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16 ($%+&#$%( %' yi!1$'& ni%+1 '%#!F#1& 3 1 #' EE 8"%' &# =%#$'%#3 $1 #'! %$& E*%! ' # #' * ##%#!# θ i3! 1$'& *'!F#1&"#&% $5& ξ #'! 1$'& '* &65&!%#'==$'&!!1*#'%&!F#1&3 ξ = ( (, ) ' ''5&%&!F#1&'!%%## > "*%! y i = f θ, ξ ) + ε ( σ + σ f ( θ, ξ )) ( i i i int er slope i i %Y ε i '! 1$'& & &!! $%%# #' E! ' %+1 '%# #%# 6!5&!*%! " # &% 5& $ & &!! %#' ## #' F&# %+1 '%#E!F &''5&F!!%#''+&!%#&#!%#%*! ε N(0, ) ~ i i I n 1$ I n &# i * '$#''*#%#K I" σ#'' σ %#'&6 *'$ $' #'! 1 #$ &*%!F&"*%!F&''%*%$ '5& σ = )"#*%!F& 'E$%==$#'1 '%#$%#' #' σ #' = )"$ #!%Y!&6 *'%#' ==#')'!$%*+#" #$*%!#%#!# E=='*6'3!*%!'&%$%**&#E'%&! #1&3*!1$'& *' θ i%&!f#1&3&'1 F&##1&E!F &' #! %&! '%#" # ' $ 1 '%# &' O' 6!5& $ $''5& #1&!!3'!!5&!FX3!63!%3!$%1 +!"%'D 3!1$'&$ $%1 +! $?!F#1& " *' &1#' & == 1 '%# #6!5&M $ 1 '%# %#' 5& #'=!F#'%&$'%# *'! '% b i" 1$'& *' θi!f#1&&'!%f6*x$e&#*%!'z$%# #1 &[3! #' &#=%#$'%#$%##&!1$'&=='=6 β $%**&#E'%&! #1&3!1$'&=='! '% bi %E!F#1&'!1$'&$%1 +! Zi > θ = Kβ33I I" # &% 5&! 1$'& =='! '% b i ~ N(0, Ω) 35& ε b i i %#' ## #' F&# &B' E!F &' ' 5& %& &# *O* #1& 3 εi ' bi %#' ## #'" Ω$%%#E! * '$1 #$=='! '%%&! 5&!!$ 5&!*#'$%%# &$%*% #'&1$'& #=#3%&'#$%*' #!*%!! 1 +!'#'/%$$ %# *'F&# &B'#'1%!1*#'3%# B%&'&# &' *'! '% κ #! *%! $%##1 & KR!%# ' #3(--7I" #3!!1*#' %#' ==$'& 16

17 ($%+&#$%( & #'*%$$ %#3!1$'& *'#1&!&&B'E!F%$$ %#31 #'EE *3 &' # F$> θ = Kβ3 3I 3 κ I"=='! '% κ %#'&%'+& #%*!*#'3*%2###&!!'* '$1 #$/$%1 #$ Γ > κ 8 K)3 ΓI" =P%# #!3! =%#$'%# #'!!& %&1#' &# *%! '= %& 6%##'!" 6*! %&&#*%! *! # $%1 +!3! 1$'& *'!F#1&E!F%$$ %#3F$'!%> θ = β + + κ %#$ %'3! 1$'& +κ %& θ = β 3$'1*#'" Ψ! 1$'& 9 *' E '* ' Ψ = ( β, vec( Ω), vec( Γ )', σ, σ )"!F%#% λ!1$'&%& #'! *' int er slope 1 #$'!5& λ J = ( KΩIJ3 KΓIJ3 σ#' 3 σ ) 3%#&'$ Ψ $%** Ψ ' = ( β ', λ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Ξ "# 8&B' &65&!%# ''+&&# %'%$%!'!*#' ξ 31 #'EE8" $&#%'%$%!!*#' ξ '=# &# #%*+!1*#' # 5&! '* &65&!! %#' ' ==$'&> ξ = ( (, ) " %'%$%! %&! '%# '!%!F#*+! 8 %'%$%!!*#' > Ξ = { } ξ,, 1 ξ N "! ' $'! 8 %'%$%!!*#' %&&##%*+ 18

19 ($%+&#$%( '%'! F%+1 '%#'!5& n = N i i=1 n " # #'3! '!& =5&#' F 1%&# %'%$%! $%*%<%&&B' 2 #'!*O*%'%$%!!*#' ξ 1$!! #'EE<" { ξ, 1 N ; 1 ξ, ; ;, 2 N2 ξ Q N Q } Ξ = %'%$%!%&! '%#F$'!%> [ ] [ ] $ 5& 8 $%%# &6 #%*+ &B' 2 #'! *O* %'%$%!!*#' ξ 1$ Q q=1 N q Q q q q=1 = N ' n = n N " #3 '*#&# %'%$%! *!5& F&# ' F#'=!'* E!!%& %& $ 5&&B'* &=#! '&$'&&%'%$%!>!#%*+%&3!#%*+!1*#' %& # 5&! %%'%# &B' %$ E $ $&# %&" $ %&&##%*+'%'!!1*#' =6"%&'$#'% '%#%#'*%' #'"# =='3'& *&! '%#%#'$%#=*5&! +! #$#'!#%*+%&3! #%*+ &B' %& '!F!!%$ '%# '*!1*#' # $ 5& %& #=!&#$ =%'*#'! $%# F'* '%# *'3 $%#& #' =% E '& ##'' +!K!/4 ## 3(--)M.%#%#3(--<I"!&3$5&'%#%#'F &' #'!&#'!!3!%5&!'&%#'$%#&' #%&! '%# '$&!'!!5&! #=#'3! %## X %& $? '#'" # +%# $%6 %'%$%! &' %#$ *!%!F==$ $''&*!*#'!#!&'5&3#*#* #'!#%*+ 19 %Y!1*#' &B'3'*%#$%N'&3##!1 #'5&FE'*#=%* '=" O*!F %$ $! 5& + &! *&! '%# *' F 1% &!' ' #' #'%&1!&%'%$%!%&! '%#M*!&&*#'3$'' %$'' $%N'&#'*$!$&!3!*' #' #!#%*+%'%$%!%&1 #'O'1!&#&#'* $$' +!" 4 &! '%! # F6#$&'! # %##%#!# $! 5& K'@#%#'%#13(--,M^!''%#? '%3(--9I3&# %$#!%&'*# %'%$%! $%#' E*#*! * '$ 1 #$ $%1 #$ F'* '%# *'" 3!%#!F#!' %/ *3!F#1! * '$F#=%* '%#'! +%# #=&! * '$ 1 #$ $%1 #$ '%&' '* '& #%# + " '%!F%'* '%# %'%$%! 1 %#$ % &!F&'! '%#! * '$ F#=%* '%# " "%(+"*'$#"+(?%'$%(+"="!#F6' F6%# #!2'5&6 $'%&! * '$F#=%* '%# #! $%#'6' *%!#%#!# E =='*6'" %** $' & +&'! $'%#

20 ($%+&#$%( $#'3! #%#!# ' & *%! '&$'&! # $ *' # *'! '*# '%# #!2'5&!!%/1 *+! #$3'%#$! * '$F#=%* '%# $!$&! E ' 1 $%# $!!/$" # %6* '%#!F6%#! * '$ F#=%* '%# ' %% #' '!"K(--9I #! $ F&#*%! F& '=%Y! & %#' &% '+&#%*!*#' *%2###&!! ' 1 #$ %*%$ '5&"''6%#'+ &&#!# '%# &*%&*%! &# 1!%*#' 2!% &'%&!F #$ =='! '%3 $%** #! *'% " '' %6* '%# *' F%+'# &# *%!!# %' &6=='! '%'$%#&'E&#6%# #!2'5&! * '$F#=%* '%# "'%&''!"K,)),I%#'! &''#&!1!%*#'! * '$##'%& #' #!1$'& *'E'* σ#' ' σ $ $' #'!*%!F&&!!" 1!%*#'%#' ' 1!& *&! '%# &&# '&!! * $%$#'5& %&! *%!$&!!F#%6 #K'%&'3,)),M'%&''#'3,))7I" * '$F#=%* '%#%&! '%#$%*%$%**! %**8* '$ F#=%* '%#!*#' > KΨ3Ξ I = 8 = ( KΨ3ξ I!%'%$%!%&! '%#'=#!F#*+! %'%$%!!*#' $ $&# #1& Ξ = {, }!!&'F$ M %&" F Q q= 1 ξ,ξ N 1 "&+#3 ( Ψ, Ξ) = N M ( Ψ, ξ )!%'%$%!%&! '%#'=# < q F q %&1!%%#!F6%#! * '$F#=%* '%#%&&#%'%$%!!*#' ξ K!F#$ ' #' &* #! ' $'' $'%# =# F!!!F$'&I" %&!%#5&!1$'&$%1 +!'%## D" * '$' #=#, 3 KΨM I. 3%Y 3 KΨM I'!!%/1 *+! #$&1$'&%+1 '%# y %& I Ψ Ψ! 1$'& *' %&! '%# Ψ " %&!# %#! *%! K Kβ33II3 ξi # ==$'& #'&#1!%*#' 2!% &*%M$1!%*#''==$'& &'%&! *%2## =='! '% 3 1!& 5& ' %#$! E )" =# '#! $%*!6' $!$&!3 #%& #F!5&%# $''!# '%# & *%! F&M $'' '$'%# ' F!!& *! E $!! ==$'& =&' # " *%! ' ''5&F$'!%> JK Kβ33II3 ξi K Kβ3)3II3 ξ I + K )I + εkσ #' + σ K Kβ3)3II3 ξii = ) 20

21 ($%+&#$%( '!!%/1 *+! #$3' %$ > (, 3 KΨM I!#, π +!# 1 + K. I1 K. I %Y.'1%#'!F #$* #!'! 1 #$* #!%## >. K I I =. K Kβ3)I3 ξi, JK Kβ33II3 ξi K Kβ3I3 ξi 1 K II 1 = Ω + σ + σ β ξ #' K K 3)3II3 I J = ) = ) 1 '%#!!%/1 *+! #$ %' &6 *' %&! '%# $%#&#'!%E!F6%#! * '$F#=%* '%#!*#' #+!%$> K. 31 I 2 K. 31 I ( KΨ3 ξi, 2 K. 31 I ) K. 31 I %Y K.1I)K.1I ' 2K.1I %#' +!%$* '$! # #'!F #$. '! 1 #$1%+1 '%#> ( ( ( ( K. 31 I ), = + β β β β 1$' = (3 3 *Kβ I 1 1 ) K. 31 I = 1 1 λ λ ( ( ( ) K. 31 I = 1 1 λ β ( ( ( ) 1$' = (3 3 *Kλ I 1$ m = 1,, dim( β ) ' = (3 3 *Kλ I #!' 1 &6 $' $**#'3! +!%$ 2 K. 31 I! * '$F#=%* '%# ' &% #&!3 "! # #$! 1 #$ %+1 '%# #! *' *%2##F' #$%*'"''*'% #! &'! '!DE" F6%#! * '$ $%*!' KD JI '!& $%*!6 & ='!F#'%&$'%# 1$%#&*%!" $%* %#$&6*'%#F 1! &$&#==#$ *%' #' #!F1!& '%#&66*!K'%&''#'3,))7I" F6%#! * '$ F#=%* '%# ' '#& E *%!'# #' $%*'!F#=!&#$$%1 +!&! * $%$#'5&'F&#1 +!'#' /%$$ %# &$%&==#'%' '*#'K'%&''#'3,))7I" #$*O*' 1!3!%#' &%%&#!# '%#'23$F'E&#!# '%#&*%! &'%& 21

22 ($%+&#$%( 1!&#1&!!=='! '%"&$&# *!% '%##F '*%#' 1$$'' # %$35&%##&!' '*! E$!!1!%$**#'* %& &#'* $!$&!!&!1 & =' *&! '%# %#'!%" # =='3 #! $ F&# #!2%##3!%##$ 5&#1&%#'#$%##&'&#'* '%#*5& + 2###&'%#$ O'==$'&"#1!& '%# ( ( Ψ ξ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

23 ($%+&#$%(!& '' 1!& %! * '$ F#=%* '%# "!& &'! # * $%$#'5& ' $!& /%'*!'3 5& *' *#*!F#1 & '*# #'! * '$ F#=%* '%# %&! '%#" # '5&3! $!$&! $ $' # #$' F#1%#* '$!! &5&*#*!F#1 & '*# #'! * '$ F#=%* '%#1#'E* 6*!'*# #'! * '$"!%'* F%'* '%# %#' #$ " #%*+&6!%'* %#' ' 1!%3 $%#& #' %' E!F%'* '%# F&# %'%$%! ' #'3 %' E!F%'* '%# F&# %'%$%!'' ''5&K%&$%#'#&I" %** $' $**#'3 &# %'%$%! Ξ %& &# '& %&! '%# # * $%$#'5& &' F$ %&! =%* Ξ = { } ξ,, 1 ξn %Y$ 5& ξi $%%# &6 %'%$%!!1*#'==$'&&!&B'3 i = 1,, N "! *O*=P%#3#%& 1%# *%#'5&F!&'F6*%&! =%* Q%& Nq &B' 2 #'!*O*%'%$%! { ξ, 1 N ; 1 ξ, ; ;, 2 N2 ξ Q N Q } Ξ =!*#' > [ ] [ ] &' # ''+&E$ 5&%&&#% '%##! m &B' mq 2 #'!*O*%'%$%! ξq''!5& {[ ξ, 1 m1 ] ; [ ξ, 2 m2] ; ; ξ, Q mq} " #!!% F&# %'%$%! #'" # Q i= 1 m q N q q = 3$%%# #'E! %%'%# N = 1" Ξ &'%#$F$%&! =%* Ξ = '=# #&#%'%$%!' ''5&"#&''#$ %'%$%!' ''5&E* 5&!$%#5&$%*#')'('1= #''%&B%& Q i= 1 m = 1" %'%$%! ' *#'! +! &5&#%&#F 1%# &##%*+ mq q #'"! =&'!% *# E&# %'%$%! #' # %$ #'! %%'%# E!F#' &&!!&%$" F&#%'%$%!' ''5& &%'%$%!#'##&# 'F==$ $'&%'%$%!5&'##!''=+!" N #!%'* 'E$'' %$'!F!%'*&%%1/^ 2##K%%13(-9,M ^2##3(-9,I31!%#'!*#' #&#$%#'6'%##%#!# $! 5&& '#& &6 *%! #%#!# E ==' *6' #' '!" K(--9I"! *'! $%#'&$'%#%'%$%!$%#'#&/%'* &6"'!%'*$%#11&#%'%$%!/ %'*!5&!5&%'!%'%$%!#'!"!&'$# #'O'!%#%& ''#! $%#1#$ $!&'!!F#*+!%'%$%!!*#' %+! &$%&' '%#" %'%# ζ 3!!' $ %'%$%!!*#' 5& &1#' O' $%*% F&##%*+!1*#' ==#'!%#!$%#' #'/=#" F!%'*&%%1/^2##'' '="%'3 Ξk!%'%$%!%+'#&E!F' '%#B"! * F 'E!F' '%#BKE$%#'&!%'%$%! Ξk + 1'!5& Ξ = ( 1 α ) Ξ + α ξ 1$ α k + 1 k + 1 k + 1 k k

24 ($%+&#$%( * $%*#')'(' ξ # ζ 3&#%'%$%!!*#' E B%&' &%'%$%!3 &5&!&# * α k + 1' ''+&#' #'&#= $'%#* ''+& &6 &'%'%$%!!*#' $%*% #' Ξ k " %'%$%!!*#' ξ E = #' #! %'%$%! * $%%# #''$!&%&!5&!! ' #$ d ( Ξk, ξ ) '* 6*!3$5&1#'E$$ ξ * '!5& ξ arg max * d ( k, ξ ) ( ( )) log det M F ( Ψ, Ξk + 1 ) =# α = Ξ " ξ ζ k + 1 αk+ 1= 0 F!%'* & %%1/^ 2## ' + &! '%* F5&1!#$ R= ' ^%!=%C'?K(-8-I5&' +!'!F5&1!#$#'! '%#&1 #'> -!%'%$%! Ξ'/%'*! - max ξ ζ d ( Ξ, ξ ) = P 1$ P!#%*+ *'E'*&*%! - Ξ*#* max ξ ζ d ( Ξ, ξ ) #3! *%#' 5&F&# %'%$%! %'*! %&!F'* '%# *' & %'5&1!#'%&! 1 #$! $'%#" '&$'&' '1!F!%'*'!%! &1 #'> (" %&#%'%$%!' ''5&#'! Ξ0 #%##","!F' B3 1$!%'%$%! Ξk 3'%&1 ξ = arg max ξ ζ d ( Ξk, ξ ) O' max ξ ζ d ( Ξk, ξ ) P + ε 1$ ε 13&#&!'%! #$'*#" * 7" #%#3 $'&!!%'%$%! Ξ k + 1 = ( 1 αk + 1 ) Ξ k + αk + 1ξ 3%Y α k + 1'$%& ] 0,1[ '5& * 6* det ( M F ( Ψ, Ξk + 1 )) 3$5&$%%#E α = P d * d ( Ξk, ξ ) P ( ( Ξk, ξ ) 1) k + 1 * '!%'*%'* #!#%*+<%'%$%!!*#' '!%%'%# &B'#$!& #$ 5&%'%$%!!*#' ξq&%'%$%!%&! '%#%'*! Ξ"$ ='3!*'! '*# '%#F&#%'%$%!%&! '%##!&=# #'&#'&$'&3$F' E&#+%##+! #$#'!#%*+%& Q E#$!&3!#%*+ nq!1*#' %&'#=#!#%*+&B'E#$!& #$ 5&%&" #1 #$3%&!F%'* '%#%'%$%!#'3!%'*$=5&%#'' 1!%3 5& #! $ %'%$%! /%'* &63 %#'!%'* F$ #" & #$' #' 'F&#%'%$%!#'!'!F *!%&$$1*#'#$ # #'&# '*!1*#' &# *!!& '*" 5& &' O'!*' #' 5& #! $ #*#'!&&'*#*&!' #'&#*!!&%!&'%#K%%13(-9,M^ 2##3 (-9,M'$!!3(-9;I"!'# '1%#''%%+ &!*O*#$3$# #' &$&#$!%'*# #''! $%#1#$1!%'%$%!%'*!" " 24

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32 STATISTICS IN MEDICINE Statist. Med. 2009; 28: Published online 5 March 2009 in W iley InterScience (w w w.interscience.w iley.com ) DO I: /sim.3573 Fisherinform ation m atrix fornonlinearm ixed effects m ultiple response m odels:evaluation ofthe appropriateness of the firstorderlinearization using a pharm acokinetic/ pharm acodynam ic m odel Caroline B azzoli 1,2,,,Sylvie R etout 1,2 and France Mentré 1,2 1 IN SERM, U 738, 16, rue H enri H uchard, Paris, France 2 U niversité Paris D iderot, U FR de Médecine, Paris, France SU MMARY W e focus on the Fisher inform ation m atrix used for design evaluation and optim ization in nonlinear m ixed effects m ultiple response m odels. W e evaluate the appropriateness of its expression com puted by linearization as proposed fora single response m odel.u sing a pharm acokinetic pharm acodynam ic (PK PD) exam ple,w e first com pare the com putation of the Fisher inform ation m atrix w ith approxim ation to one derived from the observed m atrix on a large sim ulation using the stochastic approxim ation expectation m axim ization algorithm (SAEM).The expression of the Fisher inform ation m atrix for m ultiple responses is also evaluated by com parison w ith the em pirical inform ation obtained through a replicated sim ulation study using the first-order linearization estim ation m ethods im plem ented in the NO NMEM softw are (first-order (FO ), first-order conditional estim ate (FO CE)) and the SAEM algorithm in the MO NO L IX softw are.the predicted errors given by the approxim ated inform ation m atrix are close to those given by the inform ation m atrix obtained w ithoutlinearization using SAEM and to the em piricalones obtained w ith FO CE and SAEM.The sim ulation study also illustrates the accuracy ofboth FO CE and SAEM estim ation algorithm s w hen jointly m odelling m ultiple responses and the m ajor lim itations of the FO m ethod. This study highlights the appropriateness ofthe approxim ated Fisherinform ation m atrix form ultiple responses, w hich is im plem ented in PFIM 3.0,an extension of the R function PFIM dedicated to design evaluation and optim ization. It also em phasizes the use of this com puting tool for designing population m ultiple response studies,as for instance in PK PD studies or in PK studies including the m odelling of the PK of a drug and its active m etabolite.copyright 2009 John W iley & Sons,Ltd. K EY W O R DS: nonlinearm ixed effects m odels;m ultiple responses;fisherinform ation m atrix;population design; first-order approxim ation; PFIM Correspondence to: Caroline B azzoli, Inserm U 738, 16, rue H enri H uchard, Paris, France. E-m ail: caroline.bazzoli@ inserm.fr Contract/grant sponsor: F. H offm ann La R oche Ltd, B asel, Sw itzerland Received 9 May 2008 Copyright 2009 John W iley & Sons, Ltd. Accepted 30 January 2009

33 NONLINEAR MIXED EFFECTS MULTIPLE RESPONSE MODELS INTRODUCTION Nonlinear mixed effects models (NLMEM) are widely used to analyze various biological processes described by longitudinal data. As the primary models developed by Sheiner and Wakefield [1] in pharmacokinetic (PK) and pharmacodynamic (PD), NLMEM has been widely used for modelling of biological processes. NLMEM, also called the population approach, allow estimation of the mean value of the parameters in the studied population and their inter-individual variability, or population characteristics. NLMEM are also now commonly used for the joint modelling of several biological responses such as the PK of parent drugs and of their active metabolite. NLMEM allow a sparse sampling design with few data points per individual in a large set of individuals. This can be particularly useful in studies in specific populations such as children or patients with serious diseases, where classical studies with a large number of samples are often limited for ethical or physiological reasons. Estimation of the parameters in NLMEM is commonly performed by maximum likelihood. However, due to the nonlinearity of the regression function, an analytical expression of the loglikelihood in NLMEM cannot be provided. To solve this issue several methods for estimating the parameters have been proposed, based on an approximation of the log-likelihood such as the first-order method (FO) or the first-order conditional estimate (FOCE) method proposed by Lindstrom and Bates [2]. Both methods use a linearization of the structural model either around the expectation of the random effects parameter (FO) or around the individual estimates of the random effects (FOCE). These methods have been implemented in the NONMEM software [3, 4] and also in the NLME function of Splus and R software [5]. Compared with FO, the FOCE method provides less biased estimates and, in the context of joint modelling of multiple responses, is more appropriate with fewer problems of convergence or of inter-individual variance estimation [6, 7]. Alternative methods have also been proposed to maximize the likelihood using a stochastic approximation of the integrals, such as the G aussian quadrature [8] or the Adaptative G aussian quadrature methods implemented in the NLMIXED procedure of SAS. Recently, the stochastic approximation expectation maximization algorithm (SAEM) has been developed and implemented in the MONOLIX software [9, 10]. It uses a stochastic approximation version of the standard expectation maximization (EM) algorithm [11, 12]. The convergence and the consistence of the estimates have been proved by the authors. In this algorithm, the EM algorithm is used for finding the maximum likelihood estimates of parameters in models, where the model depends on the unobserved variables corresponding to the random effects in the NLMEM. An appropriate choice of experimental design for estimating parameters in NLMEM is required. Called a population design in this framework, a design is defined as a group of elementary designs; each elementary design is composed of a set of sampling times to be performed in several individuals. Determining a population design involves identifying both the allocation of the sampling times and the whole group structure, that is to say the number of elementary designs, the number of samples per elementary design and the proportion or the number of individuals in each elementary design according to a fixed total number of samples. Simulation studies have shown that the precision of estimation of the parameters depends on the choice of the design [13, 14] and that an appropriate choice can thus substantially improve the efficiency of studies. In the context of NLMEM with sparse designs, the challenge is then to determine the trade-off between few sampling times and informative data to obtain correct parameter estimates. To evaluate population designs, the theory of optimum experimental design described for instance by Atkinson and Donev [15] or by Walter and Pronzato [16] in classical nonlinear models, has been Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim

34 1942 C. BAZZOLI, S. RETOUT AND F. MENTRÉ extended to NLMEM. This theory uses the criteria based on the Fisher information matrix (M F ). It comes from the Cramer Rao inequality; indeed, the inverse of M F is the lower bound of the variance covariance matrix of any unbiased estimators of the parameters. As the likelihood has no closed form in our framework, a linearization of the model around the expectation of the random effects has been proposed by Mentré et al. [17] and extended by Retout et al. [18] to derive an approximate expression of M F. Accuracy of this approximation was first shown by simulation of an example based on a real PK study [18, 19], and was confirmed by comparison of the predicted SE computed from this approximate M F with those given by an evaluation of M F without linearization obtained by stochastic approximation using the SAEM algorithm of MONOLIX [20]. The approximated expression of M F has been implemented in R functions PFIM and PFIMOPT for population design evaluation and optimization, respectively [21 23]. Recently, PFIM Interface 2.1, a graphical user interface version, has been developed, allowing both evaluation and optimization in the same tool [21]. However, currently, these tools only allow evaluation and optimization of population designs of single response models. For multiple response models, the same linearization method around the expectation of the random effects as for single response models has been proposed to approximate the population M F [24 27]. In those papers, illustrations of this development were provided using either a PKPD model or a joint PK model of a drug and its metabolite. However, the accuracy of the development of M F by linearization for multiple responses has not yet been evaluated. Even if the same linearization as in the single response is used, computation can become more complicated for multiple responses. Indeed, some parameters can be included in several responses and the information on those parameters is therefore obtained from each of those response profiles. This is usual in the PKPD context where PD response depends on the PK parameters. Moreover, as noted previously, the use of the linearization around the expectation of the random effects appears to be inadequate for joint estimation of multiple response models [6, 7]. The appropriateness of its use in the context of design evaluation is thus also questionable and should be investigated. The objective of this study was, therefore, to evaluate the first-order approximation to compute the Fisher information matrix in NLMEM with multiple responses. To do this, we considered a PKPD simulation example associated with a population design. Then, we compared the predicted standard errors (SE), computed from the approximated expression of M F with those given by the evaluation of M F without linearization obtained by stochastic approximation using the SAEM algorithm of MONOLIX. We also performed another evaluation by comparison of those predicted SE to the empirical ones, obtained by estimation on simulated data sets using three different estimation algorithms: FO and FOCE (with NONMEM); SAEM (with MONOLIX). Based on those simulations, we also compared the performance of those three estimation methods in the same simultaneous analysis of this PKPD model. In Section 2, we introduce the notations, describe the PKPD example and present the methodology used to evaluate M F and to compare the estimation methods. Section 3 describes the results of the evaluation and the comparison. Discussion of the results is provided in Section 4. The development of M F for multiple responses is given in detail in the Appendix Notation 2. METHODS In the nonlinear mixed effect multiple response model, an elementary design ξ i for one individual i is defined by n i sampling times. It is composed of several sub-designs such that Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim

35 NONLINEAR MIXED EFFECTS MULTIPLE RESPONSE MODELS 1943 ξ i =(ξ i1,ξ i2,...,ξ i K ), with ξ ik being the sub-design associated with the kth response, k =1,..., K. ξ ik is defined by (t ik1,t ik2,...,t iknik ), the vector of the n ik sampling times for the observations of the kth response, so that n i = K k=1 n ik. For N individuals, we define a population design composed of the N allocated elementary designs ξ i,i =1,..., N. A population design is therefore described by the N elementary designs for a total number n of observations such that n = N i=1 n i : Ξ={ξ 1,...,ξ N } (1) Usually population designs are composed of a limited number Q of groups of individuals with identical designs within each group. Each of these groups is defined by an elementary design ξ q q = 1,..., Q, which is composed, for the kth response, of n qk sampling times (t qk1,t qk2,...,t qknqk ) to be performed in a number N q of individuals. The population design can then be written as follows: Ξ= { [ξ 1, N 1 ]; [ξ 2, N 2 ];...; [ξ Q, N Q ] } (2) A nonlinear mixed effects multiple response model or a multiple response population model is defined as follows. The vector of observations Y i for the ith individual is defined as the vector of the K different responses: Y i =[y T i1, yt i1,..., yt i K ]T (3) where y ik,k =1,..., K, is the vector of observations for the kth response. Each of these responses is associated with a known function f k, which defines the nonlinear structural model. The K functions f k can be grouped in a vector of multiple response models F, such as F(θ i,ξ i )=[ f 1 (θ i,ξ i1 ) T, f 2 (θ i,ξ i2 ) T,..., f K (θ i,ξ i K ) T ] T (4) where θ i is the vector of all the individual parameters needed for all the response models in individual i. The vector of individual parameters θ i depends on β, the p-vector of the fixed effects parameters and on b i, the vector of the p random effects for individual i. The relation between θ i and (β,b i ) is modelled by a function g,θ i = g(β,b i ), which is usually additive, so that θ i =β+b i, or exponential so that θ i =βexp(b i ). It is assumed that b i N(0,Ω) with Ω defined as a p p- diagonal matrix, for which, each diagonal element ωr 2,r =1,..., p, represents the variance of the rth component of the vector b i. The statistical model is thus given by Y i = F(g(β,b i ),ξ i )+ε i (5) where ε i is the vector composed of the K vectors of residual errors ε ik,k =1,..., K, associated with the K responses. We also suppose ε ik N(0,Σ ik ) with Σ ik a n ik n ik -diagonal matrix such that Σ ik (β,b i,σ interk,σ slopek,ξ ik )=diag(σ interk +σ slopek f k (g(β,b i ),ξ ik )) 2 (6) where σ interk and σ slopek qualify the model for the variance of the residual error of the kth response. The case σ slopek =0 returns a homoscedastic error model, whereas the case σ interk =0 returns a constant coefficient of variation error model. The general case where the two parameters differ from 0 is called a combined error model. We then note Σ i (β,b i,σ inter,σ slope,ξ i ), the variance of ε i, over the K responses, such that Σ i is a n i n i -diagonal matrix composed of each diagonal Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim

36 1944 C. BAZZOLI, S. RETOUT AND F. MENTRÉ element of Σ ik with k =1,..., K. σ slope and σ inter are two vectors of the K components σ interk and σ slopek,k =1,..., K, respectively. Finally, conditionally on the value of b i, we assume that the errors ε i are independently distributed. Let Ψ be the vector of the population parameters to be estimated such as Ψ T =(β T,ω 2 1,...,ω2 p, σ T inter,σt slope ) and let λ be the vector of the variance terms λt =(ω 2 1,...,ω2 p,σt inter,σt slope ), so that Ψ T =(β T,λ T ) Population Fisher information matrix for multiple response models The population Fisher information matrix for a population design Ξ (see equation (1)), is defined as the sum of the N elementary Fisher information matrices M F (Ψ,ξ i ) for each individual i: M F (Ψ,Ξ)= N M F (Ψ,ξ i ) (7) In the case of a limited number Q of groups, as in equation (2), it is expressed by: i=1 M F (Ψ,Ξ)= Q N q M F (Ψ,ξ q ) (8) q=1 The expression of an elementary Fisher information matrix for multiple responses has been extended by Hooker and V icini [25] using the same development as for single response models with a first Taylor expansion of the model as in Mentré et al. [17] and Retout et al. [22]. Its expression is given below for one individual i and depends on the approximated marginal expectation E i and the variance V i of the observations Y i. Details of this development are given in the Appendix. with M F (Ψ,ξ i )= ET i V 1 E i + 1 ( Ψ m Ψ l 2 tr V 1 i V i V 1 V i Ψ m Ψ l ) with m and l =1,...,dim(Ψ) (9) E(Y i ) = E i = F(g(β,0),ξ i ) (10) ( Var(Y i ) F T ) ( ) (g(β,0),ξ = V i = i ) F(g(β,0),ξi ) Ω b b T +Σ(β,0,σ inter,σ slope,ξ i ) (11) This expression has been implemented in an extension of the R function PFIM, PFIM 3.0. This function has been developed for R and higher versions. The implementation of the population Fisher information matrix assumes that the variance of the observations with respect to the mean parameters is constant (see Appendix). PFIM 3.0 evaluates population designs in NLMEM with multiple responses and thus returns the expected SE, defined as the square roots of the diagonal elements of the inverse of M F, on the population parameters with the design evaluated. To use PFIM 3.0, some prior information has to be supplied by the user such as the structural model, its parameterization and a priori values of the parameters. PFIM 3.0 can also optimize population designs with different optimization options. More details are available in an extensive document that can be freely downloaded with the function PFIM 3.0 on the PFIM website [21]. Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim

