A robustness analysis method with fast estimation of dose uncertainty distributions for carbon-ion therapy treatment planning

An evaluation method of clinical impact with setup, range, and radiosensitivity uncertainties in fractionated carbon-ion therapy

In light ion therapy, the dose concentration is highly sensitive to setup and range errors. Here we propose a method for evaluating the effects of these errors by using the correlation between fractions on tumour control probability (TCP) in carbon-ion therapy. This method incorporates the concept of equivalent stochastic dose (Cranmer-Sargison and Zavgorodni 2005 Phys. Med. Biol. 50 4097-109), which was defined as a dose that gives the mean expected survival fraction (SF) for the stochastically variable dose. The mean expected SFs were calculated while considering the correlation between fractions for setup and range errors. By using this SF, equivalent stochastic clinical doses (ESCD), which are weighted by relative biological effectiveness, of lung and prostate cases with varying errors were derived. To account for spatial dose heterogeneity, equivalent uniform stochastic clinical doses (EUSCD) were obtained by using the mean expected SF in the volume of interest. TCP curves were calculated for each assumed error considering inter-patient sensitivity variation with a fractionation effect. ESCD distributions, EUSCD, and TCP curves were affected by the inter-fraction correlation and the contribution of setup and range errors. Irradiated areas that could be affected by these errors can be visualized quantitatively by using the ESCD distribution. TCP curves for the errors of various conditions converged around the TCP curve in nominal conditions by using the EUSCD. EUSCD correlated well with TCP in setup and range errors when the errors were not large and was comparatively stably insensitive to uncertain biological parameters. The proposed evaluation method with EUSCD and TCP calculations will be useful to indicate tumour doses to improve realistic dose distributions in carbon-ion therapy.

Analytical probabilistic modeling of RBE-weighted dose for ion therapy

Particle therapy is especially prone to uncertainties. This issue is usually addressed with uncertainty quantification and minimization techniques based on scenario sampling. For proton therapy, however, it was recently shown that it is also possible to use closed-form computations based on analytical probabilistic modeling (APM) for this purpose. APM yields unique features compared to sampling-based approaches, motivating further research in this context. This paper demonstrates the application of APM for intensity-modulated carbon ion therapy to quantify the influence of setup and range uncertainties on the RBE-weighted dose. In particular, we derive analytical forms for the nonlinear computations of the expectation value and variance of the RBE-weighted dose by propagating linearly correlated Gaussian input uncertainties through a pencil beam dose calculation algorithm. Both exact and approximation formulas are presented for the expectation value and variance of the RBE-weighted dose and are subsequently studied in-depth for a one-dimensional carbon ion spread-out Bragg peak. With V and B being the number of voxels and pencil beams, respectively, the proposed approximations induce only a marginal loss of accuracy while lowering the computational complexity from order [Formula: see text] to [Formula: see text] for the expectation value and from [Formula: see text] to [Formula: see text] for the variance of the RBE-weighted dose. Moreover, we evaluated the approximated calculation of the expectation value and standard deviation of the RBE-weighted dose in combination with a probabilistic effect-based optimization on three patient cases considering carbon ions as radiation modality against sampled references. The resulting global γ-pass rates (2 mm,2%) are [Formula: see text]99.15% for the expectation value and [Formula: see text]94.95% for the standard deviation of the RBE-weighted dose, respectively. We applied the derived analytical model to carbon ion treatment planning, although the concept is in general applicable to other ion species considering a variable RBE.