An application of nonlinear mixed-effects modeling to pharmacokinetic data exhibiting nonlinear and time-dependent behavior

A population pharmacokinetic (PK) analysis was performed on plasma concentrations of GW468816 observed in dogs after 10, 25, and 50 mg/kg/day repeated intravenous administration. A two-compartment model was fitted to the concentration-time data using the NONMEM program. Dose and time dependency of PK parameters was investigated. Selection of the best model was performed using a stepwise approach. A Michaelis-Menten elimination process was used to describe the PK dose dependency, whereas an interoccasion variability on V(m) (the maximum elimination rate of the Michaelis-Menten elimination process) was initially used to describe the time dependency of the PKs, and the final model included an exponential function to account for time variance on V(m). The K(m) value of the final model was 29.6 microg/mL, whereas V(m) was estimated to vary with time from 4.97 microg/h/kg at day 1 to a maximum mean value of 9.64 microg/h/kg at day 14. This approach can be applied to either rich or sparse data leading to estimates of individual parameters by using Bayesian feedback. The overall information obtained can be used to interpret toxicological and pharmacological endpoints and integrated with further in vitro-in vivo studies to supply a comprehensive PK behavior before the first time in human studies.     

Detection of a drug–drug interaction on population-based phenobarbitone clearance using nonlinear mixed-effects modeling

Objective: Nonlinear mixed-effects modeling (NONMEM) was used to estimate the effects of drug–drug interaction on phenobarbitone clearance values, using 648 serum levels gathered during the routine clinical care of 349 pediatric and adult epileptic patients (age range, 0.4–33.3 years). Patients received phenobarbitone as monotherapy or in combination with either of the antiepileptic drugs carbamazepine or valproic acid. Results: The final model describing phenobarbitone clearance was CL = 52.3 · TBW^–0.567 · CO, where CL is clearance (ml · kg⁻¹ · h⁻¹), TBW is total body weight (kg) and CO is a scaling factor for concomitant medication with a value of 1 for patients on phenobarbitone monotherapy, 46.4^(−1/TBW)for those patients receiving concomitant carbamazepine and 0.642 for those patients receiving concomitant valproic acid. Phenobarbitone CL was highest in the very young and decreased in a weight-related fashion in children, with minimal changes observed in adults. This pattern was consistent whether phenobarbitone was administered alone or coadministered with carbamazepine or valproic acid. When phenobarbitone was coadministered with carbamazepine or valproic acid, phenobarbitone CL decreased compared with that in monotherapy. Its magnitudes in the presence of carbamazepine are maximal in early childhood (about 54%) and decreased in a weight-related fashion in older children, with minimal changes observed in adults. Concomitant administration of phenobarbitone and valproic acid resulted in a 35.8% decrease of phenobarbitone CL. Received: 17 February 1997 / Accepted in revised form: 21 October 1997     

Use of nonlinear, mixed-effects modeling for population analysis of ofloxacin: effects of age on oral drug pharmacokinetics.

To compare the effect of age on pharmacokinetics of orally administered ofloxacin, two separate studies were reanalyzed using mixed-effect modeling with the program NONMEM. Subjects were male volunteers, 36 age 65 years or greater and 24 age 18-40 years. The younger group received three 100-mg tablets and the older group received two 200-mg tablets of ofloxacin. Serial blood samples obtained throughout dosing were assayed for drug concentrations using high-performance liquid chromatography. A pharmacostatistical model was developed for the data using mixed-effect modeling with NONMEM. A one-compartment open model with first-order absorption, which included the covariables weight and age, best fit the data. Mean (SE) population values were clearance/F 0.219 (0.009) L/hr/kg, volume of distribution/F 1.50 (0.071) L/kg, and absorption rate constant 2.26 (0.048) hr-1. Older subjects had a 29% lower clearance and 13% lower volume of distribution then the younger subjects.     

