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An introduction to population kinetics
Didier Concordet
NATIONAL VETERINARY SCHOOL Toulouse
Preliminaries
Definitions :
Random variable
Fixed variable
Distribution
Random or fixed ?
Definitions :
A random variable is a variable whose value changes when the experiment is begun again. The value it takes is drawn from a distribution.
A fixed variable is a variable whose value does not changewhen the experiment is begun again. The value it takes is chosen (directly or indirectly) by experimenter.
Example in kinetics
A kinetics experiment is performed on two groups of 10 dogs.
The first group of 10 dogs receives the formulation A of an active principle, the other group receives the formulation B.
The two formulations are given by IV route at time t=0.The dose is the same for the two formulations D = 10mg/kg.
For both formulations, the sampling times are t1 = 2 mn, t2= 10mn, t3= 30 mn,t4 = 1h, t5=2 h, t6 = 4 h.
t
V
Cl
V
DCt exp
Random or fixed ?
The formulation
Dose
The sampling times
The concentrations
The dogs
Fixed
Fixed
Fixed
Random
Fixed
Random
Analytical errorDeparture to kinetic model
Population kinetics
Classical kinetics
Distribution ?
The distribution of a random variable is defined by theprobability of occurrence of the all the values it takes.
Clearance0 0.1 0.2 0.3 0.48.07.8 8.2 8.4
Concentrations at t=2 mn
An example
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0 5 10 15 20 25 30 35 40 450
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0 5 10 15 20 25 30 35 40 45
30 horses
Time
Con
cent
rati
on
Step 1 : Write a PK (PK/PD) model
A statistical model
Mean model :functional relationship
Variance model :Assumptions on the residuals
Step 1 : Write a deterministic (mean) model to describe the individual kinetics
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0 10 20 30 40 50 60 70
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Step 1 : Write a deterministic (mean) model to describe the individual kinetics
t
V
Cl
V
DCt exp
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140
0 10 20 30 40 50 60 70
Step 1 : Write a deterministic (mean) model to describe the individual kinetics
residual
Step 1 : Write a model (variance) to describe the magnitude of departure to the
kinetics
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-5
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5
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25
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Time
Res
idua
l
Step 1 : Write a model (variance) to describe the magnitude of departure to the
kinetics
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Time
Res
idua
l
0 10 20 30 40 50 60 70
Step 1 : Describe the shape of departure to the kinetics
Time
Residual
Step 1 :Write an "individual" model
jijii
i
iji
i
i
iji t
V
Cl
V
Dt
V
Cl
V
DY ,,,, expexp
jiY ,
jit ,
jth concentration measured on the ith animal
jth sample time of the ith animal
residual
CVGaussian residual with unit variance
Step 2 : Describe variation between individual parameters
Distribution of clearancesPopulation of horses
Clearance0 0.1 0.2 0.3 0.4
Step 2 : Our view through a sample of animals
Sample of horses Sample of clearances
Step 2 : Two main approaches
Sample of clearances Semi-parametric approach
Step 2 : Two main approaches
Sample of clearances Semi-parametric approach(e.g. kernel estimate)
Step 2 : Semi-parametric approach
• Does require a large sample size to provide results
• Difficult to implement
• Is implemented on confidential pop PK softwares
Does not lead to bias
Step 2 : Two main approaches
Sample of clearances
0 0.1 0.2 0.3 0.4
Parametric approach
Step 2 : Parametric approach
• Easier to understand• Does not require a large sample size to provide (good or
poor) results• Easy to implement• Is implemented on the most popular pop PK softwares
(NONMEM, S+, SAS,…)
Can lead to severe bias when the pop PK is used as a simulation tool
Step 2 : Parametric approach
jijii
i
iji
i
i
iji t
V
Cl
V
Dt
V
Cl
V
DY ,,,, expexp
VVi
ClCli
i
i
V
Cl
ln
ln
CllnVln
A simple model :
Cl
V
ln Cl
ln V
Cl
V VCl,
Step 2 : Population parameters
Cl V
2
2
VVCl
VClCl
Step 2 : Population parameters
Mean parameters
Variance parameters : measure inter-individual variability
Step 2 : Parametric approach
jijii
i
iji
i
i
iji t
V
Cl
V
Dt
V
Cl
V
DY ,,,, expexp
VVi
CliiCli
i
i
V
ageBWCl
ln
ln 21
A model including covariables
CliiCli i
ageBWCl 21lnClln
BW
Age
Agei
BWi
ageBWCl 21
Cl
i
Step 2 : A model including covariables
Step 3 :Estimate the parameters of the current model
Several methods with different properties
• Naive pooled data• Two-stages• Likelihood approximations
• Laplacian expansion based methods• Gaussian quadratures
• Simulations methods
Naive pooled data : a single animal
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jjjj tV
Cl
V
Dt
V
Cl
V
DY
expexp
Does not allow to estimate inter-individual variation.
