Abstract. Electrical Impedance Tomography (EIT) is an attractive method for clinically monitoring patients during mechanical ventilation, because it can provide a non-invasive continuous image of pulmonary impedance which indicates the distribution of ventilation. However, most clinical and physiological research in lung EIT is done using older and proprietary algorithms; this is an obstacle to interpretation of EIT images because the reconstructed images are not well characterized. To address this issue, we are developing a consensus linear reconstruction algorithm for lung EIT, called GREIT (Graz consensus Reconstruction algorithm for EIT). This paper describes the unified approach to linear image reconstruction developed for GREIT. The framework for the linear reconstruction algorithm consists of: 1) detailed finite element models of a representative adult and neonatal thorax; 2) consensus on the performance figures of merit for EIT image reconstruction; and 3) a systematic approach to optimize a linear reconstruction matrix to desired performance measures. Consensus figures of merit, in order of importance, are: a) uniform amplitude response, GREIT: linear EIT image reconstruction 2 b) small and uniform position error, c) small ringing artefacts, d) uniform resolution, e) limited shape deformation, and f) high resolution. Such figures of merit must be attained while maintaining small noise amplification and small sensitivity to electrode and boundary movement. This approach represents the consensus of a large and representative group of experts in EIT algorithm design and clinical applications for pulmonary monitoring. All software and data to implement and test the algorithm has been made available under an open source license which allows free research and commercial use.

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Electromagnetic (EM) breast imaging provides low-cost, safe and potentially a more specific modality for cancer detection than conventional imaging systems. A primary difficulty in validating these EM imaging modalities is that the true dielectric property values of the particular breast being imaged are not readily available on an individual subject basis. Here, we describe our initial experience in seeking to correlate tomographic EM imaging studies with discrete point spectroscopy measurements of the dielectric properties of breast tissue. The protocol we have developed involves measurement of in vivo tissue properties during partial and full mastectomy procedures in the operating room (OR) followed by ex vivo tissue property recordings in the same locations in the excised tissue specimens in the pathology laboratory immediately after resection.We have successfully applied all of the elements of this validation protocol in a series of six women with cancer diagnoses. Conductivity and permittivity gauged from ex vivo samples over the frequency range 100 Hz-8.5 GHz are found to be similar to those reported in the literature. A decrease in both conductivity and permittivity is observed when these properties are gauged from ex vivo samples instead of in vivo. We present these results in addition to a case study demonstrating how discrete point spectroscopy measurements of the tissue can be correlated and used to validate EM imaging studies.

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Maximum a posteriori estimates in inverse problems are often based on quadratic formulations, corresponding to a least-squares fitting of the data and to the use of the L2 norm on the regularization term. While the implementation of this estimation is straightforward and usually based on the Gauss-Newton method, resulting estimates are sensitive to outliers and result in spatial distributions of the estimates that are smooth. As an alternative, the use of the L1 norm on the data term renders the estimation robust to outliers, and the use of the L1 norm on the regularization term allows the reconstruction of sharp spatial profiles. The ability therefore to use the L1 norm either on the data term, on the regularization term, or on both is desirable, though the use of this norm results in non-smooth objective functions which require more sophisticated implementations compared to quadratic algorithms. Methods for L1-norm minimization have been studied in a number of contexts, including in the recently popular total variation regularization. Different approaches have been used and methods based on primal-dual interior-point methods (PD-IPMs) have been shown to be particularly efficient. In this paper we derive a PD-IPM framework for using the L1 norm indifferently on the two terms of an inverse problem. We use electrical impedance tomography as an example inverse problem to demonstrate the implementation of the algorithms we derive, and the effect of choosing the L2 or the L1 norm on the two terms of the inverse problem. Pseudo-codes for the algorithms and a public domain implementation are provided.

Abstract-In the inverse conductivity problem, as in any illposed inverse problem, regularization techniques are necessary in order to stabilize inversion. A common way to implement regularization in electrical impedance tomography is to use Tikhonov regularization. The inverse problem is formulated as a minimization of two terms: the mismatch of the measurements against the model, and the regularization functional. Most commonly, differential operators are used as regularization functionals, leading to smooth solutions. Whenever the imaged region presents discontinuities in the conductivity distribution, such as interorgan boundaries, the smoothness prior is not consistent with the actual situation. In these cases, the reconstruction is enhanced by relaxing the smoothness constraints in the direction normal to the discontinuity. In this paper, we derive a method for generating Gaussian anisotropic regularization filters. The filters are generated on the basis of the prior structural information, allowing a better reconstruction of conductivity profiles matching these priors. When incorporating prior information into a reconstruction algorithm, the risk is of biasing the inverse solutions toward the assumed distributions. Simulations show that, with a careful selection of the regularization parameters, the reconstruction algorithm is still able to detect conductivities patterns that violate the prior information. A generalized singular-value decomposition analysis of the effects of the anisotropic filters on regularization is presented in the last sections of the paper.

Transrectal electrical impedance tomography (TREIT) has been proposed as an adjunct modality for enhancing standard clinical ultrasound (US) imaging of the prostate. The proposed TREIT probe has an array of electrodes adhered to the surface of a cylindrical US probe that is introduced inside of the imaging volume. Reconstructing TREIT images in the open-domain geometry established with this technique poses additional challenges to those encountered with closeddomain geometries, present in more conventional EIT systems, because of the rapidly decaying current densities at increasing distances from the probe surface. We developed a finite element method (FEM)-based dual-mesh reconstruction algorithm which employs an interpolation scheme for linking a fine forward mesh with a coarse grid of pixels, used for conductivity estimation. Simulation studies using the developed algorithm demonstrate the feasibility of imaging moderately contrasting inclusions at distances of three times the probe radius from the probe surface and at multiple angles about the probe's axis. The large, dense FEM meshes used here require significant computational effort. We have optimized our reconstruction algorithm with multi-core processing hardware and efficient parallelized computational software packages to achieve a speedup of 9.3 times when compared to a more traditional Matlab-based, single CPU solution. The simulation findings and computational optimization provide a state-of-the-art reconstruction platform for use in further evaluating transrectal electrical impedance tomography.

Magnetic Resonance-Electrical Properties Tomography (MR-EPT) is an imaging modality that maps the spatial distribution of the electrical conductivity and permittivity using standard MRI systems. The presence of a body within the scanner alters the RF field, and by mapping these alterations it is possible to recover the electrical properties. The field is time-harmonic, and can be described by the Helmholtz equation. Approximations to this equation have been previously used to estimate conductivity and permittivity in terms of first or second derivatives of RF field data. Using these same approximations, an inverse approach to solving the MR-EPT problem is presented here that leverages a forward model for describing the magnitude and phase of the field within the imaging domain, and a fitting approach for estimating the electrical properties distribution. The advantages of this approach are that 1) differentiation of the measured data is not required, thus reducing noise sensitivity, and 2) different regularization schemes can be adopted, depending on prior knowledge of the distribution of conductivity or permittivity, leading to improved image quality. To demonstrate the developed approach, both Quadratic (QR) and Total Variation (TV) regularization methods were implemented and evaluated through numerical simulation and experimentally acquired data. The proposed inverse approach to MR-EPT reconstruction correctly identifies contrasts and accurately reconstructs the geometry in both simulations and experiments. The TV regularized scheme reconstructs sharp spatial transitions, which are difficult to reconstruct with other, more traditional approaches.