Radiotherapy Proton Interactions in Matter

Radiotherapy protons interact with matter in three ways. Multiple collisions with atomic electrons cause them to lose energy and eventually stop. Multiple collisions with atomic nuclei cause them to scatter by a few degrees. Occasional hard scatters by nuclei or their constituents throw dose out to large distances from the beam. Unlike the first two processes, these hard scatters or ‘nuclear interactions’ do not obey any simple theory, but they are rare enough to be treated as a correction. We discuss the three interactions (stopping, multiple Coulomb scattering, and hard scatters) in turn. We begin, however, with the fundamental formula relating dose (energy deposited per unit mass) to fluence (areal density of protons) and mass stopping power (their rate of energy loss). Concerning stopping, we define range experimentally, noting that the clinical and physics definitions are different. We present in simplified form the ‘Bethe-Bloch’ theory of energy loss, how it leads to range-energy tables, how to interpolate these, and two simple 2-parameter approximations to the range-energy relation. Finally we discuss range straggling, the fact that protons in a monoenergetic beam stop at slightly different depths. Concerning multiple Coulomb scattering (MCS), we present a short reading list and point out some incorrect or unhelpful aspects of the standard literature. We outline a simplified form of Molière theory, the accepted theory for protons and correct to 1% as far as is known. We explain the Gaussian approximation to Molière theory, which suffices for almost all radiotherapy calculations, and discuss Highland’s formula, vastly simpler than the full theory. We discuss scattering power, an approximation to Molière theory required by transport calculations. We treat hard scatters by focusing on the ‘nuclear halo’ acquired by a bundle of protons (a ‘pencil beam’) as it stops in matter. We emphasize recent experimental results and their implications for the most efficient parameterization of the halo. All three interactions combine in the Bragg curve, the depth-dose distribution of a monoenergetic beam stopping in water and the signature property of charged (as distinct from neutral) radiotherapy beams. For computation, the Bragg curve should be converted to an effective mass stopping power. We discuss two limiting cases. Looking ahead, we sketch how stopping theory and MCS combine in ‘Fermi-Eyges’ transport theory when, as is normally the case, both processes occur at once. We present three very general properties of beams spreading in a homogeneous medium, found by Preston and Koehler at the dawn of proton radiotherapy. Finally, we sketch a dose algorithm which outperforms those in current use and may ultimately prove useful. Five appendices contain supporting material: acronyms and symbols, a review of 1D and 2D Gaussians, relativistic single particle kinfematics and finally, a few problems that occur in beam line design. ∗Harvard University Laboratory for Particle Physics and Cosmology, 18 Hammond St., Cambridge, MA 02138, USA, bernardgottschalk @ gmail.com 1 ar X iv :1 80 4. 00 02 2v 1 [ ph ys ic s. m ed -p h] 3 0 M ar 2 01 8