Preconditioning of Nonlinear Mixed Effects Models for Stabilisation of Variance-Covariance Matrix Computations

Bayesian estimation in NONMEM

Bayesian estimation is a powerful but underutilized tool for answering drug development questions. In this tutorial, the principles of Bayesian model development, assessment, and prior selection will be outlined. An example pharmacokinetic (PK) model will be used to demonstrate the implementation of Bayesian modeling using the nonlinear mixed-effects modeling software NONMEM.

Metaheuristics for pharmacometrics

Metaheuristics is a powerful optimization tool that is increasingly used across disciplines to tackle general purpose optimization problems. Nature-inspired metaheuristic algorithms is a subclass of metaheuristic algorithms and have been shown to be particularly flexible and useful in solving complicated optimization problems in computer science and engineering. A common practice with metaheuristics is to hybridize it with another suitably chosen algorithm for enhanced performance. This paper reviews metaheuristic algorithms and demonstrates some of its utility in tackling pharmacometric problems. Specifically, we provide three applications using one of its most celebrated members, particle swarm optimization (PSO), and show that PSO can effectively estimate parameters in complicated nonlinear mixed-effects models and to gain insights into statistical identifiability issues in a complex compartment model. In the third application, we demonstrate how to hybridize PSO with sparse grid, which is an often-used technique to evaluate high dimensional integrals, to search for D -efficient designs for estimating parameters in nonlinear mixed-effects models with a count outcome. We also show the proposed hybrid algorithm outperforms its competitors when sparse grid is replaced by its competitor, adaptive gaussian quadrature to approximate the integral, or when PSO is replaced by three notable nature-inspired metaheuristic algorithms.

Pharmacokinetic/Pharmacodynamic Modelling of Allopurinol, its Active Metabolite Oxypurinol, and Biomarkers Hypoxanthine, Xanthine and Uric Acid in Hypoxic-Ischemic Encephalopathy Neonates

Background

Allopurinol, an xanthine oxidase (XO) inhibitor, is a promising intervention that may provide neuroprotection for neonates with hypoxic-ischemic encephalopathy (HIE). Currently, a double-blind, placebo-controlled study (ALBINO, NCT03162653) is investigating the neuroprotective effect of allopurinol in HIE neonates.

Objective

The aim of the current study was to establish the pharmacokinetics (PK) of allopurinol and oxypurinol, and the pharmacodynamics (PD) of both compounds on hypoxanthine, xanthine, and uric acid in HIE neonates. The dosage used and the effect of allopurinol in this population, either or not undergoing therapeutic hypothermia (TH), were evaluated.

Methods

Forty-six neonates from the ALBINO study and two historical clinical studies were included. All doses were administered on the first day of life. In the ALBINO study (n = 20), neonates received a first dose of allopurinol 20 mg/kg, and, in the case of TH (n = 13), a second dose of allopurinol 10 mg/kg. In the historical cohorts (n = 26), neonates (all without TH) received two doses of allopurinol 20 mg/kg in total. Allopurinol and oxypurinol population PK, and their effects on inhibiting conversions of hypoxanthine and xanthine to uric acid, were assessed using nonlinear mixed-effects modelling.

Results

Allopurinol and oxypurinol PK were described by two sequential one-compartment models with an autoinhibition effect on allopurinol metabolism by oxypurinol. For allopurinol, clearance (CL) was 0.83 L/h (95% confidence interval [CI] 0.62–1.09) and volume of distribution (V_d) was 2.43 L (95% CI 2.25–2.63). For metabolite oxypurinol, CL and V_d relative to a formation fraction (f_m) were 0.26 L/h (95% CI 0.23–0.3) and 11 L (95% CI 9.9–12.2), respectively. No difference in allopurinol and oxypurinol CL was found between TH and non-TH patients. The effect of allopurinol and oxypurinol on XO inhibition was described by a turnover model of hypoxanthine with sequential metabolites xanthine and uric acid. The combined allopurinol and oxypurinol concentration at the half-maximal XO inhibition was 0.36 mg/L (95% CI 0.31–0.42).

Conclusion

The PK and PD of allopurinol, oxypurinol, hypoxanthine, xanthine, and uric acid in neonates with HIE were described. The dosing regimen applied in the ALBINO trial leads to the targeted XO inhibition in neonates treated with or without TH.

