Tracer measurements of non-equilibrium volumes of distribution

According to conventional theory the product of the transport flowrate and the mean transit time of a tracer through a system yields the equilibrium volume of distribution for the tracer, regardless of tracer kinetics or space geometry. Experimental results do not support this notion. The influence of measurement time on the volume measured with a bolus technique is addressed using systems theory to analyze a tissue-impedance form of the Sangren-Sheppard model. Assymptotic solutions show that the volume estimates are governed by a time constant, tau, related to diffusion in the tissue, to tissue capacity, and to wall permeability, and by a dimensionless ratio, f, describing a relation of tau to vascular transport time. A third parameter, g, describing the relative contributions to overall resistance to diffusion of effective permeability and of limited diffusivity in the tissue, is shown to be of less importance. The derived tau is similar to but not equivalent to the often cited "characteristic time". The "equilibrium" volume of distribution is defined as that which would be measured if equilibrium were allowed to establish. The "non-equilibrium" volume of distribution is defined as that which would be measured given finite times and is shown to approach the "equilibrium" volume as such times increase. Tracer equilibration is not required to accurately measure the "equilibrium" volume. When there is no flow limitation (f much less than 1) a measurement time of tau (plus vascular transit time) would yield a "non-equilibrium" volume only 33% of the "equilibrium" volume; a time of 2 tau would yield 55%; a time of 10 tau would yield effectively the total equilibrium volume. Finite diffusivity in tissue and permeability restrictions can have significant effects on the proportion of the volume measured.