Publication Details

Penfold, S., Schulte, R., Censor, Y., Bashkirov, V., McAllister, S., Schubert, K. & Rozenfeld, A. (2010). Block-Iterative and String-averaging projection algorithms in proton computed tomography image reconstruction. In Y. Censor, M. Jiang & G. Wang (Eds.), Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning, and Inverse Problems (pp. 347-368). United States: Medical Physics Publishing.


Proton therapy is an advantageous form of radiotherapy because it allows for the placement of a high dose peak, the Bragg peak, at any desired depth by modulating proton energy. Currently, proton therapy treatment plans are carried out using data from X-ray CT scans, an imaging modality that generates tomographical maps of scaled photon linear attenuation coefficients, commonly known as Hounsfield units. However, to perform the treatment planning, one requires knowledge of the spatial distribution of electron density within the patient. In clinical practice Hounsfield units are converted to electron densities through an empirically derived relationship generated from measurements with tissue equivalent materials (Mustafa and Jackson 1983), (Schneider, Pedroni, and Lomax 1996). The end result of this conversion is a difference, typically ranging from several millimeters up to more than 1 cm, between the proton range calculated by the treatment planning software and the true proton range within the patient, depending on the anatomical region treated and the calibration method used (Schneider, Pedroni, and Lomax 1996). Thus, because the Bragg peak depth cannot be accurately predicted, the inherent advantages of proton therapy are partially negated in such an approach. Proton computed tomography (pCT) is an imaging modality that has been suggested as a means for reducing the uncertainty of Bragg peak location in proton radiation treatments. In pCT, the spatial location of individual protons pre- and post-patient, as well as the energy lost along the path is recorded (Schulte et al. 2004). The spatial measurements are employed in a maximum likelihood proton path formalism that models multiple Coulomb scattering, i.e., multiple small-angle deflections of the proton path due to interaction with the Coulomb field of the nuclei of the medium within the patient (Schulte et al. 2008), maximizing the spatial resolution. The corresponding energy loss measurements are converted to the integral relative electron density along this predicted path with the Bethe-Bloch equation, which describes the mean energy loss of a proton per unit track length as a function of density of the medium and the proton energy. By reconstructing many such events with an algebraic reconstruction technique (ART) (Gordon, Bender, and Herman 1970) capable of handling these nonlinear paths, three dimensional electron density maps can be generated without the need for any empirical conversion. These maps can then be used in the treatment planning system to accurately predict the proton dose distribution within the patient at treatment time. It has been demonstrated by previous pCT studies (Li et al. 2006) that superior spatial resolution can be achieved by employing ART for reconstruction in comparison to transform methods, such as filtered back-projection. This is primarily because transform methods must assume the proton traveled along a straight path in the reconstruction volume. Algebraic techniques, however, are much more flexible, not only allowing proton paths to be nonlinear but also permitting the inclusion of a priori knowledge about the object to be reconstructed. This flexibility may come at the expense of computation time, however, which could be greater for some iterative techniques than that for transform methods. If pCT is to be implemented in a clinical environment, fast image reconstruction is required. It has been suggested that the image reconstruction process should take less than 15 minutes for treatment planning images and less than 5 minutes for pre-treatment patient position verification images (Schulte et al. 2004). ART, recognized to be identical with the iterative projection algorithm of Kaczmarz (Kaczmarz 1937), has been implemented in previous pCT studies, displaying promising results (Li et al. 2006). However, ART carries out image updates after each proton history and is therefore inherently sequential, meaning that the speed of the reconstruction is dependent on the speed of the computer processing unit. As an example of the infeasibility of using ART for pCT in clinical practice we have recently observed that, using general purpose processing units, three dimensional images made up of a 256×256×48 voxel reconstruction volume, reconstructed with 10 million proton histories will take approximately 1.5 hours to complete a single cycle, with the optimal image often being reached after 3-4 cycles. With the development of parallel computing, work has been dedicated to developing iterative projection algorithms that can be executed in parallel over multiple processors to enable fast algebraic reconstructions. This paper compares the performance, in terms of image quality, of a number of parallel compatible block-iterative and string-averaging algebraic reconstruction algorithms with simulated pCT projection data. Quantitative assessment of image quality is based on the normalized mean absolute distance measure as described in (Herman 1980), and a qualitative note is made about image appearance. From these results recommendations are made on which image reconstruction algorithms should be used in future studies with pCT.

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