NSF Award
2204618
In non-invasive or minimally invasive medical imaging, the goal is to form an image of internal structures of the human body based on measurements performed outside of the body without harming the patient. While the visible part of medical imaging comprising the imaging devices is based on engineering and physics, the image formation and retrieval of pertinent information relies on sophisticated mathematical models and efficient computational methods. This project will address two specific imaging problems, breast cancer screening and stroke detection and classification. Breast cancer screening by mammography is a standard process. However, it is known that, in particular, when the breast tissue is dense, which is the case in 10-40 percent of US women, the risk that a radiologist misses a cancerous lesion is significant. The project will investigate a novel computational idea of using mammography images at different pressure levels and comparing the tissue displacements to estimate the elastic properties of the tissue that are known to be affected by certain cancer types that often remain undetected. Another medical imaging problem addressed in this project is stroke classification by a portable and inexpensive electrical impedance tomography device. It is known that the prognosis of ischemic stroke depends heavily on how early the therapy can be initiated. About 15 percent of stroke patients who made it to the hospital in time were diagnosed with a brain hemorrhage. The therapy meant for ischemic stroke patients would be fatal. A portable classification method suitable for an ambulance could be crucial for diagnosis to save many lives in an emergency. The idea is not new, but the mathematical and computational problems continue to be challenging. This project will focus on the computational challenges in the problems described above and in other mathematically similar problems that cover medical imaging. The results will also be useful in non-destructive material evaluation and geophysics, including other application areas. The project will also include a strong educational component through the involvement of graduate students who will work on their doctoral dissertations on topics central to the project. <br/><br/> The project will address mathematical and computational questions associated with the inverse problems of distributed parameters in the Bayesian computational framework, a general methodology that integrates the data with other information about the unknown that may be available. Distributed parameters such as electric conductivity of the brain tissue, or elastic properties of the breast tissue, are typically represented by coefficient functions of partial differential equations (PDEs) that relate these properties to the measurements corresponding to boundary values or samples of the corresponding solution. In order to handle the mathematical model numerically, a discretization of the model is necessary. The discretization of a continuous model for computing the numerical solution is an approximation of the exact model. Therefore, it introduces a discrepancy between the ideal and computational models. These problems are sensitive to any perturbation of the data. Thus even a small modeling error may have disastrous effects on the algorithms if not properly addressed. Therefore, controlling the modeling error in inverse problems will be crucial and challenging. In the approach proposed in this research project, discretization of the continuous model will be based on an underlying metric, coupled with the unknown through a hierarchical Bayesian model. The underlying discretization metric itself will be modeled as an unknown, and its estimation will be part of the inverse problem. The two selected medical imaging applications will serve as outstanding test problems for the methodology, and because of their importance, they will justify the theoretical effort of the project.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
