Model-based analyses of bioequivalence crossover trials using the stochastic approximation expectation maximisation algorithm.

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Model-based analyses of bioequivalence crossover trials

using the stochastic approximation expectation

maximisation algorithm.

Anne Dubois, Marc Lavielle, Sandro Gsteiger, Etienne Pigeolet, France

Mentré

To cite this version:

Anne Dubois, Marc Lavielle, Sandro Gsteiger, Etienne Pigeolet, France Mentré. Model-based

anal-yses of bioequivalence crossover trials using the stochastic approximation expectation maximisation

algorithm.. Statistics in Medicine, Wiley-Blackwell, 2011, 30 (21), pp.2582-600. �10.1002/sim.4286�.

�inserm-00643832�

Trials Using the SAEM Algorithm

Anne Dubois, Mar Lavielle, Sandro Gsteiger, Etienne Pigeolet

and Fran e Mentré

INSERM UMR738,University Diderot Paris 7, Paris, Fran e, 75018

INRIA Sa lay, Orsay, Fran e, 91400

Modeling and Simulation Department, Novartis Pharma AG, Basel,

Switzerland, CH-4056

Anne Dubois is a PhD student (anne.duboisinserm.fr), INSERM UMR738,

Uni-versity Diderot Paris 7, Paris, Fran e, 75018; Mar Lavielle is Professor of

Mathe-mati s (mar .laviellemath.u-psud.fr),INRIA Sa lay, Orsay, Fran e, 91400; Sandro

Gsteigeris aSenior Statisti alModeler (sandro.gsteigernovartis. om), andEtienne

Pigeolet is a Senior Expert Pharma ology Modeler (etienne.pigeoletnovartis. om),

ModelingandSimulationDepartment,NovartisPharmaAG,Basel,Switzerland,

CH-4056;Fran eMentréisProfessorofBiostatisti s(fran e.mentreinserm.fr),INSERM

UMR738, University Diderot Paris 7, Paris, Fran e, 75018. The authors thank the

ModelingandSimulationDepartment,NovartisPharmaAG,Basel,whi hsupported

by a grant Anne Dubois during this work, and Sigrid Balser from Sandoz

Biophar-ma euti al Development, Holzkir hen, Germany who provided the data set used in

In this work, we develop a bioequivalen e analysis using nonlinear mixed ee ts

models (NLMEM) that mimi s the standard non- ompartmental analysis (NCA).

NLMEM parameters, in luding between (BSV) and within subje t (WSV)

variabil-ity, and treatment, period, and sequen e ee ts are estimated. We explain how to

perform a Wald test on a se ondary parameter and we propose an extension of the

likelihoodratio test (LRT) for bioequivalen e. TheseNLMEM-based bioequivalen e

tests are ompared to standard NCA-based tests. We evaluate by simulation the

NCA and NLMEM estimates, and the type I error of the bioequivalen e tests. For

NLMEM, we use the SAEM algorithm implemented in MONOLIX. Crossover trials

are simulated under

H

0

using dierentnumbers of subje ts and of samples per

sub-je t. WesimulatewithdierentsettingsforBSVandWSV,andfortheresidualerror

varian e. The simulationstudy illustratesthe a ura y of NLMEM-based geometri

means estimated with the SAEM algorithm,whereas the NCA estimates are biased

for sparse design. NCA-based bioequivalen etests showgoodtype Ierror ex ept for

highvariability. Forari hdesign,typeIerrorsofNLMEM-basedbioequivalen etests

(Wald test and LRT) do not dier fromthe nominal level of 5%. Type I errors are

inatedforsparsedesign. Weapply thebioequivalen eWaldtest basedonNCAand

NLMEM estimates to athree-way rossovertrial, showing that Omnitrope ®

powder

and solutionare bioequivalent to Genotropin ®

. NLMEM-based bioequivalen e tests

areanalternativetostandardNCA-basedtests. However, autionisneededforsmall

sample size and highlyvariabledrug.

Keywords: nonlinear mixed ee ts model; pharma okineti s; non- ompartmental

Pharma okineti (PK)bioequivalen estudiesareperformedto omparedierentdrug

formulations. The most ommonly used design for bioequivalen e trials is the

two-period,two-sequen e, rossoverdesign. This designisre ommended bythe Foodand

Drug Administration (FDA) (FDA 2001) and the European Medi ines Evaluation

Agen y (EMEA)(EMEA2001). FDA andEMEA re ommendtotest bioequivalen e

from the ratios of the geometri means of two parameters: the area under the urve

(

AUC

)andthemaximal on entration(

C

max

)estimatedbynon- ompartmental

anal-ysis (NCA) (Gabrielsonand Weiner 2006). Asspe iedin the regulatoryguidelines,

the bioequivalen e analysis should take into a ount sour es of variation that an

be reasonably assumed to have an ee t on the endpoints

AUC

and

C

max

.

There-fore,linearmixedee ts models(LMEM)in ludingtreatment,period,sequen e, and

subje tee ts are usuallyused toanalyse thelog-transformedindividualparameters

(Haus hke etal. 2007). Bioequivalen etests are then performed onthe estimates of

the treatment ee t.

NCA requiresfewhypothesesbut alargenumberofsamplespersubje t(usually

be-tween10and 20). PKdata analsobeanalysedusingnonlinearmixedee tsmodels

(NLMEM). This method is more omplex than NCA but has several advantages: it

takesadvantage of the knowledge a umulated onthe drug and an hara terize the

PKwithfewsamplespersubje t. Thisallowsonetoperformanalysesinpatients,the

targetpopulation,inwhompharma okineti s anbedierentfromhealthysubje ts.

Inaprevious work,Duboisetal(Duboisetal.2010) omparedthestandard analysis

of bioequivalen e rossover trials based on NCA to the same usual analysis based

on individual empiri al Bayes estimates (EBE) obtained by NLMEM. PK data of

ea h treatment groupwere analysed separatelyusing NLMEM. Linear mixed ee ts

models were then performed on individual

AUC

and

C

max

derived from the EBE.

However, this methodology annot be performed when the EBE shrinkage is above

20%. Panhard and Mentré (Panhard and Mentré 2005) developed dierent

andthe likelihoodratiotest(LRT). Forbioequivalen etests, theyproposedtheWald

test but the LRT was not developed, due to the omposite null hypothesis. They

applied these tests to two-period, one-sequen e, rossover trials. In a later work,

Panhard et al (Panhard et al. 2007) demonstrated the importan e of modelling the

between-subje t (BSV)and within-subje t(WSV)variabilityto ontrolthe ination

ofthetypeIerrorusingthesamesetsofsimulationsaspreviously. Inbothsimulation

studies, the NLMEM-based bioequivalen e Wald test was performed on

AUC

only

be ause

C

max

wasase ondary parameterofthePKmodel,asofteninPKmodelling.

The use of NLMEM is still rare to analyse bioequivalen e rossover trials. Indeed,

there are few published studies whi h use NLMEM to analyse bioequivalen e trials

(Kaniwaetal.1990;Pentikisetal.1996;Combrink etal.1997; Maieretal.1999;Hu

et al. 2003; Zhou et al. 2004; Fradette et al. 2005; Zhu et al. 2008). Seven of these

studies are previousto the dierent simulationstudies (Kaniwa et al.1990; Pentikis

etal.1996; Combrink etal.1997;Maier etal.1999; Huetal.2003; Zhouetal.2004;

Fradetteetal.2005). In sixofthese studies(Kaniwaetal.1990;Pentikis etal.1996;

Combrink et al. 1997; Maier et al. 1999; Hu et al. 2003; Fradetteet al. 2005),

bioe-quivalen e Wald tests were performed on treatment ee ts estimated by NLMEM,

as Panhard et al. However, all of these applied works used dierent statisti al

ap-proa hes. The addition of the treatment ee t on dierent PK parameters was not

always justied. Only Hu et al (Hu et al. 2003) performed bioequivalen e test on

average

AUC

and

C

max

obtained from the xed ee t estimates using Monte Carlo

simulation. Finally, none estimated the WSV or adds period or sequen e ee ts as

re ommended inthe guidelines. There is urrently no published simulationstudy or

appliedwork whi h takesintoa ount treatment,period,sequen e ee ts,BSV, and

WSV for the bioequivalen e analysis of rossovertrials by NLMEM.

