Proton Therapy: the Bragg Peak and the Spread-Out Bragg Peak

A proton-radiotherapy depth-dose playground. The Bragg-Kleeman rule R = alpha E0^p with p = 1.77 gives the proton range; the pristine depth dose is the (R - z)^{1/p - 1} stopping-power profile convolved with Gaussian range straggling, producing a low entrance dose, a sharp Bragg peak just before the end of range, and essentially no dose beyond. A spread-out Bragg peak is built as a weighted superposition of pristine peaks of decreasing energy, with the weights fitted to flatten the dose over a target plateau. The photon depth dose (build-up then near-exponential attenuation) is shown for contrast. Panel A is the depth dose against the X-ray curve; Panel B is the SOBP as a sum of weighted pristine peaks; Panel C is the dose along a patient with the tumour marked. The range-energy law, the sharp Bragg peak just before the end of range with essentially no dose beyond, and the flat spread-out Bragg peak built from weighted pristine peaks are the physical content.