# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2021066
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## Automated filtering in the nonlinear Fourier domain of systematic artifacts in 2D electrical impedance tomography

 1 Department of Mathematics, Gonzaga University, Spokane, WA 99258, USA 2 Boeing Co., 6001 S Air Depot Blvd, Oklahoma City, OK 73135, USA 3 Department of Physics & Astronomy, Michigan State University, East Lansing, MI 48824, USA 4 College of Medicine, University of Arizona, Phoenix, AZ 85004, USA

*Corresponding author: Melody Alsaker

Received  July 2021 Revised  August 2021 Early access October 2021

For patients undergoing mechanical ventilation due to respiratory failure, 2D electrical impedance tomography (EIT) is emerging as a means to provide functional monitoring of pulmonary processes. In EIT, electrical current is applied to the body, and the internal conductivity distribution is reconstructed based on subsequent voltage measurements. However, EIT images are known to often suffer from large systematic artifacts arising from various limitations and exacerbated by the ill-posedness of the inverse problem. The direct D-bar reconstruction method admits a nonlinear Fourier analysis of the EIT problem, providing the ability to process and filter reconstructions in the nonphysical frequency regime. In this work, a technique is introduced for automated Fourier-domain filtering of known systematic artifacts in 2D D-bar reconstructions. The new method is validated using three numerically simulated static thoracic datasets with induced artifacts, plus two experimental dynamic human ventilation datasets containing systematic artifacts. Application of the method is shown to significantly reduce the appearance of artifacts and improve the shape of the lung regions in all datasets.

