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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

- 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

Pharmacometrics

Nonlinear mixed effect models

Bioequivalence/ biosimilarity studies

Effect

Dose

Concentration

Pharmacokinetics

Pharmacodynamics

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

Pharmacometricians

Design

+ Disease models

- 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

- 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)

- Area under the curve (AUC)
- Algorithm: linear or log-linear trapezoidal method

- Parameters of interest

- 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

- 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

- 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’

m ?

sd ?

Estimates of

individual

parameters

?

?

#1

#2

Non linear mixed effects model

Single-stage approach (population analysis)

#n

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

- 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

- 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

Method

Simulation

Real example

- 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

- 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.

- Extension of EM algorithmwithproven convergence

- Adaptative Gaussian Quadrature

- 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.

…

- Free Matlab software implementing SAEM
- developed under supervision of Pr Marc Lavielle at INRIA
- www.monolix.org
- 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

Software for estimation in NLMEM

- 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

- Estimation with the complete model withtreatment, period and sequenceeffect on all parameters, WSV in addition to BSV
- 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 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 withTfixed to -DL or +DL
- log likelihoodL-Dor L+D

- Reject of H0 if

[10] Panhard, Samson. Biostatistics. 2009

(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%)

- Randomeffects

- 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

Sl,l

Sh,l

Sl,l

Sh,l

* : true value

Sl,l

Sh,l

Sl,l

Sh,l

- RMSE (rich design) < RMSE (sparse design)
- RMSE (4 periods) < RMSE (2 periods)
- RMSE satisfactory except for WSV on V/F for low variability

Sl,l

Sh,l

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

- 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

Method

Simulation

Real example

- 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

- 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

- 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

¶

y

¶

y

log

l(y;

)

log

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)'

i

x

Y

MF

(

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=

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¶ y

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i

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M

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- 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

(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)

- 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

- 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

- Test for H0: {b -DL or b +DL}
- Power and NSN for TOST

- 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 www.pfim.biostat.fr
- 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

- 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)

- 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

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

- Correct prediction of power
- Almost no loss of power for sparse design with ‘optimal’ sampling times

- 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

PROBLEM in DRUG DEVELOPMENT

- 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

CONCLUSION

- 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
Pharmacometricians

- 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.