Theorems and implications of a model-independent elimination/distribution function decomposition of linear and some nonlinear drug dispositions. III. Peripheral bioavailability and distribution time concepts applied to the evaluation of distribution kinetics

On the determination of the time delay in reaching the steady state drug concentration in the organ compared to plasma

A problem of substantial delay in reaching the steady state drug concentration in particular organ (compartment) compared to the time of reaching the steady state plasma concentration is considered. It is shown that the ratio of the terminal (V(beta)) and the steady state (V(ss)) volumes of distribution, V(beta)/V(ss), appears to be an indication of possible lag in reaching the steady state in the organ tissue compared to plasma. The estimations of the time of reaching the steady state drug concentration in the organ are suggested. The in vivo based pharmacokinetic model, which uses the experimentally measured drug plasma concentration time course and the appropriate equation for the kinetics of drug distribution into the tissues, is suggested. It is intended to determine the kinetic mechanism of drug distribution into the tissues. The model was applied to interpret the kinetics of drug distribution into the brain. The importance of precise measurement of drug plasma concentration at terminal phase for obtaining accurate values of V(beta) and V(ss) is emphasized: this allows predicting a possible slow plasma-tissue drug transfer and substantial difference in time of reaching the steady state by the body and plasma.

The Connection Between the Steady State (Vss) and Terminal (Vβ) Volumes of Distribution in Linear Pharmacokinetics and The General Proof That Vβ ≥ Vss

The steady state and terminal (area) volumes of distribution are important pharmacokinetic parameters defined as the ratio of the total quantity of drug in the body, A(b)(t), to drug plasma concentration C(p)(t) at steady state and the terminal phase of drug elimination, respectively. The general equations for the approach of C(p)(t), A(b)(t) and the distribution volume A(b)(t)/C(p)(t) to the steady state values (for a continuous constant rate drug infusion) are derived. It is shown that the time course of A(b)(t) near the asymptotic steady state value depends on both the terminal and steady state volumes of distribution, and an accurate equation to determine the time required to reach the steady state is obtained. For a general linear pharmacokinetic system (i.e., with possible drug elimination at any state from any compartment and drug exchange between compartments) it is proven that V(beta) >/= V(ss). A physiologically determined feature, which is the drug input into plasma for reaching the steady state or terminal phase, underlies the proof. If the steady state is reached by a continuous input of drug into some compartment other than plasma, and the terminal volume of distribution is considered after dosing of a drug in the same compartment, then both cases V(ss) <> V(beta) are possible. It is shown that the general exponential series for C(p)(t) after intravenous bolus dose may have negative pre-exponents, unlike a common assumption that all pre-exponents should be positive. Its is figured out that the commonly used equations for the estimation of V(ss) and V(beta) (V(ss) = D x AUMC/AUC(2) and V(beta) = D/(AUC x beta) may yield V(ss) > V(beta) for a linear pharmacokinetic system, contrary to the usual statement (V(ss) < V(beta)) and its seemingly simple proof, which has a flaw. It is shown that the time required to reach the steady state concentration of drug in plasma could be much shorter than a commonly used estimation of 5t(1/2), where t(1/2) is the terminal half-life obtained from the intravenous bolus drug plasma concentration time course.

Change in erythropoietin pharmacokinetics following hematopoietic transplantation

Pre-clinical studies have demonstrated that bone marrow ablation has a profound effect in decreasing erythropoietin (EPO) elimination. The study's objective was to determine in humans if EPO pharmacokinetics (PKs) are perturbed following bone marrow ablation. EPO PK studies were performed in eight subjects, aged 4 to 61 years, undergoing fully myeloablative hematopoietic stem cell transplantation. Serial PK studies using intravenous injection of recombinant human EPO (92+/-2.0 U/kg) (mean+/-SEM) were carried out during four periods of altered marrow integrity: baseline pre-ablation, post-ablation pre-transplant, early post-transplant pre-engraftment, and late post-transplant full engraftment. Compared with baseline, post-ablation pre-transplant and early post-transplant EPO PKs demonstrated declines in clearance increases in terminal elimination half-life of 36 and 95%, respectively. Clearance and half-life returned to baseline following full engraftment. The association of EPO elimination with decreased bone marrow activity in patients undergoing transplantation conclusively establishes the bone marrow as a key determinant of EPO elimination in humans.

