A spatial and temporal characterisation of single proton acoustic waves in proton beam cancer therapy

An in vivo range verification technology for proton beam cancer therapy, preferably in real-time and with submillimeter resolution, is desired to reduce the present uncertainty in dose localization. Acoustical imaging technologies exploiting possible local interactions between protons and microbubbles or nanodroplets might be an interesting option. Unfortunately, a theoretical model capable of characterising the acoustical field generated by an individual proton on nanometer and micrometer scales is still missing. In this work, such a model is presented. The proton acoustic field is generated by the adiabatic expansion of a region that is locally heated by a passing proton. To model the proton heat deposition, secondary electron production due to protons has been quantified using a semi-empirical model based on Rutherford's scattering theory, which reproduces experimentally obtained electronic stopping power values for protons in water within 10% over the full energy range. The electrons transfer energy into heat via electron-phonon coupling to atoms along the proton track. The resulting temperature increase is calculated using an inelastic thermal spike model. Heat deposition can be regarded as instantaneous, thus, stress confinement is ensured and acoustical initial conditions are set. The resulting thermoacoustic field in the nanometer and micrometer range from the single proton track is computed by solving the thermoacoustic wave equation using k-space Green's functions, yielding the characteristic amplitudes and frequencies present in the acoustic signal generated by a single proton in an aqueous medium. Wavefield expansion and asymptotic approximations are used to extend the spatial and temporal ranges of the proton acoustic field.