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.
Introduction: We developed a technique including preventing errors management method capable of dealing with the virtual source position delivered by different carbon ion energies from the pattern of spot scanning beam in this study. Materials and Methods: A homemade large-format complementary metal-oxide-semiconductor (CMOS) sensor and Gaf Chromic EBT3 films were used for the virtual source position measurement. The Gaf films were embedded in a self-designed rectangular plastic frame to tighten the films and set up on a treatment couch for irradiation in the air with the film perpendicular to the carbon ion beam at the nominal source-axis-distance (SAD) as well as upstream and downstream from the SAD. The horizontal carbon ion beam with five energies at a machine opening field size was carried out in this study. The virtual source position was determined mainly with a linear regression by back projecting the full width half maximum (FWHM) to zero at a distance upstream from the various source-film-distance and double checks additionally with a geometric convergent method to avoid any mistakes caused by manual measurement on FWHM. Results: The virtual source position for higher carbon ion energy has an obvious longer distance from the SAD since the more carbon ion beam energy, the less spreading affected by the horizontal and vertical magnetism, therefore, the distance of virtual source positions is decreased from SAD with high to low energy. Conclusion: The method for investigating the virtual source position in the carbon ion beam in this study can also be used for electrons and the proton. We have developed a technique capable of dealing with the virtual source position with a geometric convergent method to avoid any mistakes in spot scanning carbon ion beam.
Purpose. An experimental and mathematical study for determining the effective point of measurement ( P eff ) for a Farmer-type cylindrical chamber in a carbon ion passive scatter beam is presented. Methods. The ionization depth curves measured by the Bragg peak chamber were plotted according to the position of the inner surface of the entrance window, while the Farmer chamber was plotted at the tip of the cylindrical geometric center. The ionization depth curves measured by a cylindrical chamber in the 3D water phantom were then compared with a high-precision parallel-plate PTW Bragg peak chamber for inspecting the upstream shift correction of the cylindrical chamber in the carbon ion beam. A component of the vertical and horizontal integration method and the barrier model, cos φ = 1 − 2 α R L / 1 + α − R L , for analyzing the shift of effective point of measurement in different carbon ion energies and various field sizes, were studied. Results. The shift between the maximum peak of the Bragg peak chamber and the Farmer chamber in a field size of 10 cm × 10 cm with an energy of 330 MeV/u of carbon ion is 2.3 mm. This upstream shift corresponds to 0.744 ± 0.07 r , where r is the Farmer chamber inner radius of 3.05 mm. Carbon ion energy from 120 MeV/u to 400 MeV/u with different field sizes show different shifts of effective point of measurement in a range of 0.649 ± 0.02 r to 0.843 ± 0.06 r of 3 cm × 3 cm at an energy of 400 MeV/u and 10 cm × 10 cm at an energy of 120 MeV/u, respectively. The vertical and horizontal scatter analysis by the barrier model can precisely describe the shift of the effective point of measurement at different carbon ion energies with various field sizes. Conclusions. We conclude that the Farmer chamber can be used for a patient-specific dose verification check in carbon ion beam treatment if P eff is well calibrated.
Introduction: This study presents an empirical method to model the electron beam percent depth dose curve (PDD) using the primary and tail functions in radiation therapy. The modeling parameters N and n can be used to derive the depth relative stopping power of the electron energy in radiation therapy. Methods and Materials: The electrons PDD curves were modeled with the primary-tail function in this study. The primary function included exponential function and main parameters of N, µ while the tail function was composed by a sigmoid function with the main parameter of n. The PDD for five electron energies were modeled by the primary and tail function by adjusting the parameters of N, µ and n. The R50 and Rp can be derived from the modeled straight line of 80% to 20% region of PDD. The same electron energy with different cone sizes was also modeled with the primary-tail function. The stopping power for different electron energies at different depths can also be derived from the parameters of N, µ and n. Percent ionization depth curve can then be derived from the percent depth dose by dividing its depth relevant stopping power for comparing with the original water phantom measurement. Results: The main parameters N, n increase, but µ decreases in primary-tail function when electron energy increased. The relationship of parameters n, N and LN(-µ) with electron energy are n = 31.667 E0 - 88, N = 0.9975 E0 - 2.8535, LN(-µ) = -0.1355 E0 - 6.0986, respectively. Stopping power of different electron energy can be derived from n and N with the equation: stopping power = (−0.042 ln N E 0 + 1.072)e(−n−E0·5·10−5+0.0381·d), where d is the depth in water. Percent depth dose was derived from the percent reading curve by multiplying the stopping power relevant to the depth in water at certain electron energy. Conclusion: The PDD of electrons at different energies and field sizes can be modeled with an empirical model to deal with the stopping power calculation. The primary-tail equation provides a uncomplicated solution than a pencil beam or other numerical algorism for investigators to research the behavior of electron beam in radiation therapy.
