Robust linearized image reconstruction for multifrequency EIT of the breast

Electrical impedance tomography (EIT) is a developing imaging modality that is beginning to show promise for detecting and characterizing tumors in the breast. At Rensselaer Polytechnic Institute, we have developed a combined EIT-tomosynthesis system that allows for the coregistered and simultaneous analysis of the breast using EIT and X-ray imaging. A significant challenge in EIT is the design of computationally efficient image reconstruction algorithms which are robust to various forms of model mismatch. Specifically, we have implemented a scaling procedure that is robust to the presence of a thin highly-resistive layer of skin at the boundary of the breast and we have developed an algorithm to detect and exclude from the image reconstruction electrodes that are in poor contact with the breast. In our initial clinical studies, it has been difficult to ensure that all electrodes make adequate contact with the breast, and thus procedures for the use of data sets containing poorly contacting electrodes are particularly important. We also present a novel, efficient method to compute the Jacobian matrix for our linearized image reconstruction algorithm by reducing the computation of the sensitivity for each voxel to a quadratic form. Initial clinical results are presented, showing the potential of our algorithms to detect and localize breast tumors.      

Methods for Compensating for Variable Electrode Contact in EIT

Electrical impedance tomography (EIT) is an imaging modality that currently shows promise for the detection and characterization of breast cancer. A very significant problem in EIT imaging is the proper modeling of the interface between the body and the electrodes. We have found empirically that it is very difficult, in a clinical setting, to assure that all electrodes make satisfactory contact with the body. In addition, we have observed a capacitive effect at the skin/electrode boundary that is spatially heterogeneous. To compensate for these problems, we have developed a hybrid nonlinear–linear reconstruction algorithm using the complete electrode model in which we first estimate electrode surface impedances, by means of a Levenberg–Marquardt iterative optimization procedure with an analytically computed Jacobian matrix. We, subsequently, use a linearized algorithm to perform a 3-D reconstruction of perturbations in both contact impedances, and in the spatial distributions of conductivity and permittivity. Results show that, with this procedure, artifacts due to electrodes making poor contact can be greatly reduced. If the experimental apparatus physically applies voltages and measures currents, we show that it is preferable to compute the reconstruction with respect to the Dirichlet-to-Neumann map rather than the Neumann-to-Dirichlet map if there is a significant possibility that electrodes will be fully disconnected. Finally, we test our electrode compensation algorithms for a set of clinical data, showing that we can significantly improve the fit of our model to the measurements by allowing the electrode surface impedances to vary.      

Electrical impedance tomography

Electrical impedance tomography (EIT) is an imaging modality that estimates the electrical properties at the interior of an object from measurements made on its surface. Typically, currents are injected into the object through electrodes placed on its surface, and the resulting electrode voltages are measured. An appropriate set of current patterns, with each pattern specifying the value of the current for each electrode, is applied to the object, and a reconstruction algorithm uses knowledge of the applied current patterns and the measured electrode voltages to solve the inverse problem, computing the electrical conductivity and permittivity distributions in the object. This article focuses on the type of EIT called adaptive current tomography (ACT) in which currents are applied simultaneously to all the electrodes. A number of current patterns are applied, where each pattern defines the current for each electrode, and the subsequent electrode voltages are measured to generate the data required for image reconstruction. A ring of electrodes may be placed in a single plane around the object, to define a two-dimensional problem, or in several layers of such rings, to define a three-dimensional problem. The reconstruction problem is described and two algorithms are discussed, a one-step, two-dimensional (2-D) Newton-Raphson algorithm and a one-step, full three-dimensional (3-D) reconstructor. Results from experimental data are presented which illustrate the performance of the algorithms     

