## 3 Results
### Sensitivity of Results to Target Size  
Subsequently, the \(y_{D}\) values depicted in Figure 2 along with the full microdosimetric spectra \(f(y)\) and \(d(y)\) were used to determine the corresponding proton \(\overline{Q}\) based on the TDRA and ICRU Report 40 expressions Equations (28) and (30), respectively. Figure 3 presents the calculated \(\overline{Q}\) values for the different analytic models (X94, XO, Xcom) and MC data from the Geant4-DNA and TEPC simulation. The LET-based \(Q\) values recommended by ICRP Report 60 are also presented for comparison. Figure 4 presents the differences of the microdosimetric-based \(\overline{Q}\) values (TDRA, ICRU Report 40) against the LET-based ICRP Report 60 recommendations. Figure 5 presents \(y_{D}\) values for liquid water spheres of different diameters (10, 100, 1000 nm). The calculated values were from the Xcom analytical model (combination of [PERSON] et al. 1994 and [PERSON] et al. 1996 models) and the MC data for the different spheres were taken from Geant4-DNA [60]. The Xcom model represents the best available analytic model for calculations of \(y\) spectra (please see Appendix A for the fitted values of \(y_{D}\) and the \(\overline{Q}_{TDRA}\) of the Xcom model). It combines both the energy deposition and its fluctuations from ion (direct) and electron (indirect) events in the target site, as well as the effects of the finite range of \(\delta\)-rays. A form of this model has been utilised for determining \(y\) spectra in TEPC measurements [50; 51]. Similar to Figure 2, the available Geant4-DNA data for these spheres are limited up to 100 MeV. Figure 2: \(y_{D}\) values for a liquid water sphere of 1 \(\upmu\)m diameter as a function of proton energy calculated by the different analytic models examined (X94, Xcom, XO), and Monte Carlo data by the Geant4-DNA code [60] and TEPC simulations with PITS99-KURBUC [46]. Inset: Logarithmical presentation of the \(y_{D}\) values for the energy range of 1–100 MeV. Figure 4: Difference (%) of TDRA and ICRU Report 40 predictions for the proton \(\overline{Q}\) with input data calculated by different analytic models (X94, XO, Xcom) and Monte-Carlo codes (Geant4 DNA [60], TEPC simulations by PTTS99-KURBUC [46]). All microdosimetric calculations pertain to a liquid water sphere of 1 \(\upmu\)m diameter. The LETbased \(Q\) values recommended by ICRP Report 60 were used as baseline for comparison. Figure 3: Energy variation of the proton \(\overline{Q}\) based on different microdosimetric approaches (TDRA, ICRU Report 40) with input data from both analytic models (X94, XO, Xcom) and Monte-Carlo codes (Geant4-DNA [60], TEPC simulations by PTTS99-KURBUC [46]). The microdosimetric data pertain to a liquid water sphere of 1 \(\upmu\)m diameter. The LET-based \(Q\) values recommended by ICRP Report 60 are also shown for comparison. Using the \(y_{D}\) values of Figure 5, Figure 6 presents the corresponding \(\overline{Q}\) for sphere diameters of 100 nm and 10 nm using the TDRA approach, Equation (28), over the proton energy range of 1-100 MeV. The LET-based \(Q\) values from the ICRP60 recommendations are also shown for comparison. In Figure 7, the differences (%) of the TDRA-based \(\overline{Q}\) for 100 nm and 10 nm calculated by the Xcom analytic model and the Geant4-DNA MC data are presented with the LET-based \(Q\) values of ICRP Report 60 recommendations used as baseline. Figure 5: \(y_{D}\) values as a function of proton energy for liquid water spherical volumes of 10, 100 and 1000 nm diameter calculated by the Xcom analytic model (blue lines) and the MonteCarlo data of Geant4-DNA [56] (red lines). Figure 6: Energy variation of the proton \(\overline{Q}\) based on the TDRA approach with input data from the Xcom analytic model (red colour) and the Geant4 DNA Monte-Carlo code [60] (black colour). The microdosimetric data pertain to a liquid water spheres of 0.1 and 0.01 μm diameter. The LETbased \(Q\) values recommended by ICRP Report 60 are also shown for comparison (dark yellow colour).