Derivation of various NONMEM estimation methods

Various estimation methods and the lack of a systematic derivation of the core objective function implemented in NONMEM for nonlinear mixed effect modeling has caused consistent confusion and inquiry among scientists who routinely use NONMEM for data analysis. This paper provides a detailed derivation of the objective functions for the most commonly used estimation methods in NONMEM, such as the Laplacian method, the first-order conditional estimation method (FOCE) with or without interaction, and the first-order method (FO). In addition, models with homogenous or heterogeneous residual error were used to demonstrate the relationship between the objective functions derived from two different types of approximation, namely Laplacian approximation of log-likelihood and linearized model approximation. The relationship between these estimation methods and those implemented in SAS and Splus is discussed.     

Evaluation of hypothesis testing for comparing two populations using NONMEM analysis

In a simulation study of inference on population pharmacokinetic parameters, two methods of performing tests of hypotheses comparing two populations using NONMEM were evaluated. These two methods are the test based upon 95% confidence intervals and the likelihood ratio test. Data were simulated according to a monoexponential model and, in that context, power curves for each test were generated for (i)the ratio of mean clearance and (ii)the ratio of the population standard deviations of clearance. To generate the power curves, a range of these parameters was employed; other pharmacokinetic parameters were selected to reflect the variability typically present in a Phase II clinical trial. For tests comparing the means, the confidence interval tests had approximately the same power as the likelihood ratio tests and were consistently more faithful to the nominal level of significance. For comparison of the standard deviations, and when the volume of information available was relatively small, however, the likelihood ratio test was more able to detect differences between the two groups. These results were then compared to results on parameter estimation in order to gain insight into the question of power. As an example, the nonnormality of estimates of the ratio of standard deviations plays an important role in explaining the low power for the confidence interval tests. We conclude that, except for the situation of modeling standard deviations with only sparse information, NONMEM produces tests of significance that are effective at detecting clinically significant differences between two populations.Partial support from the Upjohn Company, NIH-BRSG SO RR 07066, and the Burroughs Wellcome Foundation.     

Evaluation of Bayesian estimation in comparison to NONMEM for population pharmacokinetic data analysis: Application to pefloxacin in intensive care unit patients

The pharmacokinetics of pefloxacin (PF) were investigated in a population of 74 intensive care unit patients receiving 400 mg bid as 1-hr infusion using (i) Bayesian estimation (BE) of individual patient parameters followed by multiple linear regression (MLR) analysis and (ii) NONMEM analysis. The data consisted of 3 to 9 PF plasma levels per patient measured over 1 to 3 dosage intervals (total 113) according to four different limited (suboptimal) sampling 3-point protocols. Twenty-nine covariates (including 15 comedications) were considered to explain the interpatient variability. Predicted PFCLfor a patient with median covariates values was similar in both BE/ MLR and NONMEM analysis (4.02 and 3.92 L/hr, respectively). Bilirubin level and age were identified as the major determinants of PFCLby both approaches with similar predicted magnitude of effects (about 40 and 30% decrease of median CL,respectively). Confounding effects were observed between creatinine clearance (26% decrease of PF CLin the BE/MLR model), simplified acute physiology score (a global score based on 14 biological and clinical variables) (18% decrease of median CLin the NONMEM model) and age (entered in both models) which were highly correlated in our data base. However, both models predicted similar PF CLfor actual subpopulations by using actual covariate values. Finally, the NONMEM analysis allowed identification of an effect of weight on CL(decrease of CL for weight <65 kg) whereas the BE/MLR analysis predicted an increase of CLin patients treated with phenobarbital. In conclusion, both approaches allowed identification of the major risk factors of PF pharmacokinetics in ICU patients. Their potential use at different stages of drug development is discussed.     

Evaluation of Bayesian estimation in comparison to NONMEM for population pharmacokinetic data analysis: application to pefloxacin in intensive care unit patients.

