An introduction to population kinetics Didier Concordet NATIONAL VETERINARY SCHOOL Toulouse.

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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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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Step 1 : Write a deterministic (mean) model to describe the individual kinetics

t

V

Cl

V

DCt exp

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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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Time

Res

idua

l

Step 1 : Write a model (variance) to describe the magnitude of departure to the

kinetics

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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

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cent

rati

on

Two stages method: stage 1C

once

ntra

tion

jijii

i

iji

i

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iji t

V

Cl

V

Dt

V

Cl

V

DY ,,,, expexp

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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

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once

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tions Variance reduction

Step 4 : Graphical analysis

Time

ji ,

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The PK model seems good The PK model is inappropriate

Step 4 : Graphical analysis

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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