Empirical modeling of the percent depth dose for megavoltage photon beams

Abstract

Introduction This study presents an empirical method to model the high-energy photon beam percent depth dose (PDD) curve by using the home-generated buildup function and tail function (buildup-tail function) in radiation therapy. The modeling parameters n and μ of buildup-tail function can be used to characterize the Collimator Scatter Factor (Sc) either in a square field or in the different individual upper jaw and lower jaw setting separately for individual monitor unit check. Methods and materials The PDD curves for four high-energy photon beams were modeled by the buildup and tail function in this study. The buildup function was a quadratic function in the form of dd2+n with the main parameter of d (depth in water) and n, while the tail function was in the form of e−μd and was composed by an exponential function with the main parameter of d and μ. The PDD was the product of buildup and tail function, PDD = dd2+n·e−μd. The PDD of four-photon energies was characterized by the buildup-tail function by adjusting the parameters n and μ. The Sc of 6 MV and 10 MV can then be expressed simply by the modeling parameters n and μ. Results The main parameters n increases in buildup-tail function when photon energy increased. The physical meaning of the parameter n expresses the beam hardening of photon energy in PDD. The fitting results of parameters n in the buildup function are 0.17, 0.208, 0.495, 1.2 of four-photon energies, 4 MV, 6 MV, 10 MV, 18 MV, respectively. The parameter μ can be treated as attenuation coefficient in tail function and decreases when photon energy increased. The fitting results of parameters μ in the tail function are 0.065, 0.0515, 0.0458, 0.0422 of four-photon energies, 4 MV, 6 MV, 10 MV, 18 MV, respectively. The values of n and μ obtained from the fitted buildup-tail function were applied into an analytical formula of Sc = nE(S)0.63μE to get the collimator to scatter factor Sc for 6 and 10 MV photon beam, while nE, μE, S denotes n, μ at photon energy E of field size S, respectively. The calculated Sc were compared with the measured data and showed agreement at different field sizes to within ±1.5%. Conclusions We proposed a model incorporating a two-parameter formula which can improve the fitting accuracy to be better than 1.5% maximum error for describing the PDD in different photon energies used in clinical setting. This model can be used to parameterize the Sc factors for some clinical requirements. The modeling parameters n and μ can be used to predict the Sc in either square field or individual jaws opening asymmetrically for treatment monitor unit double-check in dose calculation. The technique developed in this study can also be used for systematic or random errors in the QA program, thus improves the clinical dose computation accuracy for patient treatment.

Fitting Monte Carlo simulation results with an empirical model of megavoltage x-ray beams for rapid depth dose calculations in water

“…Consequently, formulating a universally applicable characterization of megavoltage x-ray PDDs proves challenging, given the presence of tissue inhomogeneities. Nevertheless, numerous empirical models have been proposed for megavoltage beams in water, drawing on PDD data collected from clinical LINACs (Jabbari 2011, Li et al 2022 as well as data from MC simulations. These models assume a homogeneous target material and can be later adjusted for heterogeneities using existing methods.…”

“…These models assume a homogeneous target material and can be later adjusted for heterogeneities using existing methods. As presented in Li et al (2022), a three-parameter formula, included below as equation 1, has demonstrated successful modeling of PDD curves across a range of x-ray beam energies from 4 MV to 18 MV and for square field sizes between 5 × 5 cm 2 and 40…”

“…Solutions such as these require the construction of an AI model using a combination of low-statistic, or noisy, simulations and high-statistic simulations (Sarrut et al 2021). In this study, we recruit an empirical percent depth dose (PDD) formula to fit our MC simulations, and compare the best fits with published beam data (Li et al 2022). Moreover, we perform a simple linear regression analysis on the best fits to extrapolate results to higher statistics.…”

“…The specific shape of the build-up function depends on factors such as energy, field size, and other characteristics of the LINAC setup. As demonstrated by Hounsell and Wilkinson (1999), as well as Li et al (2022), this build-up function can be determined by fitting dose measurements of clinical LINACs.…”