Non-linear mixed-effects models with stochastic differential equations: implementation of an estimation algorithm

A stochastic mixed effects model to assess treatment effects and fluctuations in home‐measured peak expiratory flow and the association with exacerbation risk in asthma

Home‐based measures of lung function, inflammation, symptoms, and medication use are frequently collected in respiratory clinical trials. However, new statistical approaches are needed to make better use of the information contained in these data‐rich variables. In this work, we use data from two phase III asthma clinical trials demonstrating the benefit of benralizumab treatment to develop a novel longitudinal mixed effects model of peak expiratory flow (PEF), a lung function measure easily captured at home using a hand‐held device. The model is based on an extension of the mixed effects modeling framework to incorporate stochastic differential equations and allows for quantification of several statistical properties of a patient's PEF data: the longitudinal trend, long‐term fluctuations, and day‐to‐day variability. These properties are compared between treatment groups and related to a patient's exacerbation risk using a repeated time‐to‐event model. The mixed effects model adequately described the observed data from the two clinical trials, and model parameters were accurately estimated. Benralizumab treatment was shown to improve a patient's average PEF level and reduce long‐term fluctuations. Both of these effects were shown to be associated with a lower exacerbation risk. The day‐to‐day variability was neither significantly affected by treatment nor associated with exacerbation risk. Our work shows the potential of a stochastic model‐based analysis of home‐based lung function measures to support better estimation and understanding of treatment effects and disease stability. The proposed analysis can serve as a complement to descriptive statistics of home‐based measures in the reporting of respiratory clinical trials.

Likelihood Function through the Delta Approximation in Mixed SDE Models

Stochastic differential equations (SDE) appropriately describe a variety of phenomena occurring in random environments, such as the growth dynamics of individual animals. Using appropriate weight transformations and a variant of the Ornstein–Uhlenbeck model, one obtains a general model for the evolution of cattle weight. The model parameters are α, the average transformed weight at maturity, β, a growth parameter, and σ, a measure of environmental fluctuations intensity. We briefly review our previous work on estimation and prediction issues for this model and some generalizations, considering fixed parameters. In order to incorporate individual characteristics of the animals, we now consider that the parameters α and β are Gaussian random variables varying from animal to animal, which results in SDE mixed models. We estimate parameters by maximum likelihood, but, since a closed-form expression for the likelihood function is usually not possible, we approximate it using our proposed delta approximation method. Using simulated data, we estimate the model parameters and compare them with existing methodologies, showing that the proposed method is a good alternative. It also overcomes the existing methodologies requirement of having all animals weighed at the same ages; thus, we apply it to real data, where such a requirement fails.

Investigating Stochastic Differential Equations Modelling for Levodopa Infusion in Patients with Parkinson's Disease

BACKGROUND AND OBJECTIVES

Levodopa concentration in patients with Parkinson's disease is frequently modelled with ordinary differential equations (ODEs). Here, we investigate a pharmacokinetic model of plasma levodopa concentration in patients with Parkinson's disease by introducing stochasticity to separate the intra-individual variability into measurement and system noise, and to account for auto-correlated errors. We also investigate whether the induced stochasticity provides a better fit than the ODE approach.

METHODS

In this study, a system noise variable is added to the pharmacokinetic model for duodenal levodopa/carbidopa gel (LCIG) infusion described by three ODEs through a standard Wiener process, leading to a stochastic differential equations (SDE) model. The R package population stochastic modelling (PSM) was used for model fitting with data from previous studies for modelling plasma levodopa concentration and parameter estimation. First, the diffusion scale parameter (σ_w), measurement noise variance, and bioavailability are estimated with the SDE model. Second, σ_w is fixed to certain values from 0 to 1 and bioavailability is estimated. Cross-validation was performed to compare the average root mean square errors (RMSE) of predicted plasma levodopa concentration.

RESULTS

Both the ODE and the SDE models estimated bioavailability to be approximately 75%. The SDE model converged at different values of σ_w that were significantly different from zero. The average RMSE for the ODE model was 0.313, and the lowest average RMSE for the SDE model was 0.297 when σ_w was fixed to 0.9, and these two values are significantly different.

CONCLUSIONS

The SDE model provided a better fit for LCIG plasma levodopa concentration by approximately 5.5% in terms of mean percentage change of RMSE.

