Electrical impedance tomography (EIT) has been the subject of quite intensive research for about 20 years but has yet to become established as a routine tool in healthcare. None the less the volume of published research work in this area is still rising. This review takes a broad look at what has been achieved and attempts to give the reader sufficient information to form an opinion as to the likely future for this interesting area of research.

Electrical impedance tomography (EIT): a review

The solution of impedance distribution in electrical impedance tomography is a nonlinear inverse problem that requires the use of a regularization method. The generalized Tikhonov regularization methods have been popular in the solution of many inverse problems. The regularization matrices that are usually used with the Tikhonov method are more or less ad hoc and the implicit prior assumptions are, thus, in many cases inappropriate. In this paper, the authors propose an approach to the construction of the regularization matrix that conforms to the prior assumptions on the impedance distribution. The approach is based on the construction of an approximating subspace for the expected impedance distributions. It is shown by simulations that the reconstructions obtained with the proposed method are better than with two other schemes of the same type when the prior is compatible with the true object. On the other hand, when the prior is incompatible with the true object, the method will still give reasonable estimates.

Tikhonov regularization and prior information in electrical impedance tomography

We review developments, issues and challenges in electrical impedance tomography (EIT) for the 4th Conference on Biomedical Applications of Electrical Impedance Tomography, held at Manchester in 2003. We focus on the necessity for three-dimensional data collection and reconstruction, efficient solution of the forward problem, and both present and future reconstruction algorithms. We also suggest common pitfalls or 'inverse crimes' to avoid.

https://iopscience.iop.org/article/10.1088/0967-3334/25/1/021/pdf

EIT reconstruction algorithms: pitfalls, challenges and recent developments

The objective of the Electrical Impedance and Diffuse Optical Reconstruction Software project is to develop freely available software that can be used to reconstruct electrical or optical material properties from boundary measurements. Nonlinear and ill posed problems such as electrical impedance and optical tomography are typically approached using a finite element model for the forward calculations and a regularized nonlinear solver for obtaining a unique and stable inverse solution. Most of the commercially available finite element programs are unsuitable for solving these problems because of their conventional inefficient way of calculating the Jacobian, and their lack of accurate electrode modelling. A complete package for the two-dimensional EIT problem was officially released by Vauhkonen et al at the second half of 2000. However most industrial and medical electrical imaging problems are fundamentally three-dimensional. To assist the development we have developed and released a free toolkit of Matlab routines which can be employed to solve the forward and inverse EIT problems in three dimensions based on the complete electrode model along with some basic visualization utilities, in the hope that it will stimulate further development. We also include a derivation of the formula for the Jacobian (or sensitivity) matrix based on the complete electrode model.

https://iopscience.iop.org/article/10.1088/0957-0233/13/12/310/pdf

A Matlab toolkit for three-dimensional electrical impedance tomography: a contribution to the Electrical Impedance and Diffuse Optical Reconstruction Software project

In electrical impedance tomography an approximation for the internal resistivity distribution is computed based on the knowledge of the injected currents and measured voltages on the surface of the body. It is often assumed that the injected currents are confined to the two-dimensional (2-D) electrode plane and the reconstruction is based on 2-D assumptions. However, the currents spread out in three dimensions and, therefore, off-plane structures have significant effect on the reconstructed images. In this paper we propose a finite element-based method for the reconstruction of three-dimensional resistivity distributions. The proposed method is based on the so-called complete electrode model that takes into account the presence of the electrodes and the contact impedances. Both the forward and the inverse problems are discussed and results from static and dynamic (difference) reconstructions with real measurement data are given. It is shown that in phantom experiments with accurate finite element computations it is possible to obtain static images that are comparable with difference images that are reconstructed from the same object with the empty (saline filled) tank as a reference.

Three-dimensional electrical impedance tomography based on the complete electrode model

In electrical impedance tomography (EIT) currents are applied through the electrodes attached on the surface of the object and the resulting voltages are measured using the same or additional electrodes. The internal admittivity distribution is estimated based on the current and voltage data. When the voltages are measured on the current carrying electrodes the contact impedance that exists in the electrode–surface interface causes a voltage drop. In some cases this effect of the electrodes is known. However, this is not always the case and the contact impedance has to be taken into account in the image reconstruction. In this paper we propose an approach for estimating the contact impedance of the electrodes simultaneously with the estimation of the admittivity of the object. The complete electrode model (CEM) is used in the estimation procedure. We compare the proposed approach to a simple method which is based on the well known definition of the sample resistivity. The proposed approach is tested with real measurements by estimating the admittivity of isotonic saline solution in a cylindrical test cell and with simulations in a three-dimensional cylindrical domain. The CEM-based approach is shown to produce results that are similar to the results obtained with the simple approach in the test cell case. The advantage of the CEM-based approach over the simple approach is that the complete electrode model does not have any geometrical constraints, which makes it possible to utilize it in EIT studies. The results show that the CEM-based approach works well and can be used in practical contact impedance estimation with real measurements. This will be further studied in part II of this paper.