37 NONLINEAR MIXED EFFECTS MULTIPLE RESPONSE MODELS PKPD simulation example In this paper, we use a simple and typical PKPD model as an example to evaluate M F by simulation. It is derived from the one used by Hooker and Vicini [25] to illustrate the development of the Fisher information matrix for a multiple response model. The PK model for drug concentration is a one compartment with bolus input and first-order elimination given as follows for the sampling time t PK : f PK (θ PK,t PK )= dose ( exp Cl ) t PK (12) V C V C where θ PK =(Cl, V C ) T is the vector of the PK parameters with Cl and V C, the clearance and the volume in the central compartment, respectively. The PD model for drug effect is a simple E max model with baseline, expressed as a function of the predicted concentrations f PK, and given as follows for the sampling times t PD : f PD (θ PK,θ PD,t PD )= E 0 + E max f PK (θ PK,t PD ) C 50 + f PK (θ PK,t PD ) where θ PD =(E 0, E max,c 50 ) T is the vector of the PD parameters with E 0, E max and C 50 being the effect at baseline, the maximum effect and the concentration needed to observe half of the maximum effect, respectively. We assumed an exponential model of the random effects for both the PK and the PD parameters. We associated a proportional error model with the PK model characterized by the parameter σ slopepk and a homoscedastic error model with the PD model characterized by the parameter σ interpd. Thus, the vector of the population parameters Ψ is described by the vector of the fixed effects β T =(β Cl,β VC,β E0,β Emax,β C50 ) and by λ T the vector composed by the variance of the random effects and by the parameters for the error models such that λ T =(ω 2 Cl,ω2 V C,ω 2 E 0,ω 2 E max,ωc 2 50,σ slopepk,σ interpd ). The dose was fixed to 1 and the parameter values used in this paper are given in Table I. We determined a population design associated with this PKPD example. This determination was empirical, without any optimization. The population design was composed of one group of N =100 individuals. They all had three sampling times at 0.166, 6 and 12 for PK and four sampling times for PD at 0.166, 6, 12 and 20 h. Therefore, we had one elementary design (ξ PK,ξ PD ) with ξ PK =(0.166,6,12) and ξ PD =(0.166,6,12,20). The population design was thus defined by Ξ={[(ξ PK,ξ PD ), N]}. The curve profiles of the PK and the PD model for the fixed effects are displayed in Figure 1; the sampling times for each response are overlaid Evaluation of M F for multiple responses Comparison of M F w ith and w ithout linearization. In this section, we propose to compare the predicted SE obtained from the approximate M F for multiple responses computed by PFIM 3.0 (13) Table I. Population parameters values used for the simulated example. Parameters β Cl β VC β E0 β Emax β C50 ω 2 Cl ω 2 V C ω 2 E 0 ω 2 E max ω 2 C 50 σ slopepk σ interpd Values Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim

38 1946 C. BAZZOLI, S. RETOUT AND F. MENTRÉ Figure 1. Concentration (left) and the effect (middle) profiles versus time for the mean parameter values used in the PKPD example. PK sampling times (left) and PD sampling times (middle) are represented by. The right-hand graph describes the effect versus concentration. with the SE obtained from more exact approaches using the SAEM estimation algorithm. This latter algorithm was used by Retout et al. [20] and Samson et al. [28] to show the appropriateness of this approximation in a single response model. This SAEM algorithm allows the observed population Fisher information matrix to be computed according to two approaches. The first approach was developed by Samson et al. [28] and has been used to evaluate an exact population Fisher information matrix using Louis s principle [29]. It does not require any linearization and can thus be considered as the true population Fisher information matrix. The second approach evaluates the Fisher information matrix using a linearization of the model around the conditional expectation of the individual parameters previously estimated by SAEM without any linearization. To perform this comparison, we first computed the predicted M F for the population design associated with the PKPD example using PFIM 3.0, based on the linearization. We then simulated a data set of PK and PD observations for individuals in order to achieve asymptotic Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim

39 NONLINEAR MIXED EFFECTS MULTIPLE RESPONSE MODELS 1947 properties using the software R To do that we used the parameter values given in Table I and the sampling times shown in Figure 1, defining the PKPD example (Section 2.3). For each individual i, we simulated a vector of random effects b i in N(0,Ω), where the diagonal elements of Ω are the variance of the random effects, and we calculated the individual parameters using θ i =βexp(b i ). We then calculated the individual PK concentrations f PK (θ PK,t PK ) predicted by the model at each time t PK of ξ PK. We also computed the individual PK concentrations at each time t PD of ξ PD to derive the concentration f PK (θ PK,t PD ) for the PD response using equation (13). PD observations f PD (θ PK,θ PD,t PD ) were then generated. Finally, for each response, we simulated the random errors ε PK and ε PD from a normal distribution with zero mean and variance derived from equation (6) using the parameters σ slopepk and σ interpd, respectively. Those errors were added to the previously generated PK and PD data to form the simulated observations for the PK and the PD response, respectively. Using MONOLIX (Version 2.1) with SAEM as the estimation algorithm, we estimated the parameters using this simulated data set and we then derived the observed population Fisher information matrix with Louis s principle procedure and the linearization method of SAEM. For these two Fisher information matrices, we then transformed the observed SE for each component of the population vector Ψ obtained with a simulation of N sim =10000 individuals into the predicted SE of a population of N =100 individuals to be adapted to the design of the example using SE N (Ψ i )=SE Nsim (Ψ i ) N sim /N, for the ith component of Ψ. For estimation with the SAEM algorithm, we used an initial set of parameters with the values (0.2, 0.05, 1.2, 5, 1.5) for the fixed effects, (1, 1, 1, 1, 0.5) for the variance of the random effects and (0.5, 0.5) for the residual errors. The default values for the algorithm were used except for the number of Markov chains, which was set to 4, and the number of iterations with two different steps sizes, which was set to 1000 and 1000 to ensure good convergence. The predicted SE obtained by linearization with PFIM 3.0 were designated as PFIM. The notations SAEM LO and SAEM LI denote the predicted SE obtained with the SAEM algorithm using Louis s principle and the linearization method, respectively Comparison of M F with the empirical information through replicated simulation. Another objective was to compare the predicted SE of M F computed from PFIM 3.0 to the empirical SE obtained by the FO method, the FOCE method and the SAEM algorithm on simulated data sets. To do that, we simulated 1000 data sets of 100 individuals with the software R using the same PKPD model and population design described previously. Data sets were simulated using a similar method as in Section 2.4.1, using the same parameter values and the same sampling times. For each simulated data file, we estimated the population parameters for the PKPD model using first the FO method and the FOCE with interaction method implemented in NONMEM software version V and then using the SAEM algorithm in MONOLIX (Version 2.1). For the estimations using the FO and FOCE methods, two sets of initial parameters were defined. The first corresponded to the value of the parameters used for the simulation (Table I). The second one was used only in the case of lack of convergence with the first set. The values of the second set of initial parameters were for the fixed effects: (0.08, 0.1, 1.5, 3, 0.8), the values of the variance of the random effects and the variance of the residual errors were the same as for the first set of initial parameters. The initial values of the parameters and the different elements required to use the SAEM algorithm were identical to those described in Section For each parameter of the PKPD model, we compared the predicted SE using the three evaluations of M F with PFIM, SAEM LO and SAEM LI with the empirical SE obtained with the Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim

40 1948 C. BAZZOLI, S. RETOUT AND F. MENTRÉ FO method, the FOCE method and the SAEM algorithm, denoted by FO, FOCE, and SAEM, respectively. These empirical SE are defined as the sample estimate of the standard deviation from the parameter estimates for each method, considering only the subset of data sets fulfilling all convergence conditions. We were also interested in comparing the distribution of the observed SE provided by each of the estimation methods to the empirical SE and to the predicted SE. In this case, we considered only the subset of data sets for which both the convergence and the variance covariance matrix of estimation were obtained. For the distribution of the SE provided by the SAEM algorithm, we considered both methods of computation of the SE, Louis s principle and the linearization Comparison of results for estimation methods with and without linearization Using the previous simulations, we also compared the three methods of estimation: FO, FOCE and the SAEM algorithm. For each parameter, the relative bias as well as the relative RMSE were computed for the S data sets fulfilling convergence conditions (S 1000), which for Ψ l, the lth parameter of the population vector Ψ are given by ( Bias(Ψ l ) = 1 S ˆΨ s ) l Ψ0 l (14) S s=1 Ψ 0 l ( RMSE(Ψ l ) = 1 S ˆΨ s ) 2 l Ψ0 l (15) S with ˆΨ s l the estimated value of Ψ l for the sth simulated data sets and Ψ 0 l the true value. s=1 Ψ 0 l 3. RESULTS 3.1. Comparison of M F with and without linearization The SE predicted through the use of the SAEM algorithm on a large data set (SAEM LI and SAEM LO) and those predicted by PFIM 3.0 are reported in Table II as relative SE, i.e. SE divided by the true value of the parameter, noted RSE and expressed in per cent. Overall, whatever the method, the RSE of the population parameters were very close for the fixed effects with a Table II. Comparison of the relative standard errors (RSE in per cent) predicted by the SAEM algorithm implemented in MONOLIX V2.1 with both methods of computation of the SE (noted SAEM LO for Louis s principle and SAEM LI for the linearization method) and predicted by PFIM for the PKPD example. Parameters Methods β Cl β VC β E0 β Emax β C50 ω 2 Cl ω 2 V C ω 2 E 0 ω 2 E max ω 2 C 50 σ slopepk σ interpd PFIM SAEM LO SAEM LI Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim

41 NONLINEAR MIXED EFFECTS MULTIPLE RESPONSE MODELS 1949 difference of at most 1.3 per cent for β C50 between the SE predicted by PFIM and the one given by SAEM LO. Regarding the variance parameters, RSE were also very close, except for ω 2 C 50 for which PFIM seemed to slightly overestimate the parameter estimate precision with a difference of about 10 per cent compared with the SAEM approaches. This evaluation shows the appropriateness of the extension of the population Fisher Information matrix for multiple response models using the first-order approximation Comparison of M F with empirical information through replicated simulation Convergence was achieved for all data sets and the variance covariance estimates were obtained for 997 data sets among the 1000 simulated data sets with the FO method. Convergence with the FOCE method of NONMEM was obtained for only 853 sets. Among those 853 sets, the variance covariance matrix of estimation was obtained for only 798 sets. Finally, with the SAEM procedure, no problem of convergence or covariance variance matrix was noted for any of the 1000 data sets. For each parameter, the empirical RSE obtained with the three estimation methods and the predicted RSE of SAEM LI, SAEM LO and PFIM are displayed in Figure 2. Concerning the FO method, the empirical RSE were much larger than the RSE of PFIM, SAEM LI and SAEM LO except for the PK parameters. This difference is above all important for the PD parameter ω 2 C 50 with a RSE close to 200 per cent. In contrast, the empirical RSE for FOCE and SAEM were very close to the RSE predicted with SAEM LI, SAEM LO and PFIM. The distribution of the observed RSE from the three estimation methods, including both methods of computation of M F with the SAEM algorithm, are reported as boxplots in Figure 3, part (A) and part (B), for the mean and the variance parameters, respectively. For the FO method, the range of Figure 2. Barplot of predicted and empirical RSE (per cent) for the fixed effects, the variances of the random effects and the residual errors of the PKPD example. The empirical RSE computed by the FO, FOCE methods and the SAEM algorithm on 1000, 853 and 1000 replicates are denoted by FO, FOCE and SAEM, respectively. The predicted RSE (per cent) computed by SAEM and by PFIM are denoted SAEM LO (for SE obtained by Louis s principle), SAEM LI (for SE obtained by linearization) and PFIM, respectively. Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim

42 1950 C. BAZZOLI, S. RETOUT AND F. MENTRÉ (A) (B ) Figure 3. Boxplots of the RSE (per cent) for fixed effects (A) and for the variance components (B) estimated from 997, 758 and 1000 replicates, respectively, by the FO, FOCE INTER methods and the SAEM algorithm. SLO denotes the observed RSE of SAEM computed by Louis s principle, whereas SLI denotes the observed RSE of SAEM computed by linearization. The dotted line represents the RSE predicted by PFIM for each parameter and the star ( ) represents the empirical RSE obtained for each method. Some outliers have been omitted with respect to the Y scale for clarity of the figure. the observed RSE was much larger than for those obtained with the FOCE method or both SAEM procedures. However, the observed RSE of FO were concordant with the empirical ones. For all the parameters, the RSE predicted with PFIM were consistent with the distribution of the RSE observed with FOCE and the two SAEM procedures but not for FO. The range of the observed RSE for FOCE, SAEM and the corresponding empirical RSE were also concordant. However, for most parameters, the RSE computed using Louis s principle of SAEM had a broader distribution than by using linearization, with values for several data sets being outliers (Figure 3). This problem occurred in particular for the RSE on ω 2 C 50 parameter. In this example, the RSE predicted by PFIM, computed by the first-order linearization, were thus concordant with the empirical ones and the observed RSE obtained from the simulation study Comparison of three estimation methods The relative bias and the relative RMSE obtained with the three estimation methods are presented in Table III. Convergence was not achieved for 15 per cent of the simulated data sets using the FOCE method whereas the FO method and the SAEM algorithm converged for all data sets. Regarding the FO method, bias and RMSE were large especially for the parameters of the PD Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim

43 Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim Table III. Relative bias (per cent) and relative RMSE (per cent) of parameter estimates for data sets with fulfilling convergence conditions (denoted S) using the FO and the FOCE INTER methods implemented in NONMEM V software and the SAEM algorithm implemented in the MONOLIX V2.1 software. Parameters Methods S β Cl β VC β E0 β Emax β C50 ω 2 Cl ω 2 V C ω 2 E 0 ω 2 E max ω 2 C 50 σ slopepk σ interpd Bias FO (per cent) FOCE SAEM RMSE FO (per cent) FOCE SAEM NONLINEAR MIXED EFFECTS MULTIPLE RESPONSE MODELS 1951

44 1952 C. BAZZOLI, S. RETOUT AND F. MENTRÉ model (fixed effects, random effects and residual errors), whereas FOCE and SAEM provided reasonable bias and RMSE for all the parameters. For the fixed effects, slightly lower bias and RMSE were observed for the SAEM procedure compared with FOCE. We observed important RMSE (>40 per cent) for the parameter ωc 2 50 whatever be the estimation method. This is in agreement with the large RSE obtained for that parameter previously. 4. DISCUSSION We evaluated the expression of the population Fisher information matrix for multiple response models using a linearization of the model [25], as for single response models. Note that our evaluation focused on the case of multiple responses, in which some parameters are involved in several responses. For cases in which parameters differ across responses, the same information would be obtained using M F either for a single response or multiple responses. The expression of M F for multiple response models has been implemented in PFIM 3.0, an extension of the R function PFIM, dedicated to population design evaluation and optimization [21 23]. Using a PKPD model, we have shown the appropriateness of the predicted RSE obtained with M F computed by PFIM 3.0 by comparison with those computed without any linearization by the SAEM algorithm implemented in the MONOLIX software (see Section 3.1). The predicted RSE were indeed all very close except for a slight discrepancy in the RSE for C 50 variability. The simulation study on the PKPD example showed the concordance between the RSE predicted by PFIM and the empirical RSE computed from the estimation results with the FOCE or the SAEM algorithm using simulated data sets (see Section 3.2). Regarding the results using the FO algorithm, the empirical RSE were much larger than those predicted by PFIM or obtained with FOCE or SAEM, in particular, regarding the variability of the PD parameters. The distribution of the RSE obtained with FO, FOCE and SAEM from the simulated data files is in accordance with their respective empirical ones. Regarding the comparison of the estimation methods for multiple response models, no problems of convergence were apparent with SAEM or the FO method for the 1000 simulated data files (see Section 3.3). However, it was difficult to fulfill convergence conditions with the FOCE method for which the convergence was observed for only 853 of 1000 data sets. The simulation study illustrated the accuracy of the SAEM algorithm in the simultaneous approach, the parameter estimates being unbiased and with small RMSE. Similar results were observed for the FOCE method, but conclusions must be made with caution due to problems of convergence, as noted previously. Regarding FO, the large bias and RMSE already observed in the context of single response models [30] were also observed in this context of multiple response models with simultaneous estimation. The FO method produced larger RSE on all PD parameters compared with those computed with the FOCE method. This is in accordance with the recommendation to use the FOCE method instead of the FO method in this simultaneous estimation context [6, 7]. This apparent difference can be explained by considering the difference between FO and FOCE approximations. The FO method approximates the likelihood by linearizing the population model in its random effects about a value of zero, whereas FOCE is defined by a linearization of the model around individual estimates of the random effects. The FOCE method uses more information than the FO method, which is an advantage for estimation in multiple response models. Although the estimation method FO performed badly, the same first-order approximation in the computation of M F to predict SE performed well. The limitations of this first-order approximation around the expectation of Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim

45 NONLINEAR MIXED EFFECTS MULTIPLE RESPONSE MODELS 1953 the random effects thus differ for design evaluation where the derivatives of log-likelihood are computed and for parameter estimation. Several studies have considered the single response, stressing the limitations of this linearization for design evaluation. Using a simple model with few random effects, Han [31] found that M F was quite different when computed by linearization than by an adaptative Gaussian quadrature method. Merlé and Tod [32] compared the Fisher information matrix computed by linearization with the one computed by stochastic simulation and showed that the linearization seems to have no impact on the population D-optimal designs obtained but only on the true efficiency of the designs. Whether via adaptative Gaussian quadrature or via stochastic simulation, the evaluation of the Fisher information matrix without linearization is computationally intensive and is also limited to a matrix of small dimensions. Finally, in the context of Bayesian design where prior distributions are used, Han and Chaloner [33] proposed an attractive solution to compute M F ; however, it is also time consuming. Another alternative to the linearization method is to compute the expected Fisher information matrix using the SAEM estimation algorithm, as proposed here. A large data set is simulated to be close to the asymptotic properties. The observed M F is then estimated for the estimation performed in this data set using one of the two methods of deriving M F proposed for SAEM. This simulation study shows a broader distribution of the observed RSE when M F is computed by Louis s principle [29] compared with the RSE observed with the linearization procedure. Nevertheless, correct results are obtained from both methods. Although the approach using the SAEM algorithm of MONOLIX version 2.1 can be applied for problems with a large number of random effects, it is time consuming (hours) compared with PFIM (seconds). It may be required in specific cases such as when design evaluation is used to predict the power of a test to detect covariates [20]. The appropriateness of the RSEs of PFIM combined with its fast execution emphasizes its advantage in cases of design optimization where a large number of M F often have to be computed. Moreover, in design evaluation and optimization for nonlinear models, some a priori values of the parameters are required. Often, they are not precisely known and therefore the need to use exact methods to predict M F is questionable. In this study, we empirically determined a population design associated with a simple PKPD example for which the dose was equal to 1. No change in the dose was envisaged because the main purpose of this work was to evaluate M F for a PKPD model associated with one population design. However, it would be interesting to study the influence of dose on the population design and thus to plan dose optimization in order to have a better understanding of the relationship between the PK and the PD model. In the present study, we considered only the case of a diagonal Ω matrix with no correlation between the random effects of the PK and the PD parameters. This could be extended by exploring the appropriateness of M F when the PK parameters are directly correlated with the PD parameters, and thus the development of M F for a full Ω matrix. This development was performed by Mentré et al. [17] for the correlation of the random effects parameters of single response models and recently by Ogungbenro et al. [26] for multiple response models. Furthermore, the population Fisher information matrix implemented in PFIM 3.0 is approximated by a block diagonal matrix assuming that the variance of the observations with respect to the mean parameters is constant (see Appendix). It would therefore be interesting to investigate the influence of this assumption on the computation of the Fisher information matrix. In conclusion, the comparison of RSE predicted by PFIM with RSE computed without any linearization by SAEM as well as with RSE obtained from the simulation study supports the Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim

46 1954 C. BAZZOLI, S. RETOUT AND F. MENTRÉ appropriateness of the approximated Fisher information matrix for multiple response models. Its implementation in PFIM 3.0 provides a useful computing tool for design evaluation and optimization in the development of PKPD or PK studies. APPENDIX A: TECHNICAL DETAILS IN THE DEVELOPMENT OF M F IN NLMEM FOR MULTIPLE RESPONSES The population Fisher information matrix M F (Ψ,ξ i ) for multiple response models for the individual i with design ξ i is given by ( ) M F (Ψ,ξ i )= E 2 L i (Ψ;Y i ) Ψ Ψ T (A1) where L i (Ψ;Y i ) is the log-likelihood of the vector of observations Y i of that individual for the population parameters Ψ. Because F is nonlinear, there is no analytical expression for the log-likelihood L i (Ψ;Y i ), and we use the first-order Taylor expansion of the model F(θ i,ξ i )= F(g(β,b i ),ξ i ), around the expectation of b i, that is to say around 0: ( F(g(β,b i ),ξ i ) F T ) (g(β,0),ξ = F(g(β,0),ξ i )+ i ) b (A2) b With this approximation, the equation (5) can be written as: ( Y i F T ) (g(β,0),ξ = F(g(β,0),ξi )+ i ) b+ε i b Therefore, the approximated marginal expectation E i and variance V i of Y i are given by: (A3) E(Y i ) = E i = F(g(β,0),ξ i ) ( Var(Y i ) F T ) (g(β,0),ξ = V i = i ) Ω b ( F(g(β,0),ξi ) b T ) +Σ(β,0,σ inter,σ slope,ξ i ) (A4) (A5) The log-likelihood L i is thus approximated by: 2L i (Ψ;Y i ) =n i ln(2π)+ln( V i )+(Y i E i ) T V 1 i (Y i E i ) (A6) Note that in this approximation, for the sake of simplicity we have assumed that the variance of the error model is not linked to the random effects of the individual but only to the mean parameters. Based on this expression of the log-likelihood L i, we can derive the expression of an elementary Fisher information matrix for a multiple response model. For the sake of simplicity, we have omitted the indice i for the individual in the following. M F is a block matrix depending on the approximated marginal expectation E and variance V of the observations: ( ) M F (Ψ,ξ) = 1 A(E, V ) C(E, V ) 2 C T (A7) (E, V ) B(E, V ) Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim

47 NONLINEAR MIXED EFFECTS MULTIPLE RESPONSE MODELS 1955 where ( ) (A(E, V )) ml = 2 ET 1 E V 1 V V +tr V V 1 β m β l β l β m with m and l =1,..., p ( ) V 1 V (B(E, V )) ml = tr V V 1 λ m λ l with m and l =1,...,dim(λ) ( ) V 1 V (C(E, V )) ml = tr V V 1 λ l β m with l =1,...,dim(λ) and m =1,..., p In this paper, we consider that the variance of the observations with respect to the mean parameters is constant, i.e. C(E, V ) ml =0. ACKNOWLEDGEMENTS Part of this work was supported by a grant from F. Hoffmann La Roche Ltd, Basel, Switzerland. The authors would like to acknowledge Pr Marc Lavielle for his help regarding the use of the MONOLIX software as well as the IFR02 of INSERM and Hervé Le Nagard for the use of the centre de biomodélisation. REFERENCES 1. Sheiner LB, Wakefield J. Population modelling in drug development. Statistical Methods in Medical Research 1999; 8: Lindstrom ML, Bates DM. Nonlinear mixed effects models for repeated measures data. Biometrics 1990; 46: Beal SL, Sheiner LB. NO NMEM Users G uide. University of California: San Francisco, Pillai GC, Mentré F, Steimer JL. Non-linear mixed effects modeling from methodology and software development to driving implementation in drug development science. Journal of Pharmacokinetics and Pharmacodynamics 2005; 32: DOI: /s y. 5. Pinheiro JC, Bates MD. Mixed-Effects Models in S and S-Plus. Springer: New York, Zhang L, Beal SL, Sheiner LB. Simultaneous vs. sequential analysis for population PK/PD data I: bestcase performance. Journal of Pharmacokinetics and Pharmacodynamics 2003; 30: DOI: / B:JOPA f. 7. Zhang L, Beal SL, Sheiner LB. Simultaneous vs. sequential analysis for population PK/PD data II: robustness of methods. Journal of Pharmacokinetics and Pharmacodynamics 2003; 30: DOI: / B:JOPA e. 8. Pinheiro JC, Bates DM. Approximations to the log-likelihood function in the nonlinear mixed effects models. Journal of Computational and G raphical Statistics 1995; 4: Samson A, Lavielle M, Mentré F. Extension of the SAEM algorithm to left-censored data in nonlinear mixed-effects model: application to HIV dynamics model. Computational Statistics and Data Analysis 2006; 51: DOI: /j.csda Delyon B, Lavielle M, Moulines E. Convergence of a stochastic approximation version of the EM algorithm. Annals of Statistics 1999; 27: DOI: /aos/ Khun E, Lavielle M. Maximum likelihood estimation in nonlinear mixed effects models. Computational Statistics and Data Analysis 2005; 49: Al-Banna MK, Kelman AW, Whiting B. Experimental design and efficient parameter estimation in population pharmacokinetics. Journal of Pharmacokinetics and Biopharmaceutics 1990; 18: DOI: / BF Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim

48 1956 C. BAZZOLI, S. RETOUT AND F. MENTRÉ 14. Jonsson EN, Wade J, Karlsson MO. Comparison of some practical sampling strategies for population pharmacokinetics studies. Journal of Pharmacokinetics and Biopharmaceutics 1996; 24: DOI: / BF Atkinson AC, Donev AN. Optimum Experimental Designs. Clarendon Press: Oxford, Walter E, Pronzato L. Identification of Parametric Models from Experimental Data. Springer: New York, Mentré F, Mallet A, Baccar D. Optimal design in random effect regression models. Biometrika 1997; 84: Retout S, Mentré F, Bruno R. Fisher information matrix for non-linear mixed-effects models: evaluation and application for optimal design of enoxaparin population pharmacokinetics. Statistics in Medicine 2002; 21: DOI: /sim Retout S, Mentré F. Further developments of the Fisher information matrix in nonlinear mixed effects models with evaluation in population pharmacokinetics. Journal of Biopharmaceutical Statistics 2003; 13: DOI: /BIP Retout S, Comets E, Samson A, Mentré F. Design in nonlinear mixed effects models: optimization using the Fedorov Wynn algorithm and power of the Wald test for binary covariates. Statistics in Medicine 2007; 26: DOI: /sim Retout S, Duffull S, Mentré F. Development and implementation of the population Fisher information matrix for the evaluation of population pharmacokinetic designs. Computer Methods and Programs in Biomedicine 2001; 65: DOI: /S (00) Retout S, Mentré F. Optimization of individual and population designs using Splus. Journal of Pharmacokinetics and Pharmacodynamics 2003; 30: DOI: /B:JOPA a. 24. Gueorguieva I, Aarons L, Ogungbenro K, Jorga KM, Rodgers T, Rowland M. Optimal design for multivariate response pharmacokinetic models. Journal of Pharmacokinetics and Pharmacodynamics 2006; 33: DOI: /s Hooker A, Vicini P. Simultaneous population optimal design for pharmacokinetic pharmacodynamic experiments. American Association of Pharmaceutical Scientists Journal 2005; 7: DOI: /aapsj Ogungbenro K, Graham G, Gueorguieva I, Aarons L. Incorporating correlation in interindividual variability for the optimal design of multiresponse pharmacokinetic experiments. Journal of Biopharmaceutical Statistics 2008; 18: DOI: / Ogungbenro K, Gueorguieva I, Majid O, Graham G, Aarons L. Optimal design for multiresponse pharmacokinetic pharmacodynamic models dealing with unbalanced designs. Journal of Pharmacokinetics and Pharmacodynamics 2007; 34: DOI: /s Samson A, Lavielle M, Mentré F. The SAEM algorithm for group comparison tests in longitudinal data analysis based on non-linear mixed-effects model. Statistics in Medicine 2007; 26: DOI: /sim Louis TA. Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society 1982; 44: Dartois C, Lemenuel-Diot A, Lavielle M, Tranchand B, Tod M, Girard P. Evaluation of uncertainty parameters estimated by different population PK software and methods. Journal of Pharmacokinetics and Pharmacodynamics 2007; 34: DOI: /s Han C. Optimal Designs for Nonlinear Regression Models with Application to HIV Dynamic Studies. University of Minnesota: Minneapolis, Merlé Y, Tod M. Impact of pharmacokinetic pharmacodynamic model linearization on the accuracy of population information matrix and optimal design. Journal of Pharmacokinetics and Pharmacodynamics 2001; 28: DOI: /A: Han C, Chaloner K. Bayesian experimental design for nonlinear mixed effects models with application to HIV dynamics. Biometrics 2004; 60: DOI: /j X x. Copyright 2009 John Wiley & Sons, Ltd. Statist. Med. 2009; 28: DOI: /sim

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51 Communications in Statistics Theory and Methods, 38: , 2009 C opyright T aylor & F rancis G roup, L L C ISSN : print/ x online D O I: / Design Optimization in Nonlinear Mixed E ffects Models U sing C ost Functions: A pplication to a Joint Model ofinfliximab and Methotrexate P harmacokinetics SY L V IE R E T O U T, 1 2 E M M A N U E L L E C O M E T S, 1 2 C A R O L IN E B A Z Z O L I, 1 2 A N D F R A N C E M E N T R É U R M 738 IN SE R M, Paris, F rance 2 U niversité Paris D iderot, Site B ichat, Paris, F rance W e address the problem of design optimization using cost functions in nonlinear mixed effects models w ith multiple responses. W e focus on the relative feasibility of the optimized designs, in term of sampling times and of number of subjects. To do that,w e extend the Fedorov W ynn algorithm a dedicated design optimization algorithm to include a cost function that penalizes less feasible designs as w ell as to take into account multiple responses. W e apply this extension to the design optimization of a joint pharmacokinetic model of infliximab and methotrexate administered in rheumatoid arthritis.w e show the benefit ofsuch an approach w hen substantial constraints on the design are imposed. K eyw ords C ost function; D esign optim ization; F edorov W ynn algorithm ; F isher inform ation m atrix, N onlinear m ixed effects m odel; PF IM. Mathematics Subject C lassification Prim ary 62; Secondary 62K 05, 68U Introduction N onlinear m ixed effects m odels (N L M E M ) are increasingly used in biological studies to analyze longitudinal data. U se of those m odels has been initiated by Sheiner etal.(1972)for pharm acokinetic (PK )analyses, w hich study the tim e course of the drug concentrations in the body after the drug adm inistration. It has been extended for pharm acodynam ic (P D ) analyses to study the relationship of drug concentrations to pharm acologic effects and it is now very popular in other kinds of longitudinal studies.n L M E M are now also used for the joint m odeling of several biological responses such as the PK and the PD of a drug or the PK of parent R eceived A pril 25, 2008; A ccepted F ebruary 18, 2009 A ddress correspondence to Sylvie R etout, IN SE R M U 738, U niversité Paris D iderot, U F R de M édecine, Site B ichat, 16, rue H enri H uchard, Paris 75018, F rance; E -m ail: sylvie. retout@ inserm.fr 3351

52 3352 Retout et al. drugs and of their active metabolite (Panhard et al., 2005). The purpose of NLMEM approach, also called the population approach, is to estimate the mean value of the parameters and their interindividual variability in the studied population. Influence of covariates on the parameters can also be determined and quantified, which can help to define groups of population with different levels of response. This methodology can deal with sparse individual data, without any need of individual parameter estimates. This allows studies in populations like children for which rich individual data cannot be obtained due to ethical or physiopathological reasons. As in all experiments, a design has first to be defined to collect the data. Estimation is then performed, usually using maximum likelihood procedures; several estimation methods are available, either based on linearization of the likelihood function (Beal and Sheiner, 1992) or stochastic approximations (Kuhn and Lavielle, 2005; Pinheiro and Bates, 1995). A review of recent advances in this field can be found in Pillai et al. (2005). The choice of the design is important because it largely influences the precision of the parameter estimates. In NLMEM, designs, also called population designs, are defined by several groups of subjects; each group is composed of a number of samples to be performed on a number of subjects at given times. Simulation studies have shown that the precision of parameter estimates depends on the balance between the design variables in the group structure (number of groups to include, number of subjects per group, and number of samples per group) and the allocation of the sampling times (Al-Banna et al., 1990; Jonsson et al., 1996). The general theory of design determination used for classical nonlinear models (Atkinson and Donev, 1992; Walter and Pronzato, 1997) has been extended to NLMEM. It relies on the Cramer Rao inequality which states that the inverse of the Fisher information matrix (M F is the lower bound of the variance covariance matrix of any unbiased maximum likelihood parameter estimator. However exact analytical expression of M F cannot be derived in NLMEM due to the lack of analytical expression of the likelihood. To circumvent this problem, Mentré et al. (1997) proposed, for single-response models, an approximation of the matrix based on first-order Taylor expansion of the model around the expectation of the random effects. This expression has been extended for more complex models including fixed effects for the influence of covariates on the response and for an additional variability of the parameters of a given individual between several periods of treatment (Retout and Mentré, 2003a). Evaluation of M F using a firstorder expansion is also performed in Fedorov et al. (2002) and Gagnon and Leonov (2005). Recently, an approximate M F has been proposed for multiple response models by Hooker and Vicini (2005) and Gueorguieva et al. (2006) using the same first-order linearization of the model. Simulations have shown the approximate M F to be appropriate for single response models (Retout and Mentré, 2003a; Retout et al., 2002, 2007) and for multiple response models (Bazzoli et al., 2007); it has been implemented in PFIM, a R function for design evaluation and optimization freely available online ((Retout and Mentré, 2003b; Retout et al., 2001); Design optimization can either be restricted to optimizing the sampling times for a given group structure (exact designs), or optimize both the group structure and the sampling times (called statistical or approximate designs) (Atkinson and Donev, 1992). We consider the latter approach in this article. The Fedorov Wynn algorithm, a design-specific optimization algorithm converging towards the D-optimal design (Fedorov, 1972; Wynn, 1972), has been implemented

53 Design in Nonlinear Mixed Effects Models 3353 for this purpose in PFIM. Its effectiveness to optimize designs in NLMEM, which often involve large number of design variables, has been shown for instance in Retout et al. (2007), by comparison to the Simplex algorithm. The implementation assumes that the acceptable sampling times are given and constitute a finite set for optimization, which can be a great advantage in clinical practice to avoid unfeasible sampling times. However, optimization is performed for a fixed total number of samples without any consideration on the relative feasibility of the sampling times or of the group structure. In practice, however, it may be difficult to keep a patient at the hospital for a long time after its drug administration for no other medical reason than to obtain a blood sample for the study. Recruiting a large number of patients can also be problematic for financial and/or recruitment reasons. This problem has been introduced for single-response models through the use of cost functions by Mentré et al. (1997), Gagnon and Leonov (2005) illustrated and highlighted the benefits of such an approach on a clinical PK study. In this work, our objectives are first to extend the Fedorov Wynn algorithm for design optimization with cost functions in NLMEM with multiple responses, and second, to apply this approach to design optimization for a joint model of infliximab and methotrexate PK administered in rheumatoid arthritis. We describe the case study in Sec. 2. The statistical methods are given in Sec. 3. We first present NLMEM; we then describe the computation of the Fisher matrix in this context; last, we explain the designs optimization with cost functions using the Fedorov Wynn algorithm. Section 3 is dedicated to the application to our case study. 2. Case Study: Joint Model of Infliximab and Methotrexate Pharmacokinetics Infliximab is a high-molecular-weight chimeric monoclonal IgG1 antibody against human tumour necrosis factor- (TNF. It is given in rheumatoid arthritis to stop the inflammatory process in the synovial joints, in combination with methotrexate, an antimetabolite of folic acid (methotrexate attenuates the formation of antibodies which can inactivate infliximab; Klotz et al., 2007). To our knowledge, no study has been performed to simultaneously model the population PK of infliximab and methotrexate. However, this modeling is important since many patients receive an association of these two drugs in treatment and because methotrexate has an effect on the PK of infliximab (Klotz et al., 2007) Pharmacokinetics of Infliximab Infliximab is given as a long-term treatment in rheumatoid arthritis since it is a chronic disease. Treatment is usually initiated with 3 infusions at week 0, 2, and 6, and a maintenance dose is given every 8 weeks thereafter. The recommended dose is given as a 2 or 3 h infusion of 3 mg/kg. The PK of infliximab has been described in patients with rheumatoid arthritis in several articles (Kavanaugh et al., 2000; Klotz et al., 2007; Maini et al., 1998; St Clair et al., 2002). It is best described by a one compartment model with first-order elimination and zero-order infusion. It can be parameterized in clearance (C l and volume of distribution V.

