On the identification of Michaelis-Menten elimination parameters from a single dose-response curve

This paper deals with aspects of the numerical identifiability of parameters of a model with a capacity-limited elimination rate using a single dose-response curve, namely, the prospects of being able to identify model parameters with any meaning from real data. The concept of linear bounds, first proposed by Tong and Metzler, is described and it is shown that if the Michaelis-Menten constant K_m is greater than all the measured concentration values, approximation by a linear model is appropriate. At the other end of the scale, if K_m is small compared with measured concentration values, the nonlinear response approximates to a zero-order curve.The work described in this paper was supported by Grant GR/B 29238 from the U.K. Science and Engineering Research Council.     

Novel method of estimating volume of distribution of a drug obeying Michaelis-Menten elimination kinetics

The novel method of estimating the volume of distribution involves (a) administering an appropriate bolus intravenous dose of the drug, (b) starting a constant-rate intravenous infusion of the drug at the same time, (c) maintaining the infusion for a given number of hours, (a) measuring the drug concentration over the entire time course, (e) computer-fitting the post-infusion data to obtain estimates of V_m and K_m, (f) estimating the total area under the concentration-time curve from zero time to infinity, and (g) iteratively solving a cubic equation to obtain the estimate of the volume of distribution. The method was applied to ethanol in the cat and yielded an average value of 635ml/kg (63.5% of body weight) with a coefficient of variation of 23.0%. This is equivalent to total body water in the cat.Work performed in the Upjohn Center for Clinical Pharmacology, The Univeristy of Michigan Medical School, Ann Arbor, Michigan 48109.     

Time to reach steady state and prediction of steady-state concentrations for drugs obeying Michaelis-Menten elimination kinetics

Using a numerical integration method, concentration-time data were simulated using the one-compartment open model both with bolus intravenous administration and oral administration (first-order absorption) after multiple doses administered at constant time intervals and for each model for five different doses. Constants used produced data very similar to those which have been reported for phenytoin in the literature. In the simulation of oral data, sufficient concentrations were recorded to allow estimation of the maximum (C _n ^max ), average (¯) C_n, and minimum (C _n ^min ) concentrations during each dosage interval, but for the intravenous data only C _n ^max and C _n ^min values were recorded. The approach to steady state was monoexponential for low doses and biexponential for higher doses. The half-life of the final first-order approach to the steady-state concentration was approximately linearly related to the final steady-state concentration. For the intravenous data the number of doses required to reach 95% of C _ss ^min was a linear function of 0.95 C _ss ^min . A simple difference plot allows any given steady-state concentration of the three to be estimated from non-steady-state concentrations. When C _n ^min values are measured, as in therapeutic drug monitoring, the fitting of C _ss ^min vs. dose rate (D/) data leads to operationally useful parameters, V _m ^app and K _m ^app , which are not the true kinetic parameters, V_m and K_m, whereas fitting of ¯C_ss vs d/ data does lead to estimation of V_m and K_m.     

Simulation for the Analysis of Distorted Pharmacodynamic Data

A simulation study was conducted to compare the performance of alternative approaches for analyzing the distorted pharmacodynamic data. The pharmacodynamic data were assumed to be obtained from the natriurertic peptide-type drug, where the diuretic effect arises from the hyperbolic (E_max) dose–response model and is biased by the dose-dependent hypotensive effect. The nonlinear mixed effect model (NONMEM) method enabled assessment of the effects of hemodynamics on the diuretic effects and also quantification of intrinsic diuretic activities, but the standard two-stage (STS) and naive pooled data (NPD) methods did not give accurate estimates. Both the STS and the NONMEM methods performed well for unbiased data arising from a one-compartment model with saturable (Michaelis–Menten) elimination, whereas the NPD method resulted in inaccurate estimates. The findings suggest that nonlinearity and/or bias problems result in poor estimation by NPD and STS analyses and that the NONMEM method is useful for analyzing such nonlinear and distorted pharmacodynamic data.     

