Synopsis

Free-water elimination allows one to reduce the bias in DTI metrics induced by partial-volume effects. Unfortunately the fitting problem for this model is ill-conditioned. However, it has been recently demonstrated that the introduction of a second dimension determined by the echo-time, leads to a well-conditioned fitting problem. In this work we investigate the experimental design and data analysis pipeline of such experiments in vivo.

Introduction

Methods

The model we adopt here assumes that the diffusion MRI signal originates from two compartments with different transverse relaxation rates and diffusivities, in the slow-exchange limit^1,7 $$S\left( T_\mathrm{E},b ,\mathbf{g}\right)=S_0\left[ f_\mathrm{w}e^{-\frac{T_\mathrm{E}}{T_{2,0}}}e^{-bD_0}+\left(1-f_\mathrm{w}\right)e^{-\frac{T_\mathrm{E}}{T_{2,\mathrm{t}}}}e^{-b\mathbf{g}^\mathrm{T}\mathbf{D}\mathbf{g}}\right]\mathrm{, (1)}$$ where S₀ is the proton density, f_w, T_2,0 and D₀ are the fraction, transverse relaxation time and diffusion coefficient of the free-water compartment and T_2,t and D are the transverse relaxation time diffusion tensor for the tissue compartment. The experimentally controlled parameters are the strength and direction of the diffusion weighting gradient, b and g, and T_E (Fig.1).

Experiments were performed on a healthy volunteer, in a 3T Siemens Trio scanner (Siemens, Erlangen, Germany). Prior written, informed consent was obtained from the volunteer. A Stejskal-Tanner pulse sequence with monopolar pulse field gradients and EPI readout was implemented for this purpose. The sequence was designed to independently control the gradient pulse duration, δ, and separation, Δ, as well as T_E. Thus, the diffusion-dimension and the relaxation-dimension are fully decoupled. Experimental parameters included (Fig. 1): 4 experiments with T_E = 70, 100, 130, 170 ms; b-values = 0 (6 repetitions), 500 (20 directions), 1000 (40 directions) s/mm²; Δ = 34 ms and δ = 19 ms. Additionally, we acquired 8 more echo-times, in the range T_E $$$\in$$$ [70, 210] ms. The experiment with T_E = 70 ms was acquired twice with opposite phase-encoding (anterior-posterior, posterior-anterior) in order to correct for EPI distortions. Other parameters were voxel-size = 2³ mm³; matrix-size = 100×100×60; repetition-time, TR = 15 s and GRAPPA acceleration factor = 2.

Eddy current and EPI distortions were corrected using the EDDY toolkit available in FSL.⁸ FWET₂ DTI parameter estimation was performed in three steps:

1. Experimental data with b = 0 s/mm² was used to generate an initial guess for T_2,t using the linear least-squares (LLS) method: $$$\hat{\gamma}_\mathrm{SE}=\left(W^{\mathrm{T}}_{\mathrm{SE}}W_\mathrm{SE}\right)^{-1}W^{\mathrm{T}}_{\mathrm{SE}}y$$$, where $$$W_{\mathrm{SE}}=\left[1,-T_\mathrm{E}\right]^\mathrm{T}$$$, $$$\gamma_{\mathrm{SE}}=\left[\ln{S^\mathrm{SE}_0},1/T_{2,\mathrm{t}}\right]^\mathrm{T}$$$ and y is the measured data. Similarly the initial guess for D was generated using the DWI experiment with T_E = 70 ms using LLS: $$$ \hat{\gamma}_\mathrm{DTI}=\left(W^\mathrm{T}_\mathrm{DTI}W_\mathrm{DTI}\right)^{-1}W^{\mathrm{T}}_{\mathrm{DTI}}y$$$, where $$$W_{\mathrm{DTI}}=\left[1,-bg^2_x,-2bg_xg_y,-2bg_xg_z,-bg^2_y,-2bg_yg_z,-bg^2_z,\right]^\mathrm{T}$$$ and $$$\gamma_{\mathrm{DTI}}=\left[\ln{S^\mathrm{DTI}_0},D_{11},D_{12}...,D_{33}\right]^\mathrm{T}$$$.
2. Having had the initial guess for D and T_2,t, an initial guess for f_w was generated using the method proposed by Hoy et al.⁵ and Neto Henriques et al.⁶ using a non-linear least-squares (NLLS) method.
3. Finally FWET₂ was fitted via NLLS minimisation using the initial guess generated in the former steps, with the help of the Levenberg-Marquardt algorithm with in-house Matlab scripts. The power images method⁹ was used to reduce the bias due to background noise prior fitting. Values for D₀ = 3×10^-3 mm²/s and T_2,0 = 500 ms were fixed according to literature values.¹⁰ Thus, the model parameters are $$$\gamma_\mathrm{FWET2}=\left[S_0,f_\mathrm{w},D_{11},D_{12},...,D_{33},T_{2,\mathrm{t}}\right]$$$. The NLLS estimator is given by $$\gamma_\mathrm{FWET2}=\arg \min \sum_{i} \left[y_i-S\left(T_{\mathrm{E},i},b_i,\mathbf{g}_i\right)\right]^2\mathrm{. (2)}$$

Results and Discussions

Conclusions

Acknowledgements

References

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