Philadelphia, PA—Over 200,000 women are diagnosed with breast cancer every year. The five-year survival rate for afflicted women is 97% if the cancer is localized and discovered before it spreads to other parts of the body.
“Currently, mammography is the technique used most often for breast cancer screening, but since it gives only two-dimensional (2D) projection information of a three-dimensional (3D) anatomical structure, inaccuracies in screening often occur,” says James Nagy, co-author of a paper on breast tomosynthesis image reconstruction published last fall in the SIAM Journal on Scientific Computing (along with Veronica Mejia Bustamante, Steve Feng, and Ioannis Sechopoulos).
While conventional x-ray mammography produces 2D projection images of 3D objects, digital technologies such as tomosynthesis can produce 3D image information of an object by using slightly modified conventional digital x-ray systems. Conventional mammography is limited by superposition of breast tissue, which can sometimes mimic or obscure malignant pathology.
In digital tomosynthesis imaging, multiple projections of an object are obtained along a range of different incident angles, which are then used to reconstruct a pseudo-3D representation of the object. The 3D reconstructions are generated by computing weight fractions of the individual materials composing the object. The underlying idea is that several 2D image projections taken at varying angles can provide more and different information about the 3D object. This improves sensitivity and specificity.
“Tomosynthesis is a technique that can produce pseudo three-dimensional reconstructions, reducing the effect of tissue superposition, and therefore allowing for more cancers to be found early,” says Nagy. “Thus, optimizing the image quality in tomosynthesis by improving the reconstruction methods used to create these images is of utmost importance to maximize the benefit of breast cancer screening.” The model is further validated with real data taken from an object with known materials that simulates an actual breast.
The polyenergetic nature of the x-ray—which means that the beams are made up of photons or light particles having a wide range of energies—is also accounted for in the model. Many previous studies that have developed algorithms for tomosynthesis image reconstruction operate on the assumption that x-rays are monoenergetic.
“All reconstruction algorithms used by clinical machines use a simple, but incorrect, assumption that the x-ray beam is monoenergetic,” explains Nagy. “The polyenergetic model attempts to provide a more accurate physical model of the x-ray projection process, which results in more accurate reconstructions of images. The tradeoff is that the mathematical problem is more complicated, and the computational costs are greater.”
Reconstruction requires solving a large-scale inverse problem, which takes into account the energy of the x-ray beam, its attenuation or reduction as it passes through and interacts with matter, features of the detector, as well as various parameters of the object being imaged. “A better understanding of the mathematical model helps to advance the development of new reconstruction algorithms to solve this very challenging problem,” Nagy says.
Future work naturally involves focusing on algorithms that are computationally more economical. “Our approach requires solving a nonlinear optimization problem which is substantially more computationally intensive than algorithms currently used in a clinical setting, which are based on a linear approximation model,” says Nagy. “More work is also needed to develop implementations that can exploit modern computer architectures, such as those with multi-core CPU and GPUs.”
Nagy’s group is currently developing an implementation with flexibility to exploit either architecture, as well as hybrid CPU/GPU systems. “In addition, this novel reconstruction algorithm will allow us to perform tomosynthesis using more advanced image acquisition methods which may result in further improvements in image quality, lower radiation dose, and material decomposition,” Nagy says. A reduction in the amount of radiation patients receive, as well as improved representation of materials composing the breast will result in an enhanced procedure.
Read a previous SIAM Nugget on a mathematical model for tomography for breast cancer detection. To read other SIAM Nuggets, explaining current high level research involving applications of mathematics in popular science terms, go to http://connect.siam.org/category/siam-nuggets/.
Veronica Mejia Bustamante, James G. Nagy, Steve S. J. Feng, and Ioannis Sechopoulos
SIAM Journal on Scientific Computing, 35(5), S192–S208 (Online publish date: October 28, 2013). The source article is available for free access at the link above until May 10, 2014.
About the authors:
Veronica Mejia Bustamante was a research assistant in applied and computational mathematics at Emory University at the time of work done on the paper. James Nagy is a professor in the department of mathematics and computer science at Emory University. Steve S.J. Feng was a research assistant in biomedical engineering at Emory University and the Georgia Institute of Technology at the time of work done on the paper. Ioannis Sechopoulos is an assistant professor of radiology at Emory University.
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