the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 20 Feb 2026
An open-access WebApp for Inverse Laplace Transform analysis of TD-NMR signals
Abstract. Over recent years, compact and low-field time-domain nuclear magnetic resonance (TD-NMR) instruments have become increasingly available, expanding their use in the characterization of biomaterials across food, plant, and agro-industrial research. In this context, the Inverse Laplace Transform (ILT) has emerged as a powerful mathematical approach for extracting relaxation time distributions from TD-NMR signals. However, despite its widespread use, ILT analysis is often restricted to proprietary software or requires advanced expertise in numerical methods, limiting its accessibility to non-specialist users. In this work, we present an open-access WebApp for performing ILT analysis of TD-NMR signals in a transparent and user-friendly manner. The implemented algorithm is based on non-negative least squares combined with Tikhonov regularization and singular value decomposition, allowing robust inversion of ill-posed relaxation data. The platform supports the main TD-NMR experiments used in practice, including Carr–Purcell–Meiboom–Gill (CPMG), Inversion Recovery, and Saturation Recovery pulse sequences, and is compatible with data from instruments of any manufacturer. In addition to describing the mathematical formulation and implementation of the algorithm, a concise methodological discussion of ILT in the context of TD-NMR is provided. The performance of the WebApp is evaluated using both simulated datasets and representative experimental signals, demonstrating that the obtained relaxation time distributions are consistent with those produced by established ILT approaches. By lowering the barrier to advanced signal processing, the proposed WebApp represents a useful open scientific tool for research and teaching in magnetic resonance applications.
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Status: open (until 20 Mar 2026)
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RC1: 'Comment on mr-2026-2', Anonymous Referee #1, 26 Feb 2026
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AC1: 'Reply on RC1', Tiago Moraes, 27 Feb 2026
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We sincerely thank the reviewer for the careful evaluation of our manuscript and for the very positive assessment of our work, particularly highlighting its usefulness for research and teaching and its potential benefit to the growing time-domain NMR community. We respectfully acknowledge the reviewer’s concern regarding the journal scope. However, according to the journal’s Aims & Scope, Magnetic Resonance explicitly welcomes “innovative advances in techniques supporting magnetic resonance experiments that may range from sample preparation to computational techniques” and also welcomes “educational articles that provide informative and original insights into topics of current interest.”
The inverse Laplace transform is a fundamental computational step in TD-NMR relaxometry, directly supporting magnetic resonance experiments by enabling the interpretation of relaxation data. The main contribution of this work is the development of a fully open-access, web-based computational platform specifically designed to support magnetic resonance data analysis, improving accessibility, reproducibility, and reliability of TD-NMR relaxometry. Furthermore, this WebApp was specifically designed not only as a computational tool but also as an educational platform, allowing users to interactively explore the effects of inversion parameters and better understand inverse Laplace transform analysis in magnetic resonance. By making TD-NMR relaxometry more accessible and easier to use, this work helps extend the use of magnetic resonance techniques to a broader audience.
We respectfully hope that the reviewer may reconsider the suitability of this contribution within the scope of Magnetic Resonance. Also, the detailed comments provided by the reviewer are greatly appreciated and will be fully addressed in the revised version of the manuscript.Citation: https://doi.org/10.5194/mr-2026-2-AC1
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AC1: 'Reply on RC1', Tiago Moraes, 27 Feb 2026
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RC2: 'Comment on mr-2026-2', Anonymous Referee #2, 27 Feb 2026
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General comments
This manuscript presents an open-source, free, web-based app for the analysis of different types of time-domain NMR relaxation data in terms of distribution of relaxation times. Overall, the manuscript reads well, although some sentences could be shorten to improve clarity. The web interface is straightforward to understand and use, and presents an improvement to the existing software. The approach for the analysis of this type of data has already been used by the authors in numerous previous studies, such that we can expect its implementation to be used by other researchers in the field. However, as such, I believe it does not present a 'significant advancement in magnetic resonance', an important aspect of manuscripts published in MR and I would recommend submission in another journal.
