An introduction to population kinetics

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An introduction to population kinetics

• Didier Concordet

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Preliminaries
Definitions
Random variable
Fixed variable
Distribution
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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 change when the experiment is begun again.
The value it takes is chosen (directly or
indirectly) by experimenter.
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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 t0. 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, t52 h, t6
4 h.
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Random or fixed ?

The formulation
Fixed
Fixed
Dose
The sampling times
Fixed
Analytical error Departure to kinetic model
The concentrations
Random
The dogs
Random
Population kinetics
Classical kinetics
Fixed
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Distribution ?
The distribution of a random variable is defined
by the probability of occurrence of the all the
values it takes.
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An example
30 horses
Concentration
Time
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Step 1 Write a PK (PK/PD) model
A statistical model
Mean model functional relationship
Variance model Assumptions on the residuals
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Step 1 Write a deterministic (mean) model to
describe the individual kinetics
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Step 1 Write a deterministic (mean) model to
describe the individual kinetics
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Step 1 Write a deterministic (mean) model to
describe the individual kinetics
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Step 1 Write a model (variance) to describe the
magnitude of departure to the kinetics
Residual
Time
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Step 1 Write a model (variance) to describe the
magnitude of departure to the kinetics
Residual
Time
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Step 1 Describe the shape of departure to the
kinetics
Residual
Time
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Step 1 Write an "individual" model
jth concentration measured on the ith animal
jth sample time of the ith animal
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Step 2 Describe variation between individual
parameters
Distribution of clearances
Population of horses
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Step 2 Our view through a sample of animals
Sample of horses
Sample of clearances
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Step 2 Two main approaches
Sample of clearances
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Step 2 Two main approaches
Sample of clearances
Semi-parametric approach (e.g. kernel estimate)
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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
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Step 2 Two main approaches
Sample of clearances
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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
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Step 2 Parametric approach
A simple model
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Step 2 Population parameters
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Step 2 Population parameters
Mean parameters
Variance parameters measure inter-individual
variability
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Step 2 Parametric approach
A model including covariables
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Step 2 A model including covariables
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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

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Naive pooled data a single animal
Does not allow to estimate inter-individual
variation.
Concentration
Time
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Two stages method stage 1
Concentration
Time
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Two stages method stage 2
Does not require a specific software Does not use
information about the distribution Leads to an
overestimation of W which tends to zero when the
number of observations per animal
increases Cannot be used with sparse data
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The Maximum Likelihood Estimator
Let
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The Maximum Likelihood Estimator
is the best estimator that can be obtained
among the consistent estimators
It is efficient (it has the smallest variance)
Unfortunately, l(y,q) cannot be computed exactly
Several approximations of l(y,q)
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Laplacian expansion based methods
First Order (FO) (Beal, Sheiner 1982)
NONMEM Linearisation about 0
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Laplacian expansion based methods
First Order Conditional Estimation (FOCE) (Beal,
Sheiner) NONMEM Non Linear Mixed Effects models
(NLME) (Pinheiro, Bates)S, SAS (Wolfinger)
Linearisation about the current prediction of the
individual parameter
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Laplacian expansion based methods
First Order Conditional Estimation (FOCE) (Beal,
Sheiner) NONMEM Non Linear Mixed Effects models
(NLME) (Pinheiro, Bates)S, SAS (Wolfinger)
Linearisation about the current prediction of the
individual parameter
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Gaussian quadratures
Approximation of the integrals by discrete sums
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Simulations methods
Simulated Pseudo Maximum Likelihood (SPML)
Minimize
simulated variance
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Properties
Criterion When Advantages
Drawbacks
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 result Does not
provide consistent estimate FOCE/NLME Rich
data/ small Give quickly a result. Biased
estimates when intra individual available on
specific sparse data and/or variance 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 long when K is large
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Step 4 Graphical analysis
Variance reduction
Predicted concentrations
Observed concentrations
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Step 4 Graphical analysis
Time
The PK model is inappropriate
The PK model seems good
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Step 4 Graphical analysis
Age
Age
BW
BW
Variance model seems good
Variance model not appropriate
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Step 4 Graphical analysis
Normality should be questioned
add other covariables or try semi-parametric model
Normality acceptable
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To Summarise
Write the PK model
Write a first model for individual parameters
without any covariable
Interpret results
Add covariables
Are there variations between individuals
parameters ? (inspection of W)
Check (at least) graphically the model Is the
model correct ?
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What you should no longer believe
Messy data can provide good results
Population PK/PD is made to analyze sparse data
No stringent assumption about the data is required
Population PK/PD is too difficult for me