Theorems and implications of a model-independent elimination/distribution function decomposition of linear and some nonlinear drug dispositions. IV. Exact relationship between the terminal log-linear slope parameter beta and drug clearance

An exact formula relating the terminal log-linear beta parameter and the drug clearance is derived. The expression is valid for drugs with a linear, polyexponential disposition kinetics. The formula is useful for calculating the clearance when the clearance has changed between drug administrations and requires only drug level data from the terminal, log-linear elimination phase in addition to data from a single separate i.v. administration in the same subject. Data from an i.v. administration are necessary in order to apply the disposition decomposition technique to isolate and uniquely define the distribution kinetics in terms of the distribution function h(t). The different clearances can then be calculated from the beta values of the log-linear terminal drug level data and the parameters of h(t). The theoretical basis of the method and its assumptions and limitations are discussed and various pertinent theorems are presented. A computer program enabling an easy implementation of the proposed method is also presented. The mathematical and computational procedures of the method are demonstrated using kinetic data from i.v. and oral administrations of cimetidine, diazepam, and pentobarbital in human subjects. The classical V.beta method of approximating the clearance as the product of volume of distribution and beta is considered for comparison. For the three drugs considered the V.beta method which assumes a single exponential disposition kinetics leads to excessive errors when applied in absolute clearance comparisons. However, when applied in relative comparisons in the form of the "beta correction" the errors cancel out to some extent depending on the magnitude of the distribution kinetic effect. Whenever possible it is advisable to apply the proposed method to avoid such errors.     

Drug absorption evaluation in the presence of changes in clearance: an algorithm and computer program for deconvolution with exact clearance correction

Most commonly drug absorption is evaluated with a reference dosing given on separate occasions. The assumption that no change in drug disposition is taking place between the drug administrations is often violated resulting in errors in the calculations. A novel deconvolution method is presented which exactly compensates for a change in drug clearance. The method is based on a model independent disposition decomposition-recomposition technique. The distribution function is obtained from an i.v. administration by disposition decomposition. This distribution function is assembled together with the elimination kinetics containing the perturbed clearance to construct the perturbed disposition function in the subsequent disposition recomposition operation. The perturbed absorption response is finally deconvolved using the corresponding perturbed disposition function. It is shown that the perturbed clearance can be obtained from the log-linear terminal disposition phase once the distribution function has been obtained from an i.v. administration. The proposed method is implemented in an algorithm and computer program DCONB and demonstrated using human cimetidine drug level data from an i.v. and oral administration. The usage of DCONB is identical to DECONV previously published. It requires only regular sums of exponentials to be fitted to drug level data. Such fittings are routinely done in pharmacokinetics thereby enabling DCONB to be implemented very simply.     

Determination of drug absorption rate in time-variant disposition by direct deconvolution using beta clearance correction and end-constrained non-parametric regression

A novel numerical deconvolution method is presented that enables the estimation of drug absorption rates under time-variant disposition conditions. The method involves two components. (1) A disposition decomposition-recomposition (DDR) enabling exact changes in the unit impulse response (UIR) to be constructed based on centrally based clearance changes iteratively determined. (2) A non-parametric, end-constrained cubic spline (ECS) input response function estimated by cross-validation. The proposed DDR-ECS method compensates for disposition changes between the test and the reference administrations by using a "beta" clearance correction based on DDR analysis. The representation of the input response by the ECS method takes into consideration the complex absorption process and also ensures physiologically realistic approximations of the response. The stability of the new method to noisy data was evaluated by comprehensive simulations that considered different UIRs, various input functions, clearance changes and a novel scaling of the input function that includes the "flip-flop" absorption phenomena. The simulated input response was also analysed by two other methods and all three methods were compared for their relative performances. The DDR-ECS method provides better estimation of the input profile under significant clearance changes but tends to overestimate the input when there were only small changes in the clearance.     