Electrical impedance tomography for imaging tissue electroporation

Electroporation is a method to introduce molecules, such as gene constructs or small drugs, into cells by temporarily permeating the cell membrane with electric pulses. In molecular medicine and biotechnology, tissue electroporation is performed with electrodes placed in the target area of the body. Currently, tissue electroporation, as with all other methods of molecular medicine, is performed without real-time control or near-term information regarding the extent and degree of electroporation. This paper expands the work from our previous study by implementing new ex vivo experimental data with "front-tracking" analysis for the image reconstruction algorithm. The experimental data is incorporated into numerical simulations of electroporation procedures and images are generated using the new reconstruction algorithm to demonstrate that electrical impedance tomography (EIT) can produce an image of the electroporated area. Combining EIT with electroporation could become an important biotechnological and medical technique to introduce therapeutic molecules into cells in tissue at predetermined areas of the body.     

Electrical impedance tomography problem with inaccurately known boundary and contact impedances

In electrical impedance tomography (EIT) electric currents are injected into a body with unknown electromagnetic properties through a set of contact electrodes at the boundary of the body. The resulting voltages are measured on the same electrodes and the objective is to reconstruct the unknown conductivity function inside the body based on these data. All the traditional approaches to the reconstruction problem assume that the boundary of the body and the electrode-skin contact impedances are known a priori. However, in clinical experiments one usually lacks the exact knowledge of the boundary and contact impedances, and therefore, approximate model domain and contact impedances have to be used in the image reconstruction. However, it has been noticed that even small errors in the shape of the computation domain or contact impedances can cause large systematic artefacts in the reconstructed images, leading to loss of diagnostically relevant information. In a recent paper (Kolehmainen , 2006), we showed how in the 2-D case the errors induced by the inaccurately known boundary can be eliminated as part of the image reconstruction and introduced a novel method for finding a deformed image of the original isotropic conductivity using the theory of TeichmÜller mappings. In this paper, the theory and reconstruction method are extended to include the estimation of unknown contact impedances. The method is implemented numerically and tested with experimental EIT data. The results show that the systematic errors caused by inaccurately known boundary and contact impedances can efficiently be eliminated by the reconstruction method.      

Enhancing impedance imaging through multimodal tomography

Several noninvasive modalities including electrical impedance tomography (EIT), magnetic induction tomography (MIT), and induced-current EIT (ICEIT) have been developed for imaging the electrical conductivity distribution within a human body. Although these modalities differ in how the excitation and detection circuitry (electrodes or coils) are implemented, they share a number of common principles not only within the image reconstruction approaches but also with respect to the basic principle of generating a current density distribution inside a body and recording the resultant electric fields. In this paper, we are interested in comparing differences between these modalities and in theoretically understanding the compromises involved, despite the increased hardware cost and complexity that such a multimodal system brings along. To systematically assess the merits of combining data, we performed 3-D simulations for each modality and for the multimodal system by combining all available data. The normalized sensitivity matrices were computed for each modality based on the finite element method, and singular value decomposition was performed on the resultant matrices. We used both global and regional quality measures to evaluate and compare different modalities. This study has shown that the condition number of the sensitivity matrix obtained from the multimodal tomography with 16-electrode and 16-coil is much lower than the condition number produced in the conventional 16-channel EIT and MIT systems, and thus, produced promising results in terms of image stability. An improvement of about 20% in image resolution can be achieved considering feasible signal-to-noise ratio levels.     

Statistical estimation of resistance/conductance by electrical impedance tomography measurements

This paper is built upon the assumption that in electrical impedance tomography, vectors of voltages and currents are linearly dependent through a resistance matrix. This linear relationship was confirmed experimentally and may be derived analytically under certain assumptions regarding electrodes (Isaacson, 1991). Given measurement data consisting of voltages and currents, we treat this relationship as a linear statistical model. Thus, our goal is not to reconstruct the image but directly estimate its electromagnetic properties reflected in the resistance and/or conductance matrix using electrical impedance tomography (EIT) measurements of voltages and currents on the periphery of the body. Since no inverse problem is involved the algorithm for estimation merely reduces to one matrix inversion. We estimate the impedance resistance matrix using well established statistical inference techniques for linear regression models. We provide a comprehensive treatment for a two-dimensional homogeneous body of a circular shape, by which many concepts of electrical impedance tomography, such as width of electrodes, the difference between voltage-current and current-voltage systems are illustrated. Our theory may be applied to various tests including EIT hardware calibration and whether the medium is homogeneous. These tests are illustrated on phantom agar data.     

