The powder average of the diffusion-weighted signal serves as the basis for many diffusion MRI techniques, for example, in the quantification of micro-anisotropy [1,2], isotropic and anisotropic kurtosis [3], and neurite density [4]. The powder average is used to emulate the signal quality of a sample that contains randomly ordered structures, and is defined as the arithmetic average of the diffusion-weighted signal across multiple encoding directions. In theory, the powder signal is independent of the object orientation, in practice, residual effects of orientation will depend on the macroscopic diffusion anisotropy of the tissue and the strength and shape of the diffusion encoding tensor [5]. This work aims to find the minimum number of encoding directions that yields a signal invariant to rotation. We conduct the experiments for varying levels of tissue anisotropy, encoding strengths, and for single and double diffusion encoding (SDE, and DDE with orthogonal encoding directions).

Diffusion-weighted signal $$$(S)$$$ was simulated for multiple configurations of the b-matrix $$$(\mathbf{B})$$$ and underlying diffusion tensor $$$(\mathbf{D})$$$, according to $$S=\exp(–\mathbf{B:D}), (1)$$ where ':' denotes the double inner product. The examined tissue was assumed to exhibit Gaussian diffusion, and was described by a prolate and axially symmetric diffusion tensor, defined by its radial and axial diffusivity (RD, AD), according to $$D=\mathbf{rr}^{\text{T}}(\text{AD–RD})+\mathbf{I}\text{ RD}, (2)$$ where $$$\mathbf{r}$$$ is a unit vector defining the tensor orientation, and $$$\mathbf{I}$$$ is the identity matrix. We define the b-matrices for SDE and DDE as $$$\mathbf{B}_{\text{SDE}}=b\mathbf{gg}^{\text{T}}$$$, and $$$\mathbf{B}_{\text{DDE}}=b/2(\mathbf{I}-\mathbf{gg}^{\text{T}})$$$, respectively, where $$$b$$$ is the encoding strength $$$(b=\text{Trace}[\mathbf{B}])$$$, and $$$\mathbf{g}$$$ is a unit vector pointing along the symmetry axis of the encoding tensor (for DDE, this is the normal of the ‘encoding plane’).

To investigate the effect of rotation, $$$\mathbf{D}$$$ was rotated along 512 different orientations, $$$\mathbf{r}$$$. Diffusion encoding was simulated along $$$n=$$$ 2, 3, 6, 10, 15, 20, 32, 40, and 64 encoding directions, $$$\mathbf{g}$$$. The rotations and each set of directions was independently generated through electrostatic repulsion [6,7]. The encoding strength was in the interval $$$b=$$$ 200–4000 s/mm². The fractional anisotropy (FA) of $$$\mathbf{D}$$$ was adjusted by changing AD and RD, while retaining a mean diffusivity of MD = 1.0 µm²/ms, in the interval FA = 0.0–1.0. The powder averaged signal $$$(P)$$$ was calculated by averaging the signal across all encoding directions. Residual rotational variance was quantified as the coefficient of variation (CV) of the powder averaged signal across all 512 rotations of the underlying object, according to $$\text{CV}=\text{SD}[P]/\text{E}[P], (3)$$ where SD[] and E[] are the standard deviation and expected value, respectively. The threshold for ‘rotational invariance’ was set to CV < 1%. The minimum number of directions necessary to meet this condition $$$(n_{\text{min}})$$$ was calculated and plotted.

An example of $$$P$$$-vs-$$$b$$$ when using SDE is shown in Figure 1. The black and red lines represent powder averaging over three and ten directions, respectively. Stronger encoding yields larger rotational variance (spread increases with b), and fewer encoding directions yield more rotational variance (spread of black lines is larger than that of red lines). Figures 2 and 3 show the minimum number of directions for any combination of anisotropy and encoding strength, for SDE and DDE, respectively. As expected, the minimum number of directions increases with higher FA and higher b, while tissue with vanishing anisotropy generates little, or no, rotational variance.We estimated the minimum number of diffusion-encoding directions required to obtain a rotationally invariant signal powder average. The results can, for example, be used to distribute the number of encoding directions across multiple b-shells, where each shell must fulfill its individual requirements. The analysis assumes that FA and MD of the examined tissue are known approximately. Here, we fixed MD at 1.0 µm²/ms, however, the results can be rescaled for other values of MD due to the reciprocity between $$$\mathbf{B}$$$ and $$$\mathbf{D}$$$ in Eq. 1. For example, the corresponding result for MD = 2.0 µm²/ms can be obtained by scaling all b-values (x-axis) by a factor of ½. A limitation of this study is the selection of an appropriate cutoff threshold for the variance induced by rotation. We note, however, that CV < 1% is likely negligible compared to other sources of variability in practical applications [8,9]. In conclusion, Figs.2 and 3 can be used to find the minimum number of encoding directions that will render a powder averaged signal that is rotationally invariant for any given combination of tissue anisotropy and diffusion encoding strength, for SDE and DDE. Importantly, DDE requires fewer encoding directions than SDE, making it more efficient for powder averaging.