Pharmacokinetics of clonidine in the rat and cat

To investigate the pharmacokinetic behavior of clonidine, rats were given clonidine intravenously at 125, 250, and 500g/kg and blood clonidine concentrations were followed for 6 hr. The disposition of clonidine in two brain regions was studied in rats after an i. v. dose of 500 g/kg. The liver clearance in rats was investigated by liver perfusion techniques. The results obtained indicate that the disposition characteristics of clonidine can be described by a two-compartment open model in both rats and cats. The penetration of clonidine into tissues is rapid, and brain levels in rats were about 1.7 times higher than blood levels. Brain tissues were found to be an indistinguisible part of the central (blood) compartment. Dose-dependent pharmacokinetic behavior was found for clonidine in rats at the doses used. This was demonstrated by a decrease of both the rate constant of distribution to the peripheral compartment and the overall elimination rate constant from the body, with increase in dose. As a consequence, the volume of distribution and the clearance both decreased with increasing dose. Possible explanations for the dose-dependent behavior of clonidine are discussed.     

A method for estimating within-individual variability in clearance and in volume of distribution from standard bioavailability studies

Bioavailability studies are commonly undertaken, and most, because they involve subjects taking repeated doses of a drug, contain information on intraindividual variability in pharmacokinetics. However, because in such studies bioavailability itself is unknown, it is difficult to resolve which pharmacokinetic parameters vary within individuals. A mathematical model is presented which permits estimation of variability in clearance and in volume of distribution. When applied to pooled data arising from five theophylline bioavailability studies, this model has given statistical evidence that clearance of theophylline is inherently more variable within individuals (coefficient of variation, 13%) than volume of distribution (8%). As a result, use of the measurement AUC · rather than AUCas a more precise index of bioavailability is justified in studies where is measured with reasonable precision. The model could be applied to estimation of withinbatch within-person variability in bioavailability.Deceased, April 4th, 1981.     

Deconvolution applied to the kinetics of extracorporal drug removal. Haemodialysis of cefsulodin

Summary A novel approach to the evaluation of the kinetics of drug removal by an extracorporal device (ECD), e.g., haemodialysis, haemofiltration, and haemoperfusion, is presented. The rate and extent of extracorporal drug removal (ECR) are determined by deconvolution. The proposed method is model independent in the sense that no specific models of corporal or extracorporal disposition are required. The estimation of various derived functions and parameters useful for describing ECR such as clearance and fractional drug removal are facilitated by the technique. The kinetics of cefsulodin elimination by haemodialysis in 3 patients were evaluated using the deconvolution approach. The results indicated that cefsulodin was dialyzable with 50% of the drug in the body removed by haemodialysis over 3–4 h.     

Method for the rapid estimation of the total body drug clearance and adjustment of dosage regimens in patients during a constant-rate intravenous infusion

A simple method is proposed to estimate the total body clearance and biological half-life of drugs in patients by determining two blood or plasma levels of drugs separated by an appropriate interval during the constant-rate intravenous infusion. The method is based on the assumptions that the drug disposition in the body can be adequately described by the linear onecompartment open model and its apparent volume of distribution can be estimated with a reasonable degree of accuracy from the literature data. The validity of this newly proposed method was demonstrated in three rabbits using theophylline as a test drug. Results of theoretical error analyses indicate that the method is potentially useful in providing an early guide for dosing adjustments for those potent drugs whose elimination rates have been shown to vary greatly in patients. The method may be particularly useful in patients with changing elimination kinetics of drugs. Effects of the accuracy in V_d estimate, plasma level assay, and sampling time interval on the error of total body clearance estimation are discussed.     

