Michaelis-Menten elimination kinetics: areas under curves, steady-state concentrations, and clearances for compartment models with different types of input

For single bolus administration, intermittent bolus administrations to steady-state, a single dose as a zero order input, intermittent zero order inputs to steady-state, and continuous zero order input to steady-state, and for both simple Michaelis-Menten elimination and parallel Michaelis-Menten and first order elimination, the appropriate equations are given for the areas, AUC 0-oo or AUC 0-tau, steady-state concentrations, and clearances. Some 20 new equations have been derived. For the case of first order input and Michaelis-Menten elimination, no solution is given but the effect of input rate on systemic availability is reported following some numerical integrations. The effect of slow input in reducing systemic bioavailability when Michaelis-Menten elimination kinetics are operative is stressed and the implications of this in the field of sustained-release medication mentioned.     

Application of Mean Residence-Time Concepts to Pharmacokinetic Systems with Noninstantaneous Input and Nonlinear Elimination

Equations describing the mean residence time (MRT) of drugs in the body are derived for drugs that are administered by first-and zero-order rates into systems with Michaelis–Menten elimination. With computer simulations, the validity of these equations, the differences between them, and the conventional approach using the AUMC/AUC or the summation of mean times are demonstrated by examining calculations of the percentage of the administered dose eliminated at the MRT and AUMC/AUC. The effects of the absorption rate on the AUC and on the approximate and true MRT values in a nonlinear pharmacokinetic system are also illustrated with computer simulations. It was previously found that the true MRT_iv = V _ss · AUC_iv/dose for an iv bolus. The total MRT (sum of input and disposition) of a drug after noninstantaneous administration was found to be a function of the MRT_iv, two values of AUC (iv and non-iv), and exactly how the drug is administered expressed as the mean absorption time (MAT). In addition, a theoretical basis is proposed for calculation of the bioavailability of drugs in both linear and nonlinear pharmacokinetic systems.     

Mean Residence Time Concepts for Pharmacokinetic Systems with Nonlinear Drug Elimination Described by the Michaelis–Menten Equation

Equations for the mean residence time (MRT) of drug in the body and related functions are derived for drugs which are intravenously administered into a one- or two-compartment system with Michaelis–Menten elimination. This MRT is a function of the steady-state volume of distribution and time-average clearance obtained from the dose and area under the curve (dose/AUC). The differences between the MRT calculated by the proposed method and by using the moment theory method (AUMC/AUC) are demonstrated both mathematically and by computer simulations. The validity of the proposed method for calculation of MRT and its relationship to the moment theory result have also been assessed by examining the percentage of the administered dose eliminated and the percentage of the total area attained at MRT and at AUMC/AUC in relation to the dose. The equations evolved should be helpful in clarifying residence time derivations and in defining the disposition characteristics and differences between linear and nonlinear systems. Direct methods are provided for calculation of Michaelis–Menten parameters based on the relationship between MRT and dose.     

New and simple method to predict dosage of drugs obeying simple Michaelis-Menten elimination kinetics and to distinguish such kinetics from simple first order and from parallel Michaelis-Menten and first order kinetics

It has been found empirically that either average or minimum steady-state plasma concentrations (Css) of drugs obeying Michaelis-Menten elimination kinetics give essentially linear plots on semilogarithmic graph paper when Css is plotted versus the maintenance dose (D) or dose rate (R). The equations of such straight lines may be converted to the following nonlinear equation: Css = abD which fits the Css,D data essentially as well as D = VmCss/(Km + Css). The parameter b is analogous to unity plus the interest fraction in logarithmic growth or compound interest calculations, and each drug appears to have a characteristic value of this parameter, with extremely small intersubject variation. From the above equation the following equation, Dn+1 = Dn + 1n(Cn+1/Cn)1n b can be derived, which forms the basis of predicting the needed dosage, Dn+1, to obtain a desired steady-state concentration, Cn+1, using one initial steady-state concentration, Cn, obtained with dose, Dn, and using a population value of b for the drug. It appears that it is the value of the "initial capital" (i.e., a in relation to the initial dose) rather than the "interest fraction" (i.e., b - 1) that causes most of the intersubject variation in Css of a given drug. Several drugs illustrate the usefulness of the method. A semilogarithmic plot also appears to be an excellent method to distinguish simple Michaelis-Menten kinetics from parallel Michaelis-Menten and first order elimination kinetics and from simple first order kinetics with steady-state data in the range 0.3-3 Km.     

