{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T06:54:05Z","timestamp":1768287245024,"version":"3.49.0"},"reference-count":54,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,1,22]],"date-time":"2021-01-22T00:00:00Z","timestamp":1611273600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Chalmers Foundation Excellence","award":["Not Applicable"],"award-info":[{"award-number":["Not Applicable"]}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["R01-CA240760"],"award-info":[{"award-number":["R01-CA240760"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper focuses on the construction of the Jacobian matrix required in tomographic reconstruction algorithms. In microwave tomography, computing the forward solutions during the iterative reconstruction process impacts the accuracy and computational efficiency. Towards this end, we have applied the discrete dipole approximation for the forward solutions with significant time savings. However, while we have discovered that the imaging problem configuration can dramatically impact the computation time required for the forward solver, it can be equally beneficial in constructing the Jacobian matrix calculated in iterative image reconstruction algorithms. Key to this implementation, we propose to use the same simulation grid for both the forward and imaging domain discretizations for the discrete dipole approximation solutions and report in detail the theoretical aspects for this localization. In this way, the computational cost of the nodal adjoint method decreases by several orders of magnitude. Our investigations show that this expansion is a significant enhancement compared to previous implementations and results in a rapid calculation of the Jacobian matrix with a high level of accuracy. The discrete dipole approximation and the newly efficient Jacobian matrices are effectively implemented to produce quantitative images of the simplified breast phantom from the microwave imaging system.<\/jats:p>","DOI":"10.3390\/s21030729","type":"journal-article","created":{"date-parts":[[2021,1,22]],"date-time":"2021-01-22T03:31:30Z","timestamp":1611286290000},"page":"729","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Expansion of the Nodal-Adjoint Method for Simple and Efficient Computation of the 2D Tomographic Imaging Jacobian Matrix"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5223-1917","authenticated-orcid":false,"given":"Samar","family":"Hosseinzadegan","sequence":"first","affiliation":[{"name":"Electrical Engineering Department, Chalmers University of Technology, 41296 Gothenburg, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9858-1688","authenticated-orcid":false,"given":"Andreas","family":"Fhager","sequence":"additional","affiliation":[{"name":"Electrical Engineering Department, Chalmers University of Technology, 41296 Gothenburg, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mikael","family":"Persson","sequence":"additional","affiliation":[{"name":"Electrical Engineering Department, Chalmers University of Technology, 41296 Gothenburg, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9578-9996","authenticated-orcid":false,"given":"Shireen","family":"Geimer","sequence":"additional","affiliation":[{"name":"Thayer School of Engineering, Dartmouth College, Hanover, NH 03755, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3852-5203","authenticated-orcid":false,"given":"Paul","family":"Meaney","sequence":"additional","affiliation":[{"name":"Thayer School of Engineering, Dartmouth College, Hanover, NH 03755, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"033502","DOI":"10.1117\/1.JMI.3.3.033502","article-title":"MARIA M4: Clinical evaluation of a prototype ultrawideband radar scanner for breast cancer detection","volume":"3","author":"Preece","year":"2016","journal-title":"J. Med. Imaging"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"R35","DOI":"10.1186\/bcr3418","article-title":"Microwave imaging for neoadjuvant chemotherapy monitoring: Initial clinical experience","volume":"15","author":"Meaney","year":"2013","journal-title":"Breast Cancer Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"894","DOI":"10.1109\/TBME.2009.2036372","article-title":"A wideband microwave tomography system with a novel frequency selection procedure","volume":"57","author":"Gilmore","year":"2009","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1109\/75.808040","article-title":"Microwave system for breast tumor detection","volume":"9","author":"Fear","year":"1999","journal-title":"IEEE Microw. Guid. Wave Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"254830","DOI":"10.1155\/2008\/254830","article-title":"Microwave tomography for brain imaging: Feasibility assessment for stroke detection","volume":"2008","author":"Semenov","year":"2008","journal-title":"Int. J. Antennas Propag."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2806","DOI":"10.1109\/TBME.2014.2330554","article-title":"Microwave-based stroke diagnosis making global prehospital thrombolytic treatment possible","volume":"61","author":"Persson","year":"2014","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"649612","DOI":"10.1155\/2012\/649612","article-title":"Bone dielectric property variation as a function of mineralization at microwave frequencies","volume":"2012","author":"Meaney","year":"2012","journal-title":"Int. J. Biomed. Imaging"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"253702","DOI":"10.1063\/1.4885087","article-title":"Complex permittivities of breast tumor tissues obtained from cancer surgeries","volume":"104","author":"Sugitani","year":"2014","journal-title":"Appl. Phys. