GDR MaMoVi 2017 Parameter estimation in Models with Random effects based on Ordinary Differential Equations: A bayesian maximum a posteriori approach. Mélanie PRAGUE, Daniel COMMENGES & Rodolphe THIÉBAUT - SISTM September 27, 2017
1Non Linear mixed effects ordinary differential equations models M. Prague - SISTM - NLME-ODE September 27, 2017-2
Available Data come from clinical trials and observational studies Longitudinal data Y ijk : patient i, time j and biomarker k M. Prague - SISTM - NLME-ODE September 27, 2017-3
Mathematical Model for mechanistic models Compartiments Biologiques Compartiment Q T T V Signification CD4 Quiescents CD4 Activés CD4 Activés Infectés Virions M. Prague - SISTM - NLME-ODE September 27, 2017-4
Mathematical Model for mechanistic models Dynamique des cellules T (CD4 infectés) dt dt = γvt µ T T Paramètre Signification µ T Taux de décès des cellules T γ Infectivité : Taux d infection des cellules T par les virions M. Prague - SISTM - NLME-ODE September 27, 2017-5
Mathematical Model for mechanistic models Target cells model dq dt = λ µ Q Q αq + ρt dt dt = αq ρt µ T T γvt dt dt = γvt µ T T dv dt = πt µ V V M. Prague - SISTM - NLME-ODE September 27, 2017-6
Statistical Model for mechanistic models Target cells model Mixte effects models on parameters ( ) ξ i = α i, λ i,..., γ i 0, µ i V ξ l i = φ l + zl i (t)β l + }{{} ωl i (t)ul i }{{} Effets fixes Effets aléatoires u i N (0, I q) M. Prague - SISTM - NLME-ODE September 27, 2017-7
Observational Model for mechanistic models Among, X(t ij, ξ i ) = (Q(t ij, ξ i ), T (t ij, ξ i ), T (t ij, ξ i ), V (t ij, ξ i )) We only observe (with measurement errors): Viral load : CD4 count : Y ij1 = log 10 (V ) + ɛ ij1 Y ij2 = (Q + T + T ) 0.25 + ɛ ij2 ɛ ijm N (0, σ 2 m) Donc, g 1(.) = log 10 (.) g 2(.) = (.) 0.25 M. Prague - SISTM - NLME-ODE September 27, 2017-8
Parameters of interest We want to estimate more than 15 parameters: θ = {λ, µ Q, α, ρ, µ T, γπ, µ T, µ V } {{ } Effet fixes, β 1,..., β r }{{} Covariates effects, σ 1,..., σ s }{{}, Σ 1,..., Σ k }{{} Random effects Measurement errors } There are sometimes problems of identifiability 1 This approach is unbiased more efficient than marginal structural models in presence of dynamic treatment regimens 2 1 [1] Guedj et al. (2010), Bull. Math. Biol. 2 [2] Prague et al. (2016), Biometrics. M. Prague - SISTM - NLME-ODE September 27, 2017-9
2Bayesian penalised likelihood estimation M. Prague - SISTM - Estimation September 27, 2017-10
Existing methods Method Ref. Software Non parametric Functional analysis [Ramsay et al. 2012] - Non Bayesian parametric FOCE [Pinheiro et Bates 1995] R Bayesian SAEM [Kuhn et al. 2005 MONOLIX Lavielle et al. 2007] Bayesian MCMC [Lunn et al 2000 WinBUGS Huang et al. 2011] Bayesian penalized likelihood [Guedj et al 2007; NIMROD 3 Prague et al. 2013] 3 [3] Prague et al. (2013), Comp. Meth. and Prog. in Biomed. M. Prague - SISTM - Estimation September 27, 2017-11
Penalization for the log-likelihood It is possible to have an approximate idea of the value of biological parameters and treatment effects, for example from previous in vitro experiment or analysis of studies. Normal approximation of the posterior of previous analysis can be used as new prior for analysis as in a sequential bayesian meta-analysis 4 : J(θ) = { 9 φj E 0 (φ } 2 j) var0 (φ j) j=1 n TRT + j=1 { βj E 0 (β j) } 2 var0 (β j) 4 [4] Prague et al. (2016) Journal de la statistique francaise M. Prague - SISTM - Estimation September 27, 2017-12
Penalized likelihood computation (1) Individual likelihood (censorship δ ij = I Yij1 <ζ) L Fi u i = { [ 1 exp 1 ( Yij1 g 1(X(t ij, ξ ) i 2 ]} 1 δij )) j,1 σ 1 (2π) 2 σ 1 { ( )} ζ g1(x(t ij, ξ i δij ) Φ σ 1 [ 1 exp 1 ( Yij2 g 2(X(t ij, ξ ) i 2 ] )) σ 2 (2π) 2 σ 2 j,2 Φ Repartition function of a Normal law. ODE Solver (dlsode Fortran) - [Radhakrishnan et Hindmarsh (1993)] M. Prague - SISTM - Estimation September 27, 2017-13
