Design in nonlinear mixed effects models: Optimization using the Fedorov–Wynn algorithm and power of the Wald test for binary covariates

PyMulSim: a method for computing node similarities between multilayer networks via graph isomorphism networks

BACKGROUND

In bioinformatics, interactions are modelled as networks, based on graph models. Generally, these support a single-layer structure which incorporates a specific entity (i.e., node) and only one type of link (i.e., edge). However, real-world biological systems consisting of biological objects belonging to heterogeneous entities, and these operate and influence each other in multiple contexts, simultaneously. Usually, node similarities are investigated to assess the relatedness between biological objects in a network of interest, and node embeddings are widely used for studying novel interaction from a topological point of view. About that, the state-of-the-art presents several methods for evaluating the node similarity inside a given network, but methodologies able to evaluate similarities between pairs of nodes belonging to different networks are missing. The latter are crucial for studies that relate different biological networks, e.g., for Network Alignment or to evaluate the possible evolution of the interactions of a little-known network on the basis of a well-known one. Existing methods are ineffective in evaluating nodes outside their structure, even more so in the context of multilayer networks, in which the topic still exploits approaches adapted from static networks. In this paper, we presented pyMulSim, a novel method for computing the pairwise similarities between nodes belonging to different multilayer networks. It uses a Graph Isomorphism Network (GIN) for the representative learning of node features, that uses for processing the embeddings and computing the similarities between the pairs of nodes of different multilayer networks.

RESULTS

Our experimentation investigated the performance of our method. Results show that our method effectively evaluates the similarities between the biological objects of a source multilayer network to a target one, based on the analysis of the node embeddings. Results have been also assessed for different noise levels, also through statistical significance analyses properly performed for this purpose.

CONCLUSIONS

PyMulSim is a novel method for computing the pairwise similarities between nodes belonging to different multilayer networks, by using a GIN for learning node embeddings. It has been evaluated both in terms of performance and validity, reporting a high degree of reliability.

The midpoint of cortical thinning between late childhood and early adulthood differs between individuals and brain regions: Evidence from longitudinal modelling in a 12-wave neuroimaging sample

Charting human brain maturation between childhood and adulthood is a fundamental prerequisite for understanding the rapid biological and psychological changes during human development. Two barriers have precluded the quantification of maturational trajectories: demands on data and demands on estimation. Using high-temporal resolution neuroimaging data of up to 12-waves in the HUBU cohort (N = 90, aged 7-21 years) we investigate changes in apparent cortical thickness across childhood and adolescence. Fitting a four-parameter logistic nonlinear random effects mixed model, we quantified the characteristic, s-shaped, trajectory of cortical thinning in adolescence. This approach yields biologically meaningful parameters, including the midpoint of cortical thinning (MCT), which corresponds to the age at which the cortex shows most rapid thinning - in our sample occurring, on average, at 14 years of age. These results show that, given suitable data and models, cortical maturation can be quantified with precision for each individual and brain region.

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.

Finding optimal design in nonlinear mixed effect models using multiplicative algorithms

BACKGROUND AND OBJECTIVES

To optimize designs for longitudinal studies analyzed by nonlinear mixed effect models (NLMEMs), the Fisher information matrix (FIM) can be used. In this work, we focused on the multiplicative algorithms, previously applied in standard individual regression, to find optimal designs for NLMEMs.

METHODS

We extended multiplicative algorithms to mixed models and implemented the algorithm both in R and in C. Then, we applied the algorithm to find D-optimal designs in two longitudinal data examples, one with continuous and one with binary outcome.

RESULTS

For these examples, we quantified the improved speed when C is used instead of R. Design optimization using the multiplicative algorithm led to designs with D-efficiency gains between 13% and 25% compared to non-optimized designs.

CONCLUSION

We found that the multiplicative algorithm can be used efficiently to design longitudinal studies.

Model-based approach to sampling optimization in studies of antibacterial drugs for infants and young children

Clinical trials for pediatric indications and new pediatric drugs face challenges, including the limited blood volume due to the patients’ small bodies. In Japan, the Evaluation Committee on Unapproved or Off‐labeled Drugs with High Medical Needs has discussed the necessity of pediatric indications against the background of a lack of Japanese pediatric data. The limited treatment options regarding antibiotics for pediatric patients are associated with the emergence of antibiotic‐resistant bacteria. Regulatory guidelines promote the use of model‐based drug development to reduce practical and ethical constraints for pediatric patients. Sampling optimization is one of the key study designs for pediatric drug development. In this simulation study, we evaluated the precision of the empirical Bayes estimates of pharmacokinetic (PK) parameters based on the sampling times optimized by published pediatric population PK models. We selected three previous PK studies of cefepime and ciprofloxacin in infants and young children as paradigms. The number of sampling times was reduced from original full sampling times to two to four sampling times based on the Fisher information matrix. We observed that the precision of empirical Bayes estimates of the key PK parameters and the predicted efficacy based on the reduced sampling times were generally comparable to those based on the original full sampling times. The model‐based approach to sampling optimization provided a maximization of PK information with a minimum burden on infants and young children for the future development of pediatric drugs.

Robust designs accounting for model uncertainty in longitudinal studies with binary outcomes

To optimize designs for longitudinal studies analyzed by mixed-effect models with binary outcomes, the Fisher information matrix can be used. Optimal design approaches, however, require a priori knowledge of the model. We aim to propose, for the first time, a robust design approach accounting for model uncertainty in longitudinal trials with two treatment groups, assuming mixed-effect logistic models. To optimize designs given one model, we compute several optimality criteria based on Fisher information matrix evaluated by the new approach based on Monte-Carlo/Hamiltonian Monte-Carlo. We propose to use the DDS-optimality criterion, as it ensures a compromise between the precision of estimation of the parameters, and hence the Wald test power, and the overall precision of parameter estimation. To account for model uncertainty, we assume candidate models with their respective weights. We compute robust design across these models using compound DDS-optimality. Using the Fisher information matrix, we propose to predict the average power over these models. Evaluating this approach by clinical trial simulations, we show that the robust design is efficient across all models, allowing one to achieve good power of test. The proposed design strategy is a new and relevant approach to design longitudinal studies with binary outcomes, accounting for model uncertainty.

