nonlinear mixed effect models for the analysis and design of bioequivalence biosimilarity studies
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Nonlinear mixed effect models for the analysis and design of bioequivalence/ biosimilarity studies. Pr France Mentré Anne Dubois, Thu-Thuy N‘Guyen, Caroline Bazzoli UMR 738: Models and Methods for Therapeutic Evaluation of Chronic Diseases INSERM – Université Paris Diderot. OUTLINE .

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nonlinear mixed effect models for the analysis and design of bioequivalence biosimilarity studies

Nonlinear mixed effect models for the analysis and design of bioequivalence/biosimilarity studies

Pr France Mentré

Anne Dubois, Thu-Thuy N‘Guyen, Caroline Bazzoli

UMR 738: Models and Methods for Therapeutic Evaluation of Chronic Diseases

INSERM – Université Paris Diderot

  • Introduction
    • Pharmacometrics
    • Nonlinear mixed effect models (NLMEM)
    • Bioequivalence/biosimilarity studies
  • Bioequivalence test in NLMEM
    • Method
    • Simulation
    • Real example
  • Optimal design in NLMEM
    • Method
    • Simulation
    • Real example
  • Pharmacometrics and drug development
1 introduction

Nonlinear mixed effect models

Bioequivalence/ biosimilarity studies







The science of quantitative clinical pharmacology

  • Clinical pharmacology = PK + PD
    • Both during drug development and in clinical care
    • Main statistical tool: Nonlinear Mixed Effect Models

Data generated during clinical trials & patient care

Rational drug development & pharmacotherapy

Knowledge extraction



+ Disease models

pk pd data
PK/PD data
  • Pharmacokinetics (PK)
    • Study of the time course of drug in the body
    • PK parameters: CL, V…
  • Pharmacodynamics (PD)
    • Study of the effects of drug in the body
    • PD parameters: Emax, EC50
  • Analysis of PK/PD data: 2 types of approach
  • Non compartmental approach (NCA)
  • Model-based approaches
pk non compartmental approach
PK: Non compartmentalapproach
  • Few hypotheses
    • >10 concentrations measurements per subject
    • Trials in healthy volunteers
  • Computation
    • Parameters of interest
      • Area under the curve (AUC)
        • Different time intervals: 0-tlast, 0-infinity
        • Extrapolation between tlast and infinity, computation of terminal slope
      • Maximum concentration (Cmax)
      • Half-life (linear regression on last log concentrations)
    • Algorithm: linear or log-linear trapezoidal method
pk model based approach
PK: Model-basedapproach
  • PK models: Human body described as a set of compartments
    • Physiological parameters: Clerance of elimination, volume of distribution, rate constants
  • Example: 1 cp model with first order absorption and elimination
advantages of modelling
Advantages of modelling
  • Quantitative summary of time-profiles in few 'physiological' parameters
  • Prediction/simulation for other doses …
  • Test of hypotheses on mechanisms of action of drugs
  • Comparison of groups of patients through parameters
  • Comparaison of response to different treatment
  • Analysis of all longitudinal data in clinical trials
  • ….
  • Experimental (rich)
    • Limited number of individuals (N = 6 to 50)
    • Numerous measures per subject (n=6 to 20)
    • Generally studies of short duration
    • Identical, balanced sampling protocols
    • Examples : in vitro, preclinical PK, phase I
    • Analysis of information of each individual separately then summary statistics or global approach
  • Population (sparse)
    • Large number of individuals (N = 50 to 2000)
    • Few measures by subject (n = 1 to 10)
    • Various and unbalanced sampling protocols
    • Repeated doses, chronic administration
    • Example : PK/PD in phase IIb, III, post AMM, NDA
    • Analysis of information on all individuals together
nonlinear mixed effect models 1
Nonlinear mixed effectmodels (1)
  • Analysis of sparse or rich data
  • Global analysis of data in all individuals
  • Parametric PKPD Model: nonlinear in parameters
  • One individual one vector of parameters
  • Set of individuals
    • Same parametric model
    • Inter-individual variability inter-parameter variability
  • Statistical model
    • PKPD parameters are random in the population
    • Mixed effects model (random + fixed)
    • Nonlinear mixed effects models
  • Also called ‘population approach’
nonlinear mixed effect models 2
m ?

sd ?