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Kaufman, Z., Paran, H., Haas, I., Malinger, P., Zehavi, T., Karni, T., Pappo, I., Sandbank, J., Diment, J., and Allweis, T. (2016). Mapping breast tissue types by miniature radio-frequency near-field spectroscopy sensor in ex-vivo freshly excised specimens. BMC Med Imaging, 16.","DOI":"10.1186\/s12880-016-0160-x"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1111\/1759-7714.12605","article-title":"Dielectric properties for non-invasive detection of normal, benign, and malignant breast tissues using microwave theories","volume":"9","author":"Cheng","year":"2018","journal-title":"Thorac. Cancer"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2271","DOI":"10.1088\/0031-9155\/41\/11\/003","article-title":"The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues","volume":"41","author":"Gabriel","year":"1996","journal-title":"Phys. Med. Biol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1080\/0265673031000140822","article-title":"Microwave thermal imaging: Initial in vivo experience with a single heating zone","volume":"19","author":"Meaney","year":"2003","journal-title":"Int. J. Hyperth."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1787","DOI":"10.1109\/TBME.2014.2307072","article-title":"Real-time microwave imaging of differential temperature for thermal therapy monitoring","volume":"61","author":"Haynes","year":"2014","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Ley, S., Schilling, S., Fiser, O., Vrba, J., Sachs, J., and Helbig, M. (2019). Ultra-wideband temperature dependent dielectric spectroscopy of porcine tissue and blood in the microwave frequency range. Sensors, 19.","DOI":"10.3390\/s19071707"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1692","DOI":"10.1109\/TAP.2009.2019856","article-title":"Radar-based breast cancer detection using a hemispherical antenna array\u2014Experimental results","volume":"57","author":"Klemm","year":"2009","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"055011","DOI":"10.1088\/0266-5611\/26\/5\/055011","article-title":"Two-dimensional near-field microwave holography","volume":"26","author":"Ravan","year":"2010","journal-title":"Inverse Probl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1109\/JERM.2018.2841380","article-title":"Real-Time Microwave Imaging of a Compressed Breast Phantom with Planar Scanning","volume":"2","author":"Tajik","year":"2018","journal-title":"IEEE J. Electromagn. RF Microwaves Med. Biol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4210","DOI":"10.1118\/1.3443569","article-title":"Three-dimensional microwave imaging of realistic numerical breast phantoms via a multiple-frequency inverse scattering technique","volume":"37","author":"Shea","year":"2010","journal-title":"Med. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"024002","DOI":"10.1088\/0266-5611\/25\/2\/024002","article-title":"3D microwave imaging via preliminary support reconstruction: Testing on the Fresnel 2008 database","volume":"25","author":"Catapano","year":"2009","journal-title":"Inverse Probl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1195","DOI":"10.1109\/TBME.2014.2381270","article-title":"Wavelet-based regularization for robust microwave imaging in medical applications","volume":"62","author":"Scapaticci","year":"2015","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2454","DOI":"10.1109\/TMTT.2007.908670","article-title":"Using a priori data to improve the reconstruction of small objects in microwave tomography","volume":"55","author":"Fhager","year":"2007","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1610","DOI":"10.1109\/LAWP.2012.2236293","article-title":"MRI-derived 3-D-printed breast phantom for microwave breast imaging validation","volume":"11","author":"Burfeindt","year":"2012","journal-title":"IEEE Antennas Wirel. Propag. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Karadima, O., Rahman, M., Sotiriou, I., Ghavami, N., Lu, P., Ahsan, S., and Kosmas, P. (2020). Experimental Validation of Microwave Tomography with the DBIM-TwIST Algorithm for Brain Stroke Detection and Classification. Sensors, 20.","DOI":"10.3390\/s20030840"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2284","DOI":"10.1109\/TMTT.2005.850459","article-title":"Microwave-tomographic imaging of the high dielectric-contrast objects using different image-reconstruction approaches","volume":"53","author":"Semenov","year":"2005","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2341","DOI":"10.1109\/TAP.2009.2024478","article-title":"Comparison of an enhanced distorted born iterative method and the multiplicative-regularized contrast source inversion method","volume":"57","author":"Gilmore","year":"2009","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1109\/22.654920","article-title":"Near-field microwave imaging of biologically-based materials using a monopole transceiver system","volume":"46","author":"Meaney","year":"1998","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4239","DOI":"10.1002\/mp.12384","article-title":"Two-step inversion in microwave imaging with a logarithmic transformation","volume":"44","author":"Meaney","year":"2017","journal-title":"Med. Phys"},{"key":"ref_28","unstructured":"Semenov, S.Y. (2018). Electromagnetic Tomography for Human Brain Imaging, IEEE CAMA."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1596","DOI":"10.1109\/36.934091","article-title":"On the local minima in a tomographic imaging technique","volume":"39","author":"Isernia","year":"2001","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1895","DOI":"10.1109\/TAP.2007.898647","article-title":"On the effect of support estimation and of a new model in 2-D inverse scattering problems","volume":"55","author":"Catapano","year":"2007","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1017\/S1759078718000363","article-title":"A multiport vector network analyzer with high-precision and realtime capabilities for brain imaging and stroke detection","volume":"10","author":"Poltschak","year":"2018","journal-title":"Int. J. Microw. Wirel. Technol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1109\/JPROC.1998.658762","article-title":"Cramming more components onto integrated circuits","volume":"86","author":"Moore","year":"1998","journal-title":"Proc. IEEE"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1088\/0266-5611\/13\/3\/012","article-title":"Some Newton-type methods for the regularization of nonlinear ill-posed problems","volume":"13","author":"Kaltenbacher","year":"1997","journal-title":"Inverse Probl."