Penalized likelihood computation (2) - Observed individual likelihood L Oi = R q L Fi u i (u)φ(u)du, with φ N (0, I q) Numerical integration: Adaptive Gaussian Quadrature Numerical integration:[genz et Keister (1996)] - Penalized log-likelihood L P O = i n log (L Oi ) J(θ) Parallel computing: Each computation L Oi are independent. M. Prague - SISTM - Estimation September 27, 2017-14
Robust-Variance Scoring (RVS) We use a Newton-Raphson-like algorithm to maximize the penalized likelihood. Score computation (Gradients approximation) ( ) n L P Oi U O(θ k ) = θ θ k ODE solver (dlsode Fortran) i=1 Sensitivity Equation of ODE systems Adaptive Gaussian Quadrature Parallel computing M. Prague - SISTM - Estimation September 27, 2017-15
Robust-Variance Scoring (RVS) Computation of G (Approximation of the Hessian H) H(θ k ) G(θ k ) = i n ( UOi (θ k )U O i (θ k ) ) ν n U(θ k)u (θ k ) + 2 J(θ) θ 2 Switch to a Marquardt-Levenberg algorithm [Marquardt, JSIAM, 1963] when the RVS algorithm does not provide maximization for multiple iterations. M. Prague - SISTM - Estimation September 27, 2017-16
Convergence criteria Stabilization of parameters estimates : Stabilization of log-likelihood : θ (k+1) θ k < η 1 L P O(θ (k+1) ) L P O(θ k ) < η 2 Relative Distance to Maximum (main) : RDM(θ k ) = U(θ k)g 1 (θ k )U (θ k ) m < η 3 M. Prague - SISTM - Estimation September 27, 2017-17
4Some Illustration M. Prague - SISTM - Illustration September 27, 2017-18
Example in pharmacokinetics Pharmacokinetics model 1 compartment: Pharmacokinetics model 2 compartments: Label GI CP GT Name Gastro-intestinal tract Plasma compartment Tissue compartment M. Prague - SISTM - Illustration September 27, 2017-19
Simulations - FOCE is not stable and less accurate (Laplace integration) - MCMC is more computationally demanding than NIMROD - NIMROD gives more efficient results than MCMC - NIMROD sometimes achieve estimation where MCMC fails Failure Time Empirical Overall Overall (%) (s) SE Abs. Bias RMSE FOCE 52 0.5 0.183 0.202 0.281 MCMC 0 233 0.174 0.060 0.195 NIMROD 0 109 0.060 0.060 0.092 M. Prague - SISTM - Illustration September 27, 2017-20
Properties of RDM 150 200 10000 300 1000 Log likelihood 400 500 600 700 800 100 10 1 0.1 0.01 RDM 900 0.001 1000 0 5 10 15 20 25 30 Number of iterations 0.0001 0.00001 M. Prague - SISTM - Illustration September 27, 2017-21
Real Data: The PUZZLE study [Raguin et al. 2004] Explain the 600 mg Amprenavir (APV) concentrations in blood (A CP ) in 39 HIV infected patients Longitudinal data {0, 1/2, 1 1/2, 2, 3, 4, 6, 8, 10} hours M. Prague - SISTM - Illustration September 27, 2017-22
5Conclusion M. Prague - SISTM - Conclusion September 27, 2017-23
Existing and perspectives - www.isped.u-bordeaux2.fr/nimrod/documentation.aspx - Increase the dimension of the mechanistic models: Limited number of inter-individual variability (random effects). - Investigate alternative algorithms: Explore Kalman filters. M. Prague - SISTM - Conclusion September 27, 2017-24
References 1. Guedj, J., Thiébaut, R., and Commenges, D. (2010). Practical identifiability of HIV dynamics models. Bulletin of mathematical biology, 69(8), 2493-2513. 2. Prague, M., Commenges, D., Gran, J. M., Ledergerber, B., Young, J., Furrer, H., and Thiébaut, R. (2016). Dynamic models for estimating the effect of HAART on CD4 in observational studies: Application to the Aquitaine Cohort and the Swiss HIV Cohort Study. Biometrics, 73(1), 294-304. 3. Prague M., Commenges D., Guedj J., Drylewicz J., Thiébaut R. (2013) NIMROD: A Program for Inference via Normal Approximation of the Posterior in Models with Random effects based on Ordinary Differential Equations. Computer methods and Programs in Biomedecine 111(2) 447-458 4. Prague M. (2016) Dynamical modeling for Optimization of treatment in HIV infected patients. Invited paper in Statistical French Society journal. 157(2), 19-38 M. Prague - SISTM - Conclusion September 27, 2017-25
MERCI SISTM Inria, Bordeaux, Sud-ouest, France melanie.prague@inria.fr