InthedierentNLMEM-basedbioequivalen eanalysis,NLMEMparameterswere

es-timated by maximum likelihood. Ex ept for the simulation by Dubois et al(Dubois

The FOCE algorithm is a widely used algorithm and orresponds to the industry

standard for model-based PK analyses. Yet, this linearization-basedmethod annot

be onsidered as fully established theoreti ally. For instan e, Vonesh (Vonesh 1996)

and Geet al (Ge et al. 2004) gave examples of spe i designs resultingin

in onsis-tentestimates,su haswhenthenumberofobservationspersubje tdoesnotin rease

faster than the number of subje ts or when the variability of random ee ts is too

large. Severalestimationmethodsof maximumlikelihoodtheoryhavebeenproposed

as alternatives to linearization algorithms like the adaptative gaussian quadrature

(AGQ)method(PinheiroandBates1995)ormethodsderivedfromthe

Expe tation-Maximisation (EM) algorithm (Dempster et al. 1977). The AGQ method requires

a su iently large number of quadrature points implying anoften slow onvergen e

and is limited to a small number of random ee ts. Monte Carlo EM algorithms

as proposed by Wei and Tanner (Wei and Tanner 1990), Walker (Walker 1996), or

Wu (Wu 2004) are very time- onsuming in omputation sin e they require a huge

amount of simulated data. Alternatively, Delyon et al (Delyon et al. 1999)

intro-du ed a sto hasti approximation version of the EM algorithm (SAEM), whi h is

moree ientintermsof omputation. Later,KuhnandLavielle(KuhnandLavielle

2004) developed an algorithm whi h ombined the SAEM algorithmwith a

Monte-Carlo pro edure. They showed the good statisti al onvergen e properties of this

algorithm. Re ently, Panhard and Samson (Panhard and Samson 2009) developed

an extension of the SAEM algorithm for NLMEM in luding the estimation of the

within-subje t variability.

The main obje tive of this work is to develop a bioequivalen e analysis based on

NLMEM that mimi s the standard bioequivalen e analysis performed on NCA

esti-mates. To do so, we use a NLMEM in luding treatment, period, sequen e ee ts,

BSV, and WSV. We also explain how to perform a Wald test on a se ondary

pa-rameter of the model (like

C

max

), and we propose anextension of the LRT for

the type I errors of the dierent bioequivalen e tests. We use the same sets of

sim-ulations than the previous study by Dubois et al (Dubois et al. 2010) whi h allows

to ompare the results. As in Dubois et al, we use dierent sampling designs and

levels of variability,investigatingtheir inuen e on the results of the bioequivalen e

tests. To estimate NLMEM parameters, we use the SAEM algorithm implemented

in MONOLIX.Then,we apply the Waldtest based onNCAand NLMEM estimates

to a three-way rossover trial omparing three formulations of somatropin. A

so-matropin is a biosyntheti version of human growth hormone (hGH) synthesised in

ba teria modied by the addition of the gene for hGH. Repla ement therapy with

somatropin isawella epted, ee tive treatment forhGHde ien y in hildrenand

adults (Fau i etal. 2008).

In Se tion2 of this paper, we des ribe the LMEM for NCA estimates, the NLMEM

on on entrations, and the bioequivalen e tests based on both approa hes. In

Se -tion 3, we present the simulation study, the estimation, and the evaluation of the

estimates and of the type I errors. We present the example inSe tion4, followed by

a dis ussion in Se tion5.

2 MODELS AND BIOEQUIVALENCE TESTS IN CROSSOVER

TRIALS

In the following, we onsider rossover pharma okineti trials with

C

treatments,

K

periods, and

Q

sequen es.

2.1 Models

Linear Mixed Ee ts Model for NCA

The standard bioequivalen e analysis re ommended by FDA and EMEA (FDA

2001; EMEA 2001) is based on NCA individual estimates of

AUC

and

C

max

. We

dene

θ

ikl

the

l

th

i

(

i = 1, · · · , N

) at period

k

(

k = 1, · · · , K

). The individual parameters are log-transformed and analysed using a linear mixed ee ts model written asfollows:

log(θ

ikl

) = ν

l

+ β

T

′

l

T

ik

+ β

P

′

l

P

k

+ β

S

′

l

S

i

+ η

il

+ ǫ

ikl

(1)

where

ν

l

is the expe ted value orresponding to the ombination of ovariate

refer-en e lasses.

β

T

l

,

β

P

l

,and

β

S

l

are theve torsof oe ientsoftreatment,period,and

sequen e ee tsfor the

l

th

individualparameter (

AUC

or

C

max

).

T

ik

,

P

k

,

S

i

are the

known ve tors of treatment, period, and sequen e ovariates of size

C

,

K

, and

Q

,

respe tively.

T

ik

is omposedofzerosex ept forthe

c

th

element(

c = 1, · · · , C

)whi h

is one when treatment

c

is given to patient

i

at period

k

. Similarly,

P

k

and

S

i

are

omposed of zeros ex ept for the

k

th

and

q

th

elements (

k = 1, · · · , K

,

q = 1, · · · , Q

)

whi h are one. We onsider that the rst treatment, period, and sequen e are the

referen e lasses. The rst elements of

β

T

l

,

β

P

l

, and

β

S

l

are xed tozero, and other

omponents are estimated. It is assumed that the randomsubje t ee t

η

il

and the

residual error

ǫ

ikl

are independently normally distributed with zero mean.

Nonlinear Mixed Ee ts Modelling

We denote by

y

ijk

the on entrationfor subje t

i

(

i = 1, · · · , N

) at samplingtime

j

(

j = 1, · · · , n

ik

)for period

k

(

k = 1, · · · , K

). Wedene

f

tobethe nonlinear

phar-ma okineti fun tion whi h links on entrations to sampling times. The nonlinear

mixed ee ts model an be writtenas:

y

ijk

= f (t

ijk

, ψ

ik

) + g(t

ijk

, ψ

ik

) ǫ

ijk

(2)

with

ψ

ik

the

p

-ve tor of pharma okineti parameters of subje t

i

for period

k

.

g(t

ijk

, ψ

ik

) ǫ

_ijk

is the residual error where

ǫ

ijk

is a Gaussian random variable with zero mean and varian e one. All

ǫ

ijk

are independent and identi ally distributed.

a + bf (t

ijk

, ψ

ik

)

.

The statisti almodel used for the individual parameters

ψ

ik

is derived from the

lin-ear mixed ee ts model used to analyse the NCA individual estimates. So, the

l

th

omponent of

ψ

ik

is dened as:

log(ψ

ikl

) = log(λ

l

) + β

T

′

l

T

ik

+ β

P

′

l

P

k

+ β

S

′

l

S

i

+ η

il

+ κ

ikl

(3)

with

λ

= (λ

l

; l = 1, · · · , p)

the

p

-ve tor of xed ee ts for the ovariate referen e

lasses. The known ve tors of the treatment, period, and sequen e ovariates,

T

ik

,

P

k

,and

S

i

,are dened as forNCA (se tion2.1).