The pharmacokinetics of pefloxacin (PF) were investigated in a population of 74 intensive care unit patients receiving 400 mg bid as 1-hr infusion using (i) Bayesian estimation (BE) of individual patient parameters followed by multiple linear regression (MLR) analysis and (ii) NONMEM analysis. The data consisted of 3 to 9 PF plasma levels per patient measured over 1 to 3 dosage intervals (total 113) according to four different limited (suboptimal) sampling 3-point protocols. Twenty-nine covariates (including 15 comedications) were considered to explain the interpatient variability. Predicted PF CL for a patient with median covariates values was similar in both BE/MLR and NONMEM analysis (4.02 and 3.92 L/hr, respectively). Bilirubin level and age were identified as the major determinants of PF CL by both approaches with similar predicted magnitude of effects (about 40 and 30% decrease of median CL, respectively). Confounding effects were observed between creatinine clearance (26% decrease of PF CL in the BE/MLR model), simplified acute physiology score (a global score based on 14 biological and clinical variables) (18% decrease of median CL in the NONMEM model) and age (entered in both models) which were highly correlated in our data base. However, both models predicted similar PF CL for actual subpopulations by using actual covariate values. Finally, the NONMEM analysis allowed identification of an effect of weight on CL (decrease of CL for weight < 65 kg) whereas the BE/MLR analysis predicted an increase of CL in patients treated with phenobarbital. In conclusion, both approaches allowed identification of the major risk factors of PF pharmacokinetics in ICU patients. Their potential use at different stages of drug development is discussed.     

NONMEM法估算中国癫痫患者卡马西平的清除率

目的　考察中国癫痫患者卡马西平的群体药动学参数。方法　癫痫病例来自上海、北京两地 4所医院 ,采集服用卡马西平的 5 92例患者的稳态血药浓度 (n =70 3)。NONMEM程序估算分析时 ,采用一级吸收和消除的药动学模型并固定吸收速率、生物利用度和表观分布体积参数。结果　体重 (TBW )、剂量 (Dose)、合用丙戊酸钠 (VPA)且其日剂量大于 2 0mg·kg-1·d-1、苯妥英 (PHT)、苯巴比妥 (PB)和年龄大于 6 5岁的老年人 (ELDER)均为卡马西平清除率(CL)的影响因素。性别、合用氯硝西泮、妥吡酯不改变卡马西平的清除率。最终模型为 :CL(CL/F) (L/h) =1 32·Dose(g·kg-1·d-1) 0 42 1·TBW (kg) 0 .691·1 2 0 VPA·1 4 3PHT·1 14 PB·0 836 ELDER。讨论　根据中国癫痫患者的群体药动学模型 ,结合患者服用的剂量、体重和合并用药可估算其清除率 ,制定给药方案     

Evaluation of hypothesis testing for comparing two populations using NONMEM analysis.

In a simulation study of inference on population pharmacokinetic parameters, two methods of performing tests of hypotheses comparing two populations using NONMEM were evaluated. These two methods are the test based upon 95% confidence intervals and the likelihood ratio test. Data were simulated according to a monoexponential model and, in that context, power curves for each test were generated for (i) the ratio of mean clearance and (ii) the ratio of the population standard deviations of clearance. To generate the power curves, a range of these parameters was employed; other pharmacokinetic parameters were selected to reflect the variability typically present in a Phase II clinical trial. For tests comparing the means, the confidence interval tests had approximately the same power as the likelihood ratio tests and were consistently more faithful to the nominal level of significance. For comparison of the standard deviations, and when the volume of information available was relatively small, however, the likelihood ratio test was more able to detect differences between the two groups. These results were then compared to results on parameter estimation in order to gain insight into the question of power. As an example, the nonnormality of estimates of the ratio of standard deviations plays an important role in explaining the low power for the confidence interval tests. We conclude that, except for the situation of modeling standard deviations with only sparse information, NONMEM produces tests of significance that are effective at detecting clinically significant differences between two populations.     

Evaluation of methods for improving precision of blood pressure measurements in phase I clinical trials

Small sample sizes are typically incorporated in early Phase I clinical studies, which may lead to insignificant changes in safety parameters such as blood pressure. Therefore, it is paramount to identify an optimal, noninvasive method of accurately measuring blood pressure and an appropriate analysis strategy yielding the smallest variability. The goals of this study were (1) to compare the variability between automated and manual blood pressure measurements, (2) to determine whether triplicate blood pressure measurements were independent of one another, and (3) to assess how the number of blood pressure readings affects variability and study sample size. Twenty healthy volunteers were enrolled in this randomized, two-way crossover study. Each subject received three incremental infusions of phenylephrine or normal saline on separate days to simulate blood pressure variability. The mean systolic blood pressure readings with the automated device were consistently higher than the manual device by 3 to 5 mmHg. Conversely, the mean diastolic blood pressure readings with the automated device were consistently 3 to 5 mmHg lower than the manual device. However, the variability and absolute change in blood pressure were essentially identical with manual and automated methods. No systematic order effects such as the first blood pressure reading always being higher were detected, suggesting that the triplicate readings were independent of one another and that an interval of 2 minutes between readings is adequate. Compared to a single measurement, collecting blood pressure in triplicate results in a 40% lower sample size needed to detect a 5-mmHg difference in systolic blood pressure.     

