Adeline Samson

A review on estimation of stochastic differential equations for pharmacokinetic/pharmacodynamic models☆

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.

Maximum likelihood estimation of long-term HIV dynamic models and antiviral response.

HIV dynamics studies, based on differential equations, have significantly improved the knowledge on HIV infection. While first studies used simplified short-term dynamic models, recent works considered more complex long-term models combined with a global analysis of whole patient data based on nonlinear mixed models, increasing the accuracy of the HIV dynamic analysis. However statistical issues remain, given the complexity of the problem. We proposed to use the SAEM (stochastic approximation expectation-maximization) algorithm, a powerful maximum likelihood estimation algorithm, to analyze simultaneously the HIV viral load decrease and the CD4 increase in patients using a long-term HIV dynamic system. We applied the proposed methodology to the prospective COPHAR2-ANRS 111 trial. Very satisfactory results were obtained with a model with latent CD4 cells defined with five differential equations. One parameter was fixed, the 10 remaining parameters (eight with between-patient variability) of this model were well estimated. We showed that the efficacy of nelfinavir was reduced compared to indinavir and lopinavir.

Extension of the SAEM algorithm for nonlinear mixed models with 2 levels of random effects

This article focuses on parameter estimation of multilevel nonlinear mixed-effects models (MNLMEMs). These models are used to analyze data presenting multiple hierarchical levels of grouping (cluster data, clinical trials with several observation periods, ...). The variability of the individual parameters of the regression function is thus decomposed as a between-subject variability and higher levels of variability (e.g. within-subject variability). We propose maximum likelihood estimates of parameters of those MNLMEMs with 2 levels of random effects, using an extension of the stochastic approximation version of expectation-maximization (SAEM)-Monte Carlo Markov chain algorithm. The extended SAEM algorithm is split into an explicit direct expectation-maximization (EM) algorithm and a stochastic EM part. Compared to the original algorithm, additional sufficient statistics have to be approximated by relying on the conditional distribution of the second level of random effects. This estimation method is evaluated on pharmacokinetic crossover simulated trials, mimicking theophylline concentration data. Results obtained on those data sets with either the SAEM algorithm or the first-order conditional estimates (FOCE) algorithm (implemented in the nlme function of R software) are compared: biases and root mean square errors of almost all the SAEM estimates are smaller than the FOCE ones. Finally, we apply the extended SAEM algorithm to analyze the pharmacokinetic interaction of tenofovir on atazanavir, a novel protease inhibitor, from the Agence Nationale de Recherche sur le Sida 107-Puzzle 2 study. A significant decrease of the area under the curve of atazanavir is found in patients receiving both treatments.

Estimation in the partially observed stochastic Morris–Lecar neuronal model with particle filter and stochastic approximation methods

Parameter estimation in multi-dimensional diffusion models with only one coordinate observed is highly relevant in many biological applications, but a statistically difficult problem. In neuro-science, the membrane potential evolution in single neurons can be measured at high frequency, but biophysical realistic models have to include the unobserved dynamics of ion channels. One such model is the stochastic Morris-Lecar model, defined by a non-linear two-dimensional stochastic differential equation. The coordinates are coupled, i.e. the unobserved coordinate is non-autonomous, the model exhibits oscillations to mimick the spiking behavior, which means it is not of gradient-type, and the measurement noise from intra-cellular recordings is typically negligible. Therefore the hidden Markov model framework is degenerate, and available methods break down. The main contributions of this paper are an approach to estimate in this ill-posed situation, and non-asymptotic convergence results for the method. Specifically, we propose a sequential Monte Carlo particle filter algorithm to impute the unobserved coordinate, and then estimate parameters maximizing a pseudo-likelihood through a stochastic version of the Expectation- Maximization algorithm. It turns out that even the rate scaling parameter governing the opening and closing of ion channels of the unobserved coordinate can be reasonably estimated. An experimental data set of intracellular recordings of the membrane potential of a spinal motoneuron of a red-eared turtle is analyzed, and the performance is further evaluated in a simulation study.