Considerations in analyzing single-trough concentrations using mixed-effects modeling

The purpose of this study was to assess the effect of trial design and data analysis choices on the bias and precision of pharmacokinetic (PK) parameter estimation. NONMEM was used to simulate and analyze plasma concentrations collected according to a dense (five samples) or sparse (single-trough samples) sampling scheme for a one-compartment open model with intravenous administration. The results indicated that the bias on estimates of CL with only single-trough data was 17% compared to less than 1% for only dense data. The estimates of CL were improved by fixing all other parameters and estimating only mean and variance of CL (-11% to 1.4%, depending on the estimation method). Adding dense data led to further improvements (-2.3% to 0.3%, depending on further improvements). In these cases, first-order conditional estimation (FOCE) methods resulted in better estimates of CL than first-order (FO) methods. These steps also improved the Bayesian estimates of CL. These studies support the following recommendations: (1) avoid collecting single-trough concentrations unless there is reasonable knowledge about the PK of the drug; (2) if collecting single-trough concentrations is inevitable, avoid estimating all parameters when modeling single-trough concentration data; (3) use prior information by modeling the single-trough concentration data along with dense data from other studies; and (4) use Bayes estimates if the PK model and its parameters are known with reasonable certainty.     

Pharmacodynamic analysis of analgesic clinical trials: nonlinear mixed-effects logistic models.

In the development of an analgesic product, placebo-controlled clinical trials in patients with defined pain are used to study the dose-time-response relationship of the drug. In such trials, the response is usually an ordered categorical variable with longitudinal and subject-specific repeated measurements. The primary causal variables are time and analgesic concentration. The response may be informative right-hand censored because remedication with a known analgesic may be given if a patient has inadequate pain relief. Mixed-effects logistic models are used to estimate the probabilities of having certain pain relief or pain severity scores. A jackknife method is proposed to estimate the standard errors of parameter estimates. Posterior estimates of these probabilities, or of the scores themselves, allow the evaluation of efficacy for an analgesic. In this evaluation, therapeutic as well as statistical significance is assessed. Two case studies, one focusing on pain relief and the other on pain severity, are used to demonstrate the approach. The level of baseline pain appears to be a determinant of the pattern of response.     

Pharmacokinetic interaction between shuanghuanglian and azithromycin injection: a nonlinear mixed-effects model analysis in rats

1. This study aimed to evaluate the pharmacokinetic interaction of shuanghuanglian (SHL) and azithromycin in rats, and to provide experimental support for rational drug use in clinics.

2. High-performance liquid chromatography with ultraviolet detection (HPLC-UV) and high-performance liquid chromatography-tandem mass spectrometry (HPLC-MS/MS) approaches were respectively developed to detect the forsythiaside (active component of SHL) and azithromycin concentrations. Both non-compartmental and compartmental analyzes were employed to calculate pharmacokinetic parameters. A nonlinear mixed-effects modeling method was applied to fit the drug concentration-time data. The influence of drug coadministration on pharmacokinetic parameters was tested using forward inclusion and backward elimination procedures.

3. After drug co-administration, areas under the drug concentration–time curve (AUC) and half-lives (T_1/2) of both azithromycin and forsythiaside increased significantly, meanwhile, the drug clearance (CL) decreased compared to single drug administration. Both forsythiaside and azithromycin exposures increased after coadministration. Two-compartment models were suitable to describe the in vivo behavior of both azithromycin and forsythiaside. The coadministration of SHL could significantly decrease the central volume of azithromycin (V_CA) and forsythiaside clearance (CL_F) decreased after co-intravenous administration of azithromycin.

4. Co-intravenous administration of forsythiaside and azithromycin could significantly increase drug exposures for both drugs. Lower dose can provide sufficient drug exposure to obtain antibacterial activity. The coadministration may be a potential method to increase therapy efficiency while decrease adverse drug reactions.     

Population pharmacokinetics of gentamicin in neonates using a nonlinear, mixed-effects model.

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.     

A novel global search algorithm for nonlinear mixed-effects models using particle swarm optimization

NONMEM is one of the most popular approaches to a population pharmacokinetics/pharmacodynamics (PK/PD) analysis in fitting nonlinear mixed-effects models. As a local optimization algorithm, NONMEM usually requires an initial value close enough to the global optimum. This paper proposes a novel global search algorithm called P-NONMEM. It combines the global search strategy by particle swarm optimization (PSO) and the local estimation strategy of NONMEM. In the proposed algorithm, initial values (particles) are generated randomly by PSO, and NONMEM is implemented for each particle to find a local optimum for fixed effects and variance parameters. P-NONMEM guarantees the global optimization for fixed effects and variance parameters. Under certain regularity conditions, it also leads to global optimization for random effects. Because P-NONMEM doesn’t run PSO search for random effect estimation, it avoids tremendous computational burden. In the simulation studies, we have shown that P-NONMEM has much improved convergence performance than NONMEM. Even when the initial values were far away from the global optima, P-NONMEM converged nicely for all fixed effects, random effects, and variance components.     

