Population Design in Nonlinear Mixed Effects Multiple Response Models: extension of PFIM and evaluation by simulation with NONMEM and MONOLIX

Transcription

1 Populaton Desgn n Nonlnear Mxed Effects Multple Response Models: extenson of PFIM and evaluaton by smulaton wth NONMEM and MONOLIX May 4th 007 Carolne Bazzol, Sylve Retout, France Mentré Inserm U738 Unversty Pars7 Pars, France PODE 1

2 Context (1) Nonlnear mxed effects models (NLEM) Handle data measured repeatedly through tme and descrbed by nonlnear models Estmaton of the mean parameters and ther ntersubject varablty n the populaton to be treated Allow sparse data Nonlnear mxed effects models for multple responses For each ndvdual : vectors of observatons n tme from K dfferent types of measure (k=1,,k) Dfferent tmes of measurement n case of dfferent tme scale for each type of measure Examples Pharmacoknetcs/Pharmacodynamcs (PK/PD) Drug and ts metabolte PODE

3 Context () Maxmum lkelhood estmaton No analytcal expresson for the lkelhood Several methods Lnearsaton of the log-lkelhood NONMEM (1) FO method (Frst Order) : lnearsaton of the model around the mean of the random effects FOCE method (Frst Order Condtonal Estmate) () : lnearsaton of the model around ndvdual values of the random effects Stochastc approach MONOLIX (3) : SAEM algorthm (4) Based on the EM algorthm Used Markov Chan Monte Carlo Estmaton by stochastc approxmaton (1) Beal SL, Shener LB. NONMEM Project Group, Unversty of Calforna, 199 () Lndstrom MJ, Bates DM. Bometrcs, 1990 (3) MONOLIX, Verson.1. (007) (4) Khun E, Lavelle M. Computatonal Statstcs and Data Analyss, 005 3

4 Context (3) Collect of data: Importance of the choce of the desgn Impact on the precson of estmaton of the populaton parameters Ξ Populaton desgn Model for one response N subjects Q groups of N q subjects wth a same elementary desgn ξ q = {t 1, t,,t nq } n q samples Allocaton n tme Ξ = {[ ξ N ]; [ ξ, N ];...;[, ]} 1, 1 ξq N Q Model for multple responses Ξ = 1 K 1 K 1 K {[ ( ξ, ξ,.., ξ ), N ]; [( ξ, ξ,.., ξ ), N ];... ;[( ξ, ξ,.., ), N ]} Q Q ξq Q Elementary desgn composed of several sub-desgn ξ, k = 1,, K, assocated wth the k th type of measurement : k q 1 K { ξ, ξ ξ } ξ =,.., PODE 4

5 Context (4) Desgn evaluaton and optmsaton Approach based on the Fsher nformaton matrx For sngle response model Lnearsaton of the model usng a frst order Taylor expanson around the expectaton of the random effects (1) Relevance of ths approach demonstrated on real data () For multple response model Extenson of M F for multple responses (3) (4) Same method as for a model wth one response Computaton of the matrx more complex Some parameters ncluded n several models Need to be consdered n the dervatves Relevance of ths extenson wth ths frst order approxmaton? (1) Mentré F, Mallet A, Baccar D. Bometrka, 1997 () Retout S, Mentré F. Bruno R. Statstcs n Medcne, 00 (3) Hooker A, Vcn P. The Amercan Assocaton of Pharmaceutcal Scentsts Journal, 005 (4) Gueorgueva I, Aarons L, Ogungbenro K, Jorga KM, Rodgers T, Rowland M. Journal of Pharmacoknetcs and Pharmacodynamcs, 006 5

6 Objectve Evaluaton of the relevance of ths frst order extenson for multple response model by smulaton PODE 6

7 PODE 7 f k descrbng nonlnear model θ k vector of ndvdual parameters Notaton Nonlnear mxed effects model for one ndvdual among N [ ] T K T T y y y Y,...,, 1 = θ = β +b or θ = β exp(b ), as θ =g(β,b ) wth ~ )), ( ( ), ( nt slope er F F Y ξ θ σ σ ε ξ θ + + = Statstcal model ( ) 0,Ω N b ε are supposed ndependent from one type of measurement to the other. = ), ( ), ( ), ( ), ( K K K f f f F ξ θ ξ θ ξ θ ξ θ M )), ( ( ), ( nt k k k slope er k k k k k f f y k k ξ θ σ σ ε ξ θ + + =

8 Choce of a model Evaluaton by smulaton : PK/PD model PK model Populaton desgn f PK PK PK Dose Cl ( θ, ξ ) = exp( ξ V V PK θ PK : Cl et V Proportonal error model ) Ξ = PK PD {[ ( ξ,ξ ), N ]} ξ PK = {0.166,6,1} ξ PD = {0.166,6,1,0} N =100 PD model f PD ( θ PK, θ PD, ξ PD ) = E E f ( θ ( θ, ξ, ξ PK PD max PK + 0 C + f 50 PK PK PD ) ) θ PD : E 0, E max et C 50 Addtve error model Exponental random effects for all the parameters PODE 8