Introduction to Stochastic Models in Biology

This chapter is concerned with continuous time processes, which are often modeled as a system of ordinary differential equations (ODEs). These models assume that the observed dynamics are driven exclusively by internal, deterministic mechanisms. However, real biological systems will always be exposed to influences that are not completely understood or not feasible to model explicitly. Ignoring these phenomena in the modeling may affect the analysis of the studied biological systems. Therefore there is an increasing need to extend the deterministic models to models that embrace more complex variations in the dynamics. A way of modeling these elements is by including stochastic influences or noise. A natural extension of a deterministic differential equations model is a system of stochastic differential equations (SDEs), where relevant parameters are modeled as suitable stochastic processes, or stochastic processes are added to the driving system equations. This approach assumes that the dynamics are partly driven by noise.

Day-specific probabilities of conception in fertile cycles resulting in spontaneous pregnancies

STUDY QUESTION When, within the female cycle, does conception occur in spontaneously fertile cycles? SUMMARY ANSWER This study provides reference values of day-specific probabilities of date of conception in ongoing pregnancies. The maximum probability of being within a 5-day fertile window was reached on Day 12 following the last menstrual period (LMP). WHAT IS KNOWN ALREADY The true date of conception is not observable and may only be estimated. Accuracy of these estimates impacts on obstetric management of ongoing pregnancies. Timing of ovulation and fertility has been extensively studied in prospective studies of non-pregnant fertile women using error-prone proxies, such as hormonal changes, body-basal temperature and ultrasound, yielding day-specific probabilities of conception and fertile windows. In pregnant women, date of conception may be retrospectively estimated from early pregnancy fetal measurement by ultrasound. STUDY DESIGN, SIZE, DURATION Retrospective analysis of consecutive pregnancies in women referred for routine first-trimester screening, over a 3-year period (2009-2011) in a single ultrasound center (n = 6323). PARTICIPANTS/MATERIALS, SETTING, METHODS Within the overall population, 5830 cases with a certain date of last menses were selected for analysis. The date of conception was estimated using a crown-rump length biometry and an equation derived from IVF/ICSI pregnancies. Day-specific probabilities of conception were estimated across several covariates, including age, cycle characteristics and ethnicity, using deconvolution methods to account for measurement error. MAIN RESULTS AND THE ROLE OF CHANCE Overall, the day-specific probability of conception sharply rises at 7 days after the LMP, reaching its maximum at 15 days and returning to zero by 25 days. Older women tend to conceive earlier within their cycle, as did women with regular cycles and white and black women compared with Asian ethnicity. The probability of being within the fertile window was 2% probability at Day 4, a maximum probability of 58% at Day 12 and a 5% probability by Day 21 of the cycle. LIMITATIONS, REASONS FOR CAUTION Although conception is believed to occur within hours following ovulation, a discrepancy is theoretically possible. However, when comparing our results to those of prospective studies, no such difference was found. The equation used for estimating the date of pregnancy was estimated in IVF/ICSI pregnancies, which could lead to potential bias in spontaneous pregnancies. However, in our population, the observed bias was negligible. Non-fertile cycles and early pregnancy losses are necessarily overlooked because of the nature of our data. WIDER IMPLICATIONS OF THE FINDINGS Because of the wider access to retrospective data and the potential bias in prospective studies of ovulation monitoring, this study should broaden the perspectives of future epidemiologic research in fertility and pregnancy monitoring. STUDY FUNDING/COMPETING INTERESTS None.

Phenomenological modeling of tumor diameter growth based on a mixed effects model.

Over the last few years, taking advantage of the linear kinetics of the tumor growth during the steady-state phase, tumor diameter-based rather than tumor volume-based models have been developed for the phenomenological modeling of tumor growth. In this study, we propose a new tumor diameter growth model characterizing early, late and steady-state treatment effects. Model parameters consist of growth rhythms, growth delays and time constants and are meaningful for biologists. Biological experiments provide in vivo longitudinal data. The latter are analyzed using a mixed effects model based on the new diameter growth function, to take into account inter-mouse variability and treatment factors. The relevance of the tumor growth mixed model is firstly assessed by analyzing the effects of three therapeutic strategies for cancer treatment (radiotherapy, concomitant radiochemotherapy and photodynamic therapy) administered on mice. Then, effects of the radiochemotherapy treatment duration are estimated within the mixed model. The results highlight the model suitability for analyzing therapeutic efficiency, comparing treatment responses and optimizing, when used in combination with optimal experiment design, anti-cancer treatment modalities.

