Lateral dose profile characterization in scanning particle therapy

Abstract

In light-ion beam dose delivery with the scanning technique the spacing between adjacent spots is an important parameter during treatment planning. In order to study the effect of spot spacing on dose conformity and robustness for single field uniform dose configurations, fundamental geometrical properties of placement of Gaussian beamlets are explored. In particular, the dependence of penumbra width and flatness on spot width and spot spacing is investigated. Infinitesimal calculus and analytical methods are used to derive simple expressions for the lateral penumbra and the flatness of one-dimensional dose profiles in continuous scanning and uniform discrete spot scanning. In the same way expressions for the fundamental modes of perturbation of the spot sequence are developed. A numerical, matrix-based approach is followed to optimize weights spot-by-spot. Generally the lateral penumbra widths lie between 1.13 sigma(b) and 1.68 sigma(b) with sigma(b) being the standard deviation of the beam spot profile. For regularly placed spots of equal weight with spot spacing lambda the lateral penumbra is given by 1.68 sigma' where sigma' results from quadratic subtraction of lambda/square root of 12 from sigma(b). The quantization error is identified as additional parameter describing the lateral dose conformity. It's variance is given by lambda2/12 for a bunch of spots with uniform weights. The matrix-based optimization of weights for a one-dimensional dose box results in a lateral penumbra of typically 1.4 sigma(b). This value reduces to about 1.3 sigma(b) if also the positions of the beam spots are optimized for the considered field size. The analytical formulas for uniform discrete scanning can be used as rough approximations of the best-case scenarios for weight-optimized dose profiles if the spot spacing is defined as effective spot spacing. The trade-off between flatness, quantization error, and robustness on the one side and penumbra width on the other side can be described analytically for equally weighted spots. Treatment planning systems often perform a least-squares optimization of the individual spot weights which results in smaller lateral penumbras and smaller quantization errors than for uniform discrete scanning. However, the benefit of this weight optimization decreases with increasing lambda (in the regime lambda > sigma(b)). The spot spacing, which is obtained from the scenario that the optimization objective is met by uniform discrete scanning, poses a sharp upper limit for the spot spacing lambda in weight optimization methods.