Overall pharmacokinetics during prolonged treatment of healthy volunteers with digoxin andβ-methyldigoxin

Summary Five healthy volunteers received digoxin 0.4 mg or -methyldigoxin 0.4 mg i. v., daily for 14 days, in a randomized cross-over arrangement. By monitoring minimal plasma concentrations during multiple dosing, it was found that the steady state pharmacokinetics of digoxin and -methyldigoxin could be estimated even better by a one-compartment than by a two-compartment model. The following mean parameters were calculated: the half life of digoxin of 1.54±0.31 days was significantly shorter than the half life of 2.29±0.34 days for -methyldigoxin. The distribution volume of 807±187 liters for digoxin was not significantly larger than the 735±227 liters for -methyldigoxin. Renal digoxin clearance of 191±25 ml/min was significantly higher than both the renal clearance of -methyldigoxin of 111±23 ml/min and also the creatinine clearance, which indicates tubular secretion of digoxin. There was a 2.8-fold accumulation of -methyldigoxin injected once a day, which was significantly higher than the 1.8-fold accumulation of digoxin.Abbreviations of parameters D dose - EST estimated disposition rate constant (day^–1) semilog curve fit - KTOT total disposition rate constant (day^–1) NONLIN fit - KREN renal disposition rate constant (day^–1) NONLIN fit - VOL volume of distribution (liters) NONLIN fit - CTOT total clearance (ml/min) - CREN renal clearance (ml/min) - BMIN body stores (µg) before next dose in steady state - CCR creatinine clearnace (ml/min) - CR/CT fraction renally excreted - CR/CCR renal drug clearance versus creatinine clearance - T/2 half life (days) - BMIN/D extent of accumulation This study was supported by Bundesministerium für Jugend, Familie und Gesundheit, Federal Republic of Germany.     

Prediction of the disposition ofβ-lactam antibiotics in humans from pharmacokinetic parameters in animals

Various pharmacokinetic parameters (disposition half-life, total body clearance, renal clearance, hepatic clearance, volume of distribution, intrinsic clearance and volume of distribution of unbound drug) of six-lactam antibiotics were compared in mouse, rat, rabbit, dog, monkey, and human. Two methods for prediction of the disposition of the-lactam antibiotics in humans by extrapolation of the animal data were evaluated. One was the Adolph-Dedrick approach, which can be used to predict clearances in humans from the relationship between intrinsic clearances and body weight in the other five species. The volume of distribution in humans was predicted from the relationship between the volume of distribution and serum unbound fraction in the five species. The other was the Boxenbaum approach, which can be used to predict the pharmacokinetic parameters of the six-lactam antibiotics in humans by using the regression lines of log-log plots of intrinsic clearance and volume of distribution of unbound drug in a single species, in this case the monkey. The half-life calculated according to the latter method had a smaller absolute error than that calculated according to the former method, but the better method for extrapolation of animal data to humans seems to be the former method, which does not require a prioriinformation regarding structure-pharmacokinetic relationships among the antibiotics.     

Gompertz Pharmacokinetic Model for Drug Disposition

Purpose. Disposition of drugs among compartments of the body usually occurs at changing rates that are commonly modeled as sums of exponential terms with different rate constants. This paper describes an alternative, Gompertz kinetics, in which the rates can change systematically. Methods. Differential equations were developed and solved that fit typical examples taken from the literature. The three or four constants required for a visually satisfactory fit to data could readily be found by successive adjustment by hand, but strategies and results are presented for computer fitting of the data. Results. In four examples, the amount remaining in the blood decreases as an exponentially declining fraction of the amount present at any moment, but the antecedent processes responsible for that amount differ as follows: (a) In simple i.v. disposition (e.g., lidocaine) concentration falls as a decelerated exponential decay. (b) Delayed i.v. disposition (e.g., hexobarbital) requires, as well, a decelerated exponential growth function. (c) In simple disposition after oral administration, the concentration in the blood initially increases at a decelerating rate. (d) In biphasic oral disposition (e.g., Li⁺ carbonate), the initial Gompertz growth is followed by decelerated exponential decay. Conclusions. Gompertz kinetics provides an accurate and parsimonious mathematical model describing drug disposition.     

