## 4 Discussion  
As can be seen from Figure 3, the new range-energy relations (Equations (14)-(16)) offer an almost excellent representation (within 1.5%) of [PERSON]'s numerical data in the 100 keV-20 MeV energy range, while they are also in good agreement (within 5%) from ICRU Report 90 data (see Figure 4). Thus, the present range-energy relations together with the Cole-Howell relations at lower energies, allow for DPK-based calculations of the absorbed fraction (and S factor) by the convolution integral (Equation (4)) over a very broad energy interval (20 eV-20 MeV), which covers most applications of radiotherapeutic interest. The application of the new relations to the case of intermediate-size target volumes relevant to micrometastasis and disseminated disease is investigated in subsequent figures. Specifically, in Figures 5 and 6, we compare the present calculations against published MC data, which cover sphere radii between 1.3 mm and 6.2 cm and electron energies from 100 keV to 4 MeV. For mm-size spheres (radius 1.3-7.8 mm), the present calculations exhibit larger differences against EGS4, ETRAN, Geant4, and MCNPX and smaller for MCNP. Specifically, when averaged over the electron energies examined (100 keV-4 MeV), the maximum differences vary between 15 and 37% for ETRAN, 15 and 36% for EGS4, and 10 and 21% for MCNP. In terms of Geant4 and MCNPX codes, the maximum differences for the available presented data are 16-23% for Geant4 and 14% for MCNPX. Interestingly, the maximum differences between EGS4 and MCNP are comparable (9-12%). In all cases, differences decrease with increasing electron energy. On the other hand, for the cm-sized spheres, the differences between the present calculations and the MC data steadily decrease for all MC codes. Specifically, when averaged over the electron energies examined (100 keV-4 MeV), the maximum differences vary between 3 and 13% for ETRAN, 1 and 11% for EGS4, 1 and 6% for MCNP, 3 and 7% for Geant4, and 0.5 and 2% for MCNPX. Note that for Geant4 and MCNPX, no data are available for some spheres and electron energies. Again, the maximum differences between EGS4 and MCNP are similar (1-7%). Thus, the differences between the present DPK-based calculations and the MC codes are comparable to the differences observed among the codes. However, although an overall good agreement is observed against the MC codes (on average better than \(\sim\)10% in most cases), the present DPK-based calculations tend to consistently underestimate the MC-based absorbed fractions. The underestimation is higher for the smaller spheres (\(\sim\)mm radius) and at energies where the range of electrons is comparable to the size of the sphere. A possible explanation is that the present DPK-based calculations are based on a range-energy relation deduced from [PERSON]'s experimental data that seem to overestimate the penetration depth of electrons, as they resemble, instead, the CSDA range of ICRU (see Figure 2). In turn, the use of the CSDA range (instead of the penetration depth) in the convolution integral, Equation (4), will underestimate the actual energy deposition in the target by allowing electrons to travel and deposit energy further away. This underestimation is smaller as the size of the sphere is increasing (or the primary electron energy is decreasing), because of significant self-absorption within the target sphere. In Figure 7, the present S factor calculations are compared against the MIRDcell Software, which adopts a similar DPK-based approach, for spheres of radius from 10 um to 5 mm. The maximum difference between the present calculations and MIRDcell is less than 3%, regardless of the sphere size. It is worth pointing out the excellent agreement (within 1.5%) between the present results and MIRDcell for energies higher than 100 keV (and for all spheres), whereby our new range-energy relations apply.