400 ISOTOPIC TRACERS AND NUCLEAR RADIATIONS [Chap. 15 



or chemical state, or it may be a specific tissue or a region, with or without 

 well-defined boundaries, in which the substance under consideration is 

 consumed, produced, transferred, modified, or stored. Despite a great 

 variety of processes that may conceivably occur, these systems can be 

 described by an integral equation already familiar to mathematicians [4,5]. 



In order to clarify the meaning of the equation, consider a single phase in 

 which the total amount of metabolite present initially (/ = 0) is M . Since 

 it is being metabolized- there remains of this original amount after a time t 

 the quantity MoF(t), where F(t) is some metabolizing function appropriate 

 to the system and a function only of time. Simultaneously, additional 

 molecules of metabolite are accumulated at the rate R(t), and in any interval 

 of time 8 to 6 + dd an amount R(d) dd is added. The accumulating metab- 

 olite also undergoes metabolism, and of the amount R dd added at time 6, 

 there remains at time / the amount R(d)F(t — 6) dd. The total quantity of 

 metabolite in the phase at time / is then the sum of MgF plus what remains 

 of the amount added during to t, giving the general integral equation 



M (/) = MoF(t) + J' R(d)F(t - d) 



dd 



The method of solving the equation depends on which of the functions 

 M (/), M , F(t), and R(t) are known or can be determined empirically. When 

 M , R(t), and F(t) are known, M{t) can, of course, be obtained by direct or 

 numerical integration. If M(t), M , and F(t) can be found, the unknown 

 function R(t) appears only in the integral and the equation becomes a Volterra 

 integral equation of the first kind in R. Finally, when all the functions but 

 F(t) are known, it becomes a Volterra integral equation of the second kind 

 in F(t). 



In certain cases the metabolizing system will permit simplifications to 

 be made in the equation above. Thus, if the system is in dynamic equi- 

 librium, the total amount M(t) of metabolite present remains constant, or 

 M(t) = M , and the integral equation then becomes 



M {\ - F{t)) = £ R(6)F(t - 6) dd 



In some instances, also, the rate R is known to be constant or, if not, it may 

 sometimes be held constant during the course of the experiment. When 

 simplifications such as these are possible, the work of determining the 

 unknown functions and solving the integral equation is greatly facilitated. 

 Empirical determination of the functions depends upon which quantities 

 are accessible to measurement without seriously disturbing the system. For 

 this purpose tracer methods are perhaps the most powerful. While the 

 labeled homologue of the normal metabolite will also be described by the 

 integral equations above, the functions M(t) and M will not, in general, be 