Estimates of







Non linear mixed effects model

Single-stage approach (population analysis)


Nonlinear mixed effectmodels (2)

From Steimer (1992) : « Population models and methods, with emphasis on pharmacokinetics », in M. Rowland and L. Aarons (eds), New strategies in drug development and clinical evaluation, the population approach

nonlinear mixed effect models 3
Nonlinear mixed effectmodels (3)
  • Increasingly used
    • in all phases of drug development for analysis of PKPD data
    • in clinical use of drug for analysis of PKPD variability and for therapeutic drug monitoring
    • for analysis of response in clinical trials and cohorts
  • Relies on several assumptions
    • structural model (model nonlinear with respect to parameters)
    • model for between-subject variability (assumption on random effects)
    • model for residual error
  • Research in estimation methods, covariate testing, optimal design, model evaluation
bioequivalence biosimilarity studies
Bioequivalence/ biosimilaritystudies
  • Trials comparing pharmacokinetics of several formulations of the same drug
    • used for generic development and for formulation of biologics
  • FDA and EMEA guidelines
    • Two-periods, two-sequences crossover trials
    • Compute AUC and Cmax by non compartmental analysis
    • Test on log parameters using linear mixed effects model
      • with treatment, period and sequence effects
  • Limitations of NCA
    • >10 samples per subject → study on healthy volunteers
    • Estimation of AUC and Cmax by NCA not appropriate for nonlinear PK or complex PKPD models (similarity of kinetics of drug effect)
    • Parameters assumed to be estimated without error
    • Omit data below quantification limit

Propose and develop

  • estimation methods and tests
  • optimal design tool with prediction of number of subjects needed

for bioequivalence/biosimilarity analysis using NLMEM

statistical model
Statistical model
  • yijk concentration for individual i =1,…,N at sampling time j=1,…,nik for period k=1,…,K
  • fik individual parameter
  • ijk residual error
  • Parameters: fixed effects, variance of random effects, a and b in error model
maximum likelihood estimation in nlmem
Maximum Likelihood Estimation in NLMEM
  • Problem: no close form for the likelihood in NLMEM
  • Severalstatisticaldevelopments and specific software
  • Linearizationalgorithms: FO, FOCE
    • Not consistent
    • Very sensitive to initial conditions
  • More recentalgorithmswithoutlinearization
    • Adaptative Gaussian Quadrature
      • Only for model withsmallnumber of randomeffects
    • Stochastic Approximation EM
      • Extension of EM algorithmwithproven convergence

Delyon, Lavielle & Moulines (1999). Convergence of a stochastic approximation version of the EM procedure. Ann Stat, 27: 94-128.

Kuhn, Lavielle (2005). Maximum likelihood estimation in nonlinear mixed effectmodels, Comput Stat Data Anal, 49: 1020-1038.

saem algorithm
SAEM algorithm
  • EM algorithm
    • E-step: expectation of the log-likelihood of the complete data
    • M-step: maximisation of the log-likelihood of the complete data
  • Mixed-effects models
    • individual random-effects = missing data
    • Problem in NLMEM: no close form for the E step
  • SAEM: decomposition of E-step in 2 steps
    • S-step: simulation of individual parameters using MCMC
    • SA-step: stochastic approximation of expected likelihood
  • Various extensions

Samson, Lavielle, Mentré (2006). Extension of the SAEM algorithm to left censored data in non-linear mixed-effects model: application to HIV dynamics model.Comput Stat Data Anal 51: 1562-74.

Samson, Lavielle, Mentré (2007). The SAEM algorithm for group comparison tests in longitudinal data analysis based on non-linear mixed-effects model: application to HIV dynamics model.Stat Med, 26: 4860-75.

monolix software
MONOLIX software
  • Free Matlab software implementing SAEM
    • developed under supervision of Pr Marc Lavielle at INRIA
    • stand-alone version using MCR
    • v1.1 available since Feb 2005
    • v3.1 released in October 2009
  • Success of MONOLIX
    • Team of 4 development engineer from INRIA
    • Grant from ANR (2005-2008)
    • Use in academia and drug companies
    • Monolix project : Support from drug companies
  • Success of SAEM
    • Now implemented in NONMEM, most used software in the area
tests in bioequivalence trials
Tests in bioequivalence trials
  • Global estimation with SAEM algorithm
    • Estimation with the complete model withtreatment, period and sequenceeffect on all parameters, WSV in addition to BSV
      • Extension of the SAEM algorithm
    • SE derivedfrom Fisher information matrix
  • T : treatmenteffect on one log-parameter
  • Bioequivalence test
    • H0 : {T≤ -DL or T≥ +DL}
    • H1 : {-DL ≤T≤ +DL}
    • Schuirmann test or TOST: unilateral test for H0,-D and H0,+D
    • Reject H0with=5%:
      • if H0,-D and H0,+Drejectedwith=5%
      • if 90%CI of Tincluded in [-DL; +DL]
wald and lrt for bioequivalence
Wald and LRT for bioequivalence
  • Wald tests
    • TOST for Tfrom SE for parameter in model (e.g. AUC)
    • For secondaryparameters (e.g. Cmax)
      • Derivation of SE by delta method or simulation
  • LRT
    • Complete model: log-likelihoodLall
    • For parameter in model: estimation withTfixed to -DL or +DL
      • log likelihoodL-Dor L+D
    • Reject of H0 if

