Model‐based dose finding under model uncertainty using general parametric models

Pharmacokinetic-Pharmacodynamic Modeling and Simulation

In the previous chapters, we discussed autoregressive linear mixed effects models. In this section, we discuss the relationships between the autoregressive linear mixed effects models and nonlinear mixed effects models, growth curves, and differential equations. The autoregressive model shows a profile approaching an asymptote, where the change is proportional to the distance remaining to the asymptote. Autoregressive models in discrete time correspond to monomolecular curves in continuous time. Autoregressive linear mixed effects models correspond to monomolecular curves with random effects in the baseline and asymptote, and special error terms. The autoregressive coefficient is a nonlinear parameter, but all random effects parameters in the model are linear. Therefore, autoregressive linear mixed effects models are nonlinear mixed effects models without nonlinear random effects and have a closed form of likelihood. When there are time-dependent covariates, autoregressive linear mixed effects models are represented by a differential equation and random effects. The monomolecular curve is one of the popular growth curves. We introduce other growth curves, such as the logistic curves and von Bertalanffy curves, and generalizations of growth curves. Re-parameterization is often performed in nonlinear models, and various representations of re-parameterization in monomolecular and other curves are provided herein.

Different mixed-effects models were compared to evaluate the population dose-response and relative potency of two albuterol inhalers. Bronchodilator response was measured after ascending doses of each inhaler in 37 asthmatic patients. A linear mixed-effects model was developed based on the approach proposed by Finney for the evaluation of bioassay data. A nonlinear mixed-effects (Emax) model with interindividual and interoccasion variability (IOV) in the different pharmacodynamic parameters was also fit to the data. Both methods produced a similar estimate of relative potency. However, the estimate of relative potency was 22% lower with the nonlinear mixed-effects model if IOV was not taken into account. Monte Carlo simulations based on a similar study design demonstrated that more biased and variable estimates of ED50 and relative potency were obtained when the nonlinear mixed-effects model ignored the presence of IOV in the data. Furthermore, the linear mixed-effects model that did not account for IOV produced confidence intervals for relative potency that were too narrow and thus could lead to erroneous conclusions. These problems were avoided when the estimation model could account for IOV. Results of the simulations were consistent with those of the experimental data. Although the linear or the nonlinear mixed-effects model may be used to evaluate population dose-response and relative potency, there are important differences in the assumptions made by each method.

The population pharmacokinetics of gentamicin in neonates was determined using a nonlinear, mixed-effects model (NONMEM). The final regression equations derived to estimate clearance (Cl) and volume of distribution (Vd) were Cl = 0.120 * (WT/2.4)1.36 L/hr and Vd = 0.429 * (WT) L. The interindividual variability (% CV) for clearance was 26.2% and for volume of distribution 15.9%. Intraindividual variability was 11.0%. In a separate group of 30 neonates, the predictive ability of the NONMEM-generated population variables was compared to the predictions from a standard two-stage population analysis. The trough concentrations predicted using NONMEM-generated parameters were significantly less biased and more precise; there were no significant differences between the methods in predicting peaks. NONMEM is a useful tool for determining population pharmacokinetics and appears to be consistent across populations using routine clinical data and limited observation.

Longitudinal data are data collected repeatedly from each subject for a particular response variable over a certain time period. Specifically, in longitudinal data analysis, the researchers are interested in changes in the response levels over time and the differences in these changes among factor levels or covariates. Because of within-subject correlations, analysis methods considering the correlations or variance-covariance structures have been developed. One of the approaches is the use of mixed effects models that take into account between-subject heterogeneity by random effects. In population pharmacokinetics, the response variable corresponds to drug concentration and is analysed typically using nonlinear mixed effects models. In this article, longitudinal data analysis with a continuous response variable is introduced focusing on population pharmacokinetics. Longitudinal data analysis, linear mixed effects models, nonlinear mixed effects models, and population pharmacokinetics are discussed from a biostatistical point of view. This article is expected to be of interest to biostatisticians, pharmacologists, pharmacokineticists, and those in related fields.

Long-Term Growth of the Neoaortic Root After Arterial Switch Operation

In this software review, we provide a brief overview of four R functions to estimate nonlinear mixed-effects programs: nlme (linear and nonlinear mixed-effects model), nlmer (from the lme4 package, linear mixed-effects models using Eigen and S4), saemix (stochastic approximation expectation maximization), and brms (Bayesian regression models using Stan). We briefly describe the approaches used, provide a sample code, and highlight strengths and weaknesses of each.

