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1}
Universität zu Lübeck, Institute of Medical Engineering, Germany

Abstract
In double diffusion encoding, information on microscopic tissue structure can be derived from the dependence of the MR signal on the angle between the two diffusion gradient directions. The signal difference between identical and opposing diffusion gradient directions approximately decays exponentially when the interweighting delay is increased. The shape of this change might provide a new approach to deriving a distribution of pore sizes in tissue.
1 Background
Double diffusion weighting [1, 2] can be used to assess the mean cell size and shape. The difference between signals acquired with identical or opposite gradients scales with the pore size. When the delay between the two weightings is increased, this difference decreases [3, 4, 5]. The shape of the decrease can possibly be exploited to obtain the distribution function of pore sizes.
For small amplitudes of q = Gδγ /(2π), where G is the diffusion gradient, the double-diffusion encoded signal S(q(1),q(2)) with vectors q(i) in the two weighting periods can be approximated for spherical pores of diameter a by [6]
$\frac{\left|S(\mathit{q},\mathit{q})-S(\mathit{q},-\mathit{q})\right|}{{S}_{0}}\propto \underset{N\to \infty}{\mathrm{lim}}{B}_{N}{G}^{2}$(1)
with known functions ${B}_{N}={\sum}_{n=1}^{N}{B}^{\prime}{\text{\hspace{0.05em}}}_{n}$ depending on a, D0, δ, Δ, and τm (pore size, free diffusion coefficient, duration and separation of gradient pulses in one weighting, and temporal separation of weighting periods, respectively), where S0 denotes the signal for G = 0.
2 Results and discussion
The terms B′n in BN with n > 1 contribute little and can be omitted to good approximation. This leads to
$\frac{\left|S(\mathit{q},\mathit{q})-S(\mathit{q},-\mathit{q})\right|}{{S}_{0}}\propto f(a,{D}_{0}){e}^{-\frac{{\alpha}_{1}^{2}{\tau}_{m}{D}_{0}}{{a}^{2}}}$(2)
with a known function f (a,D0) which also depends on the timing parameters, and some constant α1. This means that the given normalized signal difference decays exponentially with increasing τm, modulated by f (a,D0). The shape of the decay depends on the pore size, a. In a sample with a distribution of pore sizes, this dependence represents a superposition of the exponentials in Eq. (2).
Measuring the τm dependence of the signal should hence yield the distribution of pore sizes, much like deriving a distribution of diffusion coefficients by applying an inverse Laplace transform to the signal as a function of diffusion weighting [7]. The stability of the Laplace inversion remains to be investigated.
References
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[2] Mitra PP. Multiple wave-vector extensions of the NMR pulsed-field-gradient spin-echo diffusion measurement. Phys. Rev. B 51, 15074–15078 (1995). DOI: 10.1103/PhysRevB.51.15074.
[3] Koch MA, Finsterbusch J. Compartment Size Estimation With Double Wave Vector Diffusion- Weighted Imaging. Magn. Reson. Med. 60, 90–101 (2008). DOI: 10.1002/mrm.21514.
[4] KochMA, Finsterbusch J. Numerical simulation of double-wave vector experiments investigating diffusion in randomly oriented ellipsoidal pores. Magn. Reson. Med. 62, 247–254 (2009). DOI: 10.1002/mrm.21976.
[5] Shemesh N, Ozarslan E, Basser PJ, Cohen Y. Measuring small compartmental dimensions with low-q angular double-PGSE NMR: The effect of experimental parameters on signal decay. J. Magn. Reson. 198, 15–23 (2009). DOI: 10.1016/j.jmr.2009.01.004.
[6] Özarslan E, Basser PJ. Microscopic anisotropy revealed by NMR double pulsed field gradient experiments with arbitrary timing parameters. J. Chem. Phys. 128, 154511 (2008). DOI: 10.1063/1.2905765.
[7] Ronen I, Moeller S, Ugurbil K, Kim DS. Analysis of the distribution of diffusion coefficients in cat brain at 9.4 T using the inverse Laplace transformation. Magn. Reson. Imaging 24, 61–68 (2006). DOI: 10.1016/j.mri.2005.10.023.

Conference:
New dimensions in diffusion encoding, Fjälkinge, Sweden, 11 Jan - 14 Jan, 2016.

Presentation Type:
Oral presentation

Topic:
New Dimensions in Diffusion Encoding

Citation:
Koch
MA and
Ulloa
P
(2016). Determination of the pore size distribution by double
diffusion encoding.
Front. Phys.
Conference Abstract:
New dimensions in diffusion encoding.
doi: 10.3389/conf.FPHY.2016.01.00012