PRINCIPLES OF RADIOLOGICAL PHYSICS 



133 



proportional to D. Therefore the combined probabiUty Pn that all n latent 

 events have occurred rises, as a function of D, in proportion to the nth power of 

 the dose, 



PniD) = kD- if Pn « 1 (51) 



The trend of variation of Pn{D) at low doses depends only on the number n of 

 required single-hit events. The trend at higher doses depends on further details 

 of the model assumed. For example, the n single-hit events may be equivalent, 

 or rather interchangeable, like ?i breaks in the chromosomes of a cell. As men- 

 tioned in Sect. 5-2a, the probability that m such events result from a dose D is 

 given by the Poisson distribution expression, Eq. (28), Sect. 3-6a: 



F^(D) = e-«-»(aZ))-/w! 



The detectable event occurs whenever at least n single-hit events have taken 

 place. Therefore its probability equals 





{n+ 1)! ' {n + 2)! 



(52) 



Let No be the number of organisms exposed to the radiation treatment and N{D) 

 the number of organisms in which the detectable event has not occurred after a 

 dose D. The frequency of "nonoccurrence," N{D)/No, equals the probabiUty 

 that less than n single events have taken place in an organism. Therefore 



N{D)/No = e-^ [l+aD + ^ + • • • + h^y\ 



(53) 



This expression reduces to the standard exponential, or "single-hit" [Eq. (49)], 

 when n = 1. It is called a "multiple-hit" curve. Figure 1-82 shows regu- 



2 

 (a) ccD 



Fig. 1-82. Plots of 1- to 5-hit dose-action curves according to Eq. (53). (a) Linear 

 plot; {h) semilogarithmic plot. 



lar and semilogarithmic plots of Eq. (53) for various values of the "number of 

 hits," n. 



Different trends of the dose-effect curves result from different assumptions 

 regarding the latent single-hit events. Suppose that these events are clearly 

 distinct from one another, as, for example, when one chromosome break is 

 required in each one of different chromosomes of a cell. Let the rates of action 

 for the production of the different single-hit events be designated by ai, 0:2, • • • , 

 a„. The probabilities of nonoccurrence of these various events after a dose D 