A nonlinear physiologic pharmacokinetic model: I. Steady-state

The two-compartment model of Rowland et al., (2) has been extended by replacing first order elimination with Michaelis-Menten elimination kinetics. All of the equations for steady-state concentrations and clearances for zero order (constant rate) input orally (into compartment #2) and intravenously (into compartment #1) are derived and reported. The steady-state concentration in compartment #1, following intravenous administration, is shown to be a nonlinear function of maximal velocity of metabolism, Vm, the Michaelis constant, Km, and liver blood flow, Q; and, following oral administration is dependent only upon Vm and Km and is independent of Q. However, oral bioavailability is a function of Vm, Km, and Q. The model allows physiologic pharmacokinetic interpretation of both linear and nonlinear data; and, together with simple modification of the model, can explain much observed pharmacokinetic data to date particularly for first-pass drugs. Future articles in the series will be concerned with single doses, evaluation of literature data in terms of the model, application of the theory in toxicology and in clinical pharmacokinetics and therapeutics.     

Drug redistribution and mean transit time concepts for nonlinear pharmacokinetic systems

Equations for the mean number of cycles through the peripheral system (R) and the mean transit time through the central compartment (MTTc) are derived for intravenous drugs with linear distribution and linear or nonlinear central elimination. This R is a function of distribution clearance (CLD), dose, and area under the plasma concentration-time curve (AUC). The MTTc is a function of the central volume of distribution, CLD, dose, and AUC. The application of the proposed calculations of R and MTTc was illustrated by computer simulations.     

Computer Modeling of the Pharmacokinetics of Fluorouracil and Thymine and Their Kinetic Interaction in Normal Dogs

The kinetic behavior of thymine and 5-fluorouracil has been shown to be non-linear and mediated largely by saturable metabolic processes. In vivo estimates of the Michaelis-Menten parameters V_max and K_m were obtained from constant infusion data in normal dogs using a system of balance equations that equate drug input with total output at steady-state. These estimates were then successfully used to simulate both steady-state and post-infusion plasma concentration-time curves for both compounds over a range of saturating and non-saturating conditions. It has been shown previously that estimates of V_max and K_m obtained from dynamic data can be incorrect if an inappropriate compartmental model is used in the analysis. Determining the Michaelis-Menten parameters at steady-state eliminates this difficulty. Moreover, the use of steady-state derived values to simulate post-infusion data confirms the validity of this technique. The kinetic interaction between thymine and 5-fluorouracil was investigated as a case of competitive metabolic inhibition in vivo by calculating K_i values from data obtained during simultaneous constant infusions of the two compounds. These values were then used in conjunction with a series of differential equations incorporating reciprocal metabolic effects to simulate the effect of thymine on FU plasma concentration.     

Nonlinear least-squares regression analysis by a simplex method using differential equations containing Michaelis-Menten type rate constants

Computer curve fittings were carried out to observed data as well as theoretically generated plasma concentrations of several drugs, using differential equations which contained nonlinear Michaelis-Menten type rate constants to discuss problems of initial parameter estimation in pharmacokinetic analysis. Calculation based on two different algorithms, each carried out by using SIMP (simplex method) and NONLIN (modified Gauss-Newton method) produced similar results. However, occasional divergence or unreasonable solutions occurred in a later case, when assumed values of Km and Vmax were used as initial parameters. A combined use of SIMP and NONLIN in which calculated values by SIMP were used as initial values for NONLIN, was shown to be effective to analyse plasma concentration data of indocyanine green bearing difficulty in estimating initial values. It is suggested that the successive method is useful for the curve fitting of plasma concentration with nonlinear pharmacokinetic rate processes.     

Theophylline. Pooled Michaelis-Menten parameters (Vmax and Km) and implications

The literature on theophylline is confusing since in the same dose range one article will report linear kinetics while another will report non-linear kinetics. Single dose clearances and lower steady-state clearances of theophylline, recently reported in the literature, were used to estimate pooled Vmax and Km values of the Michaelis-Menten equation for 10 normal subjects. The mean Vmax was 1960 mg/day and the mean Km was 24.1 mg/L. These values were then utilised to: explain another set of different oral clearances following doses of 2 and 6 mg/kg reported in the literature; estimate relative effects of dose rate and type of input on absolute bioavailability; estimate AUC (0-infinity) as a function of single dose over the range 0 to 1500 mg; estimate the average steady-state serum concentration of theophylline (Cssav and steady-state oral clearance (CLsspo) as a function of dose rate in mg/day; illustrate how Michaelis-Menten kinetics alters the apparent first-order elimination rate constant and the half-life estimated from terminal log-linear plots at concentrations appreciably lower than the Km value; and illustrate how Michaelis-Menten kinetics affects the estimation of a zero-order absorption rate constant using the Wagner-Nelson method.     

