Analytical expressions for the determination of the effective water depth of an ionization chamber for clinical proton beam dosimetry

The knowledge of the water equivalent depth of an ionization chamber in a proton beam is important when reference dosimetry is performed in a region where depth dose gradients exist, such as in the entrance region of a non-modulated beam, or when measured depth dose curves have to be corrected. In this work, an analytical model is presented to calculate the water equivalent depth of spherical and cylindrical ionization chambers in a clinical proton beam, accounting for the ionization chamber's cavity, wall, waterproofing sleeve and central electrode. For the theoretical situation of an air cavity without wall in water, this water equivalent depth equals the depth of the effective point of measurement. The model, which is a refinement of a previous model in which only the cavity and wall were taken into account, can be applied in a region where the depth dose varies approximately linear with depth. An expression to combine the model for a spherical and a cylindrical ionization chamber is given as well. This way, values for thimble chambers are obtained. The model is verified by Monte Carlo calculations using PTRAN. A further refinement allows calculation of the effective measurement depth in a region with non-linear gradients such as in the neighborhood of the Bragg peak. It is shown that this model can be used to convert a depth dose distribution measured with a thimble ionization chamber to the true depth dose distribution.