F. Mentré
1
Nonlinear mixed effect models for
the analysis and design of
bioequivalence/
biosimilarity
studies
Pr France Mentré
Anne Dubois, Thu

Thuy N‘Guyen, Caroline Bazzoli
UMR 738: Models and Methods for Therapeutic
Evaluation of Chronic Diseases
INSERM
–
Université Paris Diderot
F. Mentré
2
OUTLINE
1.
Introduction
Pharmacometrics
Nonlinear mixed effect models (NLMEM)
Bioequivalence/
biosimilarity
studies
2.
Bioequivalence test in NLMEM
Method
Simulation
Real example
3.
Optimal design in NLMEM
Method
Simulation
Real example
4.
Pharmacometrics
and drug development
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1. INTRODUCTION
•
Pharmacometrics
•
Nonlinear mixed effect models
•
Bioequivalence/ biosimilarity studies
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PHARMACOMETRICS
Clinical pharmacology = PK + PD
Both during drug development and in clinical care
Main statistical tool:
Nonlinear Mixed Effect Models
Data generated
during clinical trials
& patient care
Rational drug
development &
pharmacotherapy
Knowledge extraction
Pharmacometricians
Dose
Concentration
Effect
Pharmacokinetics
Pharmacodynamics
Design
+ Disease models
The science of quantitative clinical pharmacology
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5
Pharmacokinetics (PK)
Study of the time course of drug in the body
PK parameters: CL, V…
Pharmacodynamics (PD)
Study of the effects of drug in the body
PD parameters: Emax, EC50
Analysis of PK/PD data: 2 types of approach
PK/PD data
Non compartmental approach
(NCA)
Model

based approaches
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PK: Non
compartmental
approach
Few hypotheses
>10 concentrations measurements per subject
Trials in healthy volunteers
Computation
Parameters of interest
–
Area under the curve (AUC)
–
Different time intervals: 0

tlast, 0

infinity
–
Extrapolation between tlast and infinity, computation of
terminal slope
–
Maximum concentration (Cmax)
–
Half

life (linear regression on last log concentrations)
Algorithm: linear or log

linear trapezoidal method
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PK models: Human body described as a set of
compartments
Physiological parameters: Clerance of elimination, volume of
distribution, rate constants
Example: 1 cp model with first order absorption and elimination
PK: Model

based
approach
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Advantages of modelling
Quantitative summary of time

profiles in few
'physiological' parameters
Prediction/simulation for other doses …
Test of hypotheses on mechanisms of action of drugs
Comparison of groups of patients through parameters
Comparaison of response to different treatment
Analysis of all longitudinal data in clinical trials
….
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DESIGNS
Experimental (rich)
Limited number of individuals (N = 6 to 50)
Numerous measures per subject (n=6 to 20)
Generally studies of short duration
Identical, balanced sampling protocols
Examples : in vitro, preclinical PK, phase I
Analysis of information of each individual separately
then summary statistics or global approach
Population (sparse)
Large number of individuals (N = 50 to 2000)
Few measures by subject (n = 1 to 10)
Various and unbalanced sampling protocols
Repeated doses, chronic administration
Example : PK/PD in phase IIb, III, post AMM, NDA
Analysis of information on all individuals together
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Analysis of sparse or rich data
Global analysis of data in all individuals
Parametric PKPD Model: nonlinear in parameters
One individual
one vector of parameters
Set of individuals
Same parametric model
Inter

individual variability
inter

parameter variability
Statistical model
PKPD parameters are random in the population
Mixed effects model (random + fixed)
Nonlinear mixed effects models
Also called ‘population approach’
Nonlinear
mixed
effect
models
(1)
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Single

stage approach
(population analysis)
Estimates of
individual
parameters
?
m ?
sd ?
#1
#2
#n
?
Non linear mixed
effects model
Nonlinear
mixed
effect
models
(2)
From Steimer (1992) : «
Population models and methods, with emphasis on
pharmacokinetics
», in M. Rowland and L. Aarons (eds),
New strategies in drug
development and clinical evaluation, the population approach
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12
Nonlinear
mixed
effect
models
(3)
Increasingly used
in all phases of drug development for analysis of PKPD data
in clinical use of drug for analysis of PKPD variability and for
therapeutic drug monitoring
for analysis of response in clinical trials and cohorts
Relies on several assumptions
structural model (model nonlinear with respect to parameters)
model for between