Citation: Melody Alsaker, Benjamin Bladow, Scott E. Campbell, Emma M. Kar. Automated filtering in the nonlinear Fourier domain of systematic artifacts in 2D electrical impedance tomography. Inverse Problems & Imaging, doi: 10.3934/ipi.2021066
##### References:
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##### References:
 [1] A. Adler, R. Guardo and Y. Berthiaume, Impedance imaging of lung ventilation: Do we need to account for chest expansion?, IEEE Tran. Biomedical Engineering, 43 (1994), 414-420.  doi: 10.1109/IEMBS.1994.411917.  Google Scholar [2] A. Adler et al., GREIT: A unified approach to 2D linear EIT reconstruction of lung images, Physiological Measurement, 30 (2009), 35pp. Google Scholar [3] G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.  doi: 10.1080/00036818808839730.  Google Scholar [4] M. Alsaker and J. Mueller, A D-bar algorithm with a priori information for 2-dimensional electrical impedance tomography, SIAM J. Imaging Sci, 9 (2016), 1619-1654.  doi: 10.1137/15M1020137.  Google Scholar [5] M. Alsaker, D. A. C. Cárdenas, S. S. Furuie and J. L. Mueller, Complementary use of priors for pulmonary imaging with electrical impedance and ultrasound computed tomography, J. Compu. Appl. Math., 395 (2021), 15pp. doi: 10.1016/j.cam.2021.113591.  Google Scholar [6] M. Alsaker and J. L. Mueller, Use of an optimized spatial prior in D-bar reconstructions of EIT tank data, Inverse Probl. Imaging, 12 (2018), 883-901.  doi: 10.3934/ipi.2018037.  Google Scholar [7] M. Alsaker, J. L. Mueller and R. Murthy, Dynamic optimized priors for D-bar reconstructions of human ventilation using electrical impedance tomography, J. Comput. Appl. Math., 362 (2019), 276-294.  doi: 10.1016/j.cam.2018.07.039.  Google Scholar [8] N. Avis and D. Barber, Incorporating a priori information into the Sheffield filtered backprojection algorithm, Physiological Measurement, 16 (1995). doi: 10.1088/0967-3334/16/3A/011.  Google Scholar [9] R. H. Bayford, Bioimpedance tomography (electrical impedance tomography), Annu. Rev. Biomed. Eng., 8 (2006), 63-91.  doi: 10.1146/annurev.bioeng.8.061505.095716.  Google Scholar [10] R. Blue, D. Isaacson and J. C. 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Imaging, 8 (2014), 1013-1031.  doi: 10.3934/ipi.2014.8.1013.  Google Scholar [19] S. Guan, A. A. Khan, S. Sikdar and P. V. Chitnis, Fully dense UNet for 2-D sparse photoacoustic tomography artifact removal, IEEE J. Biomedical and Health Informatics, 24 (2019), 568-576.  doi: 10.1109/JBHI.2019.2912935.  Google Scholar [20] R. Guardo, C. Boulay, B. Murray and M. Bertrand, An experimental study in electrical impedance tomography using backprojection reconstruction, IEEE Tran., 38 (1991), 617-627.  doi: 10.1109/10.83560.  Google Scholar [21] S. Hamilton and A. Hauptmann, Deep d-bar: Real-time electrical impedance tomography imaging with deep neural networks, IEEE Trans. Med. Imaging., 37 (2018), 2367-2377.  doi: 10.1109/TMI.2018.2828303.  Google Scholar [22] Y. 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Fan, Electrical impedance tomography in adult patients undergoing mechanical ventilation: A systematic review, J. Critical Care, 35 (2016), 33-50.  doi: 10.1016/j.jcrc.2016.04.028.  Google Scholar [27] V. Kolehmainen, M. Vauhkonen, P. Karjalainen and J. Kaipio, Assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns, Physiological Measurement, 18 (1997). doi: 10.1088/0967-3334/18/4/003.  Google Scholar [28] V. Kolehmainen, M. Lassas and P. Ola, Electrical impedance tomography problem with inaccurately known boundary and contact impedances, IEEE Trans. Med. Imaging, 27 (2006), 1404-1414.  doi: 10.1109/ISBI.2006.1625120.  Google Scholar [29] K. Lee, E. J. Woo and J. K. Seo, A fidelity-embedded regularization method for robust electrical impedance tomography, IEEE Trans. Med. Imaging, 37 (2018), 1970-1977.  doi: 10.1109/TMI.2017.2762741.  Google Scholar [30] S. Leonhardt and B. Lachmann, Electrical impedance tomography: The holy grail of ventilation and perfusion monitoring?, Intensive Care Medicine, 38 (2012), 1917-1929.  doi: 10.1007/s00134-012-2684-z.  Google Scholar [31] D. Liu and J. Du, A moving morphable components based shape reconstruction framework for electrical impedance tomography, IEEE Trans. Med. Imaging, 38 (2019), 2937-2948.  doi: 10.1109/TMI.2019.2918566.  Google Scholar [32] D. Liu, V. Kolehmainen, S. Siltanen, A.-M. Laukkanen and A. Seppänen, Nonlinear difference imaging approach to three-dimensional electrical impedance tomography in the presence of geometric modeling errors, IEEE Trans. Biomedical Engineering, 63 (2016), 1956-1965.  doi: 10.1109/TBME.2015.2509508.  Google Scholar [33] D. Liu, D. Smyl, D. Gu and J. Du, Shape-driven difference electrical impedance tomography, IEEE Trans. Med. Imaging, 39 (2020), 3801-3812.  doi: 10.1109/TMI.2020.3004806.  Google Scholar [34] S. Liu, R. Cao, Y. Huang, T. Ouypornkochagorn and J. Jia, Time sequence learning for electrical impedance tomography using bayesian spatiotemporal priors, IEEE Trans. Instrumentation and Measurement, 69 (2020), 6045-6057.  doi: 10.1109/TIM.2020.2972172.  Google Scholar [35] S. Liu, J. Jia, Y. D. Zhang and Y. Yang, Image reconstruction in electrical impedance tomography based on structure-aware sparse bayesian learning, IEEE Trans. Med. Imaging, 37 (2018), 2090-2102.  doi: 10.1109/TMI.2018.2816739.  Google Scholar [36] S. Liu, J. Jia, Y. D. Zhang and Y. Yang, Image reconstruction in electrical impedance tomography based on structure-aware sparse bayesian learning, IEEE Trans. Med. Imaging, 37 (2018), 2090-2102.  doi: 10.1109/TMI.2018.2816739.  Google Scholar [37] A. Lyckegaard, G. Johnson and P. Tafforeau, Correction of ring artifacts in X-ray tomographic images, Int. J. Tomo. Stat, 18 (2011), 1-9.   Google Scholar [38] S. Martin and C. T. M. 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Top row: a sample difference reconstruction $\sigma$ (left) of one frame from Human Dataset 1, reconstructed with no priors, and the corresponding segmented lung region (right) compared against the lung region extracted from CT data, which we take to be ground truth. Note the large conductive artifacts on the left and right sides of the reconstruction (indicated by arrows) which distort the lung boundary. Bottom row: the reconstruction $\sigma'$ which now includes a priori data, and corresponding segmented lung boundaries, showing improved sharpness of organ boundaries, but still exhibiting large artifacts
