INDICATOR SUBSTANCES AND FLOW ANALYSIS 



629 



defined as 



6 we have 



•Si A' 10 4" 02 X20 



s\ + s. 



Application of the inverse transforms yields: 



•vi = XE+ (.v,„ - .ve) e *-^i ^2'' (5) 



and similarly: 



X2 = .V£ + (.v=o - xe) e~ ^^"^^)' (6) 



In the one-compartment open system, equation 5 

 alone, with p/.S"., = o, is the general solution. Two 

 special cases may be distinguished, /) if the tracer is 

 initially present in the compartment and there is no 

 tracer in the inflow : 



-i.PIS)t 



(7) 



and 2) if the inflow is labeled but none is initially 

 present in the compartment 



XE (i - e-c'^O 



(8) 



Since in a given compartment the amounts and 

 concentrations of the tracer are directly proportional 

 to the specific activities, these other units may be 

 substituted for the x's in equations 5 and 6. It will be 

 noted, however, that in general only the specific 

 activities are equal at "equilibrium," which in this 

 context refers to isotopic ecjuililjrium and does not 

 necessarily imply thermodynamic equilibrium. 



Equations 5 and 6 serve to predict the behavior of 

 a tracer in a two-compartment system in terms of the 

 parameters of the system. Application of equations 

 5 and 6 to the inverse problem of deducing the 

 parameters of the system from the observed tracer 

 behavior is achieved as follows. The data are plotted 

 semilogarithmically, as indicated in figure \A, and 

 smooth curves are drawn to fit the data. If the data 

 from the two compartments approach the same value 

 closely enough to provide an accurate estimate of 

 xe, Xe is subtracted from the values of the curve which 

 approaches xe from above and the values of the 

 curve which approaches xe from below are subtracted 

 from Xe, giving the two lines shown as (.vi —xe) and 

 {xe — .Vo) in figure iB. If the data are consistent with 

 the assumptions made for the two-compartment 

 system, these two lines will have the same slope. 

 Alternatively, if it has not been feasible to obtain 

 data for a period sufficiently long to establish .Vg, a 

 plot of the difference between ,vi and .vj should give 

 the same slope, figure iC, since from equations 5 and 



{Xi — .Vo) = (Xio 



X20) e 



■dl + l)' 



(9) 



The latter method has the advantage of avoiding 

 the effect of an error in estimating Xe, but has the 

 disad\antage of not including the check on internal 

 consistency available if the two slopes can be com- 

 pared. 



Whichever method is used, the number most 

 easily obtained to characterize the slope is the half- 

 time 7"! 2- As is shown in figure iB, the half-time may 

 be obtained by determining the interval of time 

 required for the component's value to decrease by 

 half (as from 4 to 2 or 2 to i, etc.). For straight lines 

 on semilogarithmic paper, this time interval is a 

 constant. From T^2 the exponent or "decay factor," 

 X, is obtained by use of the relationship 



X = 



0-69315 



(10) 



In the present case X = {p/Si) -\- (p/So). 



For very steep components, it may be difficult to 

 estimate a single half-time accurately, but the appro- 

 priate factor corresponding to any number of half- 

 times may be used. In particular, it is convenient to 

 note that 10 half-times involve a diminution factor 

 of 2'° = 1024, or slightly more than three logarithmic 

 cycles (1/1024 = 0.0009766). For very gentle slopes, 

 where the T^'2 is longer than the scale on the graph 

 paper being used, a larger fraction, say 0.80, may be 

 used, provided the numerator in equation 10 is 

 changed to be the positive value of log e of the new 

 factor, which for 0.80 is 0.22314. Alternatively, the 

 value for (p/Si) + (p/Si) corresponding to any reduc- 

 tion factor may be obtained in one setting on a log-log 

 slide rule. Finally, if Si and S\ are known from the 

 initial and final dilutions of the tracer, or by other 

 methods, p may be calculated by a rearrangement of 

 equation 10. 





(II) 



It is not necessary to determine either .^i or ^'2 to 

 calculate the turnover rate constants, p S\ and p/6'2, 

 separately. Using the methods explained more fully 

 in the general treatment, it can be shown that if the 

 experimental curves are described by equations 5 and 

 6: 



P_ _ X(jrio — Xe) p _ \{xe — J^a) 



Si (aTio — Xs>) S2 (.Vio — ATjo) 



(12) 



