Electrode boundary conditions and experimental validation for BEM-based EIT forward and inverse solutions

In this paper, we present theoretical developments and experimental results for the problem of estimating the conductivity map inside a volume using electrical impedance tomography (EIT) when the boundary locations of any internal inhomogeneities are known. We describe boundary element method (BEM) implementations of advanced electrode models for the forward problem of EIT. We then use them in the inverse problem with known internal boundaries and derive the associated Jacobians. We report on the results of two EIT phantom studies, one using a homogeneous cubical tank, and one using a cylindrical tank with agar conductivity inhomogeneities. We test both the accuracy of our BEM forward model, including the electrode models, as well as our inverse solution, against the measured data. Results show good agreement between measured values and both forward-computed tank voltages and inverse-computed conductivities; for instance, in a phantom experiment, we reconstructed the conductivities of three agar objects inside a cylindrical tank with an error less than 2% of their true value.     

Electrical impedance tomography for piecewise constant domains using boundary element shape-based inverse solutions

Shape-based solutions have recently received attention for certain ill-posed inverse problems. Their advantages include implicit imposition of relevant constraints and reduction in the number of unknowns, especially important for nonlinear ill-posed problems. We apply the shape-based approach to current-injection electrical impedance tomography (EIT) reconstructions. We employ a boundary element method (BEM) based solution for EIT. We introduce two shape models, one based on modified B-splines, and the other based on spherical harmonics, for BEM modeling of shapes. These methods allow us to parameterize the geometry of conductivity inhomogeneities in the interior of the volume. We assume the general shape of piecewise constant inhomogeneities is known but their conductivities and their exact location and shape is not. We also assume the internal conductivity profile is piecewise constant, meaning that each region has a constant conductivity. We propose and test three different regularization techniques to be used with either of the shape models. The performance of our methods is illustrated via both simulations in a digital torso model and phantom experiments when there is a single internal object. We observe that in the noisy environment, either simulated noise or real sources of noise in the experimental study, we get reasonable reconstructions. Since the signal-to-noise ratio (SNR) expected in modern EIT instruments is higher than that used in this study, these reconstruction methods may prove useful in practice.     

The boundary element method in the forward and inverse problem of electrical impedance tomography

In this paper, a new formulation of the reconstruction problem of electrical impedance tomography (EIT) is proposed. Instead of reconstructing a complete two-dimensional picture, a parameter representation of the gross anatomy is formulated, of which the optimal parameters are determined by minimizing a cost function. The two great advantages of this method are that the number of unknown parameters of the inverse problem is drastically reduced and that quantitative information of interest (e.g., lung volume) is estimated directly from the data, without image segmentation steps. The forward problem of EIT is to compute the potentials at the voltage measuring electrodes, for a given set of current injection electrodes and a given conductivity geometry. In this paper, it is proposed to use an improved boundary element method (BEM) technique to solve the forward problem, in which flat boundary elements are replaced by polygonal ones. From a comparison with the analytical solution of the concentric circle model, it appears that the use of polygonal elements greatly improves the accuracy of the BEM, without increasing the computation time. In this formulation, the inverse problem is a nonlinear parameter estimation problem with a limited number of parameters. Variants of Powell's and the simplex method are used to minimize the cost function. The applicability of this solution of the EIT problem was tested in a series of simulation studies. In these studies, EIT data were simulated using a standard conductor geometry and it was attempted to find back this geometry from random starting values. In the inverse algorithm, different current injection and voltage measurement schemes and different cost functions were compared. In a simulation study, it was demonstrated that a systematic error in the assumed lung conductivity results in a proportional error in the lung cross sectional area. It appears that our parametric formulation of the inverse problem leads to a stable minimization problem, with a high reliability, provided that the signal-to-noise ratio is about ten or higher.     