subject variability (assumption on random
effects)
model for residual error
Research in estimation methods, covariate testing,
optimal design, model evaluation
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13
Bioequivalence
/
biosimilarity
studies
Trials comparing pharmacokinetics of several
formulations of the same drug
used for generic development and for formulation of biologics
FDA and EMEA guidelines
Two

periods, two

sequences crossover trials
Compute AUC and C
max
by non compartmental analysis
Test on log parameters using linear mixed effects model
–
with treatment, period and sequence effects
Limitations of NCA
>10 samples per subject
→
study on healthy volunteers
Estimation of AUC and C
max
by NCA not appropriate for nonlinear PK or
complex PKPD models (similarity of kinetics of drug effect)
Parameters assumed to be estimated without error
Omit data below quantification limit
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14
Objectives
Propose and develop
1.
estimation methods and tests
2.
optimal design tool with prediction of number of
subjects needed
for bioequivalence/biosimilarity analysis using NLMEM
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2. BIOEQUIVALENCE TEST IN
NLMEM
•
Method
•
Simulation
•
Real example
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y
ijk
concentration for individual
i =1,…,N
at sampling
time
j=1,…,n
ik
for period
k=1,…,K
f
ik
individual parameter
ijk
residual error
Parameters:
fixed effects, variance of random effects,
a and b in error model
Statistical model
ijk
ik
ijk
ijk
t
f
y
)
,
(
)
,
(
with
)
,
0
(
~
ijk
ijk
ijk
t
bf
a
N
(WSV)
riability
subject va

within
)
,
0
(
~
(BSV)
riability
subject va

between
)
,
0
(
~
)
log(
)
log(
with
)
log(
)
log(
N
N
S
P
T
ik
i
i
S
k
P
ik
T
ik
i
ik
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Maximum
Likelihood
Estimation in NLMEM
Problem
: no close
form
for the
likelihood
in NLMEM
Several
statistical
developments
and
specific
software
Linearization
algorithms
: FO, FOCE
Not consistent
Very
sensitive to initial conditions
More
recent
algorithms
without
linearization
Adaptative
Gaussian
Quadrature
–
Only
for model
with
small
number
of
random
effects
Stochastic
Approximation EM
–
Extension of EM
algorithm
with
proven
convergence
Delyon
,
Lavielle
&
Moulines
(
1999
)
.
Convergence
of
a
stochastic
approximation
version
of
the
EM
procedure
.
Ann
Stat
,
27
:
94

128
.
Kuhn,
Lavielle
(
2005
)
.
Maximum
likelihood
estimation
in
nonlinear
mixed
effect
models
,
Comput
Stat
Data
Anal
,
49
:
1020

1038
.
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SAEM
algorithm
EM
algorithm
E

step
:
expectation
of
the
log

likelihood
of
the
complete
data
M

step
:
maximisation
of
the
log

likelihood
of
the
complete
data
Mixed

effects
models
individual
random

effects
=
missing
data
Problem
in
NLMEM
:
no
close
form
for
the
E
step
SAEM
:
decomposition
of
E

step
in
2
steps
S

step
:
simulation
of
individual
parameters
using
MCMC
SA

step
:
stochastic
approximation
of
expected
likelihood
Various
extensions
Samson,
Lavielle
,
Mentré
(2006).
Extension of the SAEM algorithm to left censored data in
non

linear mixed

effects model: application to HIV dynamics model.
Comput Stat Data
Anal
51: 1562

74.
Samson,
Lavielle
,
Mentré
(2007).
The SAEM algorithm for group comparison tests in
longitudinal data analysis based on non

linear mixed

effects model: application to HIV
dynamics model.
Stat Med
, 26: 4860

75.
…
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MONOLIX software
Free Matlab software implementing SAEM
developed under supervision of Pr Marc Lavielle at INRIA
www.monolix.org
stand

alone version using MCR
v1.1 available since Feb 2005
v3.1 released in October 2009
Success of MONOLIX
Team of 4 development engineer from INRIA
Grant from ANR (2005