One-compartment model with Michaelis-Menten elimination kinetics and therapeutic window: an analytical approach

The purpose of this article is to provide the analytical solutions of one-compartment models with Michaelis-Menten elimination kinetics for three different inputs (single intravenous dose, multiple-dose bolus injection and constant). All analytical solutions obtained in present paper can be described by the well defined Lambert W function which can be easily implemented in most mathematical softwares such as Matlab and Maple. These results will play an important role in fitting the Michaelis-Menten parameters and in designing a dosing regimen to maintain steady-state plasma concentrations. In particular, the analytical periodic solution for multi-dose inputs is also given, and we note that the maximum and minimum values of the periodic solution depends on the Michaelis-Menten parameters, dose and time interval of drug administration. In practice, it is important to maintain a concentration above the minimum therapeutic level at all times without exceeding the minimum toxic concentration. Therefore, the one-compartment model with therapeutic window is proposed, and further the existence of periodic solution, analytical expression and its period are analyzed. The analytical formula of period plays a key role in designing a dose regimen to maintain the plasma concentration within a specified range over long periods of therapy. Finally, the completely analytical solution for the constant input rate is derived and discussed which depends on the relations between constant input rate and maximum rate of change of concentration.     

呈米氏消除动力学特征药物半衰期的简便估计法

目的:建立一个简便估算呈米氏消除动力学特征药物半衰期的方法.方法:呈米氏消除动力学特征的药物,其消除半衰期与血药浓度成线性关系.利用一次给药后的血药浓度—时间数据便能画出该直线的方法,结果与结论:直接从直线图上读出任意浓度的半衰期.该方法还可用来估计血药浓度下降任意分数所需时间.     

Evaluation of theophylline pharmacokinetics in a pediatric population using mixed effects models

Routine clinical pharmacokinetic data collected prospectively from pediatric patients receiving theophylline were analyzed using the NONMEM (nonlinear mixed effects model) digital computer program. A total of 314 measured serum theophylline concentrations (STCs) were obtained from 84 hospitalized patients ranging in age from 4 months to 15.2 years with the majority of patients between the ages of 1 and 8 years. Fifty-six percent were male. The race/ethnicity distribution was 71.4% Latin, 15.5% black, 11.9% Caucasian, and 1.2% (one subject) Pakistani. Of the total number of observed STCs, 16.2% reflected some degree of outpatient dosing. The pharmacokinetic model used was a one-compartment open model with either zero-order or first-order absorption and first-order elimination. Age was the most important determinant of theophylline clearance (Cl);weight was inferior to age and did not statistically improve the model (p>0.005when combined with age. Total Clincreased by 10%/year over the age range of 1 to 15 years of age. Black race and male gender were associated with higher Clvalues: for a given age, Clwas 34% higher for blacks than the reference population composed of the remaining patients, and Clfor males was 25% higher than that for females. The volume of distribution (Vdfor the population was estimated to be 0.62 L/kg. The interindividual variability in Cland Vdexpressed as coefficients of variation were 19 and 28%, respectively. The residual intraindividual error variance corresponded to a standard deviation of 2.8 g/ml. The STCs that represented some degree of outpatient dosing were 21 % lower than those reflecting only inpatient dosing. Alternate models that include weight as a determinant of theophylline clearance are also provided. The NONMEM method of determining population pharmacokinetics is well suited to the pediatric population since it does not require a large number of STCs per patient. In this study a mean of only 3.7 STCs per patient were utilized to provide information which should prove useful in the design and adjustment of theophylline dosage regimens in children.     

Theoretical considerations in the calculation of bioavailability of drugs exhibiting Michaelis-Menten elimination kinetics

Two approaches used for bioavailability determination of drugs with Michaelis-Menten elimination kinetics were examined by computer simulation. The first method involved treating the drug as though its clearance remained constant during elimination, and the conventional method of taking the ratio of areas under the curve resulting from the oral and intravenous doses was used to calculate bioavailability. The second approach involved using the Michaelis parameters, V_max and K_m,to determine concentration dependent clearance values, but based these calculations on peripheral drug concentrations rather than on concentrations entering or in the liver. We have developed a simulation method that was used to test the accuracy of the above two methods. In the simulations described, V_max, K_m,and hepatic blood flow were chosen to represent a drug with an extraction ratio of 0.9 under linear conditions, but with Michaelis-Menten kinetics occurring at the doses given. Absorption was assumed to be first-order, and metabolism was assumed to occur only in the liver. These simulations showed that the most accurate determination of bioavailability requires knowledge of the direct contribution of oral absorption to the concentration of drug entering the liver. Unexpectedly, the results also showed that if a drug has a large volume of distribution or a large absorption rate constant, or both, use of the much simpler conventional method of bioavailability determination may be appropriate even in cases where the degree of saturation is substantial.This work was supported in part by grant GM26556 from the Institute of General Medical Sciences of the National Institutes of Health.     

