Electron slowing-down spectra for physical models of radiocarcinogenesis

R. P. Hugtenburg
University Hospital Trust,
Birmingham, B15 2TH,
United Kingdom
richard.hugtenburg@university-b.wmids.nhs.uk
and
J. E. Pattison
University of South Australia
Mawson Lakes, S.A., 5059
Australia
john.pattison@unisa.edu.au

Introduction

The Monte Carlo method has been used to determine electron slowing-down spectra [1] for a range of ionising radiations. These spectra tell us the quantities and energies of electrons liberated by a particle cascade. A number of radiobiology studies suggest that cell chromosomal damage is caused by electrons that are able to deposit at least 100 eV in a volume with diameter comparable to that of the DNA molecule [2]. Slowing-down spectra enable the assessment of such models by comparing the relative biological effectiveness (RBE) of radiations over widely varying source energies. Furthermore, recent Monte Carlo studies demonstrate systematic under-estimation of the absorbed dose in cell irradiations in previous work due to the improper consideration of the scatter conditions [3].

The Japanese atomic-bomb survivors represent the most important source of human data on the late carcinogenic effects of ionising radiation and these underpin radiation risk estimates. A recent study [4] has presented free-in-air $\gamma$ spectra due to prompt and delayed radiation components received by this population. Electron slowing down spectra have been computed and matched with combinations of megavoltage photon and electron beams produced by a therapeutic linear accelerator. This simulation of the atomic-bomb exposure conditions in cell cultures will, for the first time, enable a direct link to be made between current radiobiological studies and the epidemiological data from Japan.

Methods

The code system, EGS4 [5], has been used for photon and electron transport down to 1 keV kinetic energy. Though normally a 10 keV lower bound is recommended, various extensions improve the accuracy of simulation in this energy regíme. In particular, the simplified electron boundary crossing algorithm appropriate for low energy electron transport, FIXTMS [6], the low-energy Compton scattering extension, LSCAT [7], and corrections to the change in electronic discrete interaction cross-section with energy [8], have been utilised to determine in-vivo and in-vitro electron and photon spectra above 1 keV and restricted linear-energy-transfer (LET) distributions of $\Delta$ = 1 keV. The maximum energy loss during a multi-scattering step, ESTEPE, of 10% has been used as spectra were negligibly different from calculations using an ESTEPE of 1% which have not been presented. The cut-off for the production of secondary electrons in terms of total energy, AE, is 512 keV. The influence of additional electron multi-scattering by atomic electrons which is also considered explicitly as a discrete event for the production secondary electrons above AE is negligible, i.e. the parameter FUDGEMS was reduced from the default, 1, to 0 with a negligible effect on the spectrum. Calculations have been performed on a Sun Spark Ultra II.

A free-in-air $\gamma$ spectrum, including prompt and delayed components from the atomic bomb detonated at Hiroshima has been used. The radiation is treated as an isotropic source and transported through a anthropomorphic model. The dose and electron spectrum have been computed at the colon for an average survivor using a simplified semi-infinite cylindrical geometry (figure 1). The colon has been chosen as a representative site for the induction of solid tumours. Four-component soft tissue, 10.1% Hydrogen, 11.1% Carbon, 2.6% Nitrogen and 76.2% Oxygen (by mass) [9] is used. Cross-sectional data incorporates the density corrections therein.

  

Figure 1: A model torso and colon for the Japanese survivors of the atomic bomb. A semi-infinite cylinder with dimensions determined for the average survivor has been used. Not to scale

The dose per fluence has been determined for the semi-infinite torso using reciprocity relations. Fluence from n particles to a thin band of width, $\delta z$, and diameter, l,

$$\Phi_{\mbox{air}} = \frac{n}{l\delta z},$$ (1)

passing through a region of cross-sectional area, A, of indefinite height, is equivalent to a uniform source of width, l and indefinite height. The dose in the region, is

$$\mbox{D}_{\mbox{colon}} \frac{ \sum_i\varepsilon_i}{\rho A \delta z},$$ =

$$\Phi_{\mbox{air}} \frac{l}{n\rho A} \sum_i\varepsilon_i,$$ = (2)

where $\varepsilon_i$ is the $i^{\mbox{th}}$ energy deposition. Similarly, spectra are computed using the equivalency of fluence to the ratio of the summed track-lengths, s_i, and volume of the region of interest [10].

$$\Phi_{\mbox{colon}} = \Phi_{\mbox{air}} \frac{l}{nA} \sum_i s_i.$$ (3)

  

Figure 2: The irradiation jig used for the application of medical beams. The diagram is not to scale and thickness of each of the layers is as follows 5 mm of perspex attenuator, 1 mm mylar lid, 4 mm water, 1 mm tissue, 1 mm mylar base, and 100 mm perspex base

Simulations were performed for 250 kVp and 28 kVp X-rays incident on tissue cultures in an irradiation jig. The applied radiation and jig have been approximated by a broad beam with plane-parallel conditions (figure 2). The tissue culture is treated as four-component soft tissue, the nutrient layer is treated as water and he petri dish is polyethylene terephthalate mylar.

