Calculation of mean central dose in interstitial brachytherapy using Delaunay triangulation

In 1997 the ICRU published Report 58 “Dose and Volume Specification for Reporting Interstitial Therapy” with the objective of addressing the problem of absorbed dose specification for reporting contemporary interstitial therapy. One of the concepts proposed in that report is “mean central dose.” The fundamental goal of the mean central dose (MCD) calculation is to obtain a single, readily reportable and intercomparable value which is representative of dose in regions of the implant “where the dose gradient approximates a plateau.” Delaunay triangulation (DT) is a method used in computational geometry to partition the space enclosed by the convex hull of a set of distinct points P into a set of nonoverlapping cells. In the three-dimensional case, each point of P becomes a vertex of a tetrahedron and the result of the DT is a set of tetrahedra. All treatment planning for interstitial brachytherapy inherently requires that the location of the radioactive sources, or dwell positions in the case of HDR, be known or digitized. These source locations may be regarded as a set of points representing the implanted volume. Delaunay triangulation of the source locations creates a set of tetrahedra without manual intervention. The geometric centers of these tetrahedra define a new set of points which lie “in between” the radioactive sources and which are distributed uniformly over the volume of the implant. The arithmetic mean of the dose at these centers is a three dimensional analog of the two-dimensional triangulation and inspection methods proposed for calculating MCD in ICRU 58. We demonstrate that DT can be successfully incorporated into a computerized treatment planning system and used to calculate the MCD.