Application of variance-based uncertainty and sensitivity analysis to biological modeling in carbon ion treatment...

In ion beam therapy, biological models to estimate the relative biological effectiveness (RBE) and subsequently the RBE-weighted dose (RWD) are needed in treatment planning and plan evaluation. The required biological parameters as well as their dependency on ion species and ion energy can typically not be determined directly in experiments for in vivo situations. For that reason they are often derived from in vitro data and biological modeling and subject to large uncertainties. We present a model-independent Monte Carlo (variance-) based uncertainty and Sensitivity Analysis (SA) approach to quantify the impact of different input uncertainties on a simulated carbon ion treatment plan.

The influences of different input uncertainties are examined by variance-based SA methods. In this Monte Carlo approach, a function is evaluated 10³ -10⁵ times. For each of those runs, all inputs are changed simultaneously, using random numbers according to their associated uncertainties. Variance-based statistic formalisms then rank the input parameter/uncertainty pairs according to their impact on the result of the function. The method of SA includes an uncertainty analysis and was applied to a two-field spot scanning carbon ion treatment plan for two commonly used biological models and two representative tissue parameter sets.

Based on an exemplary patient case, the application of variance-based SA for biological measures, relevant in (carbon) ion therapy, is demonstrated. A voxel-wise calculation for 2.9 · 10⁵ voxels takes ~6 h. A structure-based SA, which adds an uncertainty band to a RWD-volume histogram (RW-DVH) and shows how to decrease the uncertainty in the most effective way, can be calculated in 0.1-1.5 h (depending on the size of the structure). The uncertainties in RBE, RWD or RW-DVH are broken down to the impact of different uncertainties in the (biological) model input. Biological uncertainties have a higher impact on the resulting RBE and RWD than uncertainties in the physical dose. Excluding the physical dose from the SA only slightly decreased the overall uncertainty, emphasizing the necessity to include biological uncertainties into treatment plan evaluation.

Variance-based SA is a powerful tool to evaluate the impact of uncertainties in (carbon) ion therapy. The number of input parameters that can be examined at once is only limited by computation time. A Monte Carlo-derived, comprehensive uncertainty quantification and a corresponding sensitivity analysis are implemented and provide new information for treatment plan evaluation. A possible future application is a SA-based biologically robust treatment plan optimization using the additional uncertainty information as presented here.