Quantitative pooling of Michaelis-Menten equations in models with parallel metabolite formation paths

Modelling the loss of metabolic capacities of cultured hepatocytes: application to measurement of Michaelis-Menten kinetic parameters in in vitro systems

1. The loss of metabolic capacities during culture time constitutes a major limitation for the use of hepatocyte primary cultures in in vitro metabolism measurements. A new strategy is presented that permits one to calculate the Michaelis-Menten parameters V(max) and K(m) from extended experiments, by modelling V(max) as a variable dependent on time using exponential or sigmoidal equations. 2. This method was tested with cortisol depletion in cultured rat hepatocytes. V(max) and K(m) were used to calculate intrinsic clearance, and comparisons were made with methods already described in the literature. Intrinsic clearances given by our method were scaled to in vivo hepatic clearances that were close to those reported in the literature. 3. Our method could quantify the V(max) decrease with culture time from estimates of time parameters, t(1/2) or t(50). In our system, this V(max) decrease was in agreement with P450 cytochrome inactivation rates published for the rat liver. 4. In conclusion, we propose a convenient, simple and useful general method for both Michaelis-Menten parameter estimation and modelling of variations in the metabolic capacities observed in in vitro systems. Such an approach should improve the usefulness of hepatocytes in primary cultures for long-term metabolism experiments.

Individualization of theophylline infusion rate on the basis of a nonlinear compartmental pharmacokinetic model

In the present paper, a nonlinear compartmental model for theophylline pharmacokinetics is developed. The analytical solution of the model, in parametric form, is derived under plateau conditions for plasma metabolite concentration. The parameters are obtained from plasma and urine data using best fitting techniques and their values are used in order to calculate maintenance intravenous infusion. Numerical simulation is then performed in order to compare the drug concentration obtained by our approach with that of alternative intravenous regimens. The differences argue for individualized dosage regimens, since theophylline is a drug with a narrow therapeutic window and its concentration at the active sites strongly depends on characteristic parameters of the patient's response. Our results show that it is possible to estimate the patients' parameters during the first 8 h after intravenous administration of the drug and these parameters can be used to design an individualized dosage regimen in patients receiving theophylline intravenously.

Nonlinear pharmacokinetics of nefazodone after escalating single and multiple oral doses

The single- and multiple-dose pharmacokinetics of nefazodone and its metabolites, hydroxynefazodone, p-hydroxynefazodone, and m-chlorophenylpiperazine were investigated in two groups of 18 healthy male volunteers, employing three-period complete crossover designs. In one group, single 50-mg, 100-mg, and 200-mg oral doses of nefazodone hydrochloride were administered with a 1-week washout between treatments. In the other group, doses of 50 mg, 100 mg, and 200 mg were administered twice a day (every 12 hours) for 7.5 days (15 doses) with a 1-week washout between treatments. Serial plasma samples were obtained in both groups and assayed for nefazodone, hydroxynefazodone, m-chlorophenylpiperazine, and p-hydroxynefazodone. Cmax plasma levels of nefazodone and hydroxynefazodone were attained within 2 hours of administration of nefazodone; tmax for m-chlorophenylpiperazine was more delayed, and p-hydroxynefazodone levels were generally below the assay limit. On repeated twice-daily dosing of nefazodone, steady-state levels of the drug and its metabolites were reached within 3 days. Mean single-dose plasma half-life (t1/2) values for nefazodone increased from approximately 1 hour at a 50-mg dose to approximately 2 hours at a 200-mg dose; at steady state, t1/2 values increased from approximately 2 hours at 50 mg twice daily to approximately 3.7 hours at 200 mg twice daily. Whereas dose increased in the proportion of 1:2:4, mean single-dose AUC0-infinity for nefazodone increased in the proportion of 1:3.3:8.9 and mean steady-state AUC0-tau for nefazodone increased in the proportion of 1:4.2:16.8. Plasma levels of hydroxynefazodone paralleled those of nefazodone and were approximately 33% of nefazodone levels at each dose level. Plasma levels of m-chlorophenylpiperazine were only approximately 10% those of nefazodone. Within the dosage range of 50-200 mg of nefazodone hydrochloride, nefazodone and hydroxynefazodone exhibited nonlinear pharmacokinetics; m-chlorophenylpiperazine, a minor metabolite, appeared to exhibit linear pharmacokinetics.