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。     

Fractal Michaelis-Menten Kinetics Under Steady State Conditions: Application to Mibefradil

Purpose To provide the first application of fractal kinetics under steady state conditions to pharmacokinetics as a model for the enzymatic elimination of a drug from the body. Materials and Methods A one-compartment model following fractal Michaelis-Menten kinetics under a steady state is developed and applied to concentration-time data for the cardiac drug mibefradil in dogs. The model predicts a fractal reaction order and a power law asymptotic time-dependence of the drug concentration, therefore a mathematical relationship between the fractal reaction order and the power law exponent is derived. The goodness-of-fit of the model is assessed and compared to that of four other models suggested in the literature. Results The proposed model provided the best fit to the data. In addition, it correctly predicted the power law shape of the tail of the concentration-time curve. Conclusion A simple one-compartment model with steady state fractal Michaelis–Menten kinetics describing drug elimination from the body most accurately describes the pharmacokinetics of mibefradil in dogs. The new fractal reaction order can be explained in terms of the complex geometry of the liver, the organ responsible for eliminating the drug.     

Effect of misspecification of the absorption process on subsequent parameter estimation in population analysis

A prospective simulation study has been carried out to evaluate the effect of potential misspecification of the absorption rate constant (ka) in population pharmacokinetic analysis when few to no concentration-time data were available in the absorption phase and estimation of kawas not possible. Data were simulated for 100 subjects using a one-compartment model at steady state with first-order input. Data were generated over a range of kavalues: kawas misspecified in the NONMEM analysis by factors of 0.25, 0.5,1, 2, 3, and 4. In general, clearance (CL)was typically estimated with a small, constant underprediction, regardless of the range of misspecification of ka or whether data were present in the absorption phase. The same was not true for volume of distribution (V),values were biased and sensitive to the degree of misspecification, but only when the data contained even a little information about absorption. If studies are to be designed in which information absorption is either not required or is of no therapeutic use, then blood samples could be concentrated in the postabsorption phase and the absorption input fixed according to the best a priori information available.J. R. Wade was supported by a grant from Syntex Research, Thames House, 1 Bell Street Maidenhead, Berkshire, SL6 1BU, United Kingdom.     

美罗培南的群体药动/药效学研究进展

目的：简述美罗培南的药动学和药效学基本特点,介绍群体药动学研究基本理论,重点阐述美罗培南的群体药动/药效学研究状况。方法：查阅近年关于美罗培南群体药动/药效学文献,提取最终模型以及给药方案,分析归纳并综述。结果：美罗培南的药动学特征是二房室模型,药效学考察% T>MIC。婴幼儿和老年人的给药方案需要调整,未来的研究将关注肾功能特殊的患者以及体液体积大的患者。结论：需要研究正常脂肪体重（NFM）与美罗培南的群体药动/药效学的关系。     

群体药动学及其应用研究进展

概述了群体药动学的研究方法,着重介绍了群体药动学的原理、步骤及应用。近年来群体药动学应用范围不断拓宽,极大地推进了个体化用药的发展,已成为临床药代动力学研究的重要手段。     

Integrated equation to evaluate accumulation profiles of drugs eliminated by Michaelis-Menten kinetics

Using an equation for the calculation of plasma profiles of drug based on the zero-order input and Michaelis-Menten kinetic output in a one-compartment open model system, the times required to reach various degrees of several steady-state plasma concentrations of phenytoin are calculated. The effect of the apparent volume of distribution (due to intersubject variation or change in protein and/or tissue binding, etc.) on the time required to reach various fractions of steady-state plasma concentrations is discussed.     

The kinetics of flurbiprofen in synovial fluid

Steady state plasma and synovial fluid flurbiprofen concentrations obtained from 26 rheumatoid arthritis patients receiving 100 mg of flurbiprofen b.i.d. were analyzed using the NONMEM program. Only one synovial fluid sample per patient was available. Population estimates for the plasma parameters, clearance, volume of distribution, and elimination half-life were 1.75 L hr⁻¹, 11.9 L, and 4.8 hr, respectively, and the corresponding interindividual variances in these parameters were 29, 19, and 23%, respectively. The apparent elimination half-life from synovial fluid was 7.1 hr. After accounting for interindividual variability there was a residual variability of approximately 40% in both the plasma and synovial fluid concentrations.     

