Transcript Analyse non-compartimentale

Non compartmental analysis
Update: 13/08/2010
Non cpt analysis - 1
Statistical Moment Approach
Stochastic interpretation
• Individual particles are assumed to move
independently among kinetic spaces
according to fixed transfert probabilities
• The behaviour of drug particles is described
by the statistical moments
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Statistical Moment Approach
Synonymous
!
• Model-independent approach
• Non-compartmental analysis
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The Main Non-compartmental
Parameters
• Clearance = Dose / AUC
Dose x AUMC
• Vss =
AUC2
• MRT = Vss / Cl = AUMC / AUC
• F% = AUC EV / AUC IV
DEV = DIV
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The Mean Residence Time
(MRT system)
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Non-compartmental analysis
Principle of the method: (1)
Entry
• To measure the time each molecule
stays in the system: t1, t2, t3...tn
• MRT = mean of the different times
t1 + t2 + t3 +...tn
MRT =
n
Exit
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Non-compartmental analysis
Principle of the method
rate of absorption
2 balls / s
2 balls / s
• Clearance = flow = 2 balls/second
• MRT = t = (t1 + t2... t6)/n = (0.5 + 1 + 1.5 +…+6)/6 = 3
• Vss = Clearance x MRT = 6 balls
• Tube volume  x R2 x L =  x R2 x 12R
• Ball volume (6 x 4R3)/3
• Ratio Vballe/ Vtube = 0.67 = partition coefficient between
balls and tube
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Mean Residence Time
Principle of the method : (2)
• The random variable (RV) is the presence
time in the system
• This random variable is characterized by its
mean (MRT) and its variance (VRT)
• The plasma concentration curve provides
this information under minimal assumptions
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Non-compartmental analysis
Principle of the method: (3)
• Administration of No molecules at
t=0
• AUCtot will be proportional to No
• The molecules eliminated at t1 had a
sojourn time of t1 in the system
• Number of molecules eliminated at
t1 :
C
C1
t1
(t)
C(t1) x t
x No
AUCtot
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Non-compartmental analysis
Principle of the method: (4)
Cumulated sojourn times of
molecule which has been eliminated
during t at :
C1 x t
t1 : t1 x AUC x No
C
C1
Cn
TOT
tn : tn x Cn x t x No
t1
MRT=
tn
 t1x
(t)
C1 x t x No
AUCTOT
AUCTOT
 tn x
Cn x t x No
No
AUCTOT
MRT =  ti x Ci x t / AUCTOT =  t C(t) t /  C(t) t
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Non-compartmental analysis
Requirements to compute MRT
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Mean Residence Time
Principle of the method: (5)
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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Non-compartmental analysis
Principle of the method: (6)
1
2
• 2 exit sites
• MRT is not computable by statistical
moments applied to plasma concentration
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Computation Method
• Non-compartmental analysis
• Trapezes
• Fitting to a polyexponential equation
• Equation parameters : Yi, li
• Assuming a compartmental model
• Model parameters : kij
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Non-compartmental analysis
Computation method (1)
• The 3 statistical moments
• S0 = (ti - ti-1) (Ci + Ci-1) / 2 = AUC
• S1 = (ti - ti-1) (Ci x ti + Ci x ti -1) / 2 = AUMC
2
• S2 = (ti - ti-1) (Ci x ti + Ci x ti 2-1) / 2 = AUMMC
AUC = S0
MRT = S1 / S0
VRT = S2 / S0 - (S1 - S0)2
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Non-compartmental analysis
Computation method (2)
• The 3 centered moments (normalized
in relation to the origin)

AUC = 0 C(t) x dt


MRT = 
t x C(t) x dt / 
C(t) x dt
0
0

VRT = 
(t 0
MRT)2
x C(t) x dt /

0 C(t) x dt
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Non-compartmental analysis
