Lasse M. Heikkinen

The Bayesian approximation error approach for electrical impedance tomography—experimental results

Inverse problems can be characterized as problems that tolerate measurement and modelling errors poorly. While the measurement error issue has been widely considered as a solved problem, the modelling errors have remained largely untreated. The approximation and modelling errors can, however, be argued to dominate the measurement errors in most applications. There are several applications in which the temporal and memory requirements dictate that the computational complexity of the forward solver be radically reduced. For example, in process tomography the reconstructions have to be carried out typically in a few tens of milliseconds. Recently, a Bayesian approach for the treatment of approximation and modelling errors for inverse problems has been proposed. This approach has proven to work well in several classes of problems, but the approach has not been verified in any problem with real data. In this paper, we study two different types of modelling errors in the case of electrical impedance tomography: one related to model reduction and one concerning partially unknown geometry. We show that the approach is also feasible in practice and may facilitate the reduction of the computational complexity of the nonlinear EIT problem at least by an order of magnitude.

Compensation of errors due to discretization, domain truncation and unknown contact impedances in electrical impedance tomography

Inverse problems can be characterized as problems that tolerate measurement and modelling errors poorly. Typical sources of modelling errors include (pure) approximation errors related to numerical discretization, unknown geometry and boundary data, and possibly sensor locations. With electrical impedance tomography (EIT), the unknown contact impedances are an additional error source. Recently, a Bayesian approach to the treatment of approximation and modelling errors for inverse problems has been proposed. This approach has been shown to be applicable to a variety of modelling and approximation errors, at least with simulations. Recently, it was shown that recovery from significant model reduction and moderate mismodelling of geometry in EIT was also possible with laboratory EIT data. In this paper, we show that the errors due to the unknown contact impedances can also be compensated for by employing the approximation error approach. Furthermore, the recovery from simultaneous contact impedance, domain truncation and discretization-related errors is also feasible.

Simultaneous reconstruction of electrode contact impedances and internal electrical properties: II. Laboratory experiments

In this paper, we examine how the estimation method of the electrode contact impedances proposed earlier works in two different laboratory experiments. In the first experiment, the performance of the method was studied for a tank filled with a homogeneous liquid. In this case a single parameter for the contact impedances was used. In the second experiment the method was tested with a single target within a tank. In this case the contact impedances on all the electrodes were estimated. It was found that in the two-electrode measurement protocol the estimation of the electrode contact impedances improves reconstructions substantially. If only a single electrode contact impedance value for all the electrodes was estimated it was found that in certain cases it was not adequate.

Suitability of a PXI platform for an electrical impedance tomography system

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.

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

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.

Modelling of internal structures and electrodes in electrical process tomography

Classical electrical impedance tomography is an imaging modality in which the internal impedivity distribution is reconstructed from the known injected currents and voltages measured on the surface of the object. However, in many industrial cases there are a priori known internal structures inside the vessels which could be used as internal electrodes in tomographical imaging. In this paper we consider modelling of certain types of internal structures and using them as internal electrodes in two-dimensional electrical process tomography. We also propose an inversion approach in which directional smoothness of the resistivity distribution can be taken into account. Simulations and laboratory experiments show that, by utilizing internal structural information and using these internal structures as additional electrodes, we can achieve significant improvements in image reconstruction. Improvements are shown for the cases in which the electrodes are placed at the centre of the object and are surrounded by a resistive layer.

Detection of faults in resistive coatings with an impedance-tomography-related approach

In this paper we propose a method for detecting faults in resistive thin films and coatings with an electrical-impedance-tomography-related approach. The motivation of our study is the need to develop methods for the evaluation of coating properties and quality of implants. The method is based on the complete electrode model. We model the coating as one electrode with spatially varying contact impedance. Using a set of current and voltage measurements in a specially developed measurement cell including the coating and additional electrodes, we estimate the spatially varying contact impedance on the coating. The estimated contact impedance distribution can be regarded as a measure of the condition of the coating; that is, low contact impedance indicates a fault in the coating. The necessary modifications of the complete electrode model are given. The suggested method is tested with simulations and real measurements and it is shown that, with feasible regularization, faults in the coating can be detected with good precision.

In situ imaging of paste extrusion using electrical impedance tomography

A case study is presented where electrical impedance tomography is used to image paste during extrusion. This indicates regions of high moisture at the wall of the barrel permitting slip and so making extrusion more efficient. Multi-step Gauss–Newton image reconstruction methods are used. Techniques include (i) axi-symmetric smoothing that is coupled with ultrasound measurement to provide discretization information, (ii) full impedance imaging and (iii) full three-dimensional field modelling. These provide clear images of the paste that can be used to gauge extrudability of a paste formulation.

Electrical impedance tomography for three-dimensional drug release monitoring

Electrical impedance tomography (EIT) was adapted to monitor drug release three-dimensionally as a function of time. EIT is an electrical imaging modality in which the three-dimensional conductivity distribution inside an object is computed based on electrical measurements from the boundaries. Here, the three-dimensional concentration distribution was monitored with the help of the experimentally determined relationship between drug concentration and conductivity. The EIT monitoring was carried out with propranolol hydrochloride tablets in an apparatus similar to USP dissolution apparatus 2. The release profiles estimated using EIT matched well with the UV/VIS spectrophotometric analyses that were performed as a reference. There are several benefits conferred by three-dimensional monitoring, i.e., comprehensive information about the release process; no need to take samples during experiments; and not essential to assume homogenous concentration distribution in the drug release analysis. EIT is an in-line technique, and moreover, it is non-intrusive and non-invasive. The possibilities and the characteristics of the EIT monitoring are described in detail, and some potential drug release applications are proposed. EIT is especially encouraged to be exploited for research and development purposes.

An electrical impedance tomography-based approach to monitor in vitro sodium chloride dissolution from pharmaceutical tablets

An approach to monitor in vitro dissolution process from pharmaceutical tablets utilizing electrical impedance tomography (EIT) is introduced. In the demonstration, a tablet containing sodium chloride (NaCl) was dissolution tested using tap water as a dissolution medium within an apparatus similar to the United States Pharmacopoeia dissolution apparatus II. During the process, the three-dimensional sodium chloride concentration distribution was monitored with EIT measurements as a function of time. For EIT measurements, an array of electrodes was attached on the boundary of the dissolution vessel, a set of alternating electric currents was injected through the electrodes, and the resulting voltages were measured. With these data and by applying mathematical algorithms, an approximation for the spatial/temporal concentration distribution inside the vessel was computed. It was found that the computed distributions were relatively homogeneous. A NaCl release curve was computed by integrating the concentration distribution over the vessel volume, and the final value of the curve matched well with the reference point based on the weight loss of the tablet. Finally, EIT monitoring is suggested to be used for research and product development purposes.