Direction-averaged diffusion-weighted MRI signal using different axisymmetric B-tensor encoding schemes

Abstract

Purpose: It has been shown, theoretically and in vivo, that using the Stejskal-Tanner pulsed-gradient, or linear tensor encoding (LTE), and in tissue exhibiting a "stick-like" diffusion geometry, the direction-averaged diffusion-weighted MRI signal at high b-values ( $7000<b<10000s/{\mathrm{mm}}^2$ ) follows a power-law, decaying as $1/\sqrt{b}$ . It has also been shown, theoretically, that for planar tensor encoding (PTE), the direction-averaged diffusion-weighted MRI signal decays as 1/b. We aimed to confirm this theoretical prediction in vivo. We then considered the direction-averaged signal for arbitrary b-tensor shapes and different tissue substrates to look for other conditions under which a power-law exists.

Methods: We considered the signal decay for high b-values for encoding geometries ranging from 2-dimensional PTE, through isotropic or spherical tensor encoding to LTE. When a power-law behavior was suggested, this was tested using in silico simulations and, when appropriate, in vivo using ultra-strong (300 mT/m) gradients.

Results: Our in vivo results confirmed the predicted 1/b power law for PTE. Moreover, our analysis showed that using an axisymmetric b-tensor a power-law only exists under very specific conditions: (a) "stick-like" tissue geometry and purely LTE or purely PTE waveforms; and (b) "pancake-like" tissue geometry and a purely LTE waveform.

Conclusions: A complete analysis of the power-law dependencies of the diffusion-weighted signal at high b-values has been performed. Only three specific forms of encoding result in a power-law dependency, pure linear and pure PTE when the tissue geometry is "stick-like" and pure LTE when the tissue geometry is "pancake-like". The different exponents of these encodings could be used to provide independent validation of the presence of different tissue geometries in vivo.

Keywords: B-tensor encoding; diffusion-weighted MRI; direction-averaged diffusion signal; high b-value; power-law.

Figure 1

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

Figure 2

The gradient waveform of the planar tensor encoding

Figure 3

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

Figure 4

Simulated direction‐averaged PTE signal for $7000<b<10000\mathrm{s}/{\mathrm{mm}}^2$ and the results of the power‐law fit

Figure 5

A, The minimum number of directions for a rotationally invariant powder average signal at different b‐values, B, the changes of power‐law scaling versus number of gradient directions, C, the changes of the power‐law scaling (α) versus “still water” signal fraction, and D, the changes of the power‐law scaling (α) versus sphere signal fraction for PTE compared to LTE

Figure 6

A, Direction‐averaged diffusion signal for different b‐values (b = 7000 to 10 000  $\mathrm{s}/{\mathrm{mm}}^2$) in PTE, B, FA, C, parametric map of the exponent α. D, 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. The parameters, α and β are reported in the figure. α = 1 shows the power‐law relationship between the diffusion signal and the b‐value. E, The histogram of α values