NONMEM法分析静滴异丙酚在中国人体的群体药代动力学

目的　考察中国人静脉匀速滴注异丙酚的群体药代动力学。方法　51例腰麻-硬膜外联合麻醉病人匀速输注异丙酚直至暴发脑电抑制,以HPLC法测定异丙酚血浆浓度,用NONMEM程序分析中国人异丙酚群体药代动力学。结果　异丙酚药代动力学符合三室线性开放模型,群体参数CL(L.min-1)、Vc(L)、Q2(L.min-1)、V2(L)、Q3(L.min-1)和V3(L)的标准值分别为1.10,7.63,1.54,15.0,0.76和175;体重对CL的校正为体重除以60的0.70次方,CL和Q2年龄≥60的病人较年龄<60的分别低18.1%和32.1%;年龄对V2和Q3的校正分别为年龄除以50的-0.66次方和-0.71次方。结论　NONMEM法对以三室模型群体参数估算的血药浓度值与实测值有良好相关性,体重、年龄对参数影响较大。     

Comparison of two population pharmacokinetic programs,NONMEM and P-PHARM,for tacrolimus

OBJECTIVES: To compare the population modelling programs NONMEM and P-PHARM during investigation of the pharmacokinetics of tacrolimus in paediatric liver-transplant recipients. METHODS: Population pharmacokinetic analysis was performed using NONMEM and P-PHARM on retrospective data from 35 paediatric liver-transplant patients receiving tacrolimus therapy. The same data were presented to both programs. Maximum likelihood estimates were sought for apparent clearance (CL/F) and apparent volume of distribution (V/F). Covariates screened for influence on these parameters were weight, age, gender, post-operative day, days of tacrolimus therapy, transplant type, biliary reconstructive procedure, liver function tests, creatinine clearance, haematocrit, corticosteroid dose, and potential interacting drugs. RESULTS: A satisfactory model was developed in both programs with a single categorical covariate--transplant type--providing stable parameter estimates and small, normally distributed (weighted) residuals. In NONMEM, the continuous covariates--age and liver function tests--improved modelling further. Mean parameter estimates were CL/F (whole liver) = 16.3 l/h, CL/F (cut-down liver) = 8.5 l/h and V/F = 565 l in NONMEM, and CL/F = 8.3 l/h and V/F = 155 l in P-PHARM. Individual Bayesian parameter estimates were CL/F (whole liver) = 17.9 +/- 8.8 l/h, CL/F (cut-down liver) = 11.6 +/- 8.8 l/h and V/F = 712 +/- 792 l in NONMEM, and CL/F (whole liver) = 12.8 +/- 3.5 l/h, CL/F (cut-down liver) = 8.2 +/- 3.4 l/h and V/F = 221 +/- 164 l in P-PHARM. Marked interindividual kinetic variability (38-108%) and residual random error (approximately 3 ng/ml) were observed. P-PHARM was more user friendly and readily provided informative graphical presentation of results. NONMEM allowed a wider choice of errors for statistical modelling and coped better with complex covariate data sets. CONCLUSION: Results from parametric modelling programs can vary due to different algorithms employed to estimate parameters, alternative methods of covariate analysis and variations and limitations in the software itself.     

NONMEM法估算地高辛相对生物利用度和药动学参数

目的：一步法分析地高辛相对生物利用度和药物动力学参数。方法：用ＮＯＮＭＥＭ程度包及其新的ＰＯＳＴＨＯＣ子模块功能,估算地高辛相对生物利用度和药动学参数,并与经典的３Ｐ８７方法比较。结果：两者的相对生物利用度一致;青年健康受试者口服地高辛的主要药动学参数典型值Ｋ１０（ｈ＾－１）、Ｖｃ／Ｆ（Ｌ）和β（ｈ＾－１）分别为０．１１５,１２８和０．０１８,而老年病人的典型值Ｋ１０（ｈ＾－１）、Ｖｃ／Ｆ（Ｌ）和     

非线性混合效应模型法估算癫痫患者卡马西平的群体药动学参数

目的:建立癫痫患者卡马西平(CBZ)的群体药动学(PPK)模型。方法:采集我院服用CBZ的270例门诊癫痫患者的稳态血药浓度数据(共316个样本)以及患者相关资料数据。应用非线性混合效应模型(NONMEM)法估算癫痫患者CBZ的PPK参数值,建立PPK模型。并运用自举法(Bootstrap)验证模型的可靠性。结果:年龄(AGE)、每日服药剂量(DKG)、体质量(BW)均为CBZ清除率(CL)的影响因素。最终模型:当AGE≤14岁时,CL(L/h)=[2.55+0.013×(AGE-15)]×(DKG/0.011)0.443×(BW/40)0.392;AGE>14岁时,CL(L/h)=2.55×(DKG/0.011)0.443×(BW/40)0.392。表观分布容积(Vd)=85L。经Bootstrap法验证,本模型稳定、可靠。结论:用NONMEM软件成功建立我院癫痫患者服用CBZ的PPK模型。根据本院癫痫患者的PPK模型,结合患者DKG、BW和合并用药可估算其CL,优化临床个体化用药方案。     