A Convenient Method for Estimating the Quantity of Drug Eliminated by the Routes Other than Hepatic Metabolism and Renal Excretion and the Fraction of Drug that Reaches the “First Pass” after Oral Administration

The comparison of routine pharmacokinetic data obtained after intravenous and oral drug administration allows to figure out that some quantity of drug, which reached the systemic circulation, was eliminated from the body by routes other than hepatic metabolism and renal excretion. This quantity is equal or exceeds the certain minimum value, which can be calculated from a simple equation obtained in the article. If the minimum value is equal to zero, then the maximum possible fraction of orally administered drug, that is, absorbed into the gut wall and gets through it unchanged, can be calculated. The examples considered indicate that the quantity of drug eliminated not by liver metabolism or kidney excretion could be quite substantial (exceeds half of the dose that reached the circulation).

The influence of drug kinetics in blood on the calculation of oral bioavailability in linear pharmacokinetics: The traditional equation may considerably overestimate the true value

A common calculation of oral bioavailability is based on the comparison of the areas under the concentration-time curves after intravenous and oral drug administration. It does not take into account that after the oral dosing a drug enters the systemic circulation in different states, that is, as free fraction, protein bound and partitioned into blood cells, and plasma lipids, while after intravenous input it is introduced into the systemic circulation only as a free fraction. Consideration of this difference leads to a novel equation for the oral bioavailability. In general, the traditional calculation overestimates the oral bioavailability. For a widely applied model of a linear pharmacokinetic system with central (plasma) drug elimination it is shown that the traditional calculation of the oral bioavailability could substantially overestimate the true value. If the existence of an immediate equilibrium between different drug fractions in blood is assumed, the obtained equation becomes identical to the traditional one. Thus the deviation of oral bioavailability from the value given by a common calculation appears to be a kinetic phenomenon. The difference could be significant for the drugs with the rate constant of elimination from plasma of the same order of magnitude or greater than the dissociation rate constant of drug-protein complexes, or the off-rate constant of partitioning from the blood cells, if the blood concentration profiles were used to calculate the oral bioavailability.

Volume of Distribution at Steady State for a Linear Pharmacokinetic System with Peripheral Elimination

The problem of finding the steady-state volume of distribution V(ss) for a linear pharmacokinetic system with peripheral drug elimination is considered. A commonly used equation V(ss) = (D/AUC)*MRT is applicable only for the systems with central (plasma) drug elimination. The following equation, V(ss) = (D/AUC)*MRT(int), was obtained, where AUC is the commonly calculated area under the time curve of the total drug concentration in plasma after intravenous (iv) administration of bolus drug dose, D, and MRT(int) is the intrinsic mean residence time, which is the average time the drug spends in the body (system) after entering the systemic circulation (plasma). The value of MRT(int) cannot be found from a drug plasma concentration profile after an iv bolus drug input if a peripheral drug exit occurs. The obtained equation does not contain the assumption of an immediate equilibrium of protein and tissue binding in plasma and organs, and thus incorporates the rates of all possible reactions. If drug exits the system only through central compartment (plasma) and there is an instant equilibrium between bound and unbound drug fractions in plasma, then MRT(int) becomes equal to MRT = AUMC/AUC, which is calculated using the time course of the total drug concentration in plasma after an iv bolus injection. Thus, the obtained equation coincides with the traditional one, V(ss) = (D/AUC)*MRT, if the assumptions for validity of this equation are met. Experimental methods for determining the steady-state volume of distribution and MRT(int), as well as the problem of determining whether peripheral drug elimination occurs, are considered. The equation for calculation of the tissue-plasma partition coefficient with the account of peripheral elimination is obtained. The difference between traditionally calculated V(ss) = (D/AUC)*MRT and the true value given by (D/AUC)*MRT(int) is discussed.

Drug disposition analysis: a comparison between budesonide and fluticasone

The characterisation of distribution and elimination properties of a drug is usually done using parameters like clearance and distributional volumes. To refine this characterisation, in this paper, we use drug disposition analysis to compare the distribution and elimination of the two glucocorticosteroids budesonide and fluticasone propionate, known to differ in this respect. This gives a more detailed description of the well known differences in distributional volumes using concepts like mean residence time and fraction of dose outside the central compartment. It clearly shows that fluticasone, although having lower plasma concentrations, still resides in the body in appreciable quantities.

