436 IS0T0P1C TRACERS AND NUCLEAR RADIATIONS [Chap. 16 



applied as well to irregularly shaped masses. The entire tissue volume is 

 divided (graphically) into small volume elements AVi whose dimensions 

 are very small compared to l//x. For an arbitrarily chosen point P at which 

 the dose is to be determined, the value of g is computed by a summation of 

 the contributions from all AVi] thus 



5 



I 



e-" Ri AVi 

 -^— cm 



where Ri is the distance from the center of AVi to the point P. 



The calculations are facilitated by taking as the first term of the sum the 

 largest sphere with P as its center which lies entirely within the tissue; then 



s 



t --<X- r") + I —ft- 



1 = 1 



where a is the radius of the sphere and the sum is now taken over the volume 

 divisions lying outside the sphere but in the tissue. Further simplification 

 is possible if the organ permits convenient division into concentric spherical 

 shells of thickness AR{ surrounding the first solid sphere. These shells will 

 not lie entirely within the tissue, but by estimating the fraction /* of each 

 shell which does lie within the organ, the value of g is then computed from 



s 



4tt V 



g = — (1 - «-"°) + > e-" Ri fiARi cm 



The fraction /j is, of course, determined by estimating the surface area of the 

 shell bounded by the organ and dividing by 4tR 2 { , where Ri is the radius to 

 the mid-thickness of the shell. 



The determination of integral dose or total energy absorbed by the tissue 

 presents a more difficult problem in that the geometrical factor G which 

 must be calculated is given by a double integral 



G = I ~dVdV' cm 4 



where V is the volume containing uniformly distributed activity, V is the 

 volume for which the integral dose is to be determined, and R is the distance 

 between the volume elements dV and dV. The volumes V and V need 

 not be identical. Although V is determined, in principle, uniquely by the 

 distribution of the active isotope, the volume V may be chosen arbitrarily 

 provided that the factor e _Mii is retained. This integral does not reduce to 



