Mean energy, energy-range relationships and depth-scaling factors for clinical 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 phantom surface and their relationships with half-value depth, R(50), and the practical range, R(p), for a variety of beams from five commercial medical 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 electrons at the surface because the number of scattered lower-energy electrons varies with the machine design. However, using only direct electrons to calculate (E) over bar(o), there is a relationship which is in close agreement with that calculated using monoenergetic beams by Rogers and Bielajew [Med. Phys. 13, 687-644 (1986)]. We show that the empirical 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 energies 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 bar(d)=(E) over bar(o)(1-d/R(p)), employed in AAPM protocols and with values in the IAEA Code of Practice. The conventional relations generally 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 18 MeV beam from a Clinac 2100C. It is also found that at all depths mean energies are 1%-3% higher near the field edges than at the central axis. We calculated depth-scaling factors for plastic phantoms by scaling the depth in plastics to the water-equivalent depth where the mean energies are equal. The depth-scaling factor is constant with depth in 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 polystyrene and PMMA phantom materials. The calculated depth-scaling factor is found to be equal to R(50)(water)/R(50)(plastic) This is just the AAPM definition of effective density but there are up to 2% discrepancies 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) correction to convert from parallel beam depth-dose curves to point source depth-dose curves, especially for high-energy beams.