Finite element implementation of Maxwell's equations for image reconstruction in electrical impedance tomography.

Traditionally, image reconstruction in electrical impedance tomography (EIT) has been based on Laplace's equation. However, at high frequencies the coupling between electric and magnetic fields requires solution of the full Maxwell equations. In this paper, a formulation is presented in terms of the Maxwell equations expressed in scalar and vector potentials. The approach leads to boundary conditions that naturally align with the quantities measured by EIT instrumentation. A two-dimensional implementation for image reconstruction from EIT data is realized. The effect of frequency on the field distribution is illustrated using the high-frequency model and is compared with Laplace solutions. Numerical simulations and experimental results are also presented to illustrate image reconstruction over a range of frequencies using the new implementation. The results show that scalar/vector potential reconstruction produces images which are essentially indistinguishable from a Laplace algorithm for frequencies below 1 MHz but superior at frequencies reaching 10 MHz.

Enhancing the performance of model-based elastography by incorporating additional a priori information in the modulus image reconstruction process.

Model-based elastography is fraught with problems owing to the ill-posed nature of the inverse elasticity problem. To overcome this limitation, we have recently developed a novel inversion scheme that incorporates a priori information concerning the mechanical properties of the underlying tissue structures, and the variance incurred during displacement estimation in the modulus image reconstruction process. The information was procured by employing standard strain imaging methodology, and introduced in the reconstruction process through the generalized Tikhonov approach. In this paper, we report the results of experiments conducted on gelatin phantoms to evaluate the performance of modulus elastograms computed with the generalized Tikhonov (GTK) estimation criterion relative to those computed by employing the un-weighted least-squares estimation criterion, the weighted least-squares estimation criterion and the standard Tikhonov method (i.e., the generalized Tikhonov method with no modulus prior). The results indicate that modulus elastograms computed with the generalized Tikhonov approach had superior elastographic contrast discrimination and contrast recovery. In addition, image reconstruction was more resilient to structural decorrelation noise when additional constraints were imposed on the reconstruction process through the GTK method.

Excitation patterns in three-dimensional electrical impedance tomography.

Electrical impedance tomography (EIT) is a non-invasive technique that aims to reconstruct images of internal electrical properties of a domain, based on electrical measurements on the periphery. Improvements in instrumentation and numerical modeling have led to three-dimensional (3D) imaging. The availability of 3D modeling and imaging raises the question of identifying the best possible excitation patterns that will yield to data, which can be used to produce the best image reconstruction of internal properties. In this work, we describe our 3D finite element model of EIT. Through singular value decomposition as well as examples of reconstructed images, we show that for a homogenous female breast model with four layers of electrodes, a driving pattern where each excitation plane is a sinusoidal pattern out-of-phase with its neighboring plane produces better qualitative images. However, in terms of quantitative imaging an excitation pattern where all electrode layers are in phase produces better results.

On optimal current patterns for electrical impedance tomography.

We develop a statistical criterion for optimal patterns in planar circular electrical impedance tomography. These patterns minimize the total variance of the estimation for the resistance or conductance matrix. It is shown that trigonometric patterns (Isaacson, 1986), originally derived from the concept of distinguishability, are a special case of our optimal statistical patterns. New optimal random patterns are introduced. Recovering the electrical properties of the measured body is greatly simplified when optimal patterns are used. The Neumann-to-Dirichlet map and the optimal patterns are derived for a homogeneous medium with an arbitrary distribution of the electrodes on the periphery. As a special case, optimal patterns are developed for a practical EIT system with a finite number of electrodes. For a general nonhomogeneous medium, with no a priori restriction, the optimal patterns for the resistance and conductance matrix are the same. However, for a homogeneous medium, the best current pattern is the worst voltage pattern and vice versa. We study the effect of the number and the width of the electrodes on the estimate of resistivity and conductivity in a homogeneous medium. We confirm experimentally that the optimal patterns produce minimum conductivity variance in a homogeneous medium. Our statistical model is able to discriminate between a homogenous agar phantom and one with a 2 mm air hole with error probability (p-value) 1/1000.

Variation in breast EIT measurements due to menstrual cycle.

We conducted a short study on 8 volunteer subjects to establish whether physiological changes occurring as a result of the menstrual cycle affect tissue electrical properties. For this study subjects submitted to electrical impedance tomographic breast measurement four times, over two cycles at two different points in the cycle. Statistical analysis based on reconstructed values of conductivity and permittivity were conducted using the t-test for difference of means. The results were inconsistent, with some subjects showing a difference between the two phases and in all tests, while others showed differences only in some of the tests. At this time we can only conclude that a difference is more likely than not, although it could be a phenomenon only measurable in some individuals and not others. It seems that a larger study may be in order to establish this fact definitively.

Multi-frequency electrical impedance tomography of the breast: new clinical results.

Electrical impedance tomography (EIT) has been used in the recent past for a number of clinical applications. In this work we present recent tomographic and spectroscopic findings for breast imaging from clinical exams completed at Dartmouth. The results presented here are based on 18 normal and 24 abnormal subjects. The participants were classified as normal or abnormal using the American College of Radiology (ACR) indexing system for mammograms. The EIT data were collected for ten discrete frequencies in the range 10 kHz-1 MHz using a single array of 16 electrodes. The finite element method was used to reconstruct the images. The images were examined visually and were compared with mammograms. The results were also analyzed based on zonal averages of property values and breast tissue radiodensities. Statistical analysis showed a significance difference between the mean conductivity and permittivity values of normal and abnormal subjects for various zones defined on the reconstructed images. Tissues with high radiodensity also had increased conductivity and permittivity.

A novel data calibration scheme for electrical impedance tomography.

In electrical impedance tomography surface measurements of voltages and currents are recorded and the image reconstruction algorithm uses this set of boundary data to estimate internal electrical properties of the region under investigation. Therefore correct and accurate modelling of the current and voltage distributions (forward model) is an essential part of any reconstruction method. In this paper, we explored the root cause of a boundary layer effect in the reconstructed conductivity map and found it to be an artefact arising from 2D to 3D data-model mismatch within the imaging algorithm. We propose a data calibration scheme that improves the reconstruction results by removing these boundary or edge effects. We present both two-dimensional and three-dimensional images for agar phantoms using this data calibration scheme which are markedly better than their counterparts recovered when the measurement data are not calibrated with the procedure outlined herein.

Using multiple-electrode impedance measurements to monitor cryosurgery.

We outfitted cryoprobes with electrodes and used them in conjunction with a multiple channel electrical impedance tomography (EIT) system to record data during freezing experiments in a shallow saline tank. We made measurements using electrodes mounted on the probes and the tank's periphery. Reconstructed images based on both sets of electrodes indicate a significant improvement in the appearance of the ice ball over using tank electrodes alone. The size of the ice balls was varied by deliberately altering the cooling rate. We found a positive correlation between the measured size of the ice ball and the sizes of isocontour lines in the reconstructed impedance maps. Similarly, the shape of the ice balls was altered by circulating the saline about the probe. Two-dimensional reconstructed impedance contours indicated a deformation in agreement with the shape of the ice ball during the experiments. These findings suggest that using multielectrode impedance sensing may constitute a means for monitoring cryosurgery.