Erythropoietin production rate in phlebotomy-induced acute anemia

OBJECTIVE

To estimate the rate of erythropoietin (EPO) production under physiological, conditions and to examine the regulatory mechanism of EPO production in response to acute phlebotomy-induced anemia.

METHODS

Six sheep each underwent two phlebotomies in which the hemoglobin (Hb) was reduced to 3-4 g/dl over 4-5 h. The EPO plasma level, reticulocytes, Hb and EPO clearance were followed by frequent blood sampling. The EPO production rate was determined by a semi-parametric method based on a disposition decomposition analysis that accounts for the nonlinear disposition kinetics of EPO and corrects for time-dependent changes in the clearance.

RESULTS

The controlled drop in hemoglobin resulted in an abrupt increase in the plasma EPO concentration (peak level 812+/-40 mU/ml, mean+/-CV%) that was followed by a rapid drop 2-4 days after the phlebotomy at a time when the sheep were still anemic (Hb=4.3+/-16 g/dl). The EPO production rate at baseline was 43+/-52 U/day/kg and the amounts of EPO produced over an 8 day period resulting from the first and second phlebotomy were 2927+/-40 U/kg and 3012+/-31 U/kg, respectively.

CONCLUSIONS

The rapid reduction in the EPO plasma level observed 2-4 days following the phlebotomy cannot be explained solely by the increase in EPO clearance but also by a reduction in EPO production.

Receptor-based model accounts for phlebotomy-induced changes in erythropoietin pharmacokinetics

Previous clinical studies have demonstrated two distinctive pharmacokinetic behaviors of erythropoietin (EPO): changes in pharmacokinetics (PK) after a period of rhEPO treatment and nonlinear pharmacokinetics. The objective of this work was to study the temporal changes in EPO's PK following phlebotomy in order to propose possible mechanisms for this behavior. Five healthy adult sheep were phlebotomized on two separate occasions 4-6 weeks apart to hemoglobin levels of PK 3-4 g/dL. PK parameters were estimated from the concentration-time profiles obtained following repeated intravenous bolus PK studies using tracer doses of biologically active 125I-rhEPO. Based on the changes in clearances, a PK model was derived to provide a mechanistic receptor-based description of the observed phenomena. Phlebotomy resulted in a rapid increase in the EPO plasma concentration, which peaked at 760 +/- 430 mU/mL (mean +/- SD) at 1.8 +/- 0.65 days, and which coincided with a transient reduction in EPO clearance from prephlebotomy values, i.e., from 45.6 +/- 11.2 mL/hr/kg to 24.3 +/- 9.7 mL/hr/kg. As plasma EPO levels returned toward baseline levels in the next few days, a subsequent increase in EPO clearance was noted. EPO clearance peaked at 90.2 +/- 26.2 mL/hr/kg at 8.5 +/- 3.3 days and returned to baseline by 4-5 weeks postphlebotomy. The proposed model derived from these data includes positive feedback control of the EPO receptor (EPOR) pool. The model predicts that: 1) the initial reduction in EPO plasma clearance is due to a transient saturation of EPORs resulting from the phlebotomy-induced high EPO concentration; and 2) the EPOR pool is expandable not only to compensate for EPOR loss but also to adjust to a greater need for EPORs/progenitor cells to restore hemoglobin (Hb) concentration to normal levels.

A comparison of nonlinear pharmacokinetics of erythropoietin in sheep and humans

The primary mechanism of erythropoietin's (EPO) in vivo elimination and the tissue, or tissues, responsible are unknown. Previous studies indicating that EPO pharmacokinetic (PK) behaviour is nonlinear suggest that EPO elimination takes place by a saturable mechanism. A versatile PK system analysis, the Disposition Decomposition Analysis (DDA), capable of quantification of the Michaelis-Menten parameters, V(m) and k(m) was used to analyze and compare EPO's PK behaviour in newborn sheep and preterm infants. Lambs and infants both demonstrated nonlinear PK behaviour appropriately analyzed with DDA. Compared to preterm infants, lambs had significantly greater (p<0.05) elimination capacity as determined by the V(m) (2789+/-525 versus 1767+/-250 mU/mL per h (mean+/-S.E.), respectively), and larger extrapolated linear clearances (116+/-19.1 versus 21.3+/-1.75 mL/kg per h, respectively) (p<0.01). Lambs also demonstrated significantly larger (p<0.01) degrees of nonlinearity as judged by smaller mean k(m) values (2142+/-258 versus 6796+/-1.007 mU/mL, respectively). Of note, although the DDA does not distinguish what the mechanism of EPO elimination is, enzymatic degradation and receptor-mediated cellular internalization are two possibilities. The in vivo DDA-derived k(m) values were similar to reported in vitro binding affinity k(d) data for erythroid progenitors and cell lines having EPO-R's, i.e. 240-2400 mU/mL. The present study's demonstration that EPO's nonlinear PK behaviour in both sheep and humans can be analyzed by the DDA methodology indicates that the sheep model may be used in invasive studies needed to further characterize the mechanism of EPO elimination.

