Maryam Afzali^{1}, Santiago Aja-Fernandez^{1,2}, and Derek K Jones^{1,3}

^{1}Cardiff University Brain Research Imaging Centre (CUBRIC), School of Psychology, Cardiff University, Cardiff, United Kingdom, ^{2}Laboratorio de Procesado de Imagen, ETSI Telecomunicacion Edificio de las Nuevas Tecnologias, Campus Miguel Delibes s/n, Universidad de Valladolid, Valladolid, Spain, ^{3}Mary MacKillop Institute for Health Research, Faculty of Health Sciences, Australian Catholic University, Melbourne, Victoria, Australia

It has been shown previously that for the linear (LTE), as well as planar tensor encoding (PTE) and in tissue with 'stick-like' geometry, the diffusion-weighted signal at high b-values follows a power-law. Specifically, the signal decays as $$$1/\sqrt{b}$$$ in LTE and $$$1/b$$$ in PTE. Here, we investigate whether power-law behaviors occur with other encodings and geometries. The results show that using an axisymmetric b-tensor a power-law only exists for stick-like geometries, using LTE and PTE. Finally, using ultra-strong gradients, we confirm –for the first time in vivo– that a power-law exists for PTE in white matter of the human brain.

Our motivation is: i) to disambiguate stick-like from other geometries (LTE shows a power-law not just for stick-like geometries, but also when there is a small contribution from dot and sphere compartments); ii) to find the range of b-values over which we observe a power-law scaling; and iii) to investigate if the signal amplitude in that range of b-value (7,000<b<10,000 $$$s/mm^2$$$) is significantly higher than the noise floor.

$$S(b)=\frac{\sqrt{\pi} e^{-\frac{b}{3}(D^{\mid\mid}+2D^\perp-b_\Delta(D^{\mid\mid}- D^\perp))}\textrm{erf}(\sqrt{bb_\Delta(D^{\mid\mid}-D^\perp)})}{2\sqrt{b b_\Delta(D^{\mid\mid}-D^\perp)}} \quad (1)$$

where $$$S$$$ is the normalized diffusion signal and $$$D^{\mid\mid}$$$ and $$$D^\perp$$$ are the parallel and perpendicular diffusivities respectively.

In PTE, $$$b_\Delta = -1/2$$$ and therefore:

$$S_{ic}^{PTE}(b) = \frac{\sqrt{\pi}e^{\frac{-b {D_a}^{\mid \mid} }{2}}\textrm{erfi}(\sqrt {b {D_a}^{\mid \mid}/2})}{2\sqrt{b {D_a}^{\mid \mid}/2}} \quad (2)$$

For large b-values, $$$b {D_a}^{\mid \mid} \gg 1$$$, the diffusion signal can be approximated by: $$S_{ic}^{PTE}(b) \approx \frac{1}{b {D_a}^{\mid \mid}} \sum_{k = 0}^{N} \frac{(2k-1){!}{!}}{(b {D_a}^{\mid \mid})^k} \quad (3)$$

where $$${!}{!}$$$ denotes the double factorial and N depends on the $$$b {D_a}^{\mid \mid}$$$ value (Fig. 1). A normalized error is used to compare the original, $$$S$$$ (Eq. (2)) and the approximated signal, $$$\hat{S}$$$ (Eq. (3)):

$$\textrm{Normalized error}=\frac{|S-\hat{S}|}{S} = \left|1-\frac{\hat{S}}{S}\right|\quad(4)$$

Synthetic data were generated with 60 gradient orientations

$$S/S_0 = f_1\int_{\mathbb{S}^2} W(\mathbf{n}) S_{cyl}(\mathbf{n}) d\mathbf{n} + f_2 \int_{\mathbb{S}^2} W(\mathbf{n}) S_{ec}(\mathbf{n}) d\mathbf{n} + f_3 S_{sph}(R_s) \quad (5)$$

where $$$f_1$$$, $$$f_2$$$, $$$f_3$$$, $$$S_{ec}$$$, $$$S_{cyl}$$$, and $$$S_{sph}$$$ are the intra-axonal, extra-axonal and the sphere signal fraction and diffusion signal

Two healthy participants were scanned with the same protocol as the simulation. Twenty axial slices with a voxel size of $$$4 mm$$$ isotropic and a 64$$$\times$$$64 matrix size, TE = 88 ms, TR = 3000 ms were obtained.

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Fig. 1. The approximated signal over the original PTE signal ($$$\hat{S}/{S}$$$), for different $$$N$$$ values.

Fig. 2. Maximum $$$b{D_a}^{\mid\mid}$$$ vs SNR in the baseline. The Maximum $$$b{D_a}^{\mid\mid}$$$ value is proportional to the square of SNR, ($$$b{D_a}^{\mid\mid} \sim SNR^2$$$) for LTE, where this relationship is linear for PTE ($$$b{D_a}^{\mid\mid} \sim SNR$$$) and it is logarithmic for STE ($$$b{D_a}^{\mid\mid} \sim \ln{(SNR)}$$$)

Fig. 3. Simulated direction-averaged PTE signal for $$$7000<b<10000 s/mm^2$$$ and the results of the power-law fit.

Fig. 4. (a) The minimum number of directions for a rotationally invariant powder average signal in different b-values, (b) the changes of power-law scaling versus number of gradient directions, (c) the changes of the power-law scaling, $$$\alpha$$$, versus 'still water' signal fraction, and (d) the changes of the power-law scaling, $$$\alpha$$$, versus sphere signal fraction for PTE compared to LTE.

Fig. 5. Direction-averaged diffusion signal for different b-values (b = 7000 to 10000 $$$s/mm^2$$$) in PTE, FA, Parametric map of the exponent $$$\alpha$$$. The plot of the diffusion signal vs $$$1/b$$$ for in vivo white matter voxels using planar tensor encoding. The blue curve with the error bar shows the mean and the std of the average signal and the red line shows the power-law fit. $$$\alpha = 1$$$ shows the power-law relationship between the diffusion signal and the b-value. The histogram of $$$\alpha$$$ values and the PTE gradient waveform.