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Herzberg W, Rowe DB, Hauptmann A, Hamilton SJ. Graph Convolutional Networks for Model-Based Learning in Nonlinear Inverse Problems. IEEE TRANSACTIONS ON COMPUTATIONAL IMAGING 2021; 7:1341-1353. [PMID: 35873096 PMCID: PMC9307146 DOI: 10.1109/tci.2021.3132190] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
The majority of model-based learned image reconstruction methods in medical imaging have been limited to uniform domains, such as pixelated images. If the underlying model is solved on nonuniform meshes, arising from a finite element method typical for nonlinear inverse problems, interpolation and embeddings are needed. To overcome this, we present a flexible framework to extend model-based learning directly to nonuniform meshes, by interpreting the mesh as a graph and formulating our network architectures using graph convolutional neural networks. This gives rise to the proposed iterative Graph Convolutional Newton-type Method (GCNM), which includes the forward model in the solution of the inverse problem, while all updates are directly computed by the network on the problem specific mesh. We present results for Electrical Impedance Tomography, a severely ill-posed nonlinear inverse problem that is frequently solved via optimization-based methods, where the forward problem is solved by finite element methods. Results for absolute EIT imaging are compared to standard iterative methods as well as a graph residual network. We show that the GCNM has good generalizability to different domain shapes and meshes, out of distribution data as well as experimental data, from purely simulated training data and without transfer training.
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Affiliation(s)
- William Herzberg
- Department of Mathematical and Statistical Sciences; Marquette University, Milwaukee, WI 53233 USA
| | - Daniel B Rowe
- Department of Mathematical and Statistical Sciences; Marquette University, Milwaukee, WI 53233 USA
| | - Andreas Hauptmann
- Research Unit of Mathematical Sciences; University of Oulu, Oulu, Finland and with the Department of Computer Science; University College London, London, United Kingdom
| | - Sarah J Hamilton
- Department of Mathematical and Statistical Sciences; Marquette University, Milwaukee, WI 53233 USA
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2
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Measurement-Based Domain Parameter Optimization in Electrical Impedance Tomography Imaging. SENSORS 2021; 21:s21072507. [PMID: 33916751 PMCID: PMC8038345 DOI: 10.3390/s21072507] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/23/2021] [Revised: 03/29/2021] [Accepted: 03/31/2021] [Indexed: 12/22/2022]
Abstract
This paper discusses the optimization of domain parameters in electrical impedance tomography-based imaging. Precise image reconstruction requires accurate, well-correlated physical and numerical finite element method (FEM) models; thus, we employed the Nelder–Mead algorithm and a complete electrode model to evaluate the individual parameters, including the initial conductivity, electrode misplacement, and shape deformation. The optimization process was designed to calculate the parameters of the numerical model before the image reconstruction. The models were verified via simulation and experimental measurement with single source current patterns. The impact of the optimization on the above parameters was reflected in the applied image reconstruction process, where the conductivity error dropped by 6.16% and 11.58% in adjacent and opposite driving, respectively. In the shape deformation, the inhomogeneity area ratio increased by 11.0% and 48.9%; the imprecise placement of the 6th electrode was successfully optimized with adjacent driving; the conductivity error dropped by 12.69%; and the inhomogeneity localization exhibited a rise of 66.7%. The opposite driving option produces undesired duality resulting from the measurement pattern. The designed optimization process proved to be suitable for correlating the numerical and the physical models, and it also enabled us to eliminate imaging uncertainties and artifacts.
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Meaney P, Hartov A, Raynolds T, Davis C, Richter S, Schoenberger F, Geimer S, Paulsen K. Low Cost, High Performance, 16-Channel Microwave Measurement System for Tomographic Applications. SENSORS 2020; 20:s20185436. [PMID: 32971940 PMCID: PMC7570920 DOI: 10.3390/s20185436] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/17/2020] [Revised: 09/16/2020] [Accepted: 09/17/2020] [Indexed: 12/26/2022]
Abstract
We have developed a multichannel software defined radio-based transceiver measurement system for use in general microwave tomographic applications. The unit is compact enough to fit conveniently underneath the current illumination tank of the Dartmouth microwave breast imaging system. The system includes 16 channels that can both transmit and receive and it operates from 500 MHz to 2.5 GHz while measuring signals down to −140 dBm. As is the case with multichannel systems, cross-channel leakage is an important specification and must be lower than the noise floors for each receiver. This design exploits the isolation inherent when the individual receivers for each channel are physically separate; however, these challenging specifications require more involved signal isolation techniques at both the system design level and the individual, shielded component level. We describe the isolation design techniques for the critical system elements and demonstrate specification compliance at both the component and system level.
