**The SAEM algorithm in MONOLIX for Non-Linear Mixed Effects Models with Stochastic Differential Equations**

Maud Delattre (1), Pierre Del Moral (2), Marc Lavielle (1,3)

(1) Department of Mathematics, University Paris-Sud, (2) INRIA Bordeaux, (3) INRIA Saclay

**Objectives:** The use of stochastic differential equations in non linear mixed effects models enables the decomposition of the intra-patient variability into some residual errors and some dynamical system variability. Several authors already addressed this problem and proposed an approximation method based on the First Order Conditional Estimation (FOCE) method used in non-linear mixed effects models, with the Kalman Filter used for SDEs (see [1], [2], [3]). In [4], the authors propose a stochastic EM algorithm where the diffusion process of the SDE is considered as a component of the non observed data and is simulated at each iteration. This procedure has some appealing theoretical properties but the computational effort is prohibitive for practical applications.

The objective of this contribution is to present a new maximum likelihood estimation procedure which is computationally tractable in practical situations and which avoids the linearization of the model.

**Methods: **We propose a new algorithm based on the Stochastic Approximation EM (SAEM) method with the Kalman Filter for linear SDE systems.

We show that a linear SDE system is not relevant when the components of the stochastic system are known to be positive, which is usually the case in a biological perspective. Assuming that the diffusion process randomly perturbs the coefficients of the associated ODE system is more realistic but the SDE system is not linear any more. The extended Kalman filter for non linear SDE systems can be used in such situations.

This methodology was implemented in a working version of MONOLIX and tested on several simulated PK examples.

**Results:** We show with these simulated examples that the proposed method does not reduce to consider some correlated residual errors. Indeed, we show the ability of the method to properly decompose the intra-patient variability into several components.

**Conclusions:** A new maximum likelihood estimation method for non linear mixed effects models governed by a system of stochastic differential equations is now implemented in a working version of MONOLIX.

For nonlinear SDE systems, we aim to develop in a next future a new SAEM based method using a particle filter instead of the extended Kalman filter. This method is expected to exhibit better theoretical and practical properties.

**References:**

[1] S. Mortensen, S. Klim, B. Dammann, N. Kristensen, H. Madsen, R. Overgaard "A Matlab framework for estimation of NLME models using stochastic differential equations", Journal of Pharmacokinetics and Pharmacodynamics vol:34, pages: 623-642, 2007.

[2] R. Overgaard, E. Jonsson, C. Tornøe, H. Madsen, "Non-Linear Mixed Effects Models with Stochastic Differential Equations. Implementation of an Estimation Algorithm", PAGE 2004.

[3] C. W. Tornøe, H. Agersø, R. V. Overgaard, H. A. Nielsen, H. Madsen, E. N. Jonsson, "Stochastic differential equations in NONMEM", PAGE 2004.

[4] Donnet S, Samson A, Parametric inference for mixed models defined by stochastic differential equations, ESAIM P&S, 12:196-218, 2008.