Exact Gradients Improve Parameter Estimation in Nonlinear Mixed Effects Models with Stochastic Dynamics

Nonlinear mixed effects (NLME) modeling based on stochastic differential equations (SDEs) have evolved into a promising approach for analysis of PK/PD data. SDE-NLME models go beyond the realm of standard population modeling as they consider stochastic dynamics, thereby introducing a probabilistic perspective on the state variables. This article presents a summary of the main contributions to SDE-NLME models found in the literature. The aims of this work were to develop an exact gradient version of the first-order conditional estimation (FOCE) method for SDE-NLME models and to investigate whether it enabled faster estimation and better gradient precision/accuracy compared to the use of gradients approximated by finite differences. A simulation-estimation study was set up whereby finite difference approximations of the gradients of each level were interchanged with the exact gradients. Following previous work, the uncertainty of the state variables was accounted for using the extended Kalman filter (EKF). The exact gradient FOCE method was implemented in Mathematica 11 and evaluated on SDE versions of three common PK/PD models. When finite difference gradients were replaced by exact gradients at both FOCE levels, relative runtimes improved between 6- and 32-fold, depending on model complexity. Additionally, gradient precision/accuracy was significantly better in the exact gradient case. We conclude that parameter estimation using FOCE with exact gradients can successfully be applied to SDE-NLME models.

Pharmacometrics models with hidden Markovian dynamics

The aim of this paper is to provide an overview of pharmacometric models that involve some latent process with Markovian dynamics. Such models include hidden Markov models which may be useful for describing the dynamics of a disease state that jumps from one state to another at discrete times. On the contrary, diffusion models are continuous-time and continuous-state Markov models that are relevant for modelling non observed phenomena that fluctuate continuously and randomly over time. We show that an extension of these models to mixed effects models is straightforward in a population context. We then show how the forward-backward algorithm used for inference in hidden Markov models and the extended Kalman filter used for inference in diffusion models can be combined with standard inference algorithms in mixed effects models for estimating the parameters of the model. The use of these models is illustrated with two applications: a hidden Markov model for describing the epileptic activity of a large number of patients and a stochastic differential equation based model for describing the pharmacokinetics of theophyllin.

Modeling Variability in the Progression of Huntington's Disease A Novel Modeling Approach Applied to Structural Imaging Markers from TRACK-HD

We present a novel, general class of disease progression models for Huntington's disease (HD), a neurodegenerative disease caused by a cytosine-adenine-guanine (CAG) triplet repeat expansion on the huntingtin gene. Models are fit to a selection of structural imaging markers from the TRACK 36-month database. The models are of mixed effects type and should be useful in predicting any continuous marker of HD state as a function of age and CAG length (the genetic factor that drives HD pathology). The effects of age and CAG length are modeled using flexible regression splines. Variability not accounted for by age, CAG length, or covariates is modeled using terms that represent measurement error, population variability (random slopes/intercepts), and variability due to the dynamics of the disease process (random walk terms). A Kalman filter is used to estimate variances of the random walk terms.

Stochastic nonlinear mixed effects: a metformin case study

In nonlinear mixed effect (NLME) modeling, the intra-individual variability is a collection of errors due to assay sensitivity, dosing, sampling, as well as model misspecification. Utilizing stochastic differential equations (SDE) within the NLME framework allows the decoupling of the measurement errors from the model misspecification. This leads the SDE approach to be a novel tool for model refinement. Using Metformin clinical pharmacokinetic (PK) data, the process of model development through the use of SDEs in population PK modeling was done to study the dynamics of absorption rate. A base model was constructed and then refined by using the system noise terms of the SDEs to track model parameters and model misspecification. This provides the unique advantage of making no underlying assumptions about the structural model for the absorption process while quantifying insufficiencies in the current model. This article focuses on implementing the extended Kalman filter and unscented Kalman filter in an NLME framework for parameter estimation and model development, comparing the methodologies, and illustrating their challenges and utility. The Kalman filter algorithms were successfully implemented in NLME models using MATLAB with run time differences between the ODE and SDE methods comparable to the differences found by Kakhi for their stochastic deconvolution.

