Improving the forward solver for the complete electrode model in EIT using algebraic multigrid

Supervised Descent Learning for Thoracic Electrical Impedance Tomography

OBJECTIVE

The absolute image reconstruction problem of electrical impedance tomography (EIT) is ill-posed. Traditional methods usually solve a nonlinear least squares problem with some kind of regularization. These methods suffer from low accuracy, poor anti-noise performance, and long computation time. Besides, the integration of a priori information is not very flexible. This work tries to solve EIT inverse problem using a machine learning algorithm for the application of thorax imaging.

METHODS

We developed the supervised descent learning EIT (SDL-EIT) inversion algorithm based on the idea of supervised descent method (SDM). The algorithm approximates the mapping from measured data to the conductivity image by a series of descent directions learned from training samples. We designed a training data set in which the thorax contour, and some general structure of lungs, and heart are embedded. The algorithm is implemented in both two-, and three-dimensional cases, and is evaluated using synthetic, and measured thoracic data. Results, and conclusion: For synthetic data, SDL-EIT shows better accuracy, and anti-noise performance compared with traditional Gauss-Newton inversion (GNI) method. For measured data, the result of SDL-EIT is reasonable compared with computed tomography (CT) scan image.

SIGNIFICANCE

Using SDL-EIT, prior information can be easily integrated through the specifically designed training data set, and the image reconstruction process can be accelerated. The algorithm is effective in inverting measured thoracic data. It is a potential algorithm for human thorax imaging.

Electric field simulations for transcranial brain stimulation using FEM: an efficient implementation and error analysis

OBJECTIVE

Transcranial magnetic stimulation (TMS) and transcranial electric stimulation (TES) modulate brain activity non-invasively by generating electric fields either by electromagnetic induction or by injecting currents via skin electrodes. Numerical simulations based on anatomically detailed head models of the TMS and TES electric fields can help us to understand and optimize the spatial stimulation pattern in the brain. However, most realistic simulations are still slow, and the role of anatomical fidelity on simulation accuracy has not been evaluated in detail so far.

APPROACH

We present and validate a new implementation of the finite element method (FEM) for TMS and TES that is based on modern algorithms and libraries. We also evaluate the convergence of the simulations and estimate errors stemming from numerical and modelling aspects.

MAIN RESULTS

Comparisons with analytical solutions for spherical phantoms validate our new FEM implementation, which is three to six times faster than previous implementations. The convergence results suggest that accurately capturing the tissue geometry in addition to choosing a sufficiently accurate numerical method is of fundamental importance for accurate simulations.

SIGNIFICANCE

The new implementation allows for a substantial increase in computational efficiency of FEM TMS and TES simulations. This is especially relevant for applications such as the systematic assessment of model uncertainty and the optimization of multi-electrode TES montages. The results of our systematic error analysis allow the user to select the best tradeoff between model resolution and simulation speed for a specific application. The new FEM code is openly available as a part of our open-source software SimNIBS 3.0.

Three-Dimensional Electrical Impedance Tomography With Multiplicative Regularization

OBJECTIVE

The multiplicative regularization scheme is applied to three-dimensional electrical impedance tomography (EIT) image reconstruction problem to alleviate its ill-posedness.

METHODS

A cost functional is constructed by multiplying the data misfit functional with the regularization functional. The regularization functional is based on a weighted L²-norm with the edge-preserving characteristic. Gauss-Newton method is used to minimize the cost functional. A method based on the discrete exterior calculus (DEC) theory is introduced to formulate the discrete gradient and divergence operators related to the regularization on unstructured meshes.

RESULTS

Both numerical and experimental results show good reconstruction accuracy and anti-noise performance of the algorithm. The reconstruction results using human thoracic data show promising applications in thorax imaging.

CONCLUSION

The multiplicative regularization can be applied to EIT image reconstruction with promising applications in thorax imaging.

SIGNIFICANCE

In the multiplicative regularization scheme, there is no need to set an artificial regularization parameter in the cost functional. This helps to reduce the workload related to choosing a regularization parameter which may require expertise and many numerical experiments. The DEC-based method provides a systematic and rigorous way to formulate operators on unstructured meshes. This may help EIT image reconstructions using regularizations imposing structural or spatial constraints.

A fast parallel solver for the forward problem in electrical impedance tomography

Electrical impedance tomography (EIT) is a noninvasive imaging modality, where imperceptible currents are applied to the skin and the resulting surface voltages are measured. It has the potential to distinguish between ischaemic and haemorrhagic stroke with a portable and inexpensive device. The image reconstruction relies on an accurate forward model of the experimental setup. Because of the relatively small signal in stroke EIT, the finite-element modeling requires meshes of more than 10 million elements. To study the requirements in the forward modeling in EIT and also to reduce the time for experimental image acquisition, it is necessary to reduce the run time of the forward computation. We show the implementation of a parallel forward solver for EIT using the Dune-Fem C++ library and demonstrate its performance on many CPU's of a computer cluster. For a typical EIT application a direct solver was significantly slower and not an alternative to iterative solvers with multigrid preconditioning. With this new solver, we can compute the forward solutions and the Jacobian matrix of a typical EIT application with 30 electrodes on a 15-million element mesh in less than 15 min. This makes it a valuable tool for simulation studies and EIT applications with high precision requirements. It is freely available for download.