Model representation of salicylate pharmacokinetics using unbound plasma salicylate concentrations and metabolite urinary excretion rates following a single oral dose

The pharmacokinetics of salicylic acid (SA) and its metabolites have been studied in 5 volunteers after administration of 3 g salicylic acid (as sodium salicylate) and collection of serial samples of blood and urine. SA and its metabolites were assayed with a HPLC method specific for each species. The urinary excretion rates of individual metabolites were analyzed using unbound plasma SA concentrations and Lineweaver-Burke plots. The analysis confirmed that the formation of SA urate (SU) and SA phenolic glucuronide (SPG) metabolites are saturable processes, and showed that the Michaelis-Menten values derived are consistent with earlier estimates derived solely from urinary data. The unbound salicylate plasma concentration-time profiles were then analyzed with various models assuming either saturable clearances for metabolite formation and/or saturable protein binding. The data were best described with a model that included both saturable protein binding and saturable metabolism. The model assumed first-order absorption kinetics and instantaneous distribution into extravascular and tissue compartments. The model was validated by comparing predicted relationships between the apparent volume of distribution, clearance, and plasma salicylate concentrations with previous relationships obtained using steady state data.

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.

Steady-state arterial and hepatic venous plasma concentrations of 5-bromo-2'-deoxyuridine and 5-iodo-2'-deoxyuridine in animals--drugs which are subject to both splanchnic and extra-splanchnic elimination

We have previously shown that 5-fluorouracil (FUra) in humans obeys Michaelis-Menten elimination kinetics. In this article we show that the related bromine and iodine-containing analogs in animals obey similar kinetics. Steady-state arterial (CssA) and hepatic venous plasma (CssV) concentrations of 5-bromo-2'-deoxyuridine (BrdUrd) are reported for 7 rabbits given 5 different infusion rates of BrdUrd and 5 dogs given 4 or 5 different infusion rates of BrdUrd. Steady-state arterial and hepatic venous plasma concentrations of 5-iodo-2'-deoxyuridine (IdUrd) are reported for 5 rabbits and 2 dogs given 5 different infusion rates of IdUrd. Each set of data could be fitted by a nonlinear least squares method to the equation: (equation; see text) where Vm/Q is the maximum difference, (CssA) - (CssV), and Km is the Michaelis constant. The estimated parameter Vm/Q and Km are compared for the two drugs and different species and also with the same parameters derived in the same manner from previously published data on fluorouracil in 8 cancer patients. The infusion rate needed to saturate the splanchnic elimination system (Rs in mumol/kg/min) was also estimated. For BrdUrd the mean value of Rs in the rabbit, namely 1.23, and in the dog, namely 1.25 mumol/kg/min are essentially the same.

Degradation of aliphatic alcohols by human liver alcohol dehydrogenase: effect of ethanol and pharmacokinetic implications

We have investigated the oxidation of low molecular weight aliphatic alcohols by Class I, II, and III alcohol dehydrogenases (ADH) isolated from human liver. These alcohols are generally present as byproducts of alcoholic beverages and referred to as alcoholic congeners. At concentrations corresponding to those in the blood after ingestion of alcoholic drinks (10-100 microM), the oxidation of propanol-1, isobutanol, 2-methylbutanol-1, and 3-methylbutanol-1 was mediated mainly by the isoenzymes of class I ADH, whereas butanol-1 was metabolized by Class I and II ADH. Class II ADH showed no activity with any of the alcohols at concentrations up to 100 microM. Lineweaver-Burk plots of the Class I ADH-catalyzed oxidation of all the congeners tested were linear in the pharmacokinetically relevant concentration range between 10 and 100 microM. Ethanol at concentrations found in the blood after moderate drinking (2.5-10 mM) caused a concentration-dependent inhibition of the congener oxidation. The experimentally determined kinetic constants were used to simulate the pharmacokinetics of propanol-1 metabolism in a multicompartment model system which accounts for first-pass elimination. The results suggest, in agreement with reported data from drinking experiments, that congener alcohols undergo considerable metabolism during the first liver passage, the extent of the first-pass metabolism depending on the ethanol dose.