Liraglutide effects in a paediatric (7-11 y) population with obesity: A randomized, double-blind, placebo-controlled, short-term trial to assess safety, tolerability, pharmacokinetics, and pharmacodynamics

BACKGROUND

Childhood obesity is a major public health concern with limited treatment options.

OBJECTIVE

The aim of this study was to assess safety, tolerability, pharmacokinetics, and pharmacodynamics during short-term treatment with liraglutide in children (7-11 y) with obesity.

METHODS

In this randomized, double-blind, placebo-controlled trial, 24 children received at least one dose of once-daily subcutaneous liraglutide (n = 16) or placebo (n = 8) starting at 0.3 mg with weekly dose escalations up to 3.0 mg or maximum tolerated dose, and 20 children completed the trial (14 in the liraglutide group and six in the placebo group). The primary endpoint was the number of adverse events.

RESULTS

Baseline characteristics (mean ± standard deviation) included the following: age 9.9 ± 1.1 years, weight 71.5 ± 15.4 kg, and 62.5% male. Thirty-seven adverse events were reported in nine liraglutide-treated participants (56.3%) versus 12 events in five placebo-treated participants (62.5%). Most adverse events were mild in severity, three were of moderate severity, and none were severe. Gastrointestinal disorders were the most frequently reported events occurring in 37.5% of liraglutide-treated participants compared with placebo (12.5%). Six asymptomatic hypoglycaemic episodes occurred in five participants of whom four were liraglutide treated. Liraglutide exposure was consistent with dose proportionality. Body weight was the only covariate to significantly impact exposure. A significant reduction in body mass index (BMI) Z score from baseline to end of treatment (estimated treatment difference: -0.28; P = 0.0062) was observed.

CONCLUSION

Short-term treatment with liraglutide in children with obesity revealed a safety and tolerability profile similar to trials in adults and adolescents with obesity, with no new safety issues.

MPBPK-TMDD models for mAbs: alternative models, comparison, and identifiability issues

The aim of the present study was to evaluate model identifiability when minimal physiologically-based pharmacokinetic (mPBPK) models are integrated with target mediated drug disposition (TMDD) models in the tissue compartment. Three quasi-steady-state (QSS) approximations of TMDD dynamics were explored: on (a) antibody-target complex, (b) free target, and (c) free antibody concentrations in tissue. The effects of the QSS approximations were assessed via simulations, taking as reference the mPBPK-TMDD model with no simplifications. Approximation (a) did not affect model-derived concentrations, while with the inclusion of approximation (b) or (c), target concentration profiles alone, or both drug and target concentration profiles respectively deviated from the reference model profiles. A local sensitivity analysis was performed, highlighting the potential importance of sampling in the terminal pharmacokinetic phase and of collecting target concentration data. The a priori and a posteriori identifiability of the mPBPK-TMDD models were investigated under different experimental scenarios and designs. The reference model and QSS approximation (a) on antibody-target complex were both found to be a priori identifiable in all scenarios, while under the further inclusion of QSS approximation (b) target concentration data were needed for a priori identifiability to be preserved. The property could not be assessed for the model including all three QSS approximations. A posteriori identifiability issues were detected for all models, although improvement was observed when appropriate sampling and dose range were selected. In conclusion, this work provides a theoretical framework for the assessment of key properties of mathematical models before their experimental application. Attention should be paid when applying integrated mPBPK-TMDD models, as identifiability issues do exist, especially when rich study designs are not feasible.

PFIM 4.0, an extended R program for design evaluation and optimization in nonlinear mixed-effect models

BACKGROUND AND OBJECTIVE

Nonlinear mixed-effect models (NLMEMs) are increasingly used for the analysis of longitudinal studies during drug development. When designing these studies, the expected Fisher information matrix (FIM) can be used instead of performing time-consuming clinical trial simulations. The function PFIM is the first tool for design evaluation and optimization that has been developed in R. In this article, we present an extended version, PFIM 4.0, which includes several new features.

METHODS

Compared with version 3.0, PFIM 4.0 includes a more complete pharmacokinetic/pharmacodynamic library of models and accommodates models including additional random effects for inter-occasion variability as well as discrete covariates. A new input method has been added to specify user-defined models through an R function. Optimization can be performed assuming some fixed parameters or some fixed sampling times. New outputs have been added regarding the FIM such as eigenvalues, conditional numbers, and the option of saving the matrix obtained after evaluation or optimization. Previously obtained results, which are summarized in a FIM, can be taken into account in evaluation or optimization of one-group protocols. This feature enables the use of PFIM for adaptive designs. The Bayesian individual FIM has been implemented, taking into account a priori distribution of random effects. Designs for maximum a posteriori Bayesian estimation of individual parameters can now be evaluated or optimized and the predicted shrinkage is also reported. It is also possible to visualize the graphs of the model and the sensitivity functions without performing evaluation or optimization.

RESULTS

The usefulness of these approaches and the simplicity of use of PFIM 4.0 are illustrated by two examples: (i) an example of designing a population pharmacokinetic study accounting for previous results, which highlights the advantage of adaptive designs; (ii) an example of Bayesian individual design optimization for a pharmacodynamic study, showing that the Bayesian individual FIM can be a useful tool in therapeutic drug monitoring, allowing efficient prediction of estimation precision and shrinkage for individual parameters.

CONCLUSION

PFIM 4.0 is a useful tool for design evaluation and optimization of longitudinal studies in pharmacometrics and is freely available at http://www.pfim.biostat.fr.