Computation method (3)
• S0 by the arithmetic trapezoidal rule
C0
AUC1 =
C0 +C1
C1
2
C2
AUC1
C3
AUC2
x (t1 - t0)
extrapolation
area
AUC3
t0
t1
t2
t3
AUCTOT = S1 =  AUC1 + AUC2 ... AUCn + extrapolation area
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Non-compartmental analysis
Computation method (4)
• Computation of S1 = AUMC with the
arithmetic trapezoidal rule
C0
AUMC1 =
t0 x C0 + t1 x C1
2
C1
area to extrapolate
C2
AUMC1
AUMC2
x (t1-t0)
C3
AUMC3
t0
t3
t2
t1
AUMCTOT = S2 = AUMC1 + AUMC2 +... AUMC extrapolated
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Non-compartmental analysis
Computation method (5)
• How to extrapolate
S0 : Cz / l2
S1 : tz x Cz / lz + Cz / l2
2
S2 :
t2
Cz / lz + 2tz Cz / lz + 2Cz/lz
2
z
3
Cz : the last measured concentration at tz
Problem with
2
lz
et
3
lz
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Non-compartmental analysis
Computation method (6)
• From the parameters of a given model
n
S0 =  Yi / li
i =1
n
S1 =  Yi /li
2
i =1
n
S2 =  2Yi /li
i =1
3
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Non-compartmental analysis
Computation method (7)
• Bicompartmental model :
C(t) = Y1 exp(-l1t) + Y2 exp(-l2t)
MRTsystem =
2
Y1/l1 + Y2 / l2
2
Y1/l1 + Y2 / l2
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Non-compartmental analysis
Principle of the method:
• MRT =  t x C(t) x t  C(t) x t


• MRT =  0 t C(t) dt  0 C(t) dt
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MRT system: interpretation
Monocompartmental model (IV)
t1/2 : time to eliminate 50% of the molecules
MRT : time to eliminate 63.2% of the molecules
MRT = 1/ K10
t1/2 = 0.693 MRT
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MRT system: interpretation
Multicompartmental model
Concentration
terminal half-life vs MRT
MRT = 16 h
MRT = 4 h
t1/2 = 12 h
24
temps (h)
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MRT system
• Comparison of published results
• Author 1 : bicompartmental model: t1/2 = 6h
• Author 2 : tricompartmental model: t1/2 = 18h
• Solution : a posteriori computation of MRTsystem
MRT bicompartmental
2
Y1/l1
+ Y2 /
2
l2
Y1/l1 + Y2 / l2
MRT tricompartmental
?
=
2
Y1/l1
+ Y2 /
2
l2 + Y3
/
2
l3
Y1/l1 + Y2 / l2 + Y3 / l3
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The Mean Absorption Time
(MAT)
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The MAT
Definition : mean time for the arrival of
bioavailable drug
Ka
K10
MAT
F = 100%
Administration
MAT =
1
Ka
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The MAT
How to evaluate the MAT
Po
IV
Ka
K10
1- IV administration
MRTIV = 1 / K10
2- Oral administration
MRToral longer than MRTIV
MRToral = 1 / K10 + 1 / Ka
MAT = MRToral - MRTIV = 1 / Ka
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The MAT
MAT and bioavailability
• The MAT measures the MRT at the
administration site and not the "rate" of
drug arrival in the central compartment
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The MAT
MAT and bioavailability
• Actually, the MAT is the MRT at the
injection site
• MAT does not provide information
about the absorption process unless
F = 100%
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The MAT
MAT and bioavailability
Ka1
K10
MAT
Ka2
F = Ka1 / (Ka1 +Ka2)
1
MRT oral =
Ka1 + Ka2
!
+
1
K10
=
1
Ka
+
1
K10
MAT is influenced by all processes of
elimination (absorption, degradation,…) located
at the administration site
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The MAT
MAT and bioavailability
1
2
1.5
1
K10
MAT = 1/(1+1) = 0.5h
0.5
K10
MAT= 1/(1.5+0.5)= 0.5h
0
K10
MAT=1/(0+2)=0.5h
Conclusion : by measuring (AUMC/AUC), the same
MAT will be obtained
! This does not mean that the absorption processes
towards the central compartment are equivalent
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The MAT
MAT and bioavailability
A
1
MATA =
!