β

T

l

,

β

P

l

, and

β

S

l

are the ve tors of

oe ients of treatment, period, and sequen e ee ts for the

l

th

PK parameter. As

previously mentioned,we onsider that the rst treatment,period,and sequen e are

thereferen e lasses. Therstelementsof

β

T

l

,

β

P

l

,and

β

S

l

arexedtozero,andother

omponents are estimated.

η

i

= (η

il

; l = 1, · · · , p)

is the ve tor of randomee ts of

subje t

i

orresponding to the between-subje t variability.

κ

ik

= (κ

ikl

; l = 1, · · · , p)

istheve torofrandomee tsofsubje t

i

atperiod

k

orrespondingtothevariability

between periods of treatment for the same individual, or within-subje t variability.

These random ee ts are assumed to be normally distributed with zero mean and

ovarian ematrixof size

p × p

named

Ω

and

Γ

,respe tively. Wedene

ω

2

l

and

γ

2

l

the

varian efor BSV and WSVof the

l

th

parameter, orresponding tothe

l

th

element of

the diagonalof

Ω

and

Γ

.

η

i

,

κ

ik

,and

ǫ

ijk

are assumed to bemutually independent.

Finally, the unknown population parameters of the statisti al model are the xed

ee ts ("referen e" and ovariate ee ts) and the varian eparameters (

Ω

,

Γ

,

a

,

b

).

2.2 Two-One Sided Tests

The bioequivalen e test is performed on the

c

th

treatment ee t of the

l

th

param-eter,

β

T

c,l

(

c = 2, · · · , C

and

l = 1, 2

for NCA or

l = 1, · · · , p

for NLMEM). Its null

hypothesis is

H

0

:

{β

T

c,l

≤ −δ

or

β

T

c,l

≥ δ}

whi h is de omposed in two one-sided hy-potheses

H

0,−δ

:

{β

T

c,l

≤ −δ}

and

H

0,δ

:

{β

T

on S huirmann's two one-sided tests (TOST) pro edure (S huirmann 1987).

H

0,−δ

and

H

0,δ

are tested separately by a one-sided test. The global null hypothesis

H

0

is reje ted with a type I error

α

if both one-sided hypotheses are reje ted with a

type I error

α

. The p-value of the TOST is the maximum of both p-values of the

one-sided tests. The major issue of a bioequivalen e test is to dene

δ

. To assess

pharma okineti bioequivalen e,theguidelines(FDA2001;EMEA2001)re ommend

δ = log(1.25) ≈ 0.22

(i.e.

−δ = log(0.8)

) for

log(AUC)

and

log(C

max

)

. Due to the

linear model on log-parameters, these bounds orresponds to 80%-125% on the

pa-rameter s ale.

Wald Tests Based on NCA Estimates

Inthe following,we all

se(β

T

c,l

)

the standard errorofthe treatmentee t estimate

b

β

T

c,l

. We also dene

W

−δ

= ( b

β

c,l

T

+ δ)/se(β

c,l

T

)

and

W

δ

= ( b

β

T

c,l

− δ)/se(β

c,l

T

)

, the two

Wald statisti s for the one-sided hypotheses

H

0,−δ

and

H

0,δ

, respe tively. For the

standard NCA-based bioequivalen e analysis, we assume that

W

−δ

and

W

δ

followa

Student t-distributionwith

df

degrees offreedom under

H

0,−δ

and

H

0,δ

,respe tively.

The globalnullhypothesis

H

0

isreje tedwith atype Ierror

α

if

W

−δ

≥ t

1−α

(df )

and

W

δ

≤ −t

1−α

(df )

,where

t

1−α

(df )

isthe

(1 − α)

quantileof the Student t-distribution

with

df

degrees of freedom. For balan ed datasets (i.e. with

N

subje ts for ea h

period),

df = N − 2

(Haus hke et al.2007; Chowand Liu2000). An alternative

ap-proa htoperformabioequivalen etestisto ompute the(

1 −2α

) onden einterval

(CI) of

β

b

T

c,l

.

H

0

isreje ted if this

(1 − 2α)

CI lieswithin

[−δ; δ]

.

Wald Test Based on NLMEM Estimates

For the bioequivalen e Wald test using NLMEM estimates, we use a very

simi-lar approa h to NCA-based bioequivalen e Wald test. Same notations are used for

NLMEM-based analyses as for NCA. For NLMEM, we assume that

W

−δ

and

W

δ

follow a Gaussian distribution under

H

0,−δ

and

H

0,δ

, respe tively. The global null

hypothesis

H

0

is reje ted with a type I error

α

if

W

where

z

1−α

isthe

(1 − α)

quantileof the standardnormaldistribution. Thereje tion

of

H

0

an alsobe based onthe

(1 − 2α)

CI asdes ribed previously (se tion2.2).

Tomimi thestandardbioequivalen eanalysis,wewouldliketoperformthe

NLMEM-based bioequivalen e Wald test on

AUC

and

C

max

whi h are often se ondary

pa-rameters of the PK model. So, we propose an approa h to perform the

NLMEM-based bioequivalen e Wald on a se ondary parameter. A se ondary parameter is a

fun tion of the PK parameters of the stru tural model. Its

c

th

treatment ee t is

β

T

c,SP

= h(λ, β

T

c

)

with

h

thefun tionlinking

β

T

c,SP

tothePKparameters,

λ

the refer-en e ee ts,and

β

T

c

the

c

th

treatmentee ts. Toperformabioequivalen eWaldtest

on the

c

th

treatment ee t of a se ondary parameter,

β

T

c,SP

and its standard error,

se(β

T

c,SP

)

, should be estimated. By denition,

β

b

T

c,SP

= h(b

λ, b

β

T

c

)

. However,

se(β

T

c,SP

)

annot be dire tly omputed as

h

is usually a nonlinear fun tion. We propose to

approximate it using the delta method (Oehlert 1992) or simulations. For the delta

method, we use the partial derivatives of

h

, the xed ee t estimates

(b

λ

,

β

b

T

c

)

, and

theirestimated ovarian ematrix

Σ

b

,whi hisasubmatrixoftheinverse ofthe Fisher

information matrix estimate:

se(β

T

c,SP

) =

q

∇h(b

λ, b

β

T

c

)

′

Σ

b

∇h(b

λ, b

β

T

c

)

. To estimate

se(β

T

c,SP

)

by simulations, we simulate

β

T

c,SP

N

s

times using a Gaussian distribution withmeanthexedee testimates

(b

λ

,

β

b

T

c

)

and ovarian ematrix

Σ

b

. Then,

se(β

T

c,SP

)

is estimated as the standard deviation of the

N

s

simulated

β

T

c,SP

.

Likelihood Ratio Test Based on NLMEM Estimates

Thereis nosimple extension of the likelihoodratio test for the omposite null

hy-pothesis of a bioequivalen e test. Therefore, for a parameter of the PK model, we

develop a methodology to perform a NLMEM-based bioequivalen e LRT based on

prole likelihood methods (Bates and Watts 1988; Meeker and Es obar 1995). Let

usdene

M

all

tobethe NLMEM whereallxed ee ts are estimateedand

M

δ,c,l

the

NLMEMwhere

β

T

c,l

isxed to

δ

andallotherparameters(in ludingtheother ompo-nentsof

β

T

l

)areestimated. Theproposedapproa htestwhetherthelikelihood-based

onden e interval of

β

b

T

LRT taking into a ount

β

b

T

c,l

estimated with

M

all

, and the estimation of the log-likelihood of three models

M

all

,

M

−δ,c,l

, and

M

δ,c,l

. We dene the statisti

Λ

δ,c,l

as

follows:

Λ

δ,c,l

= −2 × (L

δ,c,l

− L

all

)

with

L

all

and

L

δ,c,l

the estimated log-likelihoods

for the models

M

all

and

M

δ,c,l

, respe tively. The null hypothesis

H

0,−δ

is reje ted

with a type I error

α

if

Λ

−δ,c,l

≥ χ

2

1

(1 − 2α)

and

−δ < b

β

T

c,l

, where

χ

2

1

(1 − 2α)

is the

(1 − 2α)

quantileof the Chi-squared distributionwith one degreeof freedom.