A minimal resting time of 25 min is needed before measuring stabilized blood pressure in subjects addressed for vascular investigations

Blood pressure (BP) measurement is a central element in clinical practice. According to international recommendations 3 to 5 minutes of resting is needed before blood pressure measurement. Surprisingly, no study has modelled the time course of BP decrease and the minimum resting-time before BP measurement. A cross-sectional bicentric observational study was performed including outpatients addressed for vascular examination. Using two automatic BP monitors we recorded the blood pressure every minute during 11 consecutive minutes. The data was analyzed by non-linear mixed effect regression. Systolic (SBP) and diastolic BPs were studied and we tested the effect of covariates on its evolution through log-likelihood ratio tests. We included 199 patients (66+/−13years old). SBP was found to decrease exponentially. Simulations based on the final model show that only half the population reaches a stabilized SBP (defined as SBP + 5 mmHg) after 5 min of resting-time while it takes 25 min to ensure 90% of the population has a stabilized SBP. In conclusion, our results and simulations suggest that 5 minutes are not enough to achieve a stabilized SBP in most patients and at least 25 minutes are required. This questions whether the diagnosis of hypertension can be reliably made during routine visits in general practitioners’ offices.

FPCA-based method to select optimal sampling schedules that capture between-subject variability in longitudinal studies

A critical component of longitudinal study design involves determining the sampling schedule. Criteria for optimal design often focus on accurate estimation of the mean profile, although capturing the between-subject variance of the longitudinal process is also important since variance patterns may be associated with covariates of interest or predict future outcomes. Existing design approaches have limited applicability when one wishes to optimize sampling schedules to capture between-individual variability. We propose an approach to derive optimal sampling schedules based on functional principal component analysis (FPCA), which separately characterizes the mean and the variability of longitudinal profiles and leads to a parsimonious representation of the temporal pattern of the variability. Simulation studies show that the new design approach performs equally well compared to an existing approach based on parametric mixed model (PMM) when a PMM is adequate for the data, and outperforms the PMM-based approach otherwise. We use the methods to design studies aiming to characterize daily salivary cortisol profiles and identify the optimal days within the menstrual cycle when urinary progesterone should be measured.

Safety, dosing, and pharmaceutical quality for studies that evaluate medicinal products (including biological products) in neonates

The study of medications among pediatric patients has increased worldwide since 1997 in response to new legislation and regulations, but these studies have not yet adequately addressed the therapeutic needs of neonates. Additionally, extant guidance developed by regulatory agencies worldwide does not fully address the specificities of neonatal drug development, especially among extremely premature newborns who currently survive. Consequently, an international consortium from Canada, Europe, Japan, and the United States was organized by the Critical Path Institute to address the content of guidance. This group included neonatologists, neonatal nurses, parents, regulators, ethicists, clinical pharmacologists, specialists in pharmacokinetics, specialists in clinical trials and pediatricians working in the pharmaceutical industry. This group has developed a comprehensive, referenced White Paper to guide neonatal clinical trials of medicines - particularly early phase studies. Key points include: the need to base product development on neonatal physiology and pharmacology while making the most of knowledge acquired in other settings; the central role of families in research; and the value of the whole neonatal team in the design, implementation and interpretation of studies. This White Paper should facilitate successful clinical trials of medicines in neonates by informing regulators, sponsors, and the neonatal community of existing good practice.

Optimum designs for non-linear mixed effects models in the presence of covariates

In this article, we present a new method for optimizing designs of experiments for non-linear mixed effects models, where a categorical factor with covariate information is a design variable combined with another design factor. The work is motivated by the need to efficiently design preclinical experiments in enzyme kinetics for a set of Human Liver Microsomes. However, the results are general and can be applied to other experimental situations where the variation in the response due to a categorical factor can be partially accounted for by a covariate. The covariate included in the model explains some systematic variability in a random model parameter. This approach allows better understanding of the population variation as well as estimation of the model parameters with higher precision.

Accelerating Monte Carlo power studies through parametric power estimation

Estimating the power for a non-linear mixed-effects model-based analysis is challenging due to the lack of a closed form analytic expression. Often, computationally intensive Monte Carlo studies need to be employed to evaluate the power of a planned experiment. This is especially time consuming if full power versus sample size curves are to be obtained. A novel parametric power estimation (PPE) algorithm utilizing the theoretical distribution of the alternative hypothesis is presented in this work. The PPE algorithm estimates the unknown non-centrality parameter in the theoretical distribution from a limited number of Monte Carlo simulation and estimations. The estimated parameter linearly scales with study size allowing a quick generation of the full power versus study size curve. A comparison of the PPE with the classical, purely Monte Carlo-based power estimation (MCPE) algorithm for five diverse pharmacometric models showed an excellent agreement between both algorithms, with a low bias of less than 1.2 % and higher precision for the PPE. The power extrapolated from a specific study size was in a very good agreement with power curves obtained with the MCPE algorithm. PPE represents a promising approach to accelerate the power calculation for non-linear mixed effect models.

Joint Model of Iron and Hepcidin During the Menstrual Cycle in Healthy Women

Hepcidin regulates serum iron levels, and its dosage is used in differential diagnostic of iron-related pathologies. We used the data collected in the HEPMEN (named after HEPcidin during MENses) study to investigate the joint dynamics of serum hepcidin and iron during the menstrual cycle in healthy women. Ninety menstruating women were recruited after a screening visit. Six fasting blood samples for determination of iron-status variables were taken in the morning throughout the cycle, starting on the second day of the period. Non-linear mixed effect models were used to describe the evolution of iron and hepcidin. Demographic and medical covariates were tested for their effect on model parameters. Parameter estimation was performed using the SAEM algorithm implemented in the Monolix software. A general pattern was observed for both hepcidin and iron, consisting of an initial decrease during menstruation, followed by a rebound and stabilising during the second half of the cycle. We developed a joint model including a menstruation-induced decrease of both molecules at the beginning of the menses and a rebound effect after menses. Iron stimulated the release of hepcidin. Several covariates, including contraception, amount of blood loss and ferritin, were found to influence the parameters. The joint model of iron and hepcidin was able to describe the fluctuations induced by blood loss from menstruation in healthy non-menopausal women and the subsequent regulation. The HEPMEN study showed fluctuations of iron-status variables during the menstrual cycle, which should be considered when using hepcidin measurements for diagnostic purposes in women of child-bearing potential.

