Sec. 15.3] THEORY OF TRACER METHODS 393 



distinguishable to the observer, their change in concentration is described 

 by the differential equation above. It is for this reason that tracers provide 

 a powerful tool for investigating rates of processes in dynamic systems. The 

 observed change in concentration of tagged molecules, or in some cases, the 

 labeling agent itself, is in each process directly related to the constant rate 

 of reaction for the untagged homologue, or more exactly, for the total tagged 

 plus untagged material. 



A great variety of first-order physical and chemical processes may be 

 described by identical mathematical expressions in which only the units 

 differ. These processes may include such apparently divergent phenomena 

 as the transfer of a labeled substance from one region or tissue to another, 

 chemical reactions in which a substance A is degraded or synthesized into a 

 series of substances A — ■> B — » C — > . . . , radioactive decay, absorption of 

 gamma rays, and combinations of these and other processes. For the sake 

 of generality, therefore, a labeling agent introduced into a system in one form 

 or another will be considered to be present in certain "phases" with the under- 

 standing that a phase may refer to a particular chemical or physical form, 

 an organ or certain tissue, or a specified volume with or without well-defined 

 geometrical form. The precise meaning and units of phase will, consequently, 

 depend upon the type of system and process under consideration. Expressed 

 in general terms, tracer techniques provide the means for identifying the 

 phases involved in the processes under consideration, determination of the 

 amounts of substance present in each phase, and the determination of the 

 rates of change of a substance from one phase to another. 



For a steady-state system containing only irreversible first-order reactions, 

 the concentration of a labeled substance as a function of time in any one phase 

 of the system can be represented by the polynomial 



x = aie klt + a%e k2t -}-••■ + a n e knt 



The parameters a,- and kj can be determined only from measurements of the 

 tagged molecules in the phase. The coefficients a ; - may take any positive 

 and negative values or may be zero. They represent the amplitude of the 

 separate terms when extrapolated back to zero time. For phases other than 

 the one in which the labeled substance is initially introduced, the coefficients 

 usually have the property that ai -f- a 2 + ■ • ■ + a n = 0. The unit of 

 dj may be microcuries or microcuries per gram when radioactive isotopes are 

 used, and atom per cent or atom per cent excess for stable isotopes. 



The parameters kj are negative or zero but not positive, since positive 

 values indicate a concentration increasing without limit. They represent 

 the fractional amounts of labeled substances entering (for dj positive) and 

 leaving (for a, negative) the phase per unit time. 



In practice the experimental data, consisting of periodic measurements of 