H

0,δ

is

reje ted if

Λ

δ,c,l

≥ χ

2

1

(1 − 2α)

and

β

b

T

c,l

< δ

. Consequently, the global nullhypothesis

H

0

is reje ted with a type I error

α

if

−δ < b

β

T

c,l

< δ

and

Λ

−δ,c,l

≥ χ

2

1

(1 − 2α)

and

Λ

δ,c,l

≥ χ

2

1

(1 − 2α)

.

3 SIMULATION STUDY

3.1 SimulationSettings

Weusethe on entrationdataoftheanti-asthmati drugtheophyllinetodenethe

populationPKmodelforthesimulationstudy. Thesedataare lassi alinpopulation

pharma okineti s (Pinheiro and Bates 2000) and have been used in previous

simu-lation studies (Panhard and Mentré 2005; Panhard et al. 2007; Dubois et al. 2010).

We assume that on entrations an be des ribed by aone- ompartment model with

rst-order absorption and rst-order elimination:

f (t, k

a

, CL/F, V /F ) =

F Dk

a

CL − V k

a

(exp( −k

a

t) − exp( −CL/V t))

(4)

where

D

is the dose,

F

the bioavailability,

k

a

the absorption rate onstant,

CL

the

learan e of the drug,and

V

the volumeof distribution.

We simulate two-treatment, two-sequen e, rossover trials with two or four periods.

Forea htwo-periodtrial,the

N/2

subje tsof therst sequen e re eive thereferen e

treatment (Ref) and the test treatment (Test) in period one and two, respe tively.

The other

N/2

subje ts allo ated to the se ond sequen e re eive treaments in the

re-twoand four. The

N/2

subje ts of the se ondsequen e re eivethe treament Test in

periodsone and three, and the treatment Ref inperiods two and four.

We onsider that sampling times are similar for all subje ts and all periods. So,

j = 1, · · · , n

, where

n

is a xed number of sampling times for ea h simulated

sam-pling design. We use fourdierent samplingdesigns, whi h are alsoused by Dubois

etal(Duboisetal.2010). Wesimulatewiththeoriginaldesignwith

N = 12

subje ts

and

n = 10

samples per subje t and per period, taken at the times of the initial

study (0.25, 0.5, 1, 2, 3.5, 5, 7, 9, 12, and 24

h

after dosing). We alsosimulate with

an intermediate design with

N = 24

subje ts and

n = 5

samples, taken at 0.25, 1.5,

3.35, 12, and 24

h

after dosing, a sparse design with

N = 40

subje ts and

n = 3

samples,taken at0.25, 3.35,and24

h

afterdosing,andari hdesign

N = 40

subje ts

and

n = 10

samples,taken at the times of the initialstudy.

For the simulation study, we assume that

α = 5%

and

δ = log(1.25)

. We x the

dose to 4mgfor allsubje ts. The ve tor ofpopulationparameters

λ

is omposed of

λ

k

a

= 1.48 h

−1

,

λ

CL/F

= 40.36 mL/h

,and

λ

V /F

= 0.48 L

forthereferen e treatment.

We assume that the bioavailability hanges between treatments, i.e., we assume the

same modi ation for

CL/F

and

V /F

. It also similarly ae ts both se ondary

pa-rameters

AUC

and

C

max

with

AUC = F D/CL

and

C

max

dened in Equation 5 of

the Appendix. In the following, as we onsider only two treatments in the

simula-tion study,weomitthesubs ript

c

;wedene

β

T

CL/F

and

β

T

V /F

the treatmentee t on

CL/F

and

V /F

forthetreatmentTest (

β

T

k

a

= 0

). AssuggestedbyLiuandWeng(Liu and Weng 1995),the type Ierrorof thebioequivalen etest anbe evaluatedfor ea h

boundary of

H

0

spa e, i.e.,

log(0.8)

and

log(1.25)

. Consequently, we simulate under

two dierent hypotheses:

β

T

CL/F

= β

V /F

T

= log(0.8)

and

β

T

CL/F

= β

V /F

T

= log(1.25)

whi h are the boundaries of

H

0,log(0.8)

and

H

0,log(1.25)

, respe tively. In the following,

we all

H

0;80%

and

H

0;125%

these two simulated hypotheses. We assume no period

ee t orsequen e ee t, and

Ω

and

Γ

are diagonal.

variability. Forthe lowlevel of variability, we x

ω

k

a

and

ω

CL/F

to0.2, and

ω

V /F

to

0.1;

γ

k

a

,

γ

CL/F

and

γ

V /F

are xedtohalfBSV forthe threeparameters. Forthe high

level,wexthethreestandard deviationsto0.5for BSV,and0.15 forWSV.Wealso

simulate with two levelsof variabilityfor the residualerror:

a = 0.1 mg/L, b = 0.10

for the low level, and

a = 1 mg/L, b = 0.25

for the high level. The high level of

residual error is only used with the high level of BSV and WSV. We all

S

l,l

the

variabilitysettingwith lowvariabilityforBSV, WSV,andfor theresidualerror.

S

h,l

isthe variabilitysetting orrespondingtohighvariabilityforBSV,WSV,andlowfor

theresidualerror. Finally,

S

h,h

isthevariabilitysettingwithhighvariabilityforBSV,

WSV, and for the residual error. In the following, we all a simulation setting the

asso iation of one design with one variabilitysetting and one simulated hypothesis.

We simulate rossover trials with 2 periods under

H

0;80%

and

H

0;125%

. In that ase,

for ea h samplingdesign, we simulate using the variability settings

S

l,l

and

S

h,l

. We

simulate using

S

h,h

only for the intermediate design. We simulate rossover trials

with 4 periods under

H

0;80%

using ri h and sparse sampling designs, and the two

variability settings

S

l,l

and

S

h,l

. All simulations are performed using the statisti al

software R 2.7.1. We use the fun tion rmvnorm of the pa kage mvtnorm, whi h is a

pseudorandom number generator for the multivariate normal distribution. For ea h

simulated trial, we spe ify the seed using the fun tion set.seed in order to make

simulations reprodu ible.

3.2 Estimation

NCA

AsinDuboisetal(Duboisetal.2010),weestimate

AUC

and

C

max

bynon

ompart-mentalanalysis (Gabrielson and Weiner 2006) using aR fun tion whi h we develop.

For a rossover trial, this fun tion provides the estimation of dierent NCA

param-eters for ea h subje t and ea h treatment group. In this study, we use the linear

trapezoidalruleto omputethe

AUC

0−last

between the timeof dose(equal to

0

)and

in-nity),we estimatethe terminalslopeequalto

CL/V

using thelogarithmofthe last

on entrations to perform a log-linear regression. To do so, we use a xed number

of on entrationswhi h depends onthenumberofsamplespersubje tinthedesign.

For the original and ri h designs where

n = 10

, we use the last four on entrations

whi h orrespond to samplingtimes 7, 9, 12 and 24

h

. For intermediate and sparse

designswhere

n = 5

and

n = 3

respe tively,weuse thelasttwo on entrationswhi h

orrespond to sampling times 12 and 24

h

for the intermediate design, and to 3.35

and 24

h

for the sparse design. For all designs,

C

max

is estimated as the maximal

observed on entration.

