Non-Gaussian diffusion MRI (dMRI) is currently an active field of research. Several multi-compartment models have been proposed to describe the diffusion-weighted MRI signal attenuation in the extended range of diffusion weightings (DW) (b-values), where deviations from the Gaussian assumption are observed^1,2. However, the intrinsic low signal-to-noise ratio is a limiting factor for subsequent data analysis. Therefore, in order to decrease the uncertainty in the parameter estimation, one needs properly optimised DW settings. In this work, a parameterisation and optimisation strategy for multi-shell DW settings using the Cramér-Rao lower-bound (CRLB) method^3,4 is proposed and tested in the context of the biexponential diffusion tensor analysis (BEDTA) approach¹.

Parameterisation. The DW settings consist of $$$N_\mathrm{b}$$$ shells, with the $$$i^\mathrm{th}$$$ shell having a radius $$$b_i$$$ and containing $$$N_i$$$ isotropically distributed gradient directions $$$(i=0,...,N_\mathrm{b}-1)$$$, generated with the DISCOBALL approach⁵. We propose to parameterise the distribution of b-values and gradient directions per shell with the following power laws: $$b_i=b_\mathrm{max}\eta_i^\beta$$ and $$N_i=N_0+\left(N_\mathrm{max}-N_0\right)\eta_i^\nu,$$ where $$$\eta_i=i/\left(N_\mathrm{b}-1\right)$$$, $$$b_\mathrm{max}$$$ is the maximum b-value, $$$N_0$$$ is the number of non-diffusion-weighted volumes and $$$N_\mathrm{max}$$$ is the number of gradient directions at the shell with $$$b=b_\mathrm{max}$$$ (determined by the total amount of volumes $$$M$$$). Here $$$\beta$$$ characterises the b-value distribution in the range $$$\left[0, b_\mathrm{max}\right]$$$. For $$$\beta=1$$$ all b-values are equidistant, for $$$\beta>1$$$ the b-values are more densely sampled at the lower range and vice versa for $$$0<\beta<1$$$ (and similarly for $$$\nu$$$). Under this parameterisation the DW settings are entirely characterised by the vector $$${\bf{P}}=\left(\beta,\nu,N_\mathrm{b},N_0, b_\mathrm{max},M\right)$$$.

Optimisation strategy. Here we seek for the optimal DW settings, $$$\bf{P}$$$, via minimisation of the normalised CRLB of the elements of the “fast” and “slow” diffusion tensors ($$$D_{nm}^{\left(\mathrm{f}\right)}$$$ and $$$D_{nm}^{\left(\mathrm{s}\right)}$$$) as well as the relative volume fraction,$$$f_\mathrm{f}$$$. The objective function we propose to minimise is:

$$H\left[{\bf{P}};\rho({\boldsymbol\theta})\right]=\sum_{k=1}^{T}\rho_k||{\bf{\Omega}}\left({\boldsymbol{\theta}}_k\right)||_2,$$ where $$$\rho_k$$$ is the relative fraction of tissue having the BEDTA properties $$$\boldsymbol{\theta}_k=\left(f_\mathrm{f},D_{11}^{\left(\mathrm{f}\right)},D_{12}^{\left(\mathrm{f}\right)},...,D_{11}^{\left(\mathrm{s}\right)},D_{12}^{\left(\mathrm{s}\right)},...\right)$$$ and $$$k = 1,...,T$$$ spans the different tissue types⁴ (here we take k = 2, i.e., white and grey matter). The elements of the vector $$$\boldsymbol{\Omega}$$$, $$$\Omega_l=\sqrt{I_l^{-1}}/{\theta_l}$$$ , contain the CRLB, $$$I_l^{-1}$$$, of the BEDTA parameters $$$\theta_l$$$, $$$l=1,...,13$$$.

In vivo experiments. Experiments were carried out using the twice-refocused bipolar spin-echo EPI pulse sequence, using the optimised DW settings for an acquisition time of 30 minutes, which corresponds to M = 275 volumes. Other protocol parameters were: TR = 6600 ms; TE = 148 ms; voxel-size = 2.4×2.4×2.4 mm³; BW = 1208 Hz/pixel; matrix-size 88×88×36. GRAPPA accel. factor = 2.

Data analysis. Eddy current, EPI and motion distortions were corrected using the EDDY toolkit available in FSL. BEDTA was carried out via non-linear least-squares minimisation using the Levenberg-Marquardt algorithm with in-house Matlab scripts.

Comparison with suboptimal DW settings. We demonstrate the improvement of the optimised, $$$\mathbf{P}_\mathrm{opt}$$$, versus suboptimal, $$$\mathbf{P}_\mathrm{so}$$$, DW settings by evaluating the following ratio⁴ $$R=\mathrm{ln}\left(\frac{H\left(\mathbf{P}_\mathrm{so}\right)}{H\left(\mathbf{P}_\mathrm{opt}\right)}\right).$$ The suboptimal DW settings include 7 b-values, b = 0, 1000, ..., 6000 s/mm², and 45 directions per shell.

Figure 1 shows the minimised value of the cost function $$$H$$$, versus $$$N_\mathrm{b}$$$ and $$$N_0$$$. One can see that the optimal number of shells is $$$N_\mathrm{b}=4$$$. For $$$N_\mathrm{b}=4$$$, there is a slight increase of $$$H$$$ with increasing $$$N_0$$$. For practical reasons, we take $$$N_0=5$$$ interspersed with the diffusion-weighted volumes to ease motion correction. Figure 2 shows the distribution of the optimal power law coefficients $$$\beta$$$ (a) and $$$\nu$$$ (b), the minimised cost function $$$H$$$ (c) and the corresponding optimal shell populations, $$$N_i$$$ (d), upon 5000 random initialisations of the optimisation algorithm. They clearly depict a unimodal distribution, proving the robustness of the optimisation framework against different initialisation values. Figure 3 shows maps of the fast relative fraction (a), mean diffusivities (b,c) and fractional anisotropies (d,e) for the fast and slow diffusion tensors, respectively, obtained using the optimised multi-shell DW settings. Figure 4 shows the ratio $$$R$$$, which demonstrates that the optimal DW settings perform better than the suboptimal ones for all voxels in the slice.We have proposed a parameterisation for multi-shell DW settings which is sufficiently flexible to cover a broad range of configurations, on the one hand, but uses only 6 degrees of freedom, on the other. Such parameterisation enables a straightforward optimisation of the DW settings by minimising the CRLB of different parameters. We have shown that our approach is very robust against different algorithm initialisation. We have demonstrated the performance of the optimal settings, in the framework of BEDTA. The optimised settings perform better in therms of the cost function, $$$H$$$, than the suboptimal design, built by following the conventional radial-like design, i.e., same amount of directions per shell.