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Affiliation(s)
- Paul Meaney
- Thayer School of Engineering, Dartmouth College, Hanover, NH 03755, USA; (A.H.); (T.R.); (S.G.); (K.P.)
- Correspondence: ; Tel.: +1-603-646-3939
| | - Alexander Hartov
- Thayer School of Engineering, Dartmouth College, Hanover, NH 03755, USA; (A.H.); (T.R.); (S.G.); (K.P.)
| | - Timothy Raynolds
- Thayer School of Engineering, Dartmouth College, Hanover, NH 03755, USA; (A.H.); (T.R.); (S.G.); (K.P.)
| | | | - Sebastian Richter
- German Federal Ministry of Defense, 2E1202 Hamburg, Germany; (S.R.); (F.S.)
| | | | - Shireen Geimer
- Thayer School of Engineering, Dartmouth College, Hanover, NH 03755, USA; (A.H.); (T.R.); (S.G.); (K.P.)
| | - Keith Paulsen
- Thayer School of Engineering, Dartmouth College, Hanover, NH 03755, USA; (A.H.); (T.R.); (S.G.); (K.P.)
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4
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Muller PA, Mueller JL, Mellenthin MM. Real-Time Implementation of Calderón's Method on Subject-Specific Domains. IEEE TRANSACTIONS ON MEDICAL IMAGING 2017; 36:1868-1875. [PMID: 28436855 DOI: 10.1109/tmi.2017.2695893] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
A real-time implementation of Calderón's method for the reconstruction of a 2-D conductivity from electrical impedance tomography data is presented, in which domain-specific modeling is taken into account. This is the first implementation of Calderón's method that accounts for correct modeling of non-symmetric domain boundaries in image reconstruction. The domain-specific Calderón's method is derived and reconstructions from experimental tank data are presented, quantifying the distortion when correct modeling is not included in the reconstruction algorithm. Reconstructions from human subject volunteers are presented, demonstrating the method's effectiveness for imaging changes due to ventilation and perfusion in the human thorax.
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Hamilton SJ. EIT Imaging of admittivities with a D-bar method and spatial prior: experimental results for absolute and difference imaging. Physiol Meas 2017; 38:1176-1192. [PMID: 28530208 DOI: 10.1088/1361-6579/aa63d7] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Abstract
Electrical impedance tomography (EIT) is an emerging imaging modality that uses harmless electrical measurements taken on electrodes at a body's surface to recover information about the internal electrical conductivity and or permittivity. The image reconstruction task of EIT is a highly nonlinear inverse problem that is sensitive to noise and modeling errors making the image reconstruction task challenging. D-bar methods solve the nonlinear problem directly, bypassing the need for detailed and time-intensive forward models, to provide absolute (static) as well as time-difference EIT images. Coupling the D-bar methodology with the inclusion of high confidence a priori data results in a noise-robust regularized image reconstruction method. In this work, the a priori D-bar method for complex admittivities is demonstrated effective on experimental tank data for absolute imaging for the first time. Additionally, the method is adjusted for, and tested on, time-difference imaging scenarios. The ability of the method to be used for conductivity, permittivity, absolute as well as time-difference imaging provides the user with great flexibility without a high computational cost.
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Affiliation(s)
- S J Hamilton
- Department of Mathematics, Statistics, and Computer Science, Marquette University, Milwaukee, WI 53233, United States of America
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Liu D, Khambampati AK, Kim S, Kim KY. Multi-phase flow monitoring with electrical impedance tomography using level set based method. NUCLEAR ENGINEERING AND DESIGN 2015. [DOI: 10.1016/j.nucengdes.2015.04.023] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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Muller PA, Isaacson D, Newell JC, Saulnier GJ. Calderón's method on an elliptical domain. Physiol Meas 2013; 34:609-22. [PMID: 23719023 DOI: 10.1088/0967-3334/34/6/609] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
One possible application for electrical impedance tomography is in medical imaging where lung and heart function may be monitored. One drawback of current algorithms is that they are implemented for use in a circular domain, but a human thorax is more elliptical than circular. In this paper, a reconstruction algorithm based on the work of Calderón (1980 Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro) pp 65-75) on the inverse conductivity problem is derived for an elliptical domain. It is explained how this reconstruction algorithm uses a transformed Dirichlet-to-Neumann map. Experimental results from an elliptical tank are given to show how correct domain modelling reduces the artefacts produced by this version of Calderón's reconstruction algorithm.