Designing a Pediatric Study for an Antimalarial Drug by Using Information from Adults

The objectives of this study were to design a pharmacokinetic (PK) study by using information about adults and evaluate the robustness of the recommended design through a case study of mefloquine. PK data about adults and children were available from two different randomized studies of the treatment of malaria with the same artesunate-mefloquine combination regimen. A recommended design for pediatric studies of mefloquine was optimized on the basis of an extrapolated model built from adult data through the following approach. (i) An adult PK model was built, and parameters were estimated by using the stochastic approximation expectation-maximization algorithm. (ii) Pediatric PK parameters were then obtained by adding allometry and maturation to the adult model. (iii) A D-optimal design for children was obtained with PFIM by assuming the extrapolated design. Finally, the robustness of the recommended design was evaluated in terms of the relative bias and relative standard errors (RSE) of the parameters in a simulation study with four different models and was compared to the empirical design used for the pediatric study. Combining PK modeling, extrapolation, and design optimization led to a design for children with five sampling times. PK parameters were well estimated by this design with few RSE. Although the extrapolated model did not predict the observed mefloquine concentrations in children very accurately, it allowed precise and unbiased estimates across various model assumptions, contrary to the empirical design. Using information from adult studies combined with allometry and maturation can help provide robust designs for pediatric studies.

Methods and software tools for design evaluation in population pharmacokinetics-pharmacodynamics studies

Population pharmacokinetic (PK)-pharmacodynamic (PKPD) models are increasingly used in drug development and in academic research; hence, designing efficient studies is an important task. Following the first theoretical work on optimal design for nonlinear mixed-effects models, this research theme has grown rapidly. There are now several different software tools that implement an evaluation of the Fisher information matrix for population PKPD. We compared and evaluated the following five software tools: PFIM, PkStaMp, PopDes, PopED and POPT. The comparisons were performed using two models, a simple-one compartment warfarin PK model and a more complex PKPD model for pegylated interferon, with data on both concentration and response of viral load of hepatitis C virus. The results of the software were compared in terms of the standard error (SE) values of the parameters predicted from the software and the empirical SE values obtained via replicated clinical trial simulation and estimation. For the warfarin PK model and the pegylated interferon PKPD model, all software gave similar results. Interestingly, it was seen, for all software, that the simpler approximation to the Fisher information matrix, using the block diagonal matrix, provided predicted SE values that were closer to the empirical SE values than when the more complicated approximation was used (the full matrix). For most PKPD models, using any of the available software tools will provide meaningful results, avoiding cumbersome simulation and allowing design optimization.

Influence of covariance between random effects in design for nonlinear mixed-effect models with an illustration in pediatric pharmacokinetics

Nonlinear mixed-effect models are used increasingly during drug development. For design, an alternative to simulations is based on the Fisher information matrix. Its expression was derived using a first-order approach, was then extended to include covariance and implemented into the R function PFIM. The impact of covariance on standard errors, amount of information, and optimal designs was studied. It was also shown how standard errors can be predicted analytically within the framework of rich individual data without the model. The results were illustrated by applying this extension to the design of a pharmacokinetic study of a drug in pediatric development.

Pilot Phase II study of mazindol in children with attention deficit/hyperactivity disorder

OBJECTIVE

Mazindol has been proposed as a potential treatment of children with attention deficit/hyperactivity disorder (ADHD). The purpose of this pilot study was to assess its pharmacokinetics, short-term efficacy, and safety.

SUBJECTS AND METHODS

A total of 24 children (aged 9-12 years) with ADHD (according to the Diagnostic and Statistical Manual of Mental Disorders, Fourth Edition, text-revision criteria) received a daily dose of 1 mg for 7 days and were followed for 3 additional weeks. Pharmacokinetic samples were collected after the first administration. ADHD symptoms were assessed using the ADHD Rating Scale (RS)-IV, Conners' Parent Rating Scale - Revised: Long (CPRS-R:L) at screening, baseline, and the end of the study. The Clinical Global Impression - Severity (CGI-S) scale was assessed at baseline, and the CGI - Improvement (CGI-I) scale was assessed at subsequent visits.

RESULTS

Twenty-one subjects (aged 10±1 years) were analyzed. Pharmacokinetic data were described by a one-compartment model with first-order absorption, elimination, and lag time. The typical apparent clearance and apparent volume of distribution were 27.9 L/h and 234 L, and increased with fat-free mass and age, respectively. The mean change in score in ADHD RS-IV after 1 week of mazindol was -24.1 (P<0.0001), greater than a 90% improvement from baseline. Reduction of CPRS-R:L and CGI-S scores were -52.1 (P<0.0001) and -2.5 (P<0.01), respectively. Adverse events were mild to moderate, decreased appetite and upper abdominal pain being the most common.

CONCLUSION

This preliminary study shows that mazindol might be an effective, well-tolerated, and long-acting (more than 8 hours) agent for the treatment of ADHD in children.

Utilization of optimal study design for maternal and fetal sheep propofol pharmacokinetics study: a preliminary study

Multiple blood samples are generally required for measurement of pharmacokinetic (PK) parameters. D-optimal design is a popular and frequently used approach for determination of sampling time points in order to minimize the number of samples, while optimizing the estimation of PK parameters. Optimal design utilizing ADAPT (v5, BSR, University of Southern California, Los Angeles) developed a sparse sampling strategy to determine measurement of propofol in pregnant sheep. Propofal was administered as supplemental anesthetic agent to inhalation anesthesia to mimic anesthesia for open fetal surgery. In our preliminary study, propofol 3 mg/kg was given as a bolus to the ewe, followed by propofol infusion at rate 450 mcg/kg/min for 60 minutes, then decreased to 75 mcg/kg/min for 90 more minutes and then ceased. A three compartment model described the PK parameters with the fetus assumed as the third compartment. Initially, sampling times were chosen from thirteen time points as previously stated in the literature. Using priori propofol PK estimates, the final 9 sample time points were proposed in an optimal design with a change in infusion rate occurring between 65 and 75 minutes and sampling proposed at 5, 15, 25, 65, 75, 100, 110, 150, and 180 minutes. D-optimal design optimized the number and timing of samplings, which led to a reduction of cost and man power in the study protocol while preserving the ability to estimate propofol PK parameters in the maternal and fetal sheep model. Initial evaluation of samples collected from three sheep using the optimal design strategy confirmed the performance of the design in obtaining effective PK parameter estimates.

