Niels Rode Kristensen

Parameter estimation in stochastic grey-box models

An efficient and flexible parameter estimation scheme for grey-box models in the sense of discretely, partially observed Ito stochastic differential equations with measurement noise is presented along with a corresponding software implementation. The estimation scheme is based on the extended Kalman filter and features maximum likelihood as well as maximum a posteriori estimation on multiple independent data sets, including irregularly sampled data sets and data sets with occasional outliers and missing observations. The software implementation is compared to an existing software tool and proves to have better performance both in terms of quality of estimates for nonlinear systems with significant diffusion and in terms of reproducibility. In particular, the new tool provides more accurate and more consistent estimates of the parameters of the diffusion term.

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.

Nonlinear analysis and control of a continuous fermentation process

Abstract Different types of nonlinear controllers are designed and compared for a simple continuous bioreactor operating near optimal productivity. This operating point is located close to a fold bifurcation point. Nonlinear analysis of stability, controllability and zero dynamics is used to investigate open-loop system properties, to explore the possible control difficulties and to select the system output to be used in the control structure. A wide range of controllers are tested including pole placement and LQ controllers, feedback and input–output linearization controllers and a nonlinear controller based on direct passivation. The comparison is based on time-domain performance and on investigating the stability region, robustness and tuning possibilities of the controllers. Controllers using partial state feedback of the substrate concentration and not directly depending on the reaction rate are recommended for the simple fermenter. Passivity based controllers have been found to be globally stable, not very sensitive to the uncertainties in the reaction rate and controller parameter but they require full nonlinear state feedback.

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 Filters one-step predictions.

Insulin degludec's ultra-long pharmacokinetic properties observed in adults are retained in children and adolescents with type 1 diabetes.

Insulin degludec (IDeg) is a basal insulin with an ultra‐long pharmacokinetic profile in adults that at steady‐state produces remarkably flat and stable insulin levels; however, no studies have yet reported on the pharmacokinetic properties of IDeg in subjects younger than 18 years of age. This was a single‐centre, randomised, single‐dose, double‐blind, two‐period crossover trial conducted in children (6–11 years), adolescents (12–17 years), and adults (18–65 years) with type 1 diabetes. Subjects received a single subcutaneous dose of 0.4 U/kg IDeg or insulin glargine (IGlar), respectively, on two separate dosing visits, with pharmacokinetic blood sampling up to 72‐h postdose. A total of 37 subjects (12 children, 13 adolescents, and 12 adults) completed the trial. Total exposure of IDeg after a single dose (AUCIDeg,0‐∞,SD) was higher in children compared to adults [estimated ratio children/adults 1.48 (95% confidence interval, CI: 0.98; 2.24)] and in adolescents compared to adults [estimated ratio adolescents/adults 1.33 (95% CI: 1.08; 1.64)]; however, the difference was only statistically significant for the latter comparison. No statistically significant difference in maximum concentration of IDeg (Cmax,IDeg,SD) was observed. Estimated ratios for Cmax,IDeg,SD were (children/adults) 1.20 (95% CI: 0.90; 1.60) and (adolescents/adults) 1.23 (95% CI: 1.00; 1.51). Simulated mean steady state pharmacokinetic profiles supported a flat and stable IDeg exposure across a 24‐h dosing interval. IDeg was detectable in serum for at least 72 h (end of blood sampling period) in all subjects following single dose. In conclusion, the ultra‐long pharmacokinetic properties of IDeg observed in adults are preserved in children and adolescents with type 1 diabetes.

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.

Longitudinal Modeling of the Relationship Between Mean Plasma Glucose and HbA1c Following Antidiabetic Treatments

Late‐phase clinical trials within diabetes generally have a duration of 12–24 weeks, where 12 weeks may be too short to reach steady‐state glycated hemoglobin (HbA1c). The main determinant for HbA1c is blood glucose, which reaches steady state much sooner. In spite of this, few publications have used individual data to assess the time course of both glucose and HbA1c, for predicting HbA1c. In this paper, we present an approach for predicting HbA1c at end‐of‐trial (24–28 weeks) using glucose and HbA1c measurements up to 12 weeks. The approach was evaluated using data from 4 trials covering 12 treatment arms (oral antidiabetic drug, glucagon‐like peptide‐1, and insulin treatment) with measurements at 24–28 weeks to evaluate predictions vs. observations. HbA1c percentage was predicted for each arm at end‐of‐trial with a mean prediction error of 0.14% [0.01;0.24]. Furthermore, end points in terms of HbA1c reductions relative to comparator were accurately predicted. The proposed model provides a good basis to optimize late‐stage clinical development within diabetes.