Evaluation of limited sampling methods for estimation of tacrolimus exposure in adult kidney transplant recipients

AIMS

To examine the predictive performance of limited sampling methods for estimation of tacrolimus exposure in adult kidney transplant recipients.

METHODS

Twenty full tacrolimus area under the concentration–time curve from 0 to 12 h post-dose (AUC_0–12) profiles (AUCf) were collected from 20 subjects. Predicted tacrolimus AUC_0–12 (AUCp) was calculated using the following: (i) 42 multiple regression-derived limited sampling strategies (LSSs); (ii) five population pharmacokinetic (PK) models in the Bayesian forecasting program TCIWorks; and (iii) a Web-based consultancy service. Correlations (r²) between C₀ and AUCf and between AUCp and AUCf were examined. Median percentage prediction error (MPPE) and median absolute percentage prediction error (MAPE) were calculated.

RESULTS

Correlation between C₀ and AUCf was 0.53. Using the 42 LSS equations, correlation between AUCp and AUCf ranged from 0.54 to 0.99. The MPPE and MAPE were <15% for 29 of 42 equations (62%), including five of eight equations based on sampling taken ≤2 h post-dose. Using the PK models in TCIWorks, AUCp derived from only C₀ values showed poor correlation with AUCf (r² = 0.27–0.54) and unacceptable imprecision (MAPE 17.5–31.6%). In most cases, correlation, bias and imprecision estimates progressively improved with inclusion of a greater number of concentration time points. When concentration measurements at 0, 1, 2 and 4 h post-dose were applied, correlation between AUCp and AUCf ranged from 0.75 to 0.93, and MPPE and MAPE were <15% for all models examined. Using the Web-based consultancy service, correlation between AUCp and AUCf was 0.74, and MPPE and MAPE were 6.6 and 9.6%, respectively.

CONCLUSIONS

Limited sampling methods better predict tacrolimus exposure compared with C₀ measurement. Several LSSs based on sampling taken 2 h or less post-dose predicted exposure with acceptable bias and imprecision. Generally, Bayesian forecasting methods required inclusion of a concentration measurement from <2 h post-dose to adequately predict exposure.     

住院病人茶碱群体药物动力学NONMEM法分析

对72名临床病人进行茶碱群体药物动力学研究,用NONMEM法分析了多种因素对茶碱药物动力学过程的影响。结果表明:在18至77a范围内,年龄对清除率(Cl)有显著性影响,每岁降低1.25％;长期多剂量服用茶碱的哮喘患者,Cl降低26.8％;OLD时Cl降低33.9％;合并用Ac-SPM时Cl略有降低,但影响不大;性别和体重对Cl无显著影响。     

Evaluation of Type I Error Rates When Modeling Ordered Categorical Data in NONMEM

The development of non-linear mixed pharmacokinetic/pharmacodynamic models for continuous variables is usually guided by graphical assessment of goodness of fit and statistical significance criteria. The latter is usually the likelihood ratio test (LR). When the variable to be modeled is categorical, on the other hand, the available graphical methods are less informative and/or more complicated to use and the modeler needs to rely more heavily on statistical significance assessment in the model development. The aim of this study was to evaluate the type I error rates, obtained from using the LR test, for inclusion of a false parameter in a non-linear mixed effects model for ordered categorical data when modeling with NONMEM. Data with four ordinal categories were simulated from a logistic model. Two nested multinomial models were fitted to the data, the model used for simulation and a model containing one additional parameter. The difference in fit (objective function value) between models was calculated. Three types of models were explored; (i) a model without interindividual variability (IIV) where the addition of a parameter describing IIV was assessed, (ii) a model with IIV where the addition of a drug effect parameter (either categorical or continuous drug exposure measure) was evaluated, and (iii) a model including IIV and drug effect where the inclusion of a random effects parameter on the drug effect was assessed. Alterations were made to the simulation conditions, for example, varying the number of individuals and the size and distribution of the IIV, to explore potential influences on the type I error rate. The estimated type I error rate for inclusion of a false random effect parameter in model (i) and (iii) were, as expected, lower than the nominal. When the additional parameter was a fixed effects parameter describing drug effect (model(II)) the estimated type I error rates were in agreement with the nominal. None of the different simulation conditions tried changed this pattern. Thus, the LR test seems appropriate for judging the statistical significance of fixed effects parameters when modeling categorical data with NONMEM.     