54 3354 Retout et al. An exponential model is used to relate individual parameters and random effects, e.g., for clearance: Cl i = Cl exp b i Cl where Cl is the mean population value and Cl i and b i Cl are, respectively, the individual clearance parameter and random effect for individual i. The plasma concentration f Infli at time t after dose at steady state can be written as follows: [ D 1 (1 e Cl T Inf Cl f Infli t = D 1 T Inf Cl t) Cl V + e Cl V 1 e [ 1 e Cl V T Inf e Cl 1 e Cl V V t T Inf ] V T Inf e Cl 1 e Cl V V t T Inf ] if t T Inf if not where D is the dose, T Inf is the duration of the infusion and is the interval between two infusions. Population parameter values are given in Table 1 (Kavanaugh et al., 2000) Pharmacokinetics of Methotrexate The usual dose of methotrexate in rheumatoid arthritis is within a range of mg/week. The PK profile of methotrexate has been described in many articles, including modeling through population approaches. It is best described by a two-compartment first order oral absorption model (Godfrey et al., 1998). It is parameterized in rate constant of absorption k a, clearance Cl, central compartment volume V C, peripheral compartment volume V p, intercompartmental clearance Q, with an exponential modeling of the random effects. To simplify the model, we neglect the oral absorption lag of 0.23 h and assume a bioavailability of 100%. Therefore, the plasma concentration f Metho at time t after dose at steady-state can be written: ( A e t B e t f Metho t = D + 1 e 1 e A + B e kat 1 e k a ) T able 1 Mean values and interpatient variability for infliximab (Kavanaugh et al., 2000) and methotrexate parameters (Godfrey et al., 1998) Parameter Mean Interpatient variability (CV% ) infliximab Cl (l/h) V (l) methotrexate k a (h 1 ) Cl (l/h) V c (l) V p (l) Q (l/h)

55 Design in Nonlinear Mixed Effects Models 3355 where D is the dose, is the interval between two doses, = = 1 2 A = k a V c Q V c + Q V p + Cl V c Q V p k a and B = k a V c Q Cl Vp Vc, ( Q + Q + Cl ) 2 4 Q Cl V c V p V c V p V c Q V p k a Population parameter values are given in Table 1 (Godfrey et al., 1998) Design for a Joint PK Modeling We aim at determining a design for a joint population analysis of infliximab and methotrexate PK at steady state. The mean kinetic profiles of both drugs are represented in Fig. 1, assuming, as we will in the rest of this article, a 3-h infusion dose of infliximab of 210 mg every 8 weeks, corresponding to a dose of 3 mg/kg for a mean weight of 70 kg, and a weekly dose of methotrexate of 7.5 mg. We assume a proportional error model for the infliximab concentrations: Var = 0 2f Infli 2 and a combined error model for methotrexate: Var = f Metho 2. Our motivation in this article is to develop optimal designs under different constraints, taking into account different relative feasibilities in clinical practice, such as the inconvenience for patients to be kept a long time at the hospital after its dose administration or the increase of cost induced by the inclusion of a new patient compared to additional samples in a patient already included. We compare the optimal designs to a design established empirically, by taking into account the very different time scales of the PK course of each drug (Fig. 1). This design involves one group of 50 subjects in whom 12 samples are taken. Figure 1. Kinetic profile of infliximab and methotrexate at steady state for the mean parameters described in Table 1. The sampling times at Week 1 and Week 8 of the empirical design are shown as *.

56 3356 Retout et al. At each sampling time, both infliximab and methotrexate concentrations are measured, yielding a total number of samples of Six samples are taken on Day 1 of Week 1, after the beginning of the infusion of infliximab at 30 min, 1, 3, 6, 8, and 24 h; the same six samples are repeated at Day 1 of Week 8, the day of the last dose of methotrexate before the next infusion of infliximab. Both periods correspond to a 24-h period after the administration of a dose of methotrexate, and should also be informative for infliximab since they are set, respectively, at the beginning and end of the dosing interval for this drug. We call this design the empirical design. The sampling times of this design are reported on the kinetic profiles of infliximab and methotrexate for the mean parameters at steady state in Fig Statistical Methods 3.1. Notations We define an elementary design as a set of sampling times. A population design is then composed of N individuals each with an associated elementary design i (i = 1 N. Let n i denote the number of sampling times in i. We write the population design = 1 N, and n = N i=1 n i the total number of observations. For multiple response model, an elementary design i is composed of several sub-designs i = i1 i2 ik with ik, k = 1 K, being the design associated with the kth response. ik is defined by t ik1 t ik2 t iknik the vector of the n ik sampling times for the observations of the kth response, so that n i = K k=1 n ik. This notation accounts for different number of samples and different sampling times across the different responses, to accommodate different response profiles. Usually, population designs are composed of a limited number Q of groups of different elementary designs q q = 1 Q, to be performed in a number N q of individuals with Q q=1 N q = N. The population design can then be noted: = K N K N 2 Q1 Q2 QK N Q A nonlinear mixed effects multiple response model or a multiple response population model is defined as follows. The vector of observations Y i for the ith individual is defined as the n i -vector of the K different responses Y i = y i1 T yt i2 yt ik T, where y ik, k = 1 K, is the n ik -vector of observations for the kth response. Each of these responses is associated with a known function f k, such that f k i ik = f k i t ik1 f k i t ik2 f k i t iknik T is a n ik -vector which describes the nonlinear model. The K functions f k can be grouped in a vector of multiple response models F, such as F i i = f 1 i i1 T f 2 i i2 T f K i ik T. i denotes the vector of individual parameters in individual i; some parameters may be shared across different model functions (e.g., parent/metabolite model or PK/PD model). i is defined by i = g b i where is the vector of the fixed effects parameters, b i the vector of the random effects for individual i and g a known function. It is assumed that b i N 0. The function g usually assumes an additive relation between the fixed effect and the random effect of each i, which are thus normally distributed. We consider here a more general g function, which allows us to consider i as the PK parameters, and not as some transformed parameters.

57 Design in Nonlinear Mixed Effects Models 3357 In condensed form, we thus write the statistical model for the nonlinear multiple response mixed-effect model: Y i = F g b i i + i where i is the vector composed of the K vectors of residual errors ik, k = 1 K, associated with the K responses. Conditionally on the value of b i, we assume that the errors i are independently distributed and that ik N 0 ik b i k ik, with k the vector of parameters characterizing the kth variance error model. We then note i = i b i i the block diagonal variance matrix of i over the K responses; i is composed of the elements of ik and the vector of the K components k. Let be the P-vector of population parameters to be estimated T = T T, where denotes the distinct elements of. We also note be the vector of variance terms T = T, so that T = T T Fisher Information Matrix for NLME Multiple Response Models The population Fisher information matrix M F for multiple response model for one individual with design is given by M F = E ( ) 2 l Y, where l Y T is the log likelihood of the vector of observations Y of that individual for the population parameters. Note that for sake of simplicity, we omit the index i for the individual in this section. Because F is nonlinear, there is no analytical expression for the log-likelihood l Y. As in Mentré et al. (1997), a first-order Taylor expansion of the structural model F = F g b around the (zero) expectation of b is used: ( ) F T g 0 F g b F g 0 + b b Assuming the individual parameters are normally distributed, this expansion is equivalent to the one around as in Gagnon and Leonov (2005). The statistical model can then be written as: Y F g 0 + ( ) F T g 0 b b +. For the sake of simplicity, we further assume that the variance of the error model does not depend on the random effects of the individual but only on the mean parameters, so that Var = 0. The log-likelihood l is then approximated by: 2l Y n ln 2 + ln V + Y E T V 1 Y E where E and V are the approximated marginal expectation and variance of Y given by: E Y E = F g 0 ( ) F T g 0 Var Y V = b ( F g 0 b T ) + 0 Based on this expression of the log-likelihood l, the expression of an elementary Fisher information matrix for multiple response model can be derived. It is a block

58 3358 Retout et al. matrix depending on the approximated marginal expectation E and variance V of the observations: M F 1 ( ) A E V C E V 2 C T E V B E V where A E V mn = 2 ET 1 E V + tr ( V 1 V V V 1) with m and n = 1 dim m n n m B E V mn = tr ( V 1 V V V 1) with m and n = 1 dim C E V m n mn = tr ( V 1 V V V 1) with n = 1 dim and m = 1 dim. n m The population Fisher information matrix for a population design is thus derived as the sum of the N elementary Fisher information matrices with i for each individual i M F = N i=1 M F i. In the case of a limited number Q of groups, this matrix is expressed as M F = Q q=1 N qm F q Fedorov Wynn Algorithm for Design Optimization Using Cost Functions Design O ptimization with Cost Functions. We consider design optimization within a finite set of possible designs S. The objective is to maximize in some sense the information matrix of the population design, M F, since the variance of the estimate is asymptotically proportional to M 1. Here, we use the D-optimality F criterion, which consists in maximizing det M F 0 where det denotes the determinant and 0 a given a priori value of the population parameters. In the following, we will drop the explicit dependency on 0 in the notation, and write M F = M F 0 for simplicity. We also consider the normalised information matrix, defined by I F = M F = Q N q N q=1 M N F q (Gagnon and Leonov, 2005). This matrix represents the average information matrix for one individual in design. For a given maximal number of subjects to be included, N, which will be attained for the optimal design, and in the absence of other constraints, the maximization problem is to find such that: = arg max ( det M F = arg max det N Q q=1 ) q M F q (1) where {( is defined by a set of Q elementary designs and their associated frequencies q = N q } N q) q = 1 Q, where 1 Q are in S, and the frequencies satisfy 0 q 1 and Q q=1 q = 1. Note that (1) is equivalent with N fixed to solving = arg max det I F. Even though the N q should technically be integers, the problem is usually solved assuming that q is continuous and rounding off the resulting N q = N q under the constraint Q q=1 N q = N. The optimization problem is therefore a maximization problem on a convex compact surface given by Q q=1 q = 1 (Atkinson and Donev, 1992). We now assume, following Mentré et al. (1997) and Gagnon and Leonov (2005), that there is a cost incurred by each elementary design, which we denote C. Instead of a maximum number of subjects, we set a maximal cost C tot, such that N Q q=1 qc q C tot. The optimal population design again satisfies the equality.

59 Design in Nonlinear Mixed Effects Models 3359 Introducing C tot and C, the criterion to be maximized in (1) can be rewritten as: ( det M F = det N Q q=1 ) ( Q ( q M F q = det w q M F q C )) tot C q where w q = N qc q and corresponds to the proportion of the total cost C C tot tot attributed to each group q. By construction, Q q=1 w q = 1 so that problem (1) is equivalent to: ( = arg max det Q C tot q=1 q=1 ) w q H F q (2) where is defined by a set of Q elementary designs and their associated frequency w q q q = 1 Q, where 1 Q are in S, the frequencies verify 0 w q 1 and Q q=1 w q = 1; H F q corresponds to the information per unit cost and is defined as H F q = M F q. C q The numbers of subjects per group are derived from the proportions of cost w q and rounded to integers, for a total cost of Q q=1 N qc q. using N q = w qc tot C q Fedorov Wynn A lgorithm. We use the Fedorov Wynn algorithm, as in Mentré et al. (1997), to optimize designs with cost functions, as formalized in Eq. (2). Design optimization problems formalized in Eq. (1) can also be solved by the same algorithm as it is a special case with constant costs. We provide a brief description of the algorithm here, referring interested readers to the book by Walter and Pronzato (1997) for details. The Fedorov Wynn algorithm relies on the Kiefer Wolfowitz equivalence theorem, which states that the three following proposals are equivalent: (i) is D-optimal, i.e., det M F is maximal within the set of population designs generated by S; (ii) max S d = P; (iii) minimizes max S d, where d is a function of a population design and an elementary design, defined below. Given a population design = w q q q = 1 Q an elementary design in S, and any w between 0 and 1, we can define a new design by adding to an elementary design with weight w and by multiplying w 1 w 2 w Q by 1 w. For simplicity, we write this as = 1 w + w, confounding the elementary design with the population design where all individuals have elementary design. The information matrix of this design is M F 1 w + w. The function d is then defined as the derivative of log det M F taken at w = 0. The equivalence theorem provides a way to construct the optimal design iteratively: 1. start with an initial guess 0 ; 2. at step k, with the current design being k, find = arg max S d k stop if max S d k P + where 1 is a predetermined tolerance;

60 3360 Retout et al. 3. otherwise, update the design to k = 1 w k + w, where w is chosen over 0 1 such that w = d k P ; P d k 1 4. optimize the weights using an active set method with a projected gradient method for the selection of the direction, as in Mallet (1986), leading to design k+1. Steps 1 4 never remove an elementary design from k. However, we know that, following the Caratheodory s theorem, there exists an optimal design which includes at most P P + 1 /2 different elementary designs (Fedorov, 1972). To control the number of elementary designs, an additional step is included in the algorithm after the optimization of the weights to remove elementary designs with a weight lower than a predetermined (we chose = 10 8 ). 4. Design Optimization for the Joint Model of Infliximab and Methotrexate Pharmacokinetics In this section, we apply the Fedorov Wynn algorithm with cost functions to design optimization for a joint PK modeling of infliximab and methotrexate. The a priori values of the population parameters are fixed to those estimated previously in the literature (Table 1) Design and Cost Function Specifications To allow comparison with the empirical design defined in Sec. 2.3, we use the same set of 12 possible times as for the empirical design. We only consider designs with the same sampling times for both drugs. The total cost C tot is fixed to 1,200 and the number of samples per patient is allowed to vary from To take into account different relative feasibilities in clinical practice, we define four different cost functions. The first cost function is a classical one, i.e., the cost of an elementary design is equal to its number of samples. For multiple responses, we define the number of samples for one elementary design as the sum of the number of samples for each response, even if several responses are measured from only one actual blood sample, reflecting the cost of the analysis rather than the cost of the act of sampling itself. We therefore define this cost function as simply C samples i = n i The second cost function, denoted C inconv measures the inconvenience for the patient by penalizing late samples during the dose interval. Patients are kept in the hospital for 3 h during the infusion of infliximab, therefore samples taken during the initial 3 h of Day 1 of Week 1 are attributed an unitary cost of 1. During the same day, samples taken at 6 or 8 h required the patient to stay for a further 3 or 5 h and are given a cost of 2. The 24-h sampling time requires the patient to return to the hospital the next day; it incurs a cost of 4. For times within Day 1 of Week 8, as patients also need to return to the hospital, we assume a cost of 3 for the visit, in addition to the cost of the samples themselves, which are set to those of Day 1 of Week 1. For each response, the second cost function is therefore computed as: C inconv i = j k C time t ikj + C visit i where C time t = 1 if t = 0 5, 1 or 3; C time t = 2 if t = 6 or 8; C time t = 4 if t = 24. C visit i = 3 if i contains at least one of the times (0.5, 1, 3, 6, 8) of the second period, 0 otherwise.

61 Design in Nonlinear Mixed Effects Models 3361 The third cost function considers the inclusion of a new patient in the design as more costly than additional blood samples in patients already in the study. In this case, the cost of an elementary design is given by the number of sampling times penalised by a constant: C patient i = n i Choosing 12 for the cost of adding a new patient is equivalent to taking 6 additional blood samples in one patient since there are 2 responses. Finally, the fourth cost function denoted C inconv patient combines both the inconvenience of each sample with the cost of additional patients in the study: C inconv patient i = j k C time t ikj + C visit i Numerical Implementation The expression of M F for a multiple response model with the first order approximation is implemented in PFIM 3.0, an extension of PFIM. The implementation considers only cases with diagonal variance matrix of the random effects and specific variance error models given as: Var k = diag 1k + 2k f k g b k 2, where 1k and 2k qualify the model for the variance of the residual error of the kth model. Furthermore, the implementation assumes that the variance of the observations with respect to the mean parameters is constant, which results in a block diagonal M F, i.e., C E V mn = 0 with n = 1 dim and m = 1 dim. The Fedorov Wynn algorithm is implemented in PFIM using a C code, and linked with R via a dynamic link library. Note that the current implementation for multiple response models as implemented in PFIM 3.0 does not require the same sampling times for the different responses, but runtimes may become prohibitively high for complex designs Designs Comparison We then perform comparisons between the optimal population designs for the different cost functions. We compare the efficiency of the empirical design to the design optimised with the classical cost function on the total number of samples. To do that, we use the information function classically used to compare efficiency between designs. is defined as the determinant standardized by P, the dimension of the parameters vector 0, = det M F 0 1/P. The relative efficiency of a population design 1 with respect to a population design 2 is given by 1 / 2. This ratio can then be considered as the geometric mean of variance decrease using 1 instead of 2. Moreover, for each cost function, we compare the total cost of the four optimized designs and we investigate the cost-efficiency relation between the designs Results The optimal designs are given in Table 2, with sampling times corresponding to measurements of the two drugs in the study. They are called O pt samples, O pt inconv, O pt patient, and O pt inconv patient corresponding to the four cost functions C samples, C inconv, C patient, and C inconv patient, respectively. The optimal designs are different according

62 3362 Retout et al. Table 2 Optimal population designs for the joint model of infliximab and methotrexate according to different cost functions (for each design, Q is the number of elementary designs and N q the rounded number of subjects with the elementary design q ). Elementary designs are shown only once, but correspond to samples taken for both drugs Elementary designs q (hr) Information value Design Q Week 1 Week 8 N q Empirical 0.5, 1, 3, 6, 8, , 1, 3, 6, 8, Opt samples , , 1, 3, 8, 24 3, 8, Opt inconv 5 0.5, 1, , 1, 3, , 1, 3, 6, 8, , , 3 1 Opt patient 1 0.5, 1, 3, 8, , 1, 3, 8, Opt inconv patient 5 0.5, 1, 3, 8 0.5, 1, , 1, 3, 6, , 1, 3, 6, 8, , 1.3, , 1, 3, to the cost functions used; they have different sampling times but also different group structures, with, for example a total number of patients ranging from 36 for Opt inconv patient to 194 for Opt samples. Opt samples involves 4 elementary designs with an unequal repartition of the number of patients and of the number of samples per group; nearly 70% of the patients have only 2 samples compared to the empirical design with 12 samples to be performed in all the patients. The relative standard errors (RSE) for the empirical design and the design Opt samples are reported in Table 3; these RSE are defined as the standard error divided by the true value of the parameter, expressed in %. Both designs allow good parameter estimate precisions for the infliximab parameters and the methotrexate fixed effects parameters (about or lower than 20%), with higher precision for Opt samples. Some variance parameters of methotrexate cannot be accurately estimated with either design, but the expected RSE are in the same range for both designs. Design Opt inconv involves five elementary designs with two groups of only one patient. More than 60% of the subjects have samples only during the first three hours of Week 1, reflecting the penalties on the cost of the late samples. Only one group of 11 patients are scheduled for one sample at an additional visit (Week 8). Note that removing the two groups with only 1 patient would simplify the group structure of this design and would not involve any major loss of information with an information value of instead of

63 Design in Nonlinear Mixed Effects Models 3363 Table 3 Relative standard errors (RSE) evaluated from the Fisher information matrix for the empirical design and the design Opt samples RSE (%) Parameters Empirical Opt samples infliximab Cl V Cl V methotrexate k a Cl V c V p Q k 2 a Cl V 2 c V 2 p Q The optimization taking into account of the difficulty of adding new patients provides an optimal design Opt patient with one group of only 37 patients, but with 5 sampling times at Week 1 to be repeated at Week 8. Last, the optimization taking into account both constraints of times allocation and patients provides a design with the smallest number of patients, and varying number of samples per patient from 4 7, mainly performed at Week 1. Comparing designs using the information function, we find the largest value for Opt samples (210.8) and the smallest for Opt inconv patient (68.4), as expected because of its substantial cost constraints. Design Opt inconv patient involves higher standard errors; they are still acceptable on the fixed effects, with values around or lower 20% except for the rate constant of absorption and the peripheral compartment volume with values around 25 and 35%, respectively. The comparison of the cost of each optimized designs using the different cost functions are reported in Fig. 2. The design Opt samples in which no cost other than the number of samples is assumed, always incurs the highest cost of the four designs, whatever the cost function, with a cost of nearly 5,500 for the cost function C inconv patient. Logically, the most constrained design Opt inconv patient always incurs the lowest cost, with a cost of 440 for the cost function C samples. The cost-efficiency relationship is represented on Fig. 3 for the cost function C inconv patient, which is the most clinically realistic. Although the cost of a design can largely exceed the allowed total cost, the corresponding efficiency does not increase proportionally, which signals a waste of resources.

64 3364 Retout et al. Figure 2. Cost of the four optimized designs under each cost function. The dotted line represents the reference cost of 1,200. Figure 3. Relative cost vs. relative efficiency for the four optimized designs considering the cost function C inconv patient. Efficiencies and costs are given relatively to the efficiency and cost of the most constrained design Opt inconv patient. The optimized designs Opt samples, Opt inconv, Opt patient, and Opt inconv patient are represented by,,, and,, respectively. The full line represents the unity line.

65 5. Discussion Design in Nonlinear Mixed Effects Models 3365 In this article, we find optimal designs for NLMEM with multiple responses using the Fedorov Wynn algorithm with cost functions. The application to a joint PK model of infliximab and methotrexate, with very different time scales, nicely shows the benefit of design optimization using cost functions, especially when substantial constraints on the design are imposed. Indeed, the optimal designs obtained with the four different cost functions are very different and reflect the penalties imposed on the times allocation and/or the group structure. Our work here shows that by combining a cost function with a user-specified set of possible sampling times, the Fedorov Wynn algorithm is a powerful tool for finding optimal designs suitable to clinical applications. The expression of M F used to optimize design is based on a linearization of the model. Moreover, in our application, we use the block diagonal version of the approximated expression of M F, assuming that the variance of the observations with respect to the mean parameters is constant. Although those are approximations, it has been shown for single response models (Retout et al., 2007) and for multiple response models (Bazzoli et al., 2007) that the approximated M F is very close to that computed by more exact methods, such as the Fisher matrix based on the Stochastic Approximation Expectation Maximization (SAEM) algorithm implemented in MONOLIX (Kuhn and Lavielle, 2005; Samson et al., 2006). Indeed, using this software, evaluation of the expected SE can be performed under asymptotic convergence assumption. To do that, a data set with a large number of subjects is simulated, the parameters are then estimated as well as the observed Fisher information matrix on the simulated data set using the Louis s principle (Louis, 1982). Expected SE are then obtained by rescaling of the observed SE to the true number of subjects. Although this exact method does not involve any linearization, it is time consuming, and it cannot then be applied to design optimization. The linearization is thus a great advantage for optimization process where a large number of designs have to be evaluated in a reasonable time-frame. In our application, we consider computation of M F only for a diagonal variance of the random effects; however, in practice, one may want to allow correlation between the random effects. Although the proposed expression of M F allows those correlations, it has not been yet implemented in PFIM. In our example, the models for infliximab and methotrexate PK do not share any parameter, but the expression of M F for multiple response models can be used for more complex modeling, including models with common parameters, such as models for the PK of a parent drug and of its metabolites, drug interactions or PK/PD models. Here, for simplicity, we assumed the same sampling times for the two drugs. In some cases, different sampling times for the two responses may be required; the Fedorov Wynn algorithm can be used under this assumption but the number of possible elementary designs will quickly become prohibitively large and can considerably increase the computation time and memory requirement can become a problem. This Fedorov Wynn algorithm has been implemented to find the optimal design with a number of groups lower than the bound of the Caratheodory s theorem. Other optimal designs with higher number of groups, although possible, are not considered here because they would be very difficult to implement in clinical settings. In our application, most of the optimized designs lead to several groups of subjects with different number of subjects for the different groups. In addition to the

66 3366 Retout et al. influence of the cost functions, this may also be related to the small intra-individual correlation, which has been shown to lead to imbalance (van Breukelen et al., 2008). The numbers of subjects per group has been rounded to the nearest integer. This was mainly to keep the total cost as close to 1,200 as possible, in order to achieve comparable designs with same total cost, but it is obvious that, in clinical practice, this number should be rounded to the nearest five or even to the nearest ten. Depending on the cost function used, this could change the total cost of the design and care should be taken to avoid a substantial cost over-run. We conclude by noting that attention has to be given to the cost function used. Indeed, a too strict cost function with high penalization may considerably decrease the efficiency of a design, as illustrated by the design Opt inconv patient compared to the design Opt samples. It is then important to strike a proper balance between the clinical constraints and the information that really needs to be collected from the study to obtain reliable results. From a decision-making point of view, design optimization including costs can be a very valuable tool to assess both the clinical consequences and the cost-effectiveness of a candidate design: we can show the efficiency of a design optimized under given constraints, and also immediately compute the cost of adding more patients or redesigning the study. This is useful because the trade-off between total cost and efficiency is now clear. Consequently, a go or no-go decision can be made with confidence. The Fedorov Wynn algorithm with the usual cost function, equal to the total number of samples, is available in PFIM 3.0 for single and multiple response models. The version including the possibility to specify user-defined cost-functions will be available in version 3.1 of the PFIM software, and in the meantime, can be obtained from the authors on request. References Al-Banna, M. K., Kelman, A. W., Whiting, B. (1990). Experimental design and efficient parameter estimation in population pharmacokinetics. J. Pharmacokin. Biopharm. 18(4): Atkinson, A. C., Donev, A. N. (1992). Optimum experimental designs. In: Copas, J. B., Eagleson, G. K., Pierce, D. A., Schervish, M. J., eds. Oxford Statistical Science Series. Oxford: Clarendon Press. Bazzoli, C., Retout, S., Mentré, F. (2007). Population design in nonlinear mixed effects multiple responses models: Extension of PFIM and evaluation by simulation with NONMEM and MONOLIX. Abstr 1176 [ 1176], p. 16. Beal, S. L., Sheiner, L. B. (1992). NONMEM users guides. In: Group, P., ed. NONMEM. San Francisco: University of California. Fedorov, V. V. (1972). Theory of Optimal Experiments. New York: Academic Press. Fedorov, V. V., Gagnon, R. C., Leonov, S. L. (2002). Design of experiments with unknown parameters in variance. Appl. Stochastic Models Bus. Industry 18(3): Gagnon, R., Leonov, S. (2005). Optimal population designs for PK models with serial sampling. J. Biopharmaceut. Statist. 15(1): Godfrey, C., Sweeney, K., Miller, K., Hamilton, R., Kremer, J. (1998). The population pharmacokinetics of long-term methotrexate in rheumatoid arthritis. Brit. J. Clin. Pharmacol. 46(4): Gueorguieva, I., Aarons, L., Ogungbenro, K., Jorga, K. M., Rodgers, T., Rowland, M. (2006). Optimal design for multivariate response pharmacokinetic models. J. Pharmacokin. Pharmacodyn. 33(2):

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71 CO M M ; N o. of Pages 1 1 ARTICLE IN PRESS computer methods and programs in biomedicine x x x ( ) xxx xxx journal homepage: www. i ntl. elsevierhealth. c om/ j ournals/ c mpb Design evaluation and optimisation in multiple response nonlinear mix ed effec t models: P F IM 3.0 Caroline Bazzoli a,b,, S y lv ie R etou t a,b, F ranc e M entré a,b a INSERM, U738, Paris, France b Univ ersité Paris D id ero t, UFR d e Mé d ecine, Paris, France a r t i c l e i n f o a b s t r a c t A rticle h isto ry : R eceived 1 1 A pril R eceived in revised form 1 6 S eptember A ccepted 1 8 S eptember K ey w o rd s: F isher information matrix N onlinear mixed effect models M ultiple response models O ptimal designs D- optimality PF IM N onlinear mixed effect models (N L M E M ) w ith multiple responses are increasingly used in pharmacometrics, one of the main examples being the joint analy sis of the pharmacok inetics (PK ) and pharmacody namics (PD) of a drug. E ffi cient tools for design evaluation and optimisation in N L M E M are necessary. T he R functions PF IM 1.2 and PF IM O PT 1.0 w ere proposed for these purposes, but accommodate only single response models. T he methodology used is based on the F isher information matrix, developed using a linearisation of the model. In this paper, w e present an extended version, PF IM 3.0, dedicated to both design evaluation and optimisation for multiple response models, using a similar method as for single response models. In addition to handling multiple response models, several features have been integrated into PF IM 3.0 for model specifi cation and optimisation. T he extension includes a library of classical analy tical pharmacok inetics models and allow s the user to describe more complex models using differential eq uations. R egarding the optimisation algorithm, an alternative to the S implex algorithm has been implemented, the F edorov W y nn algorithm to optimise more practical D- optimal design. Indeed, this algorithm optimises design among a set of sampling times specifi ed by the user. T his R function is freely available at w w.pfi m.biostat.fr. T he effi ciency of this approach and the simplicity of use of PF IM 3.0 are illustrated w ith a real example of the joint PK PD analy sis of w arfarin, an oral anticoagulant, w ith a model defi ned by ordinary differential eq uations E lsevier Ireland L td. A ll rights reserved. 1. Introduc tion N onlinear mixed effect models (N L M E M s) are increasingly used for the analy sis of longitudinal data describing a biological process. T hey allow the estimation of the mean value of parameters in population studies and their inter- individual variability. T hey are also used for the joint modelling of several biological responses, such as the joint analy sis of pharmacok inetic (PK ) and pharmacody namic (PD) data [1 ]. Pharmacok inetics deals w ith the time course of drug concentration, w hereas pharmacody namics refers to the time course of drug action in the body. In pharmacometrics, analy sis through a N L M E M is called the population approach. T o estimate parameters in N L M E M s, maximum lik elihood estimation is used primarily, although the lik elihood for these models has no analy tical solution. S pecifi c algorithms, implemented in several softw are pack ages, have therefore been proposed to perform this maximisation [2 ]. Before the estimation step, the investigator of a study is confronted w ith the choice of the design w hich is crucial for an effi cient estimation of model parameters. A design in N L M E M, also called a population design, is composed of the number of elementary designs (or groups) and the specifi ca- C o rresp o nd ing au th o r at: 1 6 rue H enri H uchard, Paris, F rance. T el.: ; fax: E - mail address: caroline.bazzoli@ inserm.fr (C. Bazzoli) /$ see front matter E lsevier Ireland L td. A ll rights reserved. doi: /j.cmpb Please cite this article in press as: C. Bazzoli, et al., Design evaluation and optimisation in multiple response nonlinear mixed effect models: PF IM 3.0, Comput. M ethods Programs Biomed. ( ), doi: /j.cmpb