Dose-dependent pharmacokinetics of probenecid in the rat

The basic pharmacokinetics of probenecid was studied by administration of three different i.v. bolus doses (50, 75, and 100 mg kg-1) to rats. The protein binding of probenecid in pooled rat serum was estimated by equilibrium dialysis. The unbound fraction was found to increase non-linearly with increasing total concentration, yielding a maximum free fraction of 49 per cent. The plasma concentration data obtained were described by a two-compartment model with Michaelis-Menten elimination. The maximal rate of elimination (Vm) remained unchanged between different doses irrespective of whether it was calculated in total or free concentrations (mean 187.2 +/- 8.3 (SD) microgram min-1). The Michaelis-Menten constant (Km) decreased slightly with increasing dose, while the unbound Michaelis-Menten constant (Km,u) did not change between the doses (mean 37.1 +/- 1.3 (SD) microgram ml-1). The volume of distribution of the central compartment (Vc) did not alter when the dose was increased from 50 to 100 mg kg-1 (mean 56.5 +/- 4.3 (SD) ml), but the unbound volume of distribution of the central compartment (Vc,u) decreased from 186.5 +/- 15.6 (SD) to 89.8 +/- 6.9 (SD) ml, which is in accordance with the reduction to be expected for drugs that only distribute in the extracellular fluid.

Single dose and steady-state pharmacokinetics of adinazolam after oral administration to man

An aqueous solution containing 1 mg of adinazolam mesylate per ml was administered orally as a single dose (40 mg) and with loading doses followed by hourly doses such that final dose rates of 1, 2, and 3 mg h-1 were administered to steady-state. Four subjects exhibited linear steady-state kinetics, while the other four exhibited Michaelis-Menten kinetics, based on measurement by HPLC of both unchanged drug and the major N-demethyl metabolite. The drug is very rapidly absorbed and has an intrinsic clearance of total (bound + free) drug which averaged 2.14 l min-1 based on the steady-state data and 1.17 l min-1 based on the single dose data, but these means do not differ significantly. The apparent metabolite clearance, CLmc/fm (where fm = fraction of adinazolam converted to the N-demethyl metabolite), averaged 0.170 l min-1 based on steady-state data and 0.179 l min-1 based on single dose data and these means do not differ significantly. Pharmacokinetic parameters, such as these clearances, had large intersubject variations. Three types of bioavailabilities were estimated from the data.

Single intravenous dose and steady-state oral dose pharmacokinetics of nicardipine in healthy subjects

Nicardipine HCl oral doses (10-40 mg) were administered sequentially to six healthy subjects. For each regimen the capsule dose was administered every 8 hours (q 8 h) for 3 days and the plasma profiles of nicardipine and its pyridine analogue (M5) were determined following the last dose on day 4. Steady-state plasma concentrations of nicardipine for each subject were fitted very well by the Michaelis-Menten equation. An intravenous tracer dose (0.885 mg nicardipine HCl) was administered simultaneously with the final oral dose on the fourth day of the 30 mg q 8 h regimen. The steady-state bioavailability of nicardipine was shown to be dose-dependent and averaged 19 per cent (10 mg), 22 per cent (20 mg), 28 per cent (30 mg), and 38 per cent (40 mg). Nicardipine undergoes linear first-pass metabolism to M5. Other metabolic pathways are responsible for the saturable first-pass metabolism observed for nicardipine.

Non-linear elimination and protein binding of probenecid

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 micrograms/min) did not differ between doses whether calculated from the total or free concentrations. The Michaelis-Menten constant constant decreased significantly from 67.1 to 55.5 micrograms/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 11.41 as the dose increased from 0.5 to 2 g, but the unbound volume of distribution decreased significantly from 164 to 99 1. Absorption was complete and was independent of the dose administered.

Influence of age, gender, and obesity on salicylate kinetics following single doses of aspirin

Salicylate kinetics following single, 650-mg intravenous and oral doses of aspirin were evaluated in humans in 2 studies. Complete conversion of aspirin to salicylate was assumed. The first study involved 25 young (25-40 years) and 21 elderly (66-89 years) healthy male and female volunteers. Mean salicylate clearance was lower in elderly females compared with that in young females; however, the difference between young men and elderly men was not significant. Salicylate free fraction in plasma increased significantly with age in men and women. After correction for free fraction, unbound mean clearance was reduced in elderly men compared with young men, and in elderly women compared with young women. Peak plasma salicylate concentrations after taking oral aspirin were not significantly influenced by age, and systemic availability of salicylate in all groups was complete. The second study compared 20 obese subjects (mean weight 113 kg) with 20 normal weight controls (mean weight 67 kg) matched for age, sex, height, and smoking habits. Small differences between obese and control groups were observed in total salicylate volume of distribution (Vd), unbound Vd, and mean clearance of total or unbound salicylate. Following normalization for total weight, however, values of total Vd and mean clearance were significantly smaller in obese subjects than in normal weight subjects. Rate and completeness of salicylate absorption were not influenced by obesity when aspirin was ingested, although peak levels were lower in obese subjects. If applied to multiple doses, the reduced unbound clearance of salicylate in the elderly would imply increased accumulation unless doses are appropriately adjusted downward. During long-term therapy, salicylate dosage for obese individuals should not be adjusted upward in proportion to total weight.