Detailed comments
Eq. 2: ck(0) should likely read ci(0)
Eq. 4 does not show a time-dependence of the noise value (en instead of en(t)).
I find Eq. 5 inconsistent with the previous notations. I believe the experimental data should not be labelled c as c is the function modelling the behaviour of the experimental data (for example, as in Eq. 4). Some phrasing should also be changed in that regard, as the functions written in Eq. 1 to 4 are not experimental data, but functions modelling experimental data.
As highlighted by reviewer 1, the effect of the noise should be discussed more thoroughly, especially as the authors note that the SNR heavily affects the ILT spectrum (line 279). The experimental data leading to the results shown in Fig. 7 should also be included.
A discussion on the effect of the regularization parameter α is included, but should be extended. For example, it is not clear what guides the decision toward choosing a value over another.
The 3D plot in Fig 5 does highlight the broadening of the distribution from the increasing value of α, but I suggest showing a contour plot instead to make the evolution more readable.
Citation: https://doi.org/10.5194/mr-2026-2-RC2 -
AC2: 'Reply on RC2', Tiago Moraes, 16 Mar 2026
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We sincerely thank the reviewer for the careful reading of the manuscript and for the constructive comments. We appreciate the positive evaluation of the web interface and the recognition that the platform provides an improvement over existing software tools for the analysis of time-domain NMR relaxation data.
Regarding the reviewer's concern about the scope of the journal, we respectfully note that, according to the Aims & Scope of Magnetic Resonance, the journal welcomes innovative advances in techniques supporting magnetic resonance experiments, including computational methods. The inverse Laplace transform is a central computational step in TD-NMR relaxometry, enabling the interpretation of relaxation measurements. Thus, the main contribution of this work is the development of a fully open-access, web-based platform designed to support the analysis of magnetic resonance relaxation data. In addition to providing a practical computational tool, the platform was also designed with an educational perspective, allowing users to interactively explore the effects of inversion parameters and better understand the principles behind ILT analysis in magnetic resonance.We appreciate the reviewer's suggestions and believe they will help improve the clarity of the manuscript. All the detailed comments provided by the reviewer will be carefully addressed and incorporated in the revised version of the manuscript when requested by the editor.
Detailed comments:
1) (Eq. 2)Response: We thank the reviewer for identifying this typographical error. This correction will be implemented in the revised version of the manuscript.
2) (Eq. 4) Response: We appreciate the reviewer’s observation. The notation will be revised to explicitly include the time dependence of the noise term in the corresponding equation.
3) (Eq. 5) Response: We agree with the reviewer that the notation used in the manuscript may lead to confusion between experimental data and the mathematical model describing the signal. The notation and the related phrasing will be revised in the final manuscript to clearly distinguish between the experimental signal and the model functions.
4) (SNR) Response: We thank the reviewer for this important suggestion. The discussion regarding the influence of signal-to-noise ratio on the ILT spectrum will be expanded in the revised manuscript to better illustrate how noise affects the stability and resolution of the relaxation time distributions.
5) (Experimental data Fig. 7) Response: The experimental data used in Fig. 7 will be included in the revised version of the manuscript.
6) (Regularization parameter) Response: The discussion regarding the selection of the regularization parameter alpha will be expanded in the revised manuscript, including additional explanation of how the parameter affects the resolution and smoothness of the relaxation time distributions.
7) (Fig. 5) Response: A contour representation will be adopted in Fig. 5.
