## 4 Discussion  
As it can be seen from Figure 2 the calculated \(y_{D}\) values for 1 um target size and for proton energies 1-250 MeV calculated by the different analytical models (X94, Xcom, XO) followed the same trend over the whole energy range. The difference between the X94 and Xcom models reached up to 10% while the XO model was consistently higher by up to 40-50% in the energy range of 50-250 MeV. This is due to the different calculation of the deposited energy of secondary electrons that affects both the indirect (electron) events and the \(\delta\)-ray escape from the volume. The secondary electron spectrum is treated differently in the XO and Xcom models; specifically, in the XO model, the full slowing-down spectrum  
Figure 10: Difference (%) of TDRA-based proton \(\overline{Q}\) between unit density gaseous and liquid water spheres of varying diameter calculated by the different analytic models. The liquid water values were used as baseline for comparison. Figure 9: Difference (%) of \(y_{D}\) for unit-density gaseous and liquid water spherical volumes of different diameter (10, 100, 1000, 3000 nm) calculated by the different analytic models. The liquid water values were used as baseline for comparison. is used. For the 1 \(\upmu\)m sphere, and combining Equations (16) and (24), the mean LET of the secondary electrons calculated by the XO model inside the sphere target is larger than the mean LET calculated by the Xcom model. Higher LET means more energy deposited inside the sphere, so there was a slight increase in the total \(y_{D}\) value for the XO model. The proton energy deposition is treated similarly in both models. Unfortunately, there are no updated references for these discrepancies among the models except the one already cited in out text [48; 49; 50; 51; 52]. To our knowledge, the present work represents the first time that all models were combined and compared with \(\overline{Q}\) calculations. Importantly, the MC literature data (Geant4-DNA, PITS99-KURBUC) on proton lineal energy has followed similar trends with the analytic calculations. Specifically, the differences between the analytical models (X94 and Xcom) and the MC data (Geant4-DNA, PITS99-KURBUC) did not exceed 10% up to 50 MeV, increasing up to 15-30% in the energy range of 50-100 MeV. The magnitude of this difference is comparable to the difference among the MC data (i.e., Geant4-DNA vs. PITS99-KURBUC), which is up to 10% for proton energies 1-50 MeV and up to 25% for the 50-100 MeV range. Extensive information about the physical models and assumptions made in the MC simulations by Geant4-DNA and PITS99-KURBUC codes can be found in the literature [46; 60]. The comparison of the microdosimetric-based proton \(\overline{Q}\) values (using TDRA and ICRU Report 40 methodologies, both at 1 \(\upmu\)m sphere) against the LET-based ICRP Report 60 recommendations showed noticeable differences (>10-20%) below about 50-75 MeV, reaching a factor of 2-3 at -5 MeV (see Figures 2 and 3). It is noteworthy that these differences increased further when MC literature data from Geant4-DNA and TEPC simulations from PITS99-KURBUC were used to calculate \(\overline{Q}\) by the TDRA. In particular, at energies relevant to the Bragg peak region (<20-30 MeV), the deviations of both the analytic and MC microdosimetric-based \(\overline{Q}\) values from ICRP Report 60 agreed above 20-50%. It follows from Figure 5 that, as the sphere diameter decreased to 100 and 10 nm, the differences in the \(y_{D}\) values between the analytical model calculations and the Geant4-DNA MC data became more pronounced at lower energies (~5-10 MeV). This is expected given that at smaller volumes, the details of how the energy-loss processes (including the effect of straggling and \(\delta\)-ray production and transport) are considered in each approach (analytical models vs. MC simulations) become more important for the energy imparted to the site.