Statistical power calculations for mixed pharmacokinetic study designs using a population approach

Simultaneous modelling of dense and sparse pharmacokinetic data is possible with a population approach. To determine the number of individuals required to detect the effect of a covariate, simulation-based power calculation methodologies can be employed. The Monte Carlo Mapped Power method (a simulation-based power calculation methodology using the likelihood ratio test) was extended in the current study to perform sample size calculations for mixed pharmacokinetic studies (i.e. both sparse and dense data collection). A workflow guiding an easy and straightforward pharmacokinetic study design, considering also the cost-effectiveness of alternative study designs, was used in this analysis. Initially, data were simulated for a hypothetical drug and then for the anti-malarial drug, dihydroartemisinin. Two datasets (sampling design A: dense; sampling design B: sparse) were simulated using a pharmacokinetic model that included a binary covariate effect and subsequently re-estimated using (1) the same model and (2) a model not including the covariate effect in NONMEM 7.2. Power calculations were performed for varying numbers of patients with sampling designs A and B. Study designs with statistical power >80% were selected and further evaluated for cost-effectiveness. The simulation studies of the hypothetical drug and the anti-malarial drug dihydroartemisinin demonstrated that the simulation-based power calculation methodology, based on the Monte Carlo Mapped Power method, can be utilised to evaluate and determine the sample size of mixed (part sparsely and part densely sampled) study designs. The developed method can contribute to the design of robust and efficient pharmacokinetic studies.

Optimal selection of individuals for repeated covariate measurements in follow-up studies

Repeated covariate measurements bring important information on the time-varying risk factors in long epidemiological follow-up studies. However, due to budget limitations, it may be possible to carry out the repeated measurements only for a subset of the cohort. We study cost-efficient alternatives for the simple random sampling in the selection of the individuals to be remeasured. The proposed selection criteria are based on forms of the D-optimality. The selection methods are compared with the simulation studies and illustrated with the data from the East-West study carried out in Finland from 1959 to 1999. The results indicate that cost savings can be achieved if the selection is focused on the individuals with high expected risk of the event and, on the other hand, on those with extreme covariate values in the previous measurements.

Utility of population pharmacokinetic modeling in the assessment of therapeutic protein-drug interactions

Assessment of pharmacokinetic (PK) based drug-drug interactions (DDI) is essential for ensuring patient safety and drug efficacy. With the substantial increase in therapeutic proteins (TP) entering the market and drug development, evaluation of TP-drug interaction (TPDI) has become increasingly important. Unlike for small molecule (e.g., chemical-based) drugs, conducting TPDI studies often presents logistical challenges, while the population PK (PPK) modeling may be a viable approach dealing with the issues. A working group was formed with members from the pharmaceutical industry and the FDA to assess the utility of PPK-based TPDI assessment including study designs, data analysis methods, and implementation strategy. This paper summarizes key issues for consideration as well as a proposed strategy with focuses on (1) PPK approach for exploratory assessment; (2) PPK approach for confirmatory assessment; (3) importance of data quality; (4) implementation strategy; and (5) potential regulatory implications. Advantages and limitations of the approach are also discussed.

Optimizing disease progression study designs for drug effect discrimination

Investigate the possibility to directly optimize a clinical trial design for statistical power to detect a drug effect and compare to optimal designs that focus on parameter precision. An improved statistic derived from the general formulation of the Wald approximation was used to predict the statistical power for given trial designs of a disease progression study. The predicted value was compared, together with the classical Wald statistic, to a type I error-corrected model-based power determined via clinical trial simulations. In a second step, a study design for maximal power was determined by directly maximizing the new statistic. The resulting power-optimal designs and their corresponding performance based on empirical power calculations were compared to designs focusing on parameter precision. Comparisons of empirically determined power and the newly developed statistic, showed excellent agreement across all scenarios investigated. This was in contrast to the classical Wald statistic, which consistently over-predicted the reference power with deviations of up to 90 %. Designs maximized using the proposed metric differed from traditional optimal designs and showed equal or up to 20 % higher power in the subsequent clinical trial simulations. Furthermore, the proposed method was used to minimize the number of individuals required to achieve 80 % power through a simultaneous optimization of study size and study design. The targeted power of 80 % was confirmed in subsequent simulation study. A new statistic was developed, allowing for the explicit optimization of a clinical trial design with respect to statistical power.

Optimal sampling times for a drug and its metabolite using SIMCYP(®) simulations as prior information

BACKGROUND

Since 2007, it is mandatory for the pharmaceutical companies to submit a Paediatric Investigation Plan to the Paediatric Committee at the European Medicines Agency for any drug in development in adults, and it often leads to the need to conduct a pharmacokinetic study in children. Pharmacokinetic studies in children raise ethical and methodological issues. Because of limitation of sampling times, appropriate methods, such as the population approach, are necessary for analysis of the pharmacokinetic data. The choice of the pharmacokinetic sampling design has an important impact on the precision of population parameter estimates. Approaches for design evaluation and optimization based on the evaluation of the Fisher information matrix (M(F)) have been proposed and are now implemented in several software packages, such as PFIM in R.

OBJECTIVES

The objectives of this work were to (1) develop a joint population pharmacokinetic model to describe the pharmacokinetic characteristics of a drug S and its active metabolite in children after intravenous drug administration from simulated plasma concentration-time data produced using physiologically based pharmacokinetic (PBPK) predictions; (2) optimize the pharmacokinetic sampling times for an upcoming clinical study using a multi-response design approach, considering clinical constraints; and (3) evaluate the resulting design taking data below the lower limit of quantification (BLQ) into account.