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Affiliation(s)
- P A Muller
- Department of Mathematics, Rensselaer Polytechnic Institute, Troy, NY 12180, USA.
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Hamilton SJ, Mueller JL. Direct EIT reconstructions of complex admittivities on a chest-shaped domain in 2-D. IEEE TRANSACTIONS ON MEDICAL IMAGING 2013; 32:757-769. [PMID: 23314771 DOI: 10.1109/tmi.2012.2237389] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Electrical impedance tomography (EIT) is a medical imaging technique in which current is applied on electrodes on the surface of the body, the resulting voltage is measured, and an inverse problem is solved to recover the conductivity and/or permittivity in the interior. Images are then formed from the reconstructed conductivity and permittivity distributions. In the 2-D geometry, EIT is clinically useful for chest imaging. In this work, an implementation of a D-bar method for complex admittivities on a general 2-D domain is presented. In particular, reconstructions are computed on a chest-shaped domain for several realistic phantoms including a simulated pneumothorax, hyperinflation, and pleural effusion. The method demonstrates robustness in the presence of noise. Reconstructions from trigonometric and pairwise current injection patterns are included.
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Affiliation(s)
- Sarah J Hamilton
- Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA.
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9
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Kolehmainen V, Lassas M, Ola P, Siltanen S. Recovering boundary shape and conductivity in electrical impedance tomography. ACTA ACUST UNITED AC 2013. [DOI: 10.3934/ipi.2013.7.217] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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10
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Kaiboriboon K, Lüders HO, Hamaneh M, Turnbull J, Lhatoo SD. EEG source imaging in epilepsy--practicalities and pitfalls. Nat Rev Neurol 2012; 8:498-507. [PMID: 22868868 DOI: 10.1038/nrneurol.2012.150] [Citation(s) in RCA: 79] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
EEG source imaging (ESI) is a model-based imaging technique that integrates temporal and spatial components of EEG to identify the generating source of electrical potentials recorded on the scalp. Recent advances in computer technologies have made the analysis of ESI data less time-consuming, and have rekindled interest in this technique as a clinical diagnostic tool. On the basis of the available body of evidence, ESI seems to be a promising tool for epilepsy evaluation; however, the precise clinical value of ESI in presurgical evaluation of epilepsy and in localization of eloquent cortex remains to be investigated. In this Review, we describe two fundamental issues in ESI; namely, the forward and inverse problems, and their solutions. The clinical application of ESI in surgical planning for patients with medically refractory focal epilepsy, and its use in source reconstruction together with invasive recordings, is also discussed. As ESI can be used to map evoked responses, we discuss the clinical utility of this technique in cortical mapping-an essential process when planning resective surgery for brain regions that are in close proximity to eloquent cortex.
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Affiliation(s)
- Kitti Kaiboriboon
- Epilepsy Center, Neurological Institute, University Hospitals Case Medical Center, Case Western Reserve University School of Medicine, 11100 Euclid Avenue, Lakeside 3200, Cleveland, OH 44106, USA. kitti.kaiboriboon@ uhhospitals.org
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Hamilton SJ, Herrera CNL, Mueller JL, Von Herrmann A. A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2-D. INVERSE PROBLEMS 2012; 28:095005. [PMID: 23641121 PMCID: PMC3638890 DOI: 10.1088/0266-5611/28/9/095005] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
A direct reconstruction algorithm for complex conductivities in W2,∞ (Ω), where Ω is a bounded, simply connected Lipschitz domain in ℝ2, is presented. The framework is based on the uniqueness proof by Francini [Inverse Problems 20 2000], but equations relating the Dirichlet-to-Neumann to the scattering transform and the exponentially growing solutions are not present in that work, and are derived here. The algorithm constitutes the first D-bar method for the reconstruction of conductivities and permittivities in two dimensions. Reconstructions of numerically simulated chest phantoms with discontinuities at the organ boundaries are included.
