2.1 Converted aperture metric
The converted aperture metric is based on the distances between the MLC leaves both parallel and opposed the MLC direction (Götstedt et al, 2015) (figure 1). The metric gives, for a MLC opening, a complexity score value between 0 (non-‐complex) and 1 (complex). The distances are measured every 5 mm.
Figure 1. For the converted aperture metric the distances between MLC leaves are measured both parallel (green solid line) and opposed (red dashed line) the MLC direction. The MLC leaves are 5 mm wide.
The conversion function, f, penalizes smaller distances compared to larger distances (eq.
1).
𝑓(𝑥) = 1 −𝑒−𝑥 (𝐸𝑞. 1)
With the distances (mm) and the equivalent field size as input arguments the function derives a value between 0 (complex) and 1 (non-‐complex). The complexity score is then given by the following equation (Götstedt et al, 2015):
𝐶𝑜𝑛𝑣𝑒𝑟𝑡𝑒𝑑 𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒 𝑚𝑒𝑡𝑟𝑖𝑐 = 1 − 𝑓 𝑑! ∙ 𝑓 𝑎!" (𝐸𝑞. 2)
where di is the measured distances and aeq is the equivalent square field size. To get a higher score for increasing complexity, and a lower score for decreasing complexity, the
score is subtracted from one. The calculations were performed in an in-‐house MatLab® software.
2.2 Edge area metric
The edge area metric depends on the length of the edge of the MLC opening (Götstedt et al, 2015). As for the converted aperture metric, the edge area metric calculates a complexity score based on the MLC opening. The MLC opening is divided into two different areas. The first area, Ae, includes 5 mm on both sides of the MLC edge (figure 2). The second area, Ao (mm2), are defined as the rest of the area of the MLC opening.
The edge area metric is given by the following equation:
𝐸𝑑𝑔𝑒 𝑎𝑟𝑒𝑎 𝑚𝑒𝑡𝑟𝑖𝑐 = 𝐴!
𝐴!+ 𝐴! (𝐸𝑞. 3)
As for the converted area metric, a higher score, between zero and one means that the field is more complex.
Figure 2. Schematic graph of the areas for the edge area metric. Area Ae (green and light green) includes 5 mm on both sides of the edges of the MLC leaves. The remaining area, Ao,
is marked as white. The MLC leves are 5 mm wide.
2.3 Pencil Beam Convolution Algorithm (PBC)
The pencil beam convolution algorithm (PBC) is implemented in Eclipse TPS (Varian medical systems, 2010) but similar algorithms are also implemented in other treatment planning systems.
The absorbed dose is calculated by convolving total energy released by unit mass (TERMA) with a pencil beam kernel. The pencil beam kernels describe the dose deposition from all photons and electrons emerging from an in-‐finite part of the beam.
In every voxel the TERMA is calculated based on the beam model. This model describes the fluence from the accelerator and is calculated once and for every unique accelerator.
It is based on measured parameters like MLC transmission factors and dose profiles.
The irradiated volume is divided into finite voxels. The voxel size is defined by the grid size chosen by the user. The dose calculation resolution i.e. grid size can be chosen to be 0.125 cm, 0.25 cm, 0.5 cm or 1.0 cm for PBC. Reduced grid size improves the dose calculation resolution and vice versa. The convolving process takes place in each voxel.
The total dose in a voxel is calculated by superposition of the smaller dose depositions.
2.4 Anisotropy Analytical Algorithm (AAA)
The anisotropic analytical algorithm (AAA) is a 3D pencil beam convolution algorithm developed by Varian medical systems (Varian medical systems, 2010). The grid size can be chosen between 0.1 cm and 0.5 cm.
The dose calculation model is divided into a beam model and a dose calculation model.
The first one describes the beam in the phase space plane by different source models.
The source models are the primary-‐, secondary-‐, wedge scattering-‐, and electron contamination source model.
Monoenergetic pencil beam kernels are transformed to polyenergetic pencil beam kernels. They are also scaled based on the electron density (heterogenities) along the central axis of the kernel and in six lateral directions.
The beam is divided into finite beamlets. Every beamlet is convolved with a kernel. This is done for every source. The total energy in every voxel is given by superposition of the doses from the different source models.
2.5 Acuros XB (AXB)
The acuros XB algorithm is implemented in the Eclipse TPS (Varian medical systems, 2010). AXB is considered to be a fast algorithm compared to MC even though the computing time compared to other clinical calculation algorithms is somewhat longer.
The grid sizes available to be chosen are between 0.1 cm and 0.3 cm. The beam model is the same as for the anisotropic analytical algorithm (AAA).
Acuros XB calculates the absorbed dose by solving the linear Boltzmann transport equation (LBTE) (Vassiliev et al, 2010). The LBTE describes macroscopically how ionizing particles for example photons and electrons interact with different matter.
LBTE is solved in Eclipse by numerical methods (Varian medical systems, 2010). In the calculations, limitations in accuracy are induced because of the discrete variables in angle, energy and space. Acuros XB calculates absorbed doses in heterogenic material in the same order of accuracy as MC calculations (Fogliata, 2011). The long computing time compared to other calculation algorithm is a disadvantage (Vassiliev et al, 2010).
2.6 Collapsed cone convolution (CC)
The collapsed cone convolution (CC) algorithm is implemented in the Oncentra TPS (Nucletron). The grid size can be chosen between 0.1 cm and 0.5 cm. The beam model is made up of two models: one that describes the primary fluence and another that takes into account the fluence of head-‐scatter components. These are stored in separate 2D-‐
matrices.
CC uses analytical point kernels to describe the dose deposition from primary photons in the medium (Ahnesjö, 1989). A point kernel gives the dose deposition distribution from primary photons and scattered photons. The absorbed dose is calculated by a 3D convolution/superposition method like the AAA and PBC algorithms. The deposited energy, within a solid angle Ω, is transported to voxels positioned along a line inside a cylindrical coordinate system. Every line defines an axis of a cone. The line is passing thought the middle point of a 3D voxel defined in a cartesian coordinate system. The total energy deposited along the line is transported to this voxel. That means that parallel lines, one for every cartesian voxel, gives the total absorbed dose in the whole radiated volume.
CC is an algorithm that effectively takes into account heterogeneities due to the use of a cylindrical coordinate system.