METHODS

Plasma concentration-time profiles were simulated in children using a PBPK model previously developed with the software SIMCYP(®) for the parent drug and its active metabolite. Data were analysed using non-linear mixed-effect models with the software NONMEM(®), using a joint model for the parent drug and its metabolite. The population pharmacokinetic design, for the future study in 82 children from 2 to 18 years old, each receiving a single dose of the drug, was then optimized using PFIM, assuming identical times for parent and metabolite concentration measurements and considering clinical constraints. Design evaluation was based on the relative standard errors (RSEs) of the parameters of interest. In the final evaluation of the proposed design, an approach was used to assess the possible effect of BLQ concentrations on the design efficiency. This approach consists of rescaling the M(F), using, at each sampling time, the probability of observing a concentration BLQ computed from Monte-Carlo simulations.

RESULTS

A joint pharmacokinetic model with three compartments for the parent drug and one for its active metabolite, with random effects on four parameters, was used to fit the simulated PBPK concentration-time data. A combined error model best described the residual variability. Parameters and dose were expressed per kilogram of bodyweight. Reaching a compromise between PFIM results and clinical constraints, the optimal design was composed of four samples at 0.1, 1.8, 5 and 10 h after drug injection. This design predicted RSE lower than 30 % for the four parameters of interest. For this design, rescaling M(F) for BLQ data had very little influence on predicted RSE.

CONCLUSION

PFIM was a useful tool to find an optimal sampling design in children, considering clinical constraints. Even if it was not forecasted initially by the investigators, this approach showed that it was really necessary to include a late sampling time for all children. Moreover, we described an approach to evaluate designs assuming expected proportions of BLQ data are omitted.

Clinical trial simulation to evaluate power to compare the antiviral effectiveness of two hepatitis C protease inhibitors using nonlinear mixed effect models: a viral kinetic approach

BACKGROUND

Models of hepatitis C virus (HCV) kinetics are increasingly used to estimate and to compare in vivo drug's antiviral effectiveness of new potent anti-HCV agents. Viral kinetic parameters can be estimated using non-linear mixed effect models (NLMEM). Here we aimed to evaluate the performance of this approach to precisely estimate the parameters and to evaluate the type I errors and the power of the Wald test to compare the antiviral effectiveness between two treatment groups when data are sparse and/or a large proportion of viral load (VL) are below the limit of detection (BLD).

METHODS

We performed a clinical trial simulation assuming two treatment groups with different levels of antiviral effectiveness. We evaluated the precision and the accuracy of parameter estimates obtained on 500 replication of this trial using the stochastic approximation expectation-approximation algorithm which appropriately handles BLD data. Next we evaluated the type I error and the power of the Wald test to assess a difference of antiviral effectiveness between the two groups. Standard error of the parameters and Wald test property were evaluated according to the number of patients, the number of samples per patient and the expected difference in antiviral effectiveness.

RESULTS

NLMEM provided precise and accurate estimates for both the fixed effects and the inter-individual variance parameters even with sparse data and large proportion of BLD data. However Wald test with small number of patients and lack of information due to BLD resulted in an inflation of the type I error as compared to the results obtained when no limit of detection of VL was considered. The corrected power of the test was very high and largely outperformed what can be obtained with empirical comparison of the mean VL decline using Wilcoxon test.

CONCLUSION

This simulation study shows the benefit of viral kinetic models analyzed with NLMEM over empirical approaches used in most clinical studies. When designing a viral kinetic study, our results indicate that the enrollment of a large number of patients is to be preferred to small population sample with frequent assessments of VL.

PopED: an extended, parallelized, nonlinear mixed effects models optimal design tool

Several developments have facilitated the practical application and increased the general use of optimal design for nonlinear mixed effects models. These developments include new methodology for utilizing advanced pharmacometric models, faster optimization algorithms and user friendly software tools. In this paper we present the extension of the optimal design software PopED, which incorporates many of these recent advances into an easily useable enhanced GUI. Furthermore, we present new solutions to problems related to the design of experiments such as: faster and more robust FIM calculations and optimizations, optimizing over cost/utility functions and diagnostic tools and plots to evaluate design performance. Examples for; (i) Group size optimization and efficiency translation, (ii) Cost/constraint optimization, (iii) Optimizations with different FIM approximations and (iv) optimization with parallel computing demonstrate the new features in PopED and underline the potential use of this tool when designing experiments.

Rapid sample size calculations for a defined likelihood ratio test-based power in mixed-effects models

Efficient power calculation methods have previously been suggested for Wald test-based inference in mixed-effects models but the only available alternative for Likelihood ratio test-based hypothesis testing has been to perform computer-intensive multiple simulations and re-estimations. The proposed Monte Carlo Mapped Power (MCMP) method is based on the use of the difference in individual objective function values (ΔiOFV) derived from a large dataset simulated from a full model and subsequently re-estimated with the full and reduced models. The ΔiOFV is sampled and summed (∑ΔiOFVs) for each study at each sample size of interest to study, and the percentage of ∑ΔiOFVs greater than the significance criterion is taken as the power. The power versus sample size relationship established via the MCMP method was compared to traditional assessment of model-based power for six different pharmacokinetic and pharmacodynamic models and designs. In each case, 1,000 simulated datasets were analysed with the full and reduced models. There was concordance in power between the traditional and MCMP methods such that for 90% power, the difference in required sample size was in most investigated cases less than 10%. The MCMP method was able to provide relevant power information for a representative pharmacometric model at less than 1% of the run-time of an SSE. The suggested MCMP method provides a fast and accurate prediction of the power and sample size relationship.

Pharmacokinetic similarity of biologics: analysis using nonlinear mixed-effects modeling

Our objective was to show, using two examples, that a pharmacokinetic (PK) similarity analysis can be performed using nonlinear mixed-effects models (NLMEM). We used two studies that compared different biosimilars: a three-way crossover trial with somatropin and a parallel-group trial with epoetin-α. For both data sets, the results of NLMEM-based analysis were compared with those of noncompartmental analysis (NCA). For the latter analysis, we performed an NLMEM-based equivalence Wald test on secondary parameters of the model: the area under the curve and the maximal concentration. Somatropin PK was described by a one-compartment model and epoetin-α PK by a two-compartment model with linear and Michaelis-Menten elimination. For both studies, similarity of PK was demonstrated by means of both NCA and NLMEM, and both methods led to similar results. Therefore, for establishing similarity, PK data can be analyzed by either of the methods. NCA is an easier approach because it does not require data modeling; however, NLMEM leads to a better understanding of the underlying biological system.

