Electrical Impedance Tomography Forward Problem Research Articles

Finite element modeling of the electrical impedance tomography technique driven by machine learning

To create a human-like skin for a robotic application, current touch sensor technologies have a few drawbacks. Electrical Impedance Tomography (EIT) is a candidate for this application due to its applicability over complex geometries; nevertheless, it has accuracy concerns. This study employs artificial neural networks (ANNs) to investigate the accuracy and capability of EIT-based touch sensors. A finite element (FE) model is utilized to solve the forward EIT problem while simultaneously determining the system’s comprehensive mechanical response. The FE model is comprised of a polyurethane (PU) foam domain, a conductive spray layer and a set of sixteen electrodes. To replicate the process of touching the sensor body, a punch of varying diameters and touch forces is utilized. The mechanical response of the sensor body is modeled using the hyperfoam material model calibrated through experimental uniaxial and shear test data, while the electric conductivity of the sprayed skin surface is obtained experimentally as function of applied strain. The viscoelastic behavior of the PU foam material is also obtained experimentally. These experimental data were implemented in the FE model through user subroutines to model the mechanical and electrical properties of the sensor in the EIT forward problem. The traditional EIT inverse problem image reconstruction was replaced utilizing ANNs as an alternative to extract mechanics based parameters. The ANNs were created to predict the spatial coordinates of the touch point, and they were proven to be extremely accurate. Using the EIT voltage readings as input, the ANNs were utilized to forecast the system’s mechanical behavior such as contact pressure, contact area, indentation depth, and touching force.

High-Dimensional Uncertainty Quantification in Electrical Impedance Tomography Forward Problem Based on Deep Neural Network

In electrical impedance tomography (EIT), the uncertainty of conductivity distribution may cause the uncertainty in the forward calculation and further affect the inverse problem. In this paper, an improved univariate dimension reduction method based on deep neural network (DNN-UDR) is proposed for the high-dimensional uncertainty quantification in EIT forward problem. Firstly, DNN is studied to build a substitute model for EIT forward problem in order to solve the high-dimensional problem. Three normalized circular finite element models are established with random uniform conductivity distribution. Then UDR is used to analyze and quantify the uncertainty in the simulation with the form of probability. Compared with Monte Carlo simulation (MCS), the probability distribution of voltage is fitted, and the quantification indicators such as mean, variance, variation coefficient and covariance, are also consistent. On the other hand, with the increase of parameter dimensions, DNN-UDR accelerates the computations obviously. This indicates that DNN-UDR is effective and has high structural stability, accurate prediction results and high computational efficiency.

Improvement of performance and sensitivity of 2D and 3D image reconstruction in EIT using EFG forward model

This paper presents a pure element-free Galerkin method (EFGM) forward model for image reconstruction in 2D and 3D electrical impedance tomography (EIT) using an adaptive current injection method. In EIT systems with the adapting current injection method, both static and dynamic images can be reconstructed; however, determination of electrode contact impedances in the complete electrode model is difficult and the Gap model is used. In this paper, in the EIT forward problem a weak form functional based on the Gap model and a pure EFGM approach are developed, and in the EIT inverse problem, Jacobian matrix is computed by the EFGM, and a fast integration technique is introduced to calculate the entries of the Jacobian matrix within an adequate computation time. The influence of increasing the density of nodes at and near the electrodes with steep electric potential gradients on the accuracy of FEM and EFGM forward solutions is investigated, and the performance of the image reconstruction algorithm with the proposed fast integration technique is examined. The numerical results reveal that the proposed EFGM forward model with the fast integration technique has an efficient performance both in terms of mean relative imaging errors and computational time.

Electrical Impedance Tomography Image Reconstruction Based on Neural Networks

electrical impedance tomography (EIT) is an imaging technique with a promising future. Several methods have been used for EIT image reconstruction, such as Simulated Annealing, Gauss Newton, Kalman filter and D-Bar. Recently, some authors solved this problem using artificial neural network (ANN) through pixel by pixel reconstruction, considering a fixed resolution for the final image. This work proposes a reconstruction based on the EIT forward problem. Two different meshes were considered: a coarse and a refined mesh. The latter was used to produce simulated potentials, which are the inputs for ANN training. The nodes conductivities, which used to create the outputs for training, defined in the coarser mesh. Therefore, the proposed method consists of training the ANN with inputs from a refined mesh and outputs from a coarse mesh. Two ANN architectures are proposed and compared: one based on the LeNet architecture, and another based on the feed-forward fully connected ANN. The obtained image is not dependent on any image resolution. The preliminary results show that the LeNet architecture has better performance.

