This project focuses on the development of computational methods for tomosynthesis breast image reconstruction, a technique used to reconstruct 3-dimensional images using slightly modified versions of conventional x-ray systems. The mathematical models addressed in this project are difficult ill-posed inverse problems; computed solutions are very sensitive to errors in the data, and implementation for large scale 3-dimensional images is nontrivial. All previous breast tomosynthesis image reconstruction algorithms use a simplified, but incorrect assumption that the source x-ray beam is comprised of photons with a constant energy; that is, the x-ray beam is assumed to be monoenergetic. The simplified monoenergetic assumption results in a linear mathematical model. This project uses the physically correct, and hence more accurate, assumption that the x-ray beam is polyenergetic. The resulting mathematical model is nonlinear, providing great challenges to the development and analysis of mathematical models, as well as for the development of computational methods. However, as this project reveals, the nonlinear model allows for reconstructing images with substantially fewer artifacts than the linear model. Moreover, the nonlinear model can incorporate parameterizations to allow for explicit decomposition of the breast into distinct materials, such as, glandular tissue, adipose tissue, calcifications, and iodinated contrast agents, thereby providing improved diagnostic information.
In the US, over 200,000 women are diagnosed with breast cancer every year, but there is a 97% five-year survival rate if the cancer is localized, and if it is discovered before it spreads to other parts of the body. Therefore, improved imaging techniques that help to detect and diagnose breast cancer early can have a profound impact on the healthcare of women. The research in this project focuses on tomosynthesis imaging, which just received FDA approval for clinical use in 2011, because it has the potential to provide substantially better screening capabilities than mammography. The new mathematical and computational approaches developed in this work are designed to unlock the potentially transformative benefits of tomosynthesis for breast cancer screening, providing significantly better diagnostic information to physicians. In addition to its application to breast cancer screening, tomosynthesis, in its whole-body implementation can be used for many other applications where standard x-ray and CT are used, such as chest imaging for detection of lung nodules, as well as non-medical imaging applications such as nuclear waste inspections and explosive detection. Thus, advances in computational methods for this application can have a very broad impact in the imaging field. Collaborations between researchers in Mathematics and Computer Science and researchers in the Department of Radiology and Imaging Sciences and Winship Cancer Institute at Emory University facilitates transitioning new software to clinical use.
To help diagnose breast cancer, a new method called Breast Tomosynthesis imaging has been developed. Breast tomosynthesis is a type of three-dimensional mammogram. It consists of taking a series of images of the breast at different angles. A computer takes the information from these images and makes a set of images that can be looked at individually or like a movie. These images show the breast in slices instead of the entire breast in one image, as mammography does. This way of imaging the breast reduces the chances that a cancer is missed or that something normal looks like cancer. Although the development of breast tomosynthesis seems to be improving the performance of screening for breast cancer, much work remains to be done to maximize its benefit. One of the aspects that can be improved is how the computer takes the series of images of the breast, at different angles, and from them creates the series of slices of the entire breast. Up to now, this process, called reconstruction, simplified this problem by assuming certain things about the image acquisition process that we know are not true. One of them is the assumption that the x-rays used during image acquisition are all the same energy, or color. We know this is not true, but it makes the mathematics behind this process a lot simpler. In this project, we developed the mathematics and the computer programs to create the breast slices without assuming that the x-rays are all the same color; instead, we take into account the actual distribution of colors included in the x-ray beam. To achieve this, we developed a more realistic mathematical description of the image acquisition process, optimized the mathematics involved in reconstructing the breast slices, and created the software program that is able to perform this process, requiring run times of only a few minutes. As a result, we have shown that the images of the breast slices from our new approach are better than the ones obtained from previously proposed methods; in particular, our reconstructions show lesions more clearly and including fewer and less severe errors in the images. In addition, due to the new mathematical processes involved in our reconstruction, we have early results that show we might also be able to improve the image acquisition process, resulting in a large reduction in the amount of radiation used to acquire a breast tomosynthesis image without losing image quality.