3567 Background: ABT-869 is an orally bioavailable, potent and specific inhibitor of all vascular endothelial growth factor and platelet derived growth factor family receptor tyrosine kinases. The objectives of this analysis were to understand the population pharmacokinetics of ABT-869 and explore the effect of several demographic/disease state covariates influencing ABT-869 disposition. Methods: A population PK analysis of 181 patients (pts) enrolled in two phase 1 (multiple types of solid tumors and AML) and three phase 2 monotherapy studies (non-small cell lung cancer, hepatocellular carcinoma [HCC] and renal cell carcinoma) was conducted. Approximately 90% of pts received ABT-869 based on body-weight dosing while the remaining pts had flat dosing. Available plasma concentrations obtained after intensive and sparse pre-dose PK sampling were analyzed by population PK using the non linear mixed effects modeling (NONMEM) approach. Potential covariates including body weight, body surface area (BSA), age, sex, creatinine clearance (CrCL) and disease state (HCC vs. non-HCC pts) were tested. Results: The mean body weight of enrolled pts was 71 kg and 57% were Asian, 36% Caucasian and 7% other races. The ABT-869 plasma concentration time profile was well described by a one-compartment model with first order absorption and elimination process. Oral clearance (CL/F) was not affected by body weight (range 35–177 kg); however, apparent volume of distribution (V/F) increased by 6L per 0.1 mg/m2 increase in BSA. CrCL (39.9–290.3 ml/min) was not a significant covariate on V/F and CL/F suggesting renally impaired pts do not require a different dose/dosing regimen. HCC pts had ∼40% lower CL/F values than pts with other malignancies suggesting a lower dose would be appropriate for HCC (Child Pugh A and B) pts. Conclusions: Population PK analysis showed that ABT-869 PK can be well described by a one-compartment model with first order absorption and elimination. Race and impaired renal function does not appear to alter PK. HCC pts had lower CL/F value therefore a lower dose may be recommended in these patients. Implications of increased V/F with increasing body size and appropriate dosing strategy are undergoing further analysis. [Table: see text]

The aim of this study was to investigate the use of nonlinear mixed effects (NLME) models in a real bioequivalence study and compare it to noncompartmental analysis (NCA) which is proposed by regulatory agencies. NCA requires few hypotheses but a large number of samples per subject. On the other hand, NLME approach is more complex than NCA but it has some advantages such as it requires few samples per subject. A real data application was provided for the study, which was get from Ege University Drug Development and Pharmacokinetics Research Center. For NLME, we used the stochastic approach to expectation-maximization (SAEM) algorithm, whereas linear trapezoidal rule was used for NCA. We estimate pharmacokinetic parameters, area under the curve (AUC0-∞) and maximum concentration (Cmax), and perform a bioequivalence tests using NCA and NLME. According to real data analysis, NLME approach has smaller within subject error, narrower confidence intervals than non-compartmental analysis. However, NLME models have some limitations because of increasing type I error. Therefore, caution is needed for small sample size and data with high variability.

Extending the code in the open-source saemix package to fit joint models of longitudinal and time-to-event data

Docetaxel, a novel anticancer agent, was given to 26 patients by short i.v. infusion (1-2 h) at various dose levels (70-115 mg/m2, the maximum tolerated dose) during 2 phase I studies. Two population analyses, one using NONMEM (nonlinear mixed-effect modeling) and the other using NPML (nonparametric maximum-likelihood), were performed sequentially to determine the structural model; estimate the mean population parameters, including clearance (Cl) and interindividual variability; and find influences of demographic covariates on them. Nine covariates were included in the analyses: age, height, weight, body surface area, sex, performance status, presence of liver metastasis, dose level, and type of formulation. A three-compartment model gave the best fit to the data, and the final NONMEM regression model for Cl was Cl = BSA(Theta1 + Theta02 x AGE), expressing Cl (in liters per hour) directly as a function of body surface area. Only these two covariates were considered in the NPML analysis to confirm the results found by NONMEM. Using NONMEM [for a patient with mean AGE (52.3 years) and mean BSA (1.68 m2)] and NPML, docetaxel Cl was estimated to be 35.6 l/h (21.2 lh-1 m-2) and 37.2 l/h with interpatient coefficients of variations (CVs) of 17.4% and 24.8%, respectively. The intraindividual CV was estimated at 23.8% by NONMEM; the corresponding variability was fixed in NPML in an additive Gaussian variance error model with a 20% CV. Discrepancies were found in the mean volume at steady state (Vss; 83.21 for NPML versus 1241 for NONMEM) and in terminal half-lives, notably the mean t1/2 gamma, which was shorter as determined by NPML (7.89 versus 12.2 h), although the interindividual CV was 89.1% and 62.7% for Vss and t1/2 gamma, respectively. However, the NPML-estimated probability density function (pdf) of t1/2 gamma was bimodal (5 and 11.4 h), probably due to the imbalance of the data. Both analyses suggest a similar magnitude of mean Cl decrease with small BSA and advanced age.

