For two surfaces (Sx) in 3D space, by S1 (x1, y1) – z1 and S2 (x2, y2) – z2 is the distance from any point on the S1 surface (x1, y1) to the other S2 surface defined as the Euclidian distance from S1 (x1, y1) to the next point on S2, as shown by the following equation: We can continue to generalize the concept of distance to the distribution of the 3D dose, because a distribution dose can also be considered a surface. For two 3D D1 and D2 dose distributions (standardized doses are always used, unless otherwise stated), the difference of one point to D1 (with a dose of D1 (x1), y1, z1)) in D2 is as follows: the k-d,19 tree, which is a data structure based on a recursive subdivision of a k-dimensional universe in separate sub-regions, was used to calculate both the surface distance and the index γ.18, 20 By defining the distance between pixels, one can quickly locate the nearest neighbor for each pixel in the structure of the trees. It should be noted that image resolution can affect the performance of the surface distance and the γ index. The effect can be even greater with the k-d-arborescence method, without interpolating the values between neighboring pixels.21 As in Eq. 10, the calculation of the γ index minimizes a cost function, with a weighting between spatial distance and dose difference, which is related to the relationship between the DTA and DD criteria, i.e.dem factor of change in the dose α. Surface distance can therefore be considered a general form of the γ index, and they are interchangeable if α is a constant, as in Eq. 5. At α constant, several observations are reported from Eqs. 3, 10: (1) Once the DTA and DD criteria (dm and DM) are established, the map of the distribution of distance with α-dm-Dm is identical to the γ map, with the exception of a `dm scale factor; (2) cards γ obtained according to different DTA and DD criteria and have the same α (e.g.

B 3 mm/3%, 2 mm/2 and 1 mm/1%) are identical, with the exception of a scale factor of and (3) The IMRT QA-Passing instruction of the criterion “at least 90% of pixels must have γ ≤ 1 for a given set of DTA/DD criteria” corresponds to the 90th percentile of the distance distribution with α-dm-Dm (D90) ≤-dm. Because of these characteristics, only the constant shape of the dose gradient factor was used in this study. The counterpart of the average absolute dose gradient in the target area (50% threshold) is the dose gradient factor. The dose gradient coefficients for IMRT, the SRS spine and the RPC H-N ghost planes and the film`s axial planar measurements are presented in Table I. The dose gradients in the measurements have always been higher than in the planes, as can be seen in Table I, a phenomenon that can be caused by measurement noises. The average dose gradient coefficients for IMRT plans and measurements were 0.956 mm/% and 0.827 mm/% respectively. These results are close to the implied dose gradient factor of 1 mm/% of the widely accepted criteria of 3 mm/3% [Eq. 12]. Similarly, the average dose gradient coefficients for the RPC phantom plans and measurements were 0.618 mm/% and 0.504 mm/% respectively, corresponding to the 4 mm/7% criteria set by THE CPP (implicit dose gradient factor – 0.571 mm/%) from one point of the agreement.