In the framework of an investigation into a possible influence of external magnetic fields on the generation of ionisation clusters in nanometric volumes [1], Monte Carlo simulations using the Geant4-DNA toolkit have been conducted in collaboration with the Australian Centre for Medical Radiation Physics (CMRP). The calculations aimed at deriving the frequency distribution of ionisation cluster size, i.e. the number of ionisations produced in a nanometric target by a particle track. The target volumes had a cylindrical shape and dimensions corresponding to the size of a DNA segment of 12 base pairs and of a nucleosome, respectively. In addition, the mean cluster size and the probability to produce a cluster size of 2 or more were calculated. The latter is supposed to be proportional to the likelihood of induction of a DNA double strand break [2].

The simulations were performed for monoenergetic electrons in the energy range between 200 eV und 10 keV and for uniform static magnetic fields of strength between 0 T and 14 T. For the calculated quantities of interest, relative deviations from the values obtained for zero magnetic field strength of up to 2 % were obtained for some values of initial energy (cf. Fig. 1). To assess whether these discrepancies from unity are significant, only statistical uncertainty contributions need to be considered, because the systematic contributions to the uncertainty for the different magnetic field strengths can be assumed to be fully correlated.

These statistical uncertainty contributions could be retrieved by data mining from the output files produced in the conducted simulations. As a simpler alternative, a method has now been developed that allows to determine the statistical uncertainty contributions for arbitrary simulated distributions using only the simulation results. Therefore, this procedure can also be used to recover the statistical uncertainty contributions in the frequently encountered case of simulations where only the quantities of interest are scored, while tallies of auxiliary quantities required for calculating the statistical uncertainty contribution are omitted.

The application of the procedure for our simulations was such that for each combination of initial electron energy and magnetic field strength the relative frequency distribution of cluster size was used to calculate the corresponding cumulative probability distribution. These cumulative probabilities were then used as bin boundaries for binning sets of random numbers that were generated by a generator based on the Wichmann-Hill AS183 algorithm [3], which yields random numbers uniformly distributed between 0 and 1. For each cluster size distribution, 100 independent sets of random numbers were used, such as to mimic 100 repetitions of the Monte Carlo simulations. Each set of random numbers contained as many members as primary particles were used in the Monte Carlo simulation for the respective combination of kinetic energy and magnetic field strength. For each set of random numbers, the relative frequency distribution of cluster size *P*_{ν}(*Q*), the mean cluster size *M*_{1} and *F*_{2}, the probability for cluster sizes of at least two, were determined. For each of these quantities, the standard deviation ui over the 100 repetitions was calculated and used as statistical uncertainty estimate. For the case of the mean cluster size for 1 keV electrons, the uncertainty estimates obtained in this way are shown as error bars in Fig. 1.

Using the uncertainty estimates derived as described above, the deviations from unity found as function of magnetic field strength were subjected to separate *χ*^{2} tests for each value of initial kinetic energy. In all cases, the likelihood for random sampling from a Gaussian distribution of expectation value 1 and standard deviation equal to the estimated standard uncertainty to result in a *χ*^{2} of at least the value obtained in the test took values between 35% and 89%. Based on the conventional 5% limit, the observed deviations from unity, therefore, had to be assessed as statistically insignificant [1].

Figure 1: Mean ionization cluster size *M*_{1} obtained in the simulations for electrons with 1 keV kinetic energy as a function of magnetic field strength, normalized to *M*_{1} at 0 T.

**Literature**

- M. U. Bug et al.:

*Effect of a magnetic field on the track structure of low-energy electrons: a Monte Carlo study*,

Eur. Phys. J. D**XXX**, p1-p8 (2010). DOI: 10.1140/epjd/e2010-00145-1. - B. Großwendt:

*Nanodosimetry, from radiation physics to radiation biology*,

Radiat. Protec. Dosim.**115**, 1-9 (2005). - B. A. Wichmann, I. D. Hill:

*Algorithm AS 183: An efficient and portable pseudo-random number generator*,

Applied Statistics**31**, 188-190 (1982).