Technical note: dose-volume histogram analysis in radiotherapy using the Gaussian error function.

A mathematical model based on the Gaussian error and complementary error functions was proposed to describe the cumulative dose-volume histogram (cDVH) for a region of interest in a radiotherapy plan. Parameters in the model (a, b, c) are related to different characteristics of the shape of a cDVH curve such as the maximum relative volume, slope and position of a curve drop off, respectively. A prostate phantom model containing a prostate, the seminal vesicle, bladder and rectum with cylindrical organ geometries was used to demonstrate the effect of interfraction prostate motion on the cDVH based on this error function model. The prostate phantom model was planned using a five-beam intensity modulated radiotherapy (IMRT), and a four-field box (4FB), technique with the clinical target volume (CTV) shifted in different directions from the center. In the case of the CTV moving out of the planning target volume (PTV), that is, the margin between the CTV and PTV is underestimated, parameter c (related to position of curve drop off) in the 4FB plan and parameters b (related to the slope of curve) and c in the IMRT plan vary significantly with CTV displacement. This shows that variation of the cDVH is present in the 4FB plan and such variation is more serious in the IMRT plan. These variations of cDVHs for 4FB and IMRT are due to the different dose gradients at the CTV edges in the anterior and posterior directions for the 4FB and IMRT plan. It is believed that a mathematical representation of the dose-volume relationship provides another viewpoint from which to illustrate problems with radiotherapy delivery such as internal organ motion that affect the dose distribution in a treatment plan.