A nonlinear mixed-effects pharmacokinetic model comparing two formulations of cyclosporine in stable renal transplant patients

Journal of Pharmacokinetics and Pharmacodynamics - A nonlinear mixed-effects model simultaneously modeled two pharmacokinetic (PK) variables in patients administered cyclosporine twice daily: (i)...     

Evaluation of bootstrap methods for estimating uncertainty of parameters in nonlinear mixed-effects models: a simulation study in population pharmacokinetics

Bootstrap methods are used in many disciplines to estimate the uncertainty of parameters, including multi-level or linear mixed-effects models. Residual-based bootstrap methods which resample both random effects and residuals are an alternative approach to case bootstrap, which resamples the individuals. Most PKPD applications use the case bootstrap, for which software is available. In this study, we evaluated the performance of three bootstrap methods (case bootstrap, nonparametric residual bootstrap and parametric bootstrap) by a simulation study and compared them to that of an asymptotic method (Asym) in estimating uncertainty of parameters in nonlinear mixed-effects models (NLMEM) with heteroscedastic error. This simulation was conducted using as an example of the PK model for aflibercept, an anti-angiogenic drug. As expected, we found that the bootstrap methods provided better estimates of uncertainty for parameters in NLMEM with high nonlinearity and having balanced designs compared to the Asym, as implemented in MONOLIX. Overall, the parametric bootstrap performed better than the case bootstrap as the true model and variance distribution were used. However, the case bootstrap is faster and simpler as it makes no assumptions on the model and preserves both between subject and residual variability in one resampling step. The performance of the nonparametric residual bootstrap was found to be limited when applying to NLMEM due to its failure to reflate the variance before resampling in unbalanced designs where the Asym and the parametric bootstrap performed well and better than case bootstrap even with stratification.     

The bayesian bias correction method of the first-order approximation of nonlinear mixed-effects models for population pharmacokinetics

Population pharmacokinetic analysis usually employs nonlinear mixed-effects models. To estimate the parameters, Beal and Sheiner (1982) proposed the first-order method that employs a first-order Taylor series expansion around the means of random individual parameters. Because of the small computational burden and the high convergence proportion of maximization of the log likelihood function, this method is often used in practice. However, it is known that the estimates are biased. This paper proposes a simple procedure to reduce the bias. The proposed method maximizes the nonapproximated log likelihood functions of each individual given estimates of the population parameters derived from the first-order method, and the derived Bayes estimates of the random individual parameters are utilized to improve the estimates of the population mean parameters. We confirmed that the proposed method reduced the bias using simulated data and actual erythropoietin concentration data.     

dOFV distributions: a new diagnostic for the adequacy of parameter uncertainty in nonlinear mixed-effects models applied to the bootstrap

Knowledge of the uncertainty in model parameters is essential for decision-making in drug development. Contrarily to other aspects of nonlinear mixed effects models (NLMEM), scrutiny towards assumptions around parameter uncertainty is low, and no diagnostic exists to judge whether the estimated uncertainty is appropriate. This work aims at introducing a diagnostic capable of assessing the appropriateness of a given parameter uncertainty distribution. The new diagnostic was applied to case bootstrap examples in order to investigate for which dataset sizes case bootstrap is appropriate for NLMEM. The proposed diagnostic is a plot comparing the distribution of differences in objective function values (dOFV) of the proposed uncertainty distribution to a theoretical Chi square distribution with degrees of freedom equal to the number of estimated model parameters. The uncertainty distribution was deemed appropriate if its dOFV distribution was overlaid with or below the theoretical distribution. The diagnostic was applied to the bootstrap of two real data and two simulated data examples, featuring pharmacokinetic and pharmacodynamic models and datasets of 20–200 individuals with between 2 and 5 observations on average per individual. In the real data examples, the diagnostic indicated that case bootstrap was unsuitable for NLMEM analyses with around 70 individuals. A measure of parameter-specific “effective” sample size was proposed as a potentially better indicator of bootstrap adequacy than overall sample size. In the simulation examples, bootstrap confidence intervals were shown to underestimate inter-individual variability at low sample sizes. The proposed diagnostic proved a relevant tool for assessing the appropriateness of a given parameter uncertainty distribution and as such it should be routinely used.     

