Rafael Neto Henriques^{1}, Ørjan Bergmann^{2}, Ariel Rokem^{3}, Ofer Pasternak^{4}, and Marta Morgado Correia^{1}

Free-water diffusion tensor imaging (fwDTI) was previously proposed to remove CSF partial volume effects of measures based on the diffusion tensor. Nevertheless, this diffusion-weighted technique is still subject to several pitfalls. In this study, an improved algorithm to fit the fwDTI to data acquired with two or more diffusion-weighting gradients is proposed. This algorithm is then used to explore the advantages and limitations of suppressing free-water in synthetic and in vivo diffusion-weighted data.

fwDTI model fitting: Similarly to previous procedures^{4}, the fwDTI model is
fitted to the diffusion-weighted signals $$$s_i$$$ using a non-linear least
squares (NLS) approach. This approach is very sensitive to the initial guess of
the model parameters $$$\gamma$$$, which can be computed by a grid search of
the best weighted linear least squares (WLLS) solution of the model parameters.
The selection of the initial guess was previously performed using the WLLS
objective function^{4}, however we suggest
using the NLS objective function:

$$F_{NLS}=\frac{1}{2}\sum_{i=1}^{m}\left[s_{i}-S_{0}f\exp(-\sum_{j=2}^{4}W_{ij}D_{iso})-(1-f)\exp(-\sum_{j=1}^{7}W_{ij}\gamma_{j})\right ]^{2}$$

where
$$$f$$$ is the fraction of free water in the voxel, $$$D_{iso}$$$ is the diffusivity
of free water, and $$$W$$$ is a matrix containing the acquisition parameters^{4}.

Simulations: To evaluate the
performance of different fwDTI fitting algorithms, multi-compartmental
simulations were first performed using the acquisition scheme and ground truth
values proposed by^{4}. To assess fwDTI
effects in the presence of restricted and hindered diffusion, diffusion
heterogeneity was incorporated by separately modelling intra- and
extra-cellular compartments for a varying number of white matter fibers^{6}. These simulations were then used to detect ground truth $$$f$$$
and FA values resulting in FA overestimation (see Fig.3 for more details).

MRI experiments: DWI data were
acquired on a Siemens 3T Trio (32-channel
coil) using a TRSE sequence, 30 directions for b-values=500,
1500s/mm^{2} and two b=0)^{4}. A second protocol had
30 directions for bvalues=1000, 2000s/mm^{2} and two b=0. To assess fwDTI-FA bias across different brain
regions, in vivo fwDTI-$$$f$$$ and DTI-FA were compared to $$$f$$$ and FA ground
truth values which results in fwDTI overestimation. Since DTI-FA and fwDTI-$$$f$$$
are always smaller than the tissue's FA and free-water volume fractions ground
truth values, the estimated bias maps indicates the worst possible fwDTI-FA
overestimation at each brain locations (see Fig.4 for more details).

Using the approach for
fwDTI initial guess estimation proposed here, and contrary to the previous approaches
(Fig.1A-C)^{4,5}, we no longer see
underestimation of $$$f$$$ values (Fig.1D-F). The results from the NLS fitting
approach that uses the proposed fwDTI initial guess estimates are shown in
Fig.1(G-I).

When restricted and hindered diffusion effects are taken into account, the fwDTI’s $$$f$$$ estimates are still proportional to the ground truth values. However, $$$f$$$ now also depends on the volume fraction of hindered diffusion (Fig.2A). For well-aligned fiber simulations, fwDTI-FA estimates seem to be invariant to free-water volume fractions up to $$$f$$$=0.6 (Fig.2B-C). When higher b-values are included in the acquisition, $$$f$$$ becomes more sensitive to the hindered compartment volume fractions (Fig.2D), while fwDTI-FA estimates remain independent up to medium range free-water contaminations (Fig.2E-F).

The lower $$$f$$$ values associated to fwDTI-FA overestimates of 1 and 5% are plotted as a function of fwDTI-FA ground truth (Fig.3B,F). In-vivo fwDTI measures are shown in Fig.4. The fwDTI bias map reveals that fwDTI-FA overestimation is lower that 5% for most white matter regions (Fig.4C,F).

[1] Pasternak O, Sochen N, Gur Y, Intrator N, Assaf Y. Free water elimination and mapping from diffusion MRI. Magn Reson Med 2009; 62(3): 717-30. doi: 10.1002/mrm.22055.

[2] O’Donnell LJ, Pasternak O. Does diffusion MRI tell us anything about the white matter? An overview of methods and pitfalls. Schizophr Res. 2015; 161(1): 133–141 doi: 10.1016/j.schres.2014.09.007

[3] Pasternak O, Westin CF, Bouix S, Seidman LJ, Goldstein JM, Woo TUW, Petryshen TL, Mesholam-Gately RI, McCarley RW, Kikinis R, Shenton ME, Kubicki M. Excessive extracellular volume reveals a neurodegenerative pattern in schizophrenia onset. J Neurosci. 2012; 32(48):17365–17372.

[4] Hoy AR, Koay CG, Kecskemeti SR, Alexander AL. Optimization of a free water elimination two-compartment model for diffusion tensor imaging. NeuroImage 2014: 103; 323-333. doi: 10.1016/j.neuroimage.2014.09.053

[5] Bergmann Ø, Westin CF, Pasternak O. Challenges in solving the two-compartment free-water diffusion MRI model. Proceedings of the 24th Annual Meeting of the International Society for Magnetic Resonance Medicine; Singapore. May 7–13, 2016.

[6] Henriques RN, Correia MM, Nunes RG, Ferreira HA. Exploring the 3D geometry of the diffusion kurtosis tensor—Impact on the development of robust tractography procedures and novel biomarkers. NeuroImage 2015; 111: 85-99. doi: 10.1016/j.neuroimage.2015.02.004.

Fig.1 - Fractional Anisotropy and free water volume fraction estimates obtained from simulations using different fwDTI fitting procedures. The upper panels show the results obtain from the initial guess estimation procedure proposed by [4]; the middle panels show the results obtain from our proposed initial guess estimation strategy; while the lower panels show the results obtain from the NLS procedures that uses as model parameters initial guess our proposed algorithm.

Fig.2 - Fractional Anisotropy and free water volume fraction estimates obtained from simulations that take into account effect of tissue heterogeneity. While the upper panels shows the results for b-values = 500/1500s.mm-2, the lower panels show the results for b-values = 1000/2000s.mm-2.

Fig. 3 - The mean fwDTI-FA estimates are first plotted as function of free-water volume contaminations for different FA ground truth values (A,C). We then measure the lowest free water volume fraction contamination capable of inducing overestimations of 1 and 5% (red and black points in A and C). These volume fractions are then plotted as function of the fwDTI for f=0 (panels B and D). Upper and lower panels correspond to data acquired for b-values=500/1500s.mm-2 and 1000/2000s.mm-2.

Fig.4 - In vivo measures of free water volume fraction (panels A, D), fwDTI-FA (panels B, E) and fwDTI-FA bias (panels C, F). The fwDTI-FA bias maps are computed by comparing the free water volume and DTI-FA estimates to the curves in Fig.4.B,D. Dark to light gray intensities are associated respectively to fwDTI-FA bias lower than 1%, lower than 5% and higher than 5%, while regions that contain mostly free water are associated to white voxels. Upper and lower panels correspond to data acquired for b-values=500/1500s.mm-2 and 1000/2000s.mm-2.