Abstract
A new method combining Anahess and Inverse Laplace Transform, for analysing one- and two-dimensional dynamic NMR data has been developed. When using the Inverse Laplace Transform (ILT) algorithm only to generate solutions from one- and two-dimensional data sets, the solutions are generally ill-posed resulting in an infinite number of possible solutions. Another approach named Anahess represents a discrete method that minimizes the number of components that is sufficient to fit the data sets properly and limits the number of solutions to a single and unique one. In this work it is shown that using the Anahess only is at the least as accurate as the ILT approach in extracting the correct T1–T2-values from a set of synthetic data. However, the Anahess is a discrete approach and does not reproduce continuous T1–T2-distributions, as is the case for many systems being investigated. Thus, a method for producing distributions of T1–T2-values from the Anahess discrete fit is provided in this work. In contrast to the ILT approach, the T1–T2-distribution produced from the Anahess discrete fit is unique as it is produced from a single set of T1–T2-values. The significant difference in performance of the two approaches, ILT and Anahess, for analysing the one- and two-dimensional dynamic NMR data is documented on synthetic data sets as well as on real data sets. It is also important to note that the analysis of the data sets using the Anahess approach does not require any user input as smoothing factor, field of view and number of grid points.
Similar content being viewed by others
Data Availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
References
S.W. Provencher, CONTIN: a general purpose constrained regularization program for inverting noisy linear algebraic and integral equations. Comput. Phys. Commun. 27(3), 229–242 (1982)
S.W. Provencher, A constrained regularization method for inverting data represented by linear algebraic or integral equations. Comput. Phys. Commun. 27(3), 213–227 (1982)
L. Venkataramanan, Y.Q. Song, M.D. Hürlimann, Solving Fredholm integrals of the first kind with tensor product structure in 2 and 2.5 dimensions. IEEE Trans. Signal Process. 50(1), 017–1026 (2002)
Y.Q. Song, L. Venkataramanan, M.D. Hürlimann, M. Flaum, P. Frulla, C. Straley, T1–T2 correlation spectra obtained using a fast two dimensional Laplace inversion. J. Magn. Reson. 154(2), 261–268 (2002)
G.C. Borgia, R.J.S. Brown, P. Fantazzini, Uniform-penalty inversion of multiexponential decay data. J. Magn. Reson 132, 65–77 (1998)
G.C. Borgia, R.J.S. Brown, P. Fantazzini, Uniform-penalty inversion of multiexponential decay data II. J. Magn. Reson. 147, 273–285 (2000)
G.H. Sørland, Dynamic Pulsed-Field_Gradient NMR (Springer, Berlin, 2014)
Å. Ukkelberg, G.H. Sørland, E.W. Hansen, H.C. Widerøe, Anahess, a new second order sum of exponential fits, compared to the Tikhonov regularization approach, with NMR applications. Int. J. Recent Res. Appl. Stud. IJRRAS 2(3) (2010)
C.L. Lawson, R.J. Hanson, Solving least squares problems. Soc. Ind. Appl. Math. (1995). https://doi.org/10.1137/1.9781611971217
A.N. Tichonov, A.S. Leonov, Ill-posed problems in natural sciences, in Proceedings of the International Conference Held in Moscow, August 19–25, 1992, VSP (1991)
A.N. Tikhonov, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic Publishers, Dordrecht, 1995)
S.W. Provencher, Inverse problems in polymer characterization: Direct analysis of polydispersity with photon correlation spectroscopy. Die Makromolekulare Chemie 180(1), 201–209 (1979)
W.H. Press, Numerical Recipes in C++: The Art of Scientific Computing (Cambridge University Press, Cambridge, 2002)
S. Gideon, Estimating the dimension of a model. Ann. Stat. 6(2), 461–464 (1978)
J.G. Seland et al., Combining PFG and CPMG NMR measurements for separate characterization of oil and water simultaneously present in a heterogeneous system. Appl. Magn. Reson. 24(1), 41–53 (2003)
P.D. Teal, C. Eccles, Adaptive truncation of matrix decompositions and efficient estimation of NMR relaxation distributions. Inverse Probl. 31(4), 045010 (2015)
Author information
Authors and Affiliations
Contributions
All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Geir Humborstad Sørland, Henrik Walbye Anthonsen, Åsmund Ukkelberg and Klaus Zick. The first draft of the manuscript was written by Geir Humborstad Sørland and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sørland, G.H., Anthonsen, H.W., Ukkelberg, Å. et al. A Robust Method for Analysing One and Two-Dimensional Dynamic NMR Data. Appl Magn Reson 53, 1345–1359 (2022). https://doi.org/10.1007/s00723-022-01479-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00723-022-01479-7