The analysis of log parameters by LMEM is then performed using the R fun tion

lme fromthe pa kage nlme. Forthe estimation of LMEM parameters (in luding the

treatmentee t and itsSE), the restri ted maximumlikelihood(REML)preo edure

isused, asre ommended intheguidelines(FDA2001;EMEA2001). Forthe original

designwhere

N = 12

,

df = 10

. Fortheri handsparsedesignwhere

N = 40

,

df = 38

.

For the intermediate design where

N = 24

,

df = 22

.

All omputations in ludingthe dataset simulation, the estimation by NCA, and the

standard bioequivalen eanalysis byLMEM are madeunderR2.7.1. The dierentR

s ripts are available upon request tothe orrespondingauthor.

NLMEM

WeestimatetheNLMEMparameters(in ludedtreatment,period,sequen eee ts,

BSV and WSV) by maximum likelihood using the SAEM algorithm (Panhard and

Samson 2009)implemented inMONOLIX (Lavielleetal. 2010). Estimationof

stan-dard errors (SE) and log-likelihoodare also needed to perform Wald and likelihood

ratio tests, respe tively. SE an be evaluated as the square root of the diagonal

ele-mentsof theinverse of the Fisherinformationmatrix estimatewhi hhas noanalyti

form. InMONOLIX, onemethodtoevaluatethismatrix istoderiveanapproximate

expression by the linearizationof the fun tion

f

around the onditional mean of the

linearization-results for SE estimation have been shown using this approa h for omputation of

the FIM (Bazzoli etal. 2009). There is no analyti al expression of the likelihood in

NLMEM. In MONOLIX, it is proposed to estimate the log-likelihood of the

obser-vations without approximation using the Importan e Sampling (IS) method (Kuhn

and Lavielle 2005; Samson et al. 2007). The IS method is a Monte-Carlo pro edure

where individual parameters are simulated at ea h iteration using an instrumental

distribution adequately hosen toredu e the varian e of the estimator.

NLMEM parameters,standarderrorsandlog-likelihoodsareestimated with

MONO-LIX 2.4., supported by MATLAB R2007a. The use of MONOLIX software for the

analysisof rossoverbioequivalen etrialsisexplainedinaonlineSupplementary

Ma-terial 1

.

3.3 Evaluation Methods

Estimates

TheSAEMalgorithmused fortheestimation ofNLMEMparametershas not been

evaluated for models in ludingtreatment, period, sequen e ee ts, BSV, and WSV.

Therefore, we evaluate the SAEM algorithmfor two and four-period rossover trials

with the ri h and sparse design. We use the

H

0;80%

simulation settings of the two

variability settings

S

l,l

and

S

h,l

. There are 8 dierent simulationsettings used (2 or

4 periods, ri h or sparse design,

S

l,l

or

S

h,l

variability). We t the statisti al model

M

all

forthe 1000trials of ea h simulationsetting of both types of rossover trials (2 or 4periods). Then, forthe 1000repli ates, we ompute the bias and the rootmean

square error (RMSE) forea h estimated parameter.

Furthermore, in the standard bioequivalen eanalysis, the geometri means of

AUC

and

C

max

are reported for ea h treatment group. We evaluate those estimates for

the referen e treatment,and for NCAand NLMEM. We alsoevaluate the treatment

ee t estimates for

AUC

and

C

max

,and forNCAand NLMEM.Lastly,good estima-1

the SE of the treatment ee t estimates, for NCA and NLMEM. To evaluate the

geometri means,thetreatmentee ts,and theirSEweuse thetwo-period rossover

trials simulated under the null hypothesis

H

0;80%

with the four dierent sampling

designs (ri h, original, intermediate and sparse), and the three variability settings

(

S

l,l

,

S

h,l

and

S

h,h

). Thereare 9dierentused simulationsettings, 4for

S

l,l

and

S

h,l

,

and 1 for

S

h,h

where only the intermediate design is simulated. For NCA, for ea h

simulated trial, the geometri mean of

AUC

(

C

max

) for the referen e treatment is

omputed from the

N

individual estimated

AUC

(

C

max

). For NLMEM, due to the

log-normaldistributionof the randomee ts, the xed ee t estimates for the

refer-en e lasses orrespondtothe geometri meanestimates for the referen e treatment.

So,theNLMEM-basedgeometri meanof

AUC

forthereferen etreatmentisdire tly

obtained from the learan e estimate as

AUC = F D/CL

. For

C

max

, the geometri

mean is omputed fromthe xed ee t estimates using Equation5 of the Appendix.

ForNCAand NLMEM estimates,the geometri meansare ompared tothe

AUC

or

C

max

omputed from the NLMEM simulated parameters. For NCA, the treatment

ee t on

AUC

and

C

max

are estimated by LMEM as explained in se tion 2.1. For

NLMEM, duetothelinear ovariatemodelonlog-parameters,

β

b

T

AU C

= −b

β

CL/F

T

. The

treatment ee t

β

b

T

C

max

is omputed from

λ

b

and

β

b

T

using Equation 6. For NCA and

NLMEM estimates, the treatment ee t estimates are ompared to the simulated

value of the treatment ee t. For NCA, standard errors of the treatment ee t are

estimated by LMEM. For NLMEM, as

β

b

T

AU C

= −b

β

CL/F

T

, their standard error are equal. The SE of

β

b

T

C

max

is estimated by the delta method and simulations (with

N

s

= 10000

). For the delta method, the expression and details are given in the

Appendix. The estimated standard errors of the treatment ee t are ompared to

the orresponding empiri al standard error for NCA and NLMEM estimates. For

one simulationsetting and oneapproa h (NCAorNLMEM), the empiri alstandard

To evaluate the type I error of the bioequivalen e tests, we use the two-period

rossover trials simulated with the four dierent sampling designs, both hypotheses

and the three variability settings. There are 18 dierent simulation settings used, 8

for

S

l,l

and

S

h,l

, and 2 for

S

h,h

where onlythe intermediatedesign is simulated. The

simulationsettings under

H

0;80%

are alsoused toevaluatethe estimates of

AUC

and

C

max

(se tion3.3).

Weperformthebioequivalen eWaldtestbasedonNCAestimateson

AUC

and

C

max

.

For NLMEM, tests on

CL/F

and

AUC

are equivalent be ause

β

b

T

AU C

= −b

β

CL/F

T

and

se(β

T

AU C

) = se(β

CL/F

T

)

. So, the NLMEM-based bioequivalen e Wald test and LRT

are performed on

AUC

. As

C

max

is a se ondary parameter of the NLMEM, only

the NLMEM-based Waldtest isperformedonthis parameter, and not the LRT. For

NLMEM, the treatmentee t

β

b

T

C

max

is omputed from

λ

b

and

β

b

T

. Its standard error

is estimated both by the delta method and by simulations (with

N

s

= 10000

). For

AUC

and

C

max

,theNLMEM-basedbioequivalen eWaldtestisperformedusing esti-matedandempiri alSE.For

C

max

, itisperformedusingestimated SEobtained from

the delta method and simulations, for omparison. For ea h one-sided hypothesis

H

0;80%

and

H

0;125%

, the type I error is estimated by the proportion of the simulated

trials for whi hthe nullhypothesis

H

0

isreje ted. The global type I error is dened

as the maximum value of both estimated type I errors (Dubois et al.2010; Panhard

and Mentré 2005). For 1000 repli ates, the 95% predi tion interval (95% PI) for a

type I error of 5% is

[3.7%; 6.4%]

.