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1654","DOI":"10.1109\/22.734548","article-title":"Microwave tomography: A two-dimensional Newton iterative scheme","volume":"46","author":"Souvorov","year":"1998","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1109\/36.905242","article-title":"Inverse scattering of two-dimensional dielectric objects buried in a lossy earth using the distorted Born iterative method","volume":"39","author":"Cui","year":"2001","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"115010","DOI":"10.1088\/0266-5611\/26\/11\/115010","article-title":"Finite-element contrast source inversion method for microwave imaging","volume":"26","author":"Zakaria","year":"2010","journal-title":"Inverse Probl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1607","DOI":"10.1088\/0266-5611\/13\/6\/013","article-title":"A contrast source inversion method","volume":"13","author":"Kleinman","year":"1997","journal-title":"Inverse Probl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2320","DOI":"10.1109\/TAP.2007.901993","article-title":"Nonlinear microwave imaging for breast-cancer screening using Gauss\u2013Newton\u2019s method and the CGLS inversion algorithm","volume":"55","author":"Meaney","year":"2007","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1109\/42.913177","article-title":"Microwave image reconstruction utilizing log-magnitude and unwrapped phase to improve high-contrast object recovery","volume":"20","author":"Meaney","year":"2001","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1742","DOI":"10.1109\/8.121595","article-title":"Inverse scattering: An iterative numerical method for electromagnetic imaging","volume":"39","author":"Joachimowicz","year":"1991","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1080\/21681163.2013.776268","article-title":"2D Fused image reconstruction approach for microwave tomography: A theoretical assessment using the FDTD model","volume":"1","author":"Bindu","year":"2013","journal-title":"Comput. Methods Biomech. Biomed. Eng. Imaging Vis."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"9014969","DOI":"10.1155\/2019\/9014969","article-title":"A Discrete Dipole Approximation Solver Based on the COCG-FFT Algorithm and Its Application to Microwave Breast Imaging","volume":"2019","author":"Hosseinzadegan","year":"2019","journal-title":"Int. J. Antennas Propag."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1109\/TAP.2009.2037691","article-title":"Viable three-dimensional medical microwave tomography: Theory and numerical experiments","volume":"58","author":"Fang","year":"2010","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"8026","DOI":"10.1364\/AO.34.008026","article-title":"Photon-measurement density functions. Part 2: Finite-element-method calculations","volume":"34","author":"Arridge","year":"1995","journal-title":"Appl. Opt."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1871","DOI":"10.1088\/0957-0233\/13\/12\/310","article-title":"A Matlab toolkit for three-dimensional electrical impedance tomography: A contribution to the Electrical Impedance and Diffuse Optical Reconstruction Software project","volume":"13","author":"Polydorides","year":"2002","journal-title":"Meas. Sci. Technol."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1109\/TMI.2004.824152","article-title":"Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation","volume":"23","author":"Fang","year":"2004","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1002\/cnm.1162","article-title":"Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction","volume":"25","author":"Dehghani","year":"2009","journal-title":"Commun. Numer. Methods Eng."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1109\/TMI.2014.2342719","article-title":"Real-time electrical impedance variations in women with and without breast cancer","volume":"34","author":"Halter","year":"2014","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_49","unstructured":"Lynch, D.R. (2004). Numerical Partial Differential Equations for Environmental Scientists and Engineers: A First Practical Course, Springer Science & Business Media."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1109\/8.371992","article-title":"Two-dimensional hybrid element image reconstruction for TM illumination","volume":"43","author":"Meaney","year":"1995","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1109\/JERM.2018.2882689","article-title":"Application of Two-Dimensional Discrete Dipole Approximation in Simulating Electric Field of a Microwave Breast Imaging System","volume":"3","author":"Hosseinzadegan","year":"2019","journal-title":"IEEE J. Electromagn. RF Microwaves Med. Biol."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"2014","DOI":"10.1118\/1.2737264","article-title":"Log transformation benefits parameter estimation in microwave tomographic imaging","volume":"34","author":"Meaney","year":"2007","journal-title":"Med. Phys."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1109\/JERM.2017.2761018","article-title":"A First Evaluation of the Realistic Supelec-Breast Phantom","volume":"1","author":"Rydholm","year":"2017","journal-title":"IEEE J. Electromagn. RF Microwaves Med. Biol."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1376","DOI":"10.1109\/TMTT.2012.2237181","article-title":"A near-field dual polarized (TE\u2013TM) microwave imaging system","volume":"61","author":"Ostadrahimi","year":"2013","journal-title":"IEEE Trans. Microw. Theory Tech."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/3\/729\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:13:49Z","timestamp":1760159629000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/3\/729"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,22]]},"references-count":54,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["s21030729"],"URL":"https:\/\/doi.org\/10.3390\/s21030729","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,22]]}}}