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Affiliation(s)
- S J Hamilton
- Department of Mathematics, Colorado State University, USA
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12
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Forsyth J, Borsic A, Halter RJ, Hartov A, Paulsen KD. Optical breast shape capture and finite-element mesh generation for electrical impedance tomography. Physiol Meas 2011; 32:797-809. [PMID: 21646711 DOI: 10.1088/0967-3334/32/7/s05] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
X-ray mammography is the standard for breast cancer screening. The development of alternative imaging modalities is desirable because mammograms expose patients to ionizing radiation. Electrical impedance tomography (EIT) may be used to determine tissue conductivity, a property which is an indicator of cancer presence. EIT is also a low-cost imaging solution and does not involve ionizing radiation. In breast EIT, impedance measurements are made using electrodes placed on the surface of the patient's breast. The complex conductivity of the volume of the breast is estimated by a reconstruction algorithm. EIT reconstruction is a severely ill-posed inverse problem. As a result, noisy instrumentation and incorrect modelling of the electrodes and domain shape produce significant image artefacts. In this paper, we propose a method that has the potential to reduce these errors by accurately modelling the patient breast shape. A 3D hand-held optical scanner is used to acquire the breast geometry and electrode positions. We develop methods for processing the data from the scanner and producing volume meshes accurately matching the breast surface and electrode locations, which can be used for image reconstruction. We demonstrate this method for a plaster breast phantom and a human subject. Using this approach will allow patient-specific finite-element meshes to be generated which has the potential to improve the clinical value of EIT for breast cancer diagnosis.
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Affiliation(s)
- J Forsyth
- Thayer School of Engineering, Dartmouth College, 8000 Cummings Hall, Hanover, NH 03755, USA.
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13
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Xu C, Dai M, You F, Shi X, Fu F, Liu R, Dong X. An optimized strategy for real-time hemorrhage monitoring with electrical impedance tomography. Physiol Meas 2011; 32:585-98. [PMID: 21478567 DOI: 10.1088/0967-3334/32/5/007] [Citation(s) in RCA: 31] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
Delayed detection of an internal hemorrhage may result in serious disabilities and possibly death for a patient. Currently, there are no portable medical imaging instruments that are suitable for long-term monitoring of patients at risk of internal hemorrhage. Electrical impedance tomography (EIT) has the potential to monitor patients continuously as a novel functional image modality and instantly detect the occurrence of an internal hemorrhage. However, the low spatial resolution and high sensitivity to noise of this technique have limited its application in clinics. In addition, due to the circular boundary display mode used in current EIT images, it is difficult for clinicians to identify precisely which organ is bleeding using this technique. The aim of this study was to propose an optimized strategy for EIT reconstruction to promote the use of EIT for clinical studies, which mainly includes the use of anatomically accurate boundary shapes, rapid selection of optimal regularization parameters and image fusion of EIT and computed tomography images. The method was evaluated on retroperitoneal and intraperitoneal bleeding piglet data. Both traditional backprojection images and optimized images among different boundary shapes were reconstructed and compared. The experimental results demonstrated that EIT images with precise anatomical information can be reconstructed in which the image resolution and resistance to noise can be improved effectively.
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Affiliation(s)
- Canhua Xu
- Department of Biomedical Engineering, Fourth Military Medical University, Xi'an 710032, People's Republic of China
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14
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Nissinen A, Kolehmainen VP, Kaipio JP. Compensation of modelling errors due to unknown domain boundary in electrical impedance tomography. IEEE TRANSACTIONS ON MEDICAL IMAGING 2011; 30:231-242. [PMID: 20840893 DOI: 10.1109/tmi.2010.2073716] [Citation(s) in RCA: 37] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
Electrical impedance tomography is a highly unstable problem with respect to measurement and modeling errors. This instability is especially severe when absolute imaging is considered. With clinical measurements, accurate knowledge about the body shape is usually not available, and therefore an approximate model domain has to be used in the computational model. It has earlier been shown that large reconstruction artefacts result if the geometry of the model domain is incorrect. In this paper, we adapt the so-called approximation error approach to compensate for the modeling errors caused by inaccurately known body shape. This approach has previously been shown to be applicable to a variety of modeling errors, such as coarse discretization in the numerical approximation of the forward model and domain truncation. We evaluate the approach with a simulated example of thorax imaging, and also with experimental data from a laboratory setting, with absolute imaging considered in both cases. We show that the related modeling errors can be efficiently compensated for by the approximation error approach. We also show that recovery from simultaneous discretization related errors is feasible, allowing the use of computationally efficient reduced order models.