2008)
Use in academia and drug companies
Monolix project : Support from drug companies
Success of SAEM
Now implemented in NONMEM, most used software in the area
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Software for estimation in NLMEM
Maximum likelihood
Bayesian
estimation
Parametric
NONMEM
(FO, FOCE, Laplace,
SAEM
)
WinNonMix
R:
nlme
(FOCE)
SAS: Proc NLMIXED
(FO,
FOCE, AGQ)
Ppharm
(ITS)
M
ONOLIX
(
SAEM
)
S

ADAPT (MCPEM)
PDX

MCPEM
PK BUGS
Nonparametric
NPML
NPEM (USC*PACK)
NONMEM
Dirichlet
process
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Tests in
bioequivalence
trials
Global estimation
with
SAEM
algorithm
Estimation
with
the
complete
model
with
treatment
,
period
and
sequence
effect
on all
parameters
, WSV
in addition to BSV
Extension of the SAEM
algorithm
SE
derived
from
Fisher information
matrix
T
:
treatment
effect
on one log

parameter
Bioequivalence
test
H
0
: {
T
≤

D
L
or
T
≥
+
D
L
}
H
1
: {

D
L
≤
T
≤
+
D
L
}
Schuirmann
test or TOST:
unilateral
test for H
0,

D
and H
0,+
D
Reject
H
0
with
=5%:
–
if H
0,

D
and H
0,+
D
rejected
with
=5%
–
if 90%CI of
T
included
in [

D
L
;
+
D
L
]
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Wald and LRT for
bioequivalence
Wald tests
TOST for
T
from
SE for
parameter
in model (
e.g
. AUC)
For
secondary
parameters
(
e.g
.
Cmax
)
–
Derivation
of SE by delta
method
or simulation
LRT
Complete model: log

likelihood
L
all
For
parameter
in model: estimation
with
T
fixed
to

D
L
or
+
D
L
–
log
likelihood
L

D
or
L
+
D
Reject
of H
0
if
D
D
T
ˆ
AND
)
1
(
)
(
2
2
1
D
all
L
L
)
1
(
)
(
2
2
1
D
all
L
L
[10] Panhard, Samson. Biostatistics. 2009
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Evaluation by simulation: design
PK model
with
one

compartment
: k
a
, V/F, CL/F
Two

periods
of four

periods
crossover
trial
Treatment
effect
on CL/F and V/F
Equivalence
limit
D
L
= 0.2
Two designs with N = 40 patients
Original n=10 , Sparse: n=3 measurements/ patient/ period
Two
levels
of
variability
Random
effects
–
Low
variability
(BSV=20%, WSV=10%):
S
l,l
–
High
variability
(BSV=50%, WSV=15%):
S
h,l
Error
model:
Low
variability
(a=0.1, b=10%)
(
Panhard
&
Mentré
,
Stat Med
, 2005; Dubois,
Gsteiger
,
Pigeolet
&
Mentré
,
Pharm
Res
, 2009)
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Evaluation by simulation:
method
1000 simulated trials under H
0,

D
and H
0,+
D
for each design
and each variability setting
Analysis by SAEM in MONOLIX v2.4
Evaluation of extension of SAEM
Computation of bias, RMSE
Designs with 2 or 4 periods for H
0,

D
Evaluation of type I error of Wald and LRT for
bioequivalence on AUC and C
max
For H
0,

D
and H
0, +
D
,
estimated by the proportion of simulated
trials for which the null hypothesis is rejected
Designs with 2 periods
)
,
max(
,
0
,
0
D
D
H
H
global
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Simulated datasets
S
l,l
S
h,l
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Boxplot
of
estimates
for CL/F, 2 or 4
periods
S
l,l
S
h,l
* : true value
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Relative RMSE
fixed
effects

