Transcript Objectives of the presentation

Non-compartmental analysis
and
The Mean Residence Time approach
A Bousquet-Mélou
1
Synonymous
Mean Residence Time approach
Statistical Moment Approach
Non-compartmental analysis
2
Statistical Moments
• Describe the distribution of a random variable :
• location, dispersion, shape ...
Standard deviation
Mean
Random variable values 3
Statistical Moment Approach
Stochastic interpretation of drug disposition
• Individual particles are considered : they are assumed to
move independently accross kinetic spaces according to
fixed transfert probabilities
• The time spent in the system by each particule is considered
as a random variable
• The statistical moments are used to describe the distribution
of this random variable, and more generally the behaviour of
drug particules in the system
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Statistical Moment Approach

n
t
  C t   dt
• n-order statistical moment
0
• zero-order :


0
 t  C t  dt   C t  dt  AUC
0
• one-order :
0

 t  C t  dt  AUMC
0
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Statistical Moment Approach
Statistical moments in pharmacokinetics.
J Pharmacokinet Biopharm. 1978 Dec;6(6):547-58.
Yamaoka K, Nakagawa T, Uno T.
Statistical moments in pharmacokinetics: models and assumptions.
J Pharm Pharmacol. 1993 Oct;45(10):871-5.
Dunne A.
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The Mean Residence Time
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Mean Residence Time
Principle of the method: (1)
Entry : time = 0, N molecules
• Evaluation of the time each molecule of a
dose stays in the system: t1, t2, t3…tN
• MRT = mean of the different times
MRT =
Exit : times t1, t2, …,tN
t1 + t2 + t3 +...tN
N
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Mean Residence Time
Principle of the method : (2)
• Under minimal assumptions, the plasma
concentration curve provides information on the
time spent by the drug molecules in the body
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Mean Residence Time
Principle of the method: (3)
Entry (exogenous, endogenous)
Central
compartment
(measure)
recirculation
exchanges
Exit (single) : excretion, metabolism
Only one exit from the measurement compartment
First-order elimination : linearity
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Mean Residence Time
Principle of the method: (4)
• N molecules administered in the system at t=0
• The molecules eliminated at t1 have a residence time in the system
equal to t1
Consequence of linearity
• AUCtot is proportional to N
C
• Number n1 of molecules eliminated at t1+ t is
proportional to AUCt:
C1
n1 =
t1
(t)
AUCt
AUCtot
XN =
C(t1) x t
XN
AUCtot
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Mean Residence Time
Principle of the method: (5)
Cumulated residence times of
molecules eliminated during t at :
C
C1
t1
MRT =
Cn
t1 : t1 x C(1) x tx N
tn
C(n) x t
tn : tn x AUC x N
TOT
AUCTOT
(t)
t1x
C1 x t x N
AUCTOT
 tn x
Cn x t x N
n1
N
AUCTOT
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Mean Residence Time
Principle of the method: (5)
MRT =
t1x
C1 x t x N
 
tn x
t1x C1 x t
 
 ti x Ci x t
MRT =
tn x Cn x t
AUCTOT
 t C(t) t
AUMC
=
AUCTOT
N
AUCTOT
AUCTOT
MRT =
Cn x t x N
=
 C(t) t
AUC
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Mean Residence Time
AUMC
MRT 
AUC

AUC   C t  dt
0

AUMC   t  C t  dt
0
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• AUC = Area Under
the zero-order
moment Curve
AUMC
AUC
• AUMC = Area
Under the firstorder Moment
Curve
From: Rowland M, Tozer TN. Clinical Pharmacokinetics – Concepts and Applications,
3rd edition, Williams and Wilkins, 1995, p. 487.
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Mean Residence Time
Limits of the method:
Central
compartment
(measure)
• 2 exit sites
• Statistical moments obtained from plasma concentration
inform only on molecules eliminated by the central
compartment
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Computational methods
• Non-compartmental analysis
Area
calculations
Trapezes
• Fitting with a poly-exponential equation
Equation parameters : Yi, li
• Analysis with a compartmental model
Model parameters : kij
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Computational methods
Area calculations by numerical integration
1. Linear trapezoidal
AUC
 t
i
i
 t i 1
 Ci  Ci1 
2
C
1
2
Concentration
t
AUMC
t i  Ci  t i1  Ci1 


