Electron dose distribution in magnetic fields
The dose distributions for 10-MeV electrons parallel to SMFs (B = 0.0, 5.0, 10.0 T) are shown in Figs. 1 (A) and (B). The track structure for 10-MeV electrons at 0.0 T is shown in Fig. 1 (C). In the case of the field parallel to the incident electron direction, the larger the SMF strength, the narrower to the x- or y-axis the electron beams is (Fig. 1 (A)). The depth-dependency of dose (y-axis) is independent of the SMF (Fig. 1 (B)). Meanwhile, the dose distributions for 1-MeV electrons perpendicular to SMFs (B = 0.0, 5.0, 10.0 T) are shown in Figs. 1 (D) and (E). The track structure for 1-MeV electrons at 0.0 T is shown in Fig. 1 (F). The dose distribution without a SMF is symmetrical with respect to x=0. However, those perpendicular to SMFs are largely biased in a large SMF strength (Fig. 1 (D)). Focusing on depth-dependencies perpendicular to the SMF, as the SMF becomes larger, the energies are deposited over less depth (Fig. 1 (E)). The change of dose distributions is analogous to the modification of track structures confirmed in supplementary data (see Fig S1). Note that the distributions calculated by etsmode were confirmed in supplementary data (see Fig. S2) and these results were in good agreement with the calculated results by EGS.
The results by PHITS simulation show that the trajectories of 10-MeV and 1-MeV electrons, which are used in radiation therapy, are significantly affected by magnetic fields. However, we confirmed that the dose distributions by low-energy electrons are not affected by the SMF as shown in supplementary data (see Fig. S3).
Impact of magnetic fields on projected ranges of electrons
Figure 2 shows the ranges of electrons perpendicular to the SMF as a function of incident electron energy. To verify the accuracy of electron track structure in the SMF simulated by etsmode, we compared the electron ranges calculated by to those by EGS. The simulation accuracy of estmode in the absence of a SMF has been extensively discussed in comparison with the recommended values of ICRU reports and experimental data reported previously.29 As shown in Fig. 2, we compared the range given by etsmode to that by EGS, further affirming the accuracy of etsmode even in magnetic fields.
The calculated ranges for monoenergetic electrons in the absence of a SMF (B = 0.0 T) are shown in Fig. 2 (A), where the incident energy range was set to be 50-1000 keV because significant SMF effects can be expected. The projected range for monoenergetic electrons (100-1000 keV) as a function of the SMF intensity (B = 0.0-10.0 T) are also shown in Fig. 2 (B); there are no significant SMF effects on projected ranges for 100-keV electrons. Meanwhile, in the case of electrons with high energies above 300 keV in the presence of SMF intensity over 3.0 T, the larger the SMF strength becomes, the shorter the projected range is. These simulation results suggest that the travelling lengths along the z-direction are shortened due to the Lorenz force within the SMF, and the macroscopic dose distributions can be modified by the SMF and projected range in the SMF is shorten compared to that in non-SMF.
Assuming the high-energy electron beams used in radiation therapy, the electrons with incident energy over 1 MeV locally deposit their energies in the region close to the departure point of incident electrons (see Fig. 1 (E)). It was therefore confirmed that the SMF effects (so called electron return effect (ERE)) for electrons in liquid water largely depend on the flight length and time. Also, the simulation shows that the dose distribution within solid tumours and normal tissues should be calculated in consideration of magnetic fields when making treatment planning.
Estimation of electron return effects in vacuum
The gyration time and radius of electrons in vacuum in the presence of a SMF (B = 0.0-10.0 T) was evaluated using DMCC. These calculations assume that the electrons are in vacuum without considering atomic interactions. Figure 3 (A) shows the relationship among incident electron energies, time and radii. The time is constant up to approximately 100 keV, then exponentially increases above 100 keV. The radius also exponentially increases in the energy range from 0.01 to 1000 keV. In a previous study20, the gyration radii for electrons (0.001-100 MeV) in vacuum applied by magnetic fields were compared with the continuous slowing down approximation (CSDA) range. This result shows that electrons with energy above 100 keV, in which the CSDA range is longer than the gyration radius, are modified by the SMF (below 10.0 T). In contrast, we show the relationship between the time and the radii of low-energy electrons in vacuum (Fig. 3 (A)). It was confirmed that the flight distance and time until attenuation due to the interaction with water is shorter than the radius and the time in one period in the energy of electrons (below a few hundreds of keV). From this relation, it was found that the electrons below a few hundreds of keV slow down immediately before they drift by SMF (i.e., a few hundreds of psec).