Mixed Effects Modeling Using Stochastic Differential Equations: Illustrated by Pharmacokinetic Data of Nicotinic Acid in Obese Zucker Rats

Inclusion of stochastic differential equations in mixed effects models provides means to quantify and distinguish three sources of variability in data. In addition to the two commonly encountered sources, measurement error and interindividual variability, we also consider uncertainty in the dynamical model itself. To this end, we extend the ordinary differential equation setting used in nonlinear mixed effects models to include stochastic differential equations. The approximate population likelihood is derived using the first-order conditional estimation with interaction method and extended Kalman filtering. To illustrate the application of the stochastic differential mixed effects model, two pharmacokinetic models are considered. First, we use a stochastic one-compartmental model with first-order input and nonlinear elimination to generate synthetic data in a simulated study. We show that by using the proposed method, the three sources of variability can be successfully separated. If the stochastic part is neglected, the parameter estimates become biased, and the measurement error variance is significantly overestimated. Second, we consider an extension to a stochastic pharmacokinetic model in a preclinical study of nicotinic acid kinetics in obese Zucker rats. The parameter estimates are compared between a deterministic and a stochastic NiAc disposition model, respectively. Discrepancies between model predictions and observations, previously described as measurement noise only, are now separated into a comparatively lower level of measurement noise and a significant uncertainty in model dynamics. These examples demonstrate that stochastic differential mixed effects models are useful tools for identifying incomplete or inaccurate model dynamics and for reducing potential bias in parameter estimates due to such model deficiencies.

Predicting Plasma Glucose From Interstitial Glucose Observations Using Bayesian Methods

One way of constructing a control algorithm for an artificial pancreas is to identify a model capable of predicting plasma glucose (PG) from interstitial glucose (IG) observations. Stochastic differential equations (SDEs) make it possible to account both for the unknown influence of the continuous glucose monitor (CGM) and for unknown physiological influences. Combined with prior knowledge about the measurement devices, this approach can be used to obtain a robust predictive model. A stochastic-differential-equation-based gray box (SDE-GB) model is formulated on the basis of an identifiable physiological model of the glucoregulatory system for type 1 diabetes mellitus (T1DM) patients. A Bayesian method is used to estimate robust parameters from clinical data. The models are then used to predict PG from IG observations from 2 separate study occasions on the same patient. First, all statistically significant diffusion terms of the model are identified using likelihood ratio tests, yielding inclusion of [Formula: see text], [Formula: see text], and [Formula: see text]. Second, estimates using maximum likelihood are obtained, but prediction capability is poor. Finally a Bayesian method is implemented. Using this method the identified models are able to predict PG using only IG observations. These predictions are assessed visually. We are also able to validate these estimates on a separate data set from the same patient. This study shows that SDE-GBs and a Bayesian method can be used to identify a reliable model for prediction of PG using IG observations obtained with a CGM. The model could eventually be used in an artificial pancreas.

Topological augmentation to infer hidden processes in biological systems

MOTIVATION

A common problem in understanding a biochemical system is to infer its correct structure or topology. This topology consists of all relevant state variables-usually molecules and their interactions. Here we present a method called topological augmentation to infer this structure in a statistically rigorous and systematic way from prior knowledge and experimental data.

RESULTS

Topological augmentation starts from a simple model that is unable to explain the experimental data and augments its topology by adding new terms that capture the experimental behavior. This process is guided by representing the uncertainty in the model topology through stochastic differential equations whose trajectories contain information about missing model parts. We first apply this semiautomatic procedure to a pharmacokinetic model. This example illustrates that a global sampling of the parameter space is critical for inferring a correct model structure. We also use our method to improve our understanding of glutamine transport in yeast. This analysis shows that transport dynamics is determined by glutamine permeases with two different kinds of kinetics. Topological augmentation can not only be applied to biochemical systems, but also to any system that can be described by ordinary differential equations.

AVAILABILITY AND IMPLEMENTATION

Matlab code and examples are available at: http://www.csb.ethz.ch/tools/index

Parameter Estimation of Population Pharmacokinetic Models with Stochastic Differential Equations: Implementation of an Estimation Algorithm

Population pharmacokinetic (PPK) models play a pivotal role in quantitative pharmacology study, which are classically analyzed by nonlinear mixed-effects models based on ordinary differential equations. This paper describes the implementation of SDEs in population pharmacokinetic models, where parameters are estimated by a novel approximation of likelihood function. This approximation is constructed by combining the MCMC method used in nonlinear mixed-effects modeling with the extended Kalman filter used in SDE models. The analysis and simulation results show that the performance of the approximation of likelihood function for mixed-effects SDEs model and analysis of population pharmacokinetic data is reliable. The results suggest that the proposed method is feasible for the analysis of population pharmacokinetic data.

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.

