## 4 Discussion
The present models (Xcom, X94, XO) are generally recommended for diameters greater than several nanometers, specifically for diameters much larger than the penetration range of electrons with energy equal to the mean excitation energy of the medium (I) which, in this case, translates to d > 2 nm. In general, for space applications, diameters in the range of 1000-3000 nm are most relevant to experimental TEPC measurements (e.g., ISS). In addition, several studies of proton (and carbon) RBE in hadron therapy have suggested that \(y_{D}\) should be determined in spheres with 10-15 nm diameter [15]. The present work reveals that the results of analytic models (and MC simulations) already diverge below 100 nm. Therefore, even larger discrepancies are expected for spheres with d < 10 nm. The influence of sphere size on the TDRA \(\overline{Q}\) values seem to be significant (>10%) only below ~50 MeV (see Figures 6-8). The \(y_{D}\) values of high-energy protons were found to be sensitive to the choice of the sphere size due to the contribution of the energetic secondary electrons (\(\delta\)-rays). Specifically, for high-energy protons, there are more indirect events (Equation (24)), and the mean LET of these electrons is often equal to or even larger than that of the high-energy proton, resulting in a strong increase in \(y_{D}\). Therefore, when calculating the TDRA quality factor for the low-energy protons, by normalizing to a reference radiation (Equation (28))--which, in this study, is a high-energy proton (100 MeV)--it results in a large variation of \(\overline{Q}\). Calculations of the \(\overline{Q}\) based on ICRU Report 40 have also been performed for nanometer targets (10-100 nm). However, only the values for 1 \(\upmu\)m are shown since Equation (29) is deduced for this particular sphere size. It is somewhat surprising that with decreasing sphere size (from 1 \(\upmu\)m to 100 and 10 nm), the \(\overline{Q}\) values by TDRA are in better agreement with the ICRP Report 60 recommendations than the \(\overline{Q}\) values by TDRA based on the conventional 1 \(\upmu\)m sphere. On the other hand, largersphere diameters (>1 \(\upmu\)m) were found to yield even higher deviations from ICRP Report 60 (not presented). The present work is, to our knowledge, the first study that systematically combines and compares the most established analytical microdosimetric models for proton \(\overline{Q}\) calculations. Proton quality factors are continuously used by various space organizations and the radiotherapy community to assess the carcinogenic risk of irradiated individuals, e.g., for long-duration manned missions and/or in the context of the organs at risks (OAR) in proton therapy, respectively. Although LET-based calculations of \(Q\) are generally preferred in practical applications due to the availability of LET values for various ions over a wide energy range, it is recognized that microdosimetric-based calculations offer a scientifically sounder approach (ICRU 40). However, the difficulties of working with microdosimetric quantities, which are stochastic in nature, has led to a situation where experimental microdosimetric measurements with TEPC are often approximated by some average LET values before their practical implementation. The combination of available analytic models discussed in the present work further supports the use of microdosimetric-based approaches to proton quality factor problems as a viable alternative to LET-based approaches. An important finding of the present study is the energy range over which LET-based and \(y\)-based proton \(\overline{Q}\) values may differ substantially. Specifically, the low-energy range (1-50 MeV) corresponds to protons that comprise the Bragg peak region, where the absorbed dose becomes maximum and variation of \(\overline{Q}\) are expected to have a major impact in absolute terms.