Sec. 15.4] THEORY OF TRACER METHODS 401 



the same for the tagged and untagged fractions. However, if it can be 

 assumed that the system does not discriminate between tagged and untagged 

 molecules, the functions R and F will be the same for both; thus, measure- 

 ment of the change in concentration of tagged molecules in the phase enables 

 a choice of functions R and F to be made which will, therefore, describe the 

 kinematic behavior of the normal metabolite. The procedure to be followed 

 may be illustrated by two applications described by Branson. The first 

 method assumes that the metabolite can be labeled and introduced as a single 

 dose at / = 0. The equation for the tagged metabolite then reduces to the 

 simple form M*(t) = M*F(t), where * denotes the labeled metabolite. 

 F(f) is determined directly from measurements of M*(t) and M*, and when 

 substituted in the integral equation 



M(t) = MoF + f* RF 



dd 



M{t) can be found by integration if M and R are known, or the equation may 

 be solved for R if M(t) and M can be determined empirically. Usually, 

 M(t) = M = constant which can be determined by the method of dilution. 

 The second case is a considerably more complex problem that arises when 

 two or more phases must be considered, as when the precursor A of the met- 

 abolite B is labeled. If a single dose of labeled A is added, the steady state 

 system A — > B — ■> is described by the equations 



A*(t) = A%F X 

 A(\ -F x ) = fi RiFtdO 



B*(t) = V R 2 F 2 dd 



B(l - F») = J l g R 2 F 2 dd = B*(t) 



An additional relation B*(t) = B*F can be used if the metabolite can be 

 tagged and introduced into phase B in the same or a similar system but in a 

 separate experiment when B* from the precursor is not present. Alter- 

 natively a second labeling agent, e.g., a stable isotope, or a different species of 

 radioactive isotope distinguishable from the first, may be used to determine 

 the relation B(i) = BJFi, where the bar indicates the metabolite tagged with 

 the second isotope. Although technically the latter procedure may be, and 

 usually is, considerably more difficult, it provides a distinct advantage in 

 that all the quantities required may be measured simultaneously and in the 

 same system. 



The general procedure to be followed in applying the integral equation 

 to complex systems of several interrelated phases is found as an extension of 

 the example above. The metabolite AU in any phase i is described by the 



