Teddy Salan^{1}, Sulaiman Sheriff^{1}, and Varan Govind^{1}

In diffusion weighted imaging (DWI) of the brain, a single voxel may contain gray matter (GM), white matter (WM), as well as free water embedded in GM, WM, and cerebrospinal fluid (CSF), each with different diffusion profiles. Free water elimination (FWE) is a method used to separate the free water components from tissue water. In this work, we extend FWE by fitting with a diffusion kurtosis imaging (DKI) model as it is a better descriptor of the non-Gaussian diffusion that occurs in tissue water.

The FWE-DTI model is described by a simple bi-exponential
expansion of DTI^{1,4}

$$$S_{i}=S_{0}\left[\left(1-f\right)exp\left(-b_{i}g_i^T\,D_{tissue}g_{i}\right)+f\,exp\left(-bD_{iso}\right)\right]$$$. (1)

Here, the measured diffusion-weighted signal $$$S_{i}$$$ is the combination of free water, with a volume fraction $$$f$$$ and
isotropic diffusion $$$D_{iso}$$$ set at a constant value of $$$3\times10^{-3}$$$mm2/s,
and tissue water, with a volume fraction $$$\left(1-f\right)$$$
and diffusion tensor $$$D_{tissue}$$$ characterized by $$$6$$$ independent elements $$$\theta_{D}=\left\{D_{ij}\right\}_{i\leq\,j\leq\,3}$$$. $$$S_{0}$$$
is the signal when no diffusion sensitization gradient is applied and $$$g_{i}$$$ is the $$$i$$$-th component of the normalized
diffusion weighting gradient direction vector $$$\bf\,g$$$ with $$$i=1,...,m$$$ for $$$m$$$ gradient directions.
We expand this formulation using the DKI model^{6}

$$$S_{i}=S_{0}\left[\left(1-f\right)exp\left[-b_{i}g_i^T\,D\,g_{i}\,+\,\frac{b_i^2}{6}\left(\sum_{i=1}^3\frac{D_{ii}}{3}\right)^2\sum_{i,j,k,l=1}^3g_{i}g_{j}g_{k}g_{l}W_{ijkl}\right]+f\,exp\left(-bD_{iso}\right)\right]$$$, (2)

where $$$W_{ijkl}$$$ represents the $$$ijkl$$$-th element of the
fully symmetric fourth-order diffusion kurtosis tensor $$$W$$$, which can be
characterized by 15 independent elements $$$\theta_{K}=\left\{W_{ijkl}\right\}_{i\leq\,j\leq\,k\leq\,l\leq3} $$$. Using the bi-exponential expression with DKI, the
fitting problem becomes difficult and ill-conditioned as we now have to find 21 parameters $$$\theta=\left[\theta_{D},\theta_{K}\right]$$$. We use a two-step approach, with a weighted least
squares (WLS) initial estimation followed by a non-linear least squares (NLS) step
to refine the solution as proposed by Hoy et al.^{4,8} The goal is to find the
$$$\left(f,\theta\right)$$$ pair that minimizes the WLS
objective function

$$$F_{WLS}=\frac