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Affiliation(s)
- Antti Nissinen
- Department of Physics and Mathematics, University of Eastern Finland, FIN-70211 Kuopio, Finland
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15
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Murphy EK, Mueller JL. Effect of domain shape modeling and measurement errors on the 2-D D-bar method for EIT. IEEE TRANSACTIONS ON MEDICAL IMAGING 2009; 28:1576-1584. [PMID: 19447702 DOI: 10.1109/tmi.2009.2021611] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
The D-bar algorithm based on Nachman's 2-D global uniqueness proof for the inverse conductivity problem (Nachman, 1996) is implemented on a chest-shaped domain. The scattering transform is computed on this chest-shaped domain using trigonometric and adjacent current patterns and the complete electrode model for the forward problem is computed with the finite element method in order to obtain simulated voltage measurements. The robustness and effectiveness of the method is demonstrated on a simulated chest with errors in input currents, output voltages, electrode placement, and domain modeling.
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17
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Seo JK, Lee J, Kim SW, Zribi H, Woo EJ. Frequency-difference electrical impedance tomography (fdEIT): algorithm development and feasibility study. Physiol Meas 2008; 29:929-44. [DOI: 10.1088/0967-3334/29/8/006] [Citation(s) in RCA: 81] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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18
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Yerworth RJ, Zhang Y, Tidswell T, Bayford RH, Holder DS. Use of statistical parametric mapping (SPM) to enhance electrical impedance tomography (EIT) image sets. Physiol Meas 2007; 28:S141-51. [PMID: 17664632 DOI: 10.1088/0967-3334/28/7/s11] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
Use of statistical parametric mapping (SPM), which is widely used in analysis of neuroimaging studies with fMRI and PET, has the potential to improve quality of EIT images for clinical use. Minimal modification to SPM is needed, but statistical analysis based on height, not extent thresholds, should be employed, due to the 20-80% variation of the point spread function, across EIT images. SPM was assessed in EIT images reconstructed with a linear time difference algorithm utilizing an anatomically realistic finite element model of the human head. Images of the average of data sets were compared with those produced using SPM over 10-40 individual image data sets without averaging. For a point disturbance, a sponge 15% of the diameter of an anatomically realistic saline-filled tank including a skull, with a contrast of 15%, and for visual evoked response data in 14 normal human volunteers, images produced with SPM were less noisy than the average images. For the human data, no consistent physiologically realistic changes were seen with either SPM or direct reconstruction; however, only a small data set was available, limiting the power of the SPM analysis. SPM may be used on EIT images and has the potential to extract improved images from clinical data series with a low signal-to-noise ratio.
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Affiliation(s)
- R J Yerworth
- Department of Medical Physics and Bioengineering, University College London, London, and Department of Clinical Neurophysiology, Addenbrooke's Hospital, Cambridge, UK
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19
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Murphy EK, Mueller JL, Newell JC. Reconstructions of conductive and insulating targets using the D-bar method on an elliptical domain. Physiol Meas 2007; 28:S101-14. [PMID: 17664628 PMCID: PMC2464779 DOI: 10.1088/0967-3334/28/7/s08] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
The D-bar algorithm based on A Nachman's 2D global uniqueness proof for the inverse conductivity problem (Nachman 1996 Ann. Math. 143 71-96) is implemented on an elliptical domain. The scattering transform is computed on an ellipse and the complete electrode model (CEM) for the forward problem is computed with the finite element method (FEM) in order to obtain static conductivity reconstructions of conductive and insulating targets in a saline-filled tank. It is demonstrated that the spatial artifacts in the image are significantly reduced when the domain is properly modeled in the reconstruction, as opposed to being modeled as a disk.
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Affiliation(s)
- E K Murphy
- Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
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20
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Yilmaz A, Akdoğan KE, Saka B. Application of conformal transformation to elliptic geometry for electric impedance tomography. Med Eng Phys 2007; 30:144-53. [PMID: 17509923 DOI: 10.1016/j.medengphy.2007.03.007] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2006] [Revised: 03/19/2007] [Accepted: 03/22/2007] [Indexed: 11/26/2022]
Abstract
Electrical impedance tomography (EIT) is a medical imaging modality that is used to compute the conductivity distribution through measurements on the cross-section of a body part. An elliptic geometry model, which defines a more general frame, ensures more accurate results in reconstruction and assessment of inhomogeneities inside. This study provides a link between the analytical solutions defined in circular and elliptical geometries on the basis of the computation of conformal mapping. The results defined as voltage distributions for the homogeneous case in elliptic and circular geometries have been compared with those obtained by the use of conformal transformation between elliptical and well-known circular geometry. The study also includes the results of the finite element method (FEM) as another approach for more complex geometries for the comparison of performance in other complex scenarios for eccentric inhomogeneities. The study emphasizes that for the elliptic case the analytical solution with conformal transformation is a reliable and useful tool for developing insight into more complex forms including eccentric inhomogeneities.