Results and discussion

Dose and fluence have been computed in the colon wall for an average survivor of the atomic bomb blast in Hiroshima at a distance of 500 m from the hypocentre. The total dose was 26.4 $\pm$ 0.1 (s.d.) Gy and not significantly different with each of the three calculation options. The example presented in this paper is a relatively high dose, the dose diminishes rapidly with distance from the hypocentre while the spectrum hardens slightly with distance.

Three variations for the calculation of the slowing-down spectrum are presented in figure 3. The calculation times are substantial due to the low value of AE. These could be improved if range rejection is incorporated in the calculation, however, care needs to be taken to ensure that bremsstrahlung photons will not be influential.

  

Figure 3: The electron slowing-down spectrum for an average survivor using EGS4/FIXTMS (bold), using the LSCAT extension (bold-dash) or with Ma and Nahum's correction to the change in discrete interaction cross-section with energy (light-dash) calculated in 17.2 hours, 19.2 hours, 20.5 hours respectively. The error (s.d.) is indicated by the width of the lines

The variation of cumulative dose with restricted LET ($\Delta$ =1 keV) have been computed for the average survivor described and in the tissue layer of an irradiation jig for a 250 kVp X-ray source [11] and a 28 kVp mammography X-ray source [12] (figure 4).

  

Figure: Cumulative dose with restricted LET ($\Delta$ = 1 keV) computed for an average survivor compared to tissue cultures irradiated with 250 kVp and 28 kVp X-rays

The dose averaged restricted LET for the three radiations 1.4, 2.7 and 3.0 keV/$\mu$m, respectively. The track-length averaged restricted LET are 0.24, 1.3 and 2.0 keV/$\mu$m, respectively. These quantities ease the comparison of the Hiroshima radiation with other medical radiation sources. The average LET for the hiroshima survivor decreases slightly with distance from the hypocentre.

Our future work will focus on determining the oncogenic RBE of tissue cultures exposed to simulated atomic bomb spectra and other radiation sources. We intend to update RBE for exposures to diagnostic radiations and for radiation workers. Our analysis, performed over a wide range of radiations, will enable us to postulate quality indices for radiation risk as a function of LET. This information will greatly assist in the interpretation of cancer induction models such as proposed by Nikjoo and Brenner (1991) [2].

Brenner and Amols[13], in a similar analysis, present calculations of risk factors for cancer induction rates in breasts, computed for 23 kVp X-rays and 250 kVp X-rays relative to those determined from the populations at Hiroshima and Nagasaki. Their risk factors are computed by convolving lineal energy distributions in dose, d_i(y), obtained experimentally for 1 $\mu$m equivalent diameter proportional counter, with a quality factors, r(y)published in ICRU Report 40 [14]. The relative index of RBE, R_i, for a range of radiation types, i, is

$$R_i = \int r(y)d_i(y)\mbox{d}y.$$ (4)

They found that the cancer risk of molybdenum target, 23 kVp, X-rays, relative to 250 kVp X-rays at equal low doses, to be 1.31. According to the same analysis, the relative cancer risk for the Hiroshima group was 0.64. The Japanese survivor data is used by the ICRP as a basis of cancer risk for all low LET radiations. Their work implies a RBE relative to the cancer induction rate amongst the survivors at Hiroshima of approximately 2 for mammographic X-rays.

Conclusion

The dose, slowing-down electron spectrum and restricted LET distributions have been computed for the colon of an average survivor of the Hiroshima atomic bomb using EGS4 with the FIXTMS and LSCAT extensions and a 1 keV cut-off. These corrections for low energy electron and photon transport, respectively, are shown to be of only minor significance when computing slowing-down spectra but may have greater influence with smaller regions of interest and higher LET sources. Corrections for changes in the electron discrete interaction cross-section with energy have been considered and were shown to be of significant influence, though errors are small in comparison with errors in the free-in-air $\gamma$ spectra used. Cumulative dose versus LET distributions and dose averaged and track averaged LET were computed and compared to several other sources relevant to medical and occupational exposures of radiation. This work is preliminary to comparison studies of the oncogenic RBE for the entire therapeutic and industrial range in relation to exposures received by the populations at Hiroshima and Nagasaki.

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Electron slowing-down spectra for physical models of radiocarcinogenesis

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