Transcript time

Lecture 4
Non-Linear and Generalized
Mixed Effects Models
Ziad Taib
Biostatistics, AZ
MV, CTH
Mars 2009
1
Date
Part II
Introduction to non-linear mixed
models in Pharmakokinetics
Typical data
Concentration
180
160
140
120
One curve per patient
100
80
60
40
20
0
0
5
10
15
20
25
30
35
Time
40
45
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
Concentrati
on
Tim
e
Pharmacokinetic models
Various types of
models
One-compartment model with rapid intravenous
administration: The pharmacokinetics parameters




Half life
Distribution volume
AUC
Tmax and Cmax
i.v.
k
D, V
D
•D: Dose
•VD: Volume
•k: Elimination rate
•Cl: Clearance
One compartment model
 General model
 Tablet
Dose k a
C (t )  F
(e  k e t  e  k a t )
V k a  ke
dC
 v in  v out
dt
 IV
C(t) , V
Vin
Ve
ka
ke
dC
  kC0
dt
Ct 
D
 Cl 
exp  
t
V
 V 
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
Fixed
Dose
Fixed
The sampling times
Fixed
The concentrations
Random
Analytical error
Departure to kinetic model
The patients
Random
Population kinetics
Fixed
Classical kinetics
An example
180
160
One PCC per patients
Concentration
140
120
100
80
60
40
Time
20
0
0
5
10
15
20
25
30
35
40
45
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
140
120
100
80
60
40
20
0
0
10
20
30
40
50
60
70
One compartment model with constant
intravenous infusion rate
D
C (t )  C0 exp  kt ; C0  ; Cl  kV
V
D
 Cl 
 C (t )  exp   t 
V
 V 
t
D
 Cl 
C  exp   t 
V
 V 
Step 1 : Write a deterministic (mean)
model to describe the individual kinetics
140
D
 Cl 
C (t )  exp   t 
V
 V 
120
100
80
60
40
20
0
0
10
20
30
40
50
60
70
Step 1 : Write a deterministic (mean)
model to describe the individual kinetics
140
120
100
residual
80
60
40
20
0
0
10
20
30
40
50
60
70
Step 1 : Write a model (variance) to
describe the magnitude of departure to
the kinetics
25
20
15
Residual
10
5
0
0
10
20
30
40
50
60
70
-5
-10
-15
-20
-25
Time
Step 1 : Write a model (variance) to
describe the magnitude of departure to
the kinetics
25
20
15
Residual
10
5
0
0
10
20
30
40
50
60
70
-5
-10
-15
-20
-25
Time
Step 1 : Describe the shape of departure
to the kinetics
Residual
0
10
20
30
40
50
60
70
Time
Step 1 :Write an "individual" model
Yi , j jth concentration measured on the ith patient
ti , j
jth sample time of the ith patient
residual
 Cli 
 Cli 
D
D
Yi , j  exp  
ti , j    exp  
ti , j  i , j
Vi
Vi
 Vi

 Vi

Gaussian residual with unit variance
Step 2 : Describe variation between
individual parameters
0
Population of patients
0.1
0.2
0.3
0.4
Distribution of clearances
Clearance
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
0
Sample of clearances
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
 Cli 
 Cli 
D
D
Yi , j  exp  
ti , j    exp  
ti , j  i , j
Vi
Vi
 Vi

 Vi

A simple model :
ln Cli  Cl   i

V
 ln Vi  V   i
Cl
ln V
ln Cl
Step 2 : Population parameters
ln V
V
 Cl,V 
V
 Cl
Cl
Mean parameters
Cl V
ln Cl
  Cl2
 Cl V   Variance parameters :

  
2
 measure inter-individual




V
 Cl V

variability
Step 2 : Parametric approach
 Cli 
 Cli 
D
D
Yi , j  exp  
ti , j    exp  
ti , j  i , j
Vi
Vi
 Vi

 Vi

A model including covariates
ln Cli   Cl  θ1 X 1i  θ2 X 2i   i

V
 ln Vi  V   i
Cl
Step 3 :Estimate the parameters of the
current model
Several methods with different properties
1. Naive pooled data
2. Two-stages
3. Likelihood approximations
1.
Laplacian expansion based metho
2.
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.
 Cl 
 Cl 
D
D
Y j  exp  
t j    exp  
t j  j
 V

