To calculate the GTV–PTV margin required at each point, delineation uncertainty must be quantified as the delineation error (ΣD). This is the standard deviation of the distances from a reference outline to each outline. It is calculated using equation 1, where di is the distance from the reference outline to the ith observer's outline, No is the number of observers and d¯ is the mean distance. Although many metrics for measuring observer variability are used [7,14,15], ΣD is the only one that can be used in a traditional PTV margin recipe to calculate PTV margins.

To measure ΣD for each point in a patient, the approach taken by Deurloo et al. [16] was followed. First, the two-dimensional contour sets were converted to three-dimensional surfaces comprising of vertices and faces. A reference surface for each patient was then generated from all the clinician's GTV surfaces. This was carried out using the simultaneous truth and performance level estimation (STAPLE) algorithm [17] at a 95% confidence level. This produced a surface that encompasses voxels that have a 95% or more probability of belonging to the GTV based on the provided outlines. These consensus surfaces were generated using Computational Environment for Radiological Research (CERR) software module [18] implemented within MatLab (The MathWorks Inc., Natick, MA, USA).

For the jth vertex on the reference surface, the reference surface normal vector, nj, was determined. The distance along that normal vector to each comparator surface was then measured and equation 1 used to give the delineation error at the jth vertex ΣD,j. The result of this process is a three-dimensional surface map with potentially varying values of ΣD.

PTV margin recipes require summary ΣD values for each sector of the GTV corresponding to each of the cardinal axes. To determine which sector each vertex on the consensus surface belongs to, vector Kj, which originated at the centre of the consensus surface and terminated at vertex vj, was created. As were six vectors originating at the centre of the volume and pointing parallel to the patient left (λL), right (λR), anterior (λA), posterior (λP), superior (λS) and inferior (λI) axes. The angles between Kj and each of the λ vectors were then measured. Finally, the vertex vj was assigned to a single sector corresponding to the vector λ with the smallest angle to Kj. This process was repeated for each vertex on the consensus surface. Figure 1 illustrates the results for a spherical target. The approach of classifying the vertices in this way is similar to one taken in previous studies [[19], [20], [21]], however those publications only measured points along the cardinal planes, unlike this study where classifications and measurements were carried out for the whole surface.

Illustrations of the vertex classifications for a spherical target as either left (blue), right (orange), anterior (purple), posterior (green), superior (yellow) or inferior (light blue). Arrows represent the axes from the centre of the volume. (A) is viewed from the superior axis, (B) from the left axis, (C) from the posterior axis and (D) is an oblique view.

The overall ΣD for a single patient was given by the geometric mean of ΣD, calculated over all vertices in that patient. The overall ΣD for all patients combined was then given by the geometric mean of ΣD from each patient. For values given for a single sector, these measurements and calculations were carried out for each sector separately. Pooling standard deviations in this way assumes that the variances between each group are equal; this assumption was tested using the methods described below. It was assumed that each case had an equal number of samples to avoid the largest GTVs dominating the value of ΣDs for all patients combined.

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