Designs in nonlinear mixedeffects for a model of HIV viral load decrease: evaluation, optimisation a

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Designs in nonlinear mixed-effects for a model of
HIV viral load decrease evaluation, optimisation
and determination of the power of a treatment
effect

• France MENTRE, Sylvie RETOUT, Adeline SAMSON
• and Emmanuelle COMETS
• INSERM U738
• Dpt of Epidemiology, Biostatistics and Clinical
  Research
• University Hospital Bichat-Claude Bernard ,
  Paris, France

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OUTLINE

• Introduction
• Model of viral load decrease
• Predicted standard error of treatment effect
• Power of the test of the treatment effect
• Design optimization with Federov-Wynn algorithm
• Conclusion

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1. Introduction

• Nonlinear mixed-effects models (NLMEM)
  increasingly used in drug development
• pharmacokinetic / pharmacodynamic studies
• analysis of longitudinal data collected during
  clinical trials
• Several algorithms for maximum likelihood
  estimation
• NONMEM, nmle (Splus/ R), Proc NLINMIX (SAS)
• based on first-order linearization of the model
• Monolix (Matlab) Stochastic approximation EM
  algorithm
• (Kuhn Lavielle, CSDA, 2005)
• MCMC method to find maximum likelihood estimates
• accurate likelihood and SE obtained by importance
  sampling
• http// mahery.math.u-psud.fr/lavielle/monolix/lo
  giciels.html

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Nonlinear mixed-effects model

• Individual model
• yi f(?i, ?i) ei
• ?i individual parameters (p components)
• e gaussian zero mean random error
• var (e) sinter2 sslope2 f(q, x) 2
• Random effects model with two treatment groups
• ?i µ b T i bi , bi N (0, ?)
• Ti 0 if group A, Ti 1 if group B
• ? diagonal wk2 Var(bik)
• Population parameters Y  (P components)
• µ, b (fixed effects)
• unknowns in ? (variance of random effects)
• sinter and/or sslope (variance of residual
  error)

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Population design

• Before the experiment choice of the population
  design
• number of patients N
• elementary design in each patient xi
• number ni of samples
• allocation of the sampling times
• General population design X x1, ..., xN
• total number of observations ntot Sni
• Grouped population design
• 1 to Q groups of elementary designs xq
• each elementary design to be performed in Nq
  patients

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Fisher Information Matrix for NLMEM

• (Mentré, Mallet Baccar, Biometrika, 1997
  Retout, Mentré Bruno, Stat Med, 2002)

• Fisher information matrix of population design X
• Nonlinear model f
• no analytical expression for MF (x, Y)
• first order expansion of f about b taken at 0
• assumption of independence between Var(bi) and
  fixed effects
• analytical expressions for MF(x, m) and MF(x, W,
  s)

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Population design evaluation and optimisation

• MC simulation
• time consuming and to be repeated for each design
• Direct evaluation and optimisation using Fisher
  Information matrix
• Designs comparison
• For a given model, a given Y, and for each X
• predict MF, det(MF)1/P
• predict SE associated with each parameter
• PFIM in Splus/R and Matlab
• http//www.bichat.inserm.fr/equipes/Emi0357/downlo
  ad.html
• Design optimisation maximisation of det(MF)
• Local planification find X for a given Y

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Optimization of exact or statistical designs

• Exact design
• Fixed group structure (Q, Nq, number of samples
  in xq)
• Optimization of the sampling times in xq, q 1,
  , Q
• General algorithms simplex, simulated annealing,
  NARS, (Duffull, Retout Mentré, Comput
  Methods Programs Biomed, 2002)
• Implementation of Simplex in Splus/R PFIMOPT
• (Retout Mentré, J Pharmacokin Pharmacodyn,
  2003)
• Statistical design
• ?q ( ? Nq/N) proportions of subjects in ?q
• Fedorov-Wynn algorithm for optimization of
• group structure (Q, ?q, nq)
• sampling times in xq among a given set (discrete
  times)
• (Mentré, Mallet Baccar, Biometrika, 1997)
• developed in C and linked with PFIMOPT

