An analytical solution to proton Bragg peak deflection in a magnetic field

The role of MR imaging for image-guided radiation therapy (IGRT) is becoming more and more important thanks to the excellent soft tissue contrast offered by MRI. Hybrid therapy devices with integrated MRI scanners are under active development for x-ray therapy. The combination of proton therapy with MRI imaging has only been investigated at the theoretical or conceptual level. Of concern is the deflection of the proton beam in the homogeneous magnetic field. A previous publication has come to the conclusion that the impact of a 0.5 T magnetic field on the dose distribution for proton therapy is very small and lateral deflections stay well below 2 mm. The purpose of this study is to provide new insights into the effects of magnetic fields on a proton beam coming to rest in a patient. We performed an analytical calculation of the lateral deflection of protons with initial energies between 50 MeV and 250 MeV, perpendicular to the beam direction and the magnetic field. We used a power-law range-energy relationship and the Lorentz force in both relativistic and non-relativistic conditions. Calculations were done for protons coming to rest in water or soft tissue, and generalized to other uniform and non-uniform media. Results were verified by comparisons with numerical calculations and Monte Carlo simulations. A key result of our calculations is that the maximum lateral deflection at the end of range is proportional to the third power of the initial energy. Accordingly, due to the strong dependence on the energy, even a relatively small magnetic field of 0.5 T will cause a deflection of the proton beam by 1 cm at the end of range of a 200 MeV beam. The maximum deflection at 200 MeV is more than 10 times larger than that of a 90 MeV beam. Relativistic corrections of the deflection are generally small but they can become non-negligible at higher energies around 200 MeV and above. Contrary to previous findings, the lateral deflection of a proton beam can be significant (1 cm and above) even in relatively small magnetic fields of 0.5 T. However, the curved path of a proton beam in a magnetic field is easily predictable and it should be possible to account for this in treatment planning.