Perturbation factors for cylindrical ionization chambers in proton beams. Part I: corrections for gradients

An analytical model is presented to calculate the effective depth in water of cylindrical ionization chambers in clinical proton or heavy ion beams in the presence of modest and linear gradients. This model is compared with Monte Carlo simulations and recommendations in IAEA TRS-398 for all ionization chambers listed in that report. A refinement of the analytical model allows its application for depth-dose curves even in the nonlinear gradient region of the Bragg peak. Combined with information from the Monte Carlo simulations it also allows one to solve the inverse problem of deriving a depth-dose curve in homogeneous water from the depth-dose response obtained with a cylindrical ionization chamber. The results show that the effective depth in water calculated analytically, calculated by Monte Carlo, and derived from experimental literature are in good agreement. The agreement with the IAEA TRS-398 recommendation is good for most ionization chambers but can be wrong by up to 1.5 mm for some ionization chamber types. The agreement between the refined analytical model and Monte Carlo simulations of depth-dose response curves is shown to be good (even in extreme situations of large ionization chambers in low-energy beams). An indication is given of situations in which a simple shift of the entire depth-dose curve is sufficiently accurate. It is demonstrated that it is possible to solve the inverse problem with this method, even for rather noisy data. This is illustrated by successfully applying it to depth-dose measurements with cylindrical ionization chambers taken from the literature.