The determination of virtual source position using convergent anti-trigonometric functions (arcCOS and arcSIN) method for scanning-passive scatter beam in carbon ion therapy

Abstract Introduction: We developed a convergent trigonometric functions technique (arcCOS, arcSIN) capable of dealing with the virtual source position delivered by different carbon ion energies from the pattern of scanning-passive scatter beam in this study. Materials and Methods: A home-made large-format CMOS sensor and Gaf Chromic EBT3 films were used for the virtual source position measurement. The Gaf films were embedded in a self-designed rectangular plastic frame to tighten the films and set up on a treatment couch for irradiation in the air with the film perpendicular to the carbon ion beam at the nominal source-axis-distance (SAD) as well as upstream and downstream from the SAD. The horizontal carbon ion beam with 5 energies at a machine opening field size was carried out in this study. The virtual source position was determined with a convergent arcCOS and arcSIN methods and compared with the linear regression by back-projecting the FWHM to zero at a distance upstream from the various source-film-distance. Results: The film FWHM measurement error of 0.5 mm (the large-format CMOS detectors was in pixel, a pixel equals 0.5 mm) leads to 1×10-3% deviation of α(cACOS and cASIN) at every assumed virtual source position. The overall uncertainty for the reproducibility of the calculated virtual source position by the assumed t in the vertical and horizontal directions amounts to 0.1%. The errors of calculated virtual source position by assumed t with back projecting FWHM to zero methods were within 1.1 ± 0.001, p = 0.033. The distance of virtual source positions is decreased from SAD with high to low energy. Conclusion: We have developed a technique capable of dealing with the virtual source position with a convergent arcCOS and arcSIN methods to avoid any manual measurement mistakes in scanning-passive scatter carbon ion beam. The method for investigating the virtual source position in the carbon ion beam in this study can also be used for external electrons and the proton.