An efficient forward solver in electrical impedance tomography by spectral element method

In electrical impedance tomography (EIT), a forward solver capable of predicting the voltages on electrodes for a given conductivity distribution is essential for reconstruction. The EIT forward solver is normally based on the conventional finite element method (FEM). One of the major problems of three-dimensional (3-D) EIT is its high demand in computing power and memory since high precision is required for obtaining a small secondary field which is typical for a small anomaly. This accuracy requirement is also set by the level of noise in the real data; although currently the noise level is still an issue, future EIT systems should significantly reduce the noise level to be capable of detecting very small anomalies. To accurately simulate the forward solution with the FEM, a mesh with large number of nodes and elements is usually needed. To overcome this problem, we proposed the spectral element method (SEM) for EIT forward problem. With the introduction of SEM, a smaller number of nodes and hence less computational time and memory are needed to achieve the same or better accuracy in the forward solution than the FEM. Numerical results demonstrate the efficiency of the SEM in 3-D EIT simulation.     

Anatomically constrained electrical impedance tomography forthree-dimensional anisotropic bodies

As shown previously for two-dimensional geometries, anisotropy effects should not be ignored in electrical impedance tomography (EIT) and structural information is important for the reconstruction of anisotropic conductivities. Here, we describe the static reconstruction of an anisotropic conductivity distribution for the more realistic three-dimensional (3-D) case. Boundaries between different conductivity regions are anatomically constrained using magnetic resonance imaging (MRI) data. The values of the conductivities are then determined using gradient-type-algorithms in a nonlinear-indirect approach. At each iteration, the forward problem is solved by the finite element method. The approach is used to reconstruct the 3-D conductivity profile of a canine torso. Both computational performance and simulated reconstruction results are presented together with a detailed study on the sensitivity of the prediction error with respect to different parameters. In particular, the use of an intracavity catheter to better extract interior conductivities is demonstrated     

Electrical impedance tomography of complex conductivitydistributions with noncircular boundary

Electrical impedance tomography (EIT) uses low-frequency current and voltage measurements made on the boundary of a body to compute the conductivity distribution within the body. Since the permittivity distribution inside the body also contributes significantly to the measured voltages, the present reconstruction algorithm images complex conductivity distributions. A finite element model (FEM) is used to solve the forward problem, using a 6017-node mesh for a piecewise-linear potential distribution. The finite element solution using this mesh is compared with the analytical solution for a homogeneous field and a maximum error of 0.05% is observed in the voltage distribution. The boundary element method (BEM) is also used to generate the voltage data for inhomogeneous conductivity distributions inside regions with noncircular boundaries. An iterative reconstruction algorithm is described for approximating both the conductivity and permittivity distributions from this data. The results for an off-centered inhomogeneity showed a 35% improvement in contrast from that seen with only one iteration, for both the conductivity and the permittivity values. It is also shown that a significant improvement in images results from accurately modeling a noncircular boundary. Both static and difference images are distorted by assuming a circular boundary and the amount of distortion increases significantly as the boundary shape becomes more elliptical. For a homogeneous field in an elliptical body with axis ratio of 0.73, an image reconstructed assuming the boundary to be circular has an artifact at the center of the image with an error of 20%. This error increased to 37% when the axis ratio was 0.64. A reconstruction algorithm which used a mesh with the same axis ratio as the elliptical boundary reduced the error in the conductivity values to within 0.5% of the actual values     

On modeling the Wilson terminal in the boundary and finite element method

In clinical electrocardiography, the zero-potential is commonly defined by the Wilson central terminal. In the electrocardiographic forward and inverse problem, the zero-potential is often defined in a different way, e.g., by the sum of all node potentials yielding zero. This study presents relatively simple to implement techniques, which enable the incorporation of the Wilson Terminal in the boundary element method (BEM) and finite element method (FEM). For the BEM, good results are obtained when properly adopting matrix deflation for modeling the Wilson terminal. Applying other zero-potential-definitions, the obtained solutions contained a remarkable offset with respect to the reference defined by the Wilson terminal. In the inverse problem (nonlinear dipole fit), errors introduced by an erroneous zero-potential-definition can lead to displacements of more than 5 mm in the computed dipole location. For the FEM, a method similar to matrix deflation is proposed in order to properly consider the Wilson central terminal. The matrix obtained from this manipulation is symmetric, sparse and positive definite enabling the application of standard FEM-solvers.     