Evaluation of Bioequivalence of Highly Variable Drugs Using Clinical Trial Simulations. II: Comparison of Single and Multiple-Dose Trials Using AUC and Cmax

Purpose. Evaluating of the effects of high intrasubject variability in clearance (CL) and volume of distribution (V), on 90% confidence intervals (CIs) for AUC (Area Under the concentration Curve) in single and multiple-dose bioequivalence studies. The main methodology was Monte Carlo simulation, and we also used deterministic simulation, and examination of clinical trials. The results are compared with those previously observed for Cmax (maximum concentration.) Methods. The time course of drug concentration in plasma was simulated using a one-compartment model with log-normal statistical distributions of intersubject and intrasubject variabilities in the pharmacokinetic parameters. Both immediate-release and prolonged-release products were simulated using several levels of intrasubject variability in single-dose and multiple-dose studies. Simulations of 2000 clinical bioequivalence trials per condition (138 conditions) with 30 subjects in each crossover trial were carried out. Simulated data were compared with data from actual bioequivalence trials. Results. The current simulations for AUC show similar probabilities of failure for single-dose and multiple-dose bioequivalence studies, even with differences in the rate of absorption or fraction absorbed. AUC values from prolonged-release scenario studies are more sensitive to changes in the first order absorption rate constant ka, and to variability in CL and V than AUC from studies of immediate-release studies. Conclusions. We showed that multiple-dose designs for highly variable drugs do not always reduce intrasubject variability in either AUC or Cmax, although the behavior of AUC differs from Cmax. Single dose AUC to the last quantifiable concentration was more reliable than either single dose AUC extrapolated to infinity, or multiple dose AUC during a steady-state interval. Multiple-dose designs may not be the best solution for assessing bioequivalence of highly variable drugs.     

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.     

Constant-Rate Intravenous Infusion Methods for Estimating Steady-State Volumes of Distribution and Mean Residence Times in the Body for Drugs Undergoing Reversible Metabolism

Equations for the steady-state volumes of distribution (V _ss) and the mean residence times in the body (MRT) are derived for a drug and its metabolite subject to reversible metabolism and separately infused intravenously at a constant rate to steady state of both compounds. The V _ss and MRT parameters are functions of the integrals of plasma concentrations, plasma concentrations at steady state, and times to reach steady state of both drug and metabolite. In addition, the MRT values are functions of the infusion rates. These equations were validated by computer simulations and comparison with IV bolus dose parameters. These relationships extend the ability to assess the pharmacokinetics of linear reversible metabolic systems.     

A nonlinear physiologic pharmacokinetic model: I. Steady-state

The two-compartment model of Rowland et al.,(2) has been extended by replacing first order elimination with Michaelis-Menten elimination kinetics. All of the equations for steady-state concentrations and clearances for zero order (constant rate) input orally (into compartment #2) and intravenously (into compartment #1) are derived and reported. The steady-state concentration in compartment #1, following intravenous administration, is shown to be a nonlinear function of maximal velocity of metabolism, V_m,the Michaelis constant, K_m,and liver blood flow, Q;and, following oral administration is dependent only upon V_m and K_m and is independent of Q.However, oral bioavailability is a function of V_m, K_m,and Q.The model allows physiologic pharmacokinetic interpretation of both linear and nonlinear data; and, together with simple modification of the model, can explain much observed pharmacokinetic data to date particularly for first-pass drugs. Future articles in the series will be concerned with single doses, evaluation of literature data in terms of the model, application of the theory in toxicology and in clinical pharmacokinetics and therapeutics.     

Dosage regimen of cimetidine reviewed. Possible drug accumulation after multiple oral doses

Summary The doses of cimetidine recommended differ in children, especially those with cystic fibrosis. These dosage regimens were derived from single-dose pharmacokinetic studies of the drug. Some authors showed, however, that after administration of repeated oral doses of cimetidine in healthy adults and children with cystic fibrosis, the elimination half-life of the drug was markedly prolonged. In view of the ability of cimetidine to inhibit metabolism of other drugs, it is suggested that the parent compound and/or its metabolite(s) may inhibit its own metabolism during a prolonged course of treatment. Enterohepatic recirculation of the drug and/or its metabolite(s) may also contribute to prolongation of its elimination. One should therefore be cautious in using single-dose pharmacokinetic parameters to calculate repeated dose regimens and expected plasma steady-state concentrations.     

