Optimizing for the Rupert property

A polyhedron is Rupert if it is possible to cut a hole in it and thread an identical polyhedron through the hole. It is known that all 5 Platonic solids, 10 of the 13 Archimedean solids, 9 of the 13 Catalan solids, and 82 of the 92 Johnson solids are Rupert. Here, a nonlinear optimization method is devised that is able to validate the previously known results in seconds. It is also used to show that 2 additional Catalan solids -- the triakis tetrahedron and the pentagonal icositetrahedron -- and 5 additional Johnson solids are Rupert

Krylov Solvers for Interior Point Methods with Applications in Radiation Therapy and Support Vector Machines

Interior point methods are widely used for different types of mathematical optimization problems. Many implementations of interior point methods in use today rely on direct linear solvers to solve systems of equations in each iteration. The need to solve ever larger optimization problems more efficiently and the rise of hardware accelerators for general purpose computing has led to a large interest in using iterative linear solvers instead, with the major issue being inevitable ill-conditioning of the linear systems arising as the optimization progresses. We investigate the use of Krylov solvers for interior point methods in solving optimization problems from radiation therapy and support vector machines. We implement a prototype interior point method using a so called doubly augmented formulation of the Karush-Kuhn-Tucker linear system of equations, originally proposed by Forsgren and Gill, and evaluate its performance on real optimization problems from radiation therapy and support vector machines. Crucially, our implementation uses a preconditioned conjugate gradient method with Jacobi preconditioning internally. Our measurements of the conditioning of the linear systems indicate that the Jacobi preconditioner improves the conditioning of the systems to a degree that they can be solved iteratively, but there is room for further improvement in that regard. Furthermore, profiling of our prototype code shows that it is suitable for GPU acceleration, which may further improve its performance in practice. Overall, our results indicate that our method can find solutions of acceptable accuracy in reasonable time, even with a simple Jacobi preconditioner

Optimizing the Traversal Time for Gantry Trajectories for Proton Arc Therapy Treatment Plans

Background: Proton arc therapy is an emerging radiation therapy technique where either the gantry or the patient continuously rotates during the irradiation treatment. One of the perceived advantages of proton arc therapy is the reduced treatment time, but it is still unclear exactly how long these treatment times will be, given that no machine capable of its delivery is available at the market at the time of writing. Purpose: We introduce the algorithm Arc Trajectory Optimization Method (ATOM), which aims to determine an efficient velocity profile for the gantry for rapid delivery of a given proton arc treatment plan. Methods: ATOM computes the trajectory with the shortest delivery time while ensuring there is enough time to deliver all spots in each energy layer and switch energy between layers. The feasibility of the dynamic gantry movement was assured by enforcing maximum and minimum limits for velocity, acceleration, and jerk. This was achieved by discretizing the gantry velocity and combining the A* algorithm with the open-source motion generation library Ruckig. The algorithm was tested on a synthetic data set as well as a liver case, a prostate case and a head and neck case. Results: Arc trajectories for plans with 360 energy layers were calculated in under a second using 256 discrete velocities. Conclusions: ATOM is an open-source C++ library with a Python interface that rapidly generates velocity profiles, making it a highly efficient tool for determining proton arc delivery times, which could be integrated into the treatment planning process

Distributed Objective Function Evaluation for Optimization of Radiation Therapy Treatment Plans

The modern workflow for radiation therapy treatment planning involves mathematical optimization to determine optimal treatment machine parameters for each patient case. The optimization problems can be computationally expensive, requiring iterative optimization algorithms to solve. In this work, we investigate a method for distributing the calculation of objective functions and gradients for radiation therapy optimization problems across computational nodes. We test our approach on the TROTS dataset -- which consists of optimization problems from real clinical patient cases -- using the IPOPT optimization solver in a leader/follower type approach for parallelization. We show that our approach can utilize multiple computational nodes efficiently, with a speedup of approximately 2-3.5 times compared to the serial version.Comment: Accepted for publication at the PPAM22 conferenc

Partitioning of discrete proton arcs into interlaced subplans can bring proton arc advances to existing proton facilities