t

t

 i i1
i
C
1
t
2
Time
2
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Computational methods
Area calculations by numerical integration
1. Linear trapezoidal
Advantages: Simple (can calculate by hand)
Disadvantages:
•Assumes straight line between data points
•If curve is steep, error may be large
•Under or over estimation, depending on whether the
curve is ascending of descending
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Computational methods
Area calculations by numerical integration
2. Log-linear trapezoidal
AUC

Ci  Ci 1 
 t i  t i1 
i
Ln

Ci

Ci 1 
C
1
C
2
Concentration
t
1
t
2
Time
AUMC
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Computational methods
Area calculations by numerical integration
2. Log-linear trapezoidal
< Linear trapezoidal
Disadvantages:
Advantages:
•Produces large errors on
•Hand calculator
an ascending curve, near
•Very accurate for monothe peak, or steeply
exponential
declining polyexponential
•Very accurate in late time points
curve
where interval between points is
substantially increased
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Computational methods
Extrapolation to infinity

AUC t

 C t  dt 

la st
tla st

AUMC t
la st

Clast
l
2
z

Clast
lz
Assumes loglinear decline
tlast  Clast
lz
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Computational methods
AUC Determination
Time (hr) C (mg/L)
0
2.55
1
2.00
3
1.13
5
0.70
7
0.43
10
0.20
18
0.025
Area (mg.hr/L)
2.275
3.13
1.83
1.13
0.945
0.900
Total 10.21
AUMC Determination
Cxt
Area
(mg/L)(hr) (mg.hr2/L)
0
2.00
1.00
3.39
5.39
3.50
6.89
3.01
6.51
2.00
7.52
0.45
9.80
37.11
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Non-compartmental analysis
The Main PK parameters can be calculated using non-compartmental
analysis
• MRT = AUMC / AUC
• Clearance = Dose / AUC
• Vss = Cl x MRT =
Dose x AUMC
AUC2
• F% = AUC EV / AUC IV
DEV = DIV
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Computational methods
• Non-compartmental analysis
Area
calculations
Trapezes
• Fitting with a poly-exponential equation Area
calculations
Equation parameters : Yi, li
• Analysis with a compartmental model
Model parameters : kij
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Fitting with a poly-exponential equation
Area calculations by mathematical integration
n
C(t)   Yi  e
 λ i t
For one compartment :
i 1
n
Yi
AUC  
i 1 λ i
n
Yi
AUMC   2
i 1 λ i
C0
AUC 
k10
C0
AUMC  2
k10
1
MRT 
k10
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Fitting with a poly-exponential equation
For two compartments :
AUC  
Yi
AUMC  
Yi
li
l
2
i
C(t)  Y1  e  λ1t  Y2  e  λ 2 t
AUC 
Y1
l1
AUMC 

Y1
l
2
1
Y2
l2

Y2
l
2
2
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Computational methods
• Non-compartmental analysis
Area
calculations
Trapezes
• Fitting with a poly-exponential equation Area
calculations
Equation parameters : Yi, li
• Analysis with a compartmental model
Model parameters : kij
Direct MRT
calculations
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Analysis with a compartmental model
Example : Two-compartments model
k12
1
2
k21
k10
dX1

dt
dX 2

dt
 k10  k12  X1
 k 21  X 2
k12  X1
- k 21  X 2
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Analysis with a compartmental model
Example : Two-compartments model
K is the 2x2 matrix of the system of differential equations
describing the drug transfer between compartments
X1
X2
dX1/dt
 k10  k12 
k 21
dX2/dt
k12
- k 21
K=
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Analysis with a compartmental model
Then the matrix (- K-1) gives the MRT in each compartment
Dosing in 1
(-K-1) =
Dosing in 2
Comp 1
MRTcomp1
MRTcomp1
Comp 2
MRTcomp2
MRTcomp2
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The Mean Residence Times
Fundamental property of MRT : ADDITIVITY
The mean residence time in the system is the sum of the
mean residence times in the compartments of the system
• Mean Absorption Time / Mean Dissolution Time
• MRT in central and peripheral compartments
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The Mean Absorption Time
(MAT)
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The Mean Absorption Time
Definition : mean time required for the drug to reach the central
compartment
IV
EV
Ka
1
A
K10
F = 100%
 AUMC 