A continuous-time adaptive particle filter for estimations under measurement time uncertainties with an application to a plasma-leucine mixed effects model

Background

When mathematical modelling is applied to many different application areas, a common task is the estimation of states and parameters based on measurements. With this kind of inference making, uncertainties in the time when the measurements have been taken are often neglected, but especially in applications taken from the life sciences, this kind of errors can considerably influence the estimation results. As an example in the context of personalized medicine, the model-based assessment of the effectiveness of drugs is becoming to play an important role. Systems biology may help here by providing good pharmacokinetic and pharmacodynamic (PK/PD) models. Inference on these systems based on data gained from clinical studies with several patient groups becomes a major challenge. Particle filters are a promising approach to tackle these difficulties but are by itself not ready to handle uncertainties in measurement times.

Results

In this article, we describe a variant of the standard particle filter (PF) algorithm which allows state and parameter estimation with the inclusion of measurement time uncertainties (MTU). The modified particle filter, which we call MTU-PF, also allows the application of an adaptive stepsize choice in the time-continuous case to avoid degeneracy problems. The modification is based on the model assumption of uncertain measurement times. While the assumption of randomness in the measurements themselves is common, the corresponding measurement times are generally taken as deterministic and exactly known. Especially in cases where the data are gained from measurements on blood or tissue samples, a relatively high uncertainty in the true measurement time seems to be a natural assumption. Our method is appropriate in cases where relatively few data are used from a relatively large number of groups or individuals, which introduce mixed effects in the model. This is a typical setting of clinical studies. We demonstrate the method on a small artificial example and apply it to a mixed effects model of plasma-leucine kinetics with data from a clinical study which included 34 patients.

Conclusions

Comparisons of our MTU-PF with the standard PF and with an alternative Maximum Likelihood estimation method on the small artificial example clearly show that the MTU-PF obtains better estimations. Considering the application to the data from the clinical study, the MTU-PF shows a similar performance with respect to the quality of estimated parameters compared with the standard particle filter, but besides that, the MTU algorithm shows to be less prone to degeneration than the standard particle filter.

Individual predictions based on nonlinear mixed modeling: application to prenatal twin growth

The assessment of growth during fetal life and childhood commonly relies upon cross'sectional reference ranges or centiles. However, individual sequential predictions may help the timewise assessment of a growth process. In twin pregnancies for example, which are at risk of growth restriction, such predictions may improve the detection of abnormal trajectories. In this article, we present a simple forecasting method, assuming that a given normal individual behaves in the same way as a reference population. We consider, as a prediction in a given individual, the forecast of a future observation conditional to any previous observation and a set of population parameters obtained by nonlinear mixed modeling in a reference population. We suggest an estimator for this prediction without resorting to linear approximation and show that it enjoys interesting asymptotics when the amount of observations increases over time. We use two independent real datasets of twin pregnancies with normal growth and outcome to illustrate the application of such predictions in prenatal growth. We consider the first dataset as a reference dataset and model it using a two'level nonlinear model. We perform illustration and validation of predictions on the second dataset.

The dose-dependent effect of S(+)-ketamine on cardiac output in healthy volunteers and complex regional pain syndrome type 1 chronic pain patients

BACKGROUND

Ketamine is used as an analgesic for treatment of acute and chronic pain. While ketamine has a stimulatory effect on the cardiovascular system, little is known about the concentration-effect relationship. We examined the effect of S(+)-ketamine on cardiac output in healthy volunteers and chronic pain patients using a pharmacokinetic-pharmacodynamic modeling approach.

METHODS

In 10 chronic pain patients (diagnosed with complex regional pain syndrome type 1 [CRPS1] with a mean age 43.2 ± 13 years, disease duration 8.4 years, range 1.1 to 21.7 years) and 12 healthy volunteers (21.3 ± 1.6 years), 7 increasing IV doses of S(+)-ketamine were given over 5 minutes at 20-minute intervals starting with 1.5 mg with 1.5-mg increments. Cardiac output (CO) was calculated from the arterial pressure curve obtained from an arterial catheter in the radial artery. Ketamine and norketamine plasma concentrations were measured. A novel pharmacokinetic-pharmacodynamic model was constructed to quantify the direct stimulatory effect of ketamine on CO and the following adaptation/inhibition.