Bayesian quantile regression for nonlinear mixed-effects joint models for longitudinal data in the presence of mismeasured covariate errors

Quantile regression (QR) models have recently received increasing attention in longitudinal studies where measurements of the same individuals are taken repeatedly over time. When continuous (longitudinal) responses follow a distribution that is quite different from a normal distribution, usual mean regression (MR)-based linear models may fail to produce efficient estimators, whereas QR-based linear models may perform satisfactorily. To the best of our knowledge, there have been very few studies on QR-based nonlinear models for longitudinal data in comparison to MR-based nonlinear models. In this article, we study QR-based nonlinear mixed-effects (NLME) joint models for longitudinal data with non-central location and outliers and/or heavy tails in response, and non-normality and measurement errors in covariate under Bayesian framework. The proposed QR-based modeling method is compared with an MR-based one by an AIDS clinical dataset and through simulation studies. The proposed QR joint modeling approach can be not only applied to AIDS clinical studies, but also may have general applications in other fields as long as relevant technical specifications are met.     

Generalized pharmacokinetic modeling for drugs with nonlinear binding: I. Theoretical framework

The following integrodifferential equation is proposed as the basis for a generalized treatment of pharmacokinetic systems in which nonlinear binding occurs $$\phi '(c_u )c'_u = - q(c_u ) + g*c_u + f$$ where c_u≡unbound plasma drug concentration, f≡drug input rate,'indicates the derivative of a function, and * indicates the convolution operation: (g* c_u)(t)=∫ ₀ ^t g(t?u)c_udu.Possible physical interpretations of the functions q, g and f are: q (c_u)≡ rate at which drug leaves the sampling compartment, g * c_u ≡ rate at which drug returns to the sampling compartment from the peripheral system (tissues that are kinetically distinct from the sampling compartment), and φ(c_u) ≡ amount of drug in the sampling compartment. The approach assumes that drug binding is sufficiently rapid that it may be treated as an equilibrium process. It may be applied to systems in which nonlinear binding occurs within the sampling compartment, i.e., in the systemic circulation or in tissues to which drug is rapidly distributed. The proposed relationship is a generalization of most existing models for drugs with nonlinear binding. It can serve as a general theoretical framework for such models or as the basis for “model-independent” methods for analyzing the pharmacokinetics of drugs with nonlinear binding. Computer programs for the numerical solution of the integrodifferential equation are presented. Methods for pharmacokinetic system characterization, prediction and bioavailability are presented and demonstrated.     

Nonlinear mixed-effects models for pharmacokinetic data analysis: assessment of the random-effects distribution

Nonlinear mixed-effects models are frequently used for pharmacokinetic data analysis, and they account for inter-subject variability in pharmacokinetic parameters by incorporating subject-specific random effects into the model. The random effects are often assumed to follow a (multivariate) normal distribution. However, many articles have shown that misspecifying the random-effects distribution can introduce bias in the estimates of parameters and affect inferences about the random effects themselves, such as estimation of the inter-subject variability. Because random effects are unobservable latent variables, it is difficult to assess their distribution. In a recent paper we developed a diagnostic tool based on the so-called gradient function to assess the random-effects distribution in mixed models. There we evaluated the gradient function for generalized liner mixed models and in the presence of a single random effect. However, assessing the random-effects distribution in nonlinear mixed-effects models is more challenging, especially when multiple random effects are present, and therefore the results from linear and generalized linear mixed models may not be valid for such nonlinear models. In this paper, we further investigate the gradient function and evaluate its performance for such nonlinear mixed-effects models which are common in pharmacokinetics and pharmacodynamics. We use simulations as well as real data from an intensive pharmacokinetic study to illustrate the proposed diagnostic tool.     

