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

Purpose 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 analyticalLETdcalculation 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. Methods InPBA‐dLET,LETvalues were assigned separately for each individual dose kernel based on thePBA. 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 constantLETdvalue for the lateral distribution of a pencil beam, the actualLETdincreases away from the beam axis, because there are more scattered and therefore lower energy protons with higher stopping powers. To reflect thisLETdbehavior, we have assumed that theLETs of primary and halo components can take different values (LETpandLEThalo), which vary only along the depth direction. The values of dual‐LETkernels were determined such that thePBA‐dLETreproduced theMCS‐generatedLETddistribution 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 theLETddistributions and meanLETd(LETd,mean) values of targets and organs at risk betweenPBA‐dLETandMCS. Both homogeneous phantom and patient geometries (prostate, liver, and lung cases) were used to validate the present method. Results In the homogeneous phantom, theLETdprofiles obtained by the dual‐LETkernels agree well with theMCSresults 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, theLETdprofiles calculated with the developed method reproducesMCSwith the similar accuracy as in the homogeneous phantom. The maximum differences inLETd,meanfor each structure between thePBA‐dLETand theMCSwere 0.06 keV/μm in homogeneous phantoms and 0.08 keV/μm in patient geometries under all tested conditions, respectively. Conclusions We confirmed that the dual‐LET‐kernel model well reproduced theMCS, not only in the homogeneous phantom but also in complex patient geometries. The accuracy of theLETdwas 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.