A reconstruction algorithm of electrical impedance tomography with optimal configuration of the driven electrodes

A reconstruction algorithm for electrical impedance tomography (EIT) is presented. The least-squares (LS) method is applied and a formulation similar to that of the perturbation method is found. The main difference from perturbation lies with the sensitivity matrix, which here is replaced by the Jacobian matrix, defined in terms of the partial derivatives of every sensing electrode pair voltage difference with respect to every element's conductivity. The mutual position between the active electrodes is chosen to give optimum sensitivity. The results shown that the algorithm presented here has a better convergence and needs fewer iterations than the perturbation method.     

Electrical impedance tomography: regularized imaging and contrast detection

Dynamic electrical impedance tomography (EIT) images changes in the conductivity distribution of a medium from low frequency electrical measurements made at electrodes on the medium surface. Reconstruction of the conductivity distribution is an under-determined and ill-posed problem, typically requiring either simplifying assumptions or regularization based on a priori knowledge. This paper presents a maximum a posteriori (MAP) approach to linearized image reconstruction using knowledge of the noise variance of the measurements and the covariance of the conductivity distribution. This approach has the advantage of an intuitive interpretation of the algorithm parameters as well as fast (near real time) image reconstruction. In order to compare this approach to existing algorithms, the authors develop figures of merit to measure the reconstructed image resolution, the noise amplification of the image reconstruction, and the fidelity of positioning in the image. Finally, the authors develop a communications systems approach to calculate the probability of detection of a conductivity contrast in the reconstructed image as a function of the measurement noise and the reconstruction algorithm used.     

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.     

A reconstruction algorithm for breast cancer imaging with electrical impedance tomography in mammography geometry

The conductivity and permittivity of breast tumors are known to differ significantly from those of normal breast tissues, and electrical impedance tomography (EIT) is being studied as a modality for breast cancer imaging to exploit these differences. At present, X-ray mammography is the primary standard imaging modality used for breast cancer screening in clinical practice, so it is desirable to study EIT in the geometry of mammography. This paper presents a forward model of a simplified mammography geometry and a reconstruction algorithm for breast tumor imaging using EIT techniques. The mammography geometry is modeled as a rectangular box with electrode arrays on the top and bottom planes. A forward model for the electrical impedance imaging problem is derived for a homogeneous conductivity distribution and is validated by experiment using a phantom tank. A reconstruction algorithm for breast tumor imaging based on a linearization approach and the proposed forward model is presented. It is found that the proposed reconstruction algorithm performs well in the phantom experiment, and that the locations of a 5-mm-cube metal target and a 6-mm-cube agar target could be recovered at a target depth of 15 mm using a 32 electrode system.     

A comparison of techniques to optimize measurement of voltage changes in electrical impedance tomography by minimizing phase shift errors

In electrical impedance tomography, errors due to stray capacitance may be reduced by optimization of the reference phase of the demodulator. Two possible methods, maximization of the demodulator output and minimization of reciprocity error have been assessed, applied to each electrode combination individually, or to all combinations as a whole. Using an EIT system with a single impedance measuring circuit and multiplexer to address the 16 electrodes, the methods were tested on resistor-capacitor networks, saline-filled tanks and humans during variation of the saline concentration of a constant fluid volume in the stomach. Optimization of each channel individually gave less error, particularly on humans, and maximization of the output of the demodulator was more robust. This method is, therefore, recommended to optimize systems and reduce systematic errors with similar EIT systems.     