The concentration-dependent disposition and kinetics of inulin

Summary The disposition of inulin was studied in 30 healthy male and 10 healthy female volunteers, and 10 patients with stable chronic renal failure (mean creatinine clearance 45 ml·min^–1) following intravenous infusion of 70 mg·kg^–1 over 5 min.Plasma concentrations fell rapidly initially but the rate of decline decreased continuously over 8 h and a linear terminal elimination phase could not be identified. Inulin was excreted rapidly by the subjects with normal renal function and 97.3% of the dose was recovered in the urine in 8 h. There was a progressive highly significant fall in the renal clearance of inulin after 2 h as plasma concentrations fell below about 150 mg·l^–1. Six to 8 h after administration the clearance was less than 50% of the initial value in the healthy volunteers and the corresponding fall in the renal patients was 33%.The concentration-dependent renal clearance of inulin was confirmed in step-up and step-down constant infusion studies in which clearances were measured at mean plasma concentrations ranging from 35.2 to 186.7 mg·l^–1. These studies virtually excluded time, changes in posture and urine flow rate as important factors. There was no statistically significant fall in clearance during the first 2 h and kinetic analysis was based on data obtained over this time. Under these conditions the mean plasma half life, volume of distribution (V_ss) and total body clearance of inulin in the healthy males, healthy females and patients with chronic renal failure were 73.2, 65.5 and 172.4 min, 10.5, 9.6 and 8.81·70 kg^–1 and 113.3, 111.5 and 43.3 ml·min^–1·70 kg^–1 respectively. There were no sex differences in any of the kinetic variables.The mechanism of the concentration-dependent clearance of inulin is unknown but the findings are consistent with saturable renal tubular reabsorption. Care is required with the use of inulin for measurement of the glomerular filtration rate by the single injection technique.     

Pharmacokinetics of nortriptyline in man after single and multiple oral doses: The predictability of steady-state plasma concentrations from single-dose plasma-level data

Summary Six healthy volunteers were given nortriptyline (NT) in single (1 mg/kg) and multiple (0.4 mg/kg) oral doses with the purpose to study the pharmacokinetics of NT in plasma. The single-dose plasma levels in 3 subjects were applied to a two-compartment open model (Model II) which might be suitable for the disposition kinetics of NT. There were significant (P<0.0005) interindividual differences in the observed mean steady-state plasma concentration,C-0304;, of NT which ranged from 51.7 to 111.0 ng/ml.C-0304; was better correlated to the single-dose plasma clearance rate of NT (r=0.98;P<0.0005) than to the single-dose plasma half-life, (t 1/2), (r=0.89;P<0.05). Thus the steady-state plasma level can be accurately predicted from single-dose studies. The apparent volume of distribution of NT at pseudodistribution equilibrium, (V _d), varied between 21.1 and 31.1 lit./kg body weight assuming complete absorption of the administered dose. There was no significant correlation (r=0.56) between the single-dose and post steady-state plasma half-lives of NT within individuals. NT does not seem to affect its own metabolismin vivo in man since the single- and multiple-dose plasma clearance rates of the drug were similar within subjects. It is concluded that there might be interindividual differences in both the apparent elimination rate and in the distribution of NT. This may be of importance in the pharmacogenetics of NT.     

Clinical pharmacokinetics of verapamil in patients with atrial fibrillation

Summary The pharmacokinetic parameters and oral bioavailability of the antiarrhythmic drug verapamil were determined in six patients with atrial fibrillation. Plasma samples were taken following i.v. injection of verapamil 10 mg (Isoptin^® 2 ml), and oral verapamil 80 mg (Isoptin^® 2 tablets of 40 mg). Verapamil and its N-demethylated metabolite, norverapamil, were analyzed to 1 ng/ml plasma by gas chromatography-mass spectrometry using deuterated standards. Following intravenous injection, the disposition of verapamil followed a biexponential pattern with a fast distribution phase and a slower elimination phase (t_1/2=5.79 h), corresponding to a plasma clearance of 0.26 1/kg/h. After oral administration, only an elimination phase was evident, with the same elimination rate (t_1/2=5.53 h). The oral bioavailability was 10.5%±7.5%. The norverapamil formed after i.v. and oral administration of verapamil had plasma half-lives of 5.86 h and 6.77 h, respectively. The elimination of verapamil in patients with atrial fibrillation was decreased compared to that in healthy young volunteers and the oral bioavailability was lower. Very good correlation between the percentage reduction in heart rate and the log plasma concentration of verapamil was found in every patient during the elimination phase, irrespective of the route of administration. There was also a high correlation when the plasma concentration — effect data from all the patients were pooled (r=0.59,n=71;p<0.0005).     

