## 2 Materials and Methods  
### MIRD Range-Energy Relations  
MIRD scheme [40] is a formalism for absorbed dose calculations of internally distributed radionuclides. The mean absorbed dose in target region \(r_{k}\) from source region \(r_{h}\) is given by the equation:  
\[\overline{D}(r_{k}\leftarrow r_{h})=\tilde{A}_{h}S(r_{k}\leftarrow r_{h}), \tag{1}\]  
where \(\tilde{A}_{h}\) is the cumulated activity in the source region \(r_{h}\) and \(S\) is the mean absorbed dose in the target region per unit of cumulative activity in the source region. The S factor can be expressed as  
\[S(r_{k}\leftarrow r_{h})=\sum_{i}\frac{\Delta_{i}\varphi_{i}(r_{k}\gets r _{h})}{m_{k}}, \tag{2}\]  
where \(\varphi_{i}\) is the absorbed fraction, which is the fraction of energy emitted in region \(r_{h}\), and it is absorbed in region \(r_{k}\); \(\Delta_{i}\) is the mean energy emitted per nuclear transition for the i-th type of emission; and \(m_{k}\) is the mass of the target. S factors are provided by MIRD Committee for most radionuclides, from organ level [9] to cellular level [41]. Also, \(S\) factors are available for monoenergetic photons [6; 8; 37] and electrons [34; 36]. For microscopic dimensions (mm to mm) relevant to sub-cellular, cellular, and multi-cellular domains, MIRD adopts the continuous slowing down approximation (CSDA) to calculate S factors using analytic range-energy relations [41]. The CSDA assumes that the energy loss of the primary particle is \"continuous\" and neglects the finite range of secondary electrons (i.e., it assumes that the energy of secondary electrons is deposited at the point of their production). Then, for a uniformly distributed radiation source inside a spherical target, the absorbed fraction that enters Equation (2) may be calculated from the DPK using the following convolution integral [1]:  
\[\varphi_{i}(r_{k}\leftarrow r_{h})=\int\Psi_{r_{k}\leftarrow r_{h}}(r)F\left( \frac{r}{r_{0}}\right)d\left(\frac{r}{r_{0}}\right), \tag{3}\]  
where \(\Psi_{rk\leftarrow rh}\) is the geometric reduction factor and \(F(r/r_{0})\) is the (scaled dimensionless) DPK. The geometric factor is the mean probability that a randomly directed vector of length \(r\) that starts from a random point within the source region \(r_{h}\) ends within the target region \(r_{k}\). For charged particles (e.g., \(\beta\)-emitters), Equation (3) may be recast in terms of the stopping power as follows [1]:  
\[\varphi_{i}(r_{k}\leftarrow r_{h})=\int\Psi_{r_{k}\leftarrow r_{h}}(r)\frac{1 }{E_{i}}\frac{dE}{dR}\bigg{|}_{R(E_{i})-r}dr, \tag{4}\]  
where \(E_{i}\) is the initial energy of the i-th particle, and \(dE/dR\mid_{R(E)+r}\) is the energy-loss rate (i.e., similar to the stopping power) evaluated at \(R(E_{i})-r\), which is the residual range of a particle with initial energy \(E_{i}\) after passing a distance of \(r\) through the medium [41]. The MIRD Committee suggests [5] the use of [PERSON]'s range-energy relations for determiningin Equation (4) the stopping power dE/dR and the residual range R(E\({}_{\rm i}\))-r [42]. [PERSON] experimentally determined the range of electrons in some low-Z solid materials with energies from 20 eV to 100 keV, and deduced the following empirical relations [42]:  
\[\rm E(R)=5.9(R+0.007)^{0.565}-\ 0.367, \tag{5}\]  
\[\rm R(E)=0.0431(E+0.367)^{1.77}-\ 0.007, \tag{6}\]  
where E is expressed in keV and R in \(\rm\mu m\). By differentiation of Equation (5), the following stopping power expression (in keV/\(\rm\mu m\)) is obtained:  
\[\rm\frac{dE}{dR}(R)=3.316(R+0.007)^{-0.435}, \tag{7}\]  
For electron energies extending from 100 keV up to 20 MeV, [PERSON] suggested another set of expressions as follows [42]:  
\[\rm E(R)=5.9(R+0.007)^{0.565}+0.00413R^{1.33}-\ 0.367, \tag{8}\]  
\[\rm\frac{dE}{dR}(R)=3.333(R+0.007)^{-0.435}+0.0055R^{0.33}, \tag{9}\]  
where E is expressed in keV and R in \(\rm\mu m\). Subsequently, [PERSON] provided improved range-energy relations that better fit [PERSON]'s experimental data for electron energies less than 0.4 keV [43].