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Affiliation(s)
- Atila Yilmaz
- Electrical & Electronics Engineering Department, Hacettepe University, 06800 Beytepe, Ankara, Turkey.
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21
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Hwan Choi M, Kao TJ, Isaacson D, Saulnier GJ, Newell JC. A reconstruction algorithm for breast cancer imaging with electrical impedance tomography in mammography geometry. IEEE Trans Biomed Eng 2007; 54:700-10. [PMID: 17405377 PMCID: PMC2759944 DOI: 10.1109/tbme.2006.890139] [Citation(s) in RCA: 64] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
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.
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Affiliation(s)
- Myoung Hwan Choi
- Department of Electrical and Electronics Engineering, Kangwon
National University, 192-1, Hyoza 2 dong, Chunchon, Kangwondo, Korea
(e-mail: )
| | - Tzu-Jen Kao
- Department of Biomedical Engineering, Rensselaer Polytechnic
Institute, Troy, NY 12180 USA (e-mail: )
| | - David Isaacson
- Department of Mathematical Sciences, Rensselaer Polytechnic
Institute, Troy, NY 12180 USA (e-mail: )
| | - Gary J. Saulnier
- Department of Electrical, Computer, and Systems Engineering,
Rensselaer Polytechnic Institute, Troy, NY 12180 USA (e-mail:
)
| | - Jonathan C. Newell
- Department of Biomedical Engineering, Rensselaer Polytechnic
Institute, Troy, NY 12180 USA (e-mail: )
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Babaeizadeh S, Brooks DH, Isaacson D. 3-D Electrical Impedance Tomography for Piecewise Constant Domains With Known Internal Boundaries. IEEE Trans Biomed Eng 2007; 54:2-10. [PMID: 17260850 DOI: 10.1109/tbme.2006.886839] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Abstract
Electrical impedance tomography (EIT) is a badly posed inverse problem, but can be stabilized if one assumes that the conductivity is piecewise constant, with a relatively small number of distinct regions, and that the region boundaries are known, for example from prior anatomical imaging. With this assumption, we introduce a three-dimensional (3-D) boundary element method (BEM) model for the forward EIT map from injected currents to measured voltages, and 3-D inverse solutions for both BEM and the finite element method (FEM) which explicitly take into account the parameterization implied by the known boundary locations. We develop expressions for the Jacobians for both methods, since they are nonlinear, to more rapidly solve the inverse problem. We show simulation results in a torso geometry with the heart and lungs as inhomogeneities. In a simulation study, we could reconstruct the conductive values of some internal organs of a human torso with more than 92% accuracy even with inaccurate internal boundary locations, a randomized rather than constant conductivity profile (with the standard deviation of the Gaussian-distributed conductivities set to 20% of their mean values), signal to measurement noise of 50 dB, and with different meshes used for the forward and inverse problems. BEM and FEM perform similarly, leading to the conclusion that the choice between them should be based on secondary considerations such as computational efficiency or the need to model conductivity anisotropies.
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Affiliation(s)
- Saeed Babaeizadeh
- Advanced Algorithm Research Center (AARC), Philips Medical, Thousand Oaks, CA 91320, USA.
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Babaeizadeh S, Brooks DH, Isaacson D, Newell JC. Electrode boundary conditions and experimental validation for BEM-based EIT forward and inverse solutions. IEEE TRANSACTIONS ON MEDICAL IMAGING 2006; 25:1180-8. [PMID: 16967803 DOI: 10.1109/tmi.2006.879957] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/11/2023]
Abstract
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.
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Affiliation(s)
- Saeed Babaeizadeh
- Advanced Algorithm Research Center (AARC), Philips Medical, Thousand Oaks, CA 91320, USA.
