Metrics of microscopic anisotropy: a comparison study

Andrada IanuČ™^{1}, Noam Shemesh^{2}, Daniel C. Alexander^{1}, and Ivana Drobnjak^{1}

For a fair comparison between the two methods, we need to adapt the original acquisition protocols in order to have sequences which are as similar as possible. To this end, we use the recently introduced double-oscillating-diffusion-encoding (DODE) gradients [10], which provide similar contrast to DDE sequences and can be easily tailored to have the same waveform as diffusion gradients with isotropic encoding necessary for estimating μFA.

*Diffusion
sequences:*
The
acquisition protocols are illustrated in Fig 1a)
and b), and have the same gradient waveform and maximum gradient
strength. In Method 1 we use DODE sequences with N=3 periods and the directional scheme presented in [1], which employs 72
gradient orientation pairs. The gradient amplitude, G_{max}, is adapted for the two types of substrates. In Method 2, for
isotropic encoding we use sequential gradients in x,y, and z
direction,
with
16 gradient amplitudes between 0 and G_{max }[2]. The directional sequences have the same waveform and 15
isotropic directions. The other parameters are pulse duration δ=60ms and mixing time τ_{m}=20ms.

*Diffusion
substrates:* Fig.
1 c) and d) illustrate the diffusion substrates used in simulations,
namely randomly oriented anisotropic pores which
exhibit restricted diffusion (corresponding G_{max}=300mT/m, b=25,780s/mm^{2}) and randomly oriented anisotropic domains
featuring non-exchanging Gaussian diffusion (corresponding G_{max}=100mT/m, b=2,865s/mm^{2}).

*FE
computation:* To
calculate FE, first we need to derive the b-values and q-values for DODE
sequences.
As
there is no clear definition of the q-value for oscillating gradients, we make the
following assumption: we write
the b-value of a DDE sequence in
terms of its q-value and diffusion time and assume
that
the equation will have the same form for a DODE sequence. Thus, for a DODE sequences we have the following formulae: $$b=\gamma^2G^2\frac{\delta^3}{6N^2},\text{ and }q=\frac{1}{2\pi}\frac{\gamma G\delta}{2\sqrt{N}}.$$ Adapting the expression of FE [1] for DODE sequences, we have: $$FE=\sqrt{\frac{\epsilon}{\epsilon+\frac{3}{5}\left(\frac{\delta}{3N}\right)^2\left(\frac{Tr(\mathbf D)}{3}\right)^2}},\text{ where }\epsilon=\frac{1}{q^4}\left(\log(\frac{1}{12}\sum S_{\parallel})-\log(\frac{1}{60}\sum S_\perp)\right)$$

*μFA
computation:* To
calculate μFA, we perform a non linear fit to the 2^{nd} order cumulant
expansion of the signal $$$\log{S}=-b\bar{D}+\frac{\mu_2}{2}b^2$$$
in
order to obtain the mean diffusion coefficient $$$\bar{D}$$$ and its variance μ_{2}.
We
fit this equation separately to the isotropically encoded measurements and directionally averaged data, assuming that the two data sets have:

• the same mean diffusivity and different variances, as assumed in [2].

• different mean diffusivities and variances.

Once we have the values for mean diffusivity and variance, we compute μFA according to: $$\mu FA=\sqrt{\frac{3}{2}}\left(1+\frac{2}{5\Delta\tilde{\mu_2}}\right)^{-1/2},$$ where $$$\Delta\tilde{\mu_2}=\frac{\mu_2^{da}}{\bar{D}^{da}}-\frac{\mu_2^{iso}}{\bar{D}^{iso}}$$$ is the difference of scaled variances for directional averaged data and isotropically encoded data.

All simulations are performed using the MISST [11] software.

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[2]. S. Lasic, F. Szczepankiewicz, S. Eriksson, M. Nilsson, , and D. Topgaard. Microanisotropy imaging: quantification of microscopic diffusion anisotropy and orientational order parameter by diffusion MRI with magic-angle spinning of the q-vector. Frontiers in Physics, 2, 2014

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[8] D.C. Alexander, Y. Cohen, I. Drobnjak, T. Dyrby, J. Finsterbusch, M.A. Koch, T. Kuder, F. Laun, M. Lawrenz, H. Lundell, P.P. Mitra, M. Nilsson, E. Ozarslan, D. Topgaard, C. F. Westin, N. Shemesh, S. N. Jespersen. Conventions and nomenclature for double diffusion encoding (DDE) NMR and MRI. Magnetic Resonance in Medicine, 2015, Early View.

[9] S. N. Jespersen, H. Lundell, C. K. Sonderby, and T. B. Dyrby. Commentary on “Microanisotropy imaging: quantification of microscopic diffusion anisotropy and orientation of order parameter by diffusion MRI with magic-angle spinning of the q-vector”, Frontiers in Physics, 2014.

[10]. N. Shemesh, A. Ianus, D. C. Alexander, and I. Drobnjak. Double oscillating diffusion encoding (dode) augments microscopic anisotropy contrast. In Proc. ISMRM, page 952, Toronto, Canada, 2015

[11] Microstructure Imaging Sequence Simulation Toolbox (MISST), http://mig.cs.ucl.ac.uk/index.php?n=Tutorial.MISST

Figure 1. Schematic
representation of diffusion sequences adapted for estimating a) FE
and b) μFA. Schematic representation of microscopically anisotropic
diffusion substrates featuring c) restricted diffusion and d)
Gaussian diffusion.

Figure 2. a)
Dependence of FE (1^{st} column) and μFA (2^{nd} and
3^{rd} columns) on pore elongation for randomly oriented finite
cylinders with different diameters (restricted diffusion). b)
Dependence of FE and μFA on the ratio between parallel and
perpendicular diffusivities in randomly oriented anisotropic domains
(Gaussian diffusion).

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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