This work aims to compare in simulation two recently introduced metrics of microscopic anisotropy, namely fractional eccentricity [1] and microscopic fractional anisotropy [2].Diffusion MRI sequences which vary the gradient orientation within one measurement can provide sensitivity to microscopic anisotropy in highly heterogeneous tissues [1-6], and several metrics of microscopic anisotropy have been proposed in the literature [1-3]. Fractional eccentricity (FE) [1] is derived from double-diffusion-encoding (DDE) [7,8] sequences with parallel and perpendicular gradients (Method 1), while microscopic fractional anisotropy (μFA) [2] is derived from a combination of sequences with isotropic and directional diffusion weighting (Method 2). A recent commentary argued that the two indices are in fact the same parameter of microscopic diffusion anisotropy [9], however, they have not been directly compared. The aim of this work is to compare FE and μFA and to investigate their behaviour in substrates featuring either restricted or Gaussian diffusion.

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/mm²) and randomly oriented anisotropic domains featuring non-exchanging Gaussian diffusion (corresponding G_max=100mT/m, b=2,865s/mm²).

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 μ₂. 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.

Fig. 2 illustrates the dependence of FE and μFA on a) pore eccentricity for randomly orientated finite cylinders featuring restricted diffusion and b) the ratio between parallel and perpendicular diffusivities of anisotropic micro-domains featuring Gaussian diffusion. For restricted diffusion, FE closely follows the ground truth values, while μFA overestimates them for pores with low eccentricity. Moreover, μFA is not a monotonously increasing function of pore eccentricity. Assuming different mean diffusivities for isotropically encoded and directionally averaged data improves the estimates of μFA for pores with low eccentricity. For substrates with Gaussian diffusion, which is the case investigated in [2], μFA is closer to the ground truth FA values then FE. This can be a consequence of using multiple b-values when deriving this metric as opposed to a single b-value for FE.The results show that FE and μFA do not provide exactly the same measure of microscopic anisotropy. For substrates featuring restriction, FE is more accurate, while for substrates featuring Gaussian diffusion μFA is closer to the ground truth. Further work is needed to investigate the effects of noise as well as more realistic tissue models.