Free-water elimination allows one to reduce the bias in DTI metrics induced by partial-volume effects. In this work, we propose a versatile approach for the optimisation of the diffusion weighting settings, given a limited acquisition time, based on a parameterised Cramér-Rao lower-bound. The optimisation shows robust convergence.
FWE DTI. The model we adopt here assumes that the normalised, diffusion MRI signal originates from two compartments in the slow-exchange limit1,2:
S\left(b,\mathbf{g},\boldsymbol{\theta}\right)=f_\mathrm{w}\mathrm{exp}\left(-bD_0\right) + \left(1-f_\text{w}\right)\mathrm{exp}\left(-b\mathbf{g}^T\mathbf{Dg}\right)\text{,}
where f_\mathrm{w} and D_0 are the fraction and the diffusion coefficient of the free-water compartment, respectively, \mathbf{D} is the diffusion tensor for the tissue compartment, \boldsymbol{\theta}\equiv\left(f_\mathrm{w},D_{11},D_{12},D_{13},D_{22},D_{23},D_{33}\right) and b and g are the strength and direction of the DW gradient, respectively. D0=3μm2/ms is fixed1,2 to the diffusion coefficient of free water at 37ºC.
Parameterisation. The DW settings are parameterised following Ref.9 They consist of N_\mathrm{sh} shells, with the i^{th} shell having a radius b_i and containing N_i isotropically distributed10 gradient directions (i=0...N_\mathrm{sh}). The distributions for b_i and N_i are given by: 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/N_\mathrm{sh}, 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 outermost shell (determined by the total amount of volumes M). Thus, the DW settings are completely determined by the vector \mathbf{P}=\left(M,N_\mathrm{sh},N_0,\boldsymbol{\beta},\boldsymbol{\nu},b_\mathrm{max}\right). In order to account for SNR reductions due to gradient strength limitations for increasing b-values, we parameterise the SNR following Ref.7
Optimisation. The optimal \mathbf{P} is found by simultaneously minimising the CRLB of f_\mathrm{w} and D_{i,j} (i,j=1,2,3)8,9. The objective function to be minimised is9:
H\left[\mathbf{P},\rho\right]=\sum_{k=1}^{T}\rho_k\|\boldsymbol{\Omega}\left(\boldsymbol{\theta_k}\right)\|_2\text{,}
where \rho_k (k=1...T spans the number of tissue types8,9) is the probability for the k-th tissue with diffusion properties \boldsymbol{\theta}_k . Here we take T=2, i.e., grey and white matter. The elements of the vector \boldsymbol{\Omega}, \Omega_l\equiv\sqrt{I_l^{-1}} , contain the CRLB, I_l^{-1}, of the parameters \theta_l (l=1,...,7). The performance of two algorithms for the minimisation of H was carried out and compared:
Algorithm 1: the local, computationally simple Nelder-Mead algorithm.
Algorithm 2: the global, computationally expensive genetic algorithm, both available in Matlab (Matlab, The MathWorks).
The upper constraint for b_\mathrm{max} was set to 1.5 ms/μm2 in order to avoid non-Gaussian effects6.
In vivo experiments. Experiments were performed on a healthy volunteer, in a 3T Trio scanner (Siemens Erlangen, Germany) using the twice-refocused bipolar spin-echo EPI pulse sequence. The optimised DW settings for an acquisition time of 8:30 minutes are shown in Table 1. Other protocol parameters were: TR=8100ms; TE=94ms; voxel-size=2×2×2mm3; BW=1446Hz/pixel; matrix-size128×128×60; GRAPPA accel. factor=2.
Data analysis. Eddy current and EPI distortions were corrected using the EDDY toolkit available in FSL11. FWE DTI parameter estimation was performed in two steps:
i) an initial guess for the tensor elements was generated by fitting conventional DTI;
ii) FWE DTI was fitted via non-linear least-squares minimisation using the initial guess generated in step i), with the help of the Levenberg-Marquardt algorithm with in-house Matlab scripts.
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