Supplementary Information

The online version contains supplementary material available at 10.1007/s40262-021-01068-0.

cLRT-Mod: An efficient methodology for pharmacometric model-based analysis of longitudinal phase II dose finding studies under model uncertainty

Within the challenging context of phase II dose-finding trials, longitudinal analyses may increase drug effect detection power compared to an end-of-treatment analysis. This work proposes cLRT-Mod, a pharmacometric adaptation of the MCP-Mod methodology, which allows the use of nonlinear mixed effect models to first detect a dose-response signal and then identify the doses for the confirmatory phase while accounting for model structure uncertainty. The method was evaluated through extensive clinical trial simulations of a hypothetical phase II dose-finding trial using different scenarios and comparing different methods such as MCP-Mod. The results show an increase in power using cLRT with longitudinal data compared to an EOT multiple contrast tests for scenarios with small sample size and weak drug effect while maintaining pre-specifiability of the models prior to data analysis and the nominal type I error. This work shows how model averaging provides better coverage probability of the drug effect in the prediction step, and avoids under-estimation of the size of the confidence interval. Finally, for illustration purpose cLRT-Mod was applied to the analysis of a real phase II dose-finding trial.

Saddle-Reset for Robust Parameter Estimation and Identifiability Analysis of Nonlinear Mixed Effects Models

Parameter estimation of a nonlinear model based on maximizing the likelihood using gradient-based numerical optimization methods can often fail due to premature termination of the optimization algorithm. One reason for such failure is that these numerical optimization methods cannot distinguish between the minimum, maximum, and a saddle point; hence, the parameters found by these optimization algorithms can possibly be in any of these three stationary points on the likelihood surface. We have found that for maximization of the likelihood for nonlinear mixed effects models used in pharmaceutical development, the optimization algorithm Broyden–Fletcher–Goldfarb–Shanno (BFGS) often terminates in saddle points, and we propose an algorithm, saddle-reset, to avoid the termination at saddle points, based on the second partial derivative test. In this algorithm, we use the approximated Hessian matrix at the point where BFGS terminates, perturb the point in the direction of the eigenvector associated with the lowest eigenvalue, and restart the BFGS algorithm. We have implemented this algorithm in industry standard software for nonlinear mixed effects modeling (NONMEM, version 7.4 and up) and showed that it can be used to avoid termination of parameter estimation at saddle points, as well as unveil practical parameter non-identifiability. We demonstrate this using four published pharmacometric models and two models specifically designed to be practically non-identifiable.

Population pharmacokinetics and covariate analysis of Sym004, an antibody mixture against the epidermal growth factor receptor, in subjects with metastatic colorectal cancer and other solid tumors

Sym004 is an equimolar mixture of two monoclonal antibodies, futuximab and modotuximab, which non-competitively block the epidermal growth factor receptor (EGFR). Sym004 has been clinically tested for treatment of solid tumors. The present work characterizes the non-linear pharmacokinetics (PK) of Sym004 and its constituent antibodies and investigates two types of covariate models for interpreting the interindividual variability of Sym004 exposure. Sym004 serum concentration data from 330 cancer patients participating in four Phase 1 and 2 trials (n = 247 metastatic colorectal cancer, n = 87 various types advanced solid tumors) were pooled for non-linear mixed effects modeling. Dose regimens of 0.4-18 mg/kg Sym004 dosed by i.v. infusion weekly or every 2nd week were explored. The PK profiles for futuximab and modotuximab were parallel, and the parameter values for their population PK models were similar. The PK of Sym004 using the sum of the serum concentrations of futuximab and modotuximab was well captured by a 2-compartment model with parallel linear and saturable, Michaelis-Menten-type elimination. The full covariate model including all plausible covariates included in a single step showed no impact on Sym004 exposure of age, Asian race, renal and hepatic function, tumor type and previous anti-EGFR treatments. The reduced covariate model contained statistically and potentially clinically significant influences of body weight, albumin, sex and baseline tumor size. Population PK modeling and covariate analysis of Sym004 were feasible using the sum of the serum concentrations of the two constituent antibodies. Full and reduced covariate models provided insights into which covariates may be clinically relevant for dose modifications and thus may need further exploration.

Lessons learned from IDeAl - 33 recommendations from the IDeAl-net about design and analysis of small population clinical trials

BACKGROUND

IDeAl (Integrated designs and analysis of small population clinical trials) is an EU funded project developing new statistical design and analysis methodologies for clinical trials in small population groups. Here we provide an overview of IDeAl findings and give recommendations to applied researchers.

