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.