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1
Non-compartmental analysisand
The Mean Residence Time approach
A Bousquet-Mélou
2
Mean Residence Time approach
Statistical Moment Approach
Non-compartmental analysis
Synonymous
3
Statistical Moments
Mean
• Describe the distribution of a random variable :• location, dispersion, shape ...
Standard deviation
Random variable values
4
Stochastic interpretation of drug disposition
Statistical Moment Approach
• The statistical moments are used to describe the distribution
of this random variable, and more generally the behaviour of
drug particules in the system
• 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
5
0
dttCt n
AUCdttCdttCt
00
0
AUMCdttCt
0
• n-order statistical moment
• zero-order :
• one-order :
Statistical Moment Approach
6
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.
Statistical Moment Approach
7
The Mean Residence Time
8
• Evaluation of the time each molecule of a dose stays in the system: t1, t2, t3…tN
• MRT = mean of the different times
MRT = N
t1 + t2 + t3 +...tN
Principle of the method: (1)Entry : time = 0, N molecules
Exit : times t1, t2, …,tN
Mean Residence Time
9
• Under minimal assumptions, the plasma
concentration curve provides information on the
time spent by the drug molecules in the body
Principle of the method : (2)
Mean Residence TimeMean Residence Time
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Only one exit from the measurement compartment
First-order elimination : linearity
Principle of the method: (3)Entry (exogenous, endogenous)
Exit (single) : excretion, metabolism
recirculationexchanges
Central compartment
(measure)
Mean Residence TimeMean Residence Time
11
Consequence of linearity
• AUCtot is proportional to N
• Number n1 of molecules eliminated at t1+ t is proportional to AUCt:
Principle of the method: (4)
C
(t)
C1
t1
Mean Residence TimeMean Residence Time
C(t1) x t
AUCtot
X Nn1 =AUCt
AUCtot
X N =
• N molecules administered in the system at t=0
• The molecules eliminated at t1 have a residence time in the system equal to t1
12
Cumulated residence times of molecules eliminated during t at :
Principle of the method: (5)
C
(t)
C1
t1
t1 : t1 x x N
tn : tn x x N
MRT = t1xtn x
N
C1 x t x N Cn x t x N
AUCTOT AUCTOT
tn
CnC(1) x t AUCTOT
C(n) x t AUCTOT
Mean Residence TimeMean Residence Time
n1
13
Principle of the method: (5)
MRT = t1xtn x
N
C1 x t x N Cn x t x N
AUCTOT AUCTOT
MRT = =
Mean Residence TimeMean Residence Time
MRT = t1xC1 x ttn x Cn x t AUCTOT
t C(t) t
C(t) t
ti x Ci x t
AUCTOT
AUMC
AUC=
14
Mean Residence TimeMean Residence Time
0
dttCAUC
0
dttCtAUMC
AUC
AUMCMRT
15From: Rowland M, Tozer TN. Clinical Pharmacokinetics – Concepts and Applications, 3rd edition, Williams and Wilkins, 1995, p. 487.
AUC
AUMC
• AUC = Area Under the zero-order moment Curve
• AUMC = Area Under the first-order Moment Curve
16
• 2 exit sites• Statistical moments obtained from plasma concentration
inform only on molecules eliminated by the central compartment
Limits of the method:
Mean Residence TimeMean Residence Time
Centralcompartment
(measure)
17
• Non-compartmental analysis
Trapezes
• Fitting with a poly-exponential equation
Equation parameters : Yi, i
• Analysis with a compartmental model Model parameters : kij
Computational methodsComputational methods
Areacalculations
18
1. Linear trapezoidal
CC
tt
1
2
1
2
Concentration
Time
Computational methodsComputational methods
Area calculations by numerical integration
2
CCtt 1ii
i1ii
2
CtCttt 1i1iii
i1ii
AUC
AUMC
19
1. Linear trapezoidal
Computational methodsComputational methods
Area calculations by numerical integration
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
20
21
2. Log-linear trapezoidalC
C
tt
1
2
1
2
Concentration
Time
Computational methodsComputational methods
Area calculations by numerical integration
1i
i
1ii
i1ii
CC
CCtt
LnAUC
AUMC
22
2. Log-linear trapezoidal
Computational methodsComputational methods
Area calculations by numerical integration
Advantages:•Hand calculator•Very accurate for mono-exponential•Very accurate in late time points where interval between points is substantially increased
Disadvantages:•Produces large errors on an ascending curve, near the peak, or steeply declining polyexponential curve
< Linear trapezoidal
23
Computational methodsComputational methods
Extrapolation to infinity
last
lastt z
lastt
CdttCAUC
Assumes log-linear decline
z
lastlast
z
lastt
CtCAUMC
last
2
24
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
