MEAN ENERGY, ENERGY-RANGE RELATIONSHIPS AND DEPTH-SCALING FACTORS FORCLINICAL ELECTRON-BEAMS

Using Monte Carlo simulations we have studied the electron mean energy , (E) over bar(o), and the most probable energy, E(o,p), at the phanto m surface and their relationships with half-value depth, R(50), and th e practical range, R(p), for a variety of beams from five commercial m edical accelerators with an energy range of 5-50 MeV. It is difficult to obtain a relation between R(50) and (E) over bar(o) for all electro ns at the surface because the number of scattered lower-energy electro ns varies with the machine design. However, using only direct electron s to calculate (E) over bar(o), there is a relationship which is in cl ose agreement with that calculated using monoenergetic beams by Rogers and Bielajew [Med. Phys. 13, 687-644 (1986)]. We show that the empiri cal formula E(o,p) = 0.22+1.98R(p)+0.0025R(p)(2) describes accurately the relationship between R(p) and E(o,p) for clinical beams of energie s from 5 to 50 MeV with an accuracy of 3%. The electron mean energy, ( E) over bar(d), is calculated as a function of depth in water as well as plastic phantoms and is compared both with the relation, (E) over b ar(d)=(E) over bar(o)(1-d/R(p)), employed in AAPM protocols and with v alues in the IAEA Code of Practice. The conventional relations general ly overestimate (E) over bar(d) over the entire therapeutic depth, e.g ., the AAPM and IAEA overestimate E(d) at d(max) by up to 20% for an 1 8 MeV beam from a Clinac 2100C. It is also found that at all depths me an energies are 1%-3% higher near the field edges than at the central axis. We calculated depth-scaling factors for plastic phantoms by scal ing the depth in plastics to the water-equivalent depth where the mean energies are equal. The depth-scaling factor is constant with depth i n a given beam but there is a small variation (<1.5%) depending on the incident beam energies. Depth-scaling factors as a function of R(50) in plastic or water are presented for clear polystyrene, white polysty rene and PMMA phantom materials. The calculated depth-scaling factor i s found to be equal to R(50)(water)/R(50)(plastic) This is just the AA PM definition of effective density but there are up to 2% discrepancie s between our calculated values and those recommended by the AAPM and the IAEA protocols. We find that the depth-scaling factors obtained by using the ratio of continuous-slowing-down ranges are inaccurate and overestimate our calculated values by 1%-2% in all cases. We also find that for accurate work, it is incorrect to use a simple 1/r(2) correc tion to convert from parallel beam depth-dose curves to point source d epth-dose curves, especially for high-energy beams.