RESULTS

Significant differences in pharmacokinetic estimates were observed between study groups with 15% and 40% larger S(+)-ketamine S(+)-norketamine concentrations in healthy volunteers compared to CRPS1 patients. S(+)-ketamine had a dose-dependent stimulatory effect on CO in patients and volunteers. After infusion an inhibitory effect on CO was observed. Pharmacodynamic model parameters did not differ between CRPS1 patients and healthy volunteers. The concentration of S(+)-ketamine causing a 1 L/min increase in CO was 243 ± 54 ng/mL with an onset/offset half-life of 1.3 ± 0.21 minutes. The inhibitory component was slow (time constant of 67.2 ± 17.0 minutes).

CONCLUSIONS

S(+)-ketamine pharmacokinetics but not pharmacodynamics differed between study populations, related to differences in disease state (CRPS1 or not) or age. The dose-dependent effect of S(+)-ketamine on CO was well described by the biphasic dynamic model. The effect of S(+)-ketamine on CO was similar between study groups with respect to its stimulatory and inhibitory components, despite group differences in age and health.

Bayesian analysis of growth curves using mixed models defined by stochastic differential equations

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.

Predictive performance for population models using stochastic differential equations applied on data from an oral glucose tolerance test

Several articles have investigated stochastic differential equations (SDEs) in PK/PD models, but few have quantitatively investigated the benefits to predictive performance of models based on real data. Estimation of first phase insulin secretion which reflects beta-cell function using models of the OGTT is a difficult problem in need of further investigation. The present work aimed at investigating the power of SDEs to predict the first phase insulin secretion (AIR (0-8)) in the IVGTT based on parameters obtained from the minimal model of the OGTT, published by Breda et al. (Diabetes 50(1):150-158, 2001). In total 174 subjects underwent both an OGTT and a tolbutamide modified IVGTT. Estimation of parameters in the oral minimal model (OMM) was performed using the FOCE-method in NONMEM VI on insulin and C-peptide measurements. The suggested SDE models were based on a continuous AR(1) process, i.e. the Ornstein-Uhlenbeck process, and the extended Kalman filter was implemented in order to estimate the parameters of the models. Inclusion of the Ornstein-Uhlenbeck (OU) process caused improved description of the variation in the data as measured by the autocorrelation function (ACF) of one-step prediction errors. A main result was that application of SDE models improved the correlation between the individual first phase indexes obtained from OGTT and AIR (0-8) (r = 0.36 to r = 0.49 and r = 0.32 to r = 0.47 with C-peptide and insulin measurements, respectively). In addition to the increased correlation also the properties of the indexes obtained using the SDE models more correctly assessed the properties of the first phase indexes obtained from the IVGTT. In general it is concluded that the presented SDE approach not only caused autocorrelation of errors to decrease but also improved estimation of clinical measures obtained from the glucose tolerance tests. Since, the estimation time of extended models was not heavily increased compared to basic models, the applied method is concluded to have high relevance not only in theory but also in practice.

Population stochastic modelling (PSM)--an R package for mixed-effects models based on stochastic differential equations

The extension from ordinary to stochastic differential equations (SDEs) in pharmacokinetic and pharmacodynamic (PK/PD) modelling is an emerging field and has been motivated in a number of articles [N.R. Kristensen, H. Madsen, S.H. Ingwersen, Using stochastic differential equations for PK/PD model development, J. Pharmacokinet. Pharmacodyn. 32 (February(1)) (2005) 109-141; C.W. Tornøe, R.V. Overgaard, H. Agersø, H.A. Nielsen, H. Madsen, E.N. Jonsson, Stochastic differential equations in NONMEM: implementation, application, and comparison with ordinary differential equations, Pharm. Res. 22 (August(8)) (2005) 1247-1258; R.V. Overgaard, N. Jonsson, C.W. Tornøe, H. Madsen, Non-linear mixed-effects models with stochastic differential equations: implementation of an estimation algorithm, J. Pharmacokinet. Pharmacodyn. 32 (February(1)) (2005) 85-107; U. Picchini, S. Ditlevsen, A. De Gaetano, Maximum likelihood estimation of a time-inhomogeneous stochastic differential model of glucose dynamics, Math. Med. Biol. 25 (June(2)) (2008) 141-155]. PK/PD models are traditionally based ordinary differential equations (ODEs) with an observation link that incorporates noise. This state-space formulation only allows for observation noise and not for system noise. Extending to SDEs allows for a Wiener noise component in the system equations. This additional noise component enables handling of autocorrelated residuals originating from natural variation or systematic model error. Autocorrelated residuals are often partly ignored in PK/PD modelling although violating the hypothesis for many standard statistical tests. This article presents a package for the statistical program R that is able to handle SDEs in a mixed-effects setting. The estimation method implemented is the FOCE(1) approximation to the population likelihood which is generated from the individual likelihoods that are approximated using the Extended Kalman Filter's one-step predictions.