METHOD

The description of the findings is broken down by the nine scientific IDeAl work packages and summarizes results from the project's more than 60 publications to date in peer reviewed journals. In addition, we applied text mining to evaluate the publications and the IDeAl work packages' output in relation to the design and analysis terms derived from in the IRDiRC task force report on small population clinical trials.

RESULTS

The results are summarized, describing the developments from an applied viewpoint. The main result presented here are 33 practical recommendations drawn from the work, giving researchers a comprehensive guidance to the improved methodology. In particular, the findings will help design and analyse efficient clinical trials in rare diseases with limited number of patients available. We developed a network representation relating the hot topics developed by the IRDiRC task force on small population clinical trials to IDeAl's work as well as relating important methodologies by IDeAl's definition necessary to consider in design and analysis of small-population clinical trials. These network representation establish a new perspective on design and analysis of small-population clinical trials.

CONCLUSION

IDeAl has provided a huge number of options to refine the statistical methodology for small-population clinical trials from various perspectives. A total of 33 recommendations developed and related to the work packages help the researcher to design small population clinical trial. The route to improvements is displayed in IDeAl-network representing important statistical methodological skills necessary to design and analysis of small-population clinical trials. The methods are ready for use.

Model selection and averaging of nonlinear mixed-effect models for robust phase III dose selection

Population model-based (pharmacometric) approaches are widely used for the analyses of phase IIb clinical trial data to increase the accuracy of the dose selection for phase III clinical trials. On the other hand, if the analysis is based on one selected model, model selection bias can potentially spoil the accuracy of the dose selection process. In this paper, four methods that assume a number of pre-defined model structure candidates, for example a set of dose-response shape functions, and then combine or select those candidate models are introduced. The key hypothesis is that by combining both model structure uncertainty and model parameter uncertainty using these methodologies, we can make a more robust model based dose selection decision at the end of a phase IIb clinical trial. These methods are investigated using realistic simulation studies based on the study protocol of an actual phase IIb trial for an oral asthma drug candidate (AZD1981). Based on the simulation study, it is demonstrated that a bootstrap model selection method properly avoids model selection bias and in most cases increases the accuracy of the end of phase IIb decision. Thus, we recommend using this bootstrap model selection method when conducting population model-based decision-making at the end of phase IIb clinical trials.

Parameter Identifiability of Fundamental Pharmacodynamic Models

Issues of parameter identifiability of routinely used pharmacodynamics models are considered in this paper. The structural identifiability of 16 commonly applied pharmacodynamic model structures was analyzed analytically, using the input-output approach. Both fixed-effects versions (non-population, no between-subject variability) and mixed-effects versions (population, including between-subject variability) of each model structure were analyzed. All models were found to be structurally globally identifiable under conditions of fixing either one of two particular parameters. Furthermore, an example was constructed to illustrate the importance of sufficient data quality and show that structural identifiability is a prerequisite, but not a guarantee, for successful parameter estimation and practical parameter identifiability. This analysis was performed by generating artificial data of varying quality to a structurally identifiable model with known true parameter values, followed by re-estimation of the parameter values. In addition, to show the benefit of including structural identifiability as part of model development, a case study was performed applying an unidentifiable model to real experimental data. This case study shows how performing such an analysis prior to parameter estimation can improve the parameter estimation process and model performance. Finally, an unidentifiable model was fitted to simulated data using multiple initial parameter values, resulting in highly different estimated uncertainties. This example shows that although the standard errors of the parameter estimates often indicate a structural identifiability issue, reasonably “good” standard errors may sometimes mask unidentifiability issues.

Disease Systems Analysis of Bone Mineral Density and Bone Turnover Markers in Response to Alendronate, Placebo, and Washout in Postmenopausal Women

A previously established mechanism-based disease systems model for osteoporosis that is based on a mathematically reduced version of a model describing the interactions between osteoclast (bone removing) and osteoblast (bone forming) cells in bone remodeling has been applied to clinical data from women (n = 1,379) receiving different doses and treatment regimens of alendronate, placebo, and washout. The changes in the biomarkers, plasma bone-specific alkaline phosphatase activity (BSAP), urinary N-telopeptide (NTX), lumbar spine bone mineral density (BMD), and total hip BMD, were linked to the underlying mechanistic core of the model. The final model gave an accurate description of all four biomarkers for the different treatments. Simulations were used to visualize the dynamics of the underlying network and the natural disease progression upon alendronate treatment and discontinuation. These results complement the previous applications of this mechanism-based disease systems model to data from various treatments for osteoporosis.