0.5
1
B
1
(1 + 1)
= 0.5 h
4
MATB =
1
= 0.28 h
(4 + 0.5)
MATB < MATA
but
Absorption clearance of B is lower
than that of A !
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the MAT
• To accurately interpret the MAT in physiological
terms it is necessary to:
• express the rate of absorption using the clearance
Ka1
concept
Vabs
Clabs = Ka1 x Vabs
Clabs
! Vabs is unknown but this approach provides a
meaning to the comparison of 2 MAT when the
bioavailability is known
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The MAT
MAT and bioavailability
• Given a MAT of 5 h with F = 100%
• Clabs = Ka1 x Vabs = 0.2 L/h
Ka1 = 0.2 h-1
Vabs = 1 L
• Given a MAT of 5 h with F = 50%
• Clabs = Ka1 x Vabs = 0.1 L/h
0.1 h-1
Vabs = 1 L
0.1 h-1
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The Mean Dissolution Time
(MDT)
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The MDT
• in vitro measurement:
• dissolution test
• statistical moments approach
• modelling approach (Weibull)
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The MDT
• in vivo measurement (1)
:
solution
tablet
digestive tract
dissolution
blood
absorption
elimination
MRTtotal = MRTdissolution + MRTabsorption + MRT elimination
What is the dissolution rate of the pellet in
the digestive tract ?
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The MDT
• in vivo measurement (2) :
IV
IV administration
MRTIV = 6 h
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The MDT
• in vivo measurement (3) :
administration
of an oral solution
digestive tract
elimination
blood
oral administration of the drug
•
•
•
•
Computation of MRTpo, solution from plasma concentrations
MRToral, solution = MRTabsorption + MRTelimination = 8 h
MAT = MRTpo - MRTIV
MAT = 8h - 6h = 2h
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The MDT
• in vivo measurement (4) :
administration
solution
Tablet administration
• computation of MRToral,tablet from plasma concentrations
• MRToral,tablet=MRTdissolution + MAT + MRTelimination=18 h
• MRT dissolution= MRToral,tablet - (MAT + MRT IV)
• MRT dissolution= 18 - (2+6) = 10h
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Mean residence time in the
central compartment (MRTc)
and in the peripheral (tissue)
compartment (MRTT)
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MRTcentral and MRTtissue
• Definition : mean time for the analyte within
the measured compartment (MRTC) or outside
the compartment (MRTT)
MRTC
MRTsystem = MRTC + MRTT
MRTT
The MRT are additive
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MRTcentral and MRTtissue
Computations
• MRTC = AUC / Co =
1
K10
• MRTT = MRTsystem - MRTC
• MRTT =
AUMC
AUC
-
=
Vc
Cl
AUC
Co
N.B. : necessary to know Co accurately
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MRTcentral and MRTtissue
Relationship with the extent of distribution
MRTsystem
MRTcentral
Vss
=
Vc
This ratio measures the affinity for the peripheral
compartment
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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
central compartment to its next exit
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The Mean Transit Time
in the measurement (central)
compartment (MTTcentral)
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The MTTcentral
Calculation :
MTTC = - C(o)
dCp/dt
for t = 0
MTTC = - C(o) C'(o)
n
MTTC =  Yi
i =1
n
 Yi li
i =1
N.B. : necessary to know Co accurately
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The MTTcentral
Computation :
example for a bicompartmental model
• C(t) = 5 exp(-0.7t) + 2 exp(-0.07t)
• MTTC = (5 + 2) / (5 x 0.7 + 2 x 0.07) = 1.428 h
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The MTTcentral and number of visits
• Definition :
• The analyte "traveled" several times between
the central and peripheral compartment
• R is the average number of times the drug
molecule returns to the central compartment
after passage through it
MRTC
R=
MTTC
-1
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The MTTcentral and number of visits
MRTC
MTTC
=R+1
When there is no recycling
(monocompartmental model) R = 0 and :
MRTC
MTTC
=1
MRTC = MTTC
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The MTTcentral and number of visits
Bicompartmental model
K12
Vc
K21
K10
MTTC = 1 / (K10 + K12)
MTTC = 1 / (Cl + Cld)
R = K12 / K10
R = Cld / Cl
MTTC describes the first pass of the analyte in the
central compartment and does not take into account
the recirculating process of the distributed fraction.
