{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:29:13Z","timestamp":1760149753087,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"18","license":[{"start":{"date-parts":[[2023,9,15]],"date-time":"2023-09-15T00:00:00Z","timestamp":1694736000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62104227"],"award-info":[{"award-number":["62104227"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Insert gradient coils with similar imaging body shapes typically have smaller dimensions and higher spatial efficiency. This often allows the gradient coils the achievement of stronger and faster gradient fields. Thus, improving existing methods to make them applicable to the design of MRI gradient coils on complex surfaces has also become a challenge. This article proposes an algorithm that smooths the implicitly expressed stream function based on the intrinsic surface Laplace\u2013Beltrami operator. This algorithm can be used to simplify the design procedure of MRI gradient coils on non-developable surfaces. The following steps are performed by the proposed algorithm: an initial design of the stream function configuration, extraction of the surface mesh, discretization of the surface smoothing operator, and a smoothing of the contour lines. To evaluate the quality of the smoothed streamline configuration, several technical parameter metrics\u2014including magnetic field accuracy, coil power consumption, theoretical minimum wire spacing, and the maximum curvature of the contour lines\u2014were evaluated. The proposed method was successfully validated in a design gradient coil on both developable and non-developable surfaces. All examples evolved from an initial value with a locally non-smooth and complex topological configuration to a smooth result while maintaining high magnetic field accuracy.<\/jats:p>","DOI":"10.3390\/s23187912","type":"journal-article","created":{"date-parts":[[2023,9,17]],"date-time":"2023-09-17T23:57:46Z","timestamp":1694995066000},"page":"7912","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Stream Function Smoothing Method for the Design of MRI Gradient Coils on Non-Developable Surfaces"],"prefix":"10.3390","volume":"23","author":[{"given":"Bohan","family":"Yang","sequence":"first","affiliation":[{"name":"Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"},{"name":"China School of Optoelectronics, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hao","family":"Ren","sequence":"additional","affiliation":[{"name":"Department of Mechanical Systems Engineering, Graduate School of Engineering, Nagoya University, Nagoya 464-8601, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tongxing","family":"Zuo","sequence":"additional","affiliation":[{"name":"Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"},{"name":"China School of Optoelectronics, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhenyu","family":"Liu","sequence":"additional","affiliation":[{"name":"Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"},{"name":"China School of Optoelectronics, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1002\/cmr.a.20163","article-title":"Theory of gradient coil design methods for magnetic resonance imaging","volume":"36A","year":"2010","journal-title":"Concepts Magn. Reson. Part A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1016\/0730-725X(93)90209-V","article-title":"Gradient coil design: A review of methods","volume":"11","author":"Turner","year":"1993","journal-title":"Magn. Reson. Imaging"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1007\/s00158-013-0992-8","article-title":"Design multiple-layer gradient coils using least-squares finite element method","volume":"49","author":"Jia","year":"2014","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1002\/cmr.b.20040","article-title":"A stream function method for gradient coil design","volume":"26B","author":"Lemdiasov","year":"2010","journal-title":"Concepts Magn. Reson. Part B Magn. Reson. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1002\/1522-2594(200103)45:3<505::AID-MRM1066>3.0.CO;2-H","article-title":"Stream function optimization for gradient coil design","volume":"45","author":"Tomasi","year":"2001","journal-title":"Magn. Reson. Med."},{"key":"ref_6","first-page":"341","article-title":"Design of super-elliptical gradient coils based on multiple objective Pareto optimization method","volume":"66","author":"Pan","year":"2017","journal-title":"Acta Phys. Sin."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"238301","DOI":"10.7498\/aps.63.238301","article-title":"Design of gradient coils on super-elliptical cylindrical surfaces","volume":"63","author":"Wang","year":"2014","journal-title":"Acta Phys. Sin."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1002\/jmri.28421","article-title":"Advancements in Gradient System Performance for Clinical and Research MRI","volume":"57","author":"Gudino","year":"2023","journal-title":"J. Magn. Reson. Imaging"},{"key":"ref_9","unstructured":"Ren, H. (2019). Design Method of Magnetic Resonance System Gradient Coil on non-Developable Surface. [Master\u2019s Thesis, University of Chinese Academy of Sciences (Changchun Institute of Optics, Fine Mechanics and Physics)]."},{"key":"ref_10","unstructured":"Tikhonov, A.N., and Arsenin, V.I.A.K. (1977). Solutions of Ill-Posed Problems, Winston Publishing."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Engl, H.W., and Ramlau, R. (2015). Regularization of Inverse Problems, Springer.","DOI":"10.1007\/978-3-540-70529-1_52"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1052","DOI":"10.1109\/TMAG.2009.2037753","article-title":"MRI Coil Design Using Boundary-Element Method with Regularization Technique: A Numerical Calculation Study","volume":"46","author":"Shou","year":"2010","journal-title":"IEEE Trans. Magn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.mrl.2022.05.001","article-title":"Accurate surface normal representation to facilitate gradient coil optimization on curved surface","volume":"3","author":"Ren","year":"2023","journal-title":"Magn. Reson. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1016\/S0377-0427(00)00414-3","article-title":"Tikhonov regularization and the L-curve for large discrete ill-posed problems","volume":"123","author":"Calvetti","year":"2000","journal-title":"J. Comput. Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Engl, H.W., Hanke, M., and Neubauer, A. (1996). Regularization of Inverse Problems, Springer Science & Business Media.","DOI":"10.1007\/978-94-009-1740-8"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1007\/BF01214002","article-title":"Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima","volume":"16","author":"Sigmund","year":"1998","journal-title":"Struct. Optim."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Desbrun, M., Meyer, M., Schr\u00f6der, P., and Barr, A.H. (1999, January 8\u201313). Implicit fairing of irregular meshes using diffusion and curvature flow. Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, Los Angeles, CA, USA.","DOI":"10.1145\/311535.311576"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1006\/gmip.1997.0442","article-title":"Intrinsic Scale Space for Images on Surfaces: The Geodesic Curvature Flow","volume":"59","author":"Kimmel","year":"1997","journal-title":"Graph. Models Image Process."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Rosenberg, S. (1997). The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds, Cambridge University Press.","DOI":"10.1017\/CBO9780511623783"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1145\/588272.588276","article-title":"Anisotropic diffusion of surfaces and functions on surfaces","volume":"22","author":"Bajaj","year":"2003","journal-title":"ACM Trans. Graph."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Crane, K. (2018). Discrete differential geometry: An applied introduction. Not. AMS Commun., 1153\u20131159.","DOI":"10.1090\/noti1578"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"948","DOI":"10.1088\/0022-3735\/21\/10\/008","article-title":"Minimum inductance coils","volume":"21","author":"Turner","year":"1988","journal-title":"J. Phys. E Sci. Instrum."},{"key":"ref_23","unstructured":"Chavhan, G.B. (2013). MRI Made Easy, JP Medical Ltd."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1007\/BF00977785","article-title":"Two algorithms for constructing a Delaunay triangulation","volume":"9","author":"Lee","year":"1980","journal-title":"Int. J. Comput. Inf. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"919","DOI":"10.1109\/TVCG.2013.253","article-title":"Surface Meshing with Curvature Convergence","volume":"20","author":"Li","year":"2014","journal-title":"IEEE Trans. Vis. Comput. Graph."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/18\/7912\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:51:45Z","timestamp":1760129505000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/18\/7912"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,15]]},"references-count":25,"journal-issue":{"issue":"18","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["s23187912"],"URL":"https:\/\/doi.org\/10.3390\/s23187912","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2023,9,15]]}}}