2 or 4
periods
S
l,l
S
h,l
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Relative RMSE variances, 2 or 4
periods
S
l,l
S
h,l
RMSE (rich design) < RMSE (sparse design)
RMSE (4 periods) < RMSE (2 periods)
RMSE satisfactory except for WSV on V/F for low variability
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Type I
error
for
bioequivalence
and 2
periods
Type I error at 5% for the rich design
Slight inflation of the type I error for the sparse design and large
variability
Close results for Wald test and LRT on AUC
S
l,l
S
h,l
W: Wald test
L: LRT
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Conclusion on estimation and test
SAEM algorithm in MONOLIX software
Accurate extension for estimation of WSV and crossover trials
analysis
Model

based
bioequivalence tests
Good tool applicable to rich and sparser design
Good statistical properties under asymptotic conditions
Wald test simpler than LRT and extended for secondary
parameters
Correction of SE needed for small sample size and large
variability
Usefulness of extension of MONOLIX as an efficient tool
for analysis of bioequivalence/
biosimilarity
trials
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3. OPTIMAL DESIGN IN NLMEM
•
Method
•
Simulation
•
Real example
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32
Design for
‘Population’
PKPD analyses
Problem beforehand: choice of ‘
population’ design
number of individuals? number of sampling times?
sampling times?
Increasingly important task for pharmacologists
Difficult to 'guess' good designs for complex models
Importance of the choice
influence the precision of parameters estimation and power of
test
poor design can
lead to
unreliable studies (complex models)
all the more important in special population (paediatric studies …)
–
severe limitations on the number of samples to be taken
–
ethical and physiological reasons
Design considerations for population PK(PD) analyses
stress out in FDA and EMEA guidelines
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Population design evaluation/optimisation
From
given cost (number of samples)
experimental constraints
statistical model and a priori values of parameters
Evaluate/compare designs
Predict standard error for each population parameter
Find best design
smallest standard errors
greatest information in the data
Two approaches
simulation studies
mathematical derivation of the Fisher Information matrix
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Population Design
N individuals i at K periods k
Elementary design
x
i
in individual i
Total of n
i
samples
Compose of the union of designs
x
ik
of each period k
–
number of samples n
ik
and sampling times: t
ik1
…t
ikn
ik
Population design
set of elementary designs
X
{
x
1
,
...,
x
N
}
number of observations
n
tot
=
S
n
i
Often few elementary designs
Q groups of N
q
individuals
same design
x
q
at each period of a total of n
q
sampling times
n
tot
=
S
N
q
n
q
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Fisher information matrix (1)
Vector
of
parameters
in NLMEM:
Y
•
Fixed
effects
:
and
•
Variance of
random
effects
:
and
•
Parameter
in
error
variance:
a
and
b
Information
Matrix
for population design
X
=
{
x
1
,
...,
x
N
}
Information Matrix for elementary design
x
i
}
)'
l(y;
log
)
l(y;
log
{
E
=
)
,
(
y
y
y
y
x
Y
i
MF
)
,
i
(
MF
N
1
i
)
(
F
M
Y
x
Y
,
X
F. Mentré
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Fisher information matrix (2)
(Mentré, Mallet & Baccar,
Biometrika
, 1997
; Retout, Mentré & Bruno,
Stat Med
,
2002; Retout & Mentré,
J Biopharm Stat
, 2002; Bazzoli, Retout & Mentré,
Stat Med
, 2009; N’Guyen, Bazzoli & Mentré,
ACOP
2009
)
Nonlinear structural models
no analytical expression for MF(
x
,
Y
)
first order expansion of f about
random
effects
taken at
0
analytical expressions for MF(
x
,
,
) and MF(
x
,
,
,
a,b
)
)
,
,
,
,
(
0
0
)
,
,
(
)
,
(
b
a
MF
MF
MF
x
x
y
x
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37
Models with discrete covariates
Additional fixed

effects to be estimated:
Evaluation of MF requires specification of
–
the
expected distribution of covariates
in the population
–
the effect size
Evaluation of the expected (mean) information matrix over
covariate distribution
prediction of the “expected” SE of each
Power of Wald comparison test
Test for
H
0
:
= 0
Compute power from SE given type I error (e.g. 5%) and
Compute Number of subject needed (NSN) for given power
Power of Wald equivalence test
Test for
H
0
: {