The pharmacokinetic properties of yohimbine in the conscious rat

Summary We used high performance liquid chromatography with fluorescence detection to measure the concentration of yohimbine in serum and brain of conscious Sprague-Dawley rats at various times after the i.v. injection of 1 mg/kg of yohimbine. The serum concentration-time profile of yohimbine was biphasic with a rapid distribution phase (t _1/2 = 0.048 h) followed by a very slow elimination phase (t _1/2 = 16.3 h). The clearance of yohimbine was 11 ml/h·kg^–1, and the volume of distribution was 259 ml/ kg. Increasing doses (0.3, 1 and 3 mg/kg, i.v.) of yohimbine produced non-linear increases in serum yohimbine concentration. Yohimbine entered the brain rapidly (5,000 ng/g at 5 min after 1 mg/kg, i.v.) and disappeared from brain with a t _1/2 of 7.7 h. In contrast to serum yohimbine concentration, increasing doses of yohimbine (0.3, 1 and 3 mg/kg) produced linear increases in brain yohimbine concentration, a phenomenon which is consistent with concentration-dependent binding of yohimbine to plasma proteins. The rapid entry of yohimbine into the brain, the slow rate of elimination of yohimbine from serum and brain and the linear relationship of brain yohimbine concentration as a function of dose should be taken into consideration whenever yohimbine is to be used as a probe of ₂-adrenoceptor function in vivo. Send offprint requests to T. Kent Keeton at the above address     

Modeling of drug response in individual subjects

Pharmacokinetic and drug response data from individual subjects are analyzed empirically by two different mathematical techniques. The drugs involved are the antiarrhythmic agent disopyramide, whose kinetics can be described by a two-compartment model, and two cardiac glycosides, digoxin and -methyl digoxin, for which three-compartment models are appropriate. The first analytical approach uses multiple linear regression to describe response in terms of the amount of drug in several kinetic compartments. The second approach describes response in terms of the drug concentration in an effect compartment. Both approaches describe the data equally well and require the same number of parameters for model specification.     

Pharmacokinetic Analysis of Mizolastine in Healthy Young Volunteers After Single Oral and Intravenous Doses: Noncompartmental Approach and Compartmental Modeling

This paper presents the analysis of the kinetics of a new antihistamine, mizolastine, in 18 healthy volunteers, from concentrations measured after an intravenous infusion and two different oral administrations: tablet and capsule. Two approaches were used to analyze these data: (i) a noncompartmental approach implemented in PHARM-NCA: (ii) a compartmental modeling approach implemented in a new S-PLUS library. NLS2, ⁵ which allows the estimation of variance parameters simultaneously with the kinetic parameters. For the compartmental modeling approach, two-compartment open models were used. According to the Akaike criterion, the best model describing the kinetics of mizolastine after oral administration was the zero-order absorption model. The kinetic parameters obtained with PHARM-NCA and NLS2 were similar. The estimated duration of absorption was greater for the tablets than for the capsules (with means equal to 1.13 hr and 0.84 hr respectively). After an intravenous infusion, the mean estimated clearance was 4.9 L/hr, the mean ₂ -phase apparent volume of distribution was 89.6 L and the mean terminal half-life was 12.9 hr.     