72 COMM-2978; No. of Pages 11 ARTICLE IN PRESS 2 computer methods and programs in biomedicine x x x ( ) xxx xxx tion of each elementary design and the associated number of subjects. In this setting, the term elementary design is used to describe a collection of subjects that have identical design characteristics defined by the number of sampling times and their allocation in time. To evaluate and compare population designs, a statistical approach based on the theory of optimum experimental design in classical nonlinear models and described for instance by Atkinson and Donev [3] or by Walter and Pronzato [4] has been extended to NLMEMs. This theory is based on the Fisher information matrix, whose inverse, according to the Cramer Rao inequality, is the lower bound of the variance covariance matrix of any unbiased estimators of the parameters. Due to the lack of an analytical expression for the likelihood in NLMEM, an exact expression of the Fisher information matrix cannot be defined. That is why an expression based on a linearisation of the model around the expectation of random effects has been proposed by Mentré et al. [5] and extended by Retout and Mentré [6] in the context of a single response model. The usefulness of this approach has been demonstrated, both by simulation [7,8] and in real pharmacokinetic studies [9,10]. To make the procedure more accessible to investigators, the approximate expression of the Fisher information matrix has been implemented in the R function PFIM 1.2, which can be used for design evaluation and comparison [11]. Regarding optimisation, two approaches can be used, either the optimisation of exact designs or of statistical designs. In the case of optimisation of exact designs, the group structure of the design is fixed: the number of elementary designs, the number of samples per design and the number of subjects per elementary design are given and the design variables used to optimise are only the sampling times. Optimisation of statistical designs consists in optimising both the allocation of the sampling times and the whole group structure, that is to say the number of elementary designs, the number of samples per elementary designs and the proportion of subjects in each elementary design. An exact design is then derived by rounding off the proportion of subjects in each elementary design. Optimisation based on the D-optimality criterion has been implemented in the R function PFIMOPT 1.0 [12], where the optimal design is the one that maximises the determinant of the Fisher information matrix. In PFIMOPT 1.0, both exact or statistical optimisation can be performed using the general Simplex algorithm [13]. It optimises the sampling times in given continuous intervals. The main limitation of PFIM 1.2 and PFIMOPT 1.0 is that both aim at evaluating and optimising population designs only for single response models. Moreover, the model had to be written using an analytic expression and thus the need to use more complex models is limited. Regarding optimisation, the Simplex algorithm is a general optimisation algorithm. However, even if its applicability has been shown in pharmacokinetic examples [8], when there are a large number of parameters to optimise, or when the model is complex, it could have difficulties in converging towards the optimal design and should sometimes be run again using the optimised design as a new initial design. Recently, the expression for the Fisher information matrix has been extended in the context of a joint estimation of a set of parameters from multiple responses using the same linearisation as for single response [14 16]. In that context, some parameters can be included in several responses, as for instance in a classical PKPD model with the PD response depending on the PK parameters That joined estimation is then more informative, obtaining information on the PK parameters from both PK and PD responses. However, it increases the complexity of the computation of the Fisher information matrix compared to its computation for each response (single response). The relevance of the use of this approximation of the Fisher information matrix for that multiple response NLMEM context has been shown by Bazzoli et al. [14] through simulation of a PKPD model; results were very similar to those obtained from a more exact computation of the information matrix, without any linearisation, using stochastic approximation through the SAEM algorithm of MONOLIX [17]. This expression has been implemented in PFIM 3.0, an extension of PFIM to handle both design evaluation and optimisation in NLMEMs with multiple responses. The extension includes a library of classical analytical pharmacokinetics models and allows more complex models to be described using a system of differential equations. Regarding the optimisation step, an alternative to the Simplex has been added in PFIM 3.0, the Fedorov Wynn algorithm. It is a specific design optimisation algorithm implemented in PFIM 3.0 for statistical optimisation [5,18] which has the property of converging towards the D-optimal design. The aim of this paper is to present PFIM 3.0. In section 2, we present a nonlinear mixed effect multiple response design and model and the expression of the population Fisher information matrix for multiple responses. Then, the structure of PFIM 3.0 and the description of its features and use are presented in Section 3. Lastly, an example of the use of PFIM 3.0 to design a new trial for the joint analysis of the pharmacokinetics and pharmacodynamics of warfarin, an oral anticoagulant, is provided in Section S tatistical meth ods 2.1. Nonlinear mixed effect multiple response design and model A design for NLMEM, i.e. a population design, is composed of N individuals to whom we allocate an elementary design i, i = 1,..., N. Each elementary design is defined by a number n i of sampling times and their allocation in time. A population design is therefore described by N elementary designs: = { 1,..., N } (1) leading to a total number n of observations. In the case of a multiple response model, an elementary design for one individual i is composed of several sub-designs, i.e. i = ( i1, i2,..., ik ), where ik is the design associated with the kth response, k = 1,..., K. ik is defined by (t ik1, t ik2,..., t iknik ), the vector of the n ik sampling times for the observations of the kth response, so that n i = K k=1 n ik. U sually, population designs are composed of a limited number Q of groups of individuals. Each group is defined by Please cite this article in press as: C. Bazzoli, et al., Design evaluation and optimisation in multiple response nonlinear mixed effect models: PFIM 3.0, Comput. Methods Programs Biomed. (2009), doi: /j.cmpb

73 COMM-2978; No. of Pages 11 ARTICLE IN PRESS computer methods and programs in biomedicine x x x ( ) xxx xxx 3 an elementary design q, q = 1,..., Q, which is composed, for the kth response, of n qk sampling times (t qk1, t qk2,..., t qknqk ) to be performed on a number N q of individuals. The population design can then be noted as follows: = {[ 1, N 1 ]; [ 2, N 2 ];... ; [ Q, N Q ]} (2) A nonlinear mixed effects multiple response model or a multiple response population model is defined as follows. The vector of observations Y i for the ith individual is defined as the vector of the K different responses: Y i = [y T i1, yt i2,..., yt ik ]T (3) where y ik, k = 1,..., K, is the vector of observations for the kth response. Each of these responses is associated with a known function f k which defines the nonlinear structural model. The K functions f k can be grouped in a vector of multiple response models F, such as: F( i, i ) = [ f 1 ( i, i1 ) T, f 2 ( i, i2 ) T,..., f K ( i, ik ) T] T where i is the vector of all the individual parameters needed for all the response models for individual i. The vector of individual parameters i depends on ˇ, the p-vector of the fixed effects parameters and on b i the vector of the p random effects for individual i. The relation between i and (ˇ, b i ) is modelled by a function g, i = g(ˇ, b i ), which is usually additive, so that i = ˇ + b i, or exponential so that i = ˇ exp(b i ). It is assumed that b i N(0, ) with defined as a p p-diagonal matrix, each diagonal element ωr 2, r = 1,..., p, representing the variance of the rth component of the vector b i. We consider here only the case of a diagonal matrix with no correlation between the random effects. The statistical model is thus given by: Y i = F(g(ˇ, b i ), i ) + ε i (5) where ε i is the vector composed of the K vectors of residual errors ε ik, k = 1,..., K, associated with the K responses. We also suppose ε ik N(0, ik ) with ik a n ik n ik -diagonal matrix such that (ˇ, b i, interk, slopek, ik ) ik = diag( interk + slopek f k (g(ˇ, b i ), ik )) 2 (6) where interk and slopek qualify the model for the variance of the residual error of the kth response. The case slopek = 0 returns a homoscedastic error model, whereas the case interk = 0 returns a constant coefficient of variation error model. The general case where the two parameters differ from 0 is called a combined error model. We then note i(ˇ, b i, inter, slope, i ) the variance of ε i, over the K responses, such that i is a n i n i -diagonal matrix composed of each diagonal element of ik with k = 1,..., K. slope and inter are two vectors of the K components interk and slopek, k = 1,..., K, respectively. Finally, depending on the value of b i, we assume that the errors ε i are independently distributed. (4) Let be the vector of population parameters to be estimated, i.e. T = (ˇT, ω1 2,..., ω2 p, T inter, T ) and let be the slope vector of variance terms T = (ω1 2,..., ω2 p, T inter, T ), so that slope T = (ˇT, T ) Fisher information matrix The population Fisher information matrix M F (, ) for multiple response models with the population design is given by: M F (, ) = E ( ) 2 L(, Y) T where L(, Y) is the log-likelihood of the vector of the whole observations Y for the population parameters. Assuming independence across individuals, this loglikelihood can be defined as the sum of the log-likelihoods of the vectors of observations for each individual: L(, Y) = N i=1 L(, Y i). Therefore, taking the second derivatives, the population Fisher information matrix for N individuals can also be defined as the sum of the N elementary information matrices M F (, i ) computed for each individual i: M F (, ) = N M F (, i ) (8) i=1 In the case of a limited number Q of groups, it is expressed as: M F (, ) = Q N q M F (, q ) (9) q=1 The expression of an elementary Fisher information matrix for multiple responses has been described in Bazzoli et al. [14] where further details are given. For the sake of simplicity, we have omitted the index i for the individual in the following. The Fisher information matrix is a block matrix depending on the approximated marginal expectation E and variance V of the observations. ( ) M F (, ) 1 A(E, V) C(E, V) = 2 C T (10) (E, V) B(E, V) where (A(E, V)) ml = 2 ET ˇm with m and l = 1,..., p 1 E V + tr ˇl ( V ( V ) 1 V (B(E, V)) ml = tr V V 1 m l ( V ) 1 V (C(E, V)) ml = tr V V 1 l ˇm ˇl with l = 1,..., dim() and m = 1,..., p ) 1 V V V 1 ˇm (7) with m and l = 1,..., dim() Please cite this article in press as: C. Bazzoli, et al., Design evaluation and optimisation in multiple response nonlinear mixed effect models: PFIM 3.0, Comput. Methods Programs Biomed. (2009), doi: /j.cmpb

74 COMM-2978; No. of Pages 11 ARTICLE IN PRESS 4 computer methods and programs in biomedicine x x x ( ) xxx xxx T ab le 1 L ist of pharmacok inetic models included in the lib rary of models of PFIM 3.0. T he tab le includes all information needed in order to use the model function chosen: the name, the ty pe of input, the numb er of compartments and the parameters used. Name Input Number of compartment Parametrisation bolus 1cpt V k IV -bolus 1 V, k bolus 1cpt V Cl IV -bolus 1 V, Cl infusion 1cpt V k IV -infusion 1 V, k infusion 1cpt V Cl IV -infusion 1 V, Cl oral1 1cpt kav k 1st order 1 ka, V, k oral1 1cpt kav Cl 1st order 1 ka, V, Cl bolus 2cpt V kk12k21 IV -bolus 2 V, k, k12, k21 bolus 2cpt ClV 1Q V 2 IV -bolus 2 Cl, V 1, Q, V 2 infusion 2cpt V kk12k21 IV -infusion 2 V, k, k12, k21 infusion 2cpt ClV 1Q V 2 IV -infusion 2 Cl, V 1, Q, V 2 oral1 2cpt kav kk12k21 1st order 2 ka, V, k, k12, k21 oral1 2cpt kaclv 1Q V 2 1st order 2 ka, Cl, V 1, Q, V 2 Note that in this work, we consider that the variance of the observations with respect to the mean parameters is constant, i.e. C(E,V) ml = 0 and A(E,V) ml = 2( E T / ˇm)V 1 ( E/ ˇl). 3. Implementation of PFIM 3.0 using R softw are The development of the population Fisher information matrix described previously has been implemented in an extension of the R function PFIM: PFIM 3.0. It allows a design evaluation and optimisation of multiple response population models. PFIM 3.0 can be used for single response models instead of the previous version PFIM 1.2 or PFIMOPT 1.0. In addition to the extension for multiple response models, options have been added for model specification and for design optimisation. The free statistical R software (R and higher versions) [19] ( is required to use PFIM 3.0. This function is freely available on the PFIM website as well as an extensive documentation and some examples to use it Model specification In PFIM 3.0, the model has to be entered in a model file called by default model.r. It must be specified by the user either in its analytical form or by using a system of differential equations. This extension to a system of differential equations requires the use of the lsoda function included in the library odesolve implemented by Setzer to solve the system [20] and the fdh ess function included in the library nlme developed by Pinheiro and Bates [21] for numerical derivatives. The lsoda function uses a function of the same name written in Fortran by Petzold and Hindmarsh [22,23]. This function solves a system of differential equations using the Adams method, a predictor corrector method for non-stiff systems; it uses the BDF (Backward Differentiation Formula) for stiff systems. Regarding the function fdh ess, it evaluates an approximate gradient of a scalar function using finite differences. Moreover, a library of standard PK models has been added. Twelve PK models are included: one or two compartment models, with first order oral, bolus or infusion administration and after a single dose, multiple doses or steady-state. Only models with first order elimination are presently available in the library of PFIM 3.0. Currently, there are no models with lag time. A list of the models and their characteristics included in the library is given in Table O ptimisation algorithms An alternative to the Simplex algorithm has been added in PFIM 3.0, the Fedorov Wynn algorithm. It is specifically dedicated to design optimisation problems and has the property of convergence towards the D-optimal design [5,24,25]. It optimises statistical designs (i.e. the group structure, the number of samples per subject and the sampling times) for a given total number of samples. Minimum and maximum number of samples per subject are specified. To optimise the design, it considers only a set of possible sampling times defined by the user. This can be an advantage in clinical practice to avoid unfeasible sampling times. Design optimisation can be constrained to the same sampling times across responses or not. It is also possible to give each elementary design sampling times from a different set of finite sampling times. For instance, it is possible to specify that each elementary design should include two points among a first set of sampling times (for instance Day 1) and a third point among a second set of sampling times (for instance Day 2). The user therefore specifies a list of sampling windows, and for each window, the set of possible sampling times and the minimum and maximum number of sampling times within that window. The Fedorov Wynn algorithm is programmed in C code and is linked to PFIM 3.0 through a dynamic library. Moreover, PFIM 3.0 uses the function combn in the R package combinat to generate all possible elementary designs from the set of pre-specified sampling times. To start the algorithm, an initial population design is required Input file The inputs to PFIM 3.0 are entered in R objects by filling in one input file called by default stdin.r. The same function PFIM 3.0 is used for the evaluation and optimisation of population designs, which is why the user has to notify it in the input file in a specific R object. Then, this input file is composed of two main sections of R objects. The first one is composed of general R objects required for evaluation and optimisation such as the model form (analytical or differential equation sys- Please cite this article in press as: C. Bazzoli, et al., Design evaluation and optimisation in multiple response nonlinear mixed effect models: PFIM 3.0, Comput. Methods Programs Biomed. (2009), doi: /j.cmpb

75 COMM-2978; No. of Pages 11 ARTICLE IN PRESS computer methods and programs in biomedicine x x x ( ) xxx xxx 5 tem), the parameterisation of the structural model, the type of inter-individual random effect model (additive or exponential), values of inter and slope for the variance error model and a priori values of the fixed effect parameters and variance parameters. The user also specifies the population design to evaluate or, for optimisation, an initial population design. They are defined as a group of elementary designs each one associated with a number of subjects. The second section is dedicated to design optimisation where the specific objects are algorithm options. First, the user specifies if the same sampling times are required for each response or not. Then, the algorithm is chosen (Simplex or Fedorov Wynn). If the Fedorov Wynn is required, the user must provide the following information: the number of sampling windows, the list of the allowed sampling times for each sampling window, the list of the number of allowed sampling times for each sampling window, the maximum and minimum total number of sampling times per subject. G raphical representations are provided automatically if graph objects are completed. Once the model file and the input file are filled in, the user can run the function PFIM 3.0 by calling the function in the R Console window: PFIM () Output file The results are written to an output file named by default stdout.r or with a name specified in the input file. PFIM 3.0 returns a summary of the input, the population design evaluated or optimised, the associated population Fisher information matrix, the standard error expected on each parameter, the corresponding coefficient of variation (% ), its determinant and a criterion defined as the determinant standardized by the dimension of the vector : () = [det(m F (,))] 1/dim( ). When design optimisation is performed, the list of the algorithm options is also specified in the output file. For instance, for the Fedorov Wynn algorithm, the set of allowed sampling times, the number of sampling times, the maximum and the minimum number of points for each response are returned. The optimised design is reported with the proportion of subjects optimised by the Fedorov Wynn algorithm and their absolute number. If a graph has been supplied in the input file, the mean profile of each response and the associated sampling times of the evaluated or optimised design are represented on an R graph. 4. Illustration of design for PK PD of warfarin We aim to illustrate the use of PFIM 3.0 for the joint PKPD modelling of the time course of total racemic warfarin plasma concentrations and effect on prothrombin complex activity (PCA), by designing a new trial for this joint population analysis. To do this, we use an example extracted from a classical pharmacology study published 40 years ago, by O Reilly et al. [26,27] on the pharmacokinetics and pharmacodynamics of warfarin. Previously, these data have been fitted by Holford for the analysis of the dose effect relationship of warfarin [28] using nonlinear mixed effects models. This example was Table 2 Population parameter values estimated by the PK PD joint modelling of warfarin. Parameters Fixed effects (ˇ) Inter-subject variability (ω 2 ) k a (h 1 ) CL (L/h) V (L) R in (% /h) IC 50 (mg/l) k out (h 1 ) slopepk 0.20 interpd (% ) 3.88 presented in [29] but using different constraints in the optimisation step P opulation design and model Warfarin is administered by a single oral dose of 100 mg. A one compartment model with first order absorption and elimination adequately describes the concentration data (PK). The PCA effect is described by a turnover model with inhibition of the input (PD). The model can be written as a system of ordinary differential equations: df PK ( PK, t) dt df PD ( PD, t) dt = k a De kat CL V f PK( PK, t) (11) ) f PK ( PD, t) = R in (1 k out f PD ( PD, t) (12) IC 50 + f PK ( PD, t) where f PK and f PD are respectively the warfarin plasma concentrations and effect model and PK and PD the parameters needed for each response. D is the dose of warfarin. The vector of the PK parameters is PK = (k a,cl,v) T with k a the absorption constant, CL the apparent clearance and V the apparent volume of distribution. PCA is controlled by the rate of input R in and the output rate constant k out of the response, and its baseline value is given by the ratio of those parameters R in /k out. The vector of PD parameters is thus defined by PD = (R in,ic 50,k out ) T where IC 50 is the drug concentration which produces 50% of maximum inhibition achieved at the effect site. The inter-individual random effects are assumed to be exponential for all parameters. We associate a proportional error for the PK model and an additive error for the PD model characterised by the parameters slopepk and interpd. Thus, the vector of population parameters is described by the vector of fixed effects ˇT = (ˇka, ˇCL, ˇV, ˇRin, ˇIC50, ˇkout ) and by T the vector composed of the variance of the random effects and of the parameters for the error models such that T = (ω 2, ω 2 k a CL, ω2 V, ω2 R, ω 2 in IC, ω 2, 50 k out slopepk, interpd ). Values of the parameters are those obtained by Holford performing the simultaneous PKPD population analysis of warfarin. The population parameters were estimated by maximum likelihood, with the NONMEM software [30]. They are given in Table D esign optimisation for a new study Using PFIM3.0, we first evaluate the design used in the previous study to determine the joint population PKPD model. We call Please cite this article in press as: C. Bazzoli, et al., Design evaluation and optimisation in multiple response nonlinear mixed effect models: PFIM 3.0, Comput. Methods Programs Biomed. (2009), doi: /j.cmpb

76 COMM-2978; No. of Pages 11 ARTICLE IN PRESS 6 computer methods and programs in biomedicine x x x ( ) xxx xxx Fig. 1 Model fi le of PFIM 3.0 for the evaluation of the empirical design emp and optimisation of both optimal designs opt iden and opt diff for the joint PKPD modelling of warfarin. this design the empirical design denoted emp. It involves one group of 32 healthy volunteers, with 13 sampling times at 0.5, 1, 2, 3, 6, 9, 12, 24, 36, 48, 72, 96, 120 h for warfarin concentration measurements and eight sampling times at 0, 24, 36, 48, 72, 96, 120, 144 h for PCA measurement. emp thus has a total of 672 measurements. We then optimise two population designs under several constraints using the Fedorov Wynn algorithm implemented Fig. 2 Input fi le of PFIM 3.0 in the context of the evaluation of the empirical design emp for the joint PKPD modelling of warfarin. Please cite this article in press as: C. Bazzoli, et al., Design evaluation and optimisation in multiple response nonlinear mixed effect models: PFIM 3.0, Comput. Methods Programs Biomed. (2009), doi: /j.cmpb

77 COMM-2978; No. of Pages 11 ARTICLE IN PRESS computer methods and programs in biomedicine x x x ( ) xxx xxx 7 Fig. 3 Input file of PFIM 3.0 in the context of the optimisation of the optimal design opt iden using the Fedorov W algorithm for the joint PKPD modelling of warfarin. ynn in PFIM 3.0. To do that, we first optimise a population design opt iden with only four sampling times per elementary design common to both responses. This optimisation step is performed on the same number of 32 volunteers and the set of admissible sampling times is the set of times used in the empirical design except times 1 and 3 for PK. We then optimise a population design opt diff, with four sampling times per elementary design, but now, the optimal sampling times can be different for each response Results Fig. 1 reports the model file used for the design evaluation and optimisation of the joint PKPD model of warfarin. The input file of PFIM 3.0 for the evaluation of emp is reported in Fig. 2. The output file of PFIM 3.0 for the optimal design opt iden and the associated R graph output are represented in Figs. 3 and 4, respectively. The optimal designs with their associated criterion are reported in Table 3. emp and both optimal designs ( opt iden, opt diff ) are different by their group structure. opt iden and opt diff are composed of two elementary designs compared to only one for emp. The Fedorov Wynn algorithm optimises proportions of subjects, to be practical those numbers are then rounded to the nearest integers. Each elementary design involves 22 and 10 subjects for opt iden and 29 and 3 for opt diff. In order to get more practical designs, opt diff is simplified in a new design opt diff simp, composed of one group of 32 subjects, meaning that the three subjects originally included in the second elementary design have been added to the first elementary design. The two optimised designs have a smaller number of samples than the empirical design and are therefore less efficient with lower criterion values (Table 3). These two designs include a sample at 144 h for the PK measurement whereas this sample is not present in the empirical design. Indeed, in population PK studies, when a proportional residual error is associated with the model, the optimal design includes the latest time among those proposed. In our case, for optimisation, we allowed the sample at 144 h both for PK and PD responses. In order to get an efficiency similar to that obtained with emp, we then derived two enlarged designs opt iden enl and opt diff simp enl from the two optimal designs by increasing the number of subjects. 55 subjects and 52 subjects are involved in opt iden enl and opt diff simp enl, meaning a total of number of sampling times of 440 and 416, respectively. The expected relative standard errors (RSE) of emp, opt iden enl and opt diff simp enl, i.e. standard error divided by the true Please cite this article in press as: C. Bazzoli, et al., Design evaluation and optimisation in multiple response nonlinear mixed effect models: PFIM 3.0, Comput. Methods Programs Biomed. (2009), doi: /j.cmpb

78 COMM-2978; No. of Pages 11 ARTICLE IN PRESS 8 computer methods and programs in biomedicine x x x ( ) xxx xxx Fig. 4 R graph output of the concentration and effect profiles versus time for the mean parameter values used in the joint PKPD modelling of warfarin using PFIM 3.0. PK sampling times and PD sampling times for the optimal design opt iden for each response are displayed using number values (i.e. 1 and 2 for samples from first and second elementary design, respectively). value of the parameter, their total number of sampling times and the associated criterion are given in Table 4. The evaluation of emp returns expected RSEs smaller than 30%, except for the variance parameters ωic 2 and ω 2, 50 k out where they are rather large (about 68% and 41%) due to the small inter-subject variability of these parameters. Comparison between both enlarged designs shows that only the coefficient of variation of the estimation of the variance parameter ωic 2 is increased compared to emp. The 50 expected RSE on the terms for residual errors increase in both optimal designs, but they remain lower than 10%. For other parameters, the results from design optimisation, in terms of precision of estimation, are very close to those obtained with the empirical design, although enlarged designs involve only 440 and 416 measurements, respectively, 35% and 38% fewer than the empirical design. This sample reduction, together with a similar efficiency, leads to a more ethical and less costly study. Using PFIM 3.0, we find new efficient designs to study the joint PKPD of warfarin fitting to different clinical constraints. 5. Discussion In this paper, we consider the R function PFIM 3.0, an extension of the R functions PFIM 1.2 and PFIMOPT 1.0, allowing evaluation and optimisation of designs for nonlinear mixed effect models with multiple responses. The methodology is based on the Fisher information matrix, developed using a linearisation of the model, as proposed for a single response by Mentré et al. [5] and extended for multiple responses [14 16]. In addition to handling multiple response models, several features have been integrated in PFIM 3.0. First, regarding the model specification, it can now be specified as a system of differential equations; this is an alternative to analytical model specification, particularly useful for complex models. For classical pharmacokinetic models, a library of 12 pharmacokinetic models is available. Concerning the optimisation algorithm, the Fedorov Wynn algorithm has been added as an alternative to the general Simplex algorithm to optimise more practical D-optimal designs [24,25]. Indeed, this algorithm allows design optimisation with fixed sampling times specified by the user which can be a great advantage with respect to medical constraints. This algorithm optimises the sampling times, which can be different or imposed to be identical across responses, but also optimises the group structure, using the approach of statistical designs. In addition to convergence issues using the Simplex algorithm compared to the Fedorov Wynn [18], optimisation within continuous intervals of times can be more questionable: the Simplex algorithm can, sometimes for a long time, pursue the optimisation by exchanging at each iteration one or several sampling times by some others which differ from the previous by only few seconds. This greatly slows down the whole optimisation process and has no relevance in clinical practice. We illustrate the different options of this new extension of PFIM using a real example of the simultaneous population analysis of the time course of warfarin concentration and its effect on the prothrombin complex activity after single dose administration. We optimise and derive two population designs using the Fedorov Wynn algorithm, either imposing the same sampling times across responses or not, and we compare them to an empirical design used in the previous study of warfarin [26,27]. Globally, the expected precision of estimations are in the same range for the three designs, even if the number of samples per patient, and thus, the total number of samples are considerably reduced (38% fewer samples in the optimised designs compare to the empirical one). In this example, we show the interest of the optimisation procedure implemented in PFIM 3.0 from a medical, economic as well as Please cite this article in press as: C. Bazzoli, et al., Design evaluation and optimisation in multiple response nonlinear mixed effect models: PFIM 3.0, Comput. Methods Programs Biomed. (2009), doi: /j.cmpb

79 Please cite this article in press as: C. Bazzoli, et al., Design evaluation and optimisation in multiple response nonlinear mixed effect models: PFIM 3.0, Comput. Methods Programs Biomed. (2009), doi: /j.cmpb Table 3 O ptimal population designs for the joint model of warfarin concentration (PK) and the effect on prothrombin complex activity (PD) according to different constraints. emp is the empirical design. opt iden and opt diff are both optimal designs. opt diff simp is a design derived from opt diff to be simpler. For each design, the number of elementary designs, the rounded number of subjects with the elementary design, the sampling times for each response, the total number of sampling times and the associated criterion value are reported. Number of elementary designs Sampling times for each response (h) PK Rounded number of subjects Total number of sampling times epm 1 0.5, 1, 2, 3, 6, 9, 12, 24, 36, 48, 72, 96,120 0, 24, 36, 48, 72, 92, 120, opt iden 2 0.5, 12, 24, , 12, 24, , 24, 120, , 24, 120, opt diff 2 0.5, 6, 9, 144 0, 12, 24, , 6, 9, 144 0, 24, 96, opt diff simp 1 0.5, 6, 9, 144 0, 24, 96, PD Criterion value, Table 4 Relative standard errors (RSE ) (% ) computed by the population Fisher information matrix for designs emp and for the enlarged designs opt iden enl and opt diff simp enl derived from the two optimal designs by increasing the number of subjects in order to get a similar efficiency obtained with emp. The total number of sampling times and the criterion value associated with each design are also reported. Expected RSE (%) ˇka ˇCL ˇV ˇRin ˇIC50 ˇkout ω 2 k a ω 2 CL ω 2 V ω 2 R in ω 2 IC 50 ω 2 k out slopepk interpd Total number of sampling times emp opt iden enl opt diff simp enl Criterion value, computer methods and programs in biomedicine x x x ( ) xxx xxx 9 COMM-2978; No. of Pages 11 ARTICLE IN PRESS

80 COMM-2978; No. of Pages 11 ARTICLE IN PRESS 10 computer methods and programs in biomedicine x x x ( ) xxx xxx ethical point of view. This approach can also be used for other complex modelling, such as the pharmacokinetics of a parent drug and its metabolite [31]. However, in some situations, due to the different time scales of the pharmacokinetic profiles between the parent drug and the metabolite, different sampling times for the two responses may be required. It should be noted however, using the Fedorov Wynn algorithm under this assumption, the number of possible elementary designs will be large and can considerably increase the computation time and the memory requirement. Limiting the number of allowed sampling times for each sampling window also helps to avoid computational issues. In the example, the prothrombin complex activity is controlled by the rate of input R in and the output rate constant k out of the response, and the initial condition is given by the ratio of these parameters R in /k out. It would be possible by using a modification of the parameterisation of the model to estimate the initial condition and R in or k out or even to fix the initial condition (for instance to 100%) and to estimate only R in or k out. The present development of the Fisher information matrix for multiple response models takes into account all the population parameters to be estimated. Another approach would have been to optimise separately one design for the PK parameters and one for the PD parameters, fixing the PK parameters to their mean in the population (sequential approach). However, this approach can be sub-optimal as no variability on the PK parameters is taken into account in the PD model. For instance, we compared this sequential approach to the one used in this paper, using the same example and similar constraints. We obtained the same optimal design for the PK model. Regarding the conditional D-optimal design for the PD, the design is quite different from opt diff simp. Indeed, it involves two elementary designs with sampling times at 0, 24, 72, 96 h and 0, 24, 36, 96 h, and with 21 and 11 subjects, respectively. Comparison of the relative standard errors shows an increase by at least a factor 2 with this design compared to those obtained with opt diff simp. Following the first theoretical work on optimal design for NLMEM, this research theme has grown rapidly both in methodological and application developments. An mailing list server PopDesign is now available for questions and comments about the experimental design of any study for which nonlinear mixed effect modelling is proposed for the analysis. Since the first tool PFIM [11], there are now several different software tools that implement an evaluation of the Fisher information matrix for population PK and PD models and propose optimisation of the experimental designs (PFIM, PopDes, PopED, WinPOPT) [32]. All software packages have ongoing development and have specific features according to the needs and goals of the research teams that developed them. PFIM is the only one using the R software. PFIM 3.0 uses an approximate Fisher information matrix to evaluate and optimise design. This approximation, using a linearisation of the model, may be a limitation. Indeed, even if the relevance of this approximation has been shown by comparison to an exact Fisher information matrix using a simulation study, the PKPD model used was very simple [14] and may not be generalised to any complexity of the models. On the other hand, use of more exact methods can be computationally intensive and thus time consuming. Their need is thus questionable for nonlinear models for which design evaluation and optimisation require some a priori values of the parameters to be estimated, which are often not known precisely. In parallel to PFIM 3.0, PFIM Interface 2.1, a graphical user interface version of PFIM 1.2 and PFIMOPT 1.0 has been recently proposed, allowing both design evaluation and optimisation but only for single response models. PFIM Interface 2.1 has the same features for model specification and optimisation algorithm as in PFIM 3.0. PFIM Interface will soon be extended to accommodate multiple response models. In addition to the new features added in PFIM 3.0, several other extensions are planned. First, the current library includes only pharmacokinetic models. Extensions to the usual pharmacokinetic pharmacodynamic models are planned, even if model specification for multiple responses is quite simple for the user as shown in the illustration using a system of differential equations. Then, the next version could integrate the computation of the Fisher matrix for a non-diagonal variance of the random effects, as, in practice, one may want to allow correlation between the random effects. Expression of the Fisher information matrix has been proposed in this case for single response in Mentré et al. [5] and for multiple responses in Ogungbenro et al. [33] but has not been yet implemented in PFIM 3.0. Afterwards, an extension of PFIM is envisaged to implement the expression of the Fisher information matrix for models including fixed effects for the infl uence of covariates on the parameters [6], and thus, to compute the predicted power of the Wald test to detect a covariate effect as well as the number of subjects needed to achieve a given power [18]. Moreover, Retout et al. [12] have recently used a modified version of PFIM for the problem of design optimisation using cost functions. They focus on the relative feasibility of the optimised designs, in terms of sampling times and of numbers of subjects. To do that the Fedorov Wynn algorithm has been extended to include cost functions to penalize less feasible designs. An illustration of this extension of design optimisation to a joint population model of infl iximab and methotrexate pharmacokinetics administered in rheumatoid arthritis was performed. The possibility of specifying userdefined cost functions will be available in the next version of PFIM. The implementation of the Fisher information matrix for multiple response models proposed in PFIM 3.0 is a relevant computing tool for the evaluation and comparison of designs in the spreading development of multiple response PK studies. PFIM 3.0 is distributed under the terms of the GNU GENERAL PUBLIC LICENSE (GNU GPL) Version 2, June 1991 and has been registered at the Agency for the Protection of Programs (APP) in 2008, at Paris, France. PFIM functions, including their extensive documentation and examples to use them, are freely available from the PFIM website: C onfl ict of interest statement The authors have no confl icts of interest that are directly relevant to the content of this work. Please cite this article in press as: C. Bazzoli, et al., Design evaluation and optimisation in multiple response nonlinear mixed effect models: PFIM 3.0, Comput. Methods Programs Biomed. (2009), doi: /j.cmpb