Population pharmacokinetic study of tacrolimus in pediatric patients with primary nephrotic syndrome: A comparison of linear and nonlinear Michaelis–Menten pharmacokinetic model

This work is motivated by a meta-analysis case study on antipsychotic medications. The Michaelis-Menten curve is employed to model the nonlinear relationship between the dose and D2 receptor occupancy across multiple studies. An intraclass correlation coefficient (ICC) is used to quantify the heterogeneity across studies. To interpret the size of heterogeneity, an accurate estimate of ICC and its confidence interval is required. The goal is to apply a recently proposed generic beta-approach for construction the confidence intervals on ICCs for linear mixed effects models to nonlinear mixed effects models using four estimation methods. These estimation methods are the maximum likelihood, second-order generalized estimating equations and two two-step procedures. The beta-approach is compared with a large sample normal approximation (delta method) and bootstrapping. The confidence intervals based on the delta method and the nonparametric percentile bootstrap with various resampling strategies failed in our settings. The beta-approach demonstrates good coverages with both two-step estimation methods and consequently, it is recommended for the computation of confidence interval for ICCs in nonlinear mixed effects models for small studies.

Rare studies have investigated the association between heterogeneity of motor progression and risk of early cognitive impairment in Parkinson's disease (PD). In this study, we aim to identify distinct trajectories of motor progression longitudinally and investigate their impact on predicting mild cognitive impairment (MCI). A 5-year cohort including 415 PD patients at baseline was collected from the Parkinson's Progression Markers Initiative. The severity of motor symptoms was evaluated using the Movement Disorder Society Unified Parkinson's Disease Rating Scale part III. The latent class trajectory model and nonlinear mixed-effects model were used to analyze and delineate the longitudinal changes in motor symptoms. Propensity score matching (PSM) was used to minimize the impact of potential confounders. Cox proportional hazard models were applied to calculate hazard ratios for MCI, and a Kaplan-Meier curve was generated using the occurrence of MCI during the follow-up as the time-to-event. Two latent trajectories were identified: a mild and remitting motor symptoms class (Class 1, 33.01%) and a severe and progressive motor symptom class (Class 2, 66.99%). Patients in Class 2 initially exhibited severe motor symptoms that worsened progressively despite receiving anti-PD medications. In comparison, patients in Class 1 exhibited milder symptoms that improved following drug therapy and a slower progression. During a 5-year follow-up, patients in Class 2 showed a higher risk of developing MCI compared to those in Class 1 before PSM (Log-Rank 28.58, p < 0.001) and after PSM (Log-Rank 8.20, p = 0.004). PD patients with severe and progressive motor symptoms are more likely to develop MCI than those with mild and stable motor symptoms.

Nonlinear mixed-effects models have received a great deal of attention in the statistical literature in recent years because of the flexibility they offer in handling the unbalanced repeated-measures data that arise in different areas of investigation, such as pharmacokinetics and economics. Several different methods for estimating the parameters in nonlinear mixed-effects model have been proposed. We concentrate here on two of them—maximum likelihood and restricted maximum likelihood. A rather complex numerical issue for (restricted) maximum likelihood estimation in nonlinear mixed-effects models is the evaluation of the log-likelihood function of the data, because it involves the evaluation of a multiple integral that, in most cases, does not have a closed-form expression. We consider here four different approximations to the log-likelihood, comparing their computational and statistical properties. We conclude that the linear mixed-effects (LME) approximation suggested by Lindstrom and Bates, the Laplacian approximation, and Gaussian quadrature centered at the conditional modes of the random effects are quite accurate and computationally efficient. Gaussian quadrature centered at the expected value of the random effects is quite inaccurate for a smaller number of abscissas and computationally inefficient for a larger number of abscissas. Importance sampling is accurate, but quite inefficient computationally.