Recalculating the "Second Event" Error

A reworking of the probabilities of Second Event damage to cells
relative to Natural Background radiation

C. C. Busby PhD

An extension of an argument contained in presentations to the International Congress on Risks to Workers and Members of the Public from Low Level Ionising Radiation, Muenster, March 19th 1998
and also to the Science and Technology Options Assessment Workshop of the European Parliament held in Brussels, February 5th 1998.

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. 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 eµ 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[-0.7td /T0.5] - exp[-0.7(tw+ td)/T0.5)]} (1)
If we put T 0.5 = 64 hours for the half-life of Y-90 and td = 10 hours for the lag and tw = 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.

Table 2. Probability of a two hit sequence from internal Sr-90--Y-90 at different overall doses from Sr-90.
Dose N decays/gm P(cell,dose)=N/(2 x 108) P(cell,dose) x P(lamda)
2mGy 1.1 x 107 0.055 5.36 x 10-4
1mGy 5.5 x 106 0.0275 2.68 x 10-4
0.1mGy 5.5 x 105 2.75 x 10-3 2.68 x 10-5
0.01mGy = 10µGy 5.5 x 104 2.75 x 10-4 2.68 x 10-6
0.001mGy = 1µGy 5500 2.75 x 10-5 2.68 x 10-7

Table 3. Excess 2nd Event hazard from internal Sr-90--Y-90 relative to external random irradiation at the same dose.
Dose External 2nd Event Hazard, Internal/External (rounded)
1mGy 30
0.1mGy 200
0.01mGy 1900
0.001mGy 9400
1 atom per gram of tissue 5 x 109

These results exclude other cells on the track. If we assume now that the track can point in any direction into 2phi steradians then the probability of overlap at each layer of cells defined by integral multiples n of the cell diameter 2R is given by the inverse of the numbers of such cells that can be fitted on each surface of concentric spheres radius 2nR. The excess probability that there will be overlap of tracks will at minimum be given by the sum of these terms divided by the total number of cells in a sphere radius equal to the decay track range.

excess probability of overlapping tracks. (1Kb)
Here, N is the total number of cells in the sphere of radius equal to the decay range of the beta particle, and n is an integer from 1 to the total number of cells in the track minus 1. In the case of the 0.5cm track from Sr-90, nmax = 499. The summation rapidly converges and for Strontium-90 the extra probability of two hit from track overlap is not more that 20%.

The rapid increase of the 2nd Event enhancement from internal seqentially decaying isotopes relative to external irradiation as the dose is reduced is expected by intuition. The extreme case of a double decay from Sr-90--Y-90 in one cell in the whole body would be an almost impossible outcome (P = 10-26) following random external irradiation at the same overall dose. Thus such a hypothetical situation would be associated with an enhancement factor of some 10-13. The origin of these enhancements is essentially an error in the use of a model which averages dose to tissue.

There are two questions that remain. The first is whether the first decay of the parent isotope can push the cell into the repair replication sequence. The second is the size of the target. In the calculations used above, I used a cell sized target, although I originally decided to look at gene sized targets and came at the problem differently, from microscopic.[2] It is not at all clear that we need to hit the same gene twice. The targets may be much larger than the chromosome, and may indeed surround the chromosome, so that the direction of emission of the beta particle is irrelevant. As early as 1962, Lea argued from target theory that the target for radiation induced division delay had a diameter much larger than the chromosomes (3). More recently, Schneidermann and Hopfer used radioiodine labelled bases to show that this target was not the chromosome itself but some structure that came close to the chromosome (4).

If the sensitive structure is one which surrounds the chromosome, a membrane perhaps, then the direction of the second emission from the Strontium daughter bound to the phosphate backbone of the DNA is largely irrelevant.

The two hit hypothesis has implications for other classes of exposure besides internal sequential beta emitters. One class is hot particles. There will exist a size of hot particle where the probability for the second event process occurring is very high. For Plutonium-239 oxide particles . The critical particle size range for such effects here is 0.1 to 1 micron diameter, which is the commonest found in the environment. There will also be implications in the field of radiotherapy and X-ray diagnosis since two diagnostic X- rays delivered ten hours apart may carry excess risk.

I want to turn now to look at studies that may have been done that support the hypothesis, apart from the obvious need to find a plausible explanation for the increases in cancer and leukemia near nuclear pollution sites. First I will summarise the theory.

Summary of the Second Event Theory

1. Dividing cells are hundreds of times more susceptible to radiation damage and mutations than quiescent cells. The critical period for mutagenesis is late in the 10 hour repair replication cycle of dividing cells. This is known.

2. Unplanned cell division preceded by DNA repair can be forced by a sub-lethal damaging first event, an ionizing radiation track. This is known.

3. A second event, or track to the same cell, occurring during the cell repair replication cycle carries a high risk of introducing a mutation. This follows from 1.

4. The process is highly improbable for external irradiation and at natural background levels. The ratio of the probability of such damage sequences occurring from internal sequential beta emitters relative to random external exposure is moderate at natural background levels of exposure and increases rapidly as the dose is reduced.

5. There is an error in the use of risk factors derived from external irradiation observations to predict effects from internal 2nd event emitters which increases rapidly as the dose decreases.

The Second Event Theory is well specified and its predictions straightforward. It would not be difficult to devise a range of experiments that would test it. Two classes of experimental investigation might be envisaged. The first would involve the extend the investigations on split doses of low LET radiation in the range of induction of cell cycle blocking effects, such as those of Miller et al.[5] The second, and more direct approach would involve the comparison of Strontium-90 and Strontium-89, the latter isotopes having a single decay scheme in the induction of various low-dose effects, particularly genetic effects in animals, like those of Luning et al (6).

(1) Goodhead, D.T (1991), ‘Biophysical features of radiation at low dose and low dose rate.’ , in C.B.Seymour and C. Mothershill (eds), New Developments in Fundamental and Applied Radiobiology London: Taylor and Francis

(2) Busby, Chris (1995), Wings of Death: Nuclear Pollution and Human Health, Aberystwyth: Green Audit

(3) Lea, D.E (1962), The Action of Radiation on Living Cells, 2nd edn, Cambridge: University Press

(4) Schneiderman, M.H and Hopfer,K.G (1980), ‘The target for radiation induced division delay.’ Radiation Research, 84: 462-76

(5) Miller, R.C, Hall, E.J, and Rossi, H.H, (1979), ‘Oncogenic Transformation in Cultured Mouse Embryo Cells with Split Doses of X-rays’, Proceedings of the National Academy of Science, 75:5755-8

(6) Luning, K.G, Frolen, H, Nelson, A and Roennbaeck, C (1963), ‘Genetic Effects of Strontium-90 injected into male mice.’ Nature, 197:304-5

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