Reconstructing population exposures to environmental chemicals from biomarkers: challenges and opportunities

A conceptual/computational framework for exposure reconstruction from biomarker data combined with auxiliary exposure-related data is presented, evaluated with example applications, and examined in the context of future needs and opportunities. This framework employs physiologically based toxicokinetic (PBTK) modeling in conjunction with numerical "inversion" techniques. To quantify the value of different types of exposure data "accompanying" biomarker data, a study was conducted focusing on reconstructing exposures to chlorpyrifos, from measurements of its metabolite levels in urine. The study employed biomarker data as well as supporting exposure-related information from the National Human Exposure Assessment Survey (NHEXAS), Maryland, while the MENTOR-3P system (Modeling ENvironment for TOtal Risk with Physiologically based Pharmacokinetic modeling for Populations) was used for PBTK modeling. Recently proposed, simple numerical reconstruction methods were applied in this study, in conjunction with PBTK models. Two types of reconstructions were studied using (a) just the available biomarker and supporting exposure data and (b) synthetic data developed via augmenting available observations. Reconstruction using only available data resulted in a wide range of variation in estimated exposures. Reconstruction using synthetic data facilitated evaluation of numerical inversion methods and characterization of the value of additional information, such as study-specific data that can be collected in conjunction with the biomarker data. Although the NHEXAS data set provides a significant amount of supporting exposure-related information, especially when compared to national studies such as the National Health and Nutrition Examination Survey (NHANES), this information is still not adequate for detailed reconstruction of exposures under several conditions, as demonstrated here. The analysis presented here provides a starting point for introducing improved designs for future biomonitoring studies, from the perspective of exposure reconstruction; identifies specific limitations in existing exposure reconstruction methods that can be applied to population biomarker data; and suggests potential approaches for addressing exposure reconstruction from such data.

Parameters of the Diffusion Leaky Integrate-and-Fire Neuronal Model for a Slowly Fluctuating Signal

Stochastic leaky integrate-and-fire (LIF) neuronal models are common theoretical tools for studying properties of real neuronal systems. Experimental data of frequently sampled membrane potential measurements between spikes show that the assumption of constant parameter values is not realistic and that some (random) fluctuations are occurring. In this article, we extend the stochastic LIF model, allowing a noise source determining slow fluctuations in the signal. This is achieved by adding a random variable to one of the parameters characterizing the neuronal input, considering each interspike interval (ISI) as an independent experimental unit with a different realization of this random variable. In this way, the variation of the neuronal input is split into fast (within-interval) and slow (between-intervals) components. A parameter estimation method is proposed, allowing the parameters to be estimated simultaneously over the entire data set. This increases the statistical power, and the average estimate over all ISIs will be improved in the sense of decreased variance of the estimator compared to previous approaches, where the estimation has been conducted separately on each individual ISI. The results obtained on real data show good agreement with classical regression methods.

Exterior exposure estimation using a one-compartment toxicokinetic model with blood sample measurements

Exposure assessment of individuals exposed to certain chemicals plays an important role in the analysis of occupational-as well as environmental-health problems. Biological monitoring, as an alternative to direct environmental measurements, may be applied to relate the exterior exposure with the amount of individual intake. In this paper, we estimate individuals' (inhalation) exposure retrospectively from their blood concentrations via a simplified one-compartment toxicokinetic model. Considering stochastic variations to the toxicokinetic model, the solution to the resultant stochastic differential equation (SDE), together with measurement error, is transformed into a dynamic linear state-space model. The unknown model parameters and the mean inhalation concentration are then estimated via Markov Chain Monte Carlo (MCMC) simulations. The proposed method is used in the analysis of the styrene data (Wang et al. in Occup Environ Med 53:601-605, 1996) to backward estimate the inhalation concentration, assuming it is unknown. The data analysis showed that the internal stochastic variations, often ignored in toxicokinetic model analysis, outweighed in standard deviation almost twice that of the measurement error. Also, the simulation results showed that the method performed relatively well to the approach considering measurement error only.

