The D-bar Method in Electrical Impedance Tomography

提出了二维和三维电阻抗成像(EIT)的重建算法。EIT是一种相对较新的医学成像技术，它将电极放置在人体表面，在电极上施加电流，并在电极上测量产生的电压。由于体内的各种器官和组织经常具有不同的导电特性，因此根据重建的电导率分布形成图像。本研究是在Nachman[Ann.数学的一部分。143(1996)]将通过研究应用于不连续电导的算法、研究电极效应对该算法的影响、将边界重建纳入数值算法、更精确地计算散射变换以及使该算法适用于非圆形几何来进一步发展。这项工作有望提高该方法的准确性，并将其带入临床实用阶段。此外，还将针对在平面电极阵列上采集的EIT数据开发三维D-BAR算法。3-D算法的具体临床应用是乳腺癌检测。这项工作将提供对D-bar方法在二维和三维方面的进一步洞察。术语“D-bar方法”源于逆散射技术，该技术导致在重建过程中要求解D-bar方程，该方程是复变量中的一类偏微分方程。二维D-bar算法是目前在EIT实验数据上成功实现的唯一一种直接(非迭代)方法，用于求解完全的非线性电导率问题。目前，还没有直接的三维EIT数值实现方法。二维EIT问题作为一种监测急性呼吸窘迫综合征(ARDS)患者的方法有着重要的应用。ARDS是一种危及生命的疾病，肺部炎症和气囊中的液体导致低血氧水平。治疗方法是机械通风，这对肺部有潜在的副作用。通过在患者的胸部周围放置电极，EIT可用于持续监测ARDS患者，提供有关他们情况的信息，并帮助确定适当的呼吸机设置，以将副作用降至最低。一种主要应用于乳腺癌检测的三维D-bar重建算法有可能比现有算法更准确地重建电导值，从而提高EIT作为乳腺癌检测方法的特异性。EIT作为一种乳腺癌检测工具还处于早期阶段，3D D-BAR算法的开发和研究将为这一应用提供更深入的EIT有效性的理解。

Reconstruction algorithms for two and three-dimensional electrical impedance tomography (EIT) are developed. EIT is a relatively new medical imaging technique in which electrodes are placed on the surface of the body, current is applied on the electrodes, and the resulting voltage is measured on the electrodes. Since the various organs and tissues in the body frequently have different conductive properties, an image is formed from the reconstructed conductivity distribution. In this research the D- bar reconstruction algorithm for 2-D EIT based on the paper by Nachman [Ann. Of Math. 143 (1996)] will be further developed through a study of the algorithm applied to discontinuous conductivities, a study of electrode effects on the algorithm, incorporating boundary reconstruction in the numerical algorithm, more accurate computations of the scattering transform, and adapting the algorithm to non-circular geometries. This work is expected to improve the accuracy of the method and bring it to a clinically useful stage. In addition, a 3-D D-bar algorithm will be developed for EIT data collected on planar electrode arrays. The particular clinical application for the 3-D algorithm is breast cancer detection.This work will provide further insight into the D-bar method in two and three dimensions. The terminology "D-bar method" comes from the techniques in inverse scattering which result in a D-bar equation, which is a certain type of partial differential equation in complex variables, to be solved in the reconstruction process. The 2-D D-bar algorithm is the only direct (non-iterative) method for EIT solving the full nonlinear conductivity problem to have been successfully implemented on experimental EIT data. At present, no direct numerically implemented method exists for EIT in three dimensions. The 2-D EIT problem has an important application as a method for monitoring patients with acute respiratory distress syndrome (ARDS). ARDS is a life-threatening condition in which inflammation of the lungs and fluid in the air sacs leads to low blood oxygen levels. The treatment is mechanical ventilation, which has potential side-effects to the lungs. By placing electrodes around the patient's thorax, EIT can be used to continually monitor ARDS patients, providing information about their condition and aiding in determining the proper ventilator settings to minimize side-effects. A 3-D D-bar reconstruction algorithm with the main application of breast cancer detection has the potential to reconstruct the conductivity values with more accuracy than existing algorithms and thus may improve the specificity of EIT as a method of breast cancer detection. EIT as a tool for breast cancer detection is in the early stages, and the development and study of a 3-D D-bar algorithm will provide a deeper understanding of the effectiveness of EIT for this application.