Segment-averaged LET concept and analytical calculation from microdosimetric quantities in proton radiation therapy.

This work introduces the concept of segment-averaged linear energy transfer (LET) as a new approach to average distributions of LET of proton beams based on a revisiting of microdosimetry theory. The concept of segment-averaged LET is then used to generate an analytical model from Monte Carlo simulations data to perform fast and accurate calculations of LET distributions for proton beams. The distribution of energy imparted by a proton beam into a representative biological structure or site is influenced by the distributions of (a) LET, (b) segment length, which is the section of the proton track in the site, and (c) energy straggling of the proton beam. The distribution of LET is thus generated by the LET of each component of the beam in the site. However, the situation when the LET of each single proton varies appreciably along its path in the site is not defined. Therefore, a new distribution can be obtained if the particle track segment is decomposed into smaller portions in which LET is roughly constant. We have called "segment distribution" of LET the one generated by the contribution of each portion. The average of that distribution is called segment-averaged LET. This quantity is obtained in the microdosimetry theory from the average and standard deviation of the distributions of energy imparted to the site, segment length, and energy imparted per collision. All this information is calculated for protons of clinically relevant energies by means of Geant4-DNA microdosimetric simulations. Finally, a set of analytical functions is proposed for each one of the previous quantities. The presented model functions are fitted to data from Geant4-DNA simulations for monoenergetic beams from 100 keV to 100 MeV and for spherical sites of 1, 5, and 10 μm in diameter. The average differences along the considered energy range between calculations based on our analytical models and MC for segment-averaged dose-averaged restricted LET are -0.2 ± 0.7 keV/μm for the 1 μm case, 0.0 ± 0.9 keV/μm for the 5 μm case, and -0.3 ± 1.1 keV/μm for the 10 μm case, respectively. All average differences are below the average standard deviation (1σ) of the MC calculations. A new way of averaging LET for a proton beam is performed to incorporate the effects produced by the variation of stopping power of each individual proton along microscopic biological structures. An analytical model based on MC simulations allows for fast and accurate calculations of segment-averaged dose-averaged restricted LET for proton beams, which otherwise would need to be calculated from exhaustive MC simulations of clinical plans.

Bertolet A ，Baratto-Roldán A ，Cortés-Giraldo MA ，Carabe-Fernandez A ... -  《-》

Dose-averaged LET calculation for proton track segments using microdosimetric Monte Carlo simulations.

There is an increasing interest in calculating linear energy transfer (LET) distributions for proton therapy treatments in order to assess the influence of this quantity in biological terms. Microdosimetric Monte Carlo (MC) simulations are useful tools to calculate dose-averaged LET, as this has been broadly proposed as the most adequate quantity to characterize these biological effects. However, a straightforward uniform sampling of the scoring site turns out to be computationally unaffordable. In contrast, some issues have been pointed out with the more efficient weighted sampling approach, frequently used in literature. Here, we address the issues associated with the latter method and propose adequate corrections to achieve reliable calculations of dose-averaged LET values from microdosimetry. Proton track structures have been simulated with Geant4-DNA considering two different approaches. One version employs a uniform sampling for placing the spherical site and is used as the reference. The other one uses a weighted sampling by considering the spatial distribution of transfer points. Some corrections are proposed for calculating a dose-averaged LET comparable to the reference case. An additional MC approach is proposed to obtain the weighted mean of the energy imparted per electronic collision of the proton within the site, the δ 2 function, related to the straggling distribution, as an intermediate step in the LET calculation. Energy imparted per event distributions are different when employing either sampling methods, due to the different geometrical randomness. We have found an agreement below (0.15 ± 0.05) keV/μm in the worst case for uniform and weighted methods in dose-averaged LET values when the weighted sampling results are corrected according to our proposal. Our analysis is restricted to spherical sites of 1 and 10 μm diameter and monoenergetic beams in the range from 2 to 90 MeV. This work shows a reliable and computational-efficient method to perform calculations of track segment dose-averaged LET using MC simulations for proton therapy beams, including the necessary considerations for obtaining the straggling distribution characteristics. The validity of this approach remains as long as the stopping power of the proton can be considered as constant along its track within the site.

