Bridging the Gap between Discrete and Continuous Partial Differential Equations in Medical imaging

在非侵入性或微创医学成像中，目标是在不伤害患者的情况下，根据在体外进行的测量形成人体内部结构的图像。由成像设备组成的医学成像的可见部分是基于工程和物理的，而图像的形成和相关信息的检索则依赖于复杂的数学模型和高效的计算方法。该项目将解决两个具体的影像学问题，乳腺癌筛查和中风检测与分类。乳房x光检查是一个标准的过程。然而，众所周知，特别是当乳房组织致密时，这是10- 40%的美国女性的情况，放射科医生错过癌症病变的风险是很大的。该项目将研究一种新的计算方法，即使用不同压力水平下的乳房x线摄影图像，并比较组织位移，以估计组织的弹性特性，这些特性已知会受到某些通常未被检测到的癌症类型的影响。另一个医学成像问题，在这个项目中解决是中风分类的便携式和廉价的电阻抗断层扫描设备。众所周知，缺血性脑卒中的预后在很大程度上取决于治疗的开始时间。在被及时送往医院的中风患者中，约有15%被诊断为脑出血。对缺血性中风患者的治疗是致命的。适合救护车的便携式分类方法对于诊断至关重要，可以在紧急情况下挽救许多生命。这个想法并不新鲜，但数学和计算问题仍然具有挑战性。这个项目将集中在上述问题中的计算挑战，以及涵盖医学成像的其他数学上类似的问题。研究结果也将用于非破坏性材料评价和地球物理学，包括其他应用领域。该项目还将包括一个强大的教育组成部分，通过研究生的参与，他们将在他们的博士论文中围绕项目的核心主题工作。该项目将解决与贝叶斯计算框架中分布参数逆问题相关的数学和计算问题，贝叶斯计算框架是一种将数据与其他可能可用的未知信息集成在一起的通用方法。分布参数，如脑组织的电导率，或乳房组织的弹性特性，通常用偏微分方程（pde）的系数函数表示，偏微分方程（pde）将这些特性与对应的边界值或相应解的样本的测量相关联。为了对数学模型进行数值处理，必须对模型进行离散化。计算数值解的连续模型的离散化是精确模型的近似。因此，它引入了理想模型和计算模型之间的差异。这些问题对数据的任何扰动都很敏感。因此，如果处理不当，即使是很小的建模错误也可能对算法产生灾难性的影响。因此，控制反问题的建模误差将是至关重要和具有挑战性的。在本研究项目中提出的方法中，连续模型的离散化将基于底层度量，并通过分层贝叶斯模型与未知相结合。潜在的离散度量本身将被建模为一个未知数，它的估计将是逆问题的一部分。这两个选定的医学成像应用程序将作为方法论的突出测试问题，并且由于它们的重要性，它们将证明该项目在理论上的努力是合理的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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. 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.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.