An Improved Method of Total Variation Superiorization Applied to Reconstruction in Proton Computed Tomography

Immunity to Increasing Condition Numbers of Linear Superiorization versus Linear Programming

Given a family of linear constraints and a linear objective function one can consider whether to apply a Linear Programming (LP) algorithm or use a Linear Superiorization (LinSup) algorithm on this data. In the LP methodology one aims at finding a point that fulfills the constraints and has the minimal value of the objective function over these constraints. The Linear Superiorization approach considers the same data as linear programming problems but instead of attempting to solve those with linear optimization methods it employs perturbation resilient feasibility-seeking algorithms and steers them toward feasible points with reduced (not necessarily minimal) objective function values. Previous studies compared LP and LinSup in terms of their respective outputs and the resources they use. In this paper we investigate these two approaches in terms of their sensitivity to condition numbers of the system of linear constraints. Condition numbers are a measure for the impact of deviations in the input data on the output of a problem and, in particular, they describe the factor of error propagation when given wrong or erroneous data. Therefore, the ability of LP and LinSup to cope with increased condition numbers, thus with ill-posed problems, is an important matter to consider which was not studied until now. We investigate experimentally the advantages and disadvantages of both LP and LinSup on examplary problems of linear programming with multiple condition numbers and different problem dimensions.

Superiorization of projection algorithms for linearly constrained inverse radiotherapy treatment planning

Objective

We apply the superiorization methodology to the constrained intensity-modulated radiation therapy (IMRT) treatment planning problem. Superiorization combines a feasibility-seeking projection algorithm with objective function reduction: The underlying projection algorithm is perturbed with gradient descent steps to steer the algorithm towards a solution with a lower objective function value compared to one obtained solely through feasibility-seeking.

Approach

Within the open-source inverse planning toolkit matRad, we implement a prototypical algorithmic framework for superiorization using the well-established Agmon, Motzkin, and Schoenberg (AMS) feasibility-seeking projection algorithm and common nonlinear dose optimization objective functions. Based on this prototype, we apply superiorization to intensity-modulated radiation therapy treatment planning and compare it with (i) bare feasibility-seeking (i.e., without any objective function) and (ii) nonlinear constrained optimization using first-order derivatives. For these comparisons, we use the TG119 water phantom, the head-and-neck and the prostate patient of the CORT dataset.

Main results

Bare feasibility-seeking with AMS confirms previous studies, showing it can find solutions that are nearly equivalent to those found by the established piece-wise least-squares optimization approach. The superiorization prototype solved the linearly constrained planning problem with similar dosimetric performance to that of a general-purpose nonlinear constrained optimizer while showing smooth convergence in both constraint proximity and objective function reduction.

Significance

Superiorization is a useful alternative to constrained optimization in radiotherapy inverse treatment planning. Future extensions with other approaches to feasibility-seeking, e.g., with dose-volume constraints and more sophisticated perturbations, may unlock its full potential for high performant inverse treatment planning.

Per-RMAP: Feasibility-Seeking and Superiorization Methods for Floorplanning with I/O Assignment

The feasibility-seeking approach provides a systematic scheme to manage and solve complex constraints for continuous problems, and we explore it for the floorplanning problems with increasingly heterogeneous constraints. The classic legality constraints can be formulated as the union of convex sets. However, the convergence of conventional projection-based algorithms is not guaranteed when the constraints sets are non-convex, which is the case with unions of convex sets. In this work, we propose a resetting strategy to greatly eliminate the divergence issue of the projection-based algorithm for the feasibility-seeking formulation. Furthermore, the superiorization methodology (SM), which lies between feasibility-seeking and constrained optimization, is firstly applied to floorplanning. The SM uses perturbations to steer the iterates of a feasibility-seeking algorithm to a feasible solution with shorter total wirelength. The proposed algorithmic flow is extendable to tackle various constraints and variants of floorplanning problems, e.g., floorplanning with I/O assignment problems. We have evaluated the proposed algorithm on the MCNC benchmarks. We can obtain legal floorplans only two times slower than the branch-and-bound method in its current prototype using MATLAB, with only 3% wirelength inferior to the optimal results. We evaluate the effectiveness of the algorithmic flow by considering the constraints of I/O assignment, and our algorithm achieves 8% improvement on wirelength.

An Iterative Least Squares Method for Proton CT Image Reconstruction

Clinically useful proton Computed Tomography images will rely on algorithms to find the three-dimensional proton stopping power distribution that optimally fits the measured proton data. We present a least squares iterative method with many features to put proton imaging into a more quantitative framework. These include the definition of a unique solution that optimally fits the protons, the definition of an iteration vector that takes into account proton measurement uncertainties, the definition of an optimal step size for each iteration individually, the ability to simultaneously optimize the step sizes of many iterations, the ability to divide the proton data into arbitrary numbers of blocks for parallel processing and use of graphical processing units, and the definition of stopping criteria to determine when to stop iterating. We find that it is possible, for any object being imaged, to provide assurance that the image is quantifiably close to an optimal solution, and the optimization of step sizes reduces the total number of iterations required for convergence. We demonstrate the use of these algorithms on real data.

