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Nov 16, 2013 (3 years and 9 months ago)

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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


F. Mentré


3

1. INTRODUCTION



Pharmacometrics



Nonlinear mixed effect models



Bioequivalence/ biosimilarity studies

F. Mentré


4


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

F. Mentré


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

F. Mentré


6

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

F. Mentré


7


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

F. Mentré


8

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


….


F. Mentré


9



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


F. Mentré


10


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)

F. Mentré


11

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

F. Mentré


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



F. Mentré


13

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

F. Mentré


14

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

F. Mentré


15

2. BIOEQUIVALENCE TEST IN
NLMEM



Method



Simulation



Real example

F. Mentré


16


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











F. Mentré


17

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
.





F. Mentré


18

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.





F. Mentré


19


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

F. Mentré


20

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

F. Mentré


21

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
]








F. Mentré


22

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

F. Mentré


23

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)

F. Mentré


24

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



F. Mentré


25

Simulated datasets



S
l,l






S
h,l



F. Mentré


26

Boxplot

of
estimates

for CL/F, 2 or 4
periods



S
l,l






S
h,l



* : true value



F. Mentré


27

Relative RMSE
fixed

effects

-

2 or 4
periods


S
l,l



S
h,l






F. Mentré


28

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


F. Mentré


29

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



F. Mentré


30

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


F. Mentré


31

3. OPTIMAL DESIGN IN NLMEM



Method



Simulation



Real example


F. Mentré


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


F. Mentré


33

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


F. Mentré


34

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

F. Mentré


35


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


36


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
F. Mentré


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





F. Mentré


38


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

F. Mentré


41

Results on SE: no treatment effect

F. Mentré


42

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


F. Mentré


43

Results on power


Correct prediction of power


Almost no loss of power for sparse
design with ‘optimal’ sampling
times

F. Mentré


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


45

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

F. Mentré


47

F. Mentré


48

F. Mentré


49

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