Bertolet A ，Baratto-Roldán A ，Barbieri S ，Baiocco G ，Carabe A ，Cortés-Giraldo MA ... -  《-》

An analytical dose-averaged LET calculation algorithm considering the off-axis LET enhancement by secondary protons for spot-scanning proton therapy.

To evaluate the biological effects of proton beams as part of daily clinical routine, fast and accurate calculation of dose-averaged linear energy transfer (LETd ) is required. In this study, we have developed the analytical LETd calculation method based on the pencil-beam algorithm (PBA) considering the off-axis enhancement by secondary protons. This algorithm (PBA-dLET) was then validated using Monte Carlo simulation (MCS) results. In PBA-dLET, LET values were assigned separately for each individual dose kernel based on the PBA. For the dose kernel, we employed a triple Gaussian model which consists of the primary component (protons that undergo the multiple Coulomb scattering) and the halo component (protons that undergo inelastic, nonelastic and elastic nuclear reaction); the primary and halo components were represented by a single Gaussian and the sum of two Gaussian distributions, respectively. Although the previous analytical approaches assumed a constant LETd value for the lateral distribution of a pencil beam, the actual LETd increases away from the beam axis, because there are more scattered and therefore lower energy protons with higher stopping powers. To reflect this LETd behavior, we have assumed that the LETs of primary and halo components can take different values (LETp and LEThalo ), which vary only along the depth direction. The values of dual-LET kernels were determined such that the PBA-dLET reproduced the MCS-generated LETd distribution in both small and large fields. These values were generated at intervals of 1 mm in depth for 96 energies from 70.2 to 220 MeV and collected in the look-up table. Finally, we compared the LETd distributions and mean LETd (LETd,mean ) values of targets and organs at risk between PBA-dLET and MCS. Both homogeneous phantom and patient geometries (prostate, liver, and lung cases) were used to validate the present method. In the homogeneous phantom, the LETd profiles obtained by the dual-LET kernels agree well with the MCS results except for the low-dose region in the lateral penumbra, where the actual dose was below 10% of the maximum dose. In the patient geometry, the LETd profiles calculated with the developed method reproduces MCS with the similar accuracy as in the homogeneous phantom. The maximum differences in LETd,mean for each structure between the PBA-dLET and the MCS were 0.06 keV/μm in homogeneous phantoms and 0.08 keV/μm in patient geometries under all tested conditions, respectively. We confirmed that the dual-LET-kernel model well reproduced the MCS, not only in the homogeneous phantom but also in complex patient geometries. The accuracy of the LETd was largely improved from the single-LET-kernel model, especially at the lateral penumbra. The model is expected to be useful, especially for proper recognition of the risk of side effects when the target is next to critical organs.

Hirayama S ，Matsuura T ，Ueda H ，Fujii Y ，Fujii T ，Takao S ，Miyamoto N ，Shimizu S ，Fujimoto R ，Umegaki K ，Shirato H ... -  《-》

A kernel-based algorithm for the spectral fluence of clinical proton beams to calculate dose-averaged LET and other dosimetric quantities of interest.

