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Ziad Taib
Biostatistics, AZ
MV, CTH
Mars 2009
Lecture 4
Non-Linear and Generalized Mixed Effects Models
1 Date
Part IIIntroduction to non-linear mixed
models in Pharmakokinetics
Typical data
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One curve per patient
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cent
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Common situation (bio)sciences:
A continuous response evolves over time (or other condition) within individuals from a population of interest
Scientific interest focuses on features or mechanisms that underlie individual time trajectories of the response and how these vary across the population.
A theoretical or empirical model for such individual profiles, typically non-linear in the parameters that may be interpreted as representing such features or mechanisms, is available.
Repeated measurements over time are available on each individual in a sample drawn from the population
Inference on the scientific questions of interest is to be made in the context of the model and its parameters
Non linear mixed effects models
Nonlinear mixed effects models: or hierarchical non-linear models
A formal statistical framework for this situation
A “hot” methodological research area in the early 1990s
Now widely accepted as a suitable approach to inference, with applications routinely reported and commercial software available
Many recent extensions, innovations
Have many applications: growth curves, pharmacokinetics, dose-response etc
PHARMACOKINETICS
A drugs can administered in many different ways: orally, by i.v. infusion, by inhalation, using a plaster etc.
Pharmacokinetics is the study of the rate processes that are responsible for the time course of the level of the drug (or any other exogenous compound in the body such as alcohol, toxins etc).
PHARMACOKINETICS
Pharmacokinetics is about what happens to the drug in the body. It involves the kinetics of drug absorption, distribution, and elimination i.e. metabolism and excretion (adme). The description of drug distribution and elimination is often termed drug disposition.
One way to model these processes is to view the body as a system with a number of compartments through which the drug is distributed at certain rates. This flow can be described using constant rates in the cases of absorbtion and elimination.
Plasma concentration curves (PCC)
The concentration of a drug in the plasma reflects many of its properties. A PCC gives a hint as to how the ADME processes interact. If we draw a PCC in a logarithmic scale after an i.v. dose, we expect to get a straight line since we assume the concentration of the drug in plasma to decrease exponentially. This is first order- or linear kinetics. The elimination rate is then proportional to the concentration in plasma. This model is approximately true for most drugs.
Plasma concentration curve
Concentration
Time
Pharmacokinetic models
Various types of models
One-compartment model with rapid intravenous
administration: The pharmacokinetics parameters
Half life
Distribution volume
AUC
Tmax and Cmax
D, VD
i.v. k
•D: Dose•VD: Volume•k: Elimination rate•Cl: Clearance
0kCdt
dC
One compartment model
General model Tablet
IV
dC
dtv in vout
)()( tktk
ea
a ae eekk
k
V
DoseFtC
Vin
C(t) , V
Ve
ka ke
t
V
Cl
V
DCt exp
Typical example in kinetics
A typical kinetics experiment is performed on a number, m, of groups of h patients.
Individuals in different groups receive the same formulation of an active principle, and different groups receive different formulations.
The formulations are given by IV route at time t=0.The dose, D, is the same for all formulations.
For all formulations, the plasma concentration is measured at certain sampling times.
Random or fixed ?
The formulation
Dose
The sampling times
The concentrations
The patients
Fixed
Fixed
Fixed
Random
Fixed
Random
Analytical errorDeparture to kinetic model
Population kinetics
Classical kinetics
An example
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One PCC per patients
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Step 1 : Write a (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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One compartment model with constant intravenous infusion rate
tV
Cl
V
DtC
kVClV
DCktCtC
exp)(
; ;exp)( 00
t
V
Cl
V
DCt exp
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Step 1 : Write a deterministic (mean) model to describe the individual kinetics
t
V
Cl
V
DtC 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
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idua
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Step 1 : Write a model (variance) to describe the magnitude of departure to the kinetics
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idua
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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 patient
jth sample time of the ith patient
residual
Gaussian residual with unit variance
Step 2 : Describe variation between individual parameters
Distribution of clearancesPopulation of patients
Clearance0 0.1 0.2 0.3 0.4
Step 2 : Our view through a sample of patients
Sample of patients Sample of clearances
Step 2 : Two main approaches:parametric and semi-parametric
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 “commercial” PK software
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 software (NONMEM, S+, SAS,…)
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 VMean parameters
2
2
VVCl
VClCl
Variance parameters :
measure inter-individualvariability
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
XθXθCl
ln
ln 2211
A model including covariates
Step 3 :Estimate the parameters of the current model
Several methods with different properties
1. Naive pooled data2. Two-stages3. Likelihood approximations
1. Laplacian expansion based methods2. Gaussian quadratures
4. Simulations methods
1. Naive pooled data : a single patient
Naïve Pooled Data combines all the data as if they came from a single reference individual and fit into a model using classical fitting procedures. It is simple, but can not investigate fixed effect sources of variability, distinguish between variability within and between individuals.
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jjjj tV
Cl
V
Dt
V
Cl
V
DY
expexp
The naïve approach does not allow to estimate inter-individual variation.
Time
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cent
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2. Two stages method: stage 1Within individual variability
Con
cent
rati
on
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 40 450
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11ˆ,ˆ VlC
22ˆ,ˆ VlC
33ˆ,ˆ VlC
nn VlC ˆ,ˆ
.
.
.
Two stages method : stage 2 Between individual variability
• Does not require a specific software
• Does not use information about the distribution
• Leads to an overestimation 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
3. 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 among the consistent estimators
•It is efficient (it has the smallest variance)
•Unfortunately, l(y,) cannot be computed exactly
•Several approximations of l(y,) are used.
3.1 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
Gaussian quadratures
N
i
P
ki
kii
i
N
iiii
yh
dyhyl
1 1
1
,,expln
,,expln,
Approximation of the integrals by discrete sums
4. 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
Model check: 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
Graphical analysis
Time
ji,
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The PK model seems good The PK model is inappropriate
Graphical analysis
Normality acceptable
Cl
iV
i
under gaussian assumption
Cl
i
V
i
Normality should be questioned
add other covariatesor try semi-parametric model
The Theophylline example
An alkaloid derived from tea or produced synthetically; it is a smooth muscle relaxant used chiefly for its bronchodilator effect in the treatment of chronic obstructive pulmonary emphysema, bronchial asthma, chronic bronchitis and bronchospastic distress. It also has myocardial stimulant, coronary vasodilator, diuretic and respiratory center stimulant effects.
http://www.tau.ac.il/cc/pages/docs/sas8/stat/chap46/sect38.htm
References
Davidian, M. and Giltinan, D.M. (1995). Nonlinear Models for Repeated Measurement Data. Chapman & Hall/CRC Press.
Davidian, M. and Giltinan, D.M. (2003). Nonlinear models for repeated measurement data: An overview and update. Journal of Agricultural, Biological, and Environmental Statistics 8, 387–419.
Davidian, M. (2009). Non-linear mixed-effects models. In Longitudinal Data Analysis, G. Fitzmaurice, M. Davidian, G. Verbeke, and G. Molenberghs (eds). Chapman & Hall/CRC Press, ch. 5, 107–141.
(An outstanding overview ) “Pharmacokinetics and pharmaco- dynamics ,” by D.M. Giltinan, in Encyclopedia of Biostatistics, 2nd edition.
Any Questions?
