New Model-Based Bioequivalence StatisticalApproaches for Pharmacokinetic Studies with

Sparse Sampling

Florence Loingeville 1,2, Thu Thuy Nguyen1, Julie Bertrand1, FranceMentré1, Andrew Babiskin3, Sun Guoying3, Stella Grosser3, Liang

Zhao3,Lanyan Fang3

1 IAME, UMR 1137 INSERM - University Paris Diderot2 Laboratoire de Biomathématiques, Faculté de Pharmacie - Université de Lille

3 Food and Drug Administration, USA

Journées du GDR Statistique & Santé, 27-28 Septembre 2018, Nantes

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Introduction Methods Simulation study Results Conclusion

Bioequivalence (BE) studies

BE: The difference in the pharmacokinetic (PK) of two formulations of agiven drug does not exceed a predefined threshold (usually δ= log(1.25))

Parallel studies preferable for drugs with long half-life: N/2 subjects receivereference treatment (R), N/2 subjects receive test treatment (T)

Non-Compartmental Approach (NCA) based BE:

Ï Individual NCA estimates of AUCi and Cmaxi

Ï Let βAUC = mean(log(AUCiT ))−mean(log(AUCiR))βCmax = mean(log(CmaxiT ))−mean(log(CmaxiR))

pros cons? Reproductible ? Require more than 10 samples per subject

? Few assumptions ?Not appropriate for complex models

Two one-sided tests (TOST) 1 at levelα=== 5% on βAUC and βCmax :

Reject of H0 (= BE significant) ifÏ (β−δ)/SE(β) ≤−z1−α and (β+δ)/SE(β) ≥ z1−α, with z1−α: (1−α)% quantile of

the normal distribution orÏ CI(β)1−2α% ∈ [−δ;δ] (CI: Confidence Interval)

1Schuirmann, J Pharmacokinet Pharmacodyn, 1987.

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Introduction Methods Simulation study Results Conclusion

Model based (MB) BE

Non linear mixed effect model (NLMEM):yij: concentration for subject i at sampling time j

yij = f (tij,φi)+g(tij,φi)εij

log(φil) = log(λl)+βlTri +ηil where

Ï βl : Test treatment effect on the log of a PK parameter l = (1, . . . ,p)

Ï λl : lth element of the vector of fixed effects

Ï Tri: vector of indicator for treatment group

Ï ηil ∼N (0,ωl): between subject random effect for parameter l

Ï εij ∼N (0,1): residual error

Ï combined error model g(tij ,φi) = a+b× f (tij ,φi)

Vector of population parameters: θ === (λ,β,ω, a, b)

Estimation of θ using SAEM algorithm 2

Estimation of SE(θ) from observed Fisher information matrix Σ

2Kuhn et Lavielle. Comput Stat Data Anal, 2005.

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Introduction Methods Simulation study Results Conclusion

MBBE

AUC and Cmax: secondary parameters of the NLMEM

Ï βAUC and βCmax functions of vectors λ and βÏ SE(βAUC ) and SE(βCmax) obtained using the delta method 3

TOST at levelα=== 5% or 90%CI on βAUC and βCmax

pros cons? Require few samples ? SE under estimated on sparse designs

per subject ⇒ Type I error inflation 4

Objectives: Development, evaluation and comparison of new model-based(MB) statistical approaches for sparse design BE studies

Ï Parametric random effect and residual bootstrapÏ Full distribution estimation using Stan 5

3Oelhert The American Statistician, 1992.4Dubois et al. Stat in Med, 2011.5Stan development team, Rstan, 2012.

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Introduction Methods Simulation study Results Conclusion

MBBE parametric random effect and residual bootstrap

Principle 6

1 Estimation of θ and Σwith saemix

2 Drawing of b = 1, . . . ,B (B=250) matrices of random effects of size N x p fromN (0,Ω)

3 Drawing of vector of residual errors of size∑N

i=1 ni from N (0,1)

4 Simulation of the B vectors of responses

5 Fit the B new data sets with saemix to get the B estimates θb and βb

6 Derive 90%CIs on βCmax and βAUC from the 5th and 95th percentiles of theserie βb

6Thai et al, J Pharmacokinet Pharmacodyn, 2014.

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Introduction Methods Simulation study Results Conclusion

MBBE full distribution using Stan

Principle 7

1 Estimation of θ and ηi with saemix

2 Full distribution in Stan 5

Ï Initialize HMC chain at estimates from step 1Ï Draw 1 chain of 1000 (including 100 burning) samples in a posteriori

distributions of λ, β,ω,a, bÏ out of the B=900 samples derive 90%CI on βCmax and βAUC

Stan model:

Default distributions on fixed effects λ,β

Non-informative priors on ω,a,b

5Stan development team, Rstan, 2012.7Ueckert et al. ACOP, 2015.

