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.

Figure 1. A proton beam deposits little dose on the way in and almost all of it in a sharp Bragg peak at a depth set by its energy through R = alpha E0^1.77, then nothing beyond; a photon beam builds up to a shallow maximum and irradiates everything past the target. Superposing pristine peaks of decreasing energy spreads the peak into a flat plateau covering a tumour. Method: the Bragg-Kleeman rule with Gaussian range straggling and weighted SOBP superposition; Canvas2D, deterministic.

proton energy (MeV)150

mode

SOBP width (%)35

WHAT TO TRY

Slide the proton energy: the Bragg peak marches deeper as R = alpha E^1.77, depositing almost nothing on the way in and stopping sharply at the end of range. Energy alone aims the dose in depth.
Switch to spread-out Bragg peak and widen the SOBP: the middle panel sums many weighted, range-shifted pristine peaks into a flat plateau that covers a whole tumour while still sparing tissue beyond.
Compare the proton strip against the X-ray strip in the patient map: the proton stops cold past the tumour, the X-ray exits the far side and dumps exit dose into healthy tissue.