We thank the reviewer again for the thoughtful comments and suggestions. We believe that addressing these points will significantly improve the clarity and completeness of the manuscript in the final revised version.Citation: https://doi.org/10.5194/mr-2026-2-AC2
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AC2: 'Reply on RC2', Tiago Moraes, 16 Mar 2026
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RC3: 'Comment on mr-2026-2', Anonymous Referee #3, 06 Mar 2026
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In this very well-written manuscript a new open-access WebApp is presented that performs Inverse Laplace Transform (ILT) analysis of time-domain NMR relaxation data in a transparent and user-friendly way. The tool implements a non-negative least squares inversion with Tikhonov regularization and singular value decomposition. It supports common TD-NMR experiments such as CPMG, inversion recovery, and saturation recovery from instruments of any manufacturer. Validation with simulated and experimental datasets is shown, demonstrating that the resulting relaxation time distributions agree with established ILT approaches.
The ILT / NNLS approach is well established and in the sense of NMR or data analysis do not “do” anything new, as other reviewers correctly remarked. However, what is new is that both the manuscript and the web app offer an excellent introduction to ILT for novice users. I feel this is an important contribution that will help students and first-time users to get into ILT TD-NMR. A multitude of other open source packages are available, as also is discussed in the manuscript, but these still represent a significant hurdle for new users of TD-NMR. Most, if not all existing packages do not only require prior knowledge of ILT, but also programming skills and a significant time investment to get them to run. The data analysis itself in most cases also is not without its pitfalls. For most TD-NMR users who do not yet have a background in NMR this is too much of a hurdle to overcome, in my experience. I therefore feel that the paper does, in fact, represent a substantial contribution within the scope of Magnetic Resonance and most certainly represents a novel tool that will change the approaches used by other groups: it will open ILT to novice users who do not happen to have access to an easy to use ILT tool embedded in their proprietary spectrometer software.
Detailed comments:
1) Sec. 3: It would be nice to see the actual function that is minimized for (including the regularization term) to get a better understanding of the concept of the ILT algorithm
2) Lines 151-152: For readers that are new to this type of analysis it might be nice to briefly discuss the general concept of regularization and already mention the effect of large and small alphas on the solution (is briefly mentioned later in text). This way the regularization concept might become less abstract at this stage.
3) I would like to compliment the authors with the clear “look” of the WebbApp, it appears very user friendly.
4) It is not clear what happens to the data after it is uploaded. Is it stored or logged, or deleted? Please clarify
5) line 26-27: please add some references of applications of TD-NMR in research and industry
6) line 59: it is not necessarily the homogeneity of the magnetic field of permanent magnets that limits spectral resolution, as Magritek has demonstrated in the last decade or so with their sub-ppm Halbach magnets. Rather, it is the lower field strength. (Although even that limitation has been overcome in the aforementioned systems).
7) Please check typos, missing equation numbers, clarity, or English in the following lines: L119; L206, L215, L249, L251, L257, L260, L274.
8) Regularization and Alpha are discussed, in similar wording, at least twice in the MS. Please remove redundancy
Citation: https://doi.org/10.5194/mr-2026-2-RC3 -
AC3: 'Reply on RC3', Tiago Moraes, 16 Mar 2026
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We sincerely thank the reviewer for the careful reading of the manuscript and for the very positive and encouraging assessment of our work. We are particularly grateful for the recognition that the WebApp and manuscript provide a clear and accessible introduction to inverse Laplace transform (ILT) analysis for novice TD-NMR users. We fully agree with the reviewer that the accessibility of ILT tools remains a barrier for many researchers entering the field, and one of the main motivations of this work was precisely to lower this barrier.
We also greatly appreciate the reviewer’s comment that the work represents a meaningful contribution within the scope of Magnetic Resonance. The goal of this platform is to make TD-NMR relaxometry analysis more accessible, transparent, and reproducible for the broader magnetic resonance community.
We thank the reviewer for the constructive suggestions to further improve the manuscript. All the points raised below will be carefully addressed and incorporated in the revised version of the manuscript when requested by the editor.
Detailed comments:
1) (function minimized) Response: We thank the reviewer for this suggestion. The explicit expression of the function minimized in the ILT inversion, including the regularization term, will be included in Section 3 in the revised manuscript in order to improve the clarity of the algorithm description.
