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Diffusion-Weighted Magnetic Resonance Signal for General Gradient Waveforms: Multiple Correlation Function Framework, Path Integrals, and Parallels Between Them

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Visualization and Processing of Higher Order Descriptors for Multi-Valued Data

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

Abstract

Effects of diffusion on the magnetic resonance (MR) signal carry a wealth of information regarding the microstructure of the medium. Characterizing such effects is immensely important for quantitative studies aiming to obtain microstructural parameters using diffusion MR acquisitions. Studies in recent years have demonstrated the potential of sophisticated gradient waveforms to provide novel information inaccessible by traditional measurements. There are mainly two approaches that can be used to incorporate the influence of restricted diffusion, particularly on experiments featuring general gradient waveforms . The multiple propagator framework essentially reduces the problem to a path integral , which can be evaluated analytically or approximated via a matrix representation . The multiple correlation function method tackles the Bloch–Torrey equation , and employs an alternative matrix formulation. In this work, we present the two techniques in a unified fashion and link the two approaches. We provide an explanation for why the multiple correlation function is computationally more efficient in the case of waveforms featuring piecewise constant gradients.

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Notes

  1. 1.

    Strictly speaking, the form of V (ξ(ν ′)) differs slightly from that in Eq. (4) due to the explicit time dependence. However, this difference doesn’t appear to violate the applicability of Kac’s theorem.

  2. 2.

    With the boundary condition that \(\mathrm{U}(0, 0) = \mathbb{I}\), where \(\mathbb{I}\) is the identity operator.

  3. 3.

    The wave vector operator, when expressed in the position basis, is a derivative, i.e., \(\langle \boldsymbol{r}\vert \mathbf{K} = -\mathrm{i}\nabla \langle \boldsymbol{r}\vert\). Its commutator with the position operator is \(\left [\mathbf{K},\mathbf{R}\right ] = -\mathrm{i}\).

  4. 4.

    Also note that for a practical implementation where the limit \(\tau \rightarrow 0\) is not actually taken, one might want to offset the argument of H by τ∕2 (like in Fig. 1) or some other amount, but we need not bother with that for our purposes.

  5. 5.

    The eigenfunction corresponding to the \(\boldsymbol{k} = 0\) eigenvalue is constant over the volume of interest: \(\langle \boldsymbol{r}\vert 0\rangle = V ^{-1/2}\). Hence \(\int \mathrm{d}\boldsymbol{r}\,\langle \boldsymbol{r}\vert \boldsymbol{k}\rangle = V ^{1/2}\int \mathrm{d}\boldsymbol{r}\,\langle 0\vert \boldsymbol{r}\rangle \langle \boldsymbol{r}\vert \boldsymbol{k}\rangle = V ^{1/2}\langle 0\vert \boldsymbol{k}\rangle = V ^{1/2}\delta _{0,\boldsymbol{k}}\). On the other hand, the initial magnetization is in equilibrium, and therefore proportional to the \(\boldsymbol{k} = 0\) eigenket, meaning \(\langle \boldsymbol{k}^{{\prime}}\vert m(0)\rangle = c\delta _{\boldsymbol{k}^{{\prime}},0}\). For convenience, we assume a normalization for \(m(\boldsymbol{r},t)\) such that \(\int \mathrm{d}\boldsymbol{r}\,m(\boldsymbol{r}, 0) = 1\), whereby c = V −1∕2.

  6. 6.

    More details can be found in [34].

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Acknowledgements

This research was supported by TÜBİTAK-EU Co-funded Brain Circulation Scheme (project number 114C015) and Boğaziçi University (project number 8521).

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  1. Cem Yolcu

    Present address: Dipartimento di Fisica e Astronomia, Università di Padova, 35131, Padova, Italy

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  1. Department of Physics, Boğaziçi University, Bebek, 34342, İstanbul, Turkey

    Cem Yolcu & Evren Özarslan

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Correspondence to Evren Özarslan .

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  1. Linköping University, Norrköping, Sweden

    Ingrid Hotz

  2. Visualization and Medical Image Analysis, University of Bonn, Bonn, Germany

    Thomas Schultz

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Yolcu, C., Özarslan, E. (2015). Diffusion-Weighted Magnetic Resonance Signal for General Gradient Waveforms: Multiple Correlation Function Framework, Path Integrals, and Parallels Between Them. In: Hotz, I., Schultz, T. (eds) Visualization and Processing of Higher Order Descriptors for Multi-Valued Data. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-319-15090-1_1

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