Influence of shape on the accuracy of grid-based volume computations

The influence of the shape of a region of interest (ROI) on the uncertainty in the sampled volume of the ROI is investigated for computations with regular Cartesian grids. Both mathematically defined volumes and clinically relevant ROIs were studied. The sampling uncertainty is shown to depend on the compactness of the ROI and on effects of grid matching and translational symmetry. In clinical ROIs without translational symmetry the estimate of the sampling uncertainty is improved up to a factor of 2.3 by taking the compactness of the ROI into account. In a spherical ROI grid-matching effects were demonstrated by means of Fourier transforms. In this type of ROI, grid-matching effects decrease as well as increase the sampling uncertainty up to a factor of 1.6. Translational symmetry is shown to cause a decrease in the sampling uncertainty convergence power from 2/3 for spherical ROIs, to 1/2 for stringlike or 1/3 for pancakelike cylinders. For clinical ROIs with translational symmetry, similar decreases were found. With the theory derived and these symmetry effects taken into account the experimental uncertainty of volume computation can be estimated for most clinical ROIs within a factor of 2.5. Special care should be taken in grid sampling of volumes inside isodose surfaces of rectangular field techniques. For the volume of a prostate an uncertainty level of 1% or 5% is obtained with less than 1050 or 80 grid points, respectively, while for such an isodose surface up to 16,000 or 500 grid points are required for the same uncertainty levels.