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Gao N, Zhu SA, He B. Estimation of electrical conductivity distribution within the human head from magnetic flux density measurement. Phys Med Biol 2005; 50:2675-87. [PMID: 15901962 DOI: 10.1088/0031-9155/50/11/016] [Citation(s) in RCA: 46] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
We have developed a new algorithm for magnetic resonance electrical impedance tomography (MREIT), which uses only one component of the magnetic flux density to reconstruct the electrical conductivity distribution within the body. The radial basis function (RBF) network and simplex method are used in the present approach to estimate the conductivity distribution by minimizing the errors between the 'measured' and model-predicted magnetic flux densities. Computer simulations were conducted in a realistic-geometry head model to test the feasibility of the proposed approach. Single-variable and three-variable simulations were performed to estimate the brain-skull conductivity ratio and the conductivity values of the brain, skull and scalp layers. When SNR = 15 for magnetic flux density measurements with the target skull-to-brain conductivity ratio being 1/15, the relative error (RE) between the target and estimated conductivity was 0.0737 +/- 0.0746 in the single-variable simulations. In the three-variable simulations, the RE was 0.1676 +/- 0.0317. Effects of electrode position uncertainty were also assessed by computer simulations. The present promising results suggest the feasibility of estimating important conductivity values within the head from noninvasive magnetic flux density measurements.
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Affiliation(s)
- Nuo Gao
- College of Electrical Engineering, Zhejiang University, Hangzhou 310027, People's Republic of China
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Saka B, Yilmaz A. Elliptic Cylinder Geometry for Distinguishability Analysis in Impedance Tomography. IEEE Trans Biomed Eng 2004; 51:126-32. [PMID: 14723501 DOI: 10.1109/tbme.2003.820335] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
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.
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Affiliation(s)
- Birsen Saka
- Department of Electrical and Electronics Engineering, Hacettepe University, 06532 Ankara, Turkey.
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Bagshaw AP, Liston AD, Bayford RH, Tizzard A, Gibson AP, Tidswell AT, Sparkes MK, Dehghani H, Binnie CD, Holder DS. Electrical impedance tomography of human brain function using reconstruction algorithms based on the finite element method. Neuroimage 2003; 20:752-64. [PMID: 14568449 DOI: 10.1016/s1053-8119(03)00301-x] [Citation(s) in RCA: 64] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2002] [Revised: 04/17/2003] [Accepted: 05/01/2003] [Indexed: 10/27/2022] Open
Abstract
Electrical impedance tomography (EIT) is a recently developed technique which enables the internal conductivity of an object to be imaged using rings of external electrodes. In a recent study, EIT during cortical evoked responses showed encouraging changes in the raw impedance measurements, but reconstructed images were noisy. A simplified reconstruction algorithm was used which modelled the head as a homogeneous sphere. In the current study, the development and validation of an improved reconstruction algorithm are described in which realistic geometry and conductivity distributions have been incorporated using the finite element method. Data from computer simulations and spherical or head-shaped saline-filled tank phantoms, in which the skull was represented by a concentric shell of plaster of Paris or a real human skull, have been reconstructed into images. There were significant improvements in image quality as a result of the incorporation of accurate geometry and extracerebral layers in the reconstruction algorithm. Image quality, assessed by blinded subjective expert observers, also improved significantly when data from the previous evoked response study were reanalysed with the new algorithm. In preliminary images collected during epileptic seizures, the new algorithm generated EIT conductivity changes which were consistent with the electrographic ictal activity. Incorporation of realistic geometry and conductivity into the reconstruction algorithm significantly improves the quality of EIT images and lends encouragement to the belief that EIT may provide a low-cost, portable functional neuroimaging system in the foreseeable future.
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Affiliation(s)
- Andrew P Bagshaw
- Department of Clinical Neurophysiology, University College London, UK
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Tang M, Wang W, Wheeler J, McCormick M, Dong X. Effects of incompatible boundary information in EIT on the convergence behavior of an iterative algorithm. IEEE TRANSACTIONS ON MEDICAL IMAGING 2002; 21:620-628. [PMID: 12166858 DOI: 10.1109/tmi.2002.800588] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
Abstract
In electrical impedance tomography, currents are applied to the body through electrodes that are attached to the surface and the corresponding surface voltages are measured. Based on these boundary measurements, the internal admittivity distribution of the body can be reconstructed. In order to improve the image quality it is necessary and useful to apply physiologically meaningful prior information into the image reconstruction. Such prior information usually can be obtained from other sources. For example, information on the object's boundary shape and internal structure can be obtained from computed tomography and magnetic resonance imaging scan. However, this type of prior information may change from time to time and from person to person. As these changes are limited anatomically and physiologically, the prior information including the possible changes can be presented in a number of variational forms. The aim of this paper is to find which form of prior information is more compatible for a specific imaged object at the time of imaging. This paper proposes a new method for selecting the most appropriate form of prior information, through the procedure of iterative image reconstruction by using the information obtained from boundary measurements. The method is based on the principle that incompatible prior information causes errors which are able to affect the image reconstruction's convergence behavior. In this method, according to the various forms of prior information available, several image reconstruction configurations are designed. Then, through monitoring the convergence behavior in an iterative image reconstruction, the configuration with compatible prior information can be found among those different configurations. As an example, the prior information regarding the imaged object's boundary shape and internal structure was studied by computer simulation. Results were shown and discussed.