Background: Proton arcs have shown potential to reduce the dose to organs at risks (OARs) by delivering the protons from many different directions. While most previous studies have been focused on dynamic arcs (delivery during rotation), an alternative approach is discrete arcs, where step-and-shoot delivery is used over a large number of beam directions. The major advantage of discrete arcs is that they can be delivered at existing proton facilities. However, this advantage comes at the expense of longer treatment times.Purpose: To exploit the dosimetric advantages of proton arcs, while achieving reasonable delivery times, we propose a partitioning approach where discrete arc plans are split into subplans to be delivered over different fractions in the treatment course.Methods: For three oropharyngeal cancer patients, four different arc plans have been created and compared to the corresponding clinical IMPT plan. The treatment plans are all planned to be delivered in 35 fractions, but with different delivery approaches over the fractions. The first arc plan (1×30) has 30 directions to be delivered every fraction, while the others are partitioned into subplans with 10 and 6 beam directions, each to be delivered every third (3×10), fifth fraction (5×6), or seventh fraction (7×10). All plans are assessed with respect to delivery time, target robustness over the treatment course, doses to OARs and NTCP for dysphagia and xerostomia.Results: The delivery time (including an additional delay of 30 s between the discrete directions to simulate manual interaction with the treatment control system) is reduced from on average 25.2 min for the 1×30 plan to 9.2 min for the 3×10 and 7×10 plans and 5.7 min for the 5×6 plans. The delivery time for the IMPT plan is 7.9 min. When accounting for the combination of delivery time, target robustness, OAR sparing, and NTCP reduction, the plans with 10 directions in each fraction are the preferred choice. Both the 3×10 and 7×10 plans show improved target robustness compared to the 1×30 plans, while keeping OAR doses and NTCP values at almost as low levels as for the 1×30 plans. For all patients the NTCP values for dysphagia are lower for the partitioned plans with 10 directions compared to the IMPT plans. NTCP reduction for xerostomia compared to IMPT is seen in two of the three patients. The best results are seen for the first patient, where the NTCP reductions for the 7×10 plan are 1.6 p.p. (grade 2 xerostomia) and 1.5 p.p. (grade 2 dysphagia). The corresponding NTCP reductions for the 1×30 plan are 2.7 p.p. (xerostomia, grade 2) and 2.0 p.p. (dysphagia, grade 2).Conclusions: Discrete proton arcs can be implemented at any proton facility with reasonable treatment times using a partitioning approach. The technique also makes the proton arc treatments more robust to changes in the patient anatomy.</p

Partitioning of discrete proton arcs into interlaced subplans can bring proton arc advances to existing proton facilities

Background: Proton arcs have shown potential to reduce the dose to organs at risks (OARs) by delivering the protons from many different directions. While most previous studies have been focused on dynamic arcs (delivery during rotation), an alternative approach is discrete arcs, where step-and-shoot delivery is used over a large number of beam directions. The major advantage of discrete arcs is that they can be delivered at existing proton facilities. However, this advantage comes at the expense of longer treatment times.Purpose: To exploit the dosimetric advantages of proton arcs, while achieving reasonable delivery times, we propose a partitioning approach where discrete arc plans are split into subplans to be delivered over different fractions in the treatment course.Methods: For three oropharyngeal cancer patients, four different arc plans have been created and compared to the corresponding clinical IMPT plan. The treatment plans are all planned to be delivered in 35 fractions, but with different delivery approaches over the fractions. The first arc plan (1×30) has 30 directions to be delivered every fraction, while the others are partitioned into subplans with 10 and 6 beam directions, each to be delivered every third (3×10), fifth fraction (5×6), or seventh fraction (7×10). All plans are assessed with respect to delivery time, target robustness over the treatment course, doses to OARs and NTCP for dysphagia and xerostomia.Results: The delivery time (including an additional delay of 30 s between the discrete directions to simulate manual interaction with the treatment control system) is reduced from on average 25.2 min for the 1×30 plan to 9.2 min for the 3×10 and 7×10 plans and 5.7 min for the 5×6 plans. The delivery time for the IMPT plan is 7.9 min. When accounting for the combination of delivery time, target robustness, OAR sparing, and NTCP reduction, the plans with 10 directions in each fraction are the preferred choice. Both the 3×10 and 7×10 plans show improved target robustness compared to the 1×30 plans, while keeping OAR doses and NTCP values at almost as low levels as for the 1×30 plans. For all patients the NTCP values for dysphagia are lower for the partitioned plans with 10 directions compared to the IMPT plans. NTCP reduction for xerostomia compared to IMPT is seen in two of the three patients. The best results are seen for the first patient, where the NTCP reductions for the 7×10 plan are 1.6 p.p. (grade 2 xerostomia) and 1.5 p.p. (grade 2 dysphagia). The corresponding NTCP reductions for the 1×30 plan are 2.7 p.p. (xerostomia, grade 2) and 2.0 p.p. (dysphagia, grade 2).Conclusions: Discrete proton arcs can be implemented at any proton facility with reasonable treatment times using a partitioning approach. The technique also makes the proton arc treatments more robust to changes in the patient anatomy.</p