A flowchart illustrating the main steps of the proposed algorithm. Fourier data is represented by colored regions plotted in the (real or imaginary) $k$-plane, with the origin at the center. The sets $\mathcal{R}_1^{{ \mbox{Re}}}$ and $\mathcal{R}_1^{{ \mbox{Im}}}$ are denoted by the general notation $\mathcal{R}_1$
Sample sequence of left and right artifact boundaries corresponding to each of 5 total iterations of Brown linear temporal smoothing, plotted on the same axes. The original extracted artifact boundaries are plotted in lightest gray, with subsequent iterations plotted in incrementally darker grayscale. The final artifact boundaries are plotted boldly, in black.
Original a priori boundaries with no artifacts (left), the reconstruction $\sigma$ containing artifacts (center), and the modified boundaries which now include automatically extracted artifacts (right), for a sample frame from Human Dataset 1
Top row: Conductivity phantoms for Simulated Datasets A, B, and C, with added artifacts, used to generate simulated data. Bottom row: the corresponding phantoms with no artifacts, considered ground truth for quantitative validation purposes. The conductivity values used in the forward solution were $\sigma_{\mathrm{heart}} = 0.75, \sigma_{\mathrm{lungs}} = 0.25, \sigma_{\mathrm{artifacts}} = 0.9, \sigma_{\mathrm{background}} = 0.5$ S/m.
FEM meshes for Simulated Datasets A, B, and C, with added artifacts. Electrode centers are plotted using dots on the boundary
Results from simulated data. Normalized reconstructions $\sigma$ (no priors or artifact filtering, top row), $\sigma'$ (with priors, no artifact filtering, center row), and $\tilde\sigma'$ (with priors and artifact filtering, bottom row) corresponding to Simulated Datasets A, B, and C left to right as labeled. The appearance of induced lateral conductive artifacts in $\sigma$ is not sufficiently improved by the addition of priors in $\sigma'$. The reconstructions $\tilde \sigma'$ exhibit visually reduced artifact appearance by comparison
Results from simulated data. NRMSE values (top row) and SSIM values (bottom row) for Simulated Datasets A, B, and C as labeled, comparing normalized reconstructions $\sigma'$ (with priors but no artifact filtering, solid blue bars) and $\tilde \sigma'$ (with artifact filtering, striped bars), computed for both the entire domain (left column) and the true lung region (right column) in each reconstruction. We take the normalized conductivity phantom without artifacts to be ground truth. Note that smaller NRMSE values and larger SSIM values are desirable
Results from simulated data. Level set segmentation of lung boundaries for reconstructions (blue) against the true lung region used in the conductivity phantom (gray), for $\sigma'$ (with priors but no artifact filtering, top row), and $\tilde\sigma'$ (with artifact filtering, bottom row), corresponding to Simulated Datasets A, B, and C left to right as labeled. Lungs regions corresponding to filtered reconstructions $\tilde\sigma'$ exhibit greater visual fidelity to the ground truth lung shape as compared to those corresponding to reconstructions $\sigma'$
Results from simulated data. NRMSE values for the segmented lung regions for Simulated Datasets A, B, and C as labeled, comparing binary images of lung regions segmented from $\sigma'$ (with priors but no artifact filtering, solid green bars) and $\tilde \sigma'$ (with artifact filtering, striped bars), computed over the entire domain for each reconstruction. We take the binary lung region used in the conductivity phantom without artifacts to be ground truth. Note that smaller NRMSE values are desirable
Archival CT scans corresponding to Human Datasets 1 (left) and 2 (right), collected upon inspiration in the plane of the electrodes immediately following the EIT scans. We take these scans to be ground truth, and they were used to extract the a priori boundaries for each dataset. Images are displayed in DICOM orientation
Results from human data. Sequence of representative difference images corresponding to one full breathing cycle from Human Datasets 1 (top) and 2 (bottom). For each dataset, the top row gives the standard regularized D-bar reconstructions $\sigma'$ with a priori sharpening methods applied. Note the appearance of lateral conductive artifacts in the reconstructions $\sigma'$, which have not been removed by inclusion of a priori data. The bottom row for each dataset gives reconstructions after artifact filtering, showing significant reduction of artifacts and improvement of lung boundary shapes
Results from human data. Plots of the segmented lung regions for the representative reconstructions $\sigma'$ and $\tilde\sigma'$ for Human Datasets 1 (top) and 2 (bottom). In each image, the segmented lung region (blue) is plotted over the lung region extracted from the CT scan (gray). Lungs regions corresponding to filtered reconstructions $\tilde\sigma'$ exhibit greater visual fidelity to the ground truth lung shape as compared to those corresponding to reconstructions $\sigma'$
Results from human data. NRMSE values for the binary images of segmented lung regions corresponding to the representative reconstructions $\sigma'$ (with priors, no artifact filtering, solid green), and $\tilde\sigma'$ (with artifact filtering, striped) for Human Datasets 1 (top) and 2 (bottom) as compared to the lung region extracted from the CT scans for each patient, which we take to be ground truth. Smaller NRMSE values are desirable. This metric provides evidence that the reconstructions with artifact filtering exhibit greater fidelity to the ground truth lung shape
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