X A " 



Sec. 15.4] THEORY OF TRACER METHODS 399 



The activity in C at any time / is then 



k,k 2 X Mi 



(k 2 -k x ){kz - k 1 )(k 3 -k 2 )M 3 



[(&3 — k 2 )e~ klt — (& 3 — ki)e~ kit — (ki — k 2 )e~ k3t ] microcuries/gm 



where k u k 2 , £3 = turnover rates of phases A, B and C 



Mi, Ms — total amounts of traced substances in phases A and C, 

 respectively 



It is seen that this formula is a form of the general expression consisting 

 of a polynomial with exponential terms. In this case the explicit expressions 

 for the coefficient ai are rather complicated because of their interdependence 

 on the turnover rates ki, k 2 , and k 3 . 



A classic example of the usefulness of the formulas outlined above for 

 describing one-, two-, and three-phase systems is provided by the experi- 

 ments of Zilversmit and his associates [6,7]. Radioactive phosphorus 

 (P 32 ) was used in these experiments to determine the turnover rate of phos- 

 pholipids in the plasma of dogs. Measurements of the specific activity of 

 the phospholipids and of their immediate precursor gave the necessary curves 

 representing the uptake and disappearance of tagged molecules from which, 

 by analysis similar to that above, the turnover rate of phospholipids could 

 be determined. 



15.4. A More General Theory of Tracer Methods. While the descriptions 

 of processes given by the simple expressions for first-order reactions are 

 probably the most widely useful, they are by no means adequate for systems 

 in which higher order reactions are involved. When it is found advantageous 

 to describe such processes in terms of mathematical expressions, it may be 

 possible, as in the case of first-order reactions, to formulate the requisite 

 differential equation and to solve for a particular solution in terms of suitable 

 initial and boundary conditions. This procedure, however, is possible only 

 when the nature of the process is already known in some detail. On the 

 other hand, general equations may be found that are valid for large classes 

 of phenomena. Thus, in first-order reactions the concentration of a tracer 



n 



as a function of time is given by the general formula x = > (ne kit . A more 



? = o 

 general equation has been pointed out by Branson [3] who showed its useful- 

 ness and great power when applied to tracer investigations of metabolizing 

 systems. In view of the importance of tracer applications to biological 

 systems, the equation and a few of its uses described by Branson are given 

 below. 



A generalized metabolizing system may be regarded as a single complex 

 phase in the sense described in the last section. The phase may be a physical 