Time
Con
cent
rati
on
Two stages method: stage 1C
once
ntra
tion
jijii
i
iji
i
i
iji t
V
Cl
V
Dt
V
Cl
V
DY ,,,, expexp
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0 5 10 15 20 25 30 35 40 45
0
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0 5 10 15 20 25 30 35 40 45
0
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180
0 5 10 15 20 25 30 35 40 45 Time
11ˆ,ˆ VlC
22ˆ,ˆ VlC
33ˆ,ˆ VlC
nn VlC ˆ,ˆ
Two stages method : stage 2
Does not require a specific softwareDoes not use information about the distribution Leads to an overestimation of
which tends to zero when the number of observations per animal increases
Cannot be used with sparse data
VVi
ClCli
i
i
V
lC
ˆln
ˆln
The Maximum Likelihood Estimator
i
N
iiii dyhyl
1
,,expln,
VCl
iii ,
Let 222 ,,,,, VClVCl
i
N
iiii dyhArg
1
,,explninfˆ
The Maximum Likelihood Estimator
is the best estimator that can be obtained amongthe consistent estimators
It is efficient (it has the smallest variance)
Unfortunately, l(y,) cannot be computed exactly
Several approximations of l(y,)
Laplacian expansion based methods
First Order (FO) (Beal, Sheiner 1982) NONMEMLinearisation about 0
jiji
V
Cl
V
Cli
Vi
Vi
Cliji
V
Cl
V
jijii
i
iji
i
i
iji
tD
ZZZtD
tV
Cl
V
Dt
V
Cl
V
DY
,,
321,
,,,,
exp
expexp
exp
exp
expexp
exp
expexp
Laplacian expansion based methods
First Order Conditional Estimation (FOCE) (Beal, Sheiner) NONMEM Non Linear Mixed Effects models (NLME) (Pinheiro, Bates)S+, SAS (Wolfinger)
jiji
i
i
i
Vi
Vi
Cli
Clii
Vi
Vii
Cli
Cliiji
i
i
i
jijii
i
iji
i
i
iji
tV
lC
V
DZ
ZZtV
lC
V
D
tV
Cl
V
Dt
V
Cl
V
DY
,,3
21,
,,,,
ˆ
ˆexp
ˆˆˆˆ,
ˆˆ,ˆˆ,ˆ
ˆexp
ˆ
expexp
Linearisation about the current prediction of the individual parameter
Laplacian expansion based methods
First Order Conditional Estimation (FOCE) (Beal, Sheiner) NONMEM Non Linear Mixed Effects models (NLME) (Pinheiro, Bates)S+, SAS (Wolfinger)
jiji
i
i
i
Vi
Vi
Cli
Clii
Vi
Vii
Cli
Cliiji
i
i
i
jijii
i
iji
i
i
iji
tV
lC
V
DZ
ZZtV
lC
V
D
tV
Cl
V
Dt
V
Cl
V
DY
,,3
21,
,,,,
ˆ
ˆexp
ˆˆˆˆ,
ˆˆ,ˆˆ,ˆ
ˆexp
ˆ
expexp
Linearisation about the current prediction of the individual parameter
Gaussian quadratures
N
i
P
ki
kii
i
N
iiii
yh
dyhyl
1 1
1
,,expln
,,expln,
Approximation of the integrals by discrete sums
Simulations methods
Simulated Pseudo Maximum Likelihood (SPML)
,,ln,1 2
,,2 1 DVDy ii
DViii
K
kjiV
V
ClCl
ClV
ji tD
KKi
Ki
Ki1,,
,
,
,exp
expexp
exp
1
iV simulated variance
Minimize
Properties
Naive pooled data Never Easy to use Does not provide consistent estimate
Two stages Rich data/ Does not require Overestimation of initial estimates a specific software variance components
FO Initial estimate quick computation Gives quickly a resultDoes not provideconsistent estimate
FOCE/NLME Rich data/ small Give quickly a result. Biased estimates whenintra individual available on specific sparse data and/orvariance softwares large intra
Gaussian Always consistent and The computation is long quadrature efficient estimates when P is large
provided P is large
SMPL Always consistent estimates The computation is longwhen K is large
Criterion When Advantages Drawbacks
Step 4 : Graphical analysis
VVi
ClCli
i
i
V
Cl
ln
ln
VVi
CliiCli
i
i
V
ageBWCl
ln
ln 21
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Observed concentrations
Pre
dict
ed c
once
ntra
tions Variance reduction
Step 4 : Graphical analysis
Time
ji ,
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1
2
3
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The PK model seems good The PK model is inappropriate
Step 4 : Graphical analysis
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Cl
i
BW
Age
BW
Age
Variance model seems goodVariance model not appropriate
Step 4 : Graphical analysis
Normality acceptable
Cl
iV
i
under gaussian assumption
Cl
i
V
i
Normality should be questionedadd other covariablesor try semi-parametric model
To Summarise
Write a first model for individual parameters without any covariable
Write the PK model
Are there variations between individuals parameters ? (inspection of )
No
Sim
plif
y th
e m
odel
Yes
Check (at least) graphically the modelIs the model correct ?
No
Yes
Add covariables
Interpret results
What you should no longer believe
Messy data can provide good results
Population PK/PD is made to analyze sparse data
Population PK/PD is too difficult for me
No stringent assumption about the data is required