Is the diaphragm motion probability density function normally distributed?

Abstract

During radiotherapy treatment planning, the margins given to the clinical target volume to form the planning target volume accounts for internal motion and set-up error. Most margin formulas assume that the underlying distributions are independent and normal. Clinical data suggests that the set-up error probability density function (pdf) can be considered to have an approximately normal distribution. However, there is evidence that internal motion does not have a normal distribution. Thus, in general, a convolution of the two pdfs should be performed to determine the total geometric error. The goals of this article were to (1) determine if the internal motion pdf due to respiration can be characterized using a normal distribution, and (2) if not, determine if the total geometric uncertainty for combining internal motion and set-up error can be characterized by a normal distribution. Sixty fluoroscopy diaphragm motion data sets were obtained using three breathing training types: free breathing, audio instruction, and visual feedback. Diaphragm motion was used as a surrogate for liver and lung cancer motion. The data were analyzed with normality tests in the following groups: (1) single motion measurements, (2) combined motion measurements for each patient, and (3) combined motion measurements for all patients. Following this analysis, the diaphragm motion pdfs were convolved with a set-up error pdf, and the standard deviation of the set-up error pdf at which the total geometric error pdf became normal was determined. At set-up error standard deviation values of at least 0.27 and 0.1 cm for free breathing, 0.57 and 0.42 cm for audio instruction, and 0.55 and 0 cm for visual feedback, for single motion measurements and combined motion measurements for each patient, respectively, total geometric error pdfs became approximately normal. When the motion measurements for all the patients were combined, diaphragm motion pdfs were approximately normal for all feedback types. Therefore, for treatment planning purposes in the absence of individual patient measurements, the diaphragm motion pdf can be considered an approximately normal distribution. However, care should be taken when determining a margin based on individual patients measurements as the total geometric error will, in general, not be normally distributed.