81 COMM-2978; No. of Pages 11 ARTICLE IN PRESS computer methods and programs in biomedicine x x x ( ) xxx xxx 11 r e f e r e n c e s [1] L. Sheiner, J. Wakefield, Population modelling in drug development, Stat. Methods Med. Res. 8 (1999) [2] G.C. Pillai, F. Mentré, J.L. Steimer, Non-linear mixed effects modelling from methodology and software development to driving implementation in drug development science, J. Pharmacokinet. Pharmacodyn. 32 (2005) [3] AC. Atkinson, A.N. Donev, Optimum Experimental Designs, Clarendon Press, Oxford, [4] E. Walter, L. Pronzato, Identification of Parametric Models from experimental Data, Springer, New Y ork, [5] F. Mentré, A. Mallet, D. Baccar, Optimal design in random effect regression models, Biometrika 84 (1997) [6] S. Retout, F. Mentré, Further developments of the Fisher information matrix in nonlinear mixed effects models with evaluation in population pharmacokinetics, J. Biopharm. Stat. 13 (2003) [7] S. Retout, F. Mentré, Optimization of individual and population designs using Splus, J. Pharmacokinet. Pharmacodyn. 30 (2003) [8] S. Retout, F. Mentré, R. Bruno, Fisher information matrix for non-linear mixed-effects models: evaluation and application for optimal design of enoxaparin population pharmacokinetics, Stat. Med. 21 (2002) [9] B. Green, S.B. Duffull, Prospective evaluation of a D-optimal designed population pharmacokinetic study, J. Pharmacokinet. Pharmacodyn. 30 (2003) [10] F. Mentré, C. Dubruc, J.P. Thenot, Population pharmacokinetic analysis and optimization of the experimental design for mizolastine solution in children, J. Pharmacokinet. Pharmacodyn. 28 (2001) [11] S. Retout, S. Duffull, F. Mentré, Development and implementation of the population Fisher information matrix for the evaluation of population pharmacokinetic designs, Comput. Methods Programs Biomed. 65 (2001) [12] S. Retout, E. Comets, C. Bazzoli, F. Mentré, Design optimisation in nonlinear mixed effects models using cost functions: application to a joint model of infliximab and methotrexate pharmacokinetics, Commun. Stat. A-Theor. 38 (2009) [13] J.A. Nelder, R. Mead, A Simplex method for function minimization, Comput. J. 1 (1965) [14] C. Bazzoli, S. Retout, F. Mentre, Fisher information matrix for nonlinear mixed effects multiple response models: evaluation of the appropriateness of the first order linearization using a pharmacokinetic/pharmacodynamic model, Stat. Med. 28 (2009) [15] I. Gueorguieva, L. Aarons, K. Ogungbenro, K.M. Jorga, T. Rodgers, M. Rowland, Optimal design for multivariate response pharmacokinetic models, J. Pharmacokinet. Pharmacodyn. 33 (2006) [16] A. Hooker, P. Vicini, Simultaneous population optimal design for pharmacokinetic pharmacodynamic experiments, AAPS J. 7 (2005) E [17] M. Lavielle, MONOLIX (MOdè les NOn LInéaires à effets mixtes), MONOLIX Group, Orsay, France, [18] S. Retout, E. Comets, A. Samson, F. Mentré, Design in nonlinear mixed effects models: optimization using the Fedorov Wynn algorithm and power of the Wald test for binary covariates, Stat. Med. 26 (2007) [19] R Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, [20] M.L. Lindstrom, D.M. Bates, Nonlinear mixed effects models for repeated measures data, Biometrics 46 (1990) [21] J. Pinheiro, D.M. Bates, Mixed-Effects Models in S and Splus, New Y ork, NY, [22] A.C. Hindmarsh, ODEPACK, A systematized collection of ODE solvers, in: R.S. Stepleman (Ed.), Scientific Computing: Applications of Mathematics and Computing to the Physical Sciences, Elsevier Science Ltd, Amsterdam, 1983, pp [23] L.R. Petzold, Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations, Siam J. Sci. Stat. Comput. 4 (1983) [24] V.V. Fedorov, Theory of Optimal Experiments, Academic Press, New Y ork, [25] H.P. Wynn, Results in the construction of D-optimum experimental designs, J. R. Statist. Soc. 34 (1972) [26] R.A. O Reilly, P.M. Aggeler, Studies on coumarin anticoagulant drugs. Initiation of warfarin therapy without a loading dose, Circulation 38 (1968) [27] R.A. O Reilly, P.M. Aggeler, L.S. Leong, Studies on the coumarin anticoagulant drugs: the pharmacodynamics of warfarin in man, J. Clin. Invest. 42 (1963) [28] N.H. Holford, Clinical pharmacokinetics and pharmacodynamics of warfarin. Understanding the dose effect relationship, Clin. Pharmacokinet. 11 (1986) [29] S. Retout, C. Bazzoli, E. Comets, H. Le Nagard, F. Mentré, Population designs evaluation and optimisation in R: the PFIM function and its new features, in: 16th Meeting of Population Group Approach in Europe (PAGE), 2007, Abstr abstract=1164. [30] S.L. Beal, L.B. Sheiner, NONMEM Users Guide, University of California, San Francisco, [31] C. Bazzoli, S. Retout, E. Rey, H. Benech, J.M. Tréluyer, D. Salmon, X. Duval, F. Mentré, COPHAR2-ANRS 111 study group, Population pharmacokinetic of AZ T and its active metabolite AZ T-TP in HIV patients (COPHAR 2 ANRS 111 trial): joint modelling and design optimisation, in: 17th Meeting of Population Group Approach in Europe (PAGE), 2008, Abstr abstract=1373. [32] F. Mentré, S. Duffull, I. Gueorguieva, A. Hooker, S. Leonov, K. Ogungbenro, S. Retout, Software for Optimal Design in Population Pharmacokinetics and Pharmacodynamics: A Comparison, Population Approach Group in Europe (PAGE), Copenhagen, Denmark, [33] K. Ogungbenro, G. Graham, I. Gueorguieva, L. Aarons, Incorporating correlation in interindividual variability for the optimal design of multiresponse pharmacokinetic experiments, J. Biopharm. Stat. 18 (2008) Please cite this article in press as: C. Bazzoli, et al., Design evaluation and optimisation in multiple response nonlinear mixed effect models: PFIM 3.0, Comput. Methods Programs Biomed. (2009), doi: /j.cmpb

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84 ********************************************************************************** PFIM 3.2 Caroline Bazzoli, Anne Dubois, Thu Thuy Nguyen, Sylvie Retout, Emanuelle Comets,France Mentré INSERM, U738, Paris, France ; Université Paris 7, Paris, France December Read me This document is an adds-on to PFIM 3.0 documentation. Thus, it only outlines the new features implemented in PFIM 3.2 and explains how to carry out to use them. This documentation version does not detail the features that were previously released in version 3.0 of PFIM. ************************************************************** PFIM 3.2 is free library of functions. The Université Paris Diderot and INSERM are the co-owners of this library of functions. Disclaimer We inform users that the PFIM 3.2 is a tool developed by the Laboratory «Models and methods of the therapeutic evaluation of the chronic diseases»- UMR-S 738, under R and GCC. PFIM 3.2 is a library of functions. The functions are published after a scientific validation. However, it may be that only extracts are published. By using this library of functions, the user accepts all the conditions of use set forth hereinafter. Licence This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. You should have received a copy of the GNU General Public License along with this program. If not, see < 1

85 THIS SOFTWARE IS PROVIDED AS IS AND ANY EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE UNIVERSITE PARIS DIDEROT OR INSERM OR ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. Redistribution and use in source and binary forms, with or without modification, are permitted under the terms of the GNU General Public Licence and provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. The end-user documentation included with the redistribution, if any, must include the following acknowledgment: "This product includes software developed by Université Paris Diderot and INSERM ( Alternately, this acknowledgment may appear in the software itself, if and wherever such third-party acknowledgments normally appear. 4. The names "PFIM" must not be used to endorse or promote products derived from this software without prior written permission. For written permission, please contact [email protected]. 5. Products derived from this software may not be called "PFIM", nor may "PFIM" appear in their name, without prior written permission of the Université Paris Diderot and INSERM. Copyright PFIM 3.2 Caroline Bazzoli, Anne Dubois, Thu Thuy Nguyen, Sylvie Retout, Emmanuelle Comets, France Mentré - Université Paris Diderot INSERM. 2

86 CONTENT 1. DESCRIPTION OF THE NEW FEATURES IN PFIM Model specification Library of pharmacokinetic models Pharmacokinetic models with a linear elimination Pharmacokinetic models with a Mickaelis-Menten elimination Library of pharmacodynamic models Immediate response pharmacodynamic models alone Pharmacodynamic models linked to pharmacokinetic model Full expression of the Fisher information matrix Inter-occasion variability (IOV) specification Discrete covariate specification Computation of power and number of subjects needed to treat Comparison test Equivalence test References USE Working directory Model writting Example 1: Single response with a PK or a PD model Example 2: Two responses defined by a PK/PD model General objects required for Evaluation and Optimisation Full or block diagonal fisher information matrix Graph option Objects required only for IOV option Objects required only for covariate option Covariates not changing with occasion Covariates changing with occasion Objects required only for computation of power and number of subjects needed for comparison test or equivalence test RESULTS 26 3

87 1. Description of the new features in PFIM 3.2 PFIM 3.2 is a new release of the R script function PFIM 3.0 [1] dedicated to design evaluation and optimisation for multiple response models. This version incorporates new features in terms of model specification and development of the expression of the Fisher information matrix (M F ). Regarding model specification, the library of standard pharmacokinetic (PK) models has been completed with three compartment models with linear elimination and models with Mickaelis-Menten elimination (one, two and three compartment models). Furthermore, a library of pharmacodynamic (PD) models is now available. Concerning the expression of the Fisher information matrix, PFIM 3.2 can handle either a block diagonal Fisher information matrix or the complete one. It is now also possible in PFIM 3.2 to use models including inter-occasion variability (IOV) with replicated designs at each occasion [2]. Last, a new feature of PFIM 3.2 is the computation of the Fisher information matrix for models including fixed effects for the influence of discrete covariates on the parameters. Specification of covariates can depend or not of the occasion. The computation of the predicted power of the Wald test for comparison or equivalence test for a given distribution of a discrete covariate as well as the number of subjects needed to achieve a given power can be computed [2, 3, 4]. The same input file, named by default stdin.r, used in PFIM 3.0 can be use in PFIM 3.2 but the new features would then not be active. PFIM 3.2 is also developed for R and higher versions Model specification Models can be specified either with their analytical form or with systems of differential equations, using the libraries of models or the user defined model option. In the later case, users can define their own model analytically or use a system of differential equations. This option has not been modified in PFIM 3.2, only the libraries of models have been completed. Compared to PFIM 3.0, three compartment models with linear elimination and models with Mickaelis-Menten elimination (one, two and three compartments models) have been added to the library of PK models. Moreover, a library of PD models is now available, supporting immediate response models (alone or linked to a pharmacokinetic model) and the turnover response models (linked to pharmacokinetic model). These libraries have been derived from the PKPD library developed by Bertrand and Mentré [5] for the MONOLIX software, and all analytical expressions are in that document. Presently, there is no model with lag time in both libraries. As in PFIM 3.0, to use the library of models, the user has to specify the path of the file in the modelfile named by default model.r Library of pharmacokinetic models Two types of PK models can now be used in PFIM 3.2, PK models with a first order linear elimination or PK models with a Mickaelis-Menten elimination. The PK models with a linear elimination are written using an analytical form whereas the PK models with a Mickaelis-Menten are written using a differential equation system. 4

88 These both types of PK model are written in the file librarypk.r available in the folder Program. Thus, the user has to specify the path of this file in the model file to use this library of models: source (paste(directory.program,dirsep, librarypk.r, sep= ) For each type of PK models, the list of the models are presented in separated tables in the following sections. These tables return every information in order to use the model function chosen. The model is described by: - a name - the type of input - the type of elimination - the number of compartments - the parameters used (parameterisation) - the type of administration (sd : single dose, md: multiple dose, ss: steady state) - for each administration type, some variables are required (or not). They are specified in the column named: Needed variables (N: number of doses, tau: interval between two doses, TInf: duration of the infusion, dose: dose) For models with infusion, the user has to specify the duration of infusion (TInf) in the needed variable. The rate of infusion is computed automatically in the function model by the expression: dose/tinf. As in PFIM 3.0, for PK models with linear elimination, the variable dose has to be specified in the input file. For example, if one uses after a multiple dose administration, the first order oral absorption with one compartment model (oral1_1cpt_kavcl with option md) from the library, the function of the model uses three parameters (ka, Cl and V) and two needed variables (N, tau): the number of doses (N) and the interval between two doses (tau). Examples of the use of the library of pharmacokinetic models are presented in section 2.2 of the present document as in section 2.1 of the PFIM 3.0 documentation Pharmacokinetic models with a linear elimination Compared to PFIM 3.0, the library of PK models with linear elimination has been completed by the three compartment models for the three types of input (bolus, infusion and first order oral absorption) and the three types of administration (single dose, multiple dose, steady state). The list of these PK models is given in Table 1. It is an update of the Table 1 presented in the documentation of PFIM

89 Table 1. Pharmacokinetic models with first order linear elimination included in the library of models Name Input Cpt Elimination Parameterisation Administration Needed Variable(s) sd - bolus_1cpt_vk IV-bolus 1 1st order V, k md N, tau ss tau sd - bolus_1cpt_vcl IV-bolus 1 1st order V, Cl md N, tau ss tau sd TInf infusion_1cpt_vk IV-infusion 1 1st order V, k md TInf, N, tau ss TInf, tau sd TInf infusion_1cpt_vcl IV-infusion 1 1st order V, Cl md TInf, N, tau ss TInf, tau sd - oral1_1cpt_kavk 1st order 1 1st order ka, V, k md N, tau ss tau sd - oral1_1cpt_kavcl 1st order 1 1st order ka, V, Cl md N, tau ss tau sd - bolus_2cpt_vkk12k21 IV-bolus 2 1st order V, k, k12, k21 md N, tau ss tau sd - bolus_2cpt_clv1qv2 IV-bolus 2 1st order Cl, V1, Q, V2 md N, tau ss tau sd TInf infusion_2cpt_vkk12k21 IV-infusion 2 1st order V, k, k12, k21 md TInf, N, tau ss TInf, tau 6

90 infusion_2cpt_clv1qv2 IV-infusion 2 1st order Cl, V1, Q, V2 oral1_2cpt_kavkk12k21 1st order 2 1st order ka, V, k, k12, k21 oral1_2cpt_kaclv1qv2 1st order 2 1st order ka, Cl, V1, Q, V2 bolus_3cpt_vkk12k21k13k31 IV-bolus 3 1st order V, k, k12, k21, k13, k31 bolus_3cpt_clv1q1v2q2v3 IV-bolus 3 1st order Cl, V1, Q1, V2, Q2, V3 infusion_3cpt_vkk12k21k13k31 IV-infusion 3 1st order V, k, k12, k21, k13, k31 infusion_3cpt_clv1q1v2q2v3 IV-infusion 3 1st order Cl, V1, Q1, V2, Q2, V3 oral1_3cpt_kavkk12k21k13k31 1st order 3 1st order ka, V, k, k12, k21, k13, k31 oral1_3cpt_kaclv1q1v2q2v3 1st order 3 1st order ka, Cl, V1, Q1, V2, Q2, V3 sd TInf md TInf, N, tau ss TInf, tau sd - md N, tau ss tau sd - md N, tau ss tau sd - md N, tau ss tau sd - md N, tau ss tau sd TInf md TInf, N, tau ss TInf, tau sd TInf md TInf, N, tau ss TInf, tau sd - md N, tau ss tau sd - md N, tau ss tau 7

91 Pharmacokinetic models with a Mickaelis-Menten elimination One, two and three-compartment models are implemented for the three types of input. For bolus input and Mickaelis-Menten elimination, only single dose models are implemented. For infusion and first order absorption input, single dose and multiple dose are implemented. There is no steady-stae conditions for PK models with Mickaelis-Menten elimination. The list of these PK models is given in Table 2. For models with a bolus input, the dose has to be specified in the input file (stdin.r by default) as the initial condition of the differential equation system. For models with infusion or first order absorption input, dose has to be specified as an argument and NOT IN THE INITIAL CONDITION OF THE MODEL IN THE INPUT FILE. 8

92 Table 2. Pharmacokinetic models with Mickaelis-Menten elimination included in the library of models Name Input Cpt Elimination Parameterisation Administration Needed Variable(s) bolus_1cpt_vvmkm IV-bolus 1 Mickaelis-Menten V, Vm, km sd - infusion_1cpt_vvmkm IV-infusion 1 Mickaelis-Menten V, Vm, km sd dose,tinf md dose,tinf, tau oral1_1cpt_kavvmkm 1st order 1 Mickaelis-Menten ka, V,Vm, km sd dose md dose,tau bolus_2cpt_vk12k21vmkm IV-bolus 2 Mickaelis-Menten V, k12, k21, Vm, km sd - bolus_2cpt_v1qv2vmkm IV-bolus 2 Mickaelis-Menten V1, Q, V2, Vm, km sd - infusion_2cpt_vk12k21vmkm IV-infusion 2 Mickaelis-Menten V, k12, k21, Vm, sd dose,tinf km md dose,tinf, tau infusion_2cpt_ V1QV2Vmkm IV-infusion 2 Mickaelis-Menten V1, Q, V2, Vm, sd dose,tinf km md dose,tinf, tau oral1_2cpt_kavk12k21vmkm 1st order 2 Mickaelis-Menten ka, V, k12, k21, sd dose Vm, km md dose, tau oral1_2cpt_kav1qv2vmkm 1st order 2 Mickaelis-Menten ka, V1, Q, V2, sd dose Vm, km md dose, tau bolus_3cpt_vk12k21k31k13vmkm IV-bolus 3 Mickaelis-Menten V, k12, k21, k13, k31, Vm, km sd - bolus_3cpt_ V1Q1V2Q2V3Vmkm IV-bolus 3 Mickaelis-Menten V1, Q1, V2, Q2, V3, Vm, km sd - infusion_3cpt_vk12k21k13k31vmkm IV-infusion 3 Mickaelis-Menten V, k12, k21, sd dose,tinf k13, k31, km, Vm md dose,tinf, tau infusion_3cpt_v1q1v2q2v3vmkm IV-infusion 3 Mickaelis-Menten V1, Q1, V2, Q2, sd dose,tinf V3, Vm, km md dose,tinf, tau oral1_3cpt_kak12k21k13k31vmkm 1st order 3 Mickaelis-Menten ka, k12, k21, sd dose k13, k31, Vm, km md dose,tau oral1_3cpt_kav1q1v2q2v3vmkm 1st order 3 Mickaelis-Menten ka, V1, Q1, V2, sd dose Q2, V3, Vm, km md dose, tau 9

93 Library of pharmacodynamic models The library of PD models supports immediate response models (alone or linked to a pharmacokinetic model) and turnover response models (linked to pharmacokinetic models). The tables presenting these models return every information in order to use the model function chosen: - a name - the parameters used (parameterisation) Examples of the use of the library of pharmacodynamic model are presented in section Immediate response pharmacodynamic models alone Linear, quadratic, logarithmic, Emax, sigmoid Emax, Imax, sigmoid Imax models with null or constant baseline are available. The list of these models is given in Table 3. These models are written with an analytical form and have to be used in the case of a model with one response (PD evaluation or optimization). They are implemented in the file librarypd.pddesign.r. Thus, the user has to specify the path of this file in the model file to use this library of models: source (paste(directory.program,dirsep, librarypd.pddesign.r, sep= ) For these models, the design variables are the concentrations or the doses instead of the sampling times. For example, if one uses a linear drug action model with a constant baseline (immed_lin_const) from the library, the model uses two parameters (Alin, S0). 10

94 Table 3. Immediate response pharmacodynamic models included in the PD library for PD alone and for PK/PD model Drug action models Null baseline Baseline Constant baseline Name Parameterisation Name Parameterisation Linear immed_lin_null Alin immed_lin_const Alin, S0 Quadratic immed_quad_null Alin, Aquad immed_quad_const Alin, Aquad, S0 Logarithmic immed_log_null Alog immed_log_const Alog, S0 Emax immed_emax_null Emax immed_emax_const Emax, S0 Sigmoid Emax immed_gammaemax_null Emax, gamma immed_gammaemax_const Emax, gamma, S0 Imax immed_imax_null Imax immed_imax_const Imax, S0 Sigmoid Imax immed_gammaimax_null Imax, gamma immed_gammaimax_const Imax, gamma, S0 11

95 Pharmacodynamic models linked to pharmacokinetic model In this section, we deal with a two response model, with one response for the PK and the other one for the PD. We thus optimise sampling times for both responses using a PK/PD model. Using the libraries of models, we have four cases to compose the PK/PD model according to the writing of each response model: either with a differential equation system (ODE) or with an analytical expression (AF). Therefore, there are four cases of PK/PD models: 1. PK model with linear elimination (AF) and immediate response PD model (AF) 2. PK model with linear elimination (AF) and turnover response PD model (ODE) 3. PK model with Mickaelis-Menten elimination (ODE) and immediate response PD model (AF) 4. PK model with Mickaelis-Menten elimination turnover (ODE) response PD model (ODE) To use PFIM for design evaluation and optimisation for a PK/PD model, it is necessary to have a PK response and a PD response implemented with a similar form. In the first case, immediate response pharmacodynamic models are written with an analytical form in the file librarypd.pkpddesign.r and thus they can be associated to pharmacokinetic models with first order linear elimination (Table 1) implemented in the file librarypk.r, which are also written with analytical expressions. In these PD functions, the expression of the PK model is given as an argument. In this case, the user has to fill in the stdin.r using analytical form options and to specify the paths of the library files in model.r: source(paste(directory.program,dirsep, librarypk.r,sep= ) source(paste(directory.program,dirsep, librarypd.pkpddesign.r,sep= ) However, for the three other cases, the PK response and the PD response are writing either with different forms or both with a differential equation system (Case 4). That is why, the user has to call a specific function in order to create a system of differential equations describing the corresponding PK/PD model. This function named Create_formED() is implemented in the file CreateModel_PKPDdesign.r and has to be used in the model file as follows: source (paste(directory.program,dirsep, librarypk.r, sep= ) create_formed(fun_pk,fun_pd,dose=na,tau=na,tinf=na) where - fun_pk and fun_pd: the names of the PK and PD models, respectively - dose: value of the dose only for a PK model with infusion or oral input (by default: NA) - tau: dosing interval to specify only for multiple dose conditions (by default: NA) - TInf: time of infusion to specify only for PK model with infusion input (by default: NA) Using this function, a new file named model_created.r is created in the directory currently used. This new file contains the complete writing of the differential equation system describing the corresponding PK/PD model 12

96 created by the function Create_formED(). This file is only created for the user and not used in the PFIM program. For these cases, the user has thus to fill in the sdtin.r using differential equation options. The list of the immediate response PD models is thus given in Table 3 plus those of Table 4. The list of the turnover response PD models is given in Table 5. For the second case where a PK model with linear elimination is associated to a turnover PD response model, the PK model is written with a differential equations system. Consequently, only some PK models from the Table 1 are implemented: - for bolus input, only single dose models; - for infusion input, single dose and multiple dose - for first order absorption input, single dose and multiple dose 13

97 Table 4. *Immediate response pharmacodynamic models linked to a pharmacokinetic model included in the library Drug action models Baseline/disease models Linear progression Exponential increase Exponential decrease Name Param. Name Param. Name Param. Linear immed_lin_lin Alin, S0, kprog immed_lin_exp Alin, S0, kprog immed_lin_dexp Alin, S0, kprog Quadratic immed_quad_lin Alin, Aquad, S0, kprog immed_quad_exp Alin, Aquad, S0, kprog immed_quad_dexp Alin, Aquad, S0, kprog Logarithmic immed_log_lin Alog, S0, kprog immed_log_exp Alog, S0, kprog immed_log_dexp Alog, S0, kprog Emax immed_emax_lin Emax, S0, kprog immed_emax_exp Emax, S0, kprog immed_emax_dexp Emax, S0, kprog Sigmoid Emax immed_gammaemax_lin Emax, gamma, S0, kprog immed_gammaemax_exp Emax, gamma, S0, kprog immed_gammaemax_dexp Emax, gamma, S0, kprog Imax immed_imax_lin Imax, S0, kprog immed_imax_exp Imax, S0, kprog immed_imax_dexp Imax, S0, kprog Sigmoid Imax immed_gammaimax_lin Imax, gamma, S0, kprog immed_gammaimax_exp Imax, gamma, S0, kprog immed_gammaimax_dexp Imax, gamma, S0, kprog * In addition to those in Table 3. 14

98 Table 5. Turnover response pharmacodynamics models linked to a pharmacokinetic model included in the library Types Models with impact on the of response Input Output a Full Imax means Imax is fixed equal to 1 Name Parameterisation Name Parameterisation Emax turn_input_emax Rin,kout,Emax,C50 turn_output_emax Rin,kout,Emax,C50 Sigmoid Emax turn_input_gammaemax Rin,kout,Emax,C50,gamma turn_output_gammaemax Rin,kout,Emax,C50,gamma Imax turn_input_imax Rin,kout,Imax,C50 turn_output_imax Rin,kout,Imax,C50 Sigmoid Imax turn_input_gammaimax Rin,kout,Imax,C50,gamma turn_output_gammaimax Rin,kout,Imax,C50,gamma Full Imax a turn_input_imaxfull Rin,kout,C50 turn_output_imaxfull Rin,kout,C50 Sigmoid full Imax a turn_input_gammaimaxfull Rin,kout,C50,gamma turn_output_gammaimaxfull Rin,kout,C50,gamma 15

99 1.2. Full expression of the Fisher information matrix The population Fisher information matrix (, ) M Ψ ξ for multiple response models, for an individual with an elementary design ξ for the vector of population parameters Ψ, is given as: M F 1 A( E, V ) C( E, V ) Ψ ξ T 2 C ( E, V ) B( E, V ) (, ) with E and V the approximated marginal expectation and the variance of the observations of the individual. The vector of population parameter Ψ is Ψ ' = µ ', λ ' with µ the p-vector of the fixed effects and λ the defined by ( ) vector of the variance terms. M F are given in [1]) with: F is given as a block matrix (more details E E V V T ( A( E, V )) ml = 2 V + tr( V V ) µ m µ l µ l µ m with m and l = 1,, p V V 1 1 ( B( E, V )) ml = tr( V V ) λm λl with m and l = 1,,dim ( λ ) V V C E V tr V V 1 1 ( (, )) ml = ( ) λ l µ m with = 1,,dim ( ) l λ and m = 1,, p In the previous versions of PFIM, the dependence of V in µ was neglected so V that = 0. Then, the population Fisher information matrix is approximated µ a block diagonal matrix that is to say the block C of the matrix was supposed to be 0 (see details in [1]). Also, the block A is simplified and is expressed as: E E ( A( E, V )) 2 µ µ T 1 ml = V m l with m and l = 1,, p In the present version, the user can now choose if a full or block diagonal information matrix is needed. However, this implementation is not developed yet for models with covariates and / or inter-occasion variability Inter-occasion variability (IOV) specification The expression of the population Fisher information matrix has been extended for model including additional random effects for inter-occasion variability (or within subject variability). The individual parameters of an individual i at occasion h are thus expressed by the following relation, which can be additive as or exponential as θ ih = µ + bi + κih θ = µ ( + κ ) 16

100 where µ is p vector of fixed effects, bi the vector of random effects associated to the individual i and the random effect associated to the κih individual i for occasion h (h=1,,h) with H the total of occasions. The and κ are independent. It is assumed that b N ( 0, Ω) and κ N ( 0, Γ) i ih with Ω and Γ are defined as diagonal matrices of size p x p. Each element ω Ω and γ of Γ represent respectively the interindividual variability of the j th component of bi and the inter-occasion variability of the j th component of κ. The size of the block C and the block B of the expression of the Fisher information matrix are thus modified to incorporate the elements of Γ. This new development was performed for any number of occasions H, it is implemented in PFIM 3.2 for the case where the same elementary designs are used at each occasion. The user can include inter-occasion variability in the model as well as covariates Discrete covariate specification The present expression of the Fisher information matrix accommodates models with parameters quantifying the influence of discrete covariates. Two or more categories can be included. In PFIM 3.2, it can be assumed either that covariates are additive on parameter if the random effect model is additive, or that covariates are additive on log parameters if the random effect model is exponential. For instance, the individual parameter i is described as the function of a discrete covariate C i, which takes K values defining K categories, with additive effect model: of K i k C i = k bi k = 2 θ = µ + β + where here with k=1 is defined as the reference group where β 1 = 0 For each covariate, the user has to specify β, the vector of covariate effect coefficients and the proportions of subjects associated to the K categories. However, it can be specified if covariates change or not through the different occasions. In the latter case, additional objects are needed: the vector of sequences of values of each covariate at each occasion and the vector of proportions of the elementary designs corresponding to each sequence of covariate values. The expected Fisher information matrix is computed for each covariate. The number of covariates, the number of parameters associated to each covariate as well as the number of categories for each covariate, are not limited. But in this version of PFIM, the distribution of the covariates are supposed independent. 17

101 1.5. Computation of power and number of subjects needed to treat Comparison test Computation of the expected power The Wald test can be used to asses the difference of a covariate effect. In PFIM, the Wald test is performed on the of each category for each covariate, a global Wald test on the vector (all effect coefficients) is not implemented. For one covariate and an effect of one category (K=2), the null hypothesis is H 0 : { =0} while the alternative hypothesis is H 1 : { 0}. The statistic of the Wald test is defined as, S W β = SE( β) with β the covariate effect estimates and SE( β) its associated standard error. Under H 1, when = 1, we then compute the power of the Wald test defined as: β 1 β 1 Pdiff = 1 φ z1 α 2 + φ z1 α 2 SE( β1) SE( β1) (1) where is the cumulative distribution function of the standard normal distribution and 2 z α is such that φ ( α ) z 2 = 1 α 2. Using the covariate effect β1 fixed by the user, the corresponding standard error SE( β1) is predicted by PFIM 3.2 for a given design and the values of population parameters. Computation of the number of subjects needed The number of subjects needed to achieve a power P to detect a covariate effect using the Wald test is also computed. First, from the equation (1), we compute the SE needed on to obtain a power of P, called NSE(P), using the following relation: NSE( P) = z α 2 β φ 1 1 ( 1 P) (2) Last, we compute the number of subjects needed to be included to obtained a power of P, called NNI(P) using ( β ) 1 NNI ( P) N (3) NSE P = SE ( ) where N is the initial number of subjects in the given design and SE ( β 1 ) corresponding predicted SE of for the given design. the 18

102 Equivalence test Computation of the expected power The Wald test can be used to asses the equivalence of a covariate effect. In PFIM, the Wald test is performed on the of each category for each covariate, a global Wald test on the vector (all effect coefficients) is not implemented. For one covariate and an effect of one category (K=2), the null hypothesis is H 0 : { - L or + L while the alternative hypothesis is H 1 : {- L + L }. H 0 is composed of two unilateral hypothesis H :{ - L } et H 0, + L :{ + L }. Equivalence between two covariate effects can be concluded if and only if the two hypothesis H0, L and H 0, + L are rejected. The two statistics of the unilateral Wald test under the null hypothesis ^ are defined as, β+ L β L S W = and S L W = with β + L SE( β) SE( β) the covariate effect estimates and, its associated standard error. Under H 1, when = 1 with 1 [- L, L ], we then compute the power of the Wald test defined as: ^ 0, L β 1 + L Pequi = 1 φ z1 α if β1 L,0 SE( β1) [ ] β1 L Pequi = φ z1 α if β1 + SE( β1) [ 0, ] L (4) (5) where is the cumulative distribution function of the standard normal distribution and z α is such that φ ( z α ) = 1 α. In equivalence test 1 is usually chosen to be zero. Using the covariate effect β1 fixed by the user, the corresponding standard error SE( β1) is predicted by PFIM 3.2 for a given design and the values of population parameters. Computation of the number of subjects needed The number of subjects needed to achieve a power P to show equivalence between two covariate effects using the Wald test is also computed. First, from equations (4) and (5), we compute the SE needed on to obtain a power of P, called NSE(P), using the following relation: ( ) NSE P ( ) NSE P ( β1 L ) ( 1 P) [ ] = β z + φ α ( β 1 + L ) z + φ ( P) if 1 1 L,0 if 1 1 0, [ ] = β + α L (6) (7) where is the cumulative distribution function of the standard normal distribution and z α is such that φ ( z α ) = 1 α. 19

