Analytical investigation of magnetic field effects on Proton lateral deflection and penetrating depth in the water phantom: A relativistic approach

Background Integrated proton therapy - MRI systems are capable of delivering high doses to the target tissues near sensitive organs and achieve better therapeutic results; however, the applied magnetic field for imaging, influences the protons path, changes the penetration depth and deflects the particles, laterally, leading to dose distribution variations. Objective To determine the effects of a magnetic field on the range and the lateral deflection of protons, analytically. Methods An analytical survey based on protons energy and range power law relation, without using small angle assumption was done. The penetration depth and lateral deflection of protons with therapeutic energy ranges 60-250 MeV in the presence of uniform magnetic fields of 0-10T intensities, were calculated analytically. Calculations were done for relativistic conditions with Mathematica software version 7.0, and MATLAB 7.0 was applied to plot curves and curve fittings. Results In the presence of a magnetic field, the depth of Bragg peak was decreased and it was shifted laterally. A second order polynomial model with power equation for its coefficients and a power model with quadratic polynomial coefficients predicted the maximum lateral deflection (ymax) and maximum penetration depth (zmax) variations with energy and magnetic field intensity, respectively. Conclusion The applied correction for deflection angle will give more reliable results in initial energy of 250 MeV and 3T magnetic field intensity. For lower energies and magnetic field intensities the differences are negligible, clinically.