Pharmacokinetics and concentration-effect analysis of intravenous RGD891, a platelet GPIIb/IIIa antagonist, using mixed-effects modeling (NONMEM)

RGD891 is a platelet GPIIb/IIIa receptor antagonist and potent inhibitor of platelet aggregation. This compound is biotransformed in vivo to RGD039, which also exhibits high affinity for the GPIIb/IIIa receptor. Pharmacokinetic/pharmacodynamic modeling was employed to describe the concentration-effect relationship of both compounds following the intravenous administration of RGD891 to healthy volunteers. The overall objectives of this work were to support the dose selection process for future intravenous RGD891 safety and efficacy studies. Various intravenous regimens of RGD891 were administered to healthy volunteers enrolled in three Phase I studies. Frequent plasma samples were collected at regular intervals for later measurement of RGD891 and RGD039 concentrations (validated LC/MS/MS methods). The pharmacokinetics of RGD891 and RGD039 were simultaneously analyzed by nonlinear mixed-effect modeling (NONMEM). Pharmacodynamic activity was assessed in all three studies by the degree to which ADP (20 microM)-induced platelet aggregation was inhibited. Population parameters describing the concentration-effect relationship of RGD891 and RGD039 were then generated using a modified competitive Emax-based model. RESULTS: Parent compound is by far the predominant active compound circulating in the plasma following intravenous administration of RGD891. The plasma RGD891 concentration-time data were best fit by a two-compartment structural model. The fit of the basic model was improved when total body weight was introduced as a covariate for RGD891 distribution. Between-subject variability in the RGD891 pharmacokinetic parameters--V1, K10, and K21--was less than 17% (coefficient of variation). Formation of the active metabolite (RGD039; Km) and its elimination (Kem) were assumed to be first-order processes (i.e., one-compartment model). The population pharmacokinetic model could only provide a rough estimate of the plasma concentration-time profile for RGD039 after administration of a given intravenous dosage regimen of RGD891 since metabolite concentrations were relatively low and highly variable. The first-order rate constant describing the formation of RGD039 from RGD891 (Km) was also associated with a substantial degree of between-subject variability (44.9%). The potency of RGD891 toward the inhibition of ADP-induced platelet aggregation was described by the population IC50 value (plasma concentration yielding 50% of maximal inhibition), which ranged from 58.0 to 95.4 ng/mL, depending on the pharmacokinetic-pharmacodynamic (PK-PD) model and the data set used. The relatively low concentrations of the active metabolite achieved following intravenous administration of RGD891 did not permit independent estimation of a population IC50 value for RGD039. Therefore, its potency was fixed at 2.2-fold greater than that of the parent compound (based on previous PK-PD analyses). Intersubject variability in the IC50 values was 30%. CONCLUSIONS: Antagonism of the platelet IIb/IIIa receptors by intravenously administered RGD891 was effective in inhibiting ADP-induced platelet aggregation in a reversible and dose-dependent manner. Pharmacodynamic activity was largely attributed to the parent compound and less to the active metabolite based on the relative potencies of both compounds and the plasma concentrations of each achieved following intravenous administration. Intravenous bolus plus maintenance infusion regimens resulted in rapid attainment of steady-state plasma RGD891 concentrations. This combination regimen also provided for a marked and sustained inhibition of platelet aggregation that reached 90% or greater (relative to baseline values) in the higher dose groups. The modified Emax model adequately described inhibition of platelet aggregation following a particular intravenous dosage regimen of RGD891 (within the range of doses administered in the present studies). (ABSTRACT TRUNCATED)     

纵向研究缺失数据多重填补及混合效应模型分析

目的阐明马尔可夫链蒙特卡罗(MCMC)多重填补与重复测量资料混合效应线性模型分析的原理,完成纵向监测数据缺失模型的软件实现。方法根据222例高血压患者纵向监测的完全数据,产生缺失比例为18.92%的随机缺失数据集。应用MCMC多重填补方法,进行缺失值填补的模拟研究以及实例分析,并实现重复测量混合效应线性模型分析。结果模拟研究和实例分析表明,样本例数200,缺失比例20%,MCMC法多重填补5次所得结果最稳健;填补前缺失数据与完全数据的混合效应模型分析结果不同,填补后完整数据与完全数据的混合效应模型分析结果相同。结论 MCMC多重填补可以充分利用缺失资料信息,是处理缺失数据模型分析的有效方法之一;针对出现缺失的重复测量资料,结合应用混合效应模型与MCMC多重填补2种方法,从而得出更为符合客观实际的结果。