{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T20:59:12Z","timestamp":1773003552540,"version":"3.50.1"},"reference-count":23,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,5,23]],"date-time":"2024-05-23T00:00:00Z","timestamp":1716422400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Commission: HORIZON-MSCA-2022-SE-01","award":["101131564"],"award-info":[{"award-number":["101131564"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computers"],"abstract":"<jats:p>Fast-Field-Cycling (FFC) Nuclear Magnetic Resonance (NMR) relaxometry is a powerful, non-destructive magnetic resonance technique that enables, among other things, the investigation of slow molecular dynamics at low magnetic field intensities. FFC-NMR relaxometry measurements provide insight into molecular motion across various timescales within a single experiment. This study focuses on a model-free approach, representing the NMRD profile R1 as a linear combination of Lorentzian functions, thereby addressing the challenges of fitting data within an ill-conditioned linear least-squares framework. Tackling this problem, we present a comprehensive review and experimental validation of three regularization approaches to implement the model-free approach to analyzing NMRD profiles. These include (1) MF-UPen, utilizing locally adapted L2 regularization; (2) MF-L1, based on L1 penalties; and (3) a hybrid approach combining locally adapted L2 and global L1 penalties. Each method\u2019s regularization parameters are determined automatically according to the Balancing and Uniform Penalty principles. Our contributions include the implementation and experimental validation of the MF-UPen and MF-MUPen algorithms, and the development of a \u201cdispersion analysis\u201d technique to assess the existence range of the estimated parameters. The objective of this work is to delineate the variance in fit quality and correlation time distribution yielded by each algorithm, thus broadening the set of software tools for the analysis of sample structures in FFC-NMR studies. The findings underline the efficacy and applicability of these algorithms in the analysis of NMRD profiles from samples representing different potential scenarios.<\/jats:p>","DOI":"10.3390\/computers13060129","type":"journal-article","created":{"date-parts":[[2024,5,23]],"date-time":"2024-05-23T09:04:25Z","timestamp":1716455065000},"page":"129","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Robust Algorithms for the Analysis of Fast-Field-Cycling Nuclear Magnetic Resonance Dispersion Curves"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9419-1965","authenticated-orcid":false,"given":"Villiam","family":"Bortolotti","sequence":"first","affiliation":[{"name":"Department of Civil, Chemical, Environmental, and Materials Engineering, University of Bologna, 40131 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2211-1225","authenticated-orcid":false,"given":"Pellegrino","family":"Conte","sequence":"additional","affiliation":[{"name":"Department of Agricultural, Food and Forest Sciences, University of Palermo, 90128 Palermo, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1250-8218","authenticated-orcid":false,"given":"Germana","family":"Landi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bologna, 40127 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5089-0556","authenticated-orcid":false,"given":"Paolo","family":"Lo Meo","sequence":"additional","affiliation":[{"name":"Department of Biological, Chemical and Pharmaceutical Sciences and Technologies, University of Palermo, 90128 Palermo, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6406-3268","authenticated-orcid":false,"given":"Anastasiia","family":"Nagmutdinova","sequence":"additional","affiliation":[{"name":"Department of Civil, Chemical, Environmental, and Materials Engineering, University of Bologna, 40131 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5996-802X","authenticated-orcid":false,"given":"Giovanni Vito","family":"Spinelli","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bologna, 40127 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3961-4037","authenticated-orcid":false,"given":"Fabiana","family":"Zama","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bologna, 40127 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/j.pnmrs.2004.03.002","article-title":"Field-cycling NMR relaxometry","volume":"44","author":"Kimmich","year":"2004","journal-title":"Prog. Nucl. Magn. Reson. Spectrosc."},{"key":"ref_2","unstructured":"Bortolotti, V., Brizi, L., Landi, G., Testa, C., and Zama, F. (2024). The Environment in a Magnet: Applications of NMR Techniques to Environmental Problems, Royal Society of Chemistry."