Design evaluation and optimisation in crossover pharmacokinetic studies analysed by nonlinear mixed effects models

Bioequivalence or interaction trials are commonly studied in crossover design and can be analysed by nonlinear mixed effects models as an alternative to noncompartmental approach. We propose an extension of the population Fisher information matrix in nonlinear mixed effects models to design crossover pharmacokinetic trials, using a linearisation of the model around the random effect expectation, including within-subject variability and discrete covariates fixed or changing between periods. We use the expected standard errors of treatment effect to compute the power for the Wald test of comparison or equivalence and the number of subjects needed for a given power. We perform various simulations mimicking crossover two-period trials to show the relevance of these developments. We then apply these developments to design a crossover pharmacokinetic study of amoxicillin in piglets and implement them in the new version 3.2 of the r function PFIM.

Design evaluation and optimization for models of hepatitis C viral dynamics

Mathematical modeling of hepatitis C viral (HCV) kinetics is widely used for understanding viral pathogenesis and predicting treatment outcome. The standard model is based on a system of five non-linear ordinary differential equations (ODE) that describe both viral kinetics and changes in drug concentration after treatment initiation. In such complex models parameter estimation is challenging and requires frequent sampling measurements on each individual. By borrowing information between study subjects, non-linear mixed effect models can deal with sparser sampling from each individual. However, the search for optimal designs in this context has been limited by the numerical difficulty of evaluating the Fisher information matrix (FIM). Using the software PFIM, we show that a linearization of the statistical model avoids most of the computational burden, while providing a good approximation to the FIM. We then compare the precision of the parameters that can be expected using five study designs from the literature. We illustrate the usefulness of rationalizing data sampling by showing that, for a given level of precision, optimal design could reduce the total number of measurements by up 50 per cent. Our approach can be used by a statistician or a clinician aiming at designing an HCV viral kinetics study.

Pharmacokinetic design optimization in children and estimation of maturation parameters: example of cytochrome P450 3A4

The aim of this work was to determine whether optimizing the study design in terms of ages and sampling times for a drug eliminated solely via cytochrome P450 3A4 (CYP3A4) would allow us to accurately estimate the pharmacokinetic parameters throughout the entire childhood timespan, while taking into account age- and weight-related changes. A linear monocompartmental model with first-order absorption was used successively with three different residual error models and previously published pharmacokinetic parameters ("true values"). The optimal ages were established by D-optimization using the CYP3A4 maturation function to create "optimized demographic databases." The post-dose times for each previously selected age were determined by D-optimization using the pharmacokinetic model to create "optimized sparse sampling databases." We simulated concentrations by applying the population pharmacokinetic model to the optimized sparse sampling databases to create optimized concentration databases. The latter were modeled to estimate population pharmacokinetic parameters. We then compared true and estimated parameter values. The established optimal design comprised four age ranges: 0.008 years old (i.e., around 3 days), 0.192 years old (i.e., around 2 months), 1.325 years old, and adults, with the same number of subjects per group and three or four samples per subject, in accordance with the error model. The population pharmacokinetic parameters that we estimated with this design were precise and unbiased (root mean square error [RMSE] and mean prediction error [MPE] less than 11% for clearance and distribution volume and less than 18% for k(a)), whereas the maturation parameters were unbiased but less precise (MPE < 6% and RMSE < 37%). Based on our results, taking growth and maturation into account a priori in a pediatric pharmacokinetic study is theoretically feasible. However, it requires that very early ages be included in studies, which may present an obstacle to the use of this approach. First-pass effects, alternative elimination routes, and combined elimination pathways should also be investigated.

Design evaluation and optimisation in multiple response nonlinear mixed effect models: PFIM 3.0

Nonlinear mixed effect models (NLMEM) with multiple responses are increasingly used in pharmacometrics, one of the main examples being the joint analysis of the pharmacokinetics (PK) and pharmacodynamics (PD) of a drug. Efficient tools for design evaluation and optimisation in NLMEM are necessary. The R functions PFIM 1.2 and PFIMOPT 1.0 were proposed for these purposes, but accommodate only single response models. The methodology used is based on the Fisher information matrix, developed using a linearisation of the model. In this paper, we present an extended version, PFIM 3.0, dedicated to both design evaluation and optimisation for multiple response models, using a similar method as for single response models. In addition to handling multiple response models, several features have been integrated into PFIM 3.0 for model specification and optimisation. The extension includes a library of classical analytical pharmacokinetics models and allows the user to describe more complex models using differential equations. Regarding the optimisation algorithm, an alternative to the Simplex algorithm has been implemented, the Fedorov-Wynn algorithm to optimise more practical D-optimal design. Indeed, this algorithm optimises design among a set of sampling times specified by the user. This R function is freely available at http://www.pfim.biostat.fr. The efficiency of this approach and the simplicity of use of PFIM 3.0 are illustrated with a real example of the joint PKPD analysis of warfarin, an oral anticoagulant, with a model defined by ordinary differential equations.