A SAEM algorithm for the estimation of template and deformation parameters in medical image sequences

This paper is about object deformations observed throughout a sequence of images. We present a statistical framework in which the observed images are defined as noisy realizations of a randomly deformed template image. In this framework, we focus on the problem of the estimation of parameters related to the template and deformations. Our main motivation is the construction of estimation framework and algorithm which can be applied to short sequences of complex and highly-dimensional images. The originality of our approach lies in the representations of the template and deformations, which are defined on a common triangulated domain, adapted to the geometry of the observed images. In this way, we have joint representations of the template and deformations which are compact and parsimonious. Using such representations, we are able to drastically reduce the number of parameters in the model. Besides, we adapt to our framework the Stochastic Approximation EM algorithm combined with a Markov Chain Monte Carlo procedure which was proposed in 2004 by Kuhn and Lavielle. Our implementation of this algorithm takes advantage of some properties which are specific to our framework. More precisely, we use the Markovian properties of deformations to build an efficient simulation strategy based on a Metropolis-Hasting-Within-Gibbs sampler. Finally, we present some experiments on sequences of medical images and synthetic data.

Routine OGTT: A Robust Model Including Incretin Effect for Precise Identification of Insulin Sensitivity and Secretion in a Single Individual

In order to provide a method for precise identification of insulin sensitivity from clinical Oral Glucose Tolerance Test (OGTT) observations, a relatively simple mathematical model (Simple Interdependent glucose/insulin MOdel SIMO) for the OGTT, which coherently incorporates commonly accepted physiological assumptions (incretin effect and saturating glucose-driven insulin secretion) has been developed. OGTT data from 78 patients in five different glucose tolerance groups were analyzed: normal glucose tolerance (NGT), impaired glucose tolerance (IGT), impaired fasting glucose (IFG), IFG+IGT, and Type 2 Diabetes Mellitus (T2DM). A comparison with the 2011 Salinari (COntinuos GI tract MOdel, COMO) and the 2002 Dalla Man (Dalla Man MOdel, DMMO) models was made with particular attention to insulin sensitivity indices ISCOMO, ISDMMO and kxgi (the insulin sensitivity index for SIMO). ANOVA on kxgi values across groups resulted significant overall (P<0.001), and post-hoc comparisons highlighted the presence of three different groups: NGT (8.62×10−5±9.36×10−5 min−1pM−1), IFG (5.30×10−5±5.18×10−5) and combined IGT, IFG+IGT and T2DM (2.09×10−5±1.95×10−5, 2.38×10−5±2.28×10−5 and 2.38×10−5±2.09×10−5 respectively). No significance was obtained when comparing ISCOMO or ISDMMO across groups. Moreover, kxgi presented the lowest sample average coefficient of variation over the five groups (25.43%), with average CVs for ISCOMO and ISDMMO of 70.32% and 57.75% respectively; kxgi also presented the strongest correlations with all considered empirical measures of insulin sensitivity. While COMO and DMMO appear over-parameterized for fitting single-subject clinical OGTT data, SIMO provides a robust, precise, physiologically plausible estimate of insulin sensitivity, with which habitual empirical insulin sensitivity indices correlate well. The kxgi index, reflecting insulin secretion dependency on glycemia, also significantly differentiates clinically diverse subject groups. The SIMO model may therefore be of value for the quantification of glucose homeostasis from clinical OGTT data.

Nonparametric estimation of random-effects densities in linear mixed-effects model

We consider a linear mixed-effects model where Y k, j =α k +β k t j +ϵ k, j is the observed value for individual k at time t j , k=1, …, N, j=0, 1, …, J. The random effects (α k , β k ) k are independent and identically distributed random variables with unknown densities f α and f β and are independent of noise. We develop nonparametric estimators of these two densities, which involve a cut-off parameter. We study their mean integrated squared risk and propose cut-off selection strategies, depending on the noise distribution assumptions. Finally, in the particular case of fixed interval between times t j , we show that a completely data-driven strategy can be implemented without any knowledge on the noise density. Intensive simulation experiments illustrate the method.