The D-bar Method in Electrical Impedance Tomography

研究了二维和三维电阻抗断层成像（EIT）的重建算法。 EIT是一种相对较新的医学成像技术，其中将电极放置在身体表面上，在电极上施加电流，并在电极上测量所得电压。 由于身体中的各种器官和组织经常具有不同的导电特性，因此根据重建的导电率分布形成图像。 在本研究中，基于Nachman的论文[Ann.OfMath.143（1996）]的用于2-D EIT的D-bar重建算法将通过应用于不连续电导率的算法的研究、电极对算法的影响的研究、在数值算法中结合边界重建、散射变换的更精确计算、以及使算法适应非圆形几何形状。 这项工作有望提高该方法的准确性，并使其进入临床有用的阶段。 此外，一个3-D的D-bar算法将开发的EIT数据收集的平面电极阵列。 三维算法的具体临床应用是乳腺癌的检测。这项工作将提供对二维和三维的D-bar方法的进一步了解。 术语“D-bar方法”来自逆散射中的技术，其导致在重建过程中求解的D-bar方程，其是复变量中的某种类型的偏微分方程。 二维D-bar算法是唯一的直接（非迭代）EIT解决完全非线性电导率问题的方法，已成功地实现了对实验EIT数据。 目前，没有直接的数值实现方法存在的三维EIT。 二维EIT问题作为一种监测急性呼吸窘迫综合征（ARDS）患者的方法具有重要的应用。 ARDS是一种危及生命的疾病，其中肺部炎症和气囊中的液体导致血氧水平降低。 治疗是机械通气，这对肺部有潜在的副作用。 通过将电极放置在患者的胸腔周围，EIT可以用于持续监测ARDS患者，提供有关其状况的信息，并帮助确定适当的呼吸机设置，以最大限度地减少副作用。 主要应用于乳腺癌检测的3-D D-bar重建算法具有比现有算法更准确地重建电导率值的潜力，因此可以提高EIT作为乳腺癌检测方法的特异性。 EIT作为乳腺癌检测的工具还处于早期阶段，三维D-bar算法的开发和研究将使人们更深入地了解EIT在这一应用中的有效性。

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.