To introduce a new analytical methodology to calculate quantities of interest in particle radiotherapy inside the treatment planning system. Models are proposed to calculate dose-averaged LET (LETd) in proton radiotherapy. A kernel-based approach for the spectral fluence of particles is developed by means of analytical functions depending on depth and lateral position. These functions are obtained by fitting them to data calculated with Monte Carlo (MC) simulations using Geant4 in liquid water for energies from 50 to 250 MeV. Contributions of primary, secondary protons and alpha particles are modeled separately. Lateral profiles and spectra are modeled as Gaussian functions to be convolved with the fluence coming from the nozzle. LETd is obtained by integrating the stopping power curves from the PSTAR and ASTAR databases weighted by the spectrum at each position. The fast MC code MCsquare is employed to benchmark the results. Considering the nine energies simulated, fits for the functions modeling the fluence in-depth provide an average R 2 equal to 0.998, 0.995 and 0.986 for each one of the particles considered. Fits for the Gaussian lateral functions yield average R 2 of 0.997, 0.982 and 0.993, respectively. Similarly, the Gaussian functions fitted to the computed spectra lead to average R 2 of 0.995, 0.938 and 0.902. LETd calculation in water shows mean differences of -0.007 ± 0.008 keV/μm with respect to MCsquare if only protons are considered and 0.022 ± 0.007 keV/μm including alpha particles. In a prostate case, mean difference for all voxels with dose >5% of prescribed dose is 0.28 ± 0.23 keV/μm. This new spectral fluence-based methodology allows for simultaneous calculations of quantities of interest in proton radiotherapy such as dose, LETd or microdosimetric quantities. The method also enables the inclusion of more particles by following an analogous process.

Bertolet A ，Cortés-Giraldo MA ，Souris K ，Carabe A ... -  《-》

Dose- rather than fluence-averaged LET should be used as a single-parameter descriptor of proton beam quality for radiochromic film dosimetry.

The dose response of Gafchromic EBT3 films exposed to proton beams depends on the dose, and additionally on the beam quality, which is often quantified with the linear energy transfer (LET) and, hence, also referred to as LET quenching. Fundamentally different methods to determine correction factors for this LET quenching effect have been reported in literature and a new method using the local proton fluence distribution differential in LET is presented. This method was exploited to investigate whether a more practical correction based on the dose- or fluence-averaged LET is feasible in a variety of clinically possible beam arrangements. The relative effectiveness (RE) was characterized within a high LET spread-out Bragg peak (SOBP) in water made up by the six lowest available energies (62.4-67.5 MeV, configuration " b 1 ") resulting in one of the highest clinically feasible dose-averaged LET distributions. Additionally, two beams were measured where a low LET proton beam (252.7 MeV) was superimposed on " b 1 ", which contributed either 50% of the initial particle fluence or 50% of the dose in the SOBP, referred to as configuration " b 2 " and " b 3 ," respectively. The proton LET spectrum was simulated with GATE/Geant4 at all measurement positions. The net optical density change differential in LET was integrated over the local proton spectrum to calculate the net optical density and therefrom the beam quality correction factor. The LET dependence of the film response was accounted for by an LET dependence of one of the three parameters in the calibration function and was determined from inverse optimization using measurement " b 1 ." This method was then validated on the measurements of " b 2 " and " b 3 " and subsequently used to calculate the RE at 900 positions in nine clinically relevant beams. The extrapolated RE set was used to derive a simple linear correction function based on dose-averaged LET ( L d ) and verify the validity in all points of the comprehensive RE set. The uncorrected film dose deviated up to 26% from the reference dose, whereas the corrected film dose agreed within 3% in all three beams in water (" b 1 ", " b 2 " and " b 3 "). The LET dependence of the calibration function started to strongly increase around 5 keV/μm and flatten out around 30 keV/μm. All REs calculated from the proton fluence in the nine simulated beams could be approximated with a linear function of dose-averaged LET (RE = 1.0258-0.0211 μm/keV L d ). However, no functional relationship of RE- and fluence-averaged LET could be found encompassing all beam energies and modulations. The film quenching was found to be nonlinear as a function of proton LET as well as of the dose-averaged LET. However, the linear relation of RE on dose-averaged LET was a good approximation in all cases. In contrast to dose-averaged LET, fluence-averaged LET could not describe the RE when multiple beams were applied.

Resch AF ，Heyes PD ，Fuchs H ，Bassler N ，Georg D ，Palmans H ... -  《-》