Abstract.  We consider N independent stochastic processes (Xi (t), t ∈  [0,Ti]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φi. The distribution of the random effect φi depends on unknown parameters which are to be estimated from the continuous observation of the processes Xi. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φi and φi has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when Ti=T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.

The reduction of viral load is frequently used as a primary endpoint in HIV clinical trials. Non-linear mixed-effects models are thus proposed to model this decrease of the viral load after initiation of treatment and to evaluate the intra-and inter-patient variability. However, left censoring due to quantification limits in the viral load measurement is an additional challenge in the analysis of longitudinal HIV data. An extension of the Stochastic Approximation Expectation-Maximization (SAEM) algorithm is proposed to estimate parameters of these models. This algorithm includes the simulation of the left-censored data in a right-truncated Gaussian distribution. Simulation results show that the proposed estimates are less biased than the usual naive methods of handling such data: omission of all censored data points, or imputation of half the quantification limit to the first point below the limit and omission of the following points. The viral load measurements obtained in the TRIANON-ANRS81 clinical trial are analyzed with this method and a significant difference is found between the two treatment groups of this trial.

This paper is a survey of existing estimation methods for pharmacokinetic/pharmacodynamic (PK/PD) models based on stochastic differential equations (SDEs). Most parametric estimation methods proposed for SDEs require high frequency data and are often poorly suited for PK/PD data which are usually sparse. Moreover, PK/PD experiments generally include not a single individual but a group of subjects, leading to a population estimation approach. This review concentrates on estimation methods which have been applied to PK/PD data, for SDEs observed with and without measurement noise, with a standard or a population approach. Besides, the adopted methodologies highly differ depending on the existence or not of an explicit transition density of the SDE solution.

Growth curve data consist of repeated measurements of a continuous growth process over time in a population of individuals. These data are classically analyzed by nonlinear mixed models. However, the standard growth functions used in this context prescribe monotone increasing growth and can fail to model unexpected changes in growth rates. We propose to model these variations using stochastic differential equations (SDEs) that are deduced from the standard deterministic growth function by adding random variations to the growth dynamics. A Bayesian inference of the parameters of these SDE mixed models is developed. In the case when the SDE has an explicit solution, we describe an easily implemented Gibbs algorithm. When the conditional distribution of the diffusion process has no explicit form, we propose to approximate it using the Euler-Maruyama scheme. Finally, we suggest validating the SDE approach via criteria based on the predictive posterior distribution. We illustrate the efficiency of our method using the Gompertz function to model data on chicken growth, the modeling being improved by the SDE approach.

SUMMARYWe extend the methodology for designs evaluation and optimization in nonlinear mixed effects models with an illustration of the decrease of human immunodeficiency virus viral load after antiretroviral treatment initiation described by a bi-exponential model. We first show the relevance of the predicted standard errors (SEs) given by the computation of the population Fisher information matrix using the R function PFIM, in comparison to those computed with the stochastic approximation expectation-maximization algorithm, implemented in the Monolix software. We then highlight the usefulness of the Fedorov-Wynn (FW) algorithm for designs optimization compared to the Simplex algorithm. From the predicted SE of PFIM, we compute the predicted power of the Wald test to detect a treatment effect as well as the number of subjects needed to achieve a given power. Using the FW algorithm, we investigate the influence of the design on the power and show that, for optimized designs with the same total number of samples, the power increases when the number of subjects increases and the number of samples per subject decreases. A simulation study is also performed with the nlme function of R to confirm this result and show the relevance of the predicted powers compared to those observed by simulation.

Summary The statistical problem of parameter estimation in partially observed hypoelliptic diffusion processes is naturally occurring in many applications. However, because of the noise structure, where the noise components of the different co‐ordinates of the multi‐dimensional process operate on different timescales, standard inference tools are ill conditioned. We propose to use a higher order scheme to approximate the likelihood, such that the different timescales are appropriately accounted for. We show consistency and asymptotic normality with non‐typical convergence rates. When only partial observations are available, we embed the approximation in a filtering algorithm for the unobserved co‐ordinates and use this as a building block in a stochastic approximation expectation–maximization algorithm. We illustrate on simulated data from three models: the harmonic oscillator, the FitzHugh–Nagumo model used to model membrane potential evolution in neuroscience and the synaptic inhibition and excitation model used for determination of neuronal synaptic input.

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Understanding how tumors develop resistance to chemotherapy is a major issue in oncology. When treated with temozolomide (TMZ), an oral alkylating chemotherapy drug, most low-grade gliomas (LGG) show an initial volume decrease but this effect is rarely long lasting. In addition, it has been suggested that TMZ may drive tumor progression in a subset of patients as a result of acquired resistance. Using longitudinal tumor size measurements from 121 patients, the aim of this study was to develop a semi-mechanistic mathematical model to determine whether resistance of LGG to TMZ was more likely to result from primary and/or from chemotherapy-induced acquired resistance that may contribute to tumor progression. We applied the model to a series of patients treated upfront with TMZ (n = 109) or PCV (procarbazine, CCNU, vincristine) chemotherapy (n = 12) and used a population mixture approach to classify patients according to the mechanism of resistance most likely to explain individual tumor growth dynamics. Our modeling results predicted acquired resistance in 51% of LGG treated with TMZ. In agreement with the different biological effects of nitrosoureas, none of the patients treated with PCV were classified in the acquired resistance group. Consistent with the mutational analysis of recurrent LGG, analysis of growth dynamics using mathematical modeling suggested that in a subset of patients, TMZ might paradoxically contribute to tumor progression as a result of chemotherapy-induced resistance. Identification of patients at risk of developing acquired resistance is warranted to better define the role of TMZ in LGG.