Current Density Based Electrical Impedance Tomography, an Emerging Hybrid Imaging Technique

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。研究者和他的合作者研究了一种新的非侵入性方法来成像人体内部的电导率。导电性的内在知识已经在各个领域得到了应用，从医学、地球物理学、环境科学到无损检测和地雷探测，仅举几例。该模型基于麦克斯韦系统，结合了电阻抗层析成像和电流密度成像的方法。新颖之处在于，在边界处保持电压时产生的电流密度场的大小的知识增强了麦克斯韦模型。后者的数据目前是通过磁场的磁共振测量获得的。在数学上，问题是研究奇异黎曼空间中的最小曲面，其度量由电流密度场的大小决定。或者，该问题可以简化为非光滑优化问题。利用反问题、偏微分方程、黎曼几何、几何测量理论和非光滑优化等领域的工具，研究者和他的合作者旨在改变现有和公认的理论，为精确和高分辨率地成像内部电导率提供数学支持和方法。本课题属于当前政府信息技术研究重点。这项研究的结果是为导电体（如人体）内部的非侵入性成像提供了新的方法。对社会的好处是重要的，因为这项研究结果的一个组成部分涉及到应用于医学成像的新方法的可负担性。更准确地说，这项新技术是对目前相当昂贵的传统磁共振成像扫描方法的补充。这项调查以两种方式结合了研究和教育。一方面，一个完整的部分是研究生和本科生的培养和指导，包括来自代表性不足的学生群体的培养和指导。另一方面，研究者正在同时设计一门专题课程，其范围基于本研究。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The investigator and his collaborators study a new non-invasive method for imaging the electrical conductivity of the inside of the body. Inner knowledge of the conductivity has found applications in various fields ranging from Medicine, Geophysics, Environmental Sciences, to Nondestructive Testing and Mine Detection, to mention only some. The model is based on the Maxwell system and combines methods from Electrical Impedance Tomography and Current Density Imaging. The novelty lies in the augmentation of the Maxwell model with the knowledge of the magnitude of the current density field generated while maintaining a voltage at the boundary. The latter data is currently obtained by using Magnetic Resonance measurements of the magnetic field. Mathematically, the problem is to study minimal surfaces in a singular Riemannian space with a metric determined by the magnitude of the current density field. Alternatively, the problem can be reduced to a Non-smooth Optimization problem. Taking advantage of the tools from the areas of Inverse Problems, Partial Differential Equations, Riemannian Geometry, Geometric Measure Theory and Non-smooth Optimization the investigator and his collaborators aim to change the existing and accepted theories and provide mathematical support and methods to image inner conductivity accurately and with high resolution.This investigation belongs to the current government research priority on Information Technology. The results of this study are to produce new methodologies for non-invasive imaging of the interior of conductive bodies, such as the human body. The benefits to society are important since one componenent of the results of this study concerns the affordability of the new methods applied to Medical Imaging. More precisely, the new technology is to complement the classical Magnetic Resonance Imaging scan, method which is fairly expensive at the moment. This investigation integrates research and education in two ways. On the one hand, an integrate part is the training and mentoring of graduate and undergraduate students, including from the under-represented categories of students. On the other hand, the investigator is concurrently designing a special topics course, whose scope is based on this study.