D
L
or
+
D
L
}
Power and NSN for TOST
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PFIM and PFIM interface
Developed initially by Sylvie Retout, France Mentré
INSERM & University Paris Diderot
Other participants:
Caroline Bazzoli
, Emmanuelle
Comets, Hervé Le Nagard, Anne Dubois, Thu

Thuy
N'Guyen
P
opulation
F
isher
I
nformation
M
atrix
Use R
Available at
www.pfim.biostat.fr
Releases of PFIM
2001: First release PFIM 1.1 similar in Splus and Matlab (S.
Duffull)
2008: PFIM 3.0 and PFIM interface 2.1
2010: PFIM 3.2
F. Mentré
39
Evaluation by simulation: design
PK model
with
one

compartment
: k
a
, V/F, CL/F
Two

periods
one

way
crossover
trial
Treatment
effect
on CL/F
Simulations for
various
treatment
effects
Two designs with N = 40 piglets
Original n=7, Sparse: n=4 measurements/ piglet/ period
Variability
Random
effects
: BSV=30%, WSV=15%
Error
model: (a=0.1, b=0)
F. Mentré
40
Evaluation by simulation:
method
Simulation
1000 simulated trials for each treatment effect and each design
Global Analysis by SAEM in MONOLIX v2.4
Derivation of empirical SE as SD of estimates
Derivation of power as proportion of trials with rejection of Wald
TOST
Predictions
Use PFIM 3.2 to predict SE for each treatment effect and each
design
Use predicted SE by PFIM 3.2 to predict power of Wald TOST
Comparison of simulations and predictions
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Results on SE: no treatment effect
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Results on SE: various treatment effect
__ :
predicted
SE
using
PFIM



:
empirical
SE
from
simulations
Histograms
of SE for
treatment
effect
for
rich
designs
Correct prediction of SE by MF for all parameters
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Results on power
Correct prediction of power
Almost no loss of power for sparse
design with ‘optimal’ sampling
times
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44
Conclusion on design
Relevance of the new development of the
population Fisher matrix for NLMEM including WSV
and treatment effect in crossover trials
Correct predictions of standard errors and of power
avoiding intensive simulations
Analysis of studies through NLMEM
Can be performed with rather sparse design with almost
no loss of power if ‘optimal’ sampling times
Usefulness of new extension of PFIM as an
efficient tool for design of bioequivalence/
biosimilarity studies
F. Mentré
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4. PHARMACOMETRICS AND
DRUG DEVELOPMENT
F. Mentré
46
Present difficulties in drug development
Increase cost and duration of drug development
Few new medical entities (NME) reach approval
Problems
For pharma industry
But also for
public health
–
life

threatening diseases, rare diseases
–
lack of 'interest' of drug pharma for some disease areas
PROBLEM in DRUG DEVELOPMENT
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F. Mentré
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F. Mentré
50
Pharmacological Modelling
Integrated Knowledge for Model

based
Drug Development
Genes … Cells … Tissues … Systems … Patients … Populations
Development
time axis
Biological Modelling
Statistical Modelling
Adapted rom JJ Orloff, Novartis, April 06
F. Mentré
51
Increasing role of quantitative analysis of all data
trough modelling in therapeutic evaluation
Main statistical tool: NLMEM
Collaborative work
Biologists, Pharmacologists, Physicians
Engineers, Mathematicians, Statisticians
Pharmacometricians
Various unsolved methodological problems
academic research needed
Training needed
Holford N, Karlsson MO. Time for quantitative clinical pharmacology: a proposal
for a pharmacometrics curriculum. Clin Pharmacol Ther. 2007;82(1):103

5.
CONCLUSION
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