Reduced Order Modelling methodologies for proton therapy applications

Abstract

In proton therapy, the calculation of the dose distribution is of great importance in order to find the best treatment plan. For a treatment plan with high quality, several error scenarios are investigated, in order to come up with a general plan that suits best in these scenarios. All of the scenarios have a different simulated error, e.g., in patient displacement and proton beam range. In order to cope with a large number of errors scenarios, a quick calculation method is needed. In this work, a method that used Reduced Order Modelling (ROM) and subsequently polynomial regression to overcome the computational problem is presented, as well as its results on a real head and neck cancer patient. Before error scenarios are simulated, using inverse optimisation a treatment plan is calculated for a perfectly known position and organ structure of the patient, i.e. the nominal scenario. Then, several error scenarios are simulated and for each of these scenarios, the real dose on tissue is calculated, using the treatment parameters that were previously set for the nominal scenario. The dose distributions of all these scenarios are stored in a matrix, on which a Singular Value Decomposition (SVD) is executed. A reduced order of singular values and vectors is used, together with regression models on the right singular vectors, to reconstruct the dose distribution matrix. This is done in order to create a computationally
cheap method for dose distribution calculations. The right singular vectors are now a function of the used parameters: positioning errors dx, dy, dz and proton beam range error dρ. The dose distribution matrix is reconstructed from the left singular vectors, first few singular values and the function for the right singular vectors. Range errors with a uniform distribution with a maximum of 3% and positioning errors with a normal distribution with a standard deviation of 3 mm in each direction were simulated. For 100 error scenarios, the dose distribution matrix could be reconstructed with an average accepted error of 1% on a voxel relative to the maximum dose for 96:5% of the voxels, using a voxel-by-voxel comparison. These results were obtained for a 17th order of the SVD and a 7th order polynomial in the regression. On unseen test data, the acceptance was 81:0% of the voxels with an allowed error of 1%. For an allowed error of 3%, 92:5% of the voxels of this test set were accepted. These results are promising and encouraging for future research. First of all, it is recommended to examine the regression of the right singular vectors more closely, especially the regression of the higher order right singular vectors. Further subjects of interest would be to perform the same procedure that is done on the dose distribution matrix on the dose influence matrix. Also, harder-to-reach error scenarios,
like organ movement and tumour deformation could be taken into account. This research was conducted at the Medical Physics and Technology section, a part of the Department of Radiation Science and Technology at Delft University of Technology. The project was part of the Bachelor program of Applied Physics at the Technical University of Delft.