432 ISOTOPIC TRACERS AND NUCLEAR RADIATIONS [Chap. 16 



for which the dose is to be calculated, the total amount of active material 

 in the organ is then just mu where m is the tissue mass. 



The activity density u is calculated from the representative, though not 

 general, equation 



■jj = f(u,t) - h(u,t) 



which states that the rate of change of activity equals the rate of uptake 

 minus the rate of elimination. The functions f(u,t) and h(u,t) are arbitrary 

 in that they are chosen to describe each case in accordance with appropriate 

 physical and chemical conditions. The function U is obtained by integration 

 of u over the interval of time to /. 



U 



= I udt microcurie sec/gm 



a. The simplest and also a common dose problem is that in which a given 

 quantity u microcuries of isotope is rapidly fixed in a tissue and subsequently 

 is neither accumulated nor eliminated. The activity density after a time / 



is 



u = — e~°- 693t/T microcuries/gm 



m 



where m = mass of tissue, gm 



T = decay half-life of isotope 



the function U for the interval of time o to / is then 



u T 

 U = k JL-, ■ (1 — e -0 - 69S ' /r ) microcurie sec/gm 



0.693m b 



When the entire quantity of isotope decays, U = u o T/0.693m. It may be 

 recalled that 3.7 X 10 4 £/ was the number of radioactive atoms originally 

 present (at time t = 0) per gram of tissue. 



b. Initially no active material is present in the tissue, but for t > it 

 is taken up and fixed in the tissue under consideration at a constant rate of 

 b microcuries per gm of tissue per second. This represents the behavior, for 

 instance, when active substances are ingested or breathed at constant rates 

 or when taken up from blood or other reservoirs maintained at a constant 

 level of activity. The activity density at any time / > is then 



u — _ ,_, (1 — e~°- G9Zt/T ) microcuries/gm 



0.693 



where T = decay half-life 



The activity in the tissue increases exponentially and approaches after 

 a long time, / » T, the constant level bT/0.693 microcurie per gm. The 

 disintegration function U for the interval to / is 



