EUD-based radiotherapy treatment plan evaluation: incorporating physical and Monte Carlo statistical dose uncertainties

The purpose of this work is to quantify the impact of dose uncertainty on radiobiologically based treatment plan evaluation. Dose uncertainties are divided into two categories: physical and statistical. Physical dose uncertainty is associated with the systematic and/or random errors incurred during treatment planning and/or delivery. The dose uncertainty associated with Monte Carlo calculated dose distributions is deemed statistical and noted as artificial with respect to the actual delivered dose. We will refer to all dose uncertainties that arise from either calculation or delivery as stochastic. Both physical and statistical dose uncertainties are considered at the intra- and inter-voxel levels. To account for voxel dose uncertainty, we calculate the mean survival fraction (SF) for the random dose deposition. Mathematically, the expression for the mean survival fraction is identical to that used by Niemierko (1997 Med. Phys. 24 103-10) in defining equivalent uniform dose (EUD). To distinguish between spatial and probabilistic dose variations, we define equivalent stochastic dose (ESD) as a voxel dose that gives the mean expected survival fraction for the randomly deposited dose. For a probability density function f(D), that represents the probabilistic voxel dose, SF(ESD) can be calculated by convolving SF(D) with f(D). In the case where the probability density function follows a Gaussian distribution, an analytic expression is derived for SF(ESD). The derived expression is verified using the Monte Carlo method and ESD values calculated with varied radiosensitivities for cases of 60 and 70 Gy at 2 Gy per fraction. The analytic expression is also extended to account for a multi-voxel dose distribution that incorporates a spatial dose heterogeneity. The results show that survival fraction increases with an increased dose uncertainty. This reduction depends on radiobiological parameters attributed to tissue and tumour. For tissue, ESD drops to 55% of the mean physical dose when the dose has a 10% intra- and inter-voxel dose uncertainty and inhomogeneity.