Propagation of measurement noise through backprojection reconstruction in electrical impedance tomography

A framework to analyze the propagation of measurement noise through backprojection reconstruction algorithms in electrical impedance tomography (EIT) is presented. Two measurement noise sources were considered: noise in the current drivers and in the voltage detectors. The influence of the acquisition system architecture (serial/semi-parallel) is also discussed. Three variants of backprojection reconstruction are studied: basic (unweighted), weighted and exponential backprojection. The results of error propagation theory have been compared with those obtained from simulated and experimental data. This comparison shows that the approach provides a good estimate of the reconstruction error variance. It is argued that the reconstruction error in EIT images obtained via backprojection can be approximately modeled as a spatially nonstationary Gaussian distribution. This methodology allows us to develop a spatial characterization of the reconstruction error in EIT images.     

Impact of model shape mismatch on reconstruction quality in electrical impedance tomography

Electrical impedance tomography (EIT) is a low-cost, noninvasive and radiation free medical imaging modality for monitoring ventilation distribution in the lung. Although such information could be invaluable in preventing ventilator-induced lung injury in mechanically ventilated patients, clinical application of EIT is hindered by difficulties in interpreting the resulting images. One source of this difficulty is the frequent use of simple shapes which do not correspond to the anatomy to reconstruct EIT images. The mismatch between the true body shape and the one used for reconstruction is known to introduce errors, which to date have not been properly characterized. In the present study we, therefore, seek to 1) characterize and quantify the errors resulting from a reconstruction shape mismatch for a number of popular EIT reconstruction algorithms and 2) develop recommendations on the tolerated amount of mismatch for each algorithm. Using real and simulated data, we analyze the performance of four EIT reconstruction algorithms under different degrees of shape mismatch. Results suggest that while slight shape mismatch is well tolerated by all algorithms, using a circular shape severely degrades their performance.     

Electrical impedance tomography using induced currents

The mathematical basis of a new imaging modality, induced current electrical impedance tomography (EIT), is investigated, The ultimate aim of this technique is the reconstruction of conductivity distribution of the human body, from voltage measurements made between electrodes placed on the surface, when currents are induced inside the body by applied time varying magnetic fields. In this study the two-dimensional problem is analyzed. A specific 9-coil system for generating nine different exciting magnetic fields (50 kHz) and 16 measurement electrodes around the object are assumed, The partial differential equation for the scaler potential function in the conductive medium is derived and finite element method (FEM) is used for its solution. Sensitivity matrix, which relates the perturbation in measurements to the conductivity perturbations, is calculated. Singular value decomposition of the sensitivity matrix shows that there are 135 independent measurements. It is found that measurements are less sensitive to changes in conductivity of the object's interior. While in this respect induced current EIT is slightly inferior to the technique of injected current EIT (using Sheffield protocol), its sensitivity matrix is better conditioned. The images obtained are found to be comparable to injected current EIT images In resolution. Design of a coil system for which parameters such as sensitivity to inner regions and condition number of the sensitivity matrix are optimum, remains to be made.     

Finite element modeling of electrode-skin contact impedance inelectrical impedance tomography

In electrical impedance tomography (EIT), the measured voltages are sensitive to electrode-skin contact impedance because the contact impedance and the current density through it are both high. Large electrodes were used to provide a more uniform current distribution and reduce the contact impedance. A large electrode differs from a point electrode in that it has shunting and edge effects that cannot be modeled by a single resistor. The finite-element method (FEM) was used to study the electric field distributions underneath an electrode, and three models were developed: a FEM model, a simplified FEM model, and a weighted load model. The FEM models considered both shunting and edge effects and closely matched the experimental measurements. It is concluded that FEM models of electrodes can be used to improve the performance of an electrical impedance tomography reconstruction algorithm     

