A New Approach to Modeling the Microdosimetry of Proton Therapy Beams

{\bf Introduction}: To revisit the formulation of the mean chord length in microdosimetry and replace it by the particle mean free path appropriate for modelings in radio-biology. {\bf Methods}: We perform a collision-by-collision following by event-by-event Geant4 Monte Carlo simulation and calculate double-averaged stepping-length, $\langle\langle l \rangle\rangle$, for a range of target sizes from mm down to $\mu$m and depth in water. We consider $\langle\langle l \rangle\rangle$ to represent the particle mean free path. {\bf Results}: We show that $\langle\langle l \rangle\rangle$ continuously drops as a function of depth and asymptotically saturates to a minimum value in low energies, where it exhibits a universal scaling behavior, independent of particle nominal beam energy. We correlate $\langle\langle l \rangle\rangle$ to linear density of DNA damage, complexities of initial lethal lesions and illustrate a relative difference between predictive RBEs in model calculations using mean chord length vs. the proposed mean free path. We demonstrate consistency between rapid increase in RBE within and beyond the Bragg peak and $\langle\langle l \rangle\rangle$, a decreasing function of depth. {\bf Discussion and conclusion}: An interplay between localities in imparted energy at nano-meter scale and subsequent physio-chemical processes, causalities and pathways in DNA damage requires substitution of geometrical chord length of cell nuclei by mean-free path of proton and charged particles to account for a mean distance among sequential collisions in DNA materials.