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, for example, 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, that is, a À DVHs, are computed. The model is subsequently evaluated on three patient cases (intracranial, paraspinal, prostate) in 30-and single-fraction 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 a À DVH, 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 AE0.2), the respective À DVHs for a={0.05,0.5,0.95} only showed small deviations in respective volume (up to AE5% 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, that is, normal or beta distributions, over DVH-points.

Cumulative Histograms under Uncertainty: An Application to Dose–Volume Histograms in Radiotherapy Treatment Planning

“…A notable drawback is the implicit nature of the mathematical transformation from input (uncertainty model) to output (probability distribution over dose or plan metrics), which obscures an explicit understanding and estimation of the resultant probability distributions [10]. This opacity has led to a scarcity of precise estimations and parameterizations of probability distributions over plan metrics [10][11][12][13]. Thus, an exploration into more accurate and computationally efficient statistical frameworks could significantly enhance the analysis of patient data and potentially lead to more reliable and optimized treatment plans.…”

“…where Θ symbolizes the Heaviside step function (e.g., [13]). In essence, only the voxels that have received a nominal dose d i ≥ d contribute with a weight of 1/V to the sum.…”

“…It is axiomatic that 0 ≤ DVH( d; d) ≤ 1 as the DVH points represent a fraction. Past research [13] illustrated how APM could be used to analytically compute the moments of DVH points, i.e., the volume fraction enveloped at least with a specific dose d. In this investigation, a multivariate normal distribution of the dose over the voxels was presupposed. However, a closer inspection reveals that this Gaussian model may be physically contentious, given the volume fraction's intrinsic constraint to positive values.…”

“…However, a closer inspection reveals that this Gaussian model may be physically contentious, given the volume fraction's intrinsic constraint to positive values. Despite this, the Gaussian model was evaluated with patient data, and comparison to Monte Carlo sampling showed satisfactory results [13].…”

Efficient uncertainty quantification for Monte Carlo dose calculations using importance (re-)weighting

“…Consequently, particle therapy demands personalized robustness analyses and mitigation. Such techniques may be based on explicit propagation of input uncertainties using probabilistic methods and statistical analysis (Bangert et al, 2013;Wahl et al, 2017Wahl et al, , 2020Kraan et al, 2013;Park et al, 2013;Perkó et al, 2016) or worst-case estimates (McGowan et al, 2015;Casiraghi et al, 2013;Lowe et al, 2016). Most of these methods then further translate to robust and probabilistic optimization to extend the conventional, generic margin approach to uncertainty mitigation (Sobotta et al, 2010;Liu et al, 2012;Fredriksson, 2012;Unkelbach et al, 2018).…”