Turnover Compartmentalization.
An Approach to Analysis of Whole-Body Retention Data
P.-E. E. BERGNER
With 2 Figures
Abstract
In the early days of biologic tracer analysis, whole-body retention curves were
frequently interpreted in terms of so-called rapidly and slowly exchanging compartments. Classic compartment analysis soon proved this approach incorrect;
but, in the following, a modified form of it is presented that can be given a precise
physico mathematical justification. The purely deductive approach is made
possible by the introduction of a specific stochastic concept: the turnover compartment. The theory is based on previously published ergodic relations for open,
heterogeneous systems.
Consider an open system: Every particle entering the system will, with
probability one, eventually leave it without loss of identity. To fix ideas let
the system be a cow, and the particles stable calcium ions. I have shown [1]
that for this kind of system a precise microphysical representation can be
constructed in terms of a stochastic language: The movement of each individual Ca 2 + through the cow is visualized as a time-homogeneous Markovian process. That is, the system (the beast) is considered as a set of "states"
on which is defined a transition probability function with appropriate
mathematical behavior (e.g. the function does not depend on the distribution of particles, i.e. the particles are statistically independent of each other).
By the steady-state distribution we mean the particle distribution that
results after there has been, for an infinitely long time, a COnstant input of
particles into the system. I have shown [2] that the steady-state distribution
is, for this type of system, a logically natural reference distribution, similar
to thermal equilibrium in statistical mechanics.
The retention probability P s (t) is the probability of a particle that
enters the system at the time zero still being in the system at the time t.
Problem: How is the transient time process {P s (t); te [0, oo)} related to
the steady-state distribution? It has been possible to demonstrate [3] that
with the mean taken with respect to the steady-state distribution. But, as
explained below, from a practical viewpoint, this result is not enough nor
satisfactory [4].
An Approach to Analysis of Whole-Body Retention Data
P.-E. E. BERGNER
With 2 Figures
Abstract
In the early days of biologic tracer analysis, whole-body retention curves were
frequently interpreted in terms of so-called rapidly and slowly exchanging compartments. Classic compartment analysis soon proved this approach incorrect;
but, in the following, a modified form of it is presented that can be given a precise
physico mathematical justification. The purely deductive approach is made
possible by the introduction of a specific stochastic concept: the turnover compartment. The theory is based on previously published ergodic relations for open,
heterogeneous systems.
Consider an open system: Every particle entering the system will, with
probability one, eventually leave it without loss of identity. To fix ideas let
the system be a cow, and the particles stable calcium ions. I have shown [1]
that for this kind of system a precise microphysical representation can be
constructed in terms of a stochastic language: The movement of each individual Ca 2 + through the cow is visualized as a time-homogeneous Markovian process. That is, the system (the beast) is considered as a set of "states"
on which is defined a transition probability function with appropriate
mathematical behavior (e.g. the function does not depend on the distribution of particles, i.e. the particles are statistically independent of each other).
By the steady-state distribution we mean the particle distribution that
results after there has been, for an infinitely long time, a COnstant input of
particles into the system. I have shown [2] that the steady-state distribution
is, for this type of system, a logically natural reference distribution, similar
to thermal equilibrium in statistical mechanics.
The retention probability P s (t) is the probability of a particle that
enters the system at the time zero still being in the system at the time t.
Problem: How is the transient time process {P s (t); te [0, oo)} related to
the steady-state distribution? It has been possible to demonstrate [3] that
with the mean taken with respect to the steady-state distribution. But, as
explained below, from a practical viewpoint, this result is not enough nor
satisfactory [4].