## 2 Materials and Methods
### Generalized Analytic Microdosimetric Models  
#### 2.2.1 The [PERSON] et al. 1994 Model  
In this model (hereafter denoted as \"X94\"), the energy deposition inside the target volume is classified as either \"direct\" or \"indirect\". The former is related to the energy deposition by a primary ion crossing the target, whereas the latter is related to the energy deposition by \(\delta\)-rays produced by an ion passing outside the target [48]. The contribution of direct (ion) and indirect (\(\delta\)-rays) energy deposition events varies with the kinetic energy of the ion and the diameter of the target volume. For example, with increasing ion energy and/or volume size, the indirect contribution is enhanced by \(\delta\)-rays become more energetic and/or have a higher probability to reach the target volume. To a first approximation, when an ion crosses a volume, the energy deposited is proportional to the product of LET and the mean chord length \(\overline{l}\). However, for insufficient (e.g., approaching the nanometer scale), energy loss-straggling, path length fluctuationsand energy carried out of the volume by \(\delta\)-rays may become pronounced. It can be shown that the \(y_{D}\) can be calculated as a function of distributions of LET, path length and single-collision energy-transfer [48]. The corresponding expressions for the direct (ion, \(y_{D,ion}\)), indirect (\(\delta\)-ray, \(y_{D,el}\)) and total events are [48]:  
\[y_{D,ion}=\frac{1}{l}(f_{ion}L_{D,ion}s_{D,ion}+\delta_{2,ion}), \tag{5}\]  
\[y_{D,el}=\frac{1}{l}(L_{D,el}s_{D,el}+\delta_{2,el}), \tag{6}\]  
\[y_{D}=f_{ion}y_{D,ion}+(1-f_{ion})y_{D,el}, \tag{7}\]  
where \(f_{ion}\) is the fraction of energy deposited in the site by the primary ion. The calculation of each parameter (\(f_{ion}\), \(\delta_{2,ion}\), \(\delta_{2,el}\), \(L_{D,ion}\), \(L_{D,el},s_{D,ion}\), \(s_{D,el}\)) entering the above expressions, Equations (5)-(7), is discussed below. When an ion traverses a site, it deposits energy and generates \(\delta\)-rays (i.e., secondary electrons) that, subsequently, may deposit only a fraction (or all) of their energy inside the site. The fraction of ion energy that is retained inside the site may be described analytically by the spatially restricted LET. Then, it may be shown that [54; 55]:  
\[f_{ion}=\frac{\ln[\frac{T_{el,max}(\Delta+\Delta1+\Delta2)}{l^{2}}]}{2\ln[ \frac{T_{el,max}}{l}]}, \tag{8}\]  
where \(l\) is the mean excitation energy of the stopping-power of the medium, and \(T_{el,max}\) is the maximum energy that an ion can transfer to a secondary electron (\(\delta\)-ray):  
\[T_{el,max}=2.179~{}T_{ion}, \tag{9}\]  
with \(T_{ion}\) expressed in \((MeV/amu)\) and \(T_{el,max}\) in keV. \(\Delta\) is the energy of an electron with range equal to the mean chord length of the site, \(\Delta 1\) accounts for the energy transferred by escaping secondary electrons to the site and \(\Delta 2\) is the electronic excitation or ionization energy present in the atoms after the interaction with the ion (i.e., binding effects) when secondary electrons migrate outside the volume. It holds that [48; 49; 54]:  
\[(\Delta 1+\Delta 2)=\left(1-\frac{\Delta}{T_{el,max}}\right)(I+\Delta). \tag{10}\]  
The dose-weighted mean-energy deposited in the site by ions in a single inelastic collision (\(\delta_{2,ion}\)) is obtained from:  
\[\delta_{2,ion}=\frac{Min[\Delta,T_{el,max}]}{2\ln[\frac{Min[\Delta,T_{el,max} ]}{l}]},Min[\Delta,T_{el,max}]\leq 1\text{keV} \tag{11}\]  
\[\delta_{2,ion}=A\Delta^{B},~{}Min[\Delta,T_{el,max}]\geq 1\text{keV}, \tag{12}\]  
with \(\delta_{2,ion}\) in units of keV. For protons in water vapor, the values \(A=0.190\) and \(B=0.621\) have been suggested [48].