Non-linear elimination and protein binding of probenecid

Summary Six healthy volunteers were given probenecid 0.5, 1 and 2 g p.o. and 0.5 g i.v. The protein binding of probenecid at different concentrations in human plasma was estimated by equilibrium dialysis. The free fraction was found to increase nonlinearly with increasing total probenecid concentration, up to a maximum free fraction of 26%. The plasma concentration-time data after the oral doses were described by a one-compartment open model with first-order absorption and Michaelis-Menten elimination. The mean absorption rate constant 0.0072 min^–1 was dose-independent, and the maximal rate of elimination (mean 1429 µg/min) did not differ between doses whether calculated from the total or free concentrations. The Michaelis-Menten constant decreased significantly from 67.1 to 55.5 µg/ml as the dose increased from 1 g to 2 g, while the unbound Michaelis-Menten constant remained unchanged. The elimination of probenecid after the 0.5 g dose was in the linear region of the Michaelis-Menten elimination when calculated from the total and the free concentrations. The volume of distribution increased only slightly from 9.5 to 11.4 l as the dose increased from 0.5 to 2 g, but the unbound volume of distribution decreased significantly from 164 to 99 l. Absorption was complete and was independent of the dose administered.     

A population analysis of the pharmacokinetics and pharmacodynamics of midazolam in the rat

The concentration-EEG effect relationship of midazolam in the rat was studied from a population perspective. Plasma concentration and EEG effect data from 27 rats were available for analysis. Effect parameters derived from aperiodic EEG analysis were used as effect parameters. The population analysis gave results that were similar to the sample theory estimates (¯xs and SDs) obtained from the fits to individual data sets. Reanalysis of the EEG data using mean population pharmacokinetic parameters as input to the pharmacodynamic model led to poorer estimation of the pharmacodynamic parameters: particularly EC₅₀.Inclusion of one observed plasma concentration per individual significantly improved the estimation of the pharmacodynamic parameters and led to results that were virtually indistinguishable from those obtained using complete pharmacokinetic data.This study was supported by Medigon grant 900-521-106.     

建立阿奇霉素在中国健康人群中的群体药代动力学模型

目的建立中国人群中阿奇霉素(大环内酯类抗生素)的群体药代动力学模型。方法对20例健康自愿者的血药浓度和生化指标,用非线性混合效应模型法进行群体药代动力学分析,估算药代动力学参数,分析固定效应的影响以及个体内/间的变异,建立群体药代动力学模型。结果口服阿奇霉素呈一级吸收的二室模型,体质量对CL1和CL2及年龄对V1均有影响。结论用非线性混合效应模型法建立的中国人群中阿奇霉素的群体药代动力学模型,结构稳定,预测准确。     

基于NONMEM法建立紫杉醇群体药动学模型

目的：建立国人紫杉醇（paclitaxel,PTX）群体药动学（population pharmacokinetic,PPK）模型,为制定个体化给药方案提供理论支持。方法：收集138例接受紫杉醇治疗的肿瘤患者（建模组105例,验证组33例）210个血样,HPLC法测定紫杉醇血药浓度,PCR-RFLP法检测MDR1 C3435T。应用非线性混合效应模型（NONMEM）法,考察MDR1 C3435T基因多态性、合并用药及病理生理因素对紫杉醇药动学参数的影响,建立紫杉醇PPK模型。对模型进行拟合优度诊断、自举法（Bootstrap）内部验证,正态预测分布误差法（NPDE）及外部验证考察模型预测能力。结果：紫杉醇清除率（CL）和表观分布容积（V_d）的群体典型值分别为64.7 L·h^-1和1 240 L,患者内生肌酐清除率（CLcr）和给药速率（RATE）显著影响紫杉醇清除率。最终模型Bootstrap法验证结果与模型计算值相符,拟合优度、准确度及精密度均优于最简模型。结论：紫杉醇PPK最终模型稳定、有效,可结合Bayesian反馈法为临床优化给药方案提供科学依据。