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

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Abstract

The problem of finding the steady-state volume of distribution Vss for a linear pharmacokinetic system with peripheral drug elimination is considered. A commonly used equation Vss = (D/AUC)*MRT is applicable only for the systems with central (plasma) drug elimination. The following equation, Vss = (D/AUC)*MRTint, 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 MRTint 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 MRTint 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 MRTint 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, Vss = (D/AUC)*MRT, if the assumptions for validity of this equation are met. Experimental methods for determining the steady-state volume of distribution and MRTint, 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 Vss = (D/AUC)*MRT and the true value given by (D/AUC)*MRTint is discussed. © 2004 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 93:1628–1640, 2004

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INTRODUCTION

The traditional equation for calculation of volume of distribution at steady state Vss from the time course of the total drug concentration in plasma after an intravenous (iv) bolus administration of dose, D, Vss = (D/AUC)*MRT has certain limitations: it is based on the assumption of drug elimination directly from plasma (or central compartment), where a drug was introduced. It also includes the assumption of an immediate equilibrium between bound and unbound drug fractions in plasma. The goal

EQUATION FOR THE STEADY-STATE VOLUME OF DISTRIBUTION WITH THE ACCOUNT OF PERIPHERAL ELIMINATION AND THE KINETICS OF THE WHOLE SYSTEM

Let us consider a linear pharmacokinetic system with drug input into the central compartment (plasma) and possible exit from any compartment. Such a system is characterized by a set of concentrations Ci = Ai/Vi, which correspond to different units of volumes Vi, where the drug is distributed, and containing the quantities of drug Ai. Vi is not necessarily the volume of the whole compartment (or body organ), but a volume of the phase with concentration Ci inside the compartment. For instance, it

EXPERIMENTAL DETERMINATION OF THE STEADY-STATE VOLUME OF DISTRIBUTION

A straightforward way to find Vss is to reach the steady state by subjecting the animal to a constant rate Ro iv drug infusion. The time required to reach the steady state, according to eq. 22, is determined by the smallest exponent λj, which is actually the terminal log-linear slope parameter β. During the interval of about five terminal half-lives (t1/2 = ln2/β), 97% level of the steady-state concentration values will be reached. After the analysis of plasma concentration Cp,ss and the total

DETERMINATION OF THE EXISTENCE OF PERIPHERAL ELIMINATION FROM PHARMACOKINETIC DATA

The time course of plasma concentration or the total quantity of drug in the body does not provide information about the exit routes of a drug. The calculated values of pharmacokinetic parameters are often interpreted as if the system eliminates a drug only from central compartment (plasma) eventually lumping together peripheral and plasma elimination. Let us consider, for instance, the system that exhibits both central and peripheral drug exit. The rate of plasma elimination, CluCu(t), is

CALCULATION OF THE PARTITION COEFFICIENT WITH THE ACCOUNT OF PERIPHERAL ELIMINATION

To calculate the steady-state volume of distribution from the physicochemical properties of drug and tissues using eq. 2, we need to obtain the plasma–tissue partition coefficient Pt-p for each organ.11,12 Let us consider a commonly used open mammillary model (Fig. 1), which considers the pharmacokinetic system as a central compartment (plasma) from which the drug can be eliminated or reversibly transferred to peripheral compartments, and each rate is the first order. The rate of possible

DISCUSSION

The obtained equation Vss = (D/AUC)*MRTint for determination of the steady-state volume of distribution from the time course of the total drug concentration in plasma after an iv bolus administration appears to be general for linear pharmacokinetics. It is based only on the assumption of linearity of the system and impulse drug input (as initial condition), and does not contain any special limitations on drug elimination routes and rates of possible reactions. The term MRTint cannot be

CONCLUSIONS

The equation for calculation of volume of distribution at steady state Vss with complete consideration of peripheral elimination and protein and tissue binding kinetics is obtained. Traditional equations for Vss = (D/AUC)*MRT and mean resident time MRT = AUMC/AUC are applicable only for the systems with central (plasma) elimination and an instant equilibrium between unbound and bound drug fractions in plasma. Experimental determination of Vss based on the obtained equations is considered. It is

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