The Mathematical Basis of the Interpretation of Tracer Experiments in Closed Steady-State Systems

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Abstract
The basic equation for the mixing of radioactive isotopes is dR/dt = I grad a, where dR/dt is the flux of radioactivity; a, the relative specific activity; and I, the interfusion constant. The equation is applied to systems of compartments and some electrical analogies are presented. As special cases of the unconstrained systems treated in an earlier paper two constrained systems are discussed in which only a single radioactive isotope is required for the study of transport phenomena. One of these (mammillary system) consists of a central compartment communicating with a number of peripheral compartments. In the other (catenary system) the compartments are arranged chainwise. Equations are derived for the n‐compartment mammillary system relating the time dependent specific activities of the compartments to the exchange rates between the peripheral and central compartments and to their content of substance being exchanged. In analyzing the effect of lumping of a series of compartments into a single equivalent one the case where the exchange rates cluster about a mean is discussed in some detail. In the catenary system the greater mathematical difficulties are indicated including the fact that the results depend on the order of the compartments. It is pointed out that in mammalian physiological experiments observations confined to only one compartment such as the circulating blood are insufficient for a complete analysis.

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