Impact of Gaussian uncertainty assumptions on probabilistic optimization in particle therapy

A spot-specific range uncertainty framework for robust optimization of proton therapy treatments

PURPOSE

An accurate estimation of range uncertainties is essential to exploit the potential of proton therapy. According to Paganetti's study, a value of 2.4% (1.5 standard deviation) is currently recommended for planning robust treatments with Monte Carlo dose engines. This number is based on a dominant contribution from the mean excitation energy of tissues. However, it was recently shown that expressing tissues as a mixture of water and "dry" material in the CT calibration process allowed for a significant reduction of this uncertainty. We thus propose an adapted framework for pencil beam scanning robust optimization. First, we move towards a spot-specific range uncertainty (SSRU) determination. Second, we use the water-based formalism to reduce range uncertainties and, potentially, to spare better the organs at risk.

METHODS

The stoichiometric calibration was adapted to provide a molecular decomposition (including water) of each voxel of the CT. The SSRU calculation was implemented in MCsquare, a fast Monte Carlo dose engine dedicated to proton therapy. For each spot, a ray-tracing method was used to propagate molecular I-values uncertainties and obtain the corresponding effective range uncertainty. These were then combined with other sources of range uncertainties, according to Paganetti's study of 2012. The method was then assessed on three head-and-neck patients. Two plans were optimized for each patient: the first one with the classical 2.4% flat range uncertainty (FRU), the second one with the variable range uncertainty. Both plans were then compared in terms of target coverage and OAR mean dose reduction. Robustness evaluations were also performed, using the SSRU for both plans in order to simulate errors as realistically as possible.

RESULTS

For patient 1, it was found that the median SSRU was 1.04% (1.5 standard deviation), yielding, therefore, a very large reduction from the 2.4% FRU. All three SSRU plans were found to have a very good robustness level at a 90% confidence interval while sparing OAR better than the classical plan. For instance, in nominal cases, average reductions in the mean dose of 15.7, 8.4, and 13.2% were observed in the left parotid, right parotid, and pharyngeal constrictor muscle, respectively. As expected, the classical plans showed a higher but unnecessary level of robustness.

CONCLUSIONS

Promising results of the SSRU framework were observed on three head-and-neck cases, and more patients should now be considered. The method could also benefit to other tumor sites and, in the long run, the variable part of the range uncertainty could be generalized to other sources of uncertainty in order to move towards more and more patient-specific treatments.

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

Objective. To present an efficient uncertainty quantification method for range and set-up errors in Monte Carlo (MC) dose calculations. Further, we show that uncertainty induced by interplay and other dynamic influences may be approximated using suitable error correlation models.Approach. We introduce an importance (re-)weighting method in MC history scoring to concurrently construct estimates for error scenarios, the expected dose and its variance from a single set of MC simulated particle histories. The approach relies on a multivariate Gaussian input and uncertainty model, which assigns probabilities to the initial phase space sample, enabling the use of different correlation models. Through modification of the phase space parameterization, accuracy can be traded between that of the uncertainty or the nominal dose estimate.Main results. The method was implemented using the MC code TOPAS and validated for proton intensity-modulated particle therapy (IMPT) with reference scenario estimates. We achieve accurate results for set-up uncertainties (γ_{2 mm/2%}≥ 99.01% (E[d]),γ_{2 mm/2%}≥ 98.04% (σ(d))) and expectedly lower but still sufficient agreement for range uncertainties, which are approximated with uncertainty over the energy distribution. Here pass rates of 99.39% (E[d])/ 93.70% (σ(d)) (range errors) and 99.86% (E[d])/ 96.64% (σ(d)) (range and set-up errors) can be achieved. Initial evaluations on a water phantom, a prostate and a liver case from the public CORT dataset show that the CPU time decreases by more than an order of magnitude.Significance. The high precision and conformity of IMPT comes at the cost of susceptibility to treatment uncertainties in particle range and patient set-up. Yet, dose uncertainty quantification and mitigation, which is usually based on sampled error scenarios, becomes challenging when computing the dose with computationally expensive but accurate MC simulations. As the results indicate, the proposed method could reduce computational effort while also facilitating the use of high-dimensional uncertainty models.

What is plan quality in radiotherapy? The importance of evaluating dose metrics, complexity, and robustness of treatment plans

Plan evaluation is a key step in the radiotherapy treatment workflow. Central to this step is the assessment of treatment plan quality. Hence, it is important to agree on what we mean by plan quality and to be fully aware of which parameters it depends on. We understand plan quality in radiotherapy as the clinical suitability of the delivered dose distribution that can be realistically expected from a treatment plan. Plan quality is commonly assessed by evaluating the dose distribution calculated by the treatment planning system (TPS). Evaluating the 3D dose distribution is not easy, however; it is hard to fully evaluate its spatial characteristics and we still lack the knowledge for personalising the prediction of the clinical outcome based on individual patient characteristics. This advocates for standardisation and systematic collection of clinical data and outcomes after radiotherapy. Additionally, the calculated dose distribution is not exactly the dose delivered to the patient due to uncertainties in the dose calculation and the treatment delivery, including variations in the patient set-up and anatomy. Consequently, plan quality also depends on the robustness and complexity of the treatment plan. We believe that future work and consensus on the best metrics for quality indices are required. Better tools are needed in TPSs for the evaluation of dose distributions, for the robust evaluation and optimisation of treatment plans, and for controlling and reporting plan complexity. Implementation of such tools and a better understanding of these concepts will facilitate the handling of these characteristics in clinical practice and be helpful to increase the overall quality of treatment plans in radiotherapy.

Analytical probabilistic modeling of dose-volume histograms

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, α - 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 α - 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 ±0.2), the respective - 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, that is, normal or beta distributions, over DVH-points.