SpringerLink
Log in

Q-Space Quantitative Diffusion MRI Measures Using a Stretched-Exponential Representation

  • Conference paper
  • First Online:
Computational Diffusion MRI

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

  • 597 Accesses

Abstract

Diffusion magnetic resonance imaging (dMRI) is a relatively modern technique used to study tissue microstructure in a non-invasive way. Non-Gaussian diffusion representation is related to the restricted diffusion and can provide information about the underlying tissue properties. In this paper, we analytically derive nth order statistics of the signal considering a stretched-exponential representation of the diffusion. Then, we retrieve the Q-space quantitative measures such as the Return-To-the-Origin Probability (RTOP), Q-space mean square displacement (QMSD), Q-space mean fourth-order displacement (QMFD). The stretched-exponential representation enables the handling of the diffusion contributions from a higher b-value regime under a non-Gaussian assumption, which can be useful in diagnosing or prognosis of neurodegenerative diseases in the early stages. Numerical implementation of the method is freely available at https://github.com/TPieciak/Stretched.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (Germany)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 149.79
Price includes VAT (Germany)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
EUR 192.59
Price includes VAT (Germany)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info
Hardcover Book
EUR 192.59
Price includes VAT (Germany)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Aja-Fernández, S., Pieciak, T., Tristán-Vega, A., Vegas-Sánchez-Ferrero, G., Molina, V., de Luis-García, R.: Scalar diffusion-MRI measures invariant to acquisition parameters: a first step towards imaging biomarkers. Magn. Reson. Imaging 54, 194–213 (2018)

    Google Scholar 

  2. Aja-Fernández, S., de Luis-García, R., Afzali, M., Molendowska, M., Pieciak, T., Tristán-Vega, A.: Micro-structure diffusion scalar measures from reduced MRI acquisitions. PLOS ONE 15(3) (2020)

    Google Scholar 

  3. Alexander, D.C.: A general framework for experiment design in diffusion MRI and its application in measuring direct tissue-microstructure features. Magn. Reson. Med. 60(2), 439–448 (2008)

    Google Scholar 

  4. Assaf, Y., Basser, P.J.: Composite hindered and restricted model of diffusion (CHARMED) MR imaging of the human brain. Neuroimage 27(1), 48–58 (2005)

    Google Scholar 

  5. Assaf, Y., Blumenfeld-Katzir, T., Yovel, Y., Basser, P.J.: AxCaliber: a method for measuring axon diameter distribution from diffusion MRI. Magn. Reson. Med. 59(6), 1347–1354 (2008)

    Google Scholar 

  6. Avram, A.V., Sarlls, J.E., Barnett, A.S., Özarslan, E., Thomas, C., Irfanoglu, M.O., Hutchinson, E., Pierpaoli, C., Basser, P.J.: Clinical feasibility of using mean apparent propagator (MAP) MRI to characterize brain tissue microstructure. Neuroimage 127, 422–434 (2016)

    Google Scholar 

  7. Basser, P.J.: Inferring microstructural features and the physiological state of tissues from diffusion-weighted images. NMR Biomed. 8(7), 333–344 (1995)

    Google Scholar 

  8. Basser, P., Pierpaoli, C.: Microstructural features measured using diffusion tensor imaging. J. Magn. Reson. B 111(3), 209–219 (1996)

    Google Scholar 

  9. Bennett, K.M., Schmainda, K.M., Bennett, R., Rowe, D.B., Lu, H., Hyde, J.S.: Characterization of continuously distributed cortical water diffusion rates with a stretched-exponential model. Magn. Reson. Med. 50(4), 727–734 (2003)

    Google Scholar 

  10. Callaghan, P.T.: Translational Dynamics and Magnetic Resonance: Principles of Pulsed Gradient Spin Echo NMR. Oxford University Press, Oxford (2011)

    Google Scholar 

  11. Clark, C.A., Le Bihan, D.: Water diffusion compartmentation and anisotropy at high b values in the human brain. Magn. Reson. Med. 44(6), 852–859 (2000)

