2.3. Kernel based algorithms for electrons created in primary photon interactions

In the Monte Carlo calculation the energy transferred from photons to secondary electrons is scored in each voxel. In the relevant range of photon energies, there are two processes where photons transfer energy to electrons: photoelectric absorption and Compton scattering. We refer to these two processes as energy transfer events (ETEs).

The dose distribution on the micrometre scale needs to take the electron energy transport into account. The electron energy absorption is calculated for each voxel individually, applying a few reasonable assumptions: within a single voxel it is assumed that the voxel material is homogeneous, the photon spectrum and the beam intensity do not change when the beam passes through the voxel. Furthermore the beam divergence may not lead to any significant changes in the microbeam pattern.

Following previous definitions, dose kernels are defined as the spatial distribution of the fractional mean energy dE absorbed per mass element dm caused by a primary particle interaction at the origin (Ahnesjö et al 1987, Bartzsch and Oelfke 2013). We refer to the electron kernel Kel3D(r) as the dose kernel of scattering electrons created in a primary photon interaction. As spectrum and material do not change, the electron kernel is also constant within the voxel.

The Monte Carlo primary dose scores ETEs of unscattered photons and hence these events occur on the initial photon beam path. Therefore, within a single voxel, ETEs of the primary dose are equally spread along the beam direction and perpendicular to the beam direction they are distributed according to the fluence profile created by the microbeam collimator. ETEs of the Monte Carlo scatter dose are equally spread throughout the voxel, since photon scattering occurs on much larger spatial scales. Hence the scatter dose leads to a homogeneous dose bath and only electrons in primary interactions contribute to the microbeam pattern. Under these conditions the dose distribution in a single voxel can be calculated via (Debus et al 2017)

where ν(r) describes the distribution of ETEs in the voxel, D_Scatter and D_Primary are the Monte Carlo primary and scatter dose contributions and  ∗  denotes the convolution operator.

Within a CT-voxel the distribution of ETEs will not change along the beam propagation direction, as there is no beam divergence and absorption does not change the beam intensity within the voxel. Hence, by choosing the coordinate system in the voxel such that the propagation direction of the microbeams points along the z-axis the distribution function ν becomes independent of z. For planar microbeams in the x–z-plane ν depends on y only. Therefore the convolution can be significantly simplified and becomes either

ν depends on x and y or

if ν depends on y only. The convolution kernels Kel1D and Kel2D can be obtained from the 3D scattering kernel Kel3D by integration,

respectively.

For electron energies between 10 keV and a few 100 keV electron scatter kernels can be derived from the Bethe–Bloch stopping power equation. A detailed derivation has been published previously (Debus et al 2017). The starting point is the approximation of the stopping power S of an electron with kinetic energy E

where K and α are constants. Fitting experimental data from Berger et al (2005) leads to α ≈ 0.415, independent of the material. Under a few assumptions, such as isotropical electron scattering and homogeneous material (Debus et al 2017), the three dimensional electron kernel can be derived,

σ denotes the electron continuous slowing down approximation range, r is the distance from the ETE, ρ is the mass density of the material and E₀ is the initial electron energy. The scatter kernel (6) is normalized to dose per single electron. Evaluating equation (4) leads to the two dimensional scatter kernel

where s stands for s=x2+y2 and defining I_2D( p) as

leads to the simple representation

of the two dimensional scatter kernel. Similarly the 1D kernel can be calculated as

which can be written in the simple representation

identifying I1Dα with

These derived kernels explicitly depend on the initial electron energy E₀ and on the electron range σ, which is material and energy dependent. Electrons produced in photoelectric absorption are assumed to receive all of the primary photon energy E_ph, neglecting the binding energy, while only a fraction p of the photon energy is transferred to electrons in Compton scattering,

The ratio of photoelectric absorption to Compton scattering interactions defined by the ratio of their scattering coefficients depends on photon energy and material; p depends on photon energy, only. The electron kernel of a polychromatic photon beam with the power contributions f(E_i) at photon energy E_i can be calculated as a weighted sum

where μ_c(E_i) and μ_p(E_i) are the energy and material dependent scattering coefficients for Compton and photoelectric effect. This formula is valid for one, two and three dimensional scattering kernels.