2) (Regularization) Response: We appreciate this suggestion. A brief conceptual explanation of regularization and the influence of the regularization parameter alpha on the solution (including the effects of small and large values) will be introduced earlier in the manuscript to make this concept clearer for readers who are new to ILT analysis.
3) (WebApp user-friendly) Response: We sincerely thank the reviewer for this encouraging comment. Ease of use and accessibility were central goals during the development of the WebApp, particularly to support students and researchers who are new to TD-NMR relaxometry.
4) (Data upload) Response: We thank the reviewer for pointing this out. A clarification will be added to the manuscript describing how uploaded data are handled by the platform: The uploaded dataset is temporarily transmitted to the server, where the ILT processing is performed and the results are generated. After the analysis is completed and the user session is closed, the server hosted at the University of São Paulo automatically discards the uploaded data. The data are not stored, logged, or used for any other purposes.
5) (Ref TD-NMR applications) Response: Additional references illustrating the applications of TD-NMR in research and industrial contexts will be included in the revised manuscript. Such as: https://doi.org/10.1016/S0066-4103(10)69003-5; https://doi.org/10.1111/1750-3841.70839 ; https://doi.org/10.1016/j.meatsci.2018.08.020 ; https://doi.org/10.1016/j.jmr.2012.11.010 ; https://doi.org/10.1016/j.colsurfa.2019.04.080 ; https://doi.org/10.1016/j.jmr.2023.107522 ; https://doi.org/10.1016/j.pnmrs.2013.09.001
6) (Magnetic field) Response: We thank the reviewer for this clarification. The corresponding sentence will be revised to more accurately reflect the relationship between magnetic field strength, magnet design, and spectral resolution in modern permanent magnet systems.
7) (Typos and clarity) Response: We appreciate the reviewer’s careful reading. The manuscript will be thoroughly revised to correct typographical errors and improve clarity in the indicated sections.
8) (Redundancy) Response: We thank the reviewer for identifying this redundancy. The relevant sections will be revised to remove repetition and improve the overall flow of the manuscript.
We thank the reviewer again for the very positive evaluation and the constructive suggestions provided. We believe these comments will help improve the clarity and pedagogical value of the manuscript in the final revised version.Citation: https://doi.org/10.5194/mr-2026-2-AC3
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AC3: 'Reply on RC3', Tiago Moraes, 16 Mar 2026
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Interactive computing environment
ILT WebApp T. B. Moraes et al. https://nmr-ilt.esalq.usp.br
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General
The manuscript reports the availability, access, and use of an open-access, on-line platform to convert NMR relaxation data into distributions or relaxation times. Despite the availability of different open-access ‘Laplace-transform’ algorithms, this platform provides a useful tool for research and teaching, as it is easy to use. The authors review the non-negative least squares algorithm with Tikhonov regularization and singular value decomposition and provide helpful advice for the beginner to achieve best results. I value this work, as it will benefit a growing community attracted to diverse incarnations of time-domain NMR. Moreover, this manuscript is well written, albeit it would still benefit from the language editing service of Copernicus Publishers. Nevertheless, I see this work outside the scope of Magnetic Resonance as it does not report a significant advance in magnetic resonance: “To be suitable for publication in MR, articles must describe substantial advancements in magnetic resonance. They should include significant innovation regarding new insights into magnetic resonance methodology, or into systems studied by magnetic resonance techniques, or expand the applicability of magnetic resonance. Routine applications of established techniques and minor technical advances are considered to be outside its scope.” Therefore, I recommend it to be submitted to a different Journal.
Detailed comments
Line 119: Equation number missing. The text reads “like equation ??”.
Line 240 (dito lines 270, 280): A validation of the algorithm with only 1% RMS Gaussian white noise added to the noise-free simulation data is a very benevolent test. It would be helpful to illustrate the limits of the algorithm, by showing how the distributions of relaxation times change with decreasing signal-to noise ratio. This would guide the unexperienced user in producing experimental data of sufficient quality.