On the concepts of dose-mean lineal energy, unrestricted and restricted dose-averaged LET in proton therapy

To calculate 3D distributions of microdosimetric-based restricted dose-averaged LET (LETd) and dose-mean lineal energy () in order to explore their similarities and differences between each other and with the traditional unrestricted LETd. Additionally, a new expression for optimum restricted LETd calculation is derived, allowing for disregarding straggling-associated functions in the classical microdosimetric theory. Restricted LETd and for polyenergetic beams can be obtained by integrating previously developed energy-dependent microdosimetric functions over the energetic spectrum of these beams. This calculation is extended to the entire calculation volume using an algorithm to determine spectral fluence. Equivalently, unrestricted LETd can be obtained integrating the stopping power curve on the spectrum. A new expression to calculate restricted LETd is also derived. Results for traditional and new formulas are compared for a clinical 100 MeV proton beam. Distributions of unrestricted LETd, restricted LETd and are analyzed for a prostate case, for microscopic spherical sites of 1 mu m and 10 mu m in diameter. Traditional and new expressions for restricted LETd remarkably agree, being the mean differences 0.05 +/- 0.04 keV mu m(-1) for the 1 mu m site and 0.05 +/- 0.02 keV mu m(-1) for the 10 mu m site. In the prostate case, the ratio between the maximum and the central value for central axis (CAX) profiles is around 2 for all the quantities, being the highest for restricted LETd for 1 mu m (2.17) and the lowest for for 1 mu m (1.78). Unrestricted LETd, restricted LETd and can be analytically computed and compared for clinical plans. Two important consequences of the calculation of are: (1) its distribution can be verified by directly measuring it in clinical beams; and (2), optimization of proton treatments based on these quantities is enabled as well as future developments of RBE models based on them.