Figure 3 (B) shows the comparisons of projected range and the gyration radii under B=3.0 (T) in liquid water and in vacuum. The projected range under B=0.0 (T) is also depicted in Fig. 3 (B). In the case of high-energy electrons (100-1000 keV), the gyration radius monotonically increases from 0.37 to 1.58 mm as the electron kinetic energy gets higher. The radius becomes closer to the projected range. In the case of low-energy electrons (0.01-90 keV), the radius is significantly larger compared to the projected length. Therefore, high energy electrons drift in the presence of a SMF. From the relations shown in Fig. 3, it was further confirmed that the electrons with higher kinetic energy than 100 keV can be strongly affected by the SMF.
DNA damage yields for monoenergetic electrons in a SMF
The DSB yield, YDSB, by monoenergetic electrons in the SMFs (B = 0.0-10.0 (T)) are shown in Fig. 4 (A). In the range of incident electron energy from 0.1 to 300 keV, there is no SMF impact on YDSB for various intensities of magnetic field in both parallel and perpendicular orientations. The DNA damage simulation based on etsmode suggests that the SMF effects do not appear at the DNA (nanometer) scale. When irradiating high-energy electrons in liquid water, uncountable secondary electrons with several dozen eV and Auger electrons with about 500 eV from the inner shells are generated. These results suggest that the secondary electrons may be intrinsically related to the induction of DSBs.
To illustrate the contribution of secondary electrons to DSB induction, we also estimated YDSB without considering the secondary electrons or Auger electrons. The result is shown in Fig. 4 (B). Focusing on 100-keV electrons, the YDSB value without any secondary electrons including Auger electrons (red circles and lines) becomes lower than that with all secondary electrons (blue circles and lines). Furthermore, the YDSB value without any secondary electrons is almost zero. The maximum value of YDSB without any secondary electrons is 1.51 in the case of 0.4-keV electrons, which is about half the value for YDSB = 3.35 when considering all secondary electrons. These results indicate that the secondary electrons including Auger electrons are major contributors to induce DSBs for high energy electrons (over 100 keV). Low-energy secondary electrons can be produced by inelastic interactions within a few fsec24,27, and the corresponding penetration length is approximately 10 nm. Also, as shown in Fig. 3, the gyration time and radius to induce SMF impact on the secondary electrons are more than several ps and several µm, respectively. The secondary electrons therefore slow down before being affected by a magnetic field. In Fig. 3B, the YDSB values of high-energy electrons (over 100 keV) were lower than that of 100-keV electrons. This is because the cross sections for higher energy part is based on vapour phase. In the vapour phase, the number of ionization events is estimated to be fewer than that in the liquid phase.
Figure 5 (A) shows that the ratio of DNA damage complexity (cDSB/DSB) for 300 keV electrons is larger than that of 100 keV electrons. The cDSB/DSB for both electron energies decreases in the case of no Auger electrons. In addition, the cDSB/DSB of 100 keV and 300 keV electrons for various intensities of the SMF is shown in Fig. 5 (B). cDSB/DSB is independent of the intensity of the SMF. These results are corelated with the fact that Auger electrons contribute to DNA damage complexity and the energy of electrons is not enough to be affected by the SMF (Fig. 3).
The present estimation for DNA damage yields based on physical processes suggests no significant biological impact caused by SMFs. This may be useful for interpreting the experimental data on surviving fractions after X-ray irradiation in the SMF.8–10 However, some reports suggest the cell-killing effects for low LET radiation increases in the direction parallel to the SMF.6,7 YDSB calculated in this study was only based on the physical interaction with electrons and biological tissues after irradiation. It is also necessary to discuss the effect of the SMF on the chemical phase or biological phase. Since the energy range of electrons in the Auger process is unaffected by the SMF, the ratio of complex DNA damage does not vary in the SMF.