A simple method for the estimation of interaction bias in crossover studies

The crossover trial is considered the most powerful means of determining the efficacy of new drugs. However this study design is frequently invalidated by treatment-by-period interaction. If, for example, the effect of the first treatment period carries on into the next one, then it influences the response to the latter period (carryover effect). A second problem is that there are no reliable statistical methods to test for this potential bias. This article takes issue with these problems and gives an alternative method for the detection of interaction simply by looking at the data. In a crossover without interaction the second period should be a true reflection of the first. If, however, the data of a treatment are better in the second period than in the first, a carryover effect is probable. If worse, a rebound phenomenon or a negative carryover effect is likely. If both treatments are better or worse, a time effect or some other external influence might be present. The authors illustrate this simple method by a summary of a few selected trials that have been published recently. This method enables not only the detection of interaction but also the differentiation between different types of interactions. Therefore, investigators are advised to use it in order to make sure that there are no unexpected problems.     

Assessment of actual significance levels for covariate effects in NONMEM

The objectives of this study were to assess the difference between actual and nominal significance levels, as judged by the likelihood ratio test, for hypothesis tests regarding covariate effects using NONMEM, and to study what factors influence these levels. Also, a strategy for obtaining closer agreement between nominal and actual significance levels was investigated. Pharmacokinetic (PK) data without covariate relationships were simulated from a one compartment iv bolus model for 50 individuals. Models with and without covariate relationships were then fitted to the data, and differences in the objective function values were calculated. Alterations were made to the simulation settings; the structural and error models, the number of individuals, the number of samples per individual and the covariate distribution. Different estimation methods in NONMEM were also tried. In addition, a strategy for estimating the actual significance levels for a specific data set, model and parameter was investigated using covariate randomization and a real data set. Under most conditions when the first-order (FO) method was used, the actual significance level for including a covariate relationship in a model was higher than the nominal significance level. Among factors with high impact were frequency of sampling and residual error magnitude. The use of the first-order conditional estimation method with interaction (FOCE-INTER) resulted in close agreement between actual and nominal significance levels. The results from the covariate randomization procedure of the real data set were in agreement with the results from the simulation study. With the FO method the actual significance levels were higher than the nominal, independent of the covariate type, but depending on the parameter influenced. When using FOCE-INTER the actual and nominal levels were similar. The most important factors influencing the actual significance levels for the FO method are the approximation of the influence of the random effects in a nonlinear model, a heteroscedastic error structure in which an existing interaction between interindividual and residual variability is not accounted for in the model, and a lognormal distribution of the residual error which is approximated by a symmetric distribution. Estimation with FOCE–INTER and the covariate randomization procedure provide means to achieve agreement between nominal and actual significance levels.     

New, simple and validated UV-spectrophotometric methods for the estimation of gatifloxacin in bulk and formulations

New, simple and cost effective UV-spectrophotometric methods were developed for the estimation of gatifloxacin in bulk and pharmaceutical formulations. Gatifloxacin was estimated at 286 nm in 100 mM phosphate buffer (pH 7.4) and 292 nm in 100 mM hydrochloric acid (pH 1.2). Linearity range was found to be 1-18 mug ml(-1) (regression equation: absorbance=0.0684 x Concentration in microg ml(-1) + 0.0050; r2 = 0.9998) in the phosphate buffer (pH 7.4) and 1-14 microg ml(-1) (regression equation: absorbance = 0.0864 x Concentration in microg ml(-1) + 0.0027; r2 = 0.9999) in hydrochloric acid medium (pH 1.2). The apparent molar absorptivity was found to be 2.62 x 10(4) l mol(-1) cm(-1) in the phosphate buffer and 3.25 x 10(4) l mol(-1) cm(-1) in hydrochloric acid media. In both the proposed methods sandell's sensitivity was found to be about 0.01 microg cm(-2)/0.001A. These methods were tested and validated for various parameters according to ICH guidelines and USP. The quantitation limits were found to be 0.312 and 0.3 microg ml(-1) in the phosphate buffer and hydrochloric acid medium, respectively. The proposed methods were successfully applied for the determination of gatifloxacin in pharmaceutical formulations (tablets, injection and ophthalmic solution). The results demonstrated that the procedure is accurate, precise and reproducible (relative standard deviation <2%), while being simple, cheap and less time consuming and can be suitably applied for the estimation of gatifloxacin in different dosage forms and dissolution studies.