    Google Scholar 

  12. Descoteaux, M., Deriche, R., Le Bihan, D., Mangin, J.F., Poupon, C.: Multiple q-shell diffusion propagator imaging. Med. Image Anal. 15(4), 603–621 (2011)

    Google Scholar 

  13. Fick, R.H., Wassermann, D., Caruyer, E., Deriche, R.: MAPL: tissue microstructure estimation using Laplacian-regularized MAP-MRI and its application to HCP data. Neuroimage 134, 365–385 (2016)

    Google Scholar 

  14. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic (2014)

    Google Scholar 

  15. Grebenkov, D.S.: Laplacian eigenfunctions in NMR. I. A numerical tool. Concepts Magn. Reson. Part A: Educ. J. 32(4), 277–301 (2008)

    Google Scholar 

  16. Hansen, B., Jespersen, S.N.: Data for evaluation of fast kurtosis strategies, b-value optimization and exploration of diffusion MRI contrast. Sci. Data 3(1), 1–5 (2016)

    Google Scholar 

  17. Jensen, J.H., Helpern, J.A., Ramani, A., Lu, H., Kaczynski, K.: Diffusional kurtosis imaging: the quantification of non-gaussian water diffusion by means of magnetic resonance imaging. Magn. Reson. Med. 53(6), 1432–1440 (2005)

    Google Scholar 

  18. Jones, D.K., Knösche, T.R., Turner, R.: White matter integrity, fiber count, and other fallacies: the do’s and don’ts of diffusion MRI. Neuroimage 73, 239–254 (2013)

    Google Scholar 

  19. Kubicki, M., McCarley, R., Westin, C.F., Park, H.J., Maier, S., Kikinis, R., Jolesz, F.A., Shenton, M.E.: A review of diffusion tensor imaging studies in schizophrenia. J. Psychiatr. Res. 41(1–2), 15–30 (2007)

    Google Scholar 

  20. Le Bihan, D.: Molecular diffusion nuclear magnetic resonance imaging. Magn. Reson. Q. 7(1), 1 (1991)

    Google Scholar 

  21. Magin, R.L., Karani, H., Wang, S., Liang, Y.: Fractional order complexity model of the diffusion signal decay in MRI. Mathematics 7(4), 348 (2019)

    Google Scholar 

  22. Ning, L., Westin, C.F., Rathi, Y.: Estimating diffusion propagator and its moments using directional radial basis functions. IEEE Trans. Med. Imaging 34(10), 2058–2078 (2015)

    Google Scholar 

  23. Novikov, D.S., Fieremans, E., Jespersen, S.N., Kiselev, V.G.: Quantifying brain microstructure with diffusion MRI: Theory and parameter estimation. NMR Biomed. 32(4) (2019)

    Google Scholar 

  24. Ozarslan, E., Koay, C., Shepherd, T., Blackb, S., Basser, P.: Simple harmonic oscillator based reconstruction and estimation for three-dimensional q-space MRI (2009)

    Google Scholar 

  25. Özarslan, E., Shepherd, T.M., Vemuri, B.C., Blackband, S.J., Mareci, T.H.: Resolution of complex tissue microarchitecture using the diffusion orientation transform (DOT). Neuroimage 31(3), 1086–1103 (2006)

    Google Scholar 

  26. Özarslan, E., Koay, C.G., Shepherd, T.M., Komlosh, M.E., İrfanoğlu, M.O., Pierpaoli, C., Basser, P.J.: Mean apparent propagator (MAP) MRI: a novel diffusion imaging method for map** tissue microstructure. Neuroimage 78, 16–32 (2013)

    Google Scholar 

  27. Pieciak, T., Bogusz, F., Tristán-Vega, A., de Luis-García, R., Aja-Fernández, S.: Single-shell return-to-the-origin probability diffusion MRI measure under a non-stationary Rician distributed noise. In: IEEE 16th International Symposium on Biomedical Imaging, pp. 131–134. IEEE (2019)

