A new framework for the optimisation of multi-shell diffusion weighting MRI settings using a parameterised Cramér-Rao lower-bound

Ezequiel Farrher^{1}, Ivan I. Maximov^{2}, Vincent Gras^{1}, Farida Grinberg^{1,3}, Rüdiger Stirnberg^{1,4}, and N. Jon Shah^{1,3}

*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 approach^{5}. 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 types^{4} (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 mm^{3}; 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 ratio^{4} $$R=\mathrm{ln}\left(\frac{H\left(\mathbf{P}_\mathrm{so}\right)}{H\left(\mathbf{P}_\mathrm{opt}\right)}\right).$$ The suboptimal DW settings include 7 *b*-values, *b* = 0, 1000, ..., 6000 s/mm^{2}, and 45 directions per shell.

1. Grinberg, F., Farrher, E., Kaffanke, J., et al. Non-Gaussian diffusion in human brain tissue at high b-factors as examined by a combined diffusion kurtosis and biexponential diffusion tensor analysis. Neuroimage 2011;57:1087-1102.

2. Zhang, H., Schneider, T., Wheeler-Kingshott, C.A., et al. NODDI: Practical in vivo neurite orientation dispersion and density imaging of the human brain. Neuroimage 2012; 61:1000-1016.

3. Alexander, C.A. A General Framework for Experiment Design in Diffusion MRI and Its Application in Measuring Direct Tissue-Microstructure Features. MRM 2008; 60:439-448.

4. Poot, D.H.J., den Dekker, A.J., Achten, E., et al. Optimal Experimental Design for Diffusion Kurtosis Imaging. IEEE 2010; 29:819-829.

5. Stirnberg, R., Stöcker, T., and Shah, N.J. A New and Versatile Gradient Encoding Scheme for DTI: a Direct Comparison with the Jones Scheme. ISMRM 2009; 3574.

Minimised value of the cost function $$$H$$$ vs. number of shells $$$(N_\mathrm{b})$$$ and number of non-DW volumes $$$(N_0)$$$.

Distribution of the optimal $$$\beta$$$ (a), $$$\nu$$$ (b), $$$H$$$ (c) and shell population, $$$N_i$$$ (d), upon 5000 random initialisations of the optimisation algorithm.

Some of the BEDTA invariants: a) fast relative volume fraction; b,c) fast and slow mean diffusivities; d,e) fast and slow fractional anisotropies.

Figure 4. Map and histogram of the ratio $$$R$$$, comparing the performance of the optimal versus suboptimal DW settings design.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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