  MRTsystem comp. A  1
 AUC  EV
 AUMC 

  MRTcomp 1
 AUC  IV
MRTcomp. A
 AUMC 
 AUMC 

 

 AUC  EV  AUC  IV
MRTcomp. A  MAT
!
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because bioavailability = 100%
The Mean Absorption Time
!
MAT and bioavailability
• Actually, the MAT calculated from plasma data is the
MRT at the injection site
• This MAT does not provide information about the
absorption process unless F = 100%
• Otherwise the real MAT is :
MAT 
MRTcomp. A
F
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The Mean Dissolution Time
• In vivo measurement of the dissolution rate in
the digestive tract
tablet
solution
absorption
dissolution
solution
digestive tract
MDT = MRTtablet - MRTsolution
blood
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Mean Residence Time in the
Central Compartment (MRTC) and in
the Peripheral (Tissues)
Compartment (MRTT)
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MRTcentral and MRTtissue
Entry
MRTC
MRTT
MRTsystem = MRTC + MRTT
Exit (single) : excretion, metabolism
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The Mean Transit Time
(MTT)
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The Mean Transit Times (MTT)
• Definition :
– Average interval of time spent by a drug particle from its
entry into the compartment to its next exit
– Average duration of one visit in the compartment
• Computation :
– The MTT in the central compartment can be calculated
for plasma concentrations after i.v.
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The Mean Residence Number
(MRN)
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The Mean Residence Number (MRN)
•Definition :
– Average number of times drug particles enter
into a compartment after their injection into the
kinetic system
– Average number of visits in the compartment
– For each compartment :
MRN =
MRT
MTT
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Stochastic interpretation of the drug
disposition in the body
R+1
IV
Mean number
of visits
R
Cldistribution
MRTC
(all the visits)
MTTC
(for a single visit)
R
number
of cycles
MRTT
(for all the visits)
MTTT
(for a single visit)
Clredistribution
Clelimination
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Stochastic interpretation of the drug
disposition in the body
Computation : intravenous administration
MRTsystem = AUMC / AUC
MRTC = AUC / C(0)
MTTC = - C(0) / C’(0)
MRTT = MRTsystem- MRTC
R+1=
MTTT =
MRTC
MTTC
MRTT
R
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Interpretation of a Compartmental Model
Determinist vs stochastic
Digoxin
21.4 e-1.99t + 0.881 e-0.017t
0.3 h
Cld = 52 L/h
MTTC : 0.5h
MRTC : 2.81h
Vc 34 L
41 h
MTTT : 10.5h
4.4
MRTT : 46h
ClR = 52 L/h VT : 551 L
stochastic
Cl = 12 L/h
Determinist
1.56 h-1
Vc : 33.7 L
VT : 551L
MRTsystem = 48.8 h
0.095 h-1
0.338 h-1
t1/2 = 41 h
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Interpretation of a Compartmental Model
Gentamicin
Determinist vs stochastic
y =5600 e-0.281t + 94.9 e-0.012t
t1/2 =3h
t1/2 =57h
stochastic
MTTC : 4.65h
MTTT : 64.5h
0.265
MRTC : 5.88h
MRTT : 17.1h
Vc : 14 L
ClR = 0.65 L/h VT : 40.8 L
Determinist
0.045 h-1
VT : 40.8L
Vc : 14 L
0.016 h-1
0.17 h-1
t1/2 = 57 h
Cld = 0.65 L/h
Clélimination = 2.39 L/h
MRTsystem = 23 h
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