The impact of path estimates in iterative ion CT reconstructions for clinical-like cases

Ion computed tomography (CT) promises to mitigate range uncertainties inherent in the conversion of x-ray Hounsfield units into ion relative stopping power (RSP) for ion beam therapy treatment planning. To improve accuracy and spatial resolution of ion CT by accounting for statistical multiple Coulomb scattering deflection of the ion trajectories from a straight line path (SLP), the most likely path (MLP) and the cubic spline path (CSP) have been proposed. In this work, we use FLUKA Monte Carlo simulations to investigate the impact of these path estimates in iterative tomographic reconstruction algorithms for proton, helium and carbon ions. To this end the ordered subset simultaneous algebraic reconstruction technique was used and coupled with a total variation superiorization (TVS). We evaluate the image quality and dose calculation accuracy in proton therapy treatment planning of cranial patient anatomies. CSP and MLP generally yielded nearly equal image quality with an average RSP relative error improvement over the SLP of 0.6%, 0.3% and 0.3% for proton, helium and carbon ion CT, respectively. Bone and low density materials have been identified as regions of largest enhancement in RSP accuracy. Nevertheless, only minor differences in dose calculation results were observed between the different models and relative range errors of better than 0.5% were obtained in all cases. Largest improvements were found for proton CT in complex scenarios with strong heterogeneities along the beam path. The additional TVS provided substantially reduced image noise, resulting in improved image quality in particular for soft tissue regions. Employing the CSP and MLP for iterative ion CT reconstructions enabled improved image quality over the SLP even in realistic and heterogeneous patient anatomy. However, only limited benefit in dose calculation accuracy was obtained even though an ideal detector system was simulated.

First application of the GPU-based software framework TIGRE for proton CT image reconstruction

In proton therapy, the knowledge of the proton stopping power, i.e. the energy deposition per unit length within human tissue, is essential for accurate treatment planning. One suitable method to directly measure the stopping power is proton computed tomography (pCT). Due to the proton interaction mechanisms in matter, pCT image reconstruction faces some challenges: the unique path of each proton has to be considered separately in the reconstruction process adding complexity to the reconstruction problem. This study shows that the GPU-based open-source software toolkit TIGRE, which was initially intended for X-ray CT reconstruction, can be applied to the pCT image reconstruction problem using a straight line approach for the proton path. This simplified approach allows for reconstructions within seconds. To validate the applicability of TIGRE to pCT, several Monte Carlo simulations modeling a pCT setup with two Catphan® modules as phantoms were performed. Ordered-Subset Simultaneous Algebraic Reconstruction Technique (OS-SART) and Adaptive-Steepest-Descent Projection Onto Convex Sets (ASD-POCS) were used for image reconstruction. Since the accuracy of the approach is limited by the straight line approximation of the proton path, requirements for further improvement of TIGRE for pCT are addressed.

The 20th Gray lecture 2019: health and heavy ions

Objective

Particle radiobiology has contributed new understanding of radiation safety and underlying mechanisms of action to radiation oncology for the treatment of cancer, and to planning of radiation protection for space travel. This manuscript will highlight the significance of precise physical and biologically effective dosimetry to this translational research for the benefit of human health.

This review provides a brief snapshot of the evolving scientific basis for, and the complex current global status, and remaining challenges of hadron therapy for the treatment of cancer. The need for particle radiobiology for risk planning in return missions to the Moon, and exploratory deep-space missions to Mars and beyond are also discussed.

Methods

Key lessons learned are summarized from an impressive collective literature published by an international cadre of multidisciplinary experts in particle physics, radiation chemistry, medical physics of imaging and treatment planning, molecular, cellular, tissue radiobiology, biology of microgravity and other stressors, theoretical modeling of biophysical data, and clinical results with accelerator-produced particle beams.

Results

Research pioneers, many of whom were Nobel laureates, led the world in the discovery of ionizing radiations originating from the Earth and the Cosmos. Six radiation pioneers led the way to hadron therapy and the study of charged particles encountered in outer space travel. Worldwide about 250,000 patients have been treated for cancer, or other lesions such as arteriovenous malformations in the brain between 1954 and 2019 with charged particle radiotherapy, also known as hadron therapy. The majority of these patients (213,000) were treated with proton beams, but approximately 32,000 were treated with carbon ion radiotherapy. There are 3500 patients who have been treated with helium, pions, neon or other ions. There are currently 82 facilities operating to provide ion beam clinical treatments. Of these, only 13 facilities located in Asia and Europe are providing carbon ion beams for preclinical, clinical, and space research. There are also numerous particle physics accelerators worldwide capable of producing ion beams for research, but not currently focused on treating patients with ion beam therapy but are potentially available for preclinical and space research. Approximately, more than 550 individuals have traveled into Lower Earth Orbit (LEO) and beyond and returned to Earth.

Conclusion

Charged particle therapy with controlled beams of protons and carbon ions have significantly impacted targeted cancer therapy, eradicated tumors while sparing normal tissue toxicities, and reduced human suffering. These modalities still require further optimization and technical refinements to reduce cost but should be made available to everyone in need worldwide. The exploration of our Universe in space travel poses the potential risk of exposure to uncontrolled charged particles. However, approaches to shield and provide countermeasures to these potential radiation hazards in LEO have allowed an amazing number of discoveries currently without significant life-threatening medical consequences. More basic research with components of the Galactic Cosmic Radiation field are still required to assure safety involving space radiations and combined stressors with microgravity for exploratory deep space travel.

Advances in knowledge

The collective knowledge garnered from the wealth of available published evidence obtained prior to particle radiation therapy, or to space flight, and the additional data gleaned from implementing both endeavors has provided many opportunities for heavy ions to promote human health.