},{"key":"ref_3","unstructured":"Sebasti\u00e3o, P. (2024, May 21). Available online: http:\/\/fitteia.org."},{"key":"ref_4","unstructured":"OriginLab Corporation (2024). Origin(Pro), OriginLab Corporation."},{"key":"ref_5","unstructured":"Vasilief, I. (2024). QtiPlot, QtiPlot."},{"key":"ref_6","unstructured":"MathWorks (2024). Curve Fitting Toolbox User\u2019s Guide, The MathWorks, Inc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/jmre.1998.1534","article-title":"Model-Free Analysis of Stretched Relaxation Dispersions","volume":"135","author":"Halle","year":"1998","journal-title":"J. Magn. Reson."},{"key":"ref_8","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_9","doi-asserted-by":"crossref","unstructured":"Tarantola, A. (2005). Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM.","DOI":"10.1137\/1.9780898717921"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1415","DOI":"10.1137\/100790756","article-title":"A Regularization Parameter for Nonsmooth Tikhonov Regularization","volume":"33","author":"Ito","year":"2011","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1088\/1361-6420\/33\/1\/015003","article-title":"Uniform Penalty Inversion of two-dimensional NMR relaxation data","volume":"33","author":"Bortolotti","year":"2016","journal-title":"Inverse Probl."},{"key":"ref_12","first-page":"127809","article-title":"An automatic L1-based regularization method for the analysis of FFC dispersion profiles with quadrupolar peaks","volume":"444","author":"Landi","year":"2023","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1215","DOI":"10.1007\/s10596-021-10049-y","article-title":"2DNMR data inversion using locally adapted multi-penalty regularization","volume":"25","author":"Bortolotti","year":"2020","journal-title":"Comput. Geosci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"8553","DOI":"10.1021\/acs.analchem.1c01264","article-title":"Heuristic Algorithm for the Analysis of Fast Field Cycling (FFC) NMR Dispersion Curves","volume":"93","author":"Terranova","year":"2021","journal-title":"Anal. Chem."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1137\/0911028","article-title":"Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank","volume":"11","author":"Hansen","year":"1990","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Golub, G.H., and Van Loan, C.F. (2013). Matrix Computations, JHU Press.","DOI":"10.56021\/9781421407944"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"606","DOI":"10.1109\/JSTSP.2007.910971","article-title":"An Interior-Point Method for Large-Scale \u21131-Regularized Least Squares","volume":"1","author":"Kim","year":"2007","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1137\/080716542","article-title":"A fast iterative shrinkage-thresholding algorithm for linear inverse problems","volume":"2","author":"Beck","year":"2009","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"054605","DOI":"10.1103\/PhysRevE.107.054605","article-title":"Nuclear spin relaxation in aqueous paramagnetic ion solutions","volume":"107","author":"Faux","year":"2023","journal-title":"Phys. Rev. E"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/S0898-8838(05)57002-8","article-title":"NMR relaxation in solution of paramagnetic complexes: Recent theoretical progress for S \u2265 1","volume":"57","author":"Kowalewski","year":"2005","journal-title":"Adv. Inorg. Chem."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"726","DOI":"10.1002\/mrc.4204","article-title":"Water dynamics in different biochar fractions","volume":"53","author":"Conte","year":"2015","journal-title":"Magn. Reson. Chem."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1211","DOI":"10.1007\/s11368-012-0489-x","article-title":"Fast field cycling NMR relaxometry characterization of biochars obtained from an industrial thermochemical process","volume":"12","author":"Marsala","year":"2012","journal-title":"J. Soils Sediments"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Smith, R.C. (2013). Uncertainty Quantification: Theory, Implementation, and Applications, SIAM.","DOI":"10.1137\/1.9781611973228"}],"container-title":["Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-431X\/13\/6\/129\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:47:12Z","timestamp":1760107632000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-431X\/13\/6\/129"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,23]]},"references-count":23,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["computers13060129"],"URL":"https:\/\/doi.org\/10.3390\/computers13060129","relation":{},"ISSN":["2073-431X"],"issn-type":[{"value":"2073-431X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,23]]}}}