3.4 Results

Evaluation of the Estimates

For the evaluated settings, all NLMEM parameters in luding treatment, period,

sequen e ee ts are estimated by the SAEMalgorithm. Boxplots of the estimates of

the learan e referen e ee t, the orresponding ovariate ee ts and the standard

periodsin reases. Forallsimulationsettingsofbothtypesoftrials,the medianofthe

xed ee ts is lose to the orresponding simulated value. For BSV and WSV, the

medianofthe estimatesis losertothe simulatedvalueforfour-periodtrialsthanfor

two-periodtrials. For the variabilitysetting

S

h,l

, BSV and WSV are slightly

under-estimated espe ially for the sparse design. Similar results (not shown) are obtained

for both PK parameters,

k

a

and

V /F

. Table 1 provides the bias (

×100

) and RMSE

(

×100

) of estimates of the referen e ee ts and the standard deviations for BSV,

WSV, and residualerror. Forall simulationsettingsand both typesof rossover

tri-als(2or4periods),thereisnobias andRMSEaresmall forthereferen e ee ts and

the residual error. For BSV and WSV, bias de reases when the number of samples

in reases. Forallparameters,RMSEde rease whenthenumberofsamplesin reases.

Furthermore, RMSE are smaller for

S

l,l

than for

S

h,l

and smaller for four-period

tri-als than for two-period trials. The same observations are made for ovariate ee ts

(results not shown).

Forea hsimulationsettingoftwo-period rossovertrialsofthehypothesis

H

0;80%

and

forNCAandNLMEM,boxplotsofthereferen e treatmentgeometri meanestimates

of

AUC

and

C

max

are displayed in Figure 2. For

AUC

and

C

max

, and for NCA and

NLMEM estimates, the distribution is narrower when the variabilityis smaller. For

NCA estimates, the median of the estimates is loser to the true simulated mean

for the ri h design, and there is a lear and very large bias of the geometri mean

estimates for sparse design. For NLMEM estimates, the median of the estimates is

lose to the true simulated mean for all simulation settings. Figure 3 displays the

boxplot of the treatment ee t estimates on

AUC

and

C

max

and their standard

er-rors for NCA and NLMEM estimates. The standard errors

se(β

T

C

max

)

are estimated bythe deltamethod,and verysimilar resultsare obtained by simulations. ForNCA

andNLMEM,forboth parametersand allsimulationsettings,themedianofthe

esti-matedtreatmentee tsis losetothesimulatedvalue. Furthermore,thedistribution

dard error is smaller when the variability de reases orwhen the number of subje ts

in reases. Forboth parameters, the medianof the estimated standard error is loser

to the empiri al one when the variability de reases. For the original design under

S

h,l

and the intermediate design under

S

h,h

, standard errors of both parameters are

underestimated for NCA and NLMEM estimates.

Evaluation of the Type I Error

Table 2 provides type I errors of bioequivalen e tests performed on the treatment

ee ts of

AUC

, and

C

max

forea h one-sided hypothesis and ea hsamplingdesign of

two-period rossover trials. Mostly, for all tests and both parameters, type I errors

of both hypotheses are lose. Only the type I errors for

C

max

and the

S

h,h

setting

are somewhatdierent. ForWaldtestsbasedonNCAestimates, andfor

S

l,l

and

S

h,l

settings, type I errors do not dier from the nominal level of 5%. For

S

h,h

setting,

the type I errors are mu htoo onservative for

AUC

, and are inated for

C

max

. For

the NLMEM-based Waldtest, typeI errors for

C

max

usingSE obtained by the delta

method or simulations are identi al. For

AUC

, type I errors of the NLMEM-based

Waldtest are lose totypeI errorsofthe LRT. Forthe ri hdesign(

N = 40

,

n = 10

),

type I errors of both tests do not dier from the nominal level of 5%. However, for

ea hsimulationsetting,thereisanin rease ofthe typeIerrorof bothtests whenthe

numberof subje ts and/or the number of samplesde reases.

The left hand side of Figure 4 displays the global type I error for

AUC

(top) and

C

max

(bottom) versus the design for ea h variabilitysetting for the Wald test based onNCA estimates. Forboth parameters,the global type I error liesinthe 95%PI of

the nominal level for all the designs of

S

l,l

and

S

h,l

settings. Forthe

S

h,h

setting, it

is too onservative for

AUC

and inated for

C

max

. The right hand side of Figure 4

displays the global typeI errorof the NLMEM-based Wald test using the estimated

or empiri al standard error, and the NLMEM-based LRT. For the Waldtests using

subje ts or the number of samples de reases and is lower for

S

l,l

than for

S

h,l

.For

the NLMEM-based Wald test using the empiri al SE, it an be seen that for both

parameters the globaltype I errors almost never dierfrom the nominal level of 5%

showing theinuen eof theunderestimation ofthestandard errorsonthe properties

of the NLMEM-based Waldtest.

4 APPLICATION

In 2005, somatropins available in the United States (and their manufa turers)

in- ludedNutropin ® (Genente h), Humatrope ® (Lilly),Genotropin ® (Pzer), Norditro-pin ®

(Novo),andSaizen ®

(Mer kSerono). In2006,theFDAapproved anew

somat-ropin alledOmnitropee ®

(Sandoz). Forthisapproval,bioequivalen e rossovertrials

wereperformed. WeanalyseoneofthemwiththestandardNCA-basedapproa hand

the proposed NLMEM-based approa h. Then, we perfom the bioequivalen e Wald

test usingNCA and NLMEM estimates.

4.1 Material and methods

A randomized, double-blind, single-dose, 3-way rossover study with three

treat-ments, three periods, and six sequen es was ondu ted to ompare the

pharma oki-neti parameters of Omnitrope ®

powder for solution for inje tion, Omnitrope ®

3.3

mg/mLsolutionforinje tion,andGenotropin ®

powderforsolutionafterasingle

sub- utaneous dose of5 mg. Thirty-six healthy au asianadults were re ruited and they

re eived o treotide for endogenous hGH suppression before ea h treatment period.

The three treatment periodswere separated by aseven day wash-out period. Blood

samplesforpharma okineti assessmentswere olle tedafterdose administrationfor

ea h periodat times1, 2,3, 4,5,6, 8,10, 12, 16, 20, and 24

h

. Con entrations were

measured by hemilumines ent immunometri assay (Iranmanesh etal.1994) witha

for the last samplingtimes.

WeanalysethedatawithNCAandNLMEMusingtheSAEMalgorithmimplemented

in MONOLIX 2.4. For NCA analysis, we use the linear trapezoid rule to estimate

AUC

0−tlast

. To obtain the total

AUC

, we ompute the terminal slope by log-linear

regression using2to4samplingtimes. Asdes ribed in2.1, thelog-transformed

indi-vidual

AUC

and

C

max

are thenanalysedusingaLMEMin ludingtreatment,period,

sequen e, and subje t ee ts. The referen e lasses are the Genotropin ®

treatment,

the rst period,and the sequen e Genotropin ®

-Omnitrope

®

powder -Omnitrope

®

solution for the treatment, period, and sequen e ovariates,respe tively.