In Vivo Impedance Imaging With Total Variation Regularization

We show that electrical impedance tomography (EIT) image reconstruction algorithms with regularization based on the total variation (TV) functional are suitable for in vivo imaging of physiological data. This reconstruction approach helps to preserve discontinuities in reconstructed profiles, such as step changes in electrical properties at interorgan boundaries, which are typically smoothed by traditional reconstruction algorithms. The use of the TV functional for regularization leads to the minimization of a nondifferentiable objective function in the inverse formulation. This cannot be efficiently solved with traditional optimization techniques such as the Newton method. We explore two implementations methods for regularization with the TV functional: the lagged diffusivity method and the primal dual–interior point method (PD-IPM). First we clarify the implementation details of these algorithms for EIT reconstruction. Next, we analyze the performance of these algorithms on noisy simulated data. Finally, we show reconstructed EIT images of in vivo data for ventilation and gastric emptying studies. In comparison to traditional quadratic regularization, TV regulariza tion shows improved ability to reconstruct sharp contrasts.      

静态阻抗断层图像重建新方法

阻抗断层图像重建是一个严重病态的非线性的逆问题,特别是在静态阻抗断层成像中,由于其图像重建模型误差和测量噪声的影响更为严重,因此常用的基于目标函数梯度信息不断迭代的改进的Newton-Raphson类重建算法,即使使用正则化技术,其稳定性仍较差,甚至发散.本文提出一种全新的静态阻抗断层图像重建方法,它利用基于生物自然选择与遗传机理的遗传算法去搜索阻抗图像重建问题的最优解,无需正则化技术,也不会象改进的Newton-Raphson类算法那样易陷入局部最优解.实验结果也表明基于遗传算法的图像重建方法重建的静态阻抗断层图像,其成像精度和空间分辨率都大大好于改进的Newton-Raphson类重建算法.     

Cardiac Mapping Instrumentation for the Instantaneous Display of Endocardial and Epicardial Activation

A real-time mapping system is described that allows instant display of both endocardial and epicardial activation as monitored by an array of transmural plunge-needle electrodes in the intact in situ heart. From the total cardiac muscle mass, 768 electrograms are recorded, with a programmable feature allowing the rapid and unrestricted selection of 110 endocardial electrograms for automated analysis. An activation analyzer (consisting of parallel analog circuitry) provides an activation image in real time, while video recorders provide both raw and processed data storage with the capability of immediate slow-motion replay. The nature of the data acquisition (110 plunge needles) limits this system to the experimental arrhythmia laboratory where its chief advantage is the enabling of the multichannel recording of endo data without the complications of open heart surgery. Equally important is the ability to automatically store the endocardial activation image locked in time to the simultaneously recorded epicardial image.     

Real-Time Reconstruction of Sensitivity Encoded Radial Magnetic Resonance Imaging Using a Graphics Processing Unit

A barrier to the adoption of non-Cartesian parallel magnetic resonance imaging for real-time applications has been the times required for the image reconstructions. These times have exceeded the underlying acquisition time thus preventing real-time display of the acquired images. We present a reconstruction algorithm for commodity graphics hardware (GPUs) to enable real time reconstruction of sensitivity encoded radial imaging (radial SENSE). We demonstrate that a radial profile order based on the golden ratio facilitates reconstruction from an arbitrary number of profiles. This allows the temporal resolution to be adjusted on the fly. A user adaptable regularization term is also included and, particularly for highly undersampled data, used to interactively improve the reconstruction quality. Each reconstruction is fully self-contained from the profile stream, i.e., the required coil sensitivity profiles, sampling density compensation weights, regularization terms, and noise estimates are computed in real-time from the acquisition data itself. The reconstruction implementation is verified using a steady state free precession (SSFP) pulse sequence and quantitatively evaluated. Three applications are demonstrated; real-time imaging with real-time SENSE 1) or $k$-$t$ SENSE 2) reconstructions, and 3) offline reconstruction with interactive adjustment of reconstruction settings.