Application of variance‐based uncertainty and sensitivity analysis to biological modeling in carbon ion treatment plans

Abstract

Purpose 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. Method 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. Results 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. Conclusion 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.

State-of-the-Art and Future Prospects of Ion Beam Therapy: Physical and Radiobiological Aspects

“…RBE modeling is frequently considered to be subject to large uncertainties, depending on the accuracy with which the model input parameters are known [114]. However, one has to keep in mind that similar uncertainties need to be considered in conventional photon therapy when e.g.…”

Variance‐based sensitivity analysis for uncertainties in proton therapy: A framework to assess the effect of simultaneous uncertainties in range, positioning, and RBE model predictions on RBE‐weighted dose distributions

“…During dose calculation, the algorithm has to select the correct neighbor raytracing to replace based on the lateral offsets and from equation 15and * ∆ ∆ (16). This method is similar to the "virtual beamlets" approach proposed by Unkelbach et al 38 , with the difference, that in our approach only the raytracing is approximated by the neighbor which is closest to the patient shift, while the lateral part in equation 12is modeled exactly according to equations (15) and (16). The approach to keep pre-calculated neighboring raytracings for every PBS spot is memory intensive, however, since only small setup-errors are expected and need to be treated in the SA, sets of only a few shifts of up to a few millimeters are sufficient.…”

“…A rapid implementation 25,26 of this model in the ij-formalism for carbon ions has already been used for a variance-based SA of biological uncertainties by Kamp et al 15 and for RWD optimization by Guan et al 40 .…”

“…While the alternative techniques mentioned have their limitations in the case of non-linear models, global, variance based SA is a modelindependent approach and is applicable for any probability distribution of the input factors. In the field of medical physics, this technique has so far only been applied to RBE modeling of carbon ion therapy, excluding range and setup uncertainties 14,15 and to nuclear medicine, in order to assess the impact of…”

“…13 While the alternative techniques mentioned have their limitations in the case of nonlinear models, global, variance-based SA is a model-independent approach and is applicable for any probability distribution of the input factors. In the field of medical physics, this technique has so far only been applied to RBE modeling of carbon ion therapy, excluding range and setup uncertainties 14,15 and to nuclear medicine, in order to assess the impact of interpatient variability on organ dose estimates. 16 In this feasibility study, we present a framework to apply the technique of global, variance-based SA to uncertainties in proton therapy, explicitly modeling RBE, range, and setup errors.…”