For NLMEM analysis, we use a one- ompartment model with rst-order absorption

with a lag time (

t

lag

) and rst-order elimination to des ribe the data. With this

model, for sampling times before

t

lag

, on entrations are null. For sampling times

after

t

lag

, on entrations are des ribed by Equation 4 repla ing

t

by

t − t

lag

. To

de-termine the stru ture of the random ee ts matri es and the residual error model,

we analyse the Genotropin ®

data. Models are ompared using the Bayesian

Infor-mation Criteria (BIC), the best statisti almodel orresponding to the smallest BIC

(Bertrand etal. 2008). Forthe stru ture of the BSV matrix, wetest diagonal,blo k

diagonal,and ompletematri es. Regardingtheerrormodel,wetestahomos edasti

(

b = 0

) and a ombined error model. For the analysis of all data, the stru ture of

the WSV matrix is hosen to be identi al tothe stru ture of the BSV matrix

deter-mined during the analysis of the Genotropin ®

data. We add treatment,period,and

sequen e ee ts on the four PK parameters. The referen e lasses are the same as

for NCA analysis. After tting the data, the model is graphi ally evaluated using

the individualweighted residuals (IWRES) and the 90% predi tion interval for ea h

formulation. For the model evaluation, from the nal statisti almodel and its

esti-mates, we simulate 200 datasets based on the stru ture of the original data (dose,

ovariates). For ea h formulation, we ompute the 5% and 95% per entiles of the

orre-We perform bioequivalen eWald tests on

AUC

and

C

max

using NCA and NLMEM

estimates with a type I error of 5%. For NLMEM, we ompute the treatment ee t

on

C

max

using xed ee ts estimates and its standard error by the delta method.

4.2 Results

Forthe analysis of the Genotropin ®

data, the best statisti al model in lude BSV

for all PK parameters with a orrelation between the learan e and the volume of

distribution, and a ombined error model. Parameter estimates (ex ept period and

sequen e ee ts) are displayed in Table 3 with their standard errors. Pre ision of

estimation is judged satisfa tory for all parameters. Con entrations of somatropin

versus time with their 90% predi tion interval and the IWRES versus time are

dis-playedinFigure5forea htreatmentgroup. Thesemodelevaluationplotsarejudged

satisfa tory.

After estimating the parameters by NCA and NLMEM, we perform bioequivalen e

Waldtests on

AUC

and

C

max

for both formulations of Omnitrope ®

. The results of

those tests are displayed in Table 4 with the ratios of

AUC

and

C

max

, the

orre-sponding 90% CI, and the p-values of the bioequivalen e Waldtests. With a type I

error of 5%,

AUC

and

C

max

of Omnitrope ®

powder and solution are bioequivalent

to those of Genotropin ®

using NCA and NLMEM bioequivalen e analysis.

5 DISCUSSION

In this study, we evaluate the type I error of NLMEM-based bioequivalen e tests

performed on the treatment ee t estimates when treatment, period, and sequen e

ee ts but alsowithin-subje tvariabilityare taken intoa ountduringthe NLMEM

estimation. This new approa h is ompared to the standard non- ompartmental

analysis where bioequivalen eWaldtests are performed onthe treatment ee t

Con erning theNLMEM-based bioequivalen etests, weshowhowWaldtests an be

performedona se ondary parameter su has

C

max

whi hallows the extension of the

standard bioequivalen e analysis based on NCA estimates to the NLMEM ontext.

Furthermore, for a parameter of the PK model, we extend the likelihood ratio test

for bioequivalen e.

AsPanhardetal(Panhardetal.2007),andDuboisetal(Duboisetal.2010),we

simu-lateunderaone- ompartmentPKmodelandestimatetheNLMEMparametersusing

the samemodel. So, wedonot study theimpa tof havingthe in orre tmodel being

used in the bioequivalen e NLMEM-based tests, and how would it ompare to the

NCAapproa hinthat ase. Nevertheless,whenbioequivalen eanalysisisperformed,

thereisalreadya umulatedinformationonthedrugandthepharma okineti model

isusually known. Furthermore, even if NCAisknown asa "model-free"approa h, it

assumes linear terminal eliminationand providesmeaningless parameters when itis

appliedtononlinearpharma okineti s. So, theproblemofestimatingwitha"wrong"

model ould exist for NCAand NLMEM.

The NLMEM-based bioequivalen e analysis requires to estimate many parameters.

So, a robustalgorithmhas tobe used. The simulationstudy illustratesthe a ura y

of the SAEM algorithm, espe ially in the ontext of bioequivalen e analysis. We

showthat biasesandRMSEobtained by theSAEMalgorithmaresatisfa toryfor all

parameters although BSV and WSV are slightlyunderestimated for large variability

and lownumberof patients. These results are similar tothose obtained by Panhard

and Samson (Panhard and Samson 2009). As expe ted, biases and RMSE de rease

when the amountof informationin reases (bythe in rease of the numberof patients

or periods). All xed ee ts are orre tly estimated with no bias, whi h is of great

interest for testing treatment ee t estimates. The good estimation of the xed

ef-fe ts using the SAEM algorithmleads to a good estimation of the geometri means

of

AUC

and

C

max

, asillustrated by our evaluation. At the opposite, this evaluation

are aboutten totwenty samplespersubje t. Thismethodisnotwellsuitedfortrials

performed in patients where the number of samples is often limited. In omparison

to model-based approa hes, the estimation of parameters through NCA has several

drawba ks. Itisgivingequalweighttoall on entrationswithouttakingintoa ount

the measurementerror. Furthermore,NCAissensitivetomissingdata,espe iallyfor

the determinationof

C

max

and the omputationof the terminalslope. Even without

missing data, the interpolation of the

AUC

between the last samplingtime and

in-nity isvery sensitive tothe number ofsamples usedto ompute the terminalslope.

However, even with biased geometri mean, the treatment ee t estimated by NCA

are not biasedwhi h partly explains the goodresults forthe type I error.

When the numberof samples persubje t is large and the variabilityisnot toohigh,

tests basedonindividualNCAestimates remainagoodapproa hsin ethey are

sim-ple and showed satisfa tory properties forboth tested parameters. For

C

max

and the

sparsedesign,weexpe tedanin reaseofthetypeIerrorbe ausethereisnosampling

time orrespondingtothemaximal on entrationwhi his loseto2

h

. Butevenwith

poor geometri mean estimates, the type I error is maintained at the nominal level

of 5%. It ould be explain by the good estimation of the treatment ee t estimate

despitethe biasedgeometri mean. Though, forsimulationwith

S

h,h

,the globaltype

I error of

AUC

is very onservative (0.8%) whi h shows the limitsof NCA for data

with high residual error.

The typeI errorof the NLMEM-based bioequivalen eWaldtest and LRT arerather

similarbutWaldtestsareeasiertoperform. Indeed,thebioequivalen eLRTrequires

to estimate the parameters and log-likelihood of three statisti al models. F

urther-more,there is urrently nomethodologytoperformaLRTonase ondaryparameter

ifthe model annotbereparameterizedusingthis parameter(e.g.

C

max

). ForaWald

test on

C

max

, the delta method orsimulations an beused toestimateits treatment

ee t standard error. Based on our simulation study, for a one- ompartment PK

are really lose and the results of the type I error are similar for both estimations.

However, theuse of the deltamethod an betri ky sin ethe analyti alexpression of

C

max

isnot always available for omplexor nonlinear PKmodels.