103 Last, we compute the number of subjects needed to be included to obtained a power of P, called NNI(P) using the equation (3) like for comparison test References [1]. Bazzoli C, Retout S, Mentré F. Fisher information matrix for nonlinear mixed effects multiple response models: evaluation of the first order linearization using a pharmacokinetic/pharmacodynamics model. Statistics in Medicine (14): [2]. Retout S, Mentré F. Further developments of the Fisher information matrix in nonlinear mixed effects models with evaluation in population pharmacokinetics. Journal of Biopharmaceutical Statistics (2) : [3]. Retout S, Comets E, Samson A, Mentré F. Design in nonlinear mixed effects models: Optimization using the Federov-Wynn algorithm and power of the Wald test for binary covariates. Statistics in Medicine (28) : [4]. Nguyen TT, Bazzoli C, Mentré F. Design evaluation and optimization in crossover pharmacokinetic studies analysed by nonlinear mixed effect models. Application to bioequivalence or interaction trials. American Conference on Pharmacometrics. 2009, Mystic, United-States. [5]. Bertrand J, Mentré F. Mathematical Expressions of the Pharmacokinetic and Pharmacodynamic Models implemented in the MONOLIX software (

104 2. Use 2.1. Working directory - Create a working directory, for example: U:\\My Documents\\PFIM 3.2_examples\\Example1 - Copy the files PFIM3.2.r, Stdin.r and model.r in this directory - In the file PFIM3.2.r, specify your working directory: directory<- U:\\My Documents\\PFIM 3.2_examples\\Example1 - Then, specify your program directory i.e. where is the folder called Program directory.program<- U:\\My Documents\\PFIM 3.2\\Program - Save the file PFIM3.2.r 2.2. Model writting Compared to PFIM 3.0, there is no change in the way to write a model using an analytical form or a differential equation system for single response multiple response models using the user defined option. The main change is to write a PK/PD model using the libraries of models. Thus, several examples are presented below with the different ways of writing models Example 1: Single response with a PK or a PD model 1. PK model using an analytical form with the library of models In this illustration, the user creates a one response model using the model function implemented in the pharmacokinetic library (Oral1_1cpt_kaVCl) describing a one compartment oral absorption after a multiple dose administration (md). N and tau are the needed variables and thus, they have to be specified by the user in the function model. Here, we have five oral administration doses with an interval between two doses equal to twelve hours. source(paste(directory.program,dirsep,"librarypk.r",sep="")) forma<-oral1_1cpt_kavcl_md(n=5,tau=12)[[1]] form<-c(forma) 2. PK model using a differential equation form with the library of models In this illustration, the user creates a one response model using the model function implemented in the pharmacokinetic library (bolus_1cpt_vvmkm) describing a one compartment bolus input with Mickaelis-Menten elimination after a single dose administration (sd). The dose is specified in a part of the R-script file stdin.r: time.condinit<-0 condinit<-expression(c(100)) # dose=100 source(paste(directory.program,dirsep,"librarypk.r",sep="")) formed<-bolus_1cpt_vvmkm() 21

105 3. PK model using a differential equation form with the library of modelsin this illustration, the user creates a one response model using the model function implemented in the pharmacokinetic library (infusion_1cpt_vvmkm) describing a one compartment infusion input with Mickaelis-Menten elimination after a single dose administration (sd). The dose is specified as an argument of the PK function in the file model.r, not in the initial condition described in a part of the R-script file stdin.r: time.condinit<-0 condinit<-expression(c(0)) source(paste(directory.program,dirsep,"librarypk.r",sep="")) formed<-infusion_1cpt_vvmkm(dose=100, Tinf=1) 4. PD model using an analytical form with the library of models In this illustration, the user creates a one response model using the model function implemented in the library (immed_lin_null) describing an immediate response model with a linear drug action and without baseline. source(paste(directory.program,dirsep,"librarypd_pddesign.r",sep="")) forma<-immed_lin_null()[[1]] form<-c(forma) Example 2: Two responses defined by a PK/PD model 5. PK model with a linear elimination + immediate response PD model In this illustration the user creates for the PK model, a one-compartment model with bolus input and first order elimination for a single dose, and for the PD model, an immediate response model with a linear drug action and no baseline is used. As shown in the example, the PK model is given as an argument of the PD model. Thus, in the PD model the drug concentration corresponds to the expression of the PK model. source(paste(directory.program,dirsep,"librarypk.r",sep="")) source(paste(directory.program,dirsep,"librarypd_pkpddesign.r",sep="")) forma<-bolus_1cpt_vk()[[1]] formb<-immedpd_lin_null(forma)[[1]] form<-c(forma, formb) 6. PK model with a linear elimination + turnover response PD model In this example, the user creates a PK/PD model with a one compartment bolus input for the PK and a turnover response model with an inhibition on the input for the PD, using the function create_formed. source(paste(directory.program,dirsep,"createmodel_pkpd.r",sep="")) create_formed(bolus_1cpt_vk,turn_input_imax) 7. PK model with a Mickaelis-Menten elimination + immediate response PD model In this example, the user creates a PK/PD model with a one compartment infusion input with Mickaelis-Menten elimination for the PK and an immediate response model with a linear drug action and no baseline for the PD, using the function create_formed. In this case, the user needs to specify the dose (here equal to 100) and the duration of infusion (here equal to 1 hour). source(paste(directory.program,dirsep,"createmodel_pkpd.r",sep="")) create_formed(infusion_1cpt_vvmkm,immedpd_lin_null,dose=100,tinf=1) 22

106 2.3. General objects required for Evaluation and Optimisation According to the new features in PFIM 3.2, only some objects have been added or modified in the input file name by default: stdin.r Full or block diagonal fisher information matrix The following object has been added: - option: integer indicating expression of the information matrix: - 1 for block diagonal - 2 for full Graph option This list of objects has been modified in the version PFIM 3.2. It allows to draw a graph with the evaluated design (evaluation step) or the optimised design (optimisation step). Compared to the version 3.0 of PFIM, the object names.data has to be replaced by the two followings objects: - names.datax: vector of character string for the names of X axis for each graph that corresponds to each type of measurement (length of this vector must equal to the number of responses). - names.datay: vector of character string for the names of Y axis for each graph that corresponds to each type of measurement (length of this vector must equal to the number of responses) Objects required only for IOV option The following list of objects is associated to the specification of the inter-occasion variability in the model. - n_occ: integer indicating the number of occasions. Example: n_occ=2 - gamma: vector of the p variances of the random effects for inter-occasion variability Objects required only for covariate option This list of objects allows to specify the inclusion of covariate effects on some parameters of the model. In the stdin.r, it appears just before the object required for the optimisation. The user can now include in the model covariates that do not change with occasion and/or covariates that change with occasion Covariates not changing with occasion If the user wants to deal with covariates which do not change with occasion, he has to specify the following object. 23

107 - covariate.model: logical value; if T, covariates are added to the model If the user has filled in by T the previous object, he has to specify the following objects: - covariate.name: list of character string indicating the name of the covariate(s) Example: covariate.name<-list(c( Gender )) - covariate.category: list of vectors of categories. Each vector is associated to one covariate and defined its corresponding categories. They can be written as character string or integer. Example: covariate.category<-list(gender=c( F, M )) - covariate.proportions: list of vectors of proportions. Each vector is associated to one covariate and defines the corresponding proportions of subjects involved in each corresponding categories. Example: covariate.proportions<-list(gender=c(0.5,0.5)) - parameter.associated: list of vectors of parameter(s) associated with each covariate. Each vector is associated to one covariate and is defined by the corresponding parameters on which is added the covariate. Example: parameter.associated<-list(gender=c(cl, V)) Name of the parameter(s) has to be identical to those entered in the object parameters. - beta.covariate: list of the values of parameters for all other categories than the reference category (for which beta=0. Example: beta.covariate<-list(gender=list(c(0.5,0.6))) Covariates changing with occasion If the user wants to deal with covariates which change with occasion, he has to specify the following object. - covariate_occ.model: logical value; if T, covariates changing with occasion are added to the model If the user has filled in by T the previous object, he has to specify the following objects: - covariate_occ.name: list of character string indicating the name of the covariate(s) Example: covariate_occ.name<-list(c( Treat )) - covariate_occ.category: list of vectors of categories. Each vector is associated to one covariate and defined its corresponding categories. They can be written as character string or integer. Example: 24

108 covariate_occ.category<-list(treat=c( A, B )) - covariate_occ.sequence: list of vectors of sequences. Each vector is associated to one sequence of values of covariates at each occasion. The size of each sequence has to be equal to the number of occasions (n_occ)for each covariate. Example: covariate_occ.sequence<list(treat=list(c( A, B ),c( B, A )) - covariate_occ.proportions: list of vectors of proportions. Each vector is associated to one covariate and defines the proportions of elementary designs corresponding to each sequence of covariate values. The size of each vector has to be equal to the number of sequences. Example: covariate_occ.proportions<list(treat=list(0.5,0.5)) - parameter_occ.associated: list of vectors of parameter(s) associated with each covariate. Each vector is associated to one covariate and is defined by the corresponding parameters on which is added the covariate. Example: parameter_occ.associated<-list(treat=c(cl)) Name of the parameter(s) has to be identical to those entered in the object parameters. - beta.covariate_occ: list of the values of parameters for all other categories than the reference category for which beta=0. Example: beta.covariate_occ<list(treat=list(c(log(1.1))) 2.6. Objects required only for computation of power and number of subjects needed for comparison test or equivalence test To compute the expected power to detect covariate effects as to compute the number of subjects needed to achieve a given power, the previous object covariate.model has to be filling in by T. Additional R objects are required to be created. The following object is needed for both options - alpha: the value of the type one error for the Wald test.example: alpha<-0.05 It is possible to compute either the expected power only or the number of subjects needed for a given power or both of them together. 25

109 - compute.power logical value, if T the expected power for comparison test is computed for each covariate.example: compute.power<-t - compute.nni logical value, if T the number of subjects needed for a given power for comparison test is computed for each covariate.example: compute.nni<-t - interval_eq vector of equivalence interval. Example: interval_eq<-c(log(0.8),log(1.25)) - compute.power_eq logical value, if T the expected power for equivalence test is computed for each covariate. Example: compute.power_eq<-t - compute.nni_eq logical value, if T the number of subjects needed for a given power for equivalence test is computed for each covariate.example: compute.nni_eq<-t - given.power the value of the given power.example: given.power< Results The results are written in the output file called by default stdout.r. This file is different when only evaluation is performed or when optimisation is performed. Compared to the one computed in PFIM 3.0, the file is only modified when inter-occasion variability and/or covariate options are added. It is only detailed below for evaluation but it is similar for optimisation. Figure 1 represents the output file from the design evaluation of a model with covariates effect and inter-occasion variability. The user can read on the Figure 1: 1 2 The name of the function used: PFIM 3.2. The name of the project and the date. 3 A summary of the input: model(s), sampling times in the elementary designs for each model(s), doses or initial conditions and subjects corresponding to those designs, residual variance error model for each model(s), random effect model, errors tolerances for the solver of differential equations system if used, a summary for inter-occasion variability, a summary of the covariate model 3a The number of occasions 26

110 3b Each covariate not changing with occasion, its (their) parameters associated, categories with their name and their corresponding proportions of subjects. 3c Each covariate changing with occasion, its (their) parameters associated, categories with their name and their corresponding proportions of subjects. The list of the sequence of values of categories for each occasion and for each covariate. 3d If the evaluation has been performed using the full or the block diagonal expression of the Fisher information matrix 4 The population Fisher information matrix, a dim*dim symmetric matrix where dim is the total number of population parameters to be estimated. 5 The value of each population parameter with the expected standard error on each parameter and the corresponding coefficient of variation. 5a The value of the variance of the random effects for interoccasion variability with the expected standard error on each parameter and the corresponding coefficient of variation. 6 The value of the determinant of the Fisher information matrix and the value of the criterion (determinant^(1/dim)) where dim is the total number of population parameters. 7 Results from the comparison test: the value and the exponential of the value of each covariate parameter with the corresponding 95% confidence interval of the parameter, the predicted value of the power of the Wald test and the number of subject needed to detect this covariate effect with the given type one error and the given power. 8 Results from the equivalence test: the value and the exponential of the value of each covariate parameter with the corresponding 90% confidence interval of the parameter, the power of the Wald test and the number of subject needed to achieve the given power for this covariate effect with the given type one error and the given interval of equivalence. 27

111 Figure 1. Example of design evaluation output file with covariate effect and interoccasion variability 1 2 3a 3b 3 3c 3d 28

112 4 29

113 5 5a 6 7 (given power=0.9) 8 (given power=0.9) 30

114 Moreover, the PFIM() function returns the following R objects: - Dose - prot: design evaluated for each response - subjects: number of subjects for each group - mfisher: the population Fisher information matrix - determinant: the determinant of the population Fisher information matrix - crit: the value of the criterion - p: the vector of the fixed effect parameters - se: the vector of the expected standard errors for each parameter - cv: the corresponding coefficients of variation, expressed in percent. - summary.exp.power: a matrix with each row corresponding to each covariate, the name of the covariates, the associated effect parameter, the 95% confidence interval, and the predicted power as columns - summary.nni: a matrix with each row corresponding to each covariate, the name of the covariate, the associated effect parameter, the 95% confidence interval and the number of subjects needed as column 31

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118 cpk CP / Review Article Bazzoli et al. Intracellular PK of Antiretrovirals Intracellular Pharmacokinetics of Antiretroviral Drugs in HIV-Infected Patients, and their Correlation with Drug Action Caroline Bazzoli, 1,2 Vincent Jullien, 3,4,5 Clotilde Le Tiec, 6 Elisabeth Rey, 3,4,5 France Mentré 1,2 and Anne-Marie Taburet 6 1 INSERM U738, Paris, France 2 Faculté de Médecine, Université Paris Diderot, Paris, France 3 Université Paris Descartes, Paris, France 4 Pharmacologie Clinique, Groupe Hospitalier Cochin Saint-Vincent de Paul, Assistance Publique Hôpitaux de Paris, Paris, France 5 INSERM U663, Paris, France 6 Pharmacie Clinique, Hôpital Bicêtre, Assistance Publique Hôpitaux de Paris, Paris, France Correspondence: Professor France Mentré, UMR 738 INSERM Université Paris Diderot, 16 rue Henri Huchard, Paris, France. [email protected]

119 Abstract In patients infected by HIV, the efficacy of highly active antiretroviral therapy through the blockade of different steps of the retrovirus life cycle is now well established. As HIV is a retrovirus that replicates within the cells of the immune system, intracellular drug concentrations are important to determine antiretroviral drug (ARV) efficacy and toxicity. Indeed, nucleoside reverse transcriptase inhibitors (NRTIs), non-nrtis (NNRTIs), newly available integrase inhibitors and protease inhibitors (PIs) act on intracellular targets. NRTIs are prodrugs that require intracellular anabolic phosphorylation to be converted into their active form of triphosphorylated NRTI metabolites, most of which have longer plasma half-lives than their parent compounds. The activity of intracellular kinases and the expression of uptake transporters, which may depend on cell functionality or their activation state, may greatly influence intracellular concentrations of triphosphorylated NRTI metabolites. In contrast, NNRTIs and PIs are not prodrugs, and they exert their activity by inhibiting enzyme targets directly. All PIs are substrates of cytochrome P450 3A, which explains why most of them display poor pharmacokinetic properties with intensive presystemic first-pass metabolism and short elimination half-lives. There is evidence that intracellular concentrations of PIs depend on P- glycoprotein and/or the activity of other efflux transporters, which is modulated by genetic polymorphism and coadministration of drugs with inhibiting or inducing properties. Adequate assay of the intracellular concentrations of ARVs is still a major technical challenge, together with the isolation and counting of peripheral blood mononuclear cells (PBMCs). Furthermore, intracellular drug could be bound to cell membranes or proteins; the amount of intracellular ARV available for ARV effectiveness is never measured, which is a limitation of all published studies. In this review, we summarize the findings of 31 studies that provided results of intracellular concentrations of ARVs in HIV-infected patients. Most studies also measured plasma concentrations, but few of them studied the relationship between plasma and intracellular concentrations. For NRTIs, most studies could not establish a significant relationship between plasma and triphosphate concentrations. Only eight published studies reported an analysis of the relationships between intracellular concentrations and the virological or immunological efficacy of ARVs in HIV patients. In prospective studies that were well designed and had a reasonable number of patients, virological efficacy was found to correlate significantly with intracellular concentrations of NRTIs but not with plasma concentrations. For PIs, the only prospectively designed trial of lopinavir found that virological efficacy was influenced by both trough plasma concentrations and intracellular concentrations. ARVs are known to cause important adverse effects through interference with cellular endogenous processes. The relationship between

120 intracellular concentrations of ARVs and their related toxicity was investigated in only four studies. For zidovudine, the relative strength of the association between a decrease in haemoglobin levels and plasma zidovudine concentrations, as compared with intracellular zidovudine triphosphate concentrations, is still unknown. Similarly, for efavirenz and neuropsychological disorders, methodological differences confound the comparison between studies. In conclusion, intracellular concentrations of ARVs play a major role in their efficacy and toxicity, and are influenced by numerous factors. However, the number of published clinical studies in that area is limited; most studies have been small and not always adequately designed. In addition, standardization of assays and PBMC counts are warranted. Larger and prospectively designed clinical studies are needed to further investigate the links between intracellular concentrations of ARVs and clinical endpoints.

121 <ARTICLE.PART> HIV is a retrovirus that replicates within the cells of the immune system. The efficacy of highly active antiretroviral therapy (HAART) is now well established and has provided extraordinary benefits to many patients with HIV infection. [1] The morbidity and mortality related to HIV infection have dramatically decreased in countries in which HAART has been made available, turning HIV infection into a manageable chronic disease. [2] Lifelong antiretroviral drug (ARV) treatment seems necessary, as viral replication and loss of CD4 cells resume when HAART is interrupted. HAART regimens have shown some limitations, the major one being the failure to eradicate HIV even after several years of therapy. One of the reasons is that despite potent ARV treatment, compartments of replication-competent virus persist, suggesting that ARVs do not reach all of the infected cells: however, there are no data to support this theoretical assumption. This review focuses on pharmacological principles that govern intracellular concentrations of ARVs and on clinical studies that have aimed to assess whether intracellular concentrations of ARVs could be a useful parameter to predict the efficacy or toxicity of an ARV regimen. The different steps of HIV replication are now well identified and understood. A number of ARVs are now available and are grouped into five pharmacological classes according to their mechanism of action. These drugs target essential receptors or enzymes at different steps of the life cycle of the virus and block the production of infectious retroviral particles from the cell. [3] However, virus eradication cannot be achieved with the available treatments because of the pool of latently infected CD4 cells. [4] HAART is the standard of care to avoid selection of viral mutations. Selection of drugs for treatment-naïve and treatment-experienced patients take into account the risk/benefit ratio and the viral genotype. Current guidelines recommend that treatment-naïve patients receive a combination of a ritonavir-boosted protease inhibitor (PI) or a non-nucleoside reverse transcriptase inhibitor (NNRTI) plus two nucleoside reverse transcriptase inhibitors (NRTIs), and that treatment-experienced patients receive a combination of at least two active ARV drugs from different classes, based on the viral genotype. [5-7] Besides entry inhibitors, which act on receptors located on the cell surface, most ARVs inhibit viral replication inside the cell, therefore intracellular concentration of ARVs should be a reliable parameter to consider when relating pharmacokinetics and efficacy. Results from several in vitro studies also exist in this area. However, because of the difficulties of extrapolating the results from in vitro to in vivo studies, this review focuses only on in vivo studies and summarizes clinical trials in which intracellular concentrations were measured and related to plasma concentrations, virological efficacy or toxicity. First, however, to understand the limitations of

122 such studies, the following topics are presented and discussed: clinical pharmacokinetics of ARVs, intracellular drug assays, and mechanisms influencing intracellular diffusion and accumulation. 1. Clinical Pharmacokinetics of ARVs The pharmacokinetic parameters of ARVs are summarized in table I. [6,8-13] Table I 1.1 Entry Inhibitors Entry inhibitors block virus attachment to receptors of the cell surface. They have an extracellular mode of action and therefore differ from other available classes of ARVs. Two drugs of this class are available, enfuvirtide and maraviroc; considering their mechanism of action, they are beyond the scope of this review, but to provide an exhaustive overview of ARVs, their pharmacological properties are briefly summarized below CCR5 Inhibitors CCR5 or CXCR4 chemokine co-receptor antagonists are promising entry inhibitors. Maraviroc is the first approved drug of this new class. Maraviroc inhibits the CCR5 chemokine co-receptor, preventing HIV from binding to the cell membrane. Its pharmacokinetic characteristics have been summarized elsewhere. [14] In brief, maraviroc is a cytochrome P450 (CYP) 3A substrate, and its dosing differs depending on coadministered drugs (150 mg twice daily with ritonavir-boosted PIs, 600 mg twice daily when combined with drugs with enzyme-inducing properties such as efavirenz, and 300 mg twice daily when combined with nucleoside analogues). Maraviroc is a p- glycoprotein (P-gp) substrate, which limits its intracellular concentrations. Its concentrations in cervico-vaginal fluid and vaginal tissue are higher than in plasma. Vicriviroc is currently in phase III studies [164] Enfuvirtide Enfuvirtide is an HIV-1 fusion inhibitor, which prevents fusion of HIV-1 and host-cell membranes. It is a synthetic peptide (4492 Da), which is not bioavailable when taken orally, and it is administered subcutaneously twice daily (90 mg twice daily), which is obviously a limitation to its long-term use. Its pharmacokinetic properties have been previously reported. [15]

123 1.2 Nucleoside and Nucleotide Analogue Inhibitors of Reverse Transcriptase (NRTIs and NTRTIs) Zidovudine is the oldest ARV, and a number of other nucleoside analogues have since been developed (zalcitabine, didanosine, stavudine, lamivudine, emtricitabine and abacavir). Tenofovir is an NTRTI obtained after drug administration of tenofovir disoproxil fumarate, its ester prodrug. Apricitabine is a new NRTI under development. Although the absolute bioavailability is unknown, bioavailability is supposed to be high for most NRTIs except for didanosine, which is degraded at acid ph, and tenofovir disoproxil fumarate. None of these drugs is highly protein bound. The parent compound is eliminated as the unchanged drug via the kidneys or non-cyp drug metabolizing enzymes, thus the potential for drug-drug interaction is low, although tenofovir disoproxil fumarate has been demonstrated to inhibit didanosine metabolism. [16] Triphosphate metabolites are the active components of all NRTIs. They also inhibit human mitochondrial polymerase to varying degrees. Phosphorylation steps occur within the cell and involve kinases, which are listed in figure 1. [17] The half-life of the active moiety is longer than the plasma half-life of the parent compound for all NRTIs. Long halflives of triphosphate metabolites favour once-daily dosing for most NRTIs, except for zidovudine and stavudine, which are administered on a twice-daily basis. Tenofovir is an NTRTI whose active form is a diphosphate. All NRTIs compete with endogenous analogues and stop DNA elongation. NRTIs such as abacavir (carbovir) and tenofovir are far less likely than stavudine to cause mitochondrial toxicity. [18] Fig Non-Nucleoside Analogue Inhibitors of Reverse Transcriptase (NNRTIs) NNRTIs do not require phosphorylation to inhibit reverse transcriptase. Nevirapine and efavirenz are the most commonly used NNRTIs. Delavirdine is available in some countries, etravirine is a new NNRTI recently approved in the EU and in the US and rilpivirine is currently in clinical development [164]. Nevirapine and efavirenz have long half-lives after single-dose

124 administration. They are metabolized through CYP3A and CYP2B6, and a genetic polymorphism has been described that explains at least part of the interindividual variability of their total clearance. They both have enzyme-inducing and autoinducing properties, which explains their drug-drug interactions and their nonlinear pharmacokinetics. [19] 1.4 Integrase Inhibitors Integrase inhibitors represent a new class of ARVs. These drugs inhibit the integration of HIV- DNA into the host genome. Raltegravir was approved in early 2008, and elvitegravir is in development. Raltegravir is rapidly absorbed, and its plasma concentrations decline with a terminal half-life of 7 12 hours, which supports twice-daily dosing. [20] Its plasma protein binding is 83%. Its biotransformation pathway involves uridine diphosphate glucuronosyltransferase 1A1, therefore its drug-drug interactions are limited. [21] Atazanavir, which inhibits UGT1A1, increases raltegravir concentrations modestly. [22] Enzyme inducers such as efavirenz, tipranavir or rifampicin (rifampin) decrease raltegravir concentrations, although the clinical consequences are currently unclear. [21] 1.5 Protease Inhibitors (PIs) PIs prevent cleavage of viral precursor proteins into the subunits required to form new virions. Approved PIs include amprenavir, fosamprenavir, atazanavir, darunavir, indinavir, lopinavir/ritonavir, nelfinavir, ritonavir, saquinavir, and tipranavir. They are all substrates and inhibitors of CYP3A, which explains part of their poor pharmacokinetic properties: presystemic first-pass metabolism, variable plasma concentrations and short half-lives in the 7- to 15-hour range. Ritonavir, which is the most potent CYP3A inhibitor, is combined with all PIs except for nelfinavir, to improve their pharmacokinetic properties, increase plasma exposure and/or decrease the required dose. [23,24] As basic organic chemicals, they are all bound to plasma proteins, 1 -acid glycoprotein and albumin. They differ in some pharmacokinetic parameters, such as the extent of first-pass metabolism and the extent of protein binding (indinavir 60%, lopinavir 98 99%), and some of them, such as amprenavir, have inducing properties that make prediction of drug-drug interactions very difficult.

125 1.6 Maturation inhibitors Bevirimat is the first representative of this novel class and is currently under investigation [164]. 2. Methodological Considerations All of the intracellular assays described to date do not discriminate between drug localized in cell membranes or in the cytoplasm, whether bound to intracellular proteins or truly unbound, which should be the effective ARV moiety. Measurement of total cell concentrations is thus of limited value. 2.1 Cell Collection Isolation of peripheral blood mononuclear cells (PBMCs) is the first step in analysis of intracellular concentrations of NRTIs, NNRTIs and/or PIs. PBMCs can be isolated using either conventional Ficoll gradient centrifugation or Vacutainer cell preparation tubes. These two procedures were compared by Becher et al., [25] using phosphorylated anabolites of two NRTIs, and were shown to give identical results. However, use of cell preparation tubes was found to be easier, less time consuming and therefore quicker, which, in the case of stavudine triphosphate, was most important, as the drug was shown to be unstable in the cell ring of the Ficoll gradient (40% loss within 40 minutes), and this led the authors to collect the ring in <10 minutes. However, before this isolation step, the stability of the phosphorylated anabolites and that of NNRTIs and PIs in blood should be considered. Regarding stavudine triphosphate, its stability has been checked in blood before isolation of PBMCs, and the authors recommend performing the isolation within 6 hours after sampling. [26] Similar results have been obtained with lamivudine triphosphate and zidovudine triphosphate. [27] It therefore seems that for phosphorylated anabolites, blood samples could be stored in cell preparation tubes for up to 6 hours before isolation, although this has not been thoroughly investigated for carbovir triphosphate and tenofovir triphosphate. Other issues during cell processing are to avoid contamination by red blood cells, which may phosphorylate some NRTIs, [28] and efflux of PIs and NNRTIs from cells. In contrast, the NRTI triphosphates are ion-trapped intracellularly. For intracellular measurement of NNRTIs and PIs, the collection of PBMCs has not been systematically studied. The authors mentioned that samples should be immediately taken to the laboratory (within 5 minutes), that all procedures should be

126 performed at 4 C to inhibit enzymatic activity, and that the time between blood sampling and the cell isolation and extraction procedure should be less than 1 hour to prevent active drug efflux. [29,30] 2.2 Estimation of Cell Numbers Since the number of cells regulates the intracellular concentration, a critical step in the process of intracellular assay is determination of the number of cells from which the compounds were quantified. In most studies where this information was provided, cell numbers were determined on a small aliquot, using a Coulter counter or a Malassez cell and a microscope. However, this last procedure may not be sufficiently accurate or precise, specifically where multiple sites are involved, which explains why a biochemical test has been developed that is based on the relationship between DNA content and the cell count by the Malassez cell. [31] This test could be performed in analytical laboratories where there is no Coulter counter available. The concentration is therefore expressed as the amount per 10 6 cells and can be converted into the amount per volume, based on the approximation that the PBMC volume is 0.4 pl, in order to compare intracellular and plasma concentrations. [32] The accuracy of this volume may be questionable, as it varies according to the state of the cells (quiescent or stimulated) or the nature of the cells (cell volume of human lymphoblast: 2.1 pl). [33] This highlights the pitfalls of the conversion. However, the 0.4 pl volume is most frequently used. [29,34] This calculation step is critical for comparison of the results from different teams, and a standardized procedure should therefore be chosen. 2.3 Analytical Methods for Intracellular Assays The approaches to the analysis of the intracellular drugs, considering the difference in their concentrations (from fmol/10 6 cells for intracellular triphosphate anabolites to pmol/10 6 cells [ng/ cells] for PIs are quite different NRTIs The major problem in measuring intracellular triphosphate anabolites is the small amount present in the cells of patients and the presence of endogenous intracellular nucleotides, which may cause interference. Thus selective and sensitive analytical methodologies should be developed.

127 In 2000, Rodriguez et al. [35] reviewed the latest information regarding intracellular in vivo quantification of NRTI triphosphates. The authors first described the methods and then pointed out all their drawbacks (lack of sensitivity, cumbersome assays, inability to differentiate NRTI triphosphates from endogenous nucleotides, lack of internal standards). More recent approaches have been based on the same first steps, i.e. separation of zidovudine anabolites using anionexchange cartridges, cleavage of the phosphate group using acid phosphatase, addition of an internal standard after enzymatic digestion, de-salting and quantification by high-performance liquid chromatography (HPLC) with tandem mass spectrometry (MS/MS). Moreover, the calibration curve was prepared from zidovudine triphosphate, in contrast to previous procedures, which used the parent compound. The limit of detection was 4.0 fmol/10 6 cells. The authors applied the same procedure to the simultaneous determination of zidovudine triphosphate and lamivudine triphosphate, [36] with limits of quantification of 0.1 pmol and 4.0 pmol, respectively. Moore et al. [37] improved this procedure, describing an analytical method that allows simultaneous measurement of intracellular lamivudine triphosphate, stavudine triphosphate and zidovudine triphosphate with HPLC-MS/MS. The limits of detection were 5, 25 and 25 pg on column for lamivudine triphosphate, stavudine triphosphate and zidovudine triphosphate, respectively. Similar methods were applied by King et al. [38] to measure tenofovir diphosphate and by Robbins et al. [39] to simultaneously measure zidovudine triphosphate, tenofovir diphosphate and lamivudine triphosphate in PBMCs, i.e. isolation by anion exchange, addition of a stable labelled isotope, dephosphorylation, de-saltation and detection by liquid chromatography (LC)-MS/MS. The lower limits of quantification were 10 fmol/10 6 cells for tenofovir diphosphate [38] and 0.11 pmol/10 6 cells, 2 fmol/10 6 cells and 3.75 fmol/10 6 cells for lamivudine triphosphate, zidovudine triphosphate and tenofovir diphosphate, respectively, for a sample size of 10 6 cells. [39] Most of these indirect methods are quite labour intensive and involve multiple steps, which may restrict their use to specialized laboratories. New methodologies have been described, which are based on direct HPLC-MS/MS determination of cellular extracts without dephosphorylation. However, these processes require the use of ion-pairing agents to circumvent the poor retention of the nucleotides, most of which are incompatible with ionisation mass spectrometry. Pruvost et al. [26] described direct determination of stavudine triphosphate and of the endogenous competitor, deoxythymidine triphosphate. Just before cell lysis, an internal standard was added. The instrument was operated in the electrospray negative-ion mode under MS/MS conditions. The lower limit of quantification was 9.8 fmol/10 6 cells, i.e. 20 pg injected for stavudine triphosphate. In this article, the authors focused on the stability of the stavudine triphosphate at the different steps (in blood, in the cell

128 ring, in dry cells at 4 C, after cell lysis at 4 C and in the injection solvent at room temperature). This procedure is very simple to perform as it does not require an extraction step. However, because of the very high ph of the mobile phase (ion pairing agent: 1,5-dimethylhexylamine), the column was changed every 2 weeks. [26] With slight modifications involving the internal standard and the chromatographic column, the same authors were able to simultaneously measure stavudine triphosphate, lamivudine triphosphate and 2',3'-dideoxyadenosine triphosphate (the active anabolite of didanosine) with their corresponding natural nucleotides in the same run. [25] However, a massive and tailing peak was observed near the retention time of zidovudine triphosphate, which precluded analysis of zidovudine triphosphate simultaneously with stavudine triphosphate, lamivudine triphosphate and 2',3'-dideoxyadenosine triphosphate. To overcome this problem, Becher et al., [41] developed specific extraction of zidovudine triphosphate, using immunoaffinity and detection of zidovudine triphosphate using LC-MS/MS. More recently, the same group improved the specificity and attained slightly better sensitivity for lamivudine triphosphate, carbovir triphosphate and tenofovir triphosphate, using a positive electrospray ionization mode. [42] Although the direct methods should be faster and more precise, King et al. [43] were unsuccessful in reproducing these methodologies. In particular, they pointed out the difficulty of analysing zidovudine triphosphate because of the large amount of adenosine triphosphate (ATP) and the interference of 2-deoxyguanosine 5-triphosphate, both of which have the same precursor ion and the same product ion. This was evidenced by Compain et al., [44] who developed an improved method of determining zidovudine triphosphate. The authors chose a minor but specific fragment ion and had to spike their sample with a constant amount of zidovudine triphosphate to allow the signal to emerge from the background in order to increase the sensitivity. HPLC-MS/MS is susceptible to matrix effects, i.e. co-eluting matrix components that affect the ionization of the target analyte, resulting in ion suppression or, in some cases, ion enhancement. [45] For intracellular assay, the main parameter to study is the influence of the number of cells in the sample, as it cannot be fixed. The matrix effect plus recovery was tested by Becher et al. [46] on stavudine triphosphate and 2',3'-dideoxyadenosine triphosphate, and the influence of cell numbers was evidenced. The use of an appropriate internal standard controlled the influence of the matrix effect between 7 and cells for simultaneous assay of stavudine triphosphate and 2',3'-dideoxyadenosine triphosphate. However, the use of a stable isotope analogue as the internal standard would be the best choice to control the influence of the matrix effect.