The inverse problem utilizing the boundary element method for a nonstandard female torso

This paper proposes a new method of rapidly deriving the transfer matrix for the boundary element method (BEM) forward problem from a tailored female torso geometry in the clinical setting. The method allows rapid calculation of epicardial potentials (EP) from body surface potentials (BSP). The use of EPs in previous studies has been shown to improve the successful detection of the life-threatening cardiac condition--acute myocardial infarction. The MRI scanning of a cardiac patient in the clinical setting is not practical and other methods are required to accurately deduce torso geometries for calculation of the transfer matrix. The new method allows the noninvasive calculation of tailored torso geometries from a standard female torso and five measurements taken from the body surface of a patient. This scaling of the torso has been successfully validated by carrying out EP calculations on 40 scaled torsos and ten female subjects. It utilizes the BEM in the calculation of the transfer matrix as the BEM depends only upon the topology of the surfaces of the torso and the heart, the former can now be accurately deduced, leaving only the latter geometry as an unknown.     

The Inverse Problem in Electrocardiography: A Model Study of the Effects of Geometry and Conductivity Parameters on the Reconstruction of Epicardial Potentials

An idealized, analytic model using spherical harmonics was developed to analyze the effects of variations in torso geometry and volume conductivity parameters on the recovery of epicardial potentials from torso potentials. The model was also used to analyze the effects of these variations on individual terms in the orthogonal series expansion. The ability to reconstruct separate, local electrical events on the epicardium was examined under the following simulated situations: 1) all conductivity and geometry parameters were known accurately, 2) the conductivity of individual torso tissue layers was varied, 3) the torso-air boundary was eliminated (the "infinite medium" assumption), 4) the heart position was not accurately known, and 5) the heart size was not accurately known. Variation in conductivity and geometry parameters was found to exert a quantitative and qualitative effect on the amplitude, resolution, and position of the reconstructed epicardial maxima and minima. Significant differences were found in the ability of the inverse procedure to recover epicardial potentials resulting from posterior as opposed to anterior myocardial sources. Important conclusions regarding the narrow allowance for error in heart size and position, and the relative contributions of the torso tissue layer conductivities can provide guidelines for inverse reconstruction of epicardial potentials with a realistic model utilizing the true geometry.     

Anatomically constrained electrical impedance tomography for anisotropic bodies via a two-step approach

Discusses the inclusion of anatomical constraints and anisotropy in static Electrical Impedance Tomography (EIT) using a two-step approach to EIT. In the first step, the boundaries between regions of different conductivities are anatomically constrained using Magnetic Resonance Imaging (MRI) data. In the second step, the conductivity values in different regions are determined. Anisotropic conductivity regions are included to allow better modeling of the muscle regions (e.g., skeletal muscle) which exhibit a greater conductivity in the direction parallel to the muscle fiber. This two-step approach is used to reconstruct the conductivity profile of a canine torso, illustrating its potential application in extracting conductivity values for bioelectric modeling.     

A high-order coupled finite element/boundary element torso model

Describes a high-order (cubic Hermite) coupled finite element/boundary element procedure for solving electrocardiographic potential problems to be ultimately used for solving forward and inverse problems on an anatomically accurate human torso. Details of both numerical procedures and the coupling between them are described. Test results, illustrating the accuracy and efficiency of this combination for both two-dimensional (2-D) and three-dimensional (3-D) problems, are also given     

Electrical impedance tomography of translationally uniform cylindrical objects with general cross-sectional boundaries

