Geometric interpretation of the gamma dose distribution comparison technique: interpolation-free calculation

The gamma dose comparison tool has been used by numerous investigators to quantitatively compare multidimensional dose distributions. The gamma tool requires the specification of dose and distance-to-agreement (DTA) criteria for acceptable variations between the dose distributions. The tool then provides a comparison that simultaneously evaluates the dose difference and distance to agreement of the two dose distributions. One of the weaknesses of the tool is that the comparison requires one of the dose distributions to have a relatively high spatial resolution, with points spaced significantly closer than the DTA criterion. The determination of gamma involves an exhaustive search process, so the computation time is significant if an accurate gamma is desired. The reason for the need for high spatial resolution lies with the fact that the gamma tool measures the closest point in one of the dose distributions (the evaluated distribution) with individual points of the other distribution (the reference distribution) when the two distributions are normalized by the dose difference and DTA criteria for the dose and spatial coordinates, respectively. The closest point in the evaluated distribution to a selected reference distribution point is the value of gamma at that reference point. If individual evaluated dose distribution points are compared, the closest point may not accurately reflect the closest value of the evaluated distribution as if it were interpolated on an infinite resolution grid. Therefore, a reinterpretation of the gamma distribution as the closest geometric distance between the two distributions is proposed. This is conducted by subdividing the evaluated distribution into simplexes; line segments, triangles, and tetrahedra for one, two, and three-dimensional (3D) dose distributions. The closest distance between any point and these simplexes can be straightforwardly computed using matrix multiplication and inversion without the need of interpolating the original evaluated distribution. While an exhaustive search is still required, not having to interpolate the evaluated distribution avoids the drastic growth of calculation time incurred by interpolation and makes the gamma tool more practical and more accurate. In our experiment, the geometric method accurately computes gamma distributions between 3D dose distributions on a 200 x 200 x 50 grid within two minutes.