129 Monitoring of the very low intracellular concentrations of these active anabolites remains an analytical challenge. All of the methods described have their drawbacks. However, they are all based on sophisticated methods that are hard to reproduce, and so each laboratory favours the analytical procedure with which it is familiar. Whatever the choice regarding the procedure, whether indirect or direct, it appears that quantification of the compounds using HPLC coupled to MS/MS is very specific and may circumvent all of the drawbacks associated with the multiple natural nucleotides that are found in the complex mixture of the intracellular medium and can interfere with the determination of intracellular phosphorylated anabolites of NRTIs NNRTIs and PIs Measurement of intracellular concentrations of NNRTIs and PIs could be achieved using HPLC-UV detection, as has been described for efavirenz [34] and for ten other drugs (nevirapine, delavirdine, amprenavir, indinavir, the M8 metabolite of nelfinavir, ritonavir, lopinavir, efavirenz, saquinavir and nelfinavir). [47] However, most reported data were obtained using LC-MS/MS methods either for one drug [48] or for simultaneous measurement of several drugs. The methods involved automated solid-phase extraction, [48] liquid-liquid extraction (amprenavir, lopinavir, saquinavir and efavirenz; [49] nevirapine; [30] and lopinavir and ritonavir [50] ) or single-step extraction (nevirapine, delavirdine, amprenavir, indinavir, the M8 metabolite of nelfinavir, ritonavir, lopinavir, efavirenz, saquinavir and nelfinavir; [47] and indinavir, amprenavir, saquinavir, ritonavir, nelfinavir, lopinavir, atazanavir and efavirenz [29] ). Few quantitative immunoassays have been published for intracellular determination of lopinavir and atazanavir. [51,52] These methods involve preparation of a polyclonal antibody obtained with a synthetic ARV derivative coupled to hemocyanin or serum albumin as the immunogen and chemical synthesis of an enzyme tracer. Obviously these methods are not available in most laboratories, which precludes their use as tools to study intracellular concentrations of NNRTIs and PIs.

130 3. Mechanisms Influencing Intracellular Accumulation 3.1 General Principles As has already been stated, most ARVs acting on cell receptors need to enter the cell to bind to ARV targets, reverse transcriptase, integrase or protease. In general, disposition from the systemic circulation and the capillary lumen to the extravascular compartment occurs by diffusion or involves active transporters. Simple diffusion is generally the most common mechanism for transmembranal movement of xenobiotics in the body. The rate of diffusion is defined by Fick s law and, accordingly, the small, lipophilic, un-ionized and unbound molecules readily diffuse across the membrane. Differences in the ph gradient between plasma and lymphocytes could explain ion trapping. As reported by Ford et al., [53] the ph gradient between plasma and lymphocytes is subject to change, depending on the membrane potential. Binding of drugs to plasma proteins may slow the diffusion rate, as only free unbound drug will cross biological membranes. However, basic drugs, which have a greater affinity for cells or tissue proteins than for plasma proteins, may very rapidly leave the bloodstream, and protein binding is not a limiting factor; for such drugs, the volume of distribution is high, the amount of the drug in plasma is small compared with the amount in tissues and cells, and small changes in plasma protein binding will not affect the amount in the extravascular compartments. [54] Membrane transporters (efflux and influx) are now recognized as playing an important role in drug absorption and disposition, and they explain, at least in part, the broad interindividual variability in intracellular concentrations of drugs. Figure 2 summarizes the different carrier proteins determining intracellular concentrations within a typical immune cell. [53,55-57] Efflux transporters that operate at the expense of ATP hydrolysis are members of ATP-binding cassette (ABC)-type transport proteins and are now well studied. P-gp was first described for its ability to reduce intracellular concentrations of anticancer compounds. Other multidrug resistance proteins (MDRs) have since been isolated. They are expressed in the apical membrane of many barrier tissues, such as the intestine, liver, kidneys, blood-brain-barrier, placenta and testes, and in immune cells. The relevance to pharmacotherapy of expression of ABC drug transporters in peripheral blood cells has been reviewed recently. [57] Breast cancer resistance protein (BCRP) was found to play a major role in nucleoside efflux. [58,59] Although influx transporters have not been as well studied, several proteins have been identified for nucleosides transport (solute carriers [SLCs]); they differ in their mechanism of action. Some are driven by Na + electrochemical gradient (concentrative nucleoside transporter [hcnt]) and others are driven by chemical gradient (equilibrative nucleoside transporter [ENT]), organic cation

131 transporters (OCTs) or organic anion transporters (OATs), although the latter protein has not been found to be expressed in immune cells. [53,60] The current knowledge of antiretroviral drugs substrates of transporters has been reviewed recently in the context of their penetration through the blood-brain barrier. [165] Fig. 2 Several factors may influence transporter expression within the cell membrane, cell subsets and their functionality, the activation state of cells, polymorphism of coding genes and development as shown recently [166]. Polymorphism in the coding region of the transporter genes has been evidenced, which led to functional changes in the encoded transporter protein and resulted in variation in drug disposition and response; however these studies have been scanty. [55] Factors that could affect intracellular concentrations of ARVs and must be taken into consideration are the drug affinity and expression of the transporters according to different cells or tissues, and the fact that many of these transporters are known to be modulated by coadministrated ARVs. 3.2 NRTIs and NTRTIs Data on cellular penetration of NRTIs (purine or pyrimidine base coupled to a sugar) are scarce. As they are more hydrophilic compounds, it has been suggested that they could be substrates of endogenous nucleoside transporters. [61,62] Although studies have demonstrated that cerebral penetration occurs mainly by passive diffusion and that the low concentration of nucleosides in the brain is the consequence of active efflux transporters, [63] expression of uptake transporters in lymphocytes could favour high intracellular concentrations. It was evidenced that zidovudine triphosphate and lamivudine triphosphate concentrations were effluxed by multidrug resistance protein (MRP)-4 and BCRP. As a nucleotide, tenofovir has an ionized phosphate group, which confers acidic properties. It has been demonstrated that tenofovir uptake in the basolateral membrane of the kidney proximal tubule is mediated via OAT1 and cellular efflux into the urine via MRP2 and MRP4. [64] Triphosphate concentrations differ according to the cell type, most likely as a consequence of influx and efflux transporter expression. In healthy subjects, lamivudine triphosphate concentrations have been shown to be similar in PBMCs and purified CD4 cells, whereas zidovudine triphosphate concentrations were lower in CD4 cells than in PBMCs. [65] Concentrations of tenofovir diphosphate have been compared in PBMCs, lymph node tissue and digestive lymphatic tissue, and were higher in PBMCs than in the other tissues. [66] These data strongly suggest that transporter localization may differ according to cell functionality. Kinase activity could also influence the intracellular concentration of triphosphates. In vitro

132 experiments have suggested that this activity varies greatly and is lower in resting cells than in activated PBMCs. [67,68] This could have important consequences, as kinase activity governs the intracellular level of both endogenous triphosphates and NRTI triphosphates, which compete at the level of HIV-reverse transcriptase. All NRTIs have been demonstrated to be more effective in monocyte-derived macrophages, which are important HIV-1 reservoirs, than in CD4+ T lymphocytes. [69] This could also explain the differences in NRTI triphosphate intracellular concentrations according to different cell types and different activation states. [28] 3.3 NNRTIs NNRTIs are weakly acidic and bind predominantly to albumin. Neither efavirenz nor nevirapine were thought to be substrates of P-gp. [70] In a limited number of patients, Almond et al. [34] demonstrated a relationship between the intracellular concentration of efavirenz and the percentage of bound efavirenz in plasma. Such data are in contrast to those for nevirapine obtained by the same team. [30] They demonstrated that intracellular concentrations of nevirapine are far below those measured in plasma. The intracellular concentration was negatively related to P-gp expression and was not related to the plasma percentage of unbound nevirapine. [30] To explain these data, the authors suggested that nevirapine could induce P-gp or co-regulated efflux transporter. Clearly, to understand all mechanisms involved in intracellular concentrations of NNRTIs, further studies are needed. 3.4 PIs The intracellular pharmacology of PIs has been carefully reviewed by Ford et al. [53] The physicochemical properties of PIs are in favour of passive transfer: The transfer is in agreement with lipophilicity measured by the n-octanol to water partition coefficient. Accumulation of PIs in lymphocytes reflects the rank order of lipophilicity: the least lipophilic PI being indinavir and the most lipophilic being nelfinavir. PIs are weak bases and are mostly un-ionized in a basic environment; intracellular sequestration depends on the ph gradient between plasma and cells. Protein binding of PIs to 1 -acid glycoprotein ranges from 60% for indinavir to 97 99% for ritonavir, lopinavir, saquinavir and nelfinavir. However protein binding per se is not a limiting factor for intracellular diffusion, as indinavir, which is 60% bound, has lower intracellular

133 concentrations than other PIs that are more highly bound. Within the cell, PIs are bound to cell proteins and HIV proteases, and therefore their relative affinity for each protein may influence their dynamic equilibrium. [53] However, active transport may play a role in intracellular accumulation. It is now well established that PIs are substrates of P-gp and other efflux transporters such as MRP1 [71] or MRP2. [72] P-gp is expressed in the gastrointestinal tract and the liver, and acts with CYP3A to reduce the bioavailability of PIs. When combined with most PIs as a pharmacological enhancer, ritonavir inhibits both CYP3A and P-gp and markedly increases the bioavailability of PIs. Such transporters are expressed on lymphocytes and may reduce cellular accumulation. Meaden et al. [73] found a relationship between combined expression of P-gp and MRP1 on PBMCs of HIVinfected patients and intracellular accumulation of saquinavir and ritonavir. In summary, there is evidence that intracellular concentrations of PIs depend on P-gp and/or other efflux transporter activity, which is modulated by genetic polymorphism and coadministration of drugs with inhibiting or inducing properties. Further studies are needed to determine how these transporters control intracellular concentrations of PIs. 3.5 Importance of Genetic Polymorphisms The role of transporters and their genetic polymorphisms in drug disposition have been reviewed recently [74] [55,75]. La Porte et al. [76] studied the relationship between ABCB1 (MDR1) genetic polymorphism, P-gp expression and saquinavir or saquinavir/ritonavir pharmacokinetics in 150 healthy subjects. No relationship was found between the C3435T, G2677T/A or C1236T polymorphisms of the ABCB1 gene and the pharmacokinetics of saquinavir or the expression and activity of P-gp in PBMCs. Seventy-one HIV-infected children treated with a nelfinavir backbone ARV regimen were evaluated for MDR1 polymorphism (MDR1 C3435T), nelfinavir plasma concentrations, CD4 cell counts and HIV-RNA. [77] Children with the C/T genotypes had higher nelfinavir concentrations 8 hours postdose and a more rapid response to HAART. Unfortunately, intracellular concentrations of PIs were not measured in these studies. In contrast, in 12 HIVinfected patients, Ford et al. [78] found no evidence of higher intracellular concentrations of nelfinavir or its M8 metabolite and lymphocyte cell surface expression of P-gp. In a cohort of 47 patients treated with PI boosted or not boosted by ritonavir, Chaillou et al. [79] demonstrated that intracellular concentrations of ritonavir were related to undetectable plasma HIV-RNA, which was not related to MDR1 gene expression.

134 Interestingly, the importance of the MRP4 carrier was evidenced by Anderson et al., [72] who demonstrated that patients carrying MRP4 T4131G had elevated lamivudine triphosphate concentrations and that patients with MRP4 G3724A had a trend for elevated zidovudine triphosphate. They also found that indinavir clearance was faster in patients expressing CYP3A5 and in patients carrying the MRP2 24C/T variant; [72] whether the latter contributes to lower intracellular concentrations remains to be established. Recently, Kiser et al. [80] demonstrated higher intracellular concentrations of tenofovir diphosphate, first with a decrease in the glomerular filtration rate and consequent decreases in total and renal clearance of tenofovir (p = 0.04), and second in the presence of the ABCC A>G variant (p = 0.04 after adjustment for race, treatment group and glomerular filtration rate). The authors pointed out the sample-size limitation of this small study of genetic association, and thus the results should be confirmed in a larger study. Whether those exploratory data could be extrapolated to intracellular concentration of tenofovir diphosphate within renal proximal tubule cells is presently unknown. The available evidence shows that the control of intracellular concentrations of ARVs is complex and depends on many factors. More work is needed in this area, taking into account the differences in cell biology. 4. Clinical Studies of Intracellular Concentrations 4.1 Relationship between Intracellular and Plasma Pharmacokinetics Clinical studies reporting intracellular concentrations in HIV-infected patients are summarized in table II. [9-13,30,34,72,78,79, ] They are listed by ARV class, and then within each class they are listed by molecule, with respect to the date of health authority approval. For each molecule, the more recent studies are presented first. All of the studies were published after 2000, except in the case of zidovudine, for which intracellular concentrations have been studied since Most studies reported both plasma and intracellular concentrations, but only a few studied the relationship between them. Table II It can be seen from table II that plasma pharmacokinetic parameters are rather similar across studies, but some differences have been observed in intracellular parameters. For NRTIs, most

135 studies did not establish a significant relationship between plasma and triphosphate concentrations. In contrast, the results for NNRTIs and PIs have been more conflicting, with some studies evidencing correlation while others did not. These results support the use of plasma concentrations of NNRTIs or PIs, but not of NRTIs, to monitor antiviral efficacy. Such results are not surprising, considering first that collection, preparation and quantification of PBMCs is not an easy task (see section 3), and second that intracellular drug penetration is influenced by many factors such as genetic polymorphism of influx and efflux carriers (see section 4). Moreover, these intracellular studies have been carried out in relatively few patients, and larger studies would be needed to consistently address the relationship between plasma and intracellular concentrations of ARVs. It is also important to note that there are potential methodological problems when studying relationships between concentrations observed at single timepoints, as has been done in a number of studies. Indeed, plasma and intracellular half-lives are very different. It is more appropriate to assess the relationship through pharmacokinetic parameters such as the area under the concentration-time curve (AUC). Surprisingly, most studies have reported concentrations at various timepoints or pharmacokinetic parameters obtained by noncompartmental analysis. Population approaches have never been used to analyse intracellular concentrations and their link with plasma concentrations, although this approach seems more appropriate as it allows analysis of sparse measurements. 4.2 Drug-Drug Interactions at the Intracellular Level From a theoretical point of view, changes in the intracellular concentrations of ARVs can be secondary to modifications of (i) the plasma concentration of the drug and/or the prodrug; (ii) activity of the enzymes responsible for drug anabolism/metabolism at the cellular level; and (iii) activity of membrane transporters involved in cellular uptake or efflux. Since the intracellular amount of the active drug is responsible for treatment efficacy, interactions leading to changes in intracellular concentrations are relevant to the virological outcome. The clinical impact of these interactions was first evidenced by the poor efficacy of therapies combining zidovudine with stavudine. [103] Though the likely mechanism of this result, competitive inhibition of stavudine phosphorylation by zidovudine, was assessed only by in vitro experiments, this phenomenon highlighted the necessity to investigate the possible alteration in intracellular concentrations of ARVs due to drug association. [104,105]

136 Potential interactions involving NRTIs at the intracellular level have therefore been investigated in several studies. Hawkins et al. [82] evaluated whether the high rate of virological failure observed in patients receiving a triple NRTI combination including tenofovir disoproxil fumarate could be explained by modifications in the intracellular anabolism of these compounds. Intracellular concentrations of tenofovir diphosphate, carbovir triphosphate and lamivudine triphosphate were measured in 15 HIV-infected patients receiving a triple NRTI combination (tenofovir disoproxil fumarate/abacavir/lamivudine or tenofovir disoproxil fumarate/abacavir/stavudine), before and after replacement of tenofovir disoproxil fumarate or abacavir by an NNRTI or a PI. No modification in the intracellular concentrations of the active anabolites of the remaining NRTIs was observed, which suggested lack of a significant interaction between the investigated drugs. Another recent study confirmed these results in 27 patients. [102] Taken together, these results suggest that the clinical failure that was observed with the triple NRTI (abacavir/tenofovir disoproxil fumarate/lamivudine) regimen was not due to drug interactions but was more likely the consequence of lack of intrinsic power. [106] This latter study also evidenced a significant 50% increase in the intracellular concentration of tenofovir diphosphate when tenofovir was combined with lopinavir/ritonavir. However, this result could simply be an intracellular reflection of the systemic interaction between these two drugs. [107] This study found no significant difference in intracellular concentrations of carbovir triphosphate and lamivudine triphosphate with respect to lopinavir/ritonavir use, despite a 46% decrease in the abacavir plasma concentration in the lopinavir/ritonavir group. Last, it was also found that nevirapine did not significantly modify intracellular concentrations of tenofovir triphosphate, carbovir triphosphate and lamivudine triphosphate. Hoggard et al. [108] investigated whether prior exposure to zidovudine could subsequently inhibit stavudine phosphorylation. The rationale for this study was the observation that treatment-naïve patients receiving a stavudine/lamivudine combination experienced a further 1 log 10 decrease in viral DNA compared with patients who had previously been treated with zidovudine. [109] Subsequent inhibition of stavudine phosphorylation due to a downregulation of thymidine kinase induced by zidovudine was one of the hypotheses raised to explain this result. However, cellular concentrations of stavudine triphosphate measured in seven zidovudine-experienced patients were no different from those measured in 20 zidovudine-naïve subjects. Furthermore, the ability of PBMCs to phosphorylate stavudine did not differ between zidovudine-experienced and zidovudine-naïve subjects. [108] Similarly, no influence on zidovudine phosphorylation of prior exposure to zidovudine was observed in 23 HIV-infected patients during a 12-month period. [83] It is therefore likely that the decrease in efficacy observed in zidovudine-experienced patients was

137 due to the acquisition of resistance mutations rather than a modification in intracellular metabolism. By measuring the triphosphate moieties of zidovudine and lamivudine in the PBMCs of eight patients, Fletcher et al. [84] found a strong correlation between intracellular concentrations of zidovudine triphosphate and lamivudine triphosphate. If this result suggested the existence of interplay between the cellular anabolism and/or metabolism of these drugs, its precise mechanism and possible consequences have still not been elucidated. Tenofovir disoproxil fumarate is known to increase the plasma concentration of didanosine, the most likely mechanism for this interaction being inhibition by tenofovir of the enzyme responsible for the hydrolysis of guanosine and adenosine analogues, the purine nucleoside phosphorylase. [16] This interaction is clinically relevant, since it is responsible for adverse effects [ ] or treatment failure, [114] which may be secondary to didanosine overexposure. Because of this, this combination is currently not recommended for initiation of HAART but is nevertheless not countraindicated for ulterior lines of treatment. [6] Pruvost et al. [85] investigated the possible consequences of this systemic interaction on intracellular concentrations of the active moieties. Intracellular concentrations of 2',3'-dideoxyadenosine triphosphate and tenofovir diphosphate were compared between 14 patients receiving the didanosine/tenofovir disoproxil fumarate (250 mg/300 mg) combination, 16 patients receiving didanosine (400 mg) without tenofovir disoproxil fumarate, and 14 patients receiving tenofovir disoproxil fumarate (300 mg) without didanosine. The measured concentrations were found to be comparable between the groups, which validated the strategy of decreasing the didanosine dose from 400 to 250 mg when it is combined with tenofovir disoproxil fumarate. [115] Apricitabine, a novel deoxycitidine analogue currently under investigation, shares with lamivudine and emtricitabine its initial phosphorylation pathway by deoxycytidine kinase. The potential interaction between apricitabine (600 mg twice daily) and lamivudine (300 mg once daily) was evaluated in a crossover study performed in 21 healthy subjects who sequentially received each drug separately and the combination of both. No significant modification in the plasma pharmacokinetics of lamivudine or in the cellular pharmacokinetics of its active triphosphate moiety was observed during combination therapy compared with the monotherapy period. However, although coadministration with lamivudine had no influence on apricitabine plasma pharmacokinetics, the intracellular concentration of apricitabine triphosphate dropped by 85% during the same period. [116] These findings strongly suggested that apricitabine should not be coadministered with deoxycitidine analogues.

138 Hydroxyurea is an antiproliferative drug, which has been shown to produce a further 0.7 log 10 reduction in plasma HIV RNA when combined with didanosine, compared with didanosine treatment alone. [117] By measuring intracellular deoxyadenosine triphosphate in 69 HIV-infected subjects, it was evidenced that patients receiving the hydroxyurea-didanosine combination achieved significantly lower deoxyadenosine triphosphate concentrations than patients receiving didanosine or hydroxyurea monotherapy, whereas no modification in the plasma pharmacokinetics of the two drugs was observed. [118] If the precise mechanism of this interaction is still unknown, the likely explanation for the enhancement of the efficacy of didanosine is the decrease in the intracellular deoxyadenosine triphosphate/2',3'-dideoxyadenosine triphosphate ratio, which would facilitate the incorporation of 2',3'-dideoxyadenosine triphosphate into the replicating viral DNA. Like hydroxyurea, mycophenolic acid (an immunosuppressive agent) is known to decrease the intracellular concentration of an endogenous nucleotide, deoxyguanosine triphosphate, which could enhance the antiviral activity of abacavir by decreasing the deoxyguanosine triphosphate/carbovir triphosphate ratio. [119] Since this ratio has not been measured to date in patients receiving the combination of mycophenolate mofetil and abacavir, this hypothesis still needs to be confirmed. Nevertheless, the lack of an influence of mycophenolic acid on lamivudine phosphorylation was suggested by the similar intracellular concentrations of lamivudine triphosphate observed in patients receiving lamivudine with or without mycophenolate mofetil. [86] Ribavirin is a nucleoside analogue used for the treatment of hepatitis C virus (HCV) infection. Although its mechanism of action is still not fully understood, it involves at least in part intracellular transformation into a triphosphate moiety. [120] Thus, because ribavirin is used in HIV/HCV co-infected patients, its potential interactions with NRTIs have been investigated in several studies. Rodriguez-Torres et al. [121] evaluated the combination of ribavirin with lamivudine, stavudine or zidovudine in HIV/HCV co-infected patients. Plasma concentrations of zidovudine, stavudine and lamivudine and intracellular concentrations of zidovudine triphosphate, stavudine triphosphate and lamivudine triphosphate were measured in 31 patients receiving concomitant ribavirin and compared with the concentrations obtained in 25 patients receiving placebo instead of ribavirin. No significant difference in plasma and cellular concentrations of the measured compounds was observed, suggesting that ribavirin does not modify the plasma pharmacokinetics and intracellular phosphorylation of zidovudine, stavudine and lamivudine. The lack of interaction between

139 ribavirine and zidovudine was confirmed in another study performed in 14 HIV-infected subjects. [122] It is worth noting that these results are discrepant with in vitro data, which evidenced inhibition of the phosphorylation of zidovudine [123,124] and stavudine [105] by ribavirin. However, it is still unexplained whether these discrepancies are due to a poor ability of in vitro models to predict in vivo phenomena or to some methodological drawbacks in the ex vivo quantification of intracellular triphosphate moieties. Conversely, ribavirin was found to potentiate the phosphorylation of didanosine in vitro via inhibition of inosine 5-monophosphate dehydrogenase. [125] However, despite its potential virological interest, this interaction is also characterized by a high risk of mitochondrial toxicity, [126,127] and so combination of ribavirin with didanosine is not recommended. With regard to PIs, the influence of atazanavir on the plasma and intracellular pharmacokinetics of saquinavir and ritonavir was investigated in nine HIV-infected patients who received the saquinavir/ritonavir (1600 mg/100 mg once daily) combination with and without atazanavir (200 mg once daily). Atazanavir was found to significantly increase both plasma and intracellular concentrations of saquinavir by a similar factor of approximately 4 but had no effect on ritonavir concentrations. [128] Interestingly, the cellular half-life of saquinavir was unaffected by atazanavir, which suggested that the increase in the intracellular concentration was secondary to the increase in the plasma concentration rather than inhibition of a cellular transporter. The possible modification of plasma and intracellular concentrations of saquinavir by quercetin, a bioflavonoid displaying inhibitory effects on CYP3A4 and P-gp, was investigated in 10 healthy adults who received saquinavir alone (1200 mg twice daily) for 11 days followed by the combination of saquinavir and quercetin (1200 mg/500 mg twice daily) for the next 3 days. [129] Although no change was observed in plasma concentrations of saquinavir, its intracellular concentrations surprisingly decreased by almost 50% when it was combined with quercetin. However, the important intra- and intersubject variability of the intracellular concentrations prevented conclusions from being drawn from this result. The accumulation ratio (equal to the cellular concentration divided by the plasma concentration) of some PIs has been found to be modified by low doses of ritonavir in HIV-infected subjects. [79] Indeed, amprenavir and indinavir accumulation ratios in the presence of 100 mg or 400 mg of ritonavir were increased 3-fold and 5-fold respectively. However, conflicting results were obtained in another study, which found that ritonavir did not increase the accumulation ratio of saquinavir and indinavir. [130]

140 Despite these discrepancies, assessment of the consequences of drug interactions at the cellular level is of great importance in order to validate new combinations. Some surprising recent results, such as the possible decrease in the efficacy of HCV therapy due to abacavir, [131] have evidenced the need for better understanding of drug interactions. 4.3 Relationship between Intracellular Concentrations and Efficacy Only eight published studies have reported analyses of the relationships between intracellular concentrations and virological or immunological efficacy of ARVs in HIV patients. Five studies examined intracellular NRTIs and three examined PIs. These studies are summarized in table III by molecule, from the most recent study to the oldest. [8,79,84,87-89,122,132] Of note, five were prospective studies and interestingly found a significant relationship between higher intracellular concentrations and the virological response. [8,84,87,88,132] The correlation with plasma concentrations was not always studied but was mainly nonsignificant or less significant. The three studies with nonsignificant results were cross-sectional studies that were not designed for that purpose. [88,89,122] Like table II, table III lists studies of NRTIs, followed by PIs; there has been no such study of NNRTIs. With NRTIs, it is important to note that the relevant determinant of the pharmacodynamic response is the ratio between the drug triphosphate and endogenous nucleoside triphosphates, rather than the absolute intracellular concentration. [133] Table III The study by Moore et al. [87] was a substudy of ACTG 862, a prospective trial in which treatment-naïve patients were started on dual NRTI therapy. A significant correlation between the change in the viral load between weeks 0 and 28 and intracellular concentrations was found for lamivudine triphosphate (R 2 = 0.62) in the 39 patients receiving either lamivudine/zidovudine or lamivudine/stavudine, and for zidovudine triphosphate (R 2 = 0.28) in the 10 patients receiving lamivudine/zidovudine. No significant correlation was found for stavudine triphosphate in the 15 patients receiving lamivudine/stavudine, and no significant relationship with changes in CD4 cells was found for any drug. The authors did not study the relationships between efficacy and plasma concentration but demonstrated a very large interpatient variability in the intracellular to plasma concentration ratio. They also showed that there was an important increase in intracellular concentrations between the day 1 (first dose) and week 28, but only for lamivudine triphosphate. The study by Aweeka et al. [122] was performed in HCV or hepatitis B virus (HBV) co-infected HIV patients, and its primary objective was to study the influence of ribavirin on zidovudine plasma and intracellular concentrations by measuring the AUC in patients before and after

141 introduction of ribavirin. Different regimens were permitted, and all patients had to receive zidovudine for at least 4 weeks. In the cross-sectional analysis performed in 13 patients before they received ribavirin, no significant relationship was found between the AUC of zidovudine triphosphate and the CD4 cell count measured the same day. The studies by Anderson et al. [8] and Fletcher et al. [84] are substudies of the well known concentration-controlled randomized trial by Fletcher et al. [134] In this trial, patients received triple therapy with zidovudine, lamivudine and indinavir, either with a fixed dose or with a dose adapted to achieve trough concentrations within a defined range. Intracellular concentrations of zidovudine triphosphate and lamivudine triphosphate were measured 2 hours after dose administration at the three pharmacokinetic visits scheduled at weeks 2, 26 and 52 and at variable times postdose at the nine bimonthly visits. For each patient, the median intracellular concentration from all measurements was considered for analysis of the link with efficacy. Unfortunately, the authors did not analyse intrapatient variability of these concentrations. In their first study, [84] only eight patients were studied, and the efficacy criteria were the change between baseline and week 24 in HIV RNA or the percentage change between baseline and week 24 in the CD4 cell count. For zidovudine triphosphate, a significant correlation with intracellular concentrations was found both for changes in HIV RNA (R 2 = 0.54) and for changes in CD4 cells (R 2 = 0.84). For lamivudine triphosphate, a significant correlation with intracellular concentrations was found for changes in HIV RNA (R 2 = 0.79) but not for changes in CD4 cells (p = 0.07, R 2 = 0.44). There was no significant correlation between efficacy and steady-state plasma concentrations of these drugs. In the study by Anderson et al. [8] 33 patients were studied; the analysis was very thorough, with various efficacy endpoints and several multivariate analyses. After an initial simple correlation study, the authors defined thresholds for intracellular concentrations as the first quartiles, which were 30 fmol/10 6 cells for zidovudine triphosphate and 7017 fmol/10 6 cells for lamivudine triphosphate. They then studied the impact on several efficacy endpoints of having a median intracellular concentration below or above these thresholds. The primary efficacy endpoint was the time to plasma HIV RNA of <50 copies/ml, using a survival analysis. For zidovudine triphosphate, the median time to plasma HIV RNA of <50 copies/ml was significantly shorter in patients with concentrations above the threshold than in those with concentrations below the threshold, as shown. [8] A significant relationship was also found for lamivudine triphosphate, but only zidovudine triphosphate remained in the multivariate analysis. The secondary efficacy endpoints were the virological status (<50 copies/ml) at weeks 24 and 52 after starting the ARV regimen. For zidovudine triphosphate, 92% of patients with concentrations above the threshold

142 had undetectable viral loads at week 24; this proportion was significantly lower (44%, p = 0.009) for patients with concentrations below the threshold, but the relationship was not significant at week 52. For lamivudine triphosphate, 96% of patients with concentrations above the threshold had undetectable viral loads at week 24, a proportion significantly higher than that of patients with concentrations below the threshold (37.5%, p = 0.002). Similarly, at week 52, 91.7% of patients with concentrations above the threshold had undetectable viral loads, a proportion significantly higher than that of patients with concentrations below the threshold (25%, p = ). Another efficacy endpoint was the time to rebound (two HIV RNA values greater than 50 copies/ml), and this time was shorter for patients with zidovudine triphosphate and lamivudine triphosphate concentrations below the threshold; in the multivariate analysis, only lamivudine triphosphate remained significant (p = 0.009). It is interesting to note that zidovudine triphosphate was mostly associated with an initial viral load decrease, whereas lamivudine triphosphate was associated with a sustained response. No significant relationship was found between CD4 cell counts and intracellular concentrations. However, the authors did not study the change in CD4 cell counts but instead performed cross-sectional analyses, looking for correlations at week 24 and week 52 between CD4 cell counts and corresponding intracellular concentrations. The link between plasma concentrations of zidovudine and lamivudine and efficacy was not studied, and no significant relationships were found with indinavir concentrations. The authors also found that intracellular concentrations were significantly higher in females than in males: 2.3-fold for zidovudine triphosphate (p < ) and 1.6-fold for lamivudine triphosphate (p < ), whereas no influence of sex on plasma concentrations was found. These results suggest that NRTI phosphorylation differs between the sexes. [8] The first published study to analyse the link between intracellular concentrations and efficacy was the pharmacokinetic trial reported by Stretcher et al., [132] in which zidovudine was given five times daily as a single therapy (500 mg/day) in 21 patients followed up for 24 weeks. The AUC of total zidovudine phosphates was evaluated at weeks 4 and 24 from five samples. Efficacy was analysed mainly through the CD4 cell count, the percentage of CD4 cells and the CD4/CD8 ratio. Unfortunately, in the analysis of the correlation between the AUC of zidovudine phosphates and immunological efficacy, the authors pooled the observations made at week 4 and week 24, not taking into account the correlation induced by the repetition within patients. Here, only results from the analyses performed separately at weeks 4 and 24 are reported. At week 4, the authors found that the AUC of zidovudine phosphates correlated significantly with both the change in the percentage of CD4 cells from baseline (R 2 = 0.06, p = 0.029) and the change in the CD4/CD8 ratio from baseline (R 2 = 0.06, p = 0.028) but not with the values measured at week 4. These

143 correlations were no longer significant at week 24. No significant correlation was found with the plasma AUC. With respect to studies with PIs, the main objective of the prospective trial by Lamotte et al. [89] was to investigate the concept of once-daily administration of the new galenic formulation softgel capsule of saquinavir in association with ritonavir in PI-experienced HIV patients. The evaluation of the link between intracellular saquinavir concentrations and virological efficacy in 13 patients was explored as one of the secondary objectives. No significant correlation was found between trough intracellular concentrations of saquinavir at weeks 2, 4 and 12 and variations in plasma HIV RNA between weeks 0 and 12. Similarly, no significant correlation was found for plasma saquinavir concentrations. The main objective of the prospective study reported by Breilh et al. [88] was to assess the impact on virological success (defined as an HIV level of <50 copies/ml) of intracellular and plasma trough concentrations of lopinavir in 38 patients receiving a lopinavir/ritonavir-based regimen. They found that trough intracellular concentrations of lopinavir at week 4 were significantly higher (12.7 g/ml) in patients achieving virological success before week 4 than in others (4.8 g/ml; p < ). Similarly, intracellular concentrations of lopinavir at week 24 were significantly higher (10.5 g/ml) in patients achieving virological success before week 24 than in others (4.6 g/ml; p < 0.002). [88] Virological success was also significantly associated with higher plasma trough concentrations at weeks 4 (p = ) and 24 (p = 0.05) and with the genotype inhibitory quotient at weeks 4 (p = ) and 24 (p = ). In a multivariate analysis of virological success at week 24, the authors found an effect of baseline lopinavir mutations, plasma concentrations at week 4 and intracellular concentrations at week 24. The authors defined thresholds of 4 g/ml and 8 g/ml for plasma and intracellular concentrations, respectively; they suggested combining the plasma and intracellular concentrations of lopinavir for therapeutic drug monitoring. The authors derived the cellular accumulation ratio, but it was not used in the analysis of the link with efficacy. The cross-sectional study by Chaillou et al. [79] included 49 patients with ARV combinations containing various PIs. The first objective was to study the relationship between MDR1 gene expression and intracellular PI concentrations and then to evaluate the correlation of intracellular PI concentrations with the virological response. Efficacy was defined as an undetectable HIV RNA load (<40 copies/ml) on the day of the study. To normalize the concentrations of the various PIs that were analysed, the authors studied the influence of the ratio of intracellular to plasma concentrations, which they defined as accumulation but which is not a measure showing the amount of drug in the body. They did not find any significant correlations for the main PIs.

