Rune Viig Overgaard

Non-Linear Mixed-Effects Models with Stochastic Differential Equations - Implementation of an Estimation Algorithm

Pharmacokinetic/pharmacodynamic modelling is most often performed using non-linear mixed-effects models based on ordinary differential equations with uncorrelated intra-individual residuals. More sophisticated residual error models as e.g. stochastic differential equations (SDEs) with measurement noise can in many cases provide a better description of the variations, which could be useful in various aspects of modelling. This general approach enables a decomposition of the intra-individual residual variation ε into system noise w and measurement noise e. The present work describes implementation of SDEs in a non-linear mixed-effects model, where parameter estimation was performed by a novel approximation of the likelihood function. This approximation is constructed by combining the First-Order Conditional Estimation (FOCE) method used in non-linear mixed-effects modelling with the Extended Kalman Filter used in models with SDEs. Fundamental issues concerning the proposed model and estimation algorithm are addressed by simulation studies, concluding that system noise can successfully be separated from measurement noise and inter-individual variability.

Stochastic differential equations in NONMEM: implementation, application, and comparison with ordinary differential equations.

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

Body composition is the main determinant for the difference in type 2 diabetes pathophysiology between Japanese and Caucasians.

OBJECTIVE This cross-sectional clinical study compared the pathophysiology of type 2 diabetes in Japanese and Caucasians and investigated the role of demographic, genetic, and lifestyle-related risk factors for insulin resistance and β-cell response. RESEARCH DESIGN AND METHODS A total of 120 Japanese and 150 Caucasians were enrolled to obtain comparable distributions of high/low BMI values across glucose tolerance states (normal glucose tolerance, impaired glucose tolerance, and type 2 diabetes), which were assessed by oral glucose tolerance tests. BMI in the two cohorts was distributed around the two regional cutoff values for obesity. RESULTS Insulin sensitivity was higher in Japanese compared with Caucasians, as indicated by the homeostatic model assessment of insulin resistance and Matsuda indices, whereas β-cell response was higher in Caucasians, as measured by homeostatic model assessment of β-cell function, the insulinogenic indices, and insulin secretion ratios. Disposition indices were similar for Japanese and Caucasians at all glucose tolerance states, indicating similar β-cell response relative to the degree of insulin resistance. The main determinants for differences in metabolic indices were measures of body composition, such as BMI and distribution of adipose tissue. Differences in β-cell response between Japanese and Caucasians were not statistically significant following adjustment by differences in BMI. CONCLUSIONS Our study showed similar disposition indices in Japanese and Caucasians and that the major part of the differences in insulin sensitivity and β-cell response between Japanese and Caucasians can be explained by differences in body composition.

Ethnic Differences in Insulin Sensitivity, β-Cell Function, and Hepatic Extraction Between Japanese and Caucasians: A Minimal Model Analysis

CONTEXT Ethnic differences have previously been reported for type 2 diabetes. OBJECTIVE We aimed at assessing the potential differences between Caucasian and Japanese subjects ranging from normal glucose tolerance (NGT) to impaired glucose tolerance (IGT) and to type 2 diabetes. DESIGN This was a cross-sectional study with oral glucose tolerance tests to assess β-cell function, hepatic insulin extraction, and insulin sensitivity. PARTICIPANTS PARTICIPANTS included 120 Japanese and 150 Caucasian subjects. MAIN OUTCOMES Measures of β-cell function, hepatic extraction, and insulin sensitivity were assessed using C-peptide, glucose, and insulin minimal models. RESULTS Basal β-cell function (Φ(b)) was lower in Japanese compared with Caucasians (P < .01). In subjects with IGT, estimates of the dynamic (Φ(d)) and static (Φ(s)) β-cell responsiveness were significantly lower in the Japanese compared with Caucasians (P < .05). In contrast, values of insulin action showed higher sensitivity in the Japanese IGT subjects. Hepatic extraction was similar in NGT and IGT groups but higher in Japanese type 2 diabetic subjects (P < .01). Despite differences in insulin sensitivity, β-cell function, and hepatic extraction, the disposition indices were similar between the 2 ethnic groups at all glucose tolerance states. Furthermore, the overall insulin sensitivity and β-cell responsiveness for all glucose tolerance states were similar in Japanese and Caucasians after accounting for differences in body mass index. CONCLUSION Our study provides evidence for a similar ability of Japanese and Caucasians to compensate for increased insulin resistance.

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.

