Analytical probabilistic modeling of dose‐volume histograms

Abstract

Purpose Radiotherapy, especially with charged particles, is sensitive to executional and preparational uncertainties that propagate to uncertainty in dose and plan quality indicators, e. g., dose‐volume histograms (DVHs). Current approaches to quantify and mitigate such uncertainties rely on explicitly computed error scenarios and are thus subject to statistical uncertainty and limitations regarding the underlying uncertainty model. Here we present an alternative, analytical method to approximate moments, in particular expectation value and (co)variance, of the probability distribution of DVH‐points, and evaluate its accuracy on patient data. Methods We use Analytical Probabilistic Modeling (APM) to derive moments of the probability distribution over individual DVH‐points based on the probability distribution over dose. By using the computed moments to parameterize distinct probability distributions over DVH‐points (here normal or beta distributions), not only the moments but also percentiles, i. e., α‐DVHs, are computed. The model is subsequently evaluated on three patient cases (intracranial, paraspinal, prostate) in 30‐ and singlefraction scenarios by assuming the dose to follow a multivariate normal distribution, whose moments are computed in closed‐form with APM. The results are compared to a benchmark based on discrete random sampling. Results The evaluation of the new probabilistic model on the three patient cases against a sampling benchmark proves its correctness under perfect assumptions as well as good agreement in realistic conditions. More precisely, ca. 90% of all computed expected DVH‐points and their standard deviations agree within 1% volume with their empirical counterpart from sampling computations, for both fractionated and single fraction treatments. When computing α‐DVHs, the assumption of a beta distribution achieved better agreement with empirical percentiles than the assumption of a normal distribution: While in both cases probabilities locally showed large deviations (up to ±0.2), the respective α ‐DVHs for α = {0:05; 0:5; 0:95} only showed small deviations in respective volume (up to ±5% volume for a normal distribution, and up to 2% for a beta distribution). A previously published model from literature, which was included for comparison, exhibited substantially larger deviations. Conclusions With APM we could derive a mathematically exact description of moments of probability distributions over DVH‐points given a probability distribution over dose. The model generalizes previous attempts and performs well for both choices of probability distributions, i. e., normal or beta distributions, over DVH‐points.