A matlab framework for estimation of NLME models using stochastic differential equations

The non-linear mixed-effects model based on stochastic differential equations (SDEs) provides an attractive residual error model, that is able to handle serially correlated residuals typically arising from structural mis-specification of the true underlying model. The use of SDEs also opens up for new tools for model development and easily allows for tracking of unknown inputs and parameters over time. An algorithm for maximum likelihood estimation of the model has earlier been proposed, and the present paper presents the first general implementation of this algorithm. The implementation is done in Matlab and also demonstrates the use of parallel computing for improved estimation times. The use of the implementation is illustrated by two examples of application which focus on the ability of the model to estimate unknown inputs facilitated by the extension to SDEs. The first application is a deconvolution-type estimation of the insulin secretion rate based on a linear two-compartment model for C-peptide measurements. In the second application the model is extended to also give an estimate of the time varying liver extraction based on both C-peptide and insulin measurements.

PKPD Model of Interleukin-21 Effects on Thermoregulation in Monkeys—Application and Evaluation of Stochastic Differential Equations

PURPOSE

To describe the pharmacodynamic effects of recombinant human interleukin-21 (IL-21) on core body temperature in cynomolgus monkeys using basic mechanisms of heat regulation. A major effort was devoted to compare the use of ordinary differential equations (ODEs) with stochastic differential equations (SDEs) in pharmacokinetic pharmacodynamic (PKPD) modelling.

METHODS

A temperature model was formulated including circadian rhythm, metabolism, heat loss, and a thermoregulatory set-point. This model was formulated as a mixed-effects model based on SDEs using NONMEM.

RESULTS

The effects of IL-21 were on the set-point and the circadian rhythm of metabolism. The model was able to describe a complex set of IL-21 induced phenomena, including 1) disappearance of the circadian rhythm, 2) no effect after first dose, and 3) high variability after second dose. SDEs provided a more realistic description with improved simulation properties, and further changed the model into one that could not be falsified by the autocorrelation function.

CONCLUSIONS

The IL-21 induced effects on thermoregulation in cynomolgus monkeys are explained by a biologically plausible model. The quality of the model was improved by the use of SDEs.

Stochastic Differential Equations in NONMEM®: Implementation, Application, and Comparison with Ordinary Differential Equations

PURPOSE

The objective of the present analysis was to explore the use of stochastic differential equations (SDEs) in population pharmacokinetic/pharmacodynamic (PK/PD) modeling.

METHODS

The intra-individual variability in nonlinear mixed-effects models based on SDEs is decomposed into two types of noise: a measurement and a system noise term. The measurement noise represents uncorrelated error due to, for example, assay error while the system noise accounts for structural misspecifications, approximations of the dynamical model, and true random physiological fluctuations. Since the system noise accounts for model misspecifications, the SDEs provide a diagnostic tool for model appropriateness. The focus of the article is on the implementation of the Extended Kalman Filter (EKF) in NONMEM for parameter estimation in SDE models.

RESULTS

Various applications of SDEs in population PK/PD modeling are illustrated through a systematic model development example using clinical PK data of the gonadotropin releasing hormone (GnRH) antagonist degarelix. The dynamic noise estimates were used to track variations in model parameters and systematically build an absorption model for subcutaneously administered degarelix.

CONCLUSIONS

The EKF-based algorithm was successfully implemented in NONMEM for parameter estimation in population PK/PD models described by systems of SDEs. The example indicated that it was possible to pinpoint structural model deficiencies, and that valuable information may be obtained by tracking unexplained variations in parameters.

Using Stochastic Differential Equations for PK/PD Model Development

A method for PK/PD model development based on stochastic differential equation models is proposed. The new method has a number of advantages compared to conventional methods. In particular, the new method avoids the exhaustive trial-and-error based search often conducted to determine the most appropriate model structure, because it allows information about the appropriate model structure to be extracted directly from data. This is accomplished through quantification of the uncertainty of the individual parts of an initial model, by means of which tools for performing model diagnostics can be constructed and guidelines for model improvement provided. Furthermore, the new method allows time-variations in key parameters to be tracked and visualized graphically, which allows important functional relationships to be revealed. Using simulated data, the performance of the new method is demonstrated by means of two examples. The first example shows how, starting from a simple assumption of linear PK, the method can be used to determine the correct nonlinear model for describing the PK of a drug following an oral dose. The second example shows how, starting from a simple assumption of no drug effect, the method can be used to determine the correct model for the nonlinear effect of a drug with known PK in an indirect response model.