Analytic and Monte Carlo calculations of dose-mean lineal energy for 1 MeV–1 GeV protons with application to radiation protection quality factor

Radiation quality for determining biological effects is commonly linked to the microdosimetric quantity lineal energy ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>y</mml:mi></mml:math> ) and to the dose-mean lineal energy ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>y</mml:mi> <mml:mtext>D</mml:mtext></mml:msub> </mml:math> ). Calculations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>y</mml:mi> <mml:mtext>D</mml:mtext></mml:msub> </mml:math> are typically performed by specialised Monte Carlo track-structure (MCTS) codes, which can be time-intensive. Thus, microdosimetry-based analytic models are potentially useful for practical calculations. Analytic model calculations of proton <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>y</mml:mi> <mml:mtext>D</mml:mtext></mml:msub> </mml:math> and radiation protection quality factor ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math> ) values in sub-micron liquid water spheres (diameter 10-1000 nm) over a broad energy range (1 MeV-1 GeV) are compared against MCTS simulations by PHITS, RITRACKS, and Geant4-DNA. Additionally, an improved analytic microdosimetry model is proposed. The original analytic model of Xapsos is refined and model parameters are updated based on Geant4-DNA physics model. Direct proton energy deposition is described by an alternative energy-loss straggling distribution and the contribution of secondary electrons is calculated using the dielectric formulation of the relativistic Born approximation. MCTS simulations of proton <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>y</mml:mi> <mml:mtext>D</mml:mtext></mml:msub> </mml:math> values using the latest versions of the PHITS, RITRACKS, and Geant4-DNA are reported along with the Monte Carlo Damage Simulation (MCDS) algorithm. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>y</mml:mi> <mml:mtext>D</mml:mtext></mml:msub> </mml:math> datasets are then used within the Theory of Dual Radiation Action (TDRA) to illustrate variations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math> with proton energy. By a careful selection of parameters, overall differences at the ~ 10% level between the proposed analytic model and the MCTS codes can be attained, significantly improving upon existing models. MCDS estimates of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>y</mml:mi> <mml:mtext>D</mml:mtext></mml:msub> </mml:math> are generally much lower than estimates from MCTS simulations. The differences of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math> among the examined methods are somewhat smaller than those of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>y</mml:mi> <mml:mtext>D</mml:mtext></mml:msub> </mml:math> . Still, estimates of proton <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math> values by the present model are in better agreement with MCTS-based estimates than the existing analytic models. An improved microdosimetry-based analytic model is presented for calculating proton <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>y</mml:mi> <mml:mtext>D</mml:mtext></mml:msub> </mml:math> values over a broad range of proton energies (1 MeV-1 GeV) and target sizes (10-1000 nm) in very good agreement with state-of-the-art MCTS simulations. It is envisioned that the proposed model might be used as an alternative to CPU-intensive MCTS simulations and advance practical microdosimetry and quality factor calculations in medical, accelerator, and space radiation applications.