Application of coloring theory to reduce intensity modulated radiotherapy dose calculations

Coloring theory is applied to reduce the dose calculations under intensity modulated radiotherapy. Intensity modulated radiotherapy varies the intensity profile across the beam. The beam face is divided into a panel of small squares or “bixels.” Each square may be opened or closed for different lengths of time by moving collimator leaves in and out of the beam. It has been shown that the distribution of dose from radiation directed through any open square depends on whether the adjacent squares are opened or closed. Taking the states of neighboring bixels into account greatly increases the required dose calculations. There are 2 8possible ways to select open or closed states for the eight neighbors of a given bixel. Each combination represents one coloring of a 3 × 3 panel, and each coloring demands a separate dose calculation. The number of calculations is reduced by considering the symmetries of a square. The 256 possible colorings can be divided into 51 distinct patterns through application of Burnside's theorem. Each pattern consists of selections of closed bixels that are the same except for a symmetric transformation of coordinates. If the symmetry between x and y coordinates is broken by collimator leaves whose ends and sides have different effects on bordering bixels, the number of patterns increases to 84. The theoretic gain in the number of calculations through the application of Burnside's theorem is fivefold if bixel borders are symmetric, and threefold if the borders are asymmetric. The results are applied to examples of generated intensity maps. The symmetry rules divide the bixel arrangements into proportionately fewer patterns as the intensity maps become larger, allowing computational gains to be achieved.