Fisher information matrix for nonlinear mixed effects multiple response models: evaluation of the appropriateness of the first order linearization using a pharmacokinetic/pharmacodynamic model

We focus on the Fisher information matrix used for design evaluation and optimization in nonlinear mixed effects multiple response models. We evaluate the appropriateness of its expression computed by linearization as proposed for a single response model. Using a pharmacokinetic-pharmacodynamic (PKPD) example, we first compare the computation of the Fisher information matrix with approximation to one derived from the observed matrix on a large simulation using the stochastic approximation expectation-maximization algorithm (SAEM). The expression of the Fisher information matrix for multiple responses is also evaluated by comparison with the empirical information obtained through a replicated simulation study using the first-order linearization estimation methods implemented in the NONMEM software (first-order (FO), first-order conditional estimate (FOCE)) and the SAEM algorithm in the MONOLIX software. The predicted errors given by the approximated information matrix are close to those given by the information matrix obtained without linearization using SAEM and to the empirical ones obtained with FOCE and SAEM. The simulation study also illustrates the accuracy of both FOCE and SAEM estimation algorithms when jointly modelling multiple responses and the major limitations of the FO method. This study highlights the appropriateness of the approximated Fisher information matrix for multiple responses, which is implemented in PFIM 3.0, an extension of the R function PFIM dedicated to design evaluation and optimization. It also emphasizes the use of this computing tool for designing population multiple response studies, as for instance in PKPD studies or in PK studies including the modelling of the PK of a drug and its active metabolite.

Pharmacogenetics and population pharmacokinetics: impact of the design on three tests using the SAEM algorithm

Pharmacogenetics is now widely investigated and health institutions acknowledge its place in clinical pharmacokinetics. Our objective is to assess through a simulation study, the impact of design on the statistical performances of three different tests used for analysis of pharmacogenetic information with nonlinear mixed effects models: (i) an ANOVA to test the relationship between the empirical Bayes estimates of the model parameter of interest and the genetic covariate, (ii) a global Wald test to assess whether estimates for the gene effect are significant, and (iii) a likelihood ratio test (LRT) between the model with and without the genetic covariate. We use the stochastic EM algorithm (SAEM) implemented in MONOLIX 2.1 software. The simulation setting is inspired from a real pharmacokinetic study. We investigate four designs with N the number of subjects and n the number of samples per subject: (i) N = 40/n = 4, similar to the original study, (ii) N = 80/n = 2 sorted in 4 groups, a design optimized using the PFIM software, (iii) a combined design, N = 20/n = 4 plus N = 80 with only a trough concentration and (iv) N = 200/n = 4, to approach asymptotic conditions. We find that the ANOVA has a correct type I error estimate regardless of design, however the sparser design was optimized. The type I error of the Wald test and LRT are moderatly inflated in the designs far from the asymptotic (<10%). For each design, the corrected power is analogous for the three tests. Among the three designs with a total of 160 observations, the design N = 80/n = 2 optimized with PFIM provides both the lowest standard error on the effect coefficients and the best power for the Wald test and the LRT while a high shrinkage decreases the power of the ANOVA. In conclusion, a correction method should be used for model-based tests in pharmacogenetic studies with reduced sample size and/or sparse sampling and, for the same amount of samples, some designs have better power than others.

Comparison of model-based tests and selection strategies to detect genetic polymorphisms influencing pharmacokinetic parameters

We evaluate by simulation three model-based methods to test the influence of a single nucleotide polymorphism on a pharmacokinetic parameter of a drug: analysis of variance (ANOVA) on the empirical Bayes estimates of the individual parameters, likelihood ratio test between models with and without genetic covariate, and Wald tests on the parameters of the model with covariate. Analyses are performed using the FO and FOCE method implemented in the NONMEM software. We compare several approaches for model selection based on tests and global criteria. We illustrate the results with pharmacokinetic data on indinavir from HIV-positive patients included in COPHAR 2-ANRS 111 to study the gene effect prospectively. Only the tests based on the EBE obtain an empirical type I error close to the expected 5%. The approximation made with the FO algorithm results in a significant inflation of the type I error of the LRT and Wald tests.

Optimisation of sampling windows design for population pharmacokinetic experiments

This paper describes an approach for optimising sampling windows for population pharmacokinetic experiments. Sampling windows designs are more practical in late phase drug development where patients are enrolled in many centres and in out-patient clinic settings. Collection of samples under the uncontrolled environment at these centres at fixed times may be problematic and can result in uninformative data. Population pharmacokinetic sampling windows design provides an opportunity to control when samples are collected by allowing some flexibility and yet provide satisfactory parameter estimation. This approach uses information obtained from previous experiments about the model and parameter estimates to optimise sampling windows for population pharmacokinetic experiments within a space of admissible sampling windows sequences. The optimisation is based on a continuous design and in addition to sampling windows the structure of the population design in terms of the proportion of subjects in elementary designs, number of elementary designs in the population design and number of sampling windows per elementary design is also optimised. The results obtained showed that optimal sampling windows designs obtained using this approach are very efficient for estimating population PK parameters and provide greater flexibility in terms of when samples are collected. The results obtained also showed that the generalized equivalence theorem holds for this approach.

How many subjects are necessary for population pharmacokinetic experiments? Confidence interval approach

OBJECTIVE

To describe an approach using simulations for determining sample size for population pharmacokinetic experiments

METHODS

We address this problem by proposing method based on the estimation of the model parameters. The power to estimate the confidence interval of a parameter of choice to a particular precision is determined for different sample sizes by making stepwise increases in the sample size until the power is achieved. The method is based on simulation using previous information about the model and parameter estimates. Two examples are presented based on one-compartment and two-compartment first-order absorption models.

RESULTS

Sample size depends on the parameter of choice, the sampling designs and the method for the analysis of the collected data among other things. For the one-compartment first-order absorption model, assuming the parameter of choice is rate of absorption, the sample sizes required to estimate the 95% confidence interval within a 20% precision level with a power of 0.9 using the FO, FOCE and FOCE/INTERACTION methods in NONMEM and WINBUGS for a design that involved sampling at 0.01, 7.75 and 12 h (the population optimal design) are 20, 30, 30 and 30 respectively. For an extensive design (sampling at 0.5, 2, 4, 8, 12 and 24 h), the sample sizes are 20, 20, 20 and 30, respectively. For the two-compartment first-order absorption model, assuming the parameter of choice is initial volume of distribution, the sample sizes required to estimate the 95% confidence interval within a 50% precision level with a power of 0.8 for FO, FOCE/INTERACTION and WINBUGS were 50, 50 and 20, respectively.

CONCLUSION

The determination of sample size using the confidence interval approach appears to be a pragmatic approach to determine the minimum number of subjects for a population pharmacokinetic experiment.