A2 k21  A1 (k10  k12 )
dt
dA2
 A1k12  A2 k21
dt

and the coefficients and exponents can be cal
if a = k10 + k12 + k21, then
a  a 2  4k10 k21
1 
2
a  a 2  4k10 k21
2 
2
k21  1
C1 
/ V1
2  1
k 
C2  21 2 / V1
1  2

Cp (t )  C1e  1t  C2e  2t * I (t )
Adv PK/PD 2006
Lecture 6
7
Three Compartment Model
dA1
 I  A2 k21  A3k31  A1 (k10  k12  k13 )
dt
dA2
 A1k12  A2 k21
dt
dA3
 A1k13  A3 k31
dt

Cp(t )  C1e
Adv PK/PD 2006
 1t
 C2e
 2t
Lecture 6
 C3e
 3t
 * I (t )
8
Three Compartment Model
1    cos   r2  a2 / 3
a0  k10 k21k31
a2  k10  k12  k13  k21  k31

2

2    cos   
3




r

a
/
3
 2 2 


p  a1  a2 / 3
3    cos   
4
3


r

a
/
3
 2 2 


a1  k10 k31  k21k31  k21k13  k10 k21  k31k12

2

q  2a2 / 27  a1a2 / 3  a0
3
k21  1 k31  1  / V
1  2 1  3  1
k21  2 k31  2 

C2 
/ V1
2  1 2  3 
k21  3 k31  3 

C3 
/ V1
3  2 3  1 
r1  ( p 3 / 27)
C1 
 q 
  arccos    / 3
 2r1 
r2  2elog( r1 ) / 3
Adv PK/PD 2006


Lecture 6
9
Population Pharmacokinetics
 General rule for compartment models:
 Estimate the volumes and clearances
 These are NOT the underlying physical reality,
by they come closer than other methods of
parameterizing the model
 Volumes and clearances are the most likely to
be normally distributed
Adv PK/PD 2006
Lecture 6
10
Volume of Distribution
at Steady State
Css
 Ass  A1,ss  A 2,ss
Vss
By definition, at steady state: C1=C2 = Css
Vss 
Ass
Css

A 1,ss  A 2,ss
A
A
C  , thus V 
V
C
Adv PK/PD 2006
C ss
Vss 
Lecture 6

A 1,ss
C1,ss
A 1,ss
C ss


A 2, ss
C2,ss
A 2,ss
C ss
 V1  V2
11
Volume of Distribution at Steady
State
However, the statement C1 = C2 = Css only holds
when elimination is from the central compartment
If elimination occurs from compartment 2, then Vss
is dependent on k20 and hence on clearance
Adv PK/PD 2006
Lecture 6
12
Saturable Elimination
1
0.1
Nonlinear
kinetics
C outflow = ½ Km
Metabolic rate/Vm
Linear
kinetics
0.01
0.001
0.001
0.01
0.1
1
10
100
1000
C outflow /Km
Adv PK/PD 2006
Lecture 6
13
Saturable distribution
 Saturable plasma
and/or tissue protein
binding eg ACE
inhibitors.
nPT Cu
C  Cu 
1  Ka  Cu

dCutissue
V
 CLu  Cuplasma  Cutissue
dt




CLu = unbound tissue clearance
Cu = unbound concentration
nPt = total number of binding sites in tissue
Ka = association constant of bound drug
Adv PK/PD 2006
Lecture 6
14

Saturable distribution


A2
k12  k12,nominal  1 

 A A 
2,50
2 

dA2
 A1k12  A2 k21
dt
Adv PK/PD 2006
Lecture 6
15
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