An algorithm is developed for electrical impedance tomography (EIT) of finite cylinders with general cross-sectional boundaries and translationally uniform conductivity distributions. The electrodes for data collection are assumed to be placed around a cross-sectional plane; therefore, the axial variation of the boundary conditions and the potential field are expanded in Fourier series. For each Fourier component a two-dimensional (2-D) partial differential equation is derived. Thus the 3-D forward problem is solved as a succession of 2-D problems, and it is shown that the Fourier series can be truncated to provide substantial savings in computation time. The finite element method is adopted and the accuracy of the boundary potential differences (gradients) thus calculated is assessed by comparison to results obtained using cylindrical harmonic expansions for circular cylinders. A 1016-element and 541-node mesh is found to be optimal. The algorithm is applied to data collected from phantoms, and the errors incurred from the several assumptions of the method are investigated.     

高压MZ—JTE终端边界元数值模拟

本文采用边界元数值方法，编制了统一的多区结终端扩展（Ｍｕｌｔｉｐｌｅ－Ｚｏｎｅ Ｊｕｎｃｔｉｏｎ Ｔｅｒ－ｍｉｎａｔｉｏｎ Ｅｘｔｅｎｓｉｏｎ，ＭＺ－ＪＴＥ）边界元程序，模拟高压ＭＺ－ＪＴＥ终端电场，电位分布，以平面结算例验证了边界元数值解精度及可靠性，定量模拟了界面态影响，取得了与国外用差分或有限元得到的结论相一致的结果。边界元方法的提出与算法，程序的成功开发，为ＭＺ－ＪＴＥ结构及其它终端结构的     

In vivo measurement of the brain and skull resistivities using an EIT-based method and realistic models for the head

In vivo measurements of equivalent resistivities of skull (rho(skull)) and brain (rho(brain)) are performed for six subjects using an electric impedance tomography (EIT)-based method and realistic models for the head. The classical boundary element method (BEM) formulation for EIT is very time consuming. However, the application of the Sherman-Morrison formula reduces the computation time by a factor of 5. Using an optimal point distribution in the BEM model to optimize its accuracy, decreasing systematic errors of numerical origin, is important because cost functions are shallow. Results demonstrate that rho(skull)/rho(brain) is more likely to be within 20 and 50 rather than equal to the commonly accepted value of 80. The variation in rho(brain)(average = 301 omega x cm, SD = 13%) and rho(skull)(average = 12230 omega x cm, SD = 18%) is decreased by half, when compared with the results using the sphere model, showing that the correction for geometry errors is essential to obtain realistic estimations. However, a factor of 2.4 may still exist between values of rho(skull)/rho(brain) corresponding to different subjects. Earlier results show the necessity of calibrating rho(brain) and rho(skull) by measuring them in vivo for each subject, in order to decrease errors associated with the electroencephalogram inverse problem. We show that the proposed method is suited to this goal.     

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.      

Studies of the Electrocardiogram Using Realistic Cardiac and Torso Models

Several aspects of the forward and inverse problems of electrocardiography are investigated through the use of digital computer models. Two forms of a fixed location, variable moment, 20-dipole cardiac model of QRS are developed from actual cardiac excitation data. One form uses time-varying orientation dipoles; the other uses fixed orientation dipoles. An electric multipole expansion (EME) cardiac model employing the dipole, quadrupole, and octupole terms is also developed and used as an equivalent forward and inverse cardiac model. Two realistically shaped torso models are used. The homogeneous torso has uniform conductivity; the inhomogeneous torso contains realistically shaped lung regions with reduced conductivity. It is found that when the EME model is used as an equivalent forward cardiac model, it can accurately represent the actual 20-dipole cardiac model in the homogeneous torso. Limb leads are accurately represented by the dipole terms alone while the precordial leads require the quadrupole and octupole terms. It is also found that while the lung regions have little effect on the ECG's produced by the models, these regions can have a significant effect on the inverse solutions for certain dipoles in the 20-dipole cardiac model. These lung regions appear to have a much smaller effect on the dipole terms in the EME model. Solutions of the inverse problem for the terms in the EME model indicate that when a limited number of measurements are used, the best results can be obtained by uniform distribution of the measurements over the torso.     

