The non-compartmental steady-state volume of distribution revisited

Abstract

The lack in the literature of a simple, yet general and complete derivation of the widely used equation for non-compartmental calculation of steady-state volume of distribution is pointed out. It is demonstrated that the most frequently cited references contain an overly simplified explanation. The logical gap consists in doubly defining the same quantities without a proof the definitions are equivalent. Two alternative solutions are proposed: analytical derivation and hydrodynamic analogy. It is shown, that the problem can be analyzed in a purely macroscopic framework by utilizing the integral mean value of the function, without the need to resort to statistical distributions.

Appendices

Appendix 1

Let $A_{e} \left( t \right)$ be an amount of drug eliminated up to time t from the system after a rapid intravenous injection of the dose D administered at t₀ = 0. If elimination occurs solely from the sampling space, then by the definition of clearance:

$$\frac{{dA_{e} \left( t \right)}}{dt} = CL \cdot C_{{{\text{iv}}}} \left( t \right)$$

or

$$dA_{e} \left( t \right) = CL \cdot C_{{{\text{iv}}}} \left( t \right)dt$$

In linear systems clearance is constant and positive, therefore $A_{e} \left( t \right)$ is a strictly increasing function of time. Because of that an inverse function, $t\left( {A_{e} } \right)$ exists. Now one can compute the integral average (or mean) value [21] of this function:

$$\bar{t} = \frac{{\int_{0}^{D} {t\left( {A_{e} } \right)dA_{e} } }}{{D - 0}} = \frac{{CL\int_{0}^{\infty } {t \cdot C_{{{\text{iv}}}} \left( t \right)dt} }}{D}$$

(5)

Recalling that $CL = {{D_{{{\text{iv}}}} } \mathord{\left/ {\vphantom {{D_{{{\text{iv}}}} } {AUC}}} \right. \kern-\nulldelimiterspace} {AUC}}$ and recognizing $\int_{0}^{\infty } {t \cdot C_{{{\text{iv}}}}(t) dt}$ as AUMC one obtains Eq. (3) again. Thus, in the macroscopic approach, one may define MRT simply as the integral mean function value of $t\left( {A_{e} } \right)$. There is no need to introduce any statistical distributions.

The first equality in Eq. (5) was probably introduced for the first time by Cutler [36]; since then it has appeared several times in the literature (e.g. in [19]), but, to our knowledge, it was always interpreted as the result of stochastic counting of molecules travelling through the system.

Appendix 2

Up to this point one need not know a specific form of $c\left( t \right)$. However, it seems to be necessary in a proof of the existence of MRT. It is known [37], that for any linear compartmental system, $c\left( t \right)$ is a linear combination of the finite number of terms having the following form:

$$t^{k} {\text{e}}^{ - \lambda t}$$

where k ≥ 0 is an integral number and λ is either a positive real number or a complex number with a positive real part. Alternatively, these terms with complex λ can also be paired so as to yield an item of the form $t^{k} e^{{ - \lambda_{r} t}} \sin \left( {\lambda_{i} t + \varphi } \right)$ in which all symbols represent real numbers and $\lambda_{r} > 0$. Each such item multiplied by t has a finite integral in the interval (0, ∞). To demonstrate this, note that, by comparison test for infinite integral [38]:

$$\left| {\int\limits_{0}^{\infty } {t^{k + 1} {\text{e}}^{{ - \lambda_{r} t}} \sin \left( {\lambda_{i} t + \varphi } \right)dt} } \right| \le \int\limits_{0}^{\infty } {\left| {t^{k + 1} {\text{e}}^{{ - \lambda_{r} t}} \sin \left( {\lambda_{i} t + \varphi } \right)} \right|} dt \le \int_{0}^{\infty } {t^{k + 1} {\text{e}}^{{ - \lambda_{r} t}} dt}$$

With a simple change of variables: $\tau = \lambda_{r} t$ and taking into account the definition and properties of Euler’s Γ function [39] one obtains:

$$\int\limits_{0}^{\infty } {t^{k + 1} } e^{{ - \lambda_{r} t}} dt = \frac{1}{{\lambda_{r}^{k + 2} }}\int\limits_{0}^{\infty } {\tau^{k + 1} e^{ - \tau } d\tau = \frac{{\Gamma \left( {k + 2} \right)}}{{\lambda_{r}^{k + 2} }}} = \frac{{\left( {k + 1} \right)!}}{{\lambda_{r}^{k + 2} }}$$

Therefore, the linear combinations of integrals that constitute AUMC and AUC are also finite, what implies the existence of finite MRT.