An instrumental electrode model for solving EIT forward problems

An instrumental electrode model (IEM) capable of describing the performance of electrical impedance tomography (EIT) systems in the MHz frequency range has been proposed. Compared with the commonly used Complete Electrode Model (CEM), which assumes ideal front-end interfaces, the proposed model considers the effects of non-ideal components in the front-end circuits. This introduces an extra boundary condition in the forward model and offers a more accurate modelling for EIT systems. We have demonstrated its performance using simple geometry structures and compared the results with the CEM and full Maxwell methods. The IEM can provide a significantly more accurate approximation than the CEM in the MHz frequency range, where the full Maxwell methods are favoured over the quasi-static approximation. The improved electrode model will facilitate the future characterization and front-end design of real-world EIT systems.

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.

Electrical impedance tomography reconstruction for three-dimensional imaging of the prostate

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 closed-domain 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.

Improving the finite element forward model of the human head by warping using elastic deformation

As the use of realistic geometry in the forward model of electrical impedance tomography (EIT) of brain function appears to improve image reconstruction, the generation of patient-specific finite element meshes has been the subject of much recent work. This paper presents a more rapid method of generating more geometrically accurate finite element meshes of the human head by warping existing meshes such that the surface boundary beneath the electrodes closely matches that of the subject with minimal degradation to the quality of the mesh. Pre-existing meshes of spheres and adult head models incorporating key internal anatomical features are warped, using elastic deformation, to match a phantom latex tank incorporating a real skull. The algorithm is described and tests are carried out to optimize the key parameters to ensure minimal degradation of mesh quality and distortion of internal features. Results show that the algorithm operating with the optimum parameters produces meshes of sound quality and could represent an important step in the timely and productive creation of forward models in clinical applications.

A narrow-band level set method applied to EIT in brain for cryosurgery monitoring

In this paper, we investigate the feasibility of applying a novel level set reconstruction technique to electrical imaging of the human brain. We focus particularly on the potential application of electrical impedance tomography (EIT) to cryosurgery monitoring. In this application, cancerous tissue is treated by a local freezing technique using a small needle-like cryosurgery probe. The interface between frozen and nonfrozen tissue can be expected to have a relatively high contrast in conductivity and we treat the inverse problem of locating and monitoring this interface during the treatment. A level set method is used as a powerful and flexible tool for tracking the propagating interfaces during the monitoring process. For calculating sensitivities and the Jacobian when deforming the interfaces we employ an adjoint formula rather than a direct differentiation technique. In particular, we are using a narrow-band technique for this procedure. This combination of an adjoint technique and a narrow-band technique for calculating Jacobians results in a computationally efficient and extremely fast method for solving the inverse problem. Moreover, due to the reduced number of unknowns in each step of the narrow-band approach compared to a pixel- or voxel-based technique, our reconstruction scheme tends to be much more stable. We demonstrate that our new method also outperforms its pixel-/voxel-based counterparts in terms of image quality in this application.

Reconstruction of Layered Biological Tissues via Electrical Impedance Tomography

The reconstruction of layered biological tissues, via electrical impedance tomography, is carried out by utilizing two alternative linear transformations, namely, Legendre polynomial expansion and Fourier Bessel transform. Both transformations are performed on a recently proposed image series expansion scheme in conjunction with the WKB approximation. The resultant spectrum posses a crucially important locality feature, assigning analytically to each spectral image term a local impedance associated with a unique layer, and thereby leading to efficient and accurate novel reconstruction procedures. The quality of the reconstruction procedures is verified through numerical simulations, revealing small reconstruction errors even in the presence of noisy data.

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.

Electrical impedance tomography imaging using a priori ultrasound data

Background

Different imaging systems (e.g. electrical, magnetic, and ultrasound) rely on a wide variety of physical properties, and the datasets obtained from such systems provide only partial information about the unknown true state. One approach is to choose complementary imaging systems, and to combine the information to achieve a better representation.

Methods

This paper discusses the combination of ultrasound and electrical impedance tomography (EIT) information. Ultrasound reflection signals are good at locating sharp acoustic density changes associated with the boundaries of objects. Some boundaries, however, may be indeterminable due to masking from intermediate boundaries or because they are outside the ultrasound beam. Conversely, the EIT data contains relatively low-quality information, but it includes the whole region enclosed by the electrodes.

Results

Results are shown from a narrowband level-set method applied to 2D and 3D EIT incorporating limited angle ultrasound time of flight data.

Conclusion

The EIT reconstruction is shown to be faster and more accurate using the additional edge information from both one and four transducer ultrasound systems.

Electrical impedance tomography method for reconstruction of biological tissues with continuous plane-stratification

A novel electrical impedance tomography method is introduced for reconstruction of layered biological tissues with continuous plane-stratification. The algorithm implements the recently proposed reconstruction scheme for piecewise constant conductivity profiles, based on an improved Prony method in conjunction with Legendre polynomial expansion (LPE). It is shown that the proposed algorithm is capable of successfully reconstructing continuous conductivity profiles with moderate (WKB) slop. Features of the presented reconstruction scheme include, an inherent linearity, achieved by the linear LPE transform, a locality feature, assigning analytically to each spectral component a local electrical impedance associated with a unique location, and effective performance even in the presence of noisy measurements.