Introduction. This study presents an empirical method to model the curve of electron beam percent depth dose (PDD) by using the primary-tail function in electron beam radiation therapy. The modeling parameters N and n can be used to predict the minimal side length when the field size is reduced below that required for lateral scatter equilibrium (LSE) in electron radiation therapy. Methods and Materials. The electrons’ PDD curves were modeled by the primary-tail function in this study. The primary function included the exponential function and the main parameters of N and μ , while the tail function was composed of a sigmoid function with the main parameter of n . The PDD of five electron energies was modeled by the primary and tail function by adjusting the parameters of N , μ , and n . The R 50 and R p can be derived from the modeled straight line of 80% to 20% region of PDD. The same electron energy with different cone sizes was also modeled by the primary-tail function. The stopping power of different electron energies in different depths can also be derived from the parameters N , μ , and n . Results. The main parameters N and n increase but μ decreases in the primary-tail function for characterizing the electron beam PDD when the electron energy increased. The relationship of parameter n , N , and ln − μ with electron energy are n = 31.667 E 0 − 88 , N = 0.9975 E 0 − 2.8535 , and ln − μ = − 0.1355 E 0 − 6.0986 , respectively. Percent depth dose was derived from the percent reading curve by multiplying the stopping power relevant to the depth in water at a certain electron energy. The stopping power of different electron energies can be derived from n and N with the following equation: stopping power = − 0.042 ln N E 0 + 1.072 e − n E 0 · 5 · 10 − 5 + 0.0381 · x , where x is the depth in water. The lateral scatter equivalence (LSE) of the clinical electron beam can be described by the parameters E 0 , n , and N in the equation of S eq = n E 0 − N E 0 0.288 / E 0 / n E 0 0.0195 . The LSE was compared with the root mean square scatter angular distribution method and shows the agreement of depth dose distributions within ±2%. Conclusions. The PDD of the electron beam at different energies and cone sizes can be modeled with an empirical model to deal with what is the minimal field size without changing the percent depth dose when approximate LSE is given in centimeters of water.
OBJECTIVE: This study aims to develop and test a new technique by using the convergent arcTAN (cATAN) method capable of dealing with the virtual source position delivered by different carbon ion energies from the pattern of scanning-passive scatter beam. MATERIALS AND METHODS: A homemade large-format CMOS sensor and Gaf Chromic EBT3 films are used for the virtual source position measurement. The Gaf films are embedded in a self-designed rectangular plastic frame to tighten films and set up on a treatment couch for irradiation in air with the film perpendicular to the carbon ion beam at the nominal source-axis-distance (SAD) as well as upstream and downstream from the SAD. The horizontal carbon ion beam with 5 energies at a machine opening field size is carried out in this study. The virtual source position is determined by using the convergent arcTAN (cATAN) method and compared with a linear regression by back projecting FWHM to zero at a distance upstream from the various source-film-distance. RESULTS: The film FWHM measurement error of 0.5 mm leads to 0.001% deviation of α (cATAN) at every assumed textend. The overall uncertainty for the reproducibility of calculated virtual source position by the assumed textend in the vertical and horizontal directions amounts to 0.1%. The errors of calculated virtual source position by assumed textend with back projecting FWHM to zero methods are within 1.1±0.001, p = 0.033. CONCLUSION: We develop a new technique capable of dealing with the virtual source position with a convergent arcTAN method to avoid any manual measurement mistakes in scanning-passive scatter carbon ion beam. The readers are encouraged to conduct the proposed cATAN method in this study to investigate the virtual source position in the Linac-based external electron beams and the proton beams.
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