An analytical approximation of the Bragg curve for therapeutic proton beams

TLDR

Proton depth‑dose curves, or Bragg curves, are essential for radiotherapy dose calculations, and an analytical representation is often preferred over measured or numerically derived data. This study delivers a closed‑form analytical approximation of the Bragg curve. The model, applicable to 10–200 MeV protons, combines a power‑law range–energy relation, a linear fluence‑reduction term, Gaussian range straggling, and a Gaussian energy‑spectrum tail, yielding a closed‑form expression using Gaussians and parabolic cylinder functions. The resulting formula fits experimental measurements within error limits and agrees closely with numerically calculated Bragg curves.

Abstract

The knowledge of proton depth‐dose curves, or “Bragg curves,” is a fundamental prerequisite for dose calculations in radiotherapy planning, among other applications. In various cases it is desirable to have an analytical representation of the Bragg curve, rather than using measured or numerically calculated data. This work provides an analytical approximation of the Bragg curve in closed form. The underlying model is valid for proton energies between about 10 and 200 MeV. Its main four constituents are: (i) a power‐law relationship describing the range‐energy dependency; (ii) a linear model for the fluence reduction due to nonelastic nuclear interactions, assuming local deposition of a fraction of the released energy; (iii) a Gaussian approximation of the range straggling distribution; and (iv) a representation of the energy spectrum of poly‐energetic beams by a Gaussian with a linear “tail.” Based on these assumptions the Bragg curve can be described in closed form using a simple combination of Gaussians and parabolic cylinder functions. The resulting expression can be fitted to measurements within the measurement error. Very good agreement is also found with numerically calculated Bragg curves.