Expected Value And Variance Of A Point Dose From Nongated Radiation Therapy Delivery

The lack of synchronization between nongated radiation therapy delivery and internal organ motion may cause fluctuations in the dose received by the target or organs at risk. These variations are particularly concerning for dynamically delivered treatments (e.g., dynamic IMRT and VMAT) when significant internal organ motion presents in the tumor target region like the liver and lungs. We conducted a theoretical study to gain more insight to the statistical nature of radiation dose received by patients from nongated radiation therapy. Two assumptions were made for mathematical analysis: a) internal organ motion is stationary and periodic, and b) no special arrangement is made to start radiation therapy delivery at any specific phase of patient internal organ motion. Expected value and variance were selected to characterize the dose received by an arbitrary internal point. Both variables are calculated through statistical ensemble averages of the point dose and the square of the fluctuation in the point dose respectively over all equally probable phases of patient internal organ motion. The Fourier analysis technique was used due to the stationary and periodicity assumptions of organ movement to facilitate the mathematical analysis. Rigorous mathematical formulae on the expected value and variance of radiation dose were derived. We managed to significantly reduce their calculation complexity from a complicated full 4D tracking process into a much more manageable time series analysis of dose rate at the point’s corresponding location on 3D CT images with a fixed phase. From our formulae, the expected value of the dose to a point was the average dose from the same radiation therapy delivery to its corresponding location on each fixed phase 3D CT images when equally sampled over a complete internal organ motion cycle. The variance of the received point dose can be conveniently expressed in terms of Fourier coefficients. The above results remain valid even when the internal organ motion significantly changes the dose distribution or point location. Closed-form or numerical solutions of the expected value and variance of the point dose were obtained for various clinical situations after applying our mathematical formulae. The mathematical formulae obtained proved to be a valuable tool in calculating the expected value and variance of radiation dose from a nongated treatment with stationary and periodic internal organ motion. It is applicable to a variety of nongated radiation therapy modalities. The results provide us new insights that we expect to play an important role in our decision-making process regarding the interplay effect of dynamically delivered nongated radiation therapy and the necessity of employing motion management in selected situations.