Serum glycoside concentrations after single or repeated intravenous doses ofβ-methyl-digoxin and digoxin

Summary The aim of the present investigation was to estimate the ratio of the intravenous doses of-methyl-digoxin and digoxin required to produce identical serum glycoside concentrations in man.20 patients on intravenous maintenance therapy were changed from-methyl-digoxin to the identical dose of digoxin or vice versa. Each drug was given for 7 days. Serum concentrations 13% higher were found during administration of-methyl-digoxin. Assuming a half life of 60 h after with drawal, the dose of digoxin producing the same minimum serum concentration was estimated to be 1.16 times higher than that of-methyl-digoxin.18 healthy volunteers received 0.4 mg -methyldigoxin, and 23 the same dose of digoxin, as an intravenous infusion over 2 h. The serum concentrations and urinary glycoside excretion were measured over a period of 32 hrs. During the first hour after the infusion the serum concentration of digoxin declined more rapidly than that of-methyl-digoxin. Thereafter, the ratio of the serum concentrations did not change appreciably up to the end of the investigation. The area under the serum concentration/time curve was about 13% greater for-methyl-digoxin than for digoxin; this difference was not significant.The average renal clearance was 96±9 ml for-methyl-digoxin, 151±13 ml for digoxin. Since the total body clearance of digoxin is only about 1.16 times higher than that of-methyl-digoxin, the lower renal clearance of-methyl-digoxin must partly be compensated by higher extrarenal clearance.From the ratios of the areas under the serum concentration/time curves after single doses of -methyldigoxin and digoxin, and the minimum serum concentrations during maintenance therapy, it was concluded that the dose of digoxin to produce the same average serum concentrations would be about 1.15 times higher than that of-methyl-digoxin. In comparison with the large variations in individual dosage of digoxin and-methyl-digoxin, this difference is too small to be of practical importance.     

Dynamics of drug distribution. I. Role of the second and third curve moments

Conventionally, the dynamics of distribution in the body is evaluated by the so-called distribution half-life (e.g., t _1/2, but then the mean time of the distribution process is underestimated due tothe influence of elimination. By contrast, information about the dynamics of distribution contained in drug disposition curves can be extracted by the second and third curve moments, parameters that are related to the variance (VDRT)and skewness (SDRT)of residence time distributions; whereas the equilibrium state characterized by the volume of distribution (V_ss), isdetermined by the mean residence time (MDRT)or the first curve moment. The approach represents a general noncompartmental analysis that is independent of a detailed structural model or a particular disposition function. Two parameters are introduced to characterize the dynamics of drug distribution: (i)the degree of departure of the system from well-mixed behavior of instantaneous distribution equilibrium (related to VDRT)and (ii)the mean time until equilibration is achieved (mean equilibration time, MEQT),which additionally depends on SDRT.Both parameters are quantitative measures of the dynamics of distribution and display explicit physical significance in terms of distribution within the corresponding noneliminating system. It is further shown that the so-called distribution phase in biexponential disposition curves is related to a monoexponential mixing curve of its corresponding noneliminating system with an equilibration or mixing half-time, t _1/2,M =t _1/2, (V/V _ss ^* ), where V _ss ^* denotes the distribution volume of the noneliminating system. The results are applied to mixing and disposition curves measured for acetaminophen in liver-ligated and intact rats, respectively.This work was supported by the Medical Research Council of Canada. M. Weiss was a Visiting Rosenstadt Professor at the University of Toronto. K. S. Pang is a recipient of the Faculty Development Award, Medical Research Council of Canada.     

Hepatic elimination of drugs with concentration-dependent serum protein binding

For a drug with concentration-dependent serum protein binding, the unbound fraction of drug decreases during the drug elimination process. The clearance of the drug at a given blood flow rate is lower than would be expected from the observed unbound fraction in venous blood from a noneliminating organ. Based on both the well-stirred and parallel tube models, simulations demonstrated that consideration of concentration-dependent binding during drug elimination is important when the intrinsic clearance is higher than the blood flow and when the unbound drug concentration is much greater than the dissociation equilibrium constant of the binding complex.Supported in part by Grant GM 28423 from the National Institutes of General Medical Sciences, NIH.