Generalisation of the SAEM algorithm to nonlinear mixed effects model defined by differential equations : application to HIV viral dynamic models
Adeline Samson (1), Xavière Panhard (1), Marc Lavielle (2) and France Mentré (1)
(1) NSERM U738, Department of Epidemiology, Biostatistics and Clinical Research, University Hospital Bichat-Claude Bernard, Paris, France. (2) University Paris-Sud, Bat. 425, Orsay, France
Objectives: We consider parametric mixed models whose regression functions are solution of an ordinary differential equation (ODE). Maximum likelihood estimation in nonlinear mixed effects models (NLMEM) cannot be directly performed as the likelihood has no close form. Kuhn and Lavielle [1] proposed the SAEM algorithm, implemented in the Matlab function MONOLIX. We adapt the SAEM algorithm to models defined by ODE. We illustrate the method on a simulated pharmacokinetic (PK) dataset, with comparison to NONMEM, and on a real dataset of HIV dynamics.
Methods: Classical ODE numerical solving methods such as Runge-Kutta can be included in SAEM. We propose an original Local Linearisation (LL) scheme to solve the ODE taking advantage of the specific structure of the MCMC algorithm included in SAEM. The LL scheme reduces significantly the computational time. We prove the convergence of this algorithm and bound the error deriving from the numerical integration of the ODE.
We simulate one dataset of 20 patients with a one-compartment PK model with first order absorption and saturable Michaelis-Menten elimination, mimicking an anti-cancerous drug [2]. We compare the estimates obtained by SAEM and NONMEM. SAEM converges and provides accurate estimates, whereas NONMEM fails to converge
Applications: We analyse simultaneously HIV viral load decrease and CD4 increase after treatment in HIV patients, using the model proposed by Perelson et al. [3]. We consider data of 32 patients until week 16 of the COPHAR2-ANRS 111 trial. We obtain a NLMEM defined by ODE, including 7 population parameters, 7 random effects, and two residual errors. The SAEM algorithm converges and provides accurate estimates of the 16 parameters.
Conclusions: The extension of SAEM to ODE model using the LL scheme is a powerful tool to provide accurate estimation on NLMEM.
References:
[1] Kuhn and Lavielle. Maximum likelihood estimation in nonlinear mixed effects model, Comput. Stat. Data Analysis, to appear.
[2] Tracewell, Trump, Vaughan, Smith and Gwilt. Population pharmacokinetics of hydroxyurea in cancer patients. Cancer Chemoter Pharmacol, 35 (1995), 417-22.
[3] Perelson, Neumann, Markowitz, Leonard and Ho. HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science, 271 (1996),1582-6.