[10] Panhard, Samson. Biostatistics. 2009

evaluation by simulation design
Evaluation by simulation: design

(Panhard & Mentré, Stat Med, 2005; Dubois, Gsteiger, Pigeolet & Mentré, Pharm Res, 2009)

  • PK model with one-compartment : ka, V/F, CL/F
  • Two-periods of four-periodscrossover trial
    • Treatmenteffect on CL/F and V/F
    • Equivalence limitDL = 0.2
  • Two designs with N = 40 patients
    • Original n=10 , Sparse: n=3 measurements/ patient/ period
  • Twolevels of variability
    • Randomeffects
      • Lowvariability (BSV=20%, WSV=10%): Sl,l
      • High variability (BSV=50%, WSV=15%): Sh,l
    • Error model: Lowvariability (a=0.1, b=10%)
evaluation by simulation method
Evaluation by simulation: method
  • 1000 simulated trials under H0,-D and H0,+D for each design and each variability setting
  • Analysis by SAEM in MONOLIX v2.4
  • Evaluation of extension of SAEM
    • Computation of bias, RMSE
    • Designs with 2 or 4 periods for H0, -D
  • Evaluation of type I error of Wald and LRT for bioequivalence on AUC and Cmax
    • For H0, -D and H0, +D ,  estimated by the proportion of simulated trials for which the null hypothesis is rejected
    • Designs with 2 periods
relative rmse variances 2 or 4 periods
Relative RMSE variances, 2 or 4 periods



  • RMSE (rich design) < RMSE (sparse design)
  • RMSE (4 periods) < RMSE (2 periods)
  • RMSE satisfactory except for WSV on V/F for low variability
type i error for bioequivalence and 2 periods
Type I error for bioequivalence and 2 periods



W: Wald testL: LRT

  • Type I error at 5% for the rich design
  • Slight inflation of the type I error for the sparse design and large variability
  • Close results for Wald test and LRT on AUC
conclusion on estimation and test
Conclusion on estimation and test
  • SAEM algorithm in MONOLIX software
    • Accurate extension for estimation of WSV and crossover trials analysis
  • Model-based bioequivalence tests
    • Good tool applicable to rich and sparser design
    • Good statistical properties under asymptotic conditions
    • Wald test simpler than LRT and extended for secondary parameters
    • Correction of SE needed for small sample size and large variability
  • Usefulness of extension of MONOLIX as an efficient tool for analysis of bioequivalence/ biosimilarity trials
3 optimal design in nlmem


Real example

design for population pkpd analyses
Design for ‘Population’ PKPD analyses
  • Problem beforehand: choice of ‘population’ design
    • number of individuals? number of sampling times?
    • sampling times?
  • Increasingly important task for pharmacologists
    • Difficult to 'guess' good designs for complex models
  • Importance of the choice
    • influence the precision of parameters estimation and power of test
    • poor design can lead to unreliable studies (complex models)
    • all the more important in special population (paediatric studies …)
      • severe limitations on the number of samples to be taken
      • ethical and physiological reasons
  • Design considerations for population PK(PD) analyses stress out in FDA and EMEA guidelines
population design evaluation optimisation
Population design evaluation/optimisation
  • From
    • given cost (number of samples)
    • experimental constraints
    • statistical model and a priori values of parameters
  • Evaluate/compare designs
    • Predict standard error for each population parameter
  • Find best design
    • smallest standard errors
    • greatest information in the data
  • Two approaches
    • simulation studies
    • mathematical derivation of the Fisher Information matrix
population design
Population Design
  • N individuals i at K periods k
  • Elementary designxiin individual i
    • Total of ni samples
    • Compose of the union of designsxik of each period k
      • number of samples nik and sampling times: tik1…tiknik
  • Population design
    • set of elementary designsX = {x1, ..., xN}
    • number of observations ntot= Sni
  • Often few elementary designs
    • Q groups of Nq individuals
    • same design xq at each period of a total of nq sampling times
    • ntot= S Nqnq
fisher information matrix 1




















¶ y

¶ y





















Fisher information matrix (1)
  • Vector of parameters in NLMEM: Y
    • Fixedeffects: m and b
    • Variance of randomeffects: W and G
    • Parameter in error variance: a and b
  • Information Matrix for population design = {1, ...,  N}
  • Information Matrix for elementary design i
fisher information matrix 2
Fisher information matrix (2)