Application of a system analysis approach to population pharmacokinetics and pharmacodynamics of nicardipine hydrochloride in healthy males

Nicardipine hydrochloride, a dihydropyridine calcium channel blocker, possesses antihypertensive and arterial vasodilator properties. A system analysis approach, which makes fewer structural assumptions than compartmental methods, is presented for determining the pharmacokinetics and pharmacodynamics of nicardipine hydrochloride in healthy males following a discontinuous infusion at four dose levels. The results indicate that the average total body clearance of nicardipine is 0.920 L/h/kg and the volume of distribution is 0.275 L/kg. Nicardipine hydrochloride has a mean residence time in the body of 1.27 h, of which 0.324 h were spent in the systemic circulation and the remainder in the periphery. The determined pharmacokinetic model was linked to a pharmacodynamic model that allowed the change in the mean arterial blood pressure and heart rate to be described and predicted. A population pharmacokinetic-pharmacodynamic model was derived and the predictive power of the proposed model was assessed with a cross-validation technique that employs a relative predictive quotient for comparing the predictions to the fitted model. The results indicate that the proposed model describes the pharmacodynamics of nicardipine in healthy males and has good predictive ability when tested with a cross-validation procedure.

Exact dosing times calculations in linear pharmacokinetics

The problem of determining the particular dosing time, tau, that results in a certain ratio between peak and trough drug levels at steady state (ss) is addressed. Calculation of tau in combination with simple dose linearity principles ensures constraint on the drug level variations at ss, contrary to calculations based on only clearance principles. It is shown that the dosing time problem is solved using a nested, single-variable rootsolving. Using a derivative-free, robust, single-variable rootsolver enables automatic, reliable calculations of tau. An algorithm and computer program, SSTATE, for the automatic calculation of tau in intra- and extravascular dosings are presented. Contrary to various approximation formulae proposed in the past, SSTATE provides solutions that are exact. SSTATE provides additional ss parameters such as time to peak, and maximum, minimum, and mean levels, and can also be executed in a simulation mode to explore various practical dosage regimen schemes. Dosage regimen calculations for quinidine are presented to demonstrate the practical utility of the proposed approach. It is also demonstrated by simulation studies that some approximation formulae previously proposed may produce excessive errors, especially when applied to extravascular dosings.

Optimal extravascular dosing intervals

An explicit formula is presented for simple calculations of the dosing time, tau, that results in a steady-state peak-to-trough ratio of 2 in extravascular dosings. Contrary to other formulae presented, the calculations are guaranteed to be well bound in the percentage error (less than 1%) for any parameter value combination. It is shown that the biexponential dosing interval problem can be transformed into a general, dimensionless problem enabling a global error analysis in the approximation. The proposed formula is demonstrated in the calculation of an "optimal" dosing interval for quinidine. An algorithm and FORTRAN computer program OPTAU for exact calculation of tau and dosing simulations is also demonstrated in the quinidine example.

Stochastic interpretation of linear pharmacokinetics: a linear system analysis approach