Application of meshless local Petrov Galerkin method (MLPG5) for EIT forward problem

This paper deals with the application of Meshless Local Petrov Galerkin method(MLPG) for solving the forward problem associated with Electrical Impedance Tomography(EIT). Out of six variants of MLPG, here we have used MLPG5 to solve the forward problem. Use of Heaviside function as test function eliminates the domain integral term from the weak formulation. Radial point interpolation method(RPIM) is used and the resulting shape functions possess Kronecker delta function property. Hence no modification is required in the weak form to impose essential boundary conditions. Aforementioned advantages have motivated us to apply this meshless method for EIT forward problem. Numerical accuracy of this method is determined by finding the relative error between numerical solution and exact solution. There are several factors that can affect the accuracy such as local sub domain size(αsub), influence domain size(αinf), nodal density, type and shape of radial basis functions used. A detailed parametric study is carried out in this paper. Radial basis functions(RBF) such as Multiquadrics(MQ), Inverse Multiquadrics(INVMQ) and Gaussian are used for function interpolation along with linear polynomial. Shape parameter of RBFs is also varied to study the effect on accuracy. The solution obtained from the meshless method and analytical method are compared with solutions from Finite Element Method using linear triangular elements. Relative error is found to be minimum when αsub lies in the range of 0.6–0.7 for MQ and INVMQ and αinf > 5 since the boundary voltage profile should be the same when current is injected into the circular medium from any angle under homogeneous conditions. For Gaussian function, minimum error is observed for αsub in the interval 0.5–0.6 and αinf > 5. Gaussian has shown the lowest error compared to MQ and INVMQ. Forward problem in EIT usually deals with nonhomogeneous medium and therefore we need to extend this study to nested domains. Since it is very difficult to get exact solution for nonhomogeneous medium, same conductivity value is given for both background(outer domain) and the inhomogeneity(inner domain) with some additional coupling conditions to be satisfied at the interface. Resulting boundary potentials are matched with the exact solution of homogeneous medium. MQ is used for MLPG5 implementation and error is calculated for different values of αsub and αinf. Minimum error is obtained at αsub = 0.6 for all values of αinf. It is better to select αinf > 5 to ensure the boundary profiles to overlap in all current injection cycles.

An element-free Galerkin forward solver for the complete-electrode model in electrical impedance tomography

An element-free Galerkin method (EFGM) is used for solving forward problem based on the complete electrode model (CEM) in electrical impedance tomography (EIT). The EFGM requires only nodal data and has the ability of providing mesh-independent solutions because no element connectivity is needed to be used in this method. However, direct imposition of Dirichlet boundary conditions for the EFGM is difficult because the shape functions employed in this method do not have the property of Dirac delta function. Solving the EIT forward problem based on the CEM by the EFGM, the effects of electrodes and contact impedances are taken into account and the complete solution of equations is provided without imposing Dirichlet boundary conditions. The numerical results are validated with experimental results obtained from a 2D circular homogeneous phantom, and the performance of the EFGM compared with the finite element method is also illustrated. Moreover, results obtained from the EFGM are presented for an inhomogeneous numerical phantom.

The EIT Forward Problem Parallelized Using a Colored pJDS Matrix Format

Electrical Impedance Tomography (EIT) is an imaging technique that attempts to reconstruct the conductivity distribution inside an object from electrical currents and potentials applied and measured at its surface. The EIT reconstruction problem is approached as an optimization problem, where the difference between the simulated and measured distributions must be minimized. This work uses the preconditioned conjugated gradients and it has two main modules: matrix-vector multiplication and the triangular solver. The main problem is the solution of a linear system with symmetric positive definite matrix. Several approaches use the compressed sparse row (CSR) format. However, the triangular solver with the CSR format has some thread divergence. In this work, the thread divergence will be eliminated by using the matrix representation called jagged diagonals storage. The proposed GPU accelerated forward solver reduces the expected computational time. Copyright ©2015 IF AC.

NUMERICAL SIMULATION OF ELECTRICAL IMPEDANCE TOMOGRAPHY PROBLEM AND STUDY OF APPROACH BASED ON FINITE VOLUME METHOD

This research has been aimed to carry out a study of peculiarities that arise in a numerical simulation of the electrical impedance tomography (EIT) problem. Static EIT image reconstruction is sensitive to a measurement noise and approximation error. A special consideration has been given to reducing of the approximation error, which originates from numerical implementation drawbacks. This paper presents in detail two numerical approaches for solving EIT forward problem. The finite volume method (FVM) on unstructured triangular mesh is introduced. In order to compare this approach, the finite element (FEM) based forward solver was implemented, which has gained the most popularity among researchers. The calculated potential distribution with the assumed initial conductivity distribution has been compared to the analytical solution of a test Neumann boundary problem and to the results of problem simulation by means of ANSYS FLUENT commercial software. Two approaches to linearized EIT image reconstruction are discussed. Reconstruction of the conductivity distribution is an ill-posed problem, typically requiring a large amount of computation and resolved by minimization techniques. The objective function to be minimized is constructed of measured voltage and calculated boundary voltage on the electrodes. A classical modified Newton type iterative method and the stochastic differential evolution method are employed. A software package has been developed for the problem under investigation. Numerical tests were conducted on simulated data. The obtained results could be helpful to researches tackling the hardware and software issues for medical applications of EIT.