(Mentré, Mallet & Baccar, Biometrika, 1997; Retout, Mentré & Bruno, Stat Med, 2002; Retout & Mentré, J Biopharm Stat, 2002; Bazzoli, Retout & Mentré, Stat Med, 2009; N’Guyen, Bazzoli & Mentré, ACOP 2009)

  • Nonlinear structural models
    • no analytical expression for MF(x, Y)
    • first order expansion of f about randomeffects taken at 0
    • analytical expressions for MF(x,m,b) and MF(x, W, G,a,b)
models with discrete covariates
Models with discrete covariates
  • Additional fixed-effects to be estimated: b
  • Evaluation of MF requires specification of
      • the expected distribution of covariates in the population
      • the effect size b
    • Evaluation of the expected (mean) information matrix over covariate distribution
    • prediction of the “expected” SE of each b
  • Power of Wald comparison test
    • Test for H0: b = 0
    • Compute power from SE given type I error (e.g. 5%) and b
    • Compute Number of subject needed (NSN) for given power
  • Power of Wald equivalence test
    • Test for H0: {b -DL or b +DL}
    • Power and NSN for TOST
pfim and pfim interface
PFIM and PFIM interface
  • Developed initially by Sylvie Retout, France Mentré
    • INSERM & University Paris Diderot
    • Other participants: Caroline Bazzoli, Emmanuelle Comets, Hervé Le Nagard, Anne Dubois, Thu-Thuy N'Guyen
  • Population Fisher Information Matrix
  • Use R
  • Available at
  • Releases of PFIM
    • 2001: First release PFIM 1.1 similar in Splus and Matlab (S. Duffull)
    • 2008: PFIM 3.0 and PFIM interface 2.1
    • 2010: PFIM 3.2
evaluation by simulation design39
Evaluation by simulation: design
  • PK model with one-compartment : ka, V/F, CL/F
  • Two-periods one-waycrossover trial
    • Treatmenteffect on CL/F
    • Simulations for varioustreatmenteffectsb
    • Two designs with N = 40 piglets
    • Original n=7, Sparse: n=4 measurements/ piglet/ period
  • Variability
    • Randomeffects: BSV=30%, WSV=15%
    • Error model: (a=0.1, b=0)
evaluation by simulation method40
Evaluation by simulation: method
  • Simulation
    • 1000 simulated trials for each treatment effect and each design
    • Global Analysis by SAEM in MONOLIX v2.4
    • Derivation of empirical SE as SD of estimates
    • Derivation of power as proportion of trials with rejection of Wald TOST
  • Predictions
    • Use PFIM 3.2 to predict SE for each treatment effect and each design
    • Use predicted SE by PFIM 3.2 to predict power of Wald TOST
  • Comparison of simulations and predictions
results on se various treatment effect
Results on SE: various treatment effect

Histograms of SE for treatmenteffect for rich designs

__ : predicted SE using PFIM

- - - : empirical SE from simulations

  • Correct prediction of SE by MF for all parameters
results on power
Results on power
  • Correct prediction of power
  • Almost no loss of power for sparse design with ‘optimal’ sampling times
conclusion on design
Conclusion on design
  • Relevance of the new development of the population Fisher matrix for NLMEM including WSV and treatment effect in crossover trials
  • Correct predictions of standard errors and of power avoiding intensive simulations
  • Analysis of studies through NLMEM
    • Can be performed with rather sparse design with almost no loss of power if ‘optimal’ sampling times
  • Usefulness of new extension of PFIM as an efficient tool for design of bioequivalence/ biosimilarity studies
  • Present difficulties in drug development
    • Increase cost and duration of drug development
    • Few new medical entities (NME) reach approval
  • Problems
    • For pharma industry
    • But also for public health
      • life-threatening diseases, rare diseases
      • lack of 'interest' of drug pharma for some disease areas
Integrated Knowledge for Model-based

Drug Development

Genes … Cells … Tissues … Systems … Patients … Populations

Developmenttime axis

Statistical Modelling

Pharmacological Modelling

Biological Modelling

Adapted rom JJ Orloff, Novartis, April 06

  • Increasing role of quantitative analysis of all data trough modelling in therapeutic evaluation
  • Main statistical tool: NLMEM
  • Collaborative work
    • Biologists, Pharmacologists, Physicians
    • Engineers, Mathematicians, Statisticians


  • Various unsolved methodological problems
    • academic research needed
  • Training needed

Holford N, Karlsson MO. Time for quantitative clinical pharmacology: a proposal for a pharmacometrics curriculum. Clin Pharmacol Ther. 2007;82(1):103-5.