Linear drug disposition is most generally defined in terms of the superposition principle. This principle is explained on the molecular level by probability principles involving stochastic, independent kinetic behavior of drug molecules. A stochastic modeling approach is presented that is more general than pharmacokinetic models typically employed in stochastic approaches. First-order microscopic transfer rate constants (Kij) are not employed or assumed in the analysis. The approach is a linear system analysis approach that makes use of the simplest possible kinetic structure that enables a differentiation of the drug disposition into elimination and distribution components. This is done by applying stochastic principles in the context of the disposition decomposition analysis (DDA). The DDA approach in its linear form is a generalization of linear pharmacokinetic systems that assume a homogeneous sampling space. Disposition kinetics is partitioned into two kinetic spaces, a homogeneous sampling space, and a heterogeneous peripheral kinetic space. A structure differentiation beyond this is difficult to justify in common situations when only the parent drug is determined from a single iv sampling site. A stochastic independent molecule (SIM) model is formulated in the structure context of DDA. The model is employed to identify core relationships by isolating elementary stochastic building blocks of the disposition kinetics and absorption kinetics. It is shown how the stochastic building blocks of the SIM-DDA model are related to various mean time parameters. Residence probability functions and drug delivery probability functions provided by the approach appear useful for extending kinetic bioavailability concepts into a purely stochastic realm. The emphasis on transit time concepts enables a kinetic differentiation and a more intrinsic characterization than possible by the use of common residence time principles. Relationships are presented that link stochastic and kinetic elements. Formulas are presented for the practical calculations of the mean time parameters and stochastic functions presented. Practical examples are given of the concepts presented using data from several drugs.

Mean time parameters in pharmacokinetics. Definition, computation and clinical implications (Part II)

Part I of this article, which appeared in the previous issue of the Journal, covered the following topics: fundamental definitions, general mean time parameter relationships and mean time parameters of classical compartmental systems. It also offered a number of examples to clarify and illustrate the various concepts.

Pharmacokinetics of low-dose methotrexate in rheumatoid arthritis patients

The pharmacokinetics and bioavailability of low-dose methotrexate (MTX) (10 mg/m2) were evaluated in 41 subjects who had definite or classical rheumatoid arthritis as defined by the American Rheumatism Association criteria. Subjects received 10 mg/m2 (to the nearest 2.5 mg) of MTX in a single oral dose and a single intravenous (iv) dose one week apart. Serum concentrations for this low-dose regimen were monitored using a radiochemical ligand binding assay. The results indicate the MTX is cleared from the plasma at a rate of 84.6 mL/min/m2. The terminal half-life was approximately 6 h. The volumes of distribution at steady state and for the central compartment were 22.2 and 13.5 L/m2, respectively. The mean residence time in the body, in the systemic circulation, and in the periphery were estimated to be 4.7, 3.0, and 1.7 h, respectively, with a peripheral single-pass mean transit time of 6.0 h and an intrinsic mean residence time in the periphery of 7.9 h. The mean absorption time was 1.2 h and the oral bioavailability was 0.70. The ratio of synovial fluid concentration to serum concentration 4 and 24 h after a dose was found to be approximately 1.0, indicating that at least within that time range serum and synovial fluid concentrations are approximately equal. Because of conflicting results and insufficient data from previous high-dose pharmacokinetic studies, it is difficult to say whether or not low-dose MTX pharmacokinetics differs from those of high-dose MTX.

Linear and nonlinear system approaches in pharmacokinetics: how much do they have to offer? I. General considerations

System approaches in pharmacokinetics are defined as generalizing and simplifying modeling approaches that mathematically model a general property of the pharmacokinetic system without modeling specifically the individual kinetic processes responsible for the general property considered. The rationale for the use of system approaches is discussed and the kinetic basis of some of the approaches is presented. An overview of the approaches is presented together with a comparison to classical approaches involving specific pharmacokinetic models. Examples are given from different application areas involving problems in linear and nonlinear pharmacokinetics and in pharmacodynamics. The advantages, disadvantages, and limitations of the system approaches are discussed. In several application areas the system approach offers some rational methods and procedures with distinct advantages over more traditional approaches.

System approaches in pharmacokinetics: II. Applications

Pharmacokinetic system approaches mathematically describe a general property of a pharmacokinetic system without modeling in specific terms the kinetic processes responsible for the general property. Certain applications of the system approach including methods for evaluating drug delivery, drug distribution, drug secretion, and biotransformation in linear and nonlinear pharmacokinetics are presented and discussed. Linear system formulae for various mean time disposition parameters and a disposition decomposition-recomposition system approach for predicting drug levels when the drug clearance changes are presented and discussed. System approaches offer certain advantages over traditional approaches for many practical applications.

System approaches in pharmacokinetics: I. Basic concepts

System approaches mathematically describe a general property of a pharmacokinetic system without modeling in specific terms the kinetic processes responsible for the general property considered. Definitions, basic concepts, kinetic basis, and rationale for system approaches as well as advantages and disadvantages of system approaches are presented and discussed. It appears that the system approaches may offer certain advantages over more traditional methods for many practical applications.