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

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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.