A Combined Wavelet-Based Mesh-Free Method for Solving the Forward Problem in Electrical Impedance Tomography

In this paper, a combined wavelet-based mesh-free method is suggested to solve the forward problem in electrical impedance tomography (EIT). EIT is a noninvasive imaging modality based on the reconstruction of the conductivity profile inside a solution domain. The forward problem simulates measurement data for comparing with experimental data in the inverse problem. The forward problem can be solved by the finite element method (FEM) or the mesh-free method. Since the solution of the FEM depends on the mesh shape and boundary condition constraints are difficult to be applied to the mesh-free method, the combined wavelet-based mesh-free approach can resolve the disadvantages of both methods in the EIT forward problem. In order to apply interface conditions between different regions, slope jump functions are entered to the set of basis functions. Two different 2-D EIT problems are modeled. The simulation results obtained by the combined wavelet-based mesh-free method are compared with the FEM in terms of accuracy and computational cost. Moreover, the effects of object movement and deformation are investigated on the accuracy of the simulated electrode voltages using the standard FEM and combined wavelet-based mesh-free method.

EIT Forward Problem Parallel Simulation Environment with Anisotropic Tissue and Realistic Electrode Models

Electrical impedance tomography (EIT) is an imaging technology based on impedance measurements. To retrieve meaningful insights from these measurements, EIT relies on detailed knowledge of the underlying electrical properties of the body. This is obtained from numerical models of current flows therein. The nonhomogeneous and anisotropic electric properties of human tissues make accurate modeling and simulation very challenging, leading to a tradeoff between physical accuracy and technical feasibility, which at present severely limits the capabilities of EIT. This work presents a complete algorithmic flow for an accurate EIT modeling environment featuring high anatomical fidelity with a spatial resolution equal to that provided by an MRI and a novel realistic complete electrode model implementation. At the same time, we demonstrate that current graphics processing unit (GPU)-based platforms provide enough computational power that a domain discretized with five million voxels can be numerically modeled in about 30 s.

Reconstruction convergence and speed enhancement in electrical impedance tomography for domains with known internal boundaries

An improved approach for electrical impedance tomography (EIT) image reconstruction, based on modifying the forward and inverse solutions, is proposed. In this approach, the EIT forward problem is solved via the finite element method (FEM) using two types of elements. The inverse problem is solved by the modified Newton–Raphson method, whereas the condition number of the Hessian matrix is being monitored. At the early stage of the reconstruction, first-order elements are used, and if the condition number exceeds the allowable limit, the algorithm restarts. Otherwise, if the reconstruction error becomes lower than a predefined threshold, second-order elements are employed in the forward solution in order to preserve the precision of the final results. The latter stage converges in very few iterations. Since the solution speed with the first-order FEM is considerably higher than the second-order FEM, the reconstruction speed improves considerably by this approach, whereas the accuracy of the results is guaranteed by the well-conditioned Hessian matrix. Numerical simulations and experiments are followed by comparisons with other reconstruction methods which demonstrate the reliability and high solution speed of this approach. According to the results, the convergence of the proposed method is significantly improved, and its speed is 2–200 times higher than the previously developed methods with the same level of precision.

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.

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.

The comparison between FVM and FEM for EIT forward problem

In this paper, the finite volume method (FVM) is introduced in detail for solving the electrical impedance tomography (EIT) forward problem. A new idea for constructing the primary and secondary elements in FVM is presented. Detailed comparisons between FVM and the finite element method (FEM), including the characteristic of the coefficient matrix and the precision of the results, are carried out under the same mesh system. It is shown that accurate estimates of the potential distribution can be obtained with an FVM solution.

3-D electrical impedance tomography forward problem with finite element method

Electrical impedance tomography (EIT) is a newly developed technique by which impedance measurements from the surface of an object are reconstructed into impedance image. The two-dimensional (2-D) EIT problem is regarded as a simplified model. As 2-D model cannot physically represent the three-dimensional (3-D) structure, the spatial information of the place where the impedance is changed by some diseases cannot be detected accurately. Therefore, 3-D EIT is necessary. In this paper, the finite element method (FEM) of 3-D EIT forward problem is presented. A sphere model is studied. The tetrahedron element is used in the meshing. Two types of sphere model, uniform model and multiplayer model are analyzed. The uniform sphere model, to which the analytical solution is applicable, is used to verify the developed FEM as well as examine the accuracy. The comparison between the numerical solution and the analytical solution shows the correctness of the developed FEM for EIT forward problem. The multiplayer model, four-layer model and three-layer model, is used to investigate the physical potential distribution inside the inhomogeneous sphere model. Reasonable potential distributions are obtained.

On a Mixed Neumann-Robin Boundary Value Problem in Electrical Impedance Tomography

Electrical Impedance Tomography (EIT) is a technique which creates images of the electrical conductivity distribution of the material in the interior of a body. In EIT mathematical models of the electrodes that include a non-zero contact impedance between them and the object under investigation predict that the current density has a singular behaviour near the edge of these electrodes. In this paper we have formulated a version of the EIT forward problem in terms of a weakly singular Fredholm integral equation of the second kind. After solving this equation numerically, we have studied the behaviour of these singularities analytically for the case of a spatially varying contact impedance. Finally we have used our analytic results to construct a class of soluble models which allows the verification of numerical algorithms for modelling the EIT forward problem.