Sec. 15.3] THEORY OF TRACER METHODS 395 



with untagged molecules of the same substance in any phase is short com- 

 pared to the time in which the concentration changes appreciably after 

 equilibrium. 



a. Simple Elimination from One Phase. The total quantity M of substance 

 in the phase is assumed to remain constant; the rates, in either grams or 

 cubic centimeters per unit time, of appearance and disappearance are equal 

 and constant. If labeled material of the same form is added to make the 

 specific activity in the phase at t = equal to X microcurie per gm, the 

 specific activity subsequently decreases as 



x = Xe 



-kt 



The parameter k, obtained from the plotted measurements, is referred to as 

 the turnover rate for the substance being traced. Physically, it is the frac- 

 tion r/M of the total amount of the substance (labeled plus unlabeled) 

 replaced per unit time. Its reciprocal 1/k = M/r, called the turnover time, 

 is the time required for the replacement of an amount equal to M. The 

 quantity r is then the actual rate, in grams or- cubic centimeters per unit time, 

 at which the substance (tagged plus untagged) enters and leaves the phase. 

 The turnover time 1/k is readily determined by inspection from the experi- 

 mental data plotted on semilog paper since 1/k = T/0.693, where T is the 

 half-time for disappearance of the tagged molecules. Thus far only the ratio 

 r/M has been found. The actual value of M, and hence of r, can however be 

 determined, as described in Sec. 15.2, by the dilution of the administered 

 active material (at / = 0). 



It is apparent that more than one route may be taken by the substance 

 when it leaves the phase, for the turnover rate may be written also as the 

 sum of rates of transfer to several different phases; e.g., 



r (fi + 7*2 + r% + ' ' * ) 

 M ~~ ~M~ 



As measured in the one phase, the dilution with time still follows the single 

 exponential expression above since only the sum of the rates is apparent. 

 The separate rates can be determined only if the labeled substance can be 

 traced to the phases that follow. 



b. Labeled Substance Accumulated in One Phase. Accumulation of sub- 

 stances normally found in a system usually does not occur over extended 

 periods when the system is in a steady state. The important exceptions to 

 this are substances eliminated from the system, usually in expired gases, 

 feces, and urine. In practice excreta are either accumulated over a suitable 

 length of time and the total content of excreted tracer is measured, or the 

 excreta may be sampled periodically for the concentration of the tracer. 

 It is usually not reasonable in such measurements to speak of specific activity, 



