Geometric tomography for measuring rectangular radiotherapy fields from six projections

Quality assurance (QA) of x-ray radiotherapy is crucial but the dosimetry of small fields is still a challenge. We are investigating the use of scintillating parallel fibers which measure the integral dose along a line, thus providing a one-dimensional (1D) parallel projection of the dose of the x-ray field in a plane of a tissue equivalent medium. By stacking a few layers of scintillating fibers, each with a different orientation, one obtains a sinogram of the dose of the x-ray field sampled with a few angles and sub-millimetric resolution of each projection. We propose a geometric approach based on projections moments to identify the few parameters defining the irradiation field (e.g., center, radius, orientation, width, length and intensity for a rectangle). The order-0 moment of at least one projection provides an estimate of the integral dose. The order-1 moments of at least two projections gives the center of mass, i.e., order-1 moments of the dose map. The order-2 moments of at least three projections identify order-2 moments of the dose map. In the specific case of a uniform rectangular field, we show that the length, the width and the orientation of the rectangle can be derived from the order-2 moments of the dose map. The formulas are extended to the case of a non-uniform field due to a lateral Gaussian penumbra. Our approach is validated on simulated data of rectangular fields with increasing levels of noise and it demonstrates an accurate, robust and rapid estimation of the rectangle parameters from six projections only. A proof of concept experiment with scintillating fibers imaged with a sCMOS camera irradiated under a 30×30 mm2 square field of a 6 MV linear accelerator gave, from six projections, 31.7×31.4 mm2 without accounting for the penumbra and 28.6×28.3 mm2 with an estimated σ = 4 mm penumbra.