144 The only parameter significantly linked with efficacy was the intracellular concentration of ritonavir (p = 0.04). Of the 19 patients receiving ritonavir as a booster, those with an undetectable HIV viral load had significantly higher ritonavir intracellular accumulation than those with detectable HIV RNA (p = 0.029). In conclusion, in prospective studies that were well designed and involved a reasonable number of patients, all authors found a significant correlation between virological efficacy and intracellular concentrations of NRTIs, with no influence of plasma concentrations. For PIs, there has been only one well designed prospective trial of lopinavir, which found influences of both trough plasma concentrations and intracellular concentrations at different timepoints after treatment initiation. From these results, it is difficult to know whether the primary association is with plasma or intracellular concentrations. These findings obtained in only 38 patients should be confirmed by other studies. 4.4 Relationship between Intracellular Concentrations and Toxicity ARVs are known to produce significant adverse effects, which are the main drawback of HAART. [135] The toxicity related to ARVs is indeed an important cause of poor compliance, which is, in turn, the main cause of treatment failure. [136] Moreover, viral rebound can be associated with the acquisition of mutation resistance by the virus, which can critically undermine the choice of the subsequent therapeutic strategy. [ ] Most toxicities displayed by ARVs are typical of a pharmacological class, except for NRTIs, which tend to have their class-specific toxicities. For instance, PIs are known to induce digestive troubles and metabolic disorders, such as hyperlipidaemia, insulin resistance, diabetes mellitus, peripheral lipodystrophy and central adiposity. [ ] NRTIs can be responsible for lipodystrophy, [144] neuropathy (didanosine, stavudine), [145] myopathy, [146] pancreatitis (didanosine, stavudine), [147] anaemia and neutropenia (zidovudine), [148] renal impairment and Fanconi syndrome (tenofovir), [149] hepatic steatosis and lactic acidosis; [144] whereas NNRTIs can cause neuropsychological disorders (efavirenz) [150] and skin or hepatic toxicity (nevirapine). [ ] To date, the mechanisms leading to these toxicities are not perfectly understood, but the main hypotheses strongly suggest interference with some cellular endogenous processes. For example, PIs could alter adipose tissue and lipid metabolism by inhibiting the heterodimeric nuclear receptor complex composed of peroxisome proliferator activated receptor and the retinoid X receptor, the cellular retinoic acid-binding protein, and the synthesis of cis-9-retinoic acid. [154] PIs could also inhibit the degradation of the sterol element-binding proteins that regulate the

145 transcription of the low density lipoprotein receptor gene. [155,156] Diabetes induced by PIs could be secondary to the direct inhibition of GLUT4, a transporter that mediates the cellular uptake of glucose stimulated by insulin. [157] Similarly, adverse effects due to NRTIs are thought to be related to michondrial damage, which is a consequence of the ability of NRTIs to inhibit mitochondrial DNA polymerase. [158] Despite these elements, the possible relationship between intracellular concentration of ARVs and their related toxicity has to date been investigated for four molecules only: zidovudine, lamivudine, tenofovir disoproxil fumarate and efavirenz. First, in a study of 13 ARV-naïve patients, Stretcher et al. [132] evaluated whether the intracellular concentration of zidovudine phosphates in PBMCs was related to markers of zidovudine-induced toxicity. A negative correlation was found between intracellular zidovudine phosphates and the decrease in haemoglobin from its baseline level. Of note, the authors found no significant correlation between plasma zidovudine and intracellular zidovudine phosphate concentrations, and did not investigate the possible relationship between plasma zidovudine and a decrease in haemoglobin. However, other studies have evidenced an association between plasma zidovudine concentration and anaemia, [159,160] and so the relative strength of the association between haemoglobin decrease and plasma zidovudine concentration compared with intracellular zidovudine phosphates concentration is still unknown. A different result was nevertheless found in adults and children. Indeed, in a study performed by Anderson et al. [8] in 33 treatment-naïve adult patients receiving a zidovudine/lamivudine/indinavir regimen, no difference in the intracellular concentrations of zidovudine triphosphate and lamivudine triphosphate was observed among the 14 patients who experienced at least a grade I biological event and the 19 patients who did not. A similar result was found in 49 neonates, as the proportion of observed haematological toxicities was not related to the intracellular concentrations of zidovudine triphosphate and lamivudine triphosphate. [99] However, differences between the pharmacokinetic criteria and the pharmacodynamic endpoints used (see table IV) [8,90,99,132] might explain the inconsistency between these studies. There are also in vitro data suggesting that zidovudine monophosphate is the cause of zidovudine-related anaemia, [161] and so toxicity relationships with zidovudine triphosphate may not be relevant. Table IV With regard to NNRTIs, in a study performed in 55 patients, Rotger at al. [90] found a significant correlation between intracellular concentrations of efavirenz and the risk of mood disorders. No significant correlation was found with the risk of sleep disorders and fatigue. In contrast with other studies, [162,163] no significant association was observed between plasma concentrations of

146 efavirenz and the neuropsychological problems that were investigated. Once again, methodological differences confounded the comparison between studies. Last, Izzedine et al. [64] found (without measuring intracellular concentrations of tenofovir diphosphate) that the risk of developing a renal proximal tubulopathy associated with treatments containing tenofovir disoproxil fumarate was significantly associated with genetic variants in the gene coding for MRP2, a transporter involved in tenofovir efflux from tubular cells to the urine. Since these variants are thought to be associated with reduced activity of the transporter, this result could indicate that an accumulation in the tubular cell due to an altered MRP2-based efflux might be responsible for tenofovir disoproxil fumarate-induced toxicity. [64] However, for obvious reasons, tenofovir concentrations in the tubular cells could not be investigated, and so this mechanistic explanation remains speculative, as do the roles of mitochondria and tenofovir diphosphate in tenofovir disoproxil toxicity. More generally, the impossibility of investigating cellular concentration of ARVs in the tissues targeted by their toxicity is a major weakness of these studies. However, it is interesting to note that the neuropsychological effect of efavirenz or the anaemia induced by zidovudine are not explained by the diffusion of these compounds to PBMCs. Significant correlations indicate that concentrations in PBMCs possibly reflect the diffusion of these drugs to other tissues such as the central nervous system or bone morrow. Measurement of concentrations of ARVs in PBMCs could therefore be an interesting tool to predict and consequently prevent the appearance of toxicities related to HAART in a clinical setting. Further studies are therefore warranted to validate PBMCs as a reliable model for investigating the relationship between intracellular concentrations of ARVs and their toxicity. 5. Conclusions In conclusion, intracellular concentrations of ARVs play a major role in their efficacy and toxicity, and are influenced by numerous factors. Although measurement of intracellular concentrations needs to be standardized, this review demonstrates that relationships between intracellular concentrations of ARVs and their efficacy have been evidenced. Such relationships should be interpreted with caution, as intracellular concentrations reflect the total amount of drug within the cell and not the effective unbound fraction. The number of clinical studies in that area is, however, rather limited, most studies being small and not always adequately designed.

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166 Fig. 1. Host cell-mediated sequential enzymatic phosphorylation steps required for activating the nucleotide- and nucleoside-analogue reverse-transcriptase inhibitors (NTRTIs and NRTIs) to the triphosphate moiety (reproduced from Anderson et al., [17] with permission). 3TC = lamivudine; 5NDPK = 5 nucleoside diphosphate kinase; 5NT = 5 nucleotidase; ABV = abacavir; AMPD = adenosine monophosphate deaminase; AMPK = adenosine monophosphate kinase (adenylate kinase); APT = adenosine phosphotransferase; CBV = carbovir; d4t = stavudine; dck = deoxycytidine kinase; dcmpk = deoxycytidine monophosphate kinase; dda = 2,3dideoxyadenosine; ddc = zalcitabine; ddi = didanosine; DP = diphosphate; FTC = emtricitabine; gk = guanylate kinase; MP = monophosphate; PMPA = tenofovir (PMPA diphosphate is a triphosphate analogue); TDF = tenofovir disoproxil fumarate; TFV = tenofovir; TP = triphosphate; ZDV = zidovudine. Fig. 2. Schematic representation of uptake and efflux transporters that may influence intracellular concentrations of antiretroviral drugs in peripheral blood cells. Transporters are named by gene and proteins (adapted from Ford et al., [53] with permission, and updated [55-57] ). ABC = adenosine triphosphate binding cassette; BCRP = breast cancer resistance protein; ENT = equilibrative nucleoside transporter; hcnt = concentrative nucleoside transporter; MRP = multidrug resistance protein; NRTI = nucleoside reverse transcriptase inhibitor; OCT = organic cation transporters; P-gp = P-glycoprotein; PI = protease inhibitor; SLC = solute carrier.

167 Table I. Summary of pharmacokinetic parameters of available antiretroviral drugs [6,8-13] Drug Bioavailability t max (h) Protein binding (%) Elimination pathway Plasma t ½ (h) Intracellular t ½ (h) (%) Entry inhibitors Enfuvirtide 70 (SC) 7 97 Peptidases amino acids 3 8 Maraviroc % renal + CYP3A 13 Nucleoside and nucleotide reverse transcriptase inhibitors Zidovudine % renal + 80% glucuronidation UGT2B Didanosine 40 1 <5 50% renal Stavudine 80 1 <5 80% renal Lamivudine 80 1 <5 80% renal Abacavir <5% renal + liver biotransformation Tenofovir <10 80% renal disoproxil fumarate Emtricitabine 90 1 <5 80% renal 9 39 Non-nucleoside reverse transcriptase inhibitors Efavirenz <1% renal + CYP2B6 50 Nevirapine <15% renal + CYP2B6 + 3A Etravirine ND <1% renal+cyp3a+cyp2c Integrase inhibitor Raltegravir ND 3 83 <5% renal + UGT1A1 9 Protease inhibitors Saquinavir <5% renal + CYP3A 5 Indinavir % renal + CYP3A Ritonavir <5% renal + CYP3A 3 5 Nelfinavir <5% renal + CYP3A 5 6 Lopinavir/rito ND 5 99 <5% renal + CYP3A 5 6 navir a Amprenavir <5% renal + CYP3A 7 12 Atazanavir ND 2 86 <10% renal + CYP3A 7 Darunavir ND <5% renal + CYP3A Tipranavir ND 3 99 <5% renal + CYP3A 6 (single dose) a Low dose ritonavir. CYP = cytochrome P450; ND = no data; SC = subcutaneous administration; t ½ = elimination half-life; t max = time to reach the maximum plasma concentration; UGT = uridine diphosphate glucuronosyltransferase.

168 Correlation References Table II. Intracellular pharmacokinetics of antiretroviral drugs and relationships with plasma pharmacokinetics Drug dose a N Design Plasma and intracellular pharmacokinetics plasma (n obs /n vis ) intracellular (n obs /n vis ) parameters b plasma intracellular ratio c NRTIs Zidovudine 8 mg/kg/day 49 1/2 1/2 C trough µg/ml fmol/10 6 cells NR Yes [99] (R 2 = NR) 300 mg bid 14 8/1 3/3 AUC µg h/ml fmol h/10 6 [81] cells NR No 300 mg bid 15 2/2 2/2 C 1 C µg/ml fmol/10 6 [87] cells NR 300 mg bid 26 5/1 C trough fmol/10 6 cells 600 mg qd 27 C trough fmol/10 6 cells C fmol/10 6 cells 200 mg tid 38 7/2 4/2 AUC 8 women µg h/ml fmol h/10 6 cells NR NR men µg h/ml fmol h/10 6 cells NR NR 300 mg tid 33 2/3 C trough C fmol/10 6 cells 300 mg bid 23 3/7 3/7 AUC 2 NR Yes wk ± 0.71 µg h/ml 110 ± 80 fmol h/10 6 cells (R 2 = NR) wk ± 1.52 µg h/ml 130 ± 110 fmol h/10 6 cells wk ± 0.41 µg h/ml 130 ± 120 fmol h/10 6 cells 200 or 300 mg bid d 8 1 2/12 1 2/12 C 2 C µg/ml fmol/10 6 cells NR No CL/F L/h/kg NR IV 1 2 mg/kg/h e 28 f >1/1 >1/1 C delivery µg/ml fmol/10 6 cells NR No CL/F L/h/kg NR 100 mg qd 10 6/1 6/1 AUC ± 0.13 µg h/ml 420 ± 420 fmol h/10 6 cells NR No 300 mg bid AUC ± 0.21 µg h/ml 610 ± 810 fmol h/10 6 cells 500 mg qd 21 6/2 6/2 AUC 8 NR No wk ± 0.31 µg h/ml 3290 ± 970 fmol h/10 6 cells wk ± 0.41 µg h/ml 2160 ±1090 fmol h/10 6 cells 500 mg qd 6 6/1 6/1 AUC ± 0.27 µg h/ml 4200 ± 2720 fmol h/10 6 cells NR No Didanosine 400 mg qd 16 4/1 C trough C fmol/10 6 cells 400 or 250 mg qd 28 1/1 1/1 C 2.5 C 28.5h µg/ml 0 23 fmol/10 6 cells NR No Stavudine 40 mg bid 19 2/2 2/2 C 1 C µg/ml fmol/10 6 cells NR 40 or 30 mg bid 28 1/1 1/1 C 2.5 C 28.5h µg/ml 0 99 fmol/10 6 cells NR Yes (R 2 = 0.46) Lamivudine 300 mg qd 15 4/1 4/1 AUC µg h/ml fmol h/10 6 cells C max µg/ml C trough µg/ml fmol/10 6 cells 4 mg/kg/day 49 1/2 1/2 C trough µg/ml fmol/10 6 cells NR Yes (R 2 = NR) 150 mg bid 14 8/1 3/3 AUC µg h/ml fmol h/10 6 cells NR NR 150 mg bid 41 2/2 2/2 C 1 C µg/ml fmol/10 6 cells NR [91] [92] [72] [83] [84] [93] [94] [12] [13] [85] [11] [87] [11] [102] [99] [81] [87] 300 mg qd 150 mg bid /1 C trough C 24 C trough fmol/10 6 cells fmol/10 6 cells fmol/10 6 cells 300 mg qd /6 C trough fmol/10 6 cells 150 mg bid 32 2/3 C trough C fmol/10 6 cells 150 mg bid e 8 1 2/12 1 2/12 C 2 C µg/ml fmol/10 6 cells CL/F µg/ml NR Abacavir 300 mg bid 600 mg qd /2 8/2 8/2 8/2 AUC 24 C max AUC 24 C max 300 mg bid 12 4/1 4/1 AUC 24 C max 7.90 ± 3.63 µg h/ml 1.84 ± 0.73 µg/ml 8.52 ± 3.66 µg h/ml 3.85 ± 1.42 µg/ml µg h/ml µg/ml 814 ± 521 fmol h/10 6 cells 58 ± 31 fmol/10 6 cells 1051 ± 746 fmol h/10 6 cells 114 ± 86 fmol/10 6 cells fmol h/10 6 cells C trough µg/ml fmol/10 6 cells 600 mg qd 8 1 2/6 C trough fmol/10 6 cells 300 mg bid 9 8/1 4/1 C trough NQ NQ C max µg/ml NR ± ± 79.9 Yes (R² = 0.66) Yes (R 2 = NR) Yes (R 2 = NR) [91] [82] [72] [84] [167] [102] [82] [86]

169 600 mg qd 5 8/1 C trough C 1 C 12 C 14 C 16 C 18 C 20 C 22 Tenofovir disoproxil fumarate 300 mg qd 27 4/1 4/1 AUC 24 C fmol/10 6 cells fmol/10 6 cells 188 fmol/10 6 cells fmol/10 6 cells fmol/10 6 cells fmol/10 6 cells fmol/10 6 cells fmol/10 6 cells µg h/ml fmol h/10 6 cells C max µg/ml C trough µg/ml fmol/10 6 cells 300 mg qd 7 1 2/6 C trough fmol/10 6 cells 300 mg qd 8 4/1 C trough C fmol/10 6 cells 300 mg qd 22 7/1 3/1 AUC µg h/ml NR NR No C 1 NR fmol/10 6 cells C 4 NR fmol/10 6 cells C µg/ml fmol/10 6 cells NR NR Emtricitabine 25 mg bid 9 8/1 5 2/2 C mg qd 8 C mg bid 8 C mg qd 8 C mg bid 8 C 1 NNRTIs Efavirenz 600 mg qd 49 1/2 1/2 C 12 wk 4 wk 24 C 4 C 4 C 4 C 4 C 4 NR NR NR NR NR NR NR NR NR NR fmol/10 6 cells fmol/10 6 cells fmol/10 6 cells fmol/10 6 cells fmol/10 6 cells fmol/10 6 cells fmol/10 6 cells fmol/10 6 cells fmol/10 6 cells fmol/10 6 cells µg/ml µg/ml µg/ml µg/ml No No 600 mg qd 10 5/1 5/1 AUC µg h/ml fmol h/10 6 cells Yes (R 2 = 0.59) 600 mg qd 55 1/1 1/1 AUC µg h/ml µg h/ml NR Yes (R² = 0.24) NVP 200 mg bid 10 5/1 5/1 AUC µg h/ml µg h/ml Yes (R 2 = 0.62) 400 mg bid 10 1/1 1/1 AUC µg h/ml µg h/ml NR NR PIs Saquinavir 1600 mg qd (+ 100 mg 12 4/1 4/1 AUC µg h/ml µg h/ml Yes ritonavir) (R 2 = 0.63) 1600 mg qd (+ 100 mg ritonavir) 13 1/3 1/3 C Yes (R 2 = 0.31) wk µg/ml µg/ml wk µg/ml µg/ml wk µg/ml µg/ml 600 mg tid (+ 100 mg ritonavir bid) 9 2/1 2/1 C trough C max Indinavir 800 mg tid 10 6/1 6/1 AUC 8 NR NR 25.1 ± 4.2 µg h/ml 7.6 ± 1.0 µg h/ml NR No C trough 10.7 ± 1.3 µg/ml 3.2 ± 0.7 µg/ml NR 800 mg bid (+ 100 mg 19 2/1 2/1 C trough NR NR 0 20 NR ritonavir) C max NR NR 1 5 NR 400 mg bid (+ 400 mg C trough NR NR NR ritonavir) C max NR NR NR 800 mg tid C trough NR NR 0 0 NR C max NR NR NR Ritonavir 100 mg bid 11 14/1 4/1 AUC µg h/ml µg h/ml NR 100 mg qd 12 4/1 4/1 AUC µg h/ml µg h/ml No 400 or 100 mg bid 22 2/1 2/1 C trough NR NR NR Nelfinavir 1250 mg bid 12 5/1 5/1 AUC µg h/ml µg h/ml NR Yes NR NR NR NR [101] [102] [82] [85] [95] [10,100] [96] [34] [90] [30] [90] [97] [89] [79] [9] [79] [98] [97] [79] [78]

170 750 mg tid 12 2/1 2/1 C trough Lopinavir 400 or 533 mg bid (+100 or 133 mg ritonavir) (R 2 = 0.45) C trough NR NR NR Yes (R 2 = 0.34) C max 38 1/2 1/2 C trough NR NR NR NR wk µg/ml µg/ml Yes (R² = 0.72) wk µg/ml µg/ml No 11 14/1 4/1 AUC µg h/ml µg h/ml NR 400 mg bid (+ 100 mg ritonavir) a All studies were performed at steady state. b Values of the pharmacokinetics parameters as published in the original article: range, IQR or mean ±SD. c Ratio defined by intracellular/plasma concentration. d Individualized regimen after the second visit. e Followed by a continuous infusion of 1 mg/kg/h until delivery. f Pregnant women. NR NR [79] [88] [98] AUC = area under the concentration-time curve; AUC x = AUC from 0 to x hours; bid = twice daily; CL/F = oral clearance; C delivery = concentration at the time of delivery; C max = maximum concentration; C trough = trough concentration; C x = concentration at x hours; IQR = interquartile range; IV = intravenous; n = number of patients;; NNRTI = non-nrti; NQ = not quantifiable; NR = not reported; NRTI = nucleoside reverse transcriptase inhibitor; n obs /n vis = no. of samples per visit/no. of visits; PI = protease inhibitor; qd = once daily; RTV = ritonavir; SC = subcutaneous administration; SD = standard deviation; tid = three times daily.

171 Table III. Relationships between intracellular concentrations and efficacy of antiretroviral drugs in patients Study Primary objective (yes/no); type of trial Intracellular moieties NRTIs Moore et al. [87] Aweeka et al. [122] Anderson et al. [8] Fletcher et al. [84] Yes; clinical trial substudy No; cross-sectional analysis Yes; clinical trial substudy Yes; clinical trial substudy Dosage regimen Patients Studied parameters from intracellular concentrations Wk 28: average of 1 h and 4 h concentrations postdose Efficacy criteria Change between wk 0 and wk 24 in: (i) log plasma HIV RNA (ii) CD4 cell count 3TC-TP 3TC 150 mg + ZDV 300 mg bid or 3TC 150 mg + d4t 40 mg bid 39 treatment-naïve (i) p < 0.02 (ii) p = NS ZDV-TP, ZDV 300 mg + 3TC 150 mg bid 10 treatment-naïve (i) p < 0.02 (ii) p = NS d4t-tp d4t 40 mg + 3TC 150 mg bid 15 treatment-naïve (i) p = NS (ii) p = NS ZDV -TP Any regimen containing ZDV 13 HCV- or HBVco-infected, with stable regimen > 4 wk ZDV-TP ZDV 300 mg bid + 3TC 150 mg bid + IDV 800 mg tid; or concentrationcontrolled ZDV-3TC-IDV regimen AUC (NCA) from 5 samples: predose and 1, 4, 6 and 8 h postdose 33 treatment-naïve Median concentration above threshold (yes/no) from samples 2 h postdose at wk 2, wk 28 and wk 56 and at 2 8 h postdose at 9 visits from wk 8 to wk 80 Thresholds: ZDV-TP: 30 fmol/10 6 3TC-TP: 7017 fmol/10 6 CD4 cell count at time of pharmacokinetic sampling (i) Time to reach <50 copies/ml of HIV RNA (ii) Undetectable HIV RNA (<50 copies/ml) at wk 24 and wk 52 (iii) Time to loss of virological response in patients achieving undetectable HIV RNA (iv) CD4 cell counts at wk 24 and wk 48 Results a p = NS (i) p = 0.01 (ii) wk 24: p = 0.009; wk 52: p = NS (iii) p = 0.02 (iv) p = NS 3TC-TP (i) p = 0.02 (ii) wk 24: p = 0.002; wk 52: p = (iii) p = (iv) p = NS ZDV-TP Stretcher Yes; PK trial Total ZDV et al. [132] phosphates PIs Lamotte et al. [89] Breilh et al. [88] Chaillou et al. [79] No; secondary objective of clinical trial Yes; observational study No; secondary objective of crosssectional trial ZDV 300 mg bid + 3TC 150 mg bid + IDV 800 mg tid; or concentrationcontrolled ZDV-3TC-IDV regimen 8 treatment-naïve Median concentration from samples 2 h postdose at wk 2, wk 28 and wk 56 and at 2 8 h postdose at 9 visits from wk 8 to wk 80 Change between wk 0 and wk 24 in: (i) log HIV RNA (ii) CD4 cell count (i) p = 0.03 (ii) p = TC-TP (i) p = (ii) p = NS ZDV only: 800 mg/day, 100 mg every 4 h while awake SQV SQV 1600 mg + RTV 100 mg qd + 2 or 3 NRTIs/NNRTIs LPV LPV 400 mg + RTV 100 mg bid + 2 or 3 NRTIs/NNRTIs NFV, IDV, APV, SQV, RTV NFV 750 mg tid or IDV 800 mg tid or APV 1200 mg bid or IDV 800 mg + RTV 100 mg bid or SQV 600 mg + RTV 100 mg bid or APV 600 mg + RTV 100 mg bid or IDV 400 mg + RTV 400 mg bid + 2 or 3 NRTIs/NNRTIs a Significant results always associated better efficacy with higher intracellular concentrations. 21 ZDV-naïve AUC (NCA) from 6 samples: predose and 1, 2, 4, 6 and 8 h postdose at wk 4 and >wk treatment-naïve C trough (24 h postdose) at wk 2, wk 4 or wk LPV-naïve C trough (12 h postdose) at wk 4 and wk experienced Ratio of intracellular to plasma C trough and C max (1.5 3 h postdose) on day of study: (i) main PI (ii) RTV Change between wk 0 and wk 4 or wk 0 and wk 24 in: (i) % CD4 cells (ii) CD4/CD8 ratio Change in plasma HIV RNA between wk 0 and wk 12 Virological success: (i) HIV RNA <50 copies/ml before wk 4 (ii) HIV RNA <50 copies/ml before wk 4 and during all follow-up until wk 24 Undetectable HIV RNA (<40 copies/ml) on day of study (i) wk 4: p = 0.029; wk 24: p = NS (ii) wk 4: p = 0.028; wk 24: p = NS p = NS for all periods of C trough measurements (i) p < (ii) p < (i) p = NS for PI ratio (ii) p = 0.04 for presence of intracellular RTV, p = with RTV ratio in 28 patients receiving RTV 3TC = lamivudine; APV = amprenavir; AUC = area under the concentration-time curve; bid = twice daily; C max = maximum concentration; C trough = trough concentration; d4t = stavudine; HBV = hepatitis B virus; HCV = hepatitis C virus; IDV = indinavir; LPV = lopinavir; NCA = noncompartmental analysis; NFV = nelfinavir; NNRTI = non-nrti; NRTI = nucleoside or nucleotide reverse transcriptase inhibitor; NS = nonsignificant; PI = protease inhibitor; PK = pharmacokinetic; qd = once daily; RTV = ritonavir; SQV = saquinavir; tid = three times daily; TP = triphosphate; ZDV = zidovudine.

172 Table IV. Relationships between intracellular concentrations and toxicity of antiretroviral drugs in patients Study Primary objective; type of trial Intracellular moieties NRTIs Anderson et al. [8] Yes; clinical trial substudy ZDV-TP, 3TC-TP Stretcher Yes; PK trial Total ZDV et al. [132] phosphates Dosage regimen Patients Studied parameters from intracellular concentrations ZDV 300 mg bid + 3TC 150 mg bid + IDV 800 mg tid; or concentration-controlled ZDV-3TC-IDV regimen ZDV only: 800 mg/day; 100 mg every 4 h while awake Durand-Gasselin et al. [99] Yes; PK trial ZDV-TP, 3TC-TP ZDV (8 mg/kg/day in 4 daily doses) ± 3TC (4 mg/kg/day in 2 daily doses) NNRTI Rotger Yes; PK trial EFV EFV + ZDV + 3TC et al. [90] EFV + abacavir + 3TC EFV + d4t + ddi ± PI (doses NR) 33 treatmentnaïve 21 treatmentnaïve a Significant results always associated increased risk of toxicity with higher intracellular concentrations Median concentration above threshold (yes/no) from samples 2 h postdose at wk 2, wk 28 and wk 56 and at 2 8 h postdose at 9 visits from wk 8 to wk 80 Thresholds: ZDV-TP: 30 fmol/10 6 3TC-TP: 7017 fmol/10 6 AUC (NCA) from 6 samples: predose and 1, 2, 4, 6 and 8 h postdose at wk 4 and wk neonates Single-point concentration (time of sampling NR) 55 Intracellular AUC obtained by Bayesian estimation (no. of samples per patient and sampling times NR) Toxicity criterion Occurrence of a grade I laboratory event (haemoglobin, absolute neutrophil count, AST, ALT) Change between wk 0 and wk 4 or wk 0 and wk 24 in: (i) neutrophils (ii) red blood cells (iii) haemoglobin Proportions of the hematological toxicity grade between neonates with intracellular concentrations above or below the observed median Presence of grade I to IV of: (i) sleep disorder (ii) mood disorder (iii) fatigue Results a NS (i) p = NS (ii) p = NS (iii) p = NS NS (i) p = NS (ii) p = 0.02 (iii) p = NS 3TC = lamivudine; AUC = area under the concentration-time curve; bid = twice daily; d4t = stavudine; ddi = didanosine; EFV = efavirenz; IDV = indinavir; NCA = noncompartmental analysis; NR = not reported; NS = nonsignificant; PI = protease inhibitors; PK = pharmacokinetic; tid = three times daily; TP = triphosphate; ZDV = zidovudine.

173 Figure 1

174 Figure 2 Uptake transporters SLC (solute carriers) Efflux transporters ABC (ATP binding cassette) SLC22A, OCT1,2 Organic cation transporters substrates: NRTIs ABCB1, P-gp Substrates : PIs SLC28, hcnt1-3 Na+ dependent substrates: NRTIs SLC29, hent1-2 Facilitated diffusion substrates: NRTIs ABCC1-5, MRP1-5 Substrates : MRP1, MRP2 PIs MRP4, MRP5 NRTIs ABCG2, BCRP Substrates: NRTIs

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