Calculation of water equivalent thickness of materials of arbitrary density, elemental composition and thickness in proton beam irradiation

In proton therapy, the radiological thickness of a material is commonly expressed in terms of water equivalent thickness (WET) or water equivalent ratio (WER). However, the WET calculations required either iterative numerical methods or approximate methods of unknown accuracy. The objective of this study was to develop a simple deterministic formula to calculate WET values with an accuracy of 1 mm for materials commonly used in proton radiation therapy. Several alternative formulas were derived in which the energy loss was calculated based on the Bragg-Kleeman rule (BK), the Bethe-Bloch equation (BB) or an empirical version of the Bethe-Bloch equation (EBB). Alternative approaches were developed for targets that were 'radiologically thin' or 'thick'. The accuracy of these methods was assessed by comparison to values from an iterative numerical method that utilized evaluated stopping power tables. In addition, we also tested the approximate formula given in the International Atomic Energy Agency's dosimetry code of practice (Technical Report Series No 398, 2000, IAEA, Vienna) and stopping power ratio approximation. The results of these comparisons revealed that most methods were accurate for cases involving thin or low-Z targets. However, only the thick-target formulas provided accurate WET values for targets that were radiologically thick and contained high-Z material.