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The Mean Transit Time in the
peripheral (tissue)
compartment (MTTtissue)
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The MTTtissue (MTTT)
Computation
• MTTT = MRTtissue / R
• MTTT =
MRTsystem - MRTcentral
R (visit)
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The MTTtissue
Computation : bicompartmental model
MTTT =
K12
1 = Vss - Vc
Cld
K21
=
Vt
Cld
K21
K10
MTTT : - does not rely on clearance
- measures drug affinity for
Jusko.J.Pharm.Sci 1988.7: 157 peripheral tissues
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Application of the MRT concept
• Interpretation of drug kinetics (1)
Gentamicin
5600e-0.218t + 94.9e-0.012t
Clearance (L/h)
2.39
12.0
Cld
(L/h)
Vss
(L)
54.8
585
Vc
(L)
14.0
33.7
VT
(L)
40.8
551.0
time : h
0.632
Digoxin
21.4e-1.99t + 0.881e-0.017t
52.4
concentration : mg l-1
Jusko.J.Pharm.Sci 1988.7: 157
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Application of the MRT concept
• Interpretation of drug kinetics (2)
Gentamicin
Digoxin
K12 (h-1)
0.045
1.56
K21 (h-1)
0.016
0.095
K10 (h-1)
0.170
0.338
R
0.265
4.37
Jusko.J.Pharm.Sci 1988.7: 157
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Application of the MRT concept
Interpretation of the mean times
Gentamicin
Digoxin
MTTcentral (transit time.
central comp)
4.65
0.532
MRTC (residence time.
central comp.)
5.88
2.81
MTTtissue (transit time
peripheral comp.)
64.5
10.5
MRTtissue (residence time
peripheral comp.)
17.1
46.0
MRTsystem (total)
23.0
48.8
Jusko.J.Pharm.Sci 1988.7: 157
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Stochastic interpretation of a kinetic
relationship
Cldistribution
MRTC
(all the visits)
MTTC
(for a single visit)
R
number
de visits
MRTT
(for all the visits)
MTTT
(for a single visit)
Clredistribution
Clelimination
MRTsystem = MRTC + MRTT
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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
4.4
ClR = 52 L/h
MTTT : 10.5h
MRTT : 46h
VT : 551 L
stochastic
Cl = 12 L/h
Determinist
1.56
h-1
Vc : 33.7 L
MRTsystem = 48.8 h
VT : 551L
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
0.265
MRTC : 5.88h
Vc : 14 L
ClR = 0.65 L/h
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
MTTT : 64.5h
MRTT : 17.1h
VT : 40.8 L
Clélimination = 2.39 L/h
MRTsystem = 23 h
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Interpretation determinist vs stochastic
Gentamicin vs digoxin
Digoxin
Determinist
Gentamicin
0.56 h-1
0.045 h-1
VT = 40.8 L
Vc = 14 L
0.016
0.17 h
h-1
0.095 h-1
0.338 h-1
t1/2 distribution : 0.3h
t1/2 : 4 h
t1/2 distribution : 3h
t1/2 : 57 h
MTTC: 4.65h Cld:0.65 L/h MTTT: 64.5h
MRTT:17.1h
0.26
MRTC: 5.88h
VT : 40.8 h
0.65 L/h
Vc = 14 L
Cl = 2.39 L/h
VT = 551 L
Vc = 34 L
MR system: 23 h
MTTC: 0.5h
MRTC: 2.81h
Vc = 34 L
Cld:52 L/h
4.4
ClR:52 L/h
Cl = 12 L/h
MR system: 48.8 h
MTTT: 10.5h
MRTT:46h
VT : 551 h
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MRTsystem
Computation
• Statistical moments
• Parameters from compartmental model
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Mean Residence Time
t1/2
MRT
=
0.693 Varea
Vss
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