Internal, sequentially decaying beta emitters:
Enhanced Mutagenicity from 2nd Event Effects'
by Chris Busby,
contributed at Muenster,
Low Level Radiation Symposium,
March 1998

Calculating the Second Event Error

Evolution has provided the DNA repair systems and the repair-replication response to reduce the effects of external natural background radiation. The system did not need to cope with Second Event damage, except as a remote possibility, because the probability of two hits separated by ten hours occurring on the same cell is very unlikely. At natural background exposure levels of 2mSv, cells on average receive 1 hit per year [1]. Using this as a basis and assuming a number of different doses of external radiation, or more exactly, equivalent chance that any cell is hit, we can calculate the External Second Event Probability for two hits separated by ten hours, with a 1 hour window for the second decay. We first obtain the average number of hits per cell per year for the given dose and then use his value as m in the Poisson equation to calculate the probability of two hits per year at this dose. We then multiply the result by 1/(365 x 24) to ensure that the second dose goes into the one hour window specified, in this case after a lag of ten hours. Results for the range 2mGy to 1µGy are given in Table 1 below.

Table 1. Poisson probability of two hits separated by 10 hours on a cell subjected to random external irradiation at different doses.

(P(2 hits) = e^-µ. µ²/2!)

Dose	Average hits/yr	P(2 hits/yr)	P(2 hits/10 hrs)a
2mGy	1	0.18	2.05 x 10-5
1mGy	0.5	0.076	8.66 x 10-6
0.1mGy	0.05	1.19 x 10-3	1.36 x 10-7
0.01mGy = 10µGy	0.005	1.24 x 10-5	1.41 x 10-9
1µGy	0.0005	2.5 x 10-7	2.9 x 10-11

a obtained by multiplying P(2hits /y) by 1/(365 x 24)

Note that for more than two hits, we generally can ignore the extra probability as second order because:

P( 2 ...n hits) = SIGMA µⁿ/ n! since e^µ approaches 1

For the immobilised internal isotope, we will compare the same doses from Strontium-90 and calculate the probability of the same sequence occurring from Sr-90 - Y-90.
At a given dose, how many disintegrations of Sr-90 are there in a gram of tissue? For a decay energy of 546keV there are 8.96 x 10^-14 Joules per disintegration. At 10mGy which is 10 x 10^-9 Joules per gram there are 11.16 x 10⁴ disintegrations of Sr-90 inside the aggregate of 2 x 10⁸ cells which comprise the 1 gram of tissue. Thus the probability of a primary decay per cell in this gram of tissue is

P(cell, 10µGy) = 11.16 x 10⁴)/ (2 x 10⁸) = 5.53 x 10^-4

This ignores the possibility of track overlap which we will look at below. We can, in passing, calculate the approximate dose to each of the 500 cells in the track. Using CSDA approximation the dose per cell of 10µm diameter is

0.56 MeV/500 = 1120 eV or 0.345µSv.per cell in track.

The probability of a second decay from the Y90 occurring in the cell already containing the Sr90 inside the ten hour period of induced repair replication sequence brought about by the first 0.345mSv decay is given by Equation (1) below:

P(lamda) ={exp[-0.7t_d /T_0.5] - exp[-0.7(t_w+ t_d)/T_0.5)]} (1)

If we put T _0.5 = 64 hours for the half-life of Y-90 and t_d = 10 hours for the lag and t_w = 1 hour for a decay into a 1 hour time window at the end of ten hours then

P(lamda) = 9.75 x 10^-3.

So the total probability P( total, internal) that any cell in the 1 gram will receive a second hit defined by this 10 hour delay and 1 hour window is given by the product of these two probabilities, P(cell,dose) x P(lamda).

Results for different doses are given in Table 2, below.

We can now calculate the excess hazard by dividing, at each dose, the probability of a cell receiving a 2nd event sequence from internal Sr-90--Y-90 relative to external irradiation. This is given in Table 3.