    Google Scholar 

  28. Rovaris, M., Filippi, M.: Diffusion tensor MRI in multiple sclerosis. J. Neuroimaging 17, 27S–30S (2007)

    Google Scholar 

  29. Tian, Q., Yang, G., Leuze, C., Rokem, A., Edlow, B.L., McNab, J.A.: Generalized diffusion spectrum magnetic resonance imaging (GDSI) for model-free reconstruction of the ensemble average propagator. Neuroimage 189, 497–515 (2019)

    Google Scholar 

  30. Tuch, D.S., Reese, T.G., Wiegell, M.R., Wedeen, V.J.: Diffusion MRI of complex neural architecture. Neuron 40(5), 885–895 (2003)

    Google Scholar 

  31. Westin, C.F., Maier, S.E., Mamata, H., Nabavi, A., Jolesz, F.A., Kikinis, R.: Processing and visualization for diffusion tensor MRI. Med. Imag. Anal. 6(2), 93–108 (2002)

    Google Scholar 

  32. Williams, G., Watts, D.C.: Non-symmetrical dielectric relaxation behaviour arising from a simple empirical decay function. Trans. Faraday Soc. 66, 80–85 (1970)

    Google Scholar 

  33. Wu, Y.C., Field, A.S., Alexander, A.L.: Computation of diffusion function measures in \(q\)-space using magnetic resonance hybrid diffusion imaging. IEEE Trans. Med. Imaging. 27(6), 858–865 (2008)

    Google Scholar 

  34. Zhang, H., Schneider, T., Wheeler-Kingshott, C.A., Alexander, D.C.: NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain. Neuroimage 61(4), 1000–1016 (2012)

    Google Scholar 

  35. Zucchelli, M., Brusini, L., Méndez, C.A., Daducci, A., Granziera, C., Menegaz, G.: What lies beneath? Diffusion EAP-based study of brain tissue microstructure. Med. Imag Anal. 32, 145–156 (2016)

    Google Scholar 

Download references

Acknowledgements

Tomasz Pieciak acknowledges the Polish National Agency for Academic Exchange for grant PN/BEK/2019/1/00421 under the Bekker programme, the Ministry of Science and Higher Education (Poland) under the scholarship for outstanding young scientists (692/STYP/13/2018) and AGH Science and Technology, Kraków (16.16.120.773). Maryam Afzali and Derek K. Jones were supported by a Wellcome Trust Investigator Award (096646/Z/11/Z) and a Wellcome Trust Strategic Award (104943/Z/14/Z). Santiago Aja-Fernández was supported by Ministerio de Ciencia e Innovación of Spain (RTI2018-094569-B-I00).

Author information

Authors and Affiliations

  1. LPI, ETSI Telecomunicación, Universidad de Valladolid, Valladolid, Spain

    Tomasz Pieciak & Santiago Aja-Fernández

  2. AGH University of Science and Technology, Kraków, Poland

    Tomasz Pieciak & Fabian Bogusz

  3. School of Psychology, Cardiff University Brain Research Imaging Centre (CUBRIC), Cardiff University, Cardiff, UK

    Maryam Afzali & Derek K. Jones

Authors
  1. Tomasz Pieciak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maryam Afzali
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fabian Bogusz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Santiago Aja-Fernández
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Derek K. Jones
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tomasz Pieciak .

Editor information

Editors and Affiliations

  1. University College London, London, UK

    Noemi Gyori

  2. Centre for Medical Engineering, King's College London, London, UK

    Jana Hutter

  3. Nvidia, Nashville, TN, USA

    Vishwesh Nath

  4. University College London, London, UK

    Marco Palombo

  5. Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kgs. Lyngby, Denmark

    Marco Pizzolato

  6. Laboratory of Mathematics in Imaging, Harvard Medical School, Boston, MA, USA

    Fan Zhang

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Pieciak, T., Afzali, M., Bogusz, F., Aja-Fernández, S., Jones, D.K. (2021). Q-Space Quantitative Diffusion MRI Measures Using a Stretched-Exponential Representation. In: Gyori, N., Hutter, J., Nath, V., Palombo, M., Pizzolato, M., Zhang, F. (eds) Computational Diffusion MRI. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-030-73018-5_10

Download citation

Publish with us

Policies and ethics

Search

Navigation