9 Evaluaton : Method (1) Implementaton of ths extenson n PFIM Computaton of the predcted standard errors wth the extenson of PFIM Relatve standard errors : RSE PFIM Comparson to the predcted SE obtaned wth an exact method Computaton of M F by the SAEM algorthm (MONOLIX) Lous method Exact method wthout lnearsaton gold standard Smulaton of one data set wth subjects Asymptotc propertes of M F Rescale of the SE for N=100 subjects PODE 9

10 Predcted RSE (%): PFIM/SAEM Emprcal FO (1000) FOCE (853) SAEM (1000) Predcted RSE PFIM RSE SAEM PODE 10

11 Evaluaton : Method () Comparson to the emprcal RSE (NONMEM et MONOLIX) Smulaton of 1000 data sets (R software) Estmaton of the populaton parameters NONMEM (FO et FOCE) MONOLIX : SAEM For each method of estmaton: Computaton of the emprcal RSE : standard-devaton on the 1000 estmates of each parameter PODE 11

12 Data sets convergence On the 1000 data sets Convergence obtaned for FO : 1000 data sets FOCE : 853 data sets SAEM : 1000 data sets PODE 1

13 Emprcal RSE ( %) Emprcal FO (1000) FOCE (853) SAEM (1000) Predcted RSE PFIM RSE SAEM PODE 13

14 Emprcal and predcted RSE ( %) Emprcal FO (1000) FOCE (853) SAEM (1000) Predcted RSE PFIM RSE SAEM PODE 14

15 Evaluaton : Method () Comparson to the dstrbuton of the relatve standard errors obtaned on each data set for each parameter NONMEM (FO et FOCE) MONOLIX : SAEM Computaton of the SE Lnearsaton around the ndvdual parameters estmated by SAEM wthout lnearsaton Lous method Comparson wth RSE PFIM and the emprcal RSE PODE 15

16 Data sets convergence (%) On the 1000 data sets Convergence obtaned for : FO : 1000 data sets FOCE : 853 data sets SAEM : 1000 data sets Varance covarance matrx obtaned for : FO : 997 data sets FOCE : 798 data sets SAEM : 1000 data sets PODE 16

17 RSE (%) RSE PFIM RSE emp β Cl β Emax β C50 FO 3 outlers (86,9%, 80,7%, 131,7%) FOCE outlers (64,4%,466%) 1 outler (30,5%) 1 outler(84,8%) RSE (%) Cl Emax FO FOCE oulters(98.8%) SA_Lo 1 outler (1.1%) 1 outler(09,4%) C50 σ slopepk 3 outlers(53.%,596.6%,100%) σ nterpd

18 RSE (%) RSE PFIM RSE emp β Cl β Emax β C50 FOCE outlers (64.4%,466,0%) 1 oulter (45%) outlers (84.6%,43.%) SA_Lo 17 fles wth NA + outlers (1.9%,13.7%) 7 outlers [10.6% : 41.8%] RSE (%) Cl Emax C50 σ slopepk σ nterpd FOCE 1 outler (98;8%) 1 oulter (13;%) outlers(45.6%,138.8%) 1 outler (48.4%) SA_Lo 1 outler (1,1%) 1 outler(09,4%) 5 outlers [500%-10^18%] 1 outler (109,5%) outlers (153.3%,163.3%)

19 Concluson Relevance of the frst order extenson Predcted RSE by PFIM equvalent to the RSE predcted by SAEM close to the emprcal RSE of FOCE and SAEM concordant wth the dstrbuton of the RSE obtaned wth SAEM (lnearsaton) and FOCE Although the extenson of M F for multple response s based on a frst order approxmaton the predcted RSE are close to those computed by FOCE and not by FO Extenson n PFIM and PFIMOPT for K responses : PFIM 3.0 /PFIMOPT 3.0 PODE 19

20 PODE 0

21 Bas and RMSE: FO, FOCE and SAEM PODE 1

22 Bas (%) and RMSE (%) Bas (%) Bas (%) β Cl β V β Emax β E0 β C50 Cl V E 0 E max C σ 50 slopepk σ nt erpd FO (1000) FOCE (853) SAEM (1000) RMSE (%) RMSE (%) β Cl β V β Emax β E0 β C50 Cl V E 0 E max C σ 50 slopepk σ nt erpd PODE

23 RMSE (%) Bas (%) Bas (%) and RMSE ( %) β Cl β V β Emax β E0 β C50 Cl V E 0 E max C σ 50 slopepk σ nt erpd FOCE (853) SAEM (1000) β Cl β V β E0 β β Emax C50 Cl V E 0 E max C σ 50 slopepk σnterpd PODE 3