We study the time dependence of the grain size distribution N͑r , t͒ during crystallization of a d-dimensional solid. A partial differential equation, including a source term for nuclei and a growth law for grains, is solved analytically for any dimension d. We discuss solutions obtained for processes described by the KolmogorovAvrami-Mehl-Johnson model for random nucleation and growth ͑RNG͒. Nucleation and growth are set on the same footing, which leads to a time-dependent decay of both effective rates. We analyze in detail how model parameters, the dimensionality of the crystallization process, and time influence the shape of the distribution. The calculations show that the dynamics of the effective nucleation and effective growth rates play an essential role in determining the final form of the distribution obtained at full crystallization. We demonstrate that for one class of nucleation and growth rates, the distribution evolves in time into the logarithmic-normal ͑lognor-mal͒ form discussed earlier by Bergmann and Bill ͓J. Cryst. Growth 310, 3135 ͑2008͔͒. We also obtain an analytical expression for the finite maximal grain size at all times. The theory allows for the description of a variety of RNG crystallization processes in thin films and bulk materials. Expressions useful for experimental data analysis are presented for the grain size distribution and the moments in terms of fundamental and measurable parameters of the model.

The small fields and sharp gradients typically encountered in proton radiosurgery require high spatial resolution dosimetric measurements, especially below 1–2 cm diameters. Radiochromic film provides high resolution, but requires postprocessing and special handling. Promising alternatives are diode detectors with small sensitive volumes (SV) that are capable of high resolution and real‐time dose acquisition. In this study we evaluated the PTW PR60020 proton dosimetry diode using radiation fields and beam energies relevant to radiosurgery applications. Energies of 127 and 157 MeV (9.7 to 15 cm range) and initial diameters of 8, 10, 12, and 20 mm were delivered using single‐stage scattering and four modulations (0, 15, 30, and 60 mm) to a water tank in our treatment room. Depth dose and beam profile data were compared with PTW Markus N23343 ionization chamber, EBT2 Gafchromic film, and Monte Carlo simulations. Transverse dose profiles were measured using the diode in "edge‐on" orientation or EBT2 film. Diode response was linear with respect to dose, uniform with dose rate, and showed an orientation‐dependent (i.e., beam parallel to, or perpendicular to, detector axis) response of less than 1%. Diode vs. Markus depth‐dose profiles, as well as Markus relative dose ratio vs. simulated dose‐weighted average lineal energy plots, suggest that any LET‐dependent diode response is negligible from particle entrance up to the very distal portion of the SOBP for the energies tested. Finally, while not possible with the ionization chamber due to partial volume effects, accurate diode depth‐dose measurements of 8, 10, and 12 mm diameter beams were obtained compared to Monte Carlo simulations. Because of the small SV that allows measurements without partial volume effects and the capability of submillimeter resolution (in edge‐on orientation) that is crucial for small fields and high‐dose gradients (e.g., penumbra, distal edge), as well as negligible LET dependence over nearly the full the SOBP, the PTW proton diode proved to be a useful high‐resolution, real‐time metrology device for small proton field radiation measurements such as would be encountered in radiosurgery applications.PACS numbers: 87.56.‐v, 87.56.jf, 87.56.Fc

The purpose of this study was to evaluate the effectiveness of full three‐dimensional (3D) gamma algorithm for spot scanning proton fields, also referred to as pencil beam scanning (PBS) fields. The difference between the full 3D gamma algorithm and a simplified two‐dimensional (2D) version was presented. Both 3D and 2D gamma algorithms are used for dose evaluations of clinical proton PBS fields. The 3D gamma algorithm was implemented in an in‐house software program without resorting to 2D interpolations perpendicular to the proton beams at the depths of measurement. Comparison between calculated and measured dose points was carried out directly using Euclidian distance in 3D space and the dose difference as a fourth dimension. Note that this 3D algorithm faithfully implemented the original concept proposed by Low et al. (1998) who described gamma criterion using 3D Euclidian distance and dose difference. Patient‐specific proton PBS plans are separated into two categories, depending on their optimization method: single‐field optimization (SFO) or multifield optimized (MFO). A total of 195 measurements were performed for 58 SFO proton fields. A MFO proton plan with four fields was also calculated and measured, although not used for treatment. Typically three different depths were selected from each field for measurements. Each measurement was analyzed by both 3D and 2D gamma algorithms. The resultant 3D and 2D gamma passing rates are then compared and analyzed. Comparison between 3D and 2D gamma passing rates of SFO fields showed that 3D algorithm does show higher passing rates than its 2D counterpart toward the distal end, while little difference is observed at depths away from the distal end. Similar phenomenon in the lateral penumbra was well documented in photon radiation therapy, and in fact brought about the concept of gamma criterion. Although 2D gamma algorithm has been shown to suffice in addressing dose comparisons in lateral penumbra for photon intensity‐modulation radiation therapy (IMRT) plans, results here showed that a full 3D algorithm is required for proton dose comparisons due to the existence of Bragg peaks and distal penumbra. A MFO proton plan with four fields was also measured and analyzed. Sharp dose gradients exist in MFO proton fields, both in the middle of the modulation and toward the most distal layers. Decreased 2D gamma passing rates at locations of high dose gradient are again observed as in the SFO fields. Results confirmed that a full 3D algorithm for gamma criterion is needed for proton PBS plan's dose comparisons. The 3D gamma algorithm is implemented by an in‐house software program. Patient‐specific proton PBS plans are measured and analyzed using both 3D and 2D gamma algorithms. For measurements performed at depths with large dose gradients along the beam direction, gamma comparison passing rates using 2D algorithm is lower than those obtained with the full 3D algorithm.PACS number: 87.53.Bn, 87.53.Jw, 87.55.de, 87.55.kd, 87.55.ne, 87.55.Qr