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2. Model of viral load decrease

• (Wu, Ding de Gruttola, Stat Med, 1998 Wu
  Ding, Biometrical J, 2002)

• Viral load decreases after initiation of
  antiretroviral treatment in HIV1-infected
  patients
• Decrease can be described by a bi-exponential
  model
• f(?, t) Log10 (P1 exp(- l1 t) P2 exp(- l2 t)
  )
• y log viral load, e with constant variance
• Four parameters with additive random effects on
  their log
• Fixed effect b for treatment effect on first
  slope
• log (l1) B log (l1) A b
• Parameter values
• Fixed effects log(P1)12, log(P2)8, log(l1)
  -0.7, log(l2) -3.0
• Variance of random effects 0.3
• SD of residual error 0.065

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Treatment effect on first slope

• First rate-constant 30 greater in group B vs A
• b 0.262
• Similar design in all patients 6 samples
• weeks 1, 3, 7, 14, 28, 56 after treatment
  initiation

Group B 40 patients
Group A 40 patients
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3. Predicted standard error of treatment effect

• Evaluation of SE of b under H0 b 0
• Two groups of 100 patients, same elementary
  design
• Predicted SE two approaches
• Extension of PFIM for covariates (in R)
• (Retout Mentré, J Biopharm Stat, 2003)
• Monolix (SAEM in Matlab)
• one simulation of 5 000 patients per group
  (closer to asymptotic)
• estimated SE divided by 501/2
• Empirical SE from replicated simulations under H0
  
• 100 simulations of 100 patients per group
• Distribution and SD of estimated b
• Estimation algorithm 1. nlme (in R) and 2.
  Monolix

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Results on SE of b

• Predicted SE
• PFIM 0.079
• Monolix 0.078
• Close results of PFIM (with linearization) and
  SAEM (exact approach but time consuming)
• Empirical SE (on 100 replications)
• nlme 0.085
• Monolix 0.086
• Close results (on this example) of the two
  estimation algorithms
• Predicted SE smaller than empirical SE
• MF-1 lower bound of estimation variance

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Results on other estimation CV ()
Predicted
Empirical
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4. Power of the test of the treatment effect

• Wald test performed to test H0 b 0
• For a given population design
• compute PFIM predicted SE under H1
• evaluate the power of Wald test under H1 (upper
  bound)
• number of subjects needed for a given power
• SE proportional to N1/2

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5. Design optimization with FW algorithm
Total of 480 samples in set 0,1,2,3,5,7,10,14,21,
28,42,56
b Rounded optimal statistical design
a Previous non-optimized design
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Evaluation of the power by simulation
Total of 480 samples in set 0,1,2,3,5,7,10,14,21,
28,42,56
b Rounded optimal statistical design
a Previous non-optimized design
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Example of PFIMOPT 2.0 output with Fedorov-Wynn
algorithm (1)
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Example of PFIMOPT 2.0 output with Fedorov-Wynn
algorithm (2)
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6. Conclusion

• Optimal design in NLMEM using FIM
• Interesting and growing field with great
  potential applications
• Avoid extensive repeated simulations
• Power of Wald test and number of patients needed
• Optimal allocation of sampling times
• Main limitations
• Based on asymptotic results
• evaluate best designs by simulation
• Local planification based on Y
• sensitivity analysis
• robust or Bayesian planification (Han Chaloner,
  Biometrics, 2004)
• Global D-optimality for all parameters in Y
• Ds optimality on subset of parameters
• optimal compound design

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Future developments

• Optimisation with IOV
• balance number of occasions / number of samples
  per occasion
• Optimisation with covariates
• given distribution
• optimal designs across patients
• optimal designs with respect to covariates values
• optimisation of distribution
• find best designs and best covariate distribution
  
• Optimisation in PK/PD (more than one variable)
• Optimal design for subset of parameters
  (DS-optimality)
• Optimal compound designs
• Need for more exact evaluation of FIM witha SAEM?