https://iopscience.iop.org/article/10.1088/0957-0233/13/12/307/pdf

Simultaneous reconstruction of electrode contact impedances and internal electrical properties: I. Theory

There are many different electrical impedance tomography (EIT) systems which are either non-commercial (in-house products) or commercial products. However, these systems are usually designed for specific applications and therefore the functionality of the systems might be limited. Nowadays there are commercially available many low-cost, efficient and accurate multifunctional components for data acquisition and signal processing. Therefore, it should be possible to construct an EIT system which is mainly built from commercially available components. The main goal of this work was to study the performance of a PXI-based EIT system. In this work, a PXI-based EIT system with 16 independent current injection channels and 80 independent measurement channels was constructed and tested. The results indicate that an EIT system can be constructed using a PXI platform which decreases the construction time of the system. Moreover, the system is efficient, accurate, modular, and it is not limited to any predetermined measurement protocols.

https://iopscience.iop.org/article/10.1088/0957-0233/20/1/015503/pdf

Suitability of a PXI platform for an electrical impedance tomography system

In many industrial applications the aim is to obtain information on three-dimensional (3D) material distribution within the process vessels. With standard two-dimensional (2D) techniques only vague cross-sectional information can be obtained. It could be possible to carry out several 2D reconstructions on different layers and in this way to obtain 3D information. However, in this approach errors are induced since no real 3D information is utilized in the image reconstruction. In this paper we describe an approach to measure, reconstruct and visualize three-dimensional electrical impedance tomography images in real time. The reconstruction is based on a difference imaging scheme. An efficient current injection and voltage measurement protocol is used in order to increase the sensitivity and reduce the data collection time. The proposed approach can produce and visualize up to 15 3D EIT images per second when 80 measurement electrodes are used. Imaging results from a stirred vessel and a flow loop will be shown. The reconstructions show, for example, that 3D air/liquid distribution in the stirred vessel can reliably be visualized in real time and material flow can be monitored in a 3D section of the flow loop. Reconstructions can be visualized and analysed in many different ways in order to produce essential information on the behaviour of the processes.

https://iopscience.iop.org/article/10.1088/0957-0233/17/8/005/pdf

Real time three-dimensional electrical impedance tomography applied in multiphase flow imaging

Electrical impedance tomography (EIT) is a diffuse imaging modality in which the resistivity distribution inside the object is estimated based on electrical measurements made on the boundary. Several applications can be found in geophysics, medicine and industry. Image reconstruction is an iterative procedure in which the norm between the computed and measured voltages is minimized. Also, an additional regularization term is included in the minimized functional due to the ill-posedness of the problem. In the reconstruction process, the geometry of the object is assumed to be known. Geometry is known in many cases but there are various situations in which the shape of the domain is unknown. For example, in medical applications the shape of the domain, a part of the human body on which the measurement electrodes are attached, is unknown unless some other imaging modality is used for receiving the shape. In industrial applications, such as in the imaging of a stirrer vessel for detecting air distribution or detecting large air bubbles in pipelines, the free surface between the liquid and air is unknown and should be estimated. Within the domain we may also have 'voids' having zero conductivity and we might be interested in detecting the shapes and locations of the voids. An example could be the detection of corrosion faults in metallic plates. In this paper, two approaches for shape and free surface estimation are proposed. The approaches taken here are based on the idea used in shape optimization problems. In the first approach, the unknown shape is parametrized using the mesh nodes as parameters. In the second approach, we define new 'design variables' which are used as parameters. These design variables are the coefficients of a Bezier curve that defines the shape of the surface. In this paper, we show results of the free surface estimation from both computer simulations and tank measurements. Also, results of simultaneous reconstruction of the resistivity distribution and the free surface are shown. Comparison of the results between these two approaches will be given. The comparison shows better performance in the Bezier curve approach.

http://iopscience.iop.org/article/10.1088/0957-0233/15/7/024/pdf

T. Savolainen

Papers

Three-dimensional electrical impedance tomography based on the complete electrode model

Simultaneous reconstruction of electrode contact impedances and internal electrical properties: I. Theory

Suitability of a PXI platform for an electrical impedance tomography system

Real time three-dimensional electrical impedance tomography applied in multiphase flow imaging

Estimating shapes and free surfaces with electrical impedance tomography