Radiographic film dosimetry suffers from its energy dependence in proton dosimetry. This study sought to develop a method of measuring proton beams by the film and to evaluate film response to proton beams for the constancy check of depth dose (DD). It also evaluated the film for profile measurements. To achieve this goal, from DDs measured by film and ion chamber (IC), calibration factors (ratios of dose measured by IC to film responses) as a function of depth in a phantom were obtained. These factors imply variable slopes (with proton energy and depth) of linear characteristic curves that relate film response to dose. We derived a calibration method that enables utilization of the factors for acquisition of dose from film density measured at later dates by adapting to a potentially altered processor condition. To test this model, the characteristic curve was obtained by using EDR2 film and in‐phantom film dosimetry in parallel with a 149.65 MeV proton beam, using the method. An additional validation of the model was performed by concurrent film and IC measurement perpendicular to the beam at various depths. Beam profile measurements by the film were also evaluated at the center of beam modulation. In order to interpret and ascertain the film dosimetry, Monte Carlos simulation of the beam was performed, calculating the proton fluence spectrum along depths and off‐axis distances. By multiplying respective stopping powers to the spectrum, doses to film and water were calculated. The ratio of film dose to water dose was evaluated. Results are as follows. The characteristic curve proved the assumed linearity. The measured DD approached that of IC, but near the end of the spread‐out Bragg peak (SOBP), a spurious peak was observed due to the mismatch of distal edge between the calibration and measurement films. The width of SOBP and the proximal edge were both reproducible within a maximum of 5 mm; the distal edge was reproducible within 1 mm. At 5 cm depth, the dose was reproducible within 10%. These large discrepancies were identified to have been contributed by film processor uncertainty across a layer of film and the misalignment of film edge to the frontal phantom surface. The deviations could drop from 5 to 2 mm in SOBP and from 10% to 4.5% at 5 cm depth in a well‐controlled processor condition (i.e., warm up). In addition to the validation of the calibration method done by the DD measurements, the concurrent film and IC measurement independently validated the model by showing the constancy of depth‐dependent calibration factors. For profile measurement, the film showed good agreement with ion chamber measurement. In agreement with the experimental findings, computationally obtained ratio of film dose to water dose assisted understanding of the trend of the film response by revealing relatively large and small variances of the response for DD and beam profile measurements, respectively. Conclusions are as follows. For proton beams, radiographic film proved to offer accurate beam profile measurements. The adaptiv...

Earlier detection of cancerous lesions due to improved imaging technology and patient screening is fueling a trend in radiation medicine to irradiate increasingly smaller targets. Proton radiosurgery has advantages compared to other external-beam radiation modalities when treating small lesions, including the delivery of high doses to target volumes with three to four treatment beams. However, for small-diameter beams (<1.0 cm), beam broadening due multiple Coulomb scattering (MCS) leads to degradation of the Bragg peak that manifests itself as lower peak-to-entrance dose (P/E) ratios and reduced rates of dose delivery to the radiation target. The effects of beam degradation are typically ameliorated by employing additional treatment beams, but this leads to increased integral dose and even longer treatment times that work against the advantages of proton radiosurgery. Magnetic focusing immediately upstream from the patient could help compensate for the effects of MCS, potentially leading to radiosurgical treatments with lower entrance doses, fewer beams, reduced integral doses, improved therapeutic ratios, and decreased treatment times (McAuley et al 2013(McAuley et al , 2015a(McAuley et al , 2018.