Analysis of hepatic metabolism affecting pharmacokinetics of propranolol in humans

We examined the metabolic kinetics of propranolol, constructed from saturable and non-saturable components, using liver microsomes. The metabolic activity in rat microsomes was much higher than that in human microsomes within the clinically observed plasma range. Using the physiologically based pharmacokinetic (PBPK) model incorporating the obtained metabolic parameters, the plasma kinetics of propranolol was well correlated with reported values, and then used to analyze the effect of hepatic first-pass metabolism on propranolol plasma pharmacokinetics in clinical doses. The simulated plasma concentrations and AUC values of propranolol increased proportionally to its dose; these levels were almost equivalent to intrinsic clearance (CLint1), presumed to be non-saturable. When Michaelis-Menten parameters were decreased to one twentieth, plasma concentrations slightly increased after 160 mg dosing. A similar result was obtained with steady-state plasma levels after repeated administration. On the other hand, the first-order absorption rate constant of propranolol did not affect AUC values. The dose-normalized AUC value started to increase about 10(3)mg dosing. When the dose exceed 10(6)mg dose, the CLint1 component hardly contributed to propranolol pharmacokinetics. Accordingly, under the conditions of the PBPK model, propranolol pharmacokinetics was considered to be dose-independent within the clinical dose range.     

Bioequivalence of carbamazepine chewable and conventional tablets: single-dose and steady-state studies

Single-dose and steady-state studies were carried out on separate occasions to examine the bioequivalence of the newly formulated carbamazepine chewable tablet. In the single-dose study, the plasma levels resulting from 2 X 200-mg conventional tablets (CT), 4 X 100-mg chewable tablets swallowed whole (SW), and 4 X 100-mg chewable tablets chewed before swallowing (CHEW) were compared. A randomized 3 X 3 Latin-square design balanced for residual effects, with a 3-week washout period, was used (n = 6). Plasma samples were analyzed by a specific GC method for carbamazepine. The following parameters were used for evaluation: AUC, Cmax, tmax, and t1/2. None of the parameters were significantly different except Cmax and t1/2 values for CHEW and CT. The Cmax was 25% higher and t1/2 was 11% shorter for CHEW than CT. The impact of differences in the peak plasma levels at steady state were examined by pharmacokinetic projection (400 mg b.i.d.) based on the single-dose data and with simulated induction equal to a 50% reduction in t1/2. The projected steady-state CT and CHEW plasma concentrations were similar, with a difference of only 4%. The results demonstrate the bioequivalence of the dosage forms with respect to the extent of absorption, and similar steady-state concentrations of carbamazepine in plasma can be expected. To test the conclusion from the projected study, a separate bioequivalence study to compare CHEW relative to CT was performed at steady state in normal volunteers (200 mg b.i.d.).(ABSTRACT TRUNCATED AT 250 WORDS)     

Michaelis-Menten metabolite formation kinetics: equations relating area under the curve and metabolite recovery to the administered dose

A computational approach which concomitantly determines the capacity-limited rate constants of parent drug elimination and metabolite formation is presented. The approach applies both the presently derived total excretory recovery versus dose relationships of the metabolite and the AUC versus dose relationships of the parent drug to identify the parameters. Three parent drug elimination conditions were assessed: pooled first-order, pooled Michaelis-Menten, and parallel first-order and pooled Michaelis-Menten kinetics. Model and parameter identification criteria are discussed. Literature data for theophylline and two of its metabolites in rats were examined to reveal pooled Michaelis-Menten elimination kinetics of theophylline and capacity-limited formation of the metabolites. The proposed technique is useful for quantitating commonly obtained nonlinear drug disposition data such as AUC and amount of metabolites excreted.     

Loratadine: multiple-dose pharmacokinetics

The steady-state pharmacokinetics of loratadine (L), a new long-acting antihistamine devoid of CNS activity, was investigated in 12 healthy male volunteers. Each volunteer received 40-mg L capsules q24h for ten days. Blood samples were collected at various times on day 1, 5, 7, and 10 and assayed for L by radioimmunoassay (RIA) and for descarboethoxyloratadine (DCL), a known active metabolite, by high-performance liquid chromatography (HPLC). The plasma L and DCL concentration-time data in the disposition phases were fitted to a biexponential equation for pharmacokinetic analysis. Steady-state plasma L Cmax concentrations were reached at 1.5 hour (Tmax) after each dose. DCL steady-state Cmax values ranged 26 to 29 ng/mL at a Tmax ranging from 1.8 to 3 hours. The AUC at steady state, AUC tau, was 80 to 96 and 349 to 421 h X ng/mL for L and DCL, respectively. The accumulation indexes (Ra) based on AUC tau ratios, did not change for either compound after day 5. Ra values for L and DCL after the fifth dose were 1.4 and 1.9, respectively, indicating that there is little accumulation of either L or DCL after a multiple (once-a-day) dosage regimen. The t1/2 beta at steady state were 14.4 and 18.7 hours for L and DCL, respectively, which were similar to those reported following a single-dose L administration. Observed plasma drug concentrations were in good agreement with predicted values derived for pharmacokinetic parameters.