## 2 Materials and Methods
### Calculations of LET\({}_{\text{D}}\)  
Empirical models use the macroscopic quantity LET to characterize the radiation quality. In practice, LET is commonly calculated through the electronic stopping power (\(S_{el}\)), which is defined as the mean energy loss of a charged particle per unit path length due to electronic collisions. For a clinical beam which consists of many particles with different energies, the fluence-averaged LET (\(\text{LET}_{\text{F}}\)) or the dose-averaged LET (\(\text{LET}_{\text{D}}\)) are used instead, and can be calculated as follows:  
\[\text{LET}_{\text{F}}=\int_{0}^{+\infty}S_{el}(E)\ \varphi(E)\ dE, \tag{3}\]  
\[\text{LET}_{\text{D}}=\frac{\int_{0}^{+\infty}S_{el}^{2}(E)\ \varphi(E)dE}{\int_{0}^{+\infty}S_{el}(E)\ \varphi(E)\ dE}=\frac{1}{\text{ LET}_{\text{F}}}\int_{0}^{+\infty}S_{el}^{2}(E)\ \varphi(E)dE, \tag{4}\]  
where \(\varphi(E)\) represents the local energy spectrum. For monoenergetic proton beams, the energy spectrum is narrow; therefore, \(\text{LET}_{\text{F}}\approx\text{LET}_{\text{D}}\)[33]. In clinical applications, there exist various methodological approaches for calculating average LET (i.e., \(\text{LET}_{\text{F}}\) or \(\text{LET}_{\text{D}}\)) which are based either on experimental data and analytic models or the interpretation and analysis of measurements through Monte Carlo simulations. For assessments of biological radiation damage, it is the dose-averaged LET (\(\text{LET}_{\text{D}}\)) that is most relevant. Different methods of interpretation can substantially influence the reported values of average LET. One significant difference among these methods lies in the types of particles that they consider in the calculation of \(\text{LET}_{\text{F}}\) or \(\text{LET}_{\text{D}}\). Some studies take into account only the primary particles, while others include the secondary ejected electrons, secondary protons or/and heavier particles produced from nuclear reactions, like hydrogen or helium isotopes [34; 35]. Other differences arise from the choice of medium for calculating the average LET. Most calculations pertain to liquid water, and therefore using other tissue-equivalent material may slightly affect the LET spectrum. If other non-unit density materials are examined, the corresponding averaged LET values must be normalized to unit density in order to be suitable for clinical applications. Additionally, if MC codes are employed, discrepancies arise, since the different simulation procedures and physics lists vary among the different MC codes and impact the calculations of LET [34; 35]. To eliminate the methodological uncertainties surrounding the calculation of \(\text{LET}_{\text{D}}\)[34; 35], which will further obscure the influence of the choice between \(\text{LET}_{\text{D}}\) and \(y_{\text{D}}\), in the present work we determine \(\text{LET}_{\text{D}}\) from \(y_{\text{D}}\) using the approach of [PERSON] [33], which was further refined by [PERSON] [36]. Microdosimetric Calculation of \(\text{LET}_{\text{D}}\)  
For a first approximation, the average energy deposited inside a microscopic volume from an ion that crosses it (usually referred to as a direct event) depends on the LET of the incident ion and the average chord length of the volume (\(\bar{l}\)):  
\[\bar{z}= \overline{\text{LET}\ \bar{l}}. \tag{5}\]To account for the fraction of the ion's energy loss that is deposited outside the volume due to the escape of secondary electrons, the above expression was refined by [PERSON] as [28; 29; 37]:  
\[\overline{\epsilon}=f_{ion}\overline{\text{LET}}\,\overline{l}\,, \tag{6}\]  
where the coefficient (\(f_{ion}\)) represents the fraction of the ion's energy loss that is imparted in the volume of interest. Fluctuations of the deposited energy can be observed due to LET variations or the different path lengths that the particles may follow inside of the volume of interest. However, ions crossing the volume with the same LET and path length may still deposit different energy values due to the energy loss straggling.