Pharmacokinetics of an anti-TFPI monoclonal antibody (concizumab) blocking the TFPI interaction with the active site of FXa in Cynomolgus monkeys after iv and sc administration

INTRODUCTION Concizumab (mAb 2021) is a monoclonal IgG4 antibody (mAb) that binds to the Kunitz-type protease inhibitor (KPI) 2 domain of TFPI thereby blocking the interaction of this domain with the active site of FXa. The objective of the present study was to characterize the pharmacokinetics of concizumab in Cynomolgus monkeys after intravenous (iv) and subcutaneous (sc) administration. METHODS Data from two studies were included in the modelling, all in all data from 52 monkeys distributed into 9 groups. Three groups received three escalating sc doses of concizumab with a one week dosing interval, two groups were administered a single dose, and four groups received multiple doses over 13 weeks of concizumab. The plasma concentration was measured using a standard ELISA, and pharmacokinetic data were analysed using NONMEM. RESULTS The pharmacokinetics of concizumab were characterised by a high bioavailability (93%) after sc administration. The time course of the elimination of concizumab from the circulation was well described by the proposed target mediated drug disposition (TMDD) model. The clearance of concizumab was estimated to be 0.14 ml/h/kg, the target clearance was characterized by a 50% saturation level of 0.54 μg/ml (Km), and the clearance at target saturation was estimated to be 11 μg/h/kg. CONCLUSION Concizumab displays a typical TMDD profile with important implications for a putative treatment regime in haemophilia patients. Compared to current standard haemophilia treatment, concizumab has a high bioavailability after sc administration and may provide a viable alternative to intravenous dosing for the treatment of haemophilia.

An Integrated Disease/Pharmacokinetic/ Pharmacodynamic Model Suggests Improved Interleukin-21 Regimens Validated Prospectively for Mouse Solid Cancers

Interleukin (IL)-21 is an attractive antitumor agent with potent immunomodulatory functions. Yet thus far, the cytokine has yielded only partial responses in solid cancer patients, and conditions for beneficial IL-21 immunotherapy remain elusive. The current work aims to identify clinically-relevant IL-21 regimens with enhanced efficacy, based on mathematical modeling of long-term antitumor responses. For this purpose, pharmacokinetic (PK) and pharmacodynamic (PD) data were acquired from a preclinical study applying systemic IL-21 therapy in murine solid cancers. We developed an integrated disease/PK/PD model for the IL-21 anticancer response, and calibrated it using selected “training” data. The accuracy of the model was verified retrospectively under diverse IL-21 treatment settings, by comparing its predictions to independent “validation” data in melanoma and renal cell carcinoma-challenged mice (R2>0.90). Simulations of the verified model surfaced important therapeutic insights: (1) Fractionating the standard daily regimen (50 µg/dose) into a twice daily schedule (25 µg/dose) is advantageous, yielding a significantly lower tumor mass (45% decrease); (2) A low-dose (12 µg/day) regimen exerts a response similar to that obtained under the 50 µg/day treatment, suggestive of an equally efficacious dose with potentially reduced toxicity. Subsequent experiments in melanoma-bearing mice corroborated both of these predictions with high precision (R2>0.89), thus validating the model also prospectively in vivo. Thus, the confirmed PK/PD model rationalizes IL-21 therapy, and pinpoints improved clinically-feasible treatment schedules. Our analysis demonstrates the value of employing mathematical modeling and in silico-guided design of solid tumor immunotherapy in the clinic.

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.

Establishing Good Practices for Exposure-Response Analysis of Clinical Endpoints in Drug Development.

This tutorial aims at promoting good practices for exposure–response (E‐R) analyses of clinical endpoints in drug development. The focus is on practical aspects of E‐R analyses to assist modeling scientists with a process of performing such analyses in a consistent manner across individuals and projects and tailored to typical clinical drug development decisions. This includes general considerations for planning, conducting, and visualizing E‐R analyses, and how these are linked to key questions.

Anti-NKG2D monoclonal antibody (NNC0142-0002) in active Crohn's disease: a randomised controlled trial

Objective Anti-NKG2D (NNC0142-0002) is an antagonising human immunoglobulin G4 monoclonal antibody that binds to natural killer group 2 member D (NKG2D) receptors, which are expressed by T cells and innate lymphoid cells, and may be linked to mucosal damage in Crohns disease (CD). Design Seventy-eight patients (aged ≥18 and ≤75 years) with CD for ≥3 months, Crohns Disease Activity Index (CDAI) ≥220 and ≤450 and either C-reactive protein ≥10 mg/L or endoscopic evidence of inflammation, were randomised 1:1 to a single subcutaneous (SC) dose of 2 mg/kg anti-NKG2D or placebo. Primary endpoint was change in CDAI (ΔCDAI) from baseline to week 4. Prespecified significance level was 10% for CDAI endpoints. A futility analysis was instituted due to slow recruitment. Results Primary endpoint was not significantly different between anti-NKG2D and placebo (week 4 ΔCDAI=–16); however, there was a significant difference by week 12 (ΔCDAI=–55; p≤0.10). Significant improvements were noted in the non-failure to biologics subgroup (treated with anti-NKG2D (n=28)) from week 1 onward. Greater effects of anti-NKG2D were also observed in patients with baseline CDAI ≥330. Frequencies of adverse events (AEs) were comparable between anti-NKG2D and placebo. Most AEs were mild (49%) or moderate (43%). No antidrug antibodies were observed. Conclusions A single SC dose of 2 mg/kg anti-NKG2D did not reduce disease activity at week 4 versus placebo, but the difference was significant at week 12, and effects were evident in key subgroups. These data support further development of anti-NKG2D in IBD. Trial registration number NCT01203631.