Numerical solutions of the γ-index in two and three dimensions

The γ-index is used routinely to establish correspondence between two dose distributions. The definition of the γ-index can be written with a single equation but solving this equation at millions of points is computationally expensive, especially in three dimensions. Our goal is to extend the vector-equation method in Bakai et al (2003 Phys. Med. Biol.48 3543-53) to higher order for better accuracy and, as important, to determine the magnitude of accuracy in a higher order solution. We construct a numerical framework for calculating the γ-index in two and three dimensions and present an efficient method for calculating the γ-index with zeroth-, first- and second-order methods using tricubic spline interpolation. For an intensity-modulated radiation therapy example with 1.78 × 10⁶ voxels, the zeroth-order, first-order, first-order iterations and semi-second-order methods calculate the three-dimensional γ-index in 1.5, 4.7, 34.7 and 35.6 s with 36.7%, 1.1%, 0.2% and 0.8% accuracy, respectively. The accuracy of linear interpolation with this example is 1.0%. We present efficient numerical methods for calculating the three-dimensional γ-index with tricubic spline interpolation. The first-order method with iterations is the most accurate and fastest choice of the numerical methods if the dose distributions may have large second-order gradients. Furthermore, the difference between iterations can be used to determine the accuracy of the method.