For NLMEM-based Wald tests and LRT, we found an ination of the type I error

when the onditions move away from asymptoti , i.e. for small sample size and/or

data with high variability. The use of NLMEM-based bioequivalen e analysis in its

urrent proposed form would be questionable for regulatory agen ies in these ases

dueto on ernsaboutpotentialtypeIerrorination. ForNLMEM-basedWaldtests,

the underestimationofthe standarderrorsare responsibleof theinationofthe type

Ierror. Indeed,thereisnoinationwhentheempiri alstandarderrorisusedinstead

of theestimated. The empiri alstandard error an beused inpra ti ebut not easily

be ause of the omputing time. It requires rst to estimate the parameters using

the data of the lini altrialof interest, then tosimulate trials with the same design

as the original dataset and nally to re-estimate the parameters for ea h simulated

trial. This approa h also assumes that the underlying stru tural model is orre t

whi h is usually the ase when bioequivalen e analysis is performed, as previously

mentionned. In our simulation, the number of subje ts is more inuential on the

ination of the type I error than the number of samples. Indeed, there is a slight

ination of the type I error for the sparse design (

N = 40, n = 3

) ompared to the

ri h(

N = 40, n = 10

, same

N

)whereas the inationishigherfor theoriginaldesign

(

N = 12, n = 10

)also omparedtotheri h(same

n

). ForNLMEM-basedWaldtest,

this isexplainedby the slighter underestimationof thestandard errorsfor thesparse

design. The ination of the type I error for NLMEM-based Wald tests and LRT is

not spe i to bioequivalen e tests. It is due to the asymptoti properties of these

tests andwasalsodemonstrated for omparisontests byPanhardetal(Panhardand

Mentré 2005) andWälhbyetal(Wählbyet al.2001). Similarly,the underestimation

of thestandard errorswasalsorelatedtothe inationofthe type Ierror for

by in reasing the number of patients. Furthermore, dierent approa hes ould be

explored to orre tthe type I error inationof NLMEM-based bioequivalen etests.

For NLMEM-basedWaldtests,the underestimation ofBSV and WSV ouldexplain

the underestimation of the standard errors. Even though maximum likelihood

es-timation is the standard approa h in NLMEM, the varian e omponents are often

underestimatedforsmallsamplesizeandhighvariability. Inlinearmixedee ts

mod-els, the REML estimation iswidely implemented,but in NLMEMit has been barely

studied, althoughthe REMLpro eduremay improvetheestimationofvarian e

om-ponents in NLMEM. Meza et al (Meza et al. 2007) developped a REML estimation

pro edure for the standard SAEM algorithm. They showed that the SAEM-REML

algorithm redu es bias and RMSE of the varian e parameter estimates in a

simu-lation study on a simple NLMEM. Further work is needed to propose the REML

estimation pro edure forthe extended SAEM algorithmdeveloped for rossovertrial

analysis.By improvingthe estimation of varian e parameters,the REML estimation

pro edure should improve the bioequivalen eWaldtest. As explained in se tion 2.1

and 3.2, for bioequivalen e Waldtests based onNCA estimates, the LMEM

param-eters are estimated by REMLand both test statisti sfollowaStudent t-distribution

with degrees of freedom depending on the number of subje ts. So, we perform the

NLMEM-based bioequivalen e Wald tests assuming a Student t-distribution under

H

0

with the same number of degrees of freedom as the NCA-based bioequivalen e Wald tests (unshown results). For all simulation settings, the type error de reases

ompared to the NLMEM-based Wald test with a Gaussian distribution but there

is still a slight inationof the type I error while the use of empiri al SE orre ts it.

Toour knowledge,there isnotheoriti al developmentorevaluationof the degrees of

freedominthe ontextof NLMEM.Thedistributionweuse ismoreorlessempiri al,

and further work is needed.

Other approa hes ould be studied su h as the orre tion of the nominallevel using

perform-havenotyetbeenproperlystudiedinNLMEM.InNLMEM ontext, thepaired

boot-strapisusuallyusedbut withouttakingintoa ountthedierentlevelsofvariability

of the NLMEM. Furthermore, there is no theoriti al or simulation result to justify

its appli ation. To our knowledge, only two published studies adress the issue of

bootstrapin NLMEM (Das and Krishen1999; O aña etal. 2005). O añaet al(Das

andKrishen1999)proposed abootstrapapproa hresamplingtherandomee tsand

residualerrors. Theyevaluateditbysimulationbuttheyperformeditusingtwo-stage

tting pro edure (Steimer et al. 1984) where "population" mean parameters are

es-timated from individual parameters obtained after separate tting of ea h subje t

data. Furthersimulationsstudiesareneeded toreallyunderstandbootstrapmethods

properties in NLMEM. So, we would favor a orre tion of the tests by degrees of

freedom, whi his alsoa less omputer intensive method.

The analysisof the rossovertrialofthreesomatropin formulationsshows the ability

toperform aNLMEM-basedbioequivalen eanalysisusing theSAEMalgorithmona

real data set. Even with forty xed ee ts and ten varian eparametersinthe

statis-ti al model, the SAEM algorithm onverges. Furthermore,the SAEM algorithm an

handledatabelowthelimitofquanti ation ontrarytoNCA.ThePKparameter

es-timatesforGenotropin ®

aresimilartothosefoundbyStanhopeetal(Stanhopeetal.

2010). We perform NLMEM-based bioequivalen e Waldtests and not LRT be ause

results onWaldtestsand LRTare similarinthe simulationstudy,and wewould like

toperformtests onthe treatmentee ts of

AUC

and

C

max

,whi hisnot possible by

LRT.Theresultsofthe bioequivalen eanalysisbasedonNCAandNLMEM are

sim-ilar. In both ases, we assess the bioequivalen e ofboth Omnitropee ®

formulations.

Bioequivalen e tests based on NLMEM allow one to de rease the number of

sam-ples persubje t, whi his of great interest fortrials performedin patients. However,

aution is needed for small sample size and data with high variability. With sparse

sampling,the hoi eofdesignisimportantnotablytoimprovethepropertiesoftests.

geneti ee t in a one-period trial. Design optimisation algorithms for models with

dis rete ovariates and dierent periods of treatment ould be used for rossover

studies. They are now available in the version 3.2 of PFIM software (Bazzoli et al.

APPENDIX: DELTA METHOD FOR

C

M AX

For a one- ompartment model with rst-order absorption and rst-order elimation

C

max

isa fun tion of the three PKparameters:

C

max

=

F D

V

exp



CL log(k

a

) − log(CL/V )

V k

a

− CL



(5) So,

β

T

C

max

is afun tion

h

of

λ

k

a

,

λ

V /F

,

λ

CL/F

,

β

T

k

a

,

β

T

V /F

,and

β

T

CL/F

:

β

_C

T

_max

=h(λ

k

a

, λ

V /F

, λ

CL/F

, β

T

k

a

, β

T

V /F

, β

CL/F

T

)

= − β

V /F

T

− A

2

λ

CL/F

exp(β

_CL/F

T

)

A

1

+

λ

_√

CL/F

A

6

log



_λ

k

a

λ

V /F

λ

CL/F



(6)

The ve tor of partial derivativesof

h

is:

∇h =



1

λ

ka



A

4

A

3

−

A

5

A

6



,

1

λ

V /F



A

4

A

3

−

A

5

A

6



,

1

λ

CL/F



−A

4

A

3

+

A

5

A

6



,

A

4

A

3

,

A

4

A

3

− 1,

−A

4

A

3



′

(7) where

A

1

=λ

k

a

λ

V /F

exp(β

T

k

a

+ β

T

V /F

) − λ

CL/F

exp(β

CL/F

T

)

A

2

= log



_λ

k

a

λ

V /F

λ

CL/F



+ β

_k

T

_a

+ β

_{V /F}

T

_{− β}

_CL/F

T

A

3

= −A

1

λ

CL/F

exp(β

CL/F

T

)

2

A

4

=A

3

+ A

2

λ

k

a

λ

V /F

λ

CL/F

exp(β

T

k

a

+ β

T

V /F

+ β

CL/F

T

)

A

5

=λ

k

a

λ

V /F

λ

CL/F

log(λ

k

a

λ

V /F

/λ

CL/F

) + λ

CL/F

(λ

CL/F

− λ

k

a

λ

V /F

)

A

6

= λ

k

a

λ

V /F

− λ

CL/F

2

(8)

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