A 3D model to calculate water-to-air stopping power ratio in therapeutic carbon ion fields

Air-filled ionization chambers (ICs) are extensively used in the dosimetry of charged particle radiotherapy [1]. The calibration procedure of ionization chambers for the determination of absorbed dose to water, which is the standard quantity used for dose determination in external radiotherapy [2] is known as ND,w formalism. In this formalism, the readout of the chamber is converted into absorbed dose to water via two factors: the calibration factor of the chamber, and a quality factor that accounts for the specificity of the beam. The water-to-air stopping power ratio, or sw,air, is one of the main components of these quality factors, and, in the case of carbon ion beams, its biggest source of uncertainty [2]. In a previous work by our group [3], an expression was proposed to calculate sw,air for carbon ion beams at different residual ranges, based on a set of Monte Carlo calculations and experimental measurements, namely: (1) where Rres is expressed in cm and calculated using a practical range at the 50% dose level Rres(z) = R50 – z, where z is the depth in water. This expression is based on a 1D analysis of dose and sw,air distributions, which is enough to model the variations in sw,air for homogeneous dose distributions, like the ones mostly used for calibration and quality assurance (QA) purposes. However, this 1D description might be insufficient in some cases. An example of this would be treatment plan verification with a matrix of ionization chambers [4], a protocol often used in scanning-beam facilities where a patient plan is shot into a water phantom and the deposited dose is measured at several points (see Fig. ​Fig.1).1). In such a case, the residual range Rres is not defined at every point, so the application of equation (1) is not possible.