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.

Analytical approximations of sensitivities of steady state predictions to errors in parameter estimation: II. Michaelis-Menten kinetics

Linear sensitivity theory is used to estimate the reliability of predictions of the minimum and maximum concentrations at steady state in the Michaelis-Menten model with i.v. bolus. The dependence of the relative errors in the predictions on the errors in the pharmacokinetic parameters is derived in an analytical form. It is shown that the quality of the predictions is not equally sensitive to all errors in parameters, and that the sensitivity factors vary with the degree of saturation of the system. An example of application for a drug, such as phenytoin, is discussed. It is suggested that sensitivity analysis may be useful in design of pharmacokinetic experiments aimed at the control of steady state levels for drugs with Michaelis-Menten kinetics.

In vivo evaluation of the Michaelis-Menten constant for a medium extraction ratio drug: application to cinromide in the rhesus monkey

The dose-dependent nonlinearity of the clearance of cinromide, a medium extraction ratio drug, has been established in two monkeys. Special problems encountered in evaluation of nonlinearity of such drugs were resolved by the experimental design: cinromide was infused to steady state via the portal vein. A linearized form of the Michaelis-Menten equation was used to determine Vmax and Km. In addition, cinromide was administered to one of the monkeys via a femoral vein to verify the overestimation of Km by administration at a peripheral venous site.

Pharmacokinetics of ibuprofen in man--III: Plasma protein binding

Plasma protein binding of ibuprofen was measured by equilibrium dialysis on 406 plasma samples collected from 15 normal volunteers following doses of 400, 800, and 1200 mg of ibuprofen as tablets (N = 102, 100, 104, respectively) and 420 mg as an aqueous solution (N = 100). Individual subject bound concentration at dialysis equilibrium (Cbd) vs. free concentration at dialysis equilibrium (Cfd) were well fitted via computer to the Scatchard equation with one class of binding sites. The binding capacity averaged 1231 microM (range 848-1658 microM), and the association constant averaged 1.76 X 10(5) M-1 (range 1.15 X 10(5) to 2.73 X 10(5) M-1). Distributional analysis was performed on the free fraction (fd) and bound/free ratios (Cbd/Cfd = 1/fd-1) at dialysis equilibrium for each treatment. Using pooled data of all four treatments, distributional analysis was also performed on the free fractions (f) and bound/free ratios (Cb/Cf = 1/f-1) corresponding to the plasma drug concentrations in blood as it was withdrawn from the subjects. The bound/free ratios were normally distributed, whereas the distributions of the free fractions were skewed towards higher values.

A new blood-flow pharmacokinetic model for ethanol

A multicompartmental mathematical model, taking into account observed vascular concentration gradients and blood flow rate limitations, has been proposed. Based on blood flows and blood and tissue volumes reported in the literature and in personal communications, an appropriate model was elaborated for ethanol in the dog. The relatively high values of r2 and a correlation coefficient determined for the simultaneous computer fitting of mean observed blood ethanol concentration, time data from two administered doses, support the proposed model. While the surgery is prohibitive for human experimentation, correlations may be extended to man. The elaboration of a more complete multicompartmental model for ethanol should prove beneficial in revealing the relationships among the doses of alcohol, the circulating blood ethanol concentrations, and physiological and psychomotor test parameters.

Pharmacokinetics of ethanol: a review

The pharmacokinetics of ethanol in man are reviewed from a historical perspective from the earliest attempts at kinetic analysis of urinary data in 1899 to the present nonlinear analysis of blood alcohol concentration (BAC) and time data. Review of the various kinetic theories that have been utilized to describe the kinetics of alcohol metabolism is provided. Extensive review is made of recent investigations supporting the application of Michaelis-Menten enzyme kinetics to describe alcohol metabolism. Results of direct, nonlinear least-squares computer fitting of BAC following intravenous and oral feeding of alcohol both in the fasting and fed states are presented with appropriate theory. The kinetics of the oral absorption of alcohol and the relationship among stomach emptying rate, the apparent absorption rate, and the area under the BAC-time curve are discussed and data presented. The kinetics of multiple Michaelis-Menten pathways are discussed with application to the (potential) contributions of the MEOS and/or ADH systems to the observed BAC curve and resultant kinetic parameters. Several methods of obtaining pharmacokinetic (Michaelis-Menten) parameters from BAC curves and their interpretation and usage in comparative studies are presented.