 V

V
V




Concentration
180
The naïve approach does not
allow to estimate interindividual variation.
160
140
120
100
80
60
40
20
0
0
5
10
15
20
25
30
35
40
Time
45
Concentration
2. Two stages method: stage 1
Within individual variability
 Cli 
 Cli 
D
D
Yi , j  exp  
ti , j    exp  
ti , j  i , j
Vi
Vi
180
 Vi

 Vi

160
160
Cˆ l1 ,Vˆ1
140
Cˆ l2 , Vˆ2
120
Cˆ l ,Vˆ



100
100
3
3



.
.
.
80
60
60
Cˆl ,Vˆ 
40
n
20
20
n
0
00
55
10
10
15
20
20
25
30
35
40
45
Time
Two stages method : stage 2
Between individual variability
ln Cˆ li  Cl   Cl
i
 ˆ
V
ln
V





i
V
i
• 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
3. The Maximum Likelihood Estimator
Let
  Cl , V ,  ,  ,  , 
   ,
i
Cl
V
i
i

N
l  y,    ln
i 1
ˆ  Arg inf 
2
Cl
2
V
2

 exp  h  , y , d
i
i
i
i
N
 ln  exp  h  , y , d
i 1
i
i
i
i
The Maximum Likelihood Estimator
ˆ
•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) NONMEM
Linearisation about 0
 Cli 
 Cli 
D
D
Yi , j  exp  
ti , j    exp  
ti , j  i , j
Vi
Vi
 Vi

 Vi

 exp  Cl  
D

exp  
ti , j   Z1  iCl  Z 2  iV  Z 3  iViCl
exp V 
 exp V  
 exp  Cl  
D

exp  
ti , j  i , j
exp V 
 exp V  
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
 Cli 
 Cli 
D
D
Yi , j  exp  
ti , j    exp  
ti , j  i , j
Vi
Vi
 Vi

 Vi

 Cˆ li 
D
 exp  
ti , j   Z1  ,ˆi  iCl  ˆiCl  Z 2  ,ˆi  iV  ˆiV
ˆ
Vˆi
V
i


 Cˆ li 
D
Cl
Cl
V
V
 Z 3  ,ˆi  i  ˆi i  ˆi   exp  
ti , j  i , j
ˆ
Vˆi
 Vi








Gaussian quadratures
Approximation of the integrals by discrete sums
N
l  y,    ln  exp  hi i , yi , di
i 1
  ln  exp  hi ik , yi , 
N
P
i 1
k 1
4. Simulations methods
Simulated Pseudo Maximum Likelihood (SPML)
Minimize
1
2
i  2 yi  i  , D  Vi1  ,D,   ln Vi  , D, 
Cl
K


exp



1
D
Cl


i , j    
exp
t
i
,
j
 exp V  V

K k 1 exp V   Cl



i ,K



Vi   simulated variance
i ,K
i ,K


Properties
Criterion
When
Advantages
Drawbacks
Naive pooled data
Never
Easy to use
Does not provide
consistent estimate
Two stages
Rich data/
initial estimates
Does not require
a specific software
Overestimation of
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.
intra individual available on specific
variance
softwares
Biased estimates when
sparse data and/or
large intra
Gaussian
quadrature
Always
consistent and
efficient estimates
provided P is large
The computation is long
when P is large
SMPL
Always
consistent estimates
The computation is long
when K is large
Model check: Graphical analysis
Predicted concentrations
ln Cli  Cl   Cli

V
ln
V





i
V
i
ln Cli  Cl  1BWi   2 agei   Cli

V
ln
V





i
V
i
180
160
160
140
140
Variance reduction
120
120
100
100
80
80
60
60
40
40
20
20
0
0
0
20
40
60
80
100
120
140
0
20
40
Observed concentrations
60
80
100
120
140
Graphical analysis
ˆi, j
3
3
2
2
1
1
0
0
10
20
30
40
50
0
0
5
10
15
20
25
30
35
40
45
-1
-1
-2
-2
-3
-4
The PK model seems good
Time
-3
The PK model is inappropriate
Graphical analysis
̂V
i
̂ Cl
i
under gaussian assumption
̂ Cl
i
̂V
i
Normality should be questioned
Normality acceptable
add other covariates
or 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