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Introduction Methods Simulation study Results Conclusion

PK model

PK model of concentrations of the anti-asthmatic drug theophylline 2

Parallel design: NR=NT =20

One-compartment model with first-order absorption and first-orderelimination. Dose=4mg

Covariate effects in test treatment group: βV =βCl = log(1.25), βKa = 0

0 5 10 15 20 25 30

01

23

45

67

Reference group

Time

Con

cent

ratio

n

AUC=100

Cmax=6.78

Tmax=2.06

0 5 10 15 20 25 300

12

34

56

7

Test group

Time

Con

cent

ratio

nAUC=80

Cmax=5.42

Tmax=2.06

Residual variability: a=0.1 mg/L, b=10%

Random effects for BSV only:

ωka (%) ωV /F (%) ωCL/F (%)22 11 22

2Dubois et al. Stat in Med, 2011.

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Introduction Methods Simulation study Results Conclusion

Simulation scenarios

500 simulated data sets for each of the 4 scenarios

Design N Sampling times (hours) Hypothesis βCL =βV

Rich 40n=10,t = (0.25,0.5,1,2,3.5,5,7,9,12,24)

H0 log(1.25)H1 log(1)

Sparse 40n=3,t = (0.25,3.35,24)

H0 log(1.25)H1 log(1)

Under H0: βAUC =βCmax = log(0.8),Under H1: βAUC =βCmax = log(1) = 0

Evaluation:

Ï Type I error (IP95%(0.05) = [0.033;0.073])Ï PowerÏ Computing times

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Introduction Methods Simulation study Results Conclusion

Type I error and power estimatesNCA MB Asymptotic MB Bootstrap MB Full distribution

SE CI CI

Mean computing 15 sec 15 sec 43 min 5 mintime for 1 data set

Rich designNCA MB Asymptotic MB Bootstrap MB Full distribution

SE CI CI

Type I error βAUC 0.058 0.056 0.062 0.040Type I error βCmax 0.062 0.058 0.064 0.044

Power βAUC 0.814 0.830 0.832 0.762Power βCmax 0.998 1.000 1.000 0.962

Sparse designNCA MB Asymptotic MB Bootstrap MB Full distribution

SE CI CI

Type I error βAUC - 0.076 0.074 0.060Type I error βCmax - 0.066 0.068 0.050

Power βAUC - 0.910 0.916 0.868Power βCmax - 1.000 1.000 0.992

IP95%(0.05) = [0.033;0.073] with 500 simulated data sets9/15

Introduction Methods Simulation study Results Conclusion

90% CI(β) on sparse designβAUC βCmax

data set

β AU

C

MB AsymptoticMB Full distribution (Stan) MB Bootstrap

−0.

8−

0.6

−0.

4−

0.2

0.0

0.2

0.4

1 2 3 4 5 6 7 8 9 10

data set

β Cm

ax

−0.

8−

0.6

−0.

4−

0.2

0.0

0.2

0.4

1 2 3 4 5 6 7 8 9 10

H0

data set

β AU

C

−0.

8−

0.6

−0.

4−

0.2

0.0

0.2

0.4

1 2 3 4 5 6 7 8 9 10

data set

β Cm

ax

−0.

8−

0.6

−0.

4−

0.2

0.0

0.2

0.4

1 2 3 4 5 6 7 8 9 10

H1

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Introduction Methods Simulation study Results Conclusion

90% CI(β) on sparse design, under H0

data set

β AU

C

MB AsymptoticMB Full distribution (Stan) MB Bootstrap

−0.

8−

0.6

−0.

4−

0.2

0.0

0.2

0.4

1 2 3 4 5 6 7 8 9 10

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Introduction Methods Simulation study Results Conclusion

Conclusion

Implementation of new methods for MBBE using parametric bootstrap andfull distribution sampled in Stan

Correction of MB TOST with asymptotic SE on sparse design using full

distribution sampled in Stan

Ï faster than bootstrap

Perspective: implementation of the methods for 2-period 2-sequencecrossover BE studies (accounting for within-subjects variability)

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Introduction Methods Simulation study Results Conclusion

Thank you

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Introduction Methods Simulation study Results Conclusion

Simulation scenarios

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Introduction Methods Simulation study Results Conclusion

90% CI(β) on rich designβAUC βCmax

data set

β AU

C

NCAMB AsymptoticMB Full distribution (Stan) MB Bootstrap

−0.

8−

0.6

−0.

4−

0.2

0.0

0.2

0.4

1 2 3 4 5 6 7 8 9 10

data set

β Cm

ax

−0.

8−

0.6

−0.

4−

0.2

0.0

0.2

0.4

1 2 3 4 5 6 7 8 9 10

H0

data set

β AU

C

−0.

8−

0.6

−0.

4−

0.2

0.0

0.2

0.4

1 2 3 4 5 6 7 8 9 10

data set

β Cm

ax

−0.

8−

0.6

−0.

4−

0.2

0.0

0.2

0.4

1 2 3 4 5 6 7 8 9 10

H1

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