A common formalism for the integral formulations of the forward EEG problem

The forward electroencephalography (EEG) problem involves finding a potential V from the Poisson equation inverted Delta x (sigma inverted Delta V) f, in which f represents electrical sources in the brain, and sigma the conductivity of the head tissues. In the piecewise constant conductivity head model, this can be accomplished by the boundary element method (BEM) using a suitable integral formulation. Most previous work uses the same integral formulation, corresponding to a double-layer potential. In this paper we present a conceptual framework based on a well-known theorem (Theorem 1) that characterizes harmonic functions defined on the complement of a bounded smooth surface. This theorem says that such harmonic functions are completely defined by their values and those of their normal derivatives on this surface. It allows us to cast the previous BEM approaches in a unified setting and to develop two new approaches corresponding to different ways of exploiting the same theorem. Specifically, we first present a dual approach which involves a single-layer potential. Then, we propose a symmetric formulation, which combines single- and double-layer potentials, and which is new to the field of EEG, although it has been applied to other problems in electromagnetism. The three methods have been evaluated numerically using a spherical geometry with known analytical solution, and the symmetric formulation achieves a significantly higher accuracy than the alternative methods. Additionally, we present results with realistically shaped meshes. Beside providing a better understanding of the foundations of BEM methods, our approach appears to lead also to more efficient algorithms.     

An iterative FEM for scattering by a 3-D cavity-backed aperture

A finite-element method (FEM)-based hybrid method (or iterative FEM) is successfully applied to a three-dimensional (3-D) scattering problem without the effect of internal resonance. With only a small number of meshes around a 3-D scatterer, this FEM is shown to give an accurate result through several iterative updates of the boundary conditions. To confirm the efficiency of this method, scattering from a 3-D cavity-backed aperture is analyzed and the results obtained are compared with the same obtained by another conventional method     

Elliptic cylinder geometry for distinguishability analysis in impedance tomography

Electrical impedance tomography (EIT) is a technique that computes the cross-sectional impedance distribution within the body by using current and voltage measurements made on the body surface. It has been reported that the image reconstruction is distorted considerably when the boundary shape is considered to be more elliptical than circular as a more realistic shape for the measurement boundary. This paper describes an alternative framework for determining the distinguishability region with a finite measurement precision for different conductivity distributions in a body modeled by elliptic cylinder geometry. The distinguishable regions are compared in terms of modeling error for predefined inhomogeneities with elliptical and circular approaches for a noncircular measurement boundary at the body surface. Since most objects investigated by EIT are noncircular in shape, the analytical solution for the forward problem for the elliptical cross section approach is shown to be useful in order to reach a better assessment of the distinguishability region defined in a noncircular boundary. This paper is concentrated on centered elliptic inhomogeneity in the elliptical boundary and an analytic solution for this type of forward problem. The distinguishability performance of elliptical cross section with cosine injected current patterns is examined for different parameters of elliptical geometry.     

Use of the finite element method to determine epicardial from body surface potentials under a realistic torso model

This paper presents a new method of solution for the inverse problem in electrocardiography using the finite element procedure. It is an application of the authors' earlier work which derived a solution method by means of an integral equation under a generalized configuration of geometry and conductivity of the torso. Based on prior geometry information, the human torso region is discretized into a series offinite elements and, then, electric fields are computed when a set of linearly independent functions chosen as a basis is imposed on the epicardial surface. The set of these forward solutions defines the forward transfer coefficients which relate epicardial to body surface potentials. By the use of the forward transfer coefficients, a constrained least-squares estimate of the epicardial potential distribution can be obtained from measured body surface potentials. The solution method is examined through numerical experiments carried out for a realistic model of the human torso. It is demonstrated that the rapid decrease in voltage far from the heart generator makes this inverse problem ill conditioned and, as a result, the accuracy of the inverse epicardial potentials calculated depends greatly upon both the signal-to-noise ratio and the number of lead points in measuring the body surface potentials.