Modeling of 24-hour glucose and insulin profiles in patients with type 2 diabetes mellitus treated with biphasic insulin aspart.

Insulin therapy for diabetes patients is designed to mimic the endogenous insulin response of healthy subjects and thereby generate normal blood glucose levels. In order to control the blood glucose in insulin‐treated diabetes patients, it is important to be able to predict the effect of exogenous insulin on blood glucose. A pharmacokinetic/pharmacodynamic model for glucose homoeostasis describing the effect of exogenous insulin would facilitate such prediction. Thus the aim of this work was to extend the previously developed integrated glucose–insulin (IGI) model to predict 24‐hour glucose profiles for patients with Type 2 diabetes following exogenous insulin administration. Clinical data from two trials were included in the analysis. In both trials, 24‐hour meal tolerance tests were used as the experimental setup, where exogenous insulin (biphasic insulin aspart) was administered in relation to meals. The IGI model was successfully extended to include the effect of exogenous insulin. Circadian variations in glucose homeostasis were assessed on relevant parameters, and a significant improvement was achieved by including a circadian rhythm on the endogenous glucose production in the model. The extended model is a useful tool for clinical trial simulation and for elucidating the effect profile of new insulin products.

Clearance of rFVIIa and NN1731 after intravenous administration to Beagle dogs

AIM NN1731 is a recombinant activated factor VII (rFVIIa) analogue with enhanced activity. The objective of the present study was to evaluate the clearance mechanisms of rFVIIa and NN1731 after intravenous administration to Beagle dogs. METHODS The study was performed in Beagle dogs administered with a single dose of 5.4 nmol/kg rFVIIa or NN1731 intravenously. Plasma samples collected up to 12-h post-administration were analysed using three different assays to determine FVIIa clot activity (FVIIa:C), total FVIIa antigen, and levels of FVIIa-antithrombin (AT) complexes. Pharmacokinetic parameters were determined by use of standard non-compartmental and non-linear mixed effects methods. RESULTS For both compounds, complex formation with AT accounted for the observed difference between the activity and the antigen curves and constituted 60-70% of the total clearance. The clearance of rFVIIa and NN1731 was estimated to be 73 and 214 mL/h/kg, respectively, accordingly, AT complex formation occurred around three times faster for NN1731. The difference in activity observed in the initial phase, resulting in distribution half-lives of 0.71 and 0.22 h for rFVIIa and NN1731, was mainly caused by the 3-fold difference in clearance. The terminal half-life of rFVIIa and NN1731 was estimated to be 2.1 and 2.5 h, respectively. The non-compartmental analysis resulted in almost identical parameters. CONCLUSION The present study demonstrates that the difference between the activity and the antigen profiles of rFVIIa and NN1731 in Beagle dogs is the result of complex formation with AT which constitutes a major pathway for the clearance of rFVIIa activity.

Time series analysis in astronomy: Limits and potentialities

In this paper we consider the problem of the limits concerning the physical information that can be extracted from the analysis of one or more time series (light curves) typical of astrophysical objects. On the basis of theoretical considerations and numerical simulations, we show that with no a priori physical model there are not many possibilities to obtain interpretable results. For this reason, the practice to develop more and more sophisticated statistical methods of time series analysis is not productive. Only techniques of data analysis developed in a specific physical context can be expected to provide useful results. The field of stochastic dynamics appears to be an interesting framework for such an approach. In particular, it is shown that modelling the experimental time series by means of the stochastic differential equations (SDE) represents a valuable tool of analysis. For example, besides a more direct connection between data analysis and theoretical models, in principle the use of SDE permits the analysis of a continuous signal independent of the characteristics (e.g., frequency, regularity, ...) of the sampling with which the experimental time series were obtained. In this respect, an efficient approach based on the extended Kalman filter technique is presented. Its performances and limits are discussed and tested through numerical experiments. Freely downloadable software is made available.