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Affiliation(s)
- Mengxing Tang
- 3D Imaging/Biomedical Engineering, Faculty of Computing Science and Engineering, De Montfort University, Leicester, UK.
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Levy S, Adam D, Bresler Y. Electromagnetic impedance tomography (EMIT): a new method for impedance imaging. IEEE TRANSACTIONS ON MEDICAL IMAGING 2002; 21:676-687. [PMID: 12166865 DOI: 10.1109/tmi.2002.800573] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
Abstract
We propose a new impedance imaging method, electromagnetic impedance tomography (EMIT), in which the boundary electric potential measurements in electrical impedance tomography (EIT) are augmented by measurements of the exterior magnetic field induced by the currents excited in the object by the standard EIT procedures. These magnetic measurements can be obtained reliably and inexpensively by simple pickup coils located around the imaged cross section. We derive expressions for the forward problem and for the Jacobian of the measurements, and propose an iterative reconstruction algorithm using a squared error cost function. The performance of EMIT and EIT is compared in numerical simulations using a finite-element model for the conductivity distribution of several phantoms. Evaluation of the rank and condition of the Jacobian demonstrates that the additional magnetic measurements provided by a few pickup coils in EMIT turn an underdetermined EIT problem into a well-posed one with reasonable condition, or significantly improve the conditioning of the EIT problem when it is already fully determined. Reconstructions of various phantoms reveal that EMIT provides particularly significant visual and quantitative improvement (threefold to tenfold reduction in the root-mean-squared error) in the sensitivity at the center of the object, which is the area most difficult to image using EIT.
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Affiliation(s)
- Shai Levy
- Department of Bioengineering, Technion, Israel Institute of Technology, Haifa
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Molinari M, Cox SJ, Blott BH, Daniell GJ. Comparison of algorithms for non-linear inverse 3D electrical tomography reconstruction. Physiol Meas 2002; 23:95-104. [PMID: 11876245 DOI: 10.1088/0967-3334/23/1/309] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
Non-linear electrical impedance tomography reconstruction algorithms usually employ the Newton-Raphson iteration scheme to image the conductivity distribution inside the body. For complex 3D problems, the application of this method is not feasible any more due to the large matrices involved and their high storage requirements. In this paper we demonstrate the suitability of an alternative conjugate gradient reconstruction algorithm for 3D tomographic imaging incorporating adaptive mesh refinement and requiring less storage space than the Newton-Raphson scheme. We compare the reconstruction efficiency of both algorithms for a simple 3D head model. The results show that an increase in speed of about 30% is achievable with the conjugate gradient-based method without loss of accuracy.
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Affiliation(s)
- Marc Molinari
- Department of Electronics and Computer Science, University of Southampton, UK
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Abstract
Electrical impedance tomography is a technology for producing images of internal body structures based upon electrical measurements made from electrodes on the body surface. Typically a single plane of electrodes is used, seeking to reconstruct a cross section of the body. Yet the majority of image reconstruction algorithms ignore the three-dimensional (3D) characteristics of the current flow in the body. Actually, a substantial amount of current flows out of the electrode plane, creating distortions in the resulting images. This paper describes a reconstruction algorithm, ToDLeR, for solving a linearized 3D inverse problem in impedance imaging. The algorithm models the body as a homogeneous cylinder and accounts for the 3D current flow in the body by analytically solving for the current flow from one or more layers of electrodes on the surface of the cylinder. The algorithm was implemented on the ACT3 real-time imaging system and data were collected from a 3D test phantom using one, two and four layers of electrodes. By using multiple planes of electrodes, improved accuracy in any particular electrode plane was obtained, with decreased sensitivity to out-of-plane objects. A cylindrical target located vertically more than 8 cm below a single layer of 16 electrodes, and positioned radially midway between the centre and the boundary, produced an image that had 35% of the value obtained when the target was in the electrode plane. By adding an additional layer of 16 electrodes below the first electrode plane, and using 3D current patterns, this artefact was reduced to less than 10% of the peak value. We conclude that the 3D algorithm, used with multiple planes of electrodes, reduces the distortions from out-of-plane structures in the body.
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Affiliation(s)
- R S Blue
- Biomedical Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA.
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