AUC Determination
Area (mg.hr/L)-2.2753.131.831.130.9450.900
Total 10.21
AUMC Determination C x t(mg/L)(hr) 0 2.00 3.39 3.50 3.01 2.00 0.45
Area(mg.hr2/L) - 1.00 5.39 6.89 6.51 7.52 9.80 37.11
Computational methodsComputational methods
25
• MRT = AUMC / AUC
• Clearance = Dose / AUC
• Vss = Cl x MRT =
• F% = AUC EV / AUC IV DEV = DIV
Dose x AUMCAUC2
The Main PK parameters can be calculated using non-compartmental analysis
Non-compartmental analysisNon-compartmental analysis
26
• Non-compartmental analysis
Trapezes
• Fitting with a poly-exponential equation
Equation parameters : Yi, i
• Analysis with a compartmental model Model parameters : kij
Computational methodsComputational methods
Areacalculations
Areacalculations
27
Fitting with a poly-exponential equationFitting with a poly-exponential equation
Area calculations by mathematical integration
n
1i
tλi
ieC(t) Y
n
1i i
i
λ
YAUC
n
1i2i
i
λ
YAUMC
For one compartment :
10
0
k
CAUC
210
0
k
CAUMC
10k
1MRT
28
Fitting with a poly-exponential equationFitting with a poly-exponential equation
For two compartments :
2
2
1
1 YY
AUCY
AUCi
i
22
221
12
YY
AUMCY
AUMCi
i
tλ2
tλ1
21 eeC(t) YY
29
• Non-compartmental analysis
Trapezes
• Fitting with a poly-exponential equation
Equation parameters : Yi, i
• Analysis with a compartmental model Model parameters : kij
Computational methodsComputational methods
Direct MRTcalculations
Areacalculations
Areacalculations
30
Example : Two-compartments model
Analysis with a compartmental modelAnalysis with a compartmental model
1
k12
k21
k10
2
dt
dX1 11210 Xkk 221 Xk
dt
dX2112 Xk 221 Xk-
31
dX1/dt
X1 X2
Example : Two-compartments model
K is the 2x2 matrix of the system of differential equations describing the drug transfer between compartments
Analysis with a compartmental modelAnalysis with a compartmental model
dX2/dtK =
1210 kk
12k
21k
21k-
32
MRTcomp1
Dosing in 1
MRTcomp2
MRTcomp1
MRTcomp2
Analysis with a compartmental modelAnalysis with a compartmental model
(-K-1) =
Then the matrix (- K-1) gives the MRT in each compartment
Dosing in 2
Comp 1
Comp 2
33
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
The Mean Residence Times
34
The Mean Absorption Time(MAT)
35
Definition : mean time required for the drug to reach the central compartment
1 A comp. systemEV
MRTAUC
AUMC
A 1
Ka
F = 100%
K10
The Mean Absorption TimeThe Mean Absorption Time
1 compIV
MRTAUC
AUMC
IVEV
IVEVA comp. AUC
AUMC
AUC
AUMCMRT
!MATMRT A comp.
because bioavailability = 100%
36
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 :
!
The Mean Absorption TimeThe Mean Absorption Time
F
MRTMAT A comp.
37
• In vivo measurement of the dissolution rate in the digestive tract
blood
solution
digestive tract
MDT = MRTtablet - MRTsolution
dissolution absorption
tablet solution
The Mean Dissolution TimeThe Mean Dissolution Time
38
Mean Residence Time in the Central Compartment (MRTC) and in
the Peripheral (Tissues) Compartment (MRTT)
39
MRTC MRTTMRTsystem = MRTC + MRTT
MRTcentral and MRTtissue
Entry
Exit (single) : excretion, metabolism
40
The Mean Transit Time(MTT)
41
• 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.
The Mean Transit Times (MTT)
42
The Mean Residence Number(MRN)
43
•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 :
The Mean Residence Number (MRN)
MRN =MRT
MTT
44
Stochastic interpretation of the drug disposition in the body
MRTC
(all the visits)MTTC
(for a single visit)
MRTT
(for all the visits)MTTT
(for a single visit)
Cldistribution
Rnumber
of cycles
Clelimination
Clredistribution
Mean numberof visits
RR+1IV
45
Stochastic interpretation of the drug disposition in the body
Computation : intravenous administration
MRTsystem = AUMC / AUC
MRTC = AUC / C(0)
MTTC = - C(0) / C’(0) R + 1 = MRTC
MTTC
MRTT = MRTsystem- MRTC
MTTT =MRTT
R
46
Interpretation of a Compartmental Model
Determinist vs stochasticDigoxin
stochastic
MTTC : 0.5hMRTC : 2.81hVc 34 L
Cld = 52 L/h
4.4
ClR = 52 L/h
MTTT : 10.5hMRTT : 46hVT : 551 L
Cl = 12 L/h
MRTsystem = 48.8 h
Determinist
Vc : 33.7 L1.56 h-1
VT : 551L0.095 h-1
0.338 h-1
t1/2 = 41 h
21.4 e-1.99t + 0.881 e-0.017t
0.3 h
41 h
47
Determinist vs stochasticGentamicin
stochastic
MTTC : 4.65hMRTC : 5.88hVc : 14 L
Cld = 0.65 L/h
0.265
ClR = 0.65 L/h
MTTT : 64.5hMRTT : 17.1hVT : 40.8 L
Clélimination = 2.39 L/h
MRTsystem = 23 h
Determinist
Vc : 14 L0.045 h-1
VT : 40.8L0.016 h-1
0.17 h-1
t1/2 = 57 h
y =5600 e-0.281t + 94.9 e-0.012t
t1/2 =3h
t1/2 =57h
Interpretation of a Compartmental Model