An empirical formula to obtain tissue-phantom ratios from percentage depth-dose curves for small fields

For small photon fields, accurate values of a tissue-phantom ratio (TPR) are difficult to obtain either by direct measurement or by the conventional method of converting from measured percentage depth doses (%dd). This study aims to develop an empirical method to accurately obtain TPRs from %dd curves for small radiosurgery beams. The Monte Carlo simulation codes BEAMnrc/DOSXYZnrc were used to simulate the accelerator head and small, collimated fields including the circular cone accessory. The Monte Carlo directly calculated TPR values as a function of depth were compared with TPRs converted from %dd curves in a water phantom for field sizes ranging from 4 mm diameter to 10 × 10 cm(2) fields. Direct measurements of TPRs were performed with the detector remaining fixed at a SAD of 100 cm and increasing the detector depth by adding water. The %dd curves were measured at 100 cm SSD in a 50 × 50 × 50 cm(3) water tank. Using the Monte Carlo values, we developed an empirical formula to obtain TPRs from %dd and validated its accuracy. The conventional method of obtaining TPRs from %dd underestimate TPR by 3.4% and 0.6% at a depth 1.5 cm and overestimate TPR by 6.4% and 1.7% at a depth of 25 cm for 4 mm and 30 mm diameter circular fields, respectively. The empirical formula is derived from realistic Monte Carlo simulations using field sizes ranging from 4 to 30 mm and depth ranging from 1.5 to 25 cm. TPRs calculated using this function deviate from TPRs directly calculated from Monte Carlo by less than 0.5%. The accuracy of this empirical formula is validated against the directly measured TPRs in water. The developed empirical method has the potential to greatly simply the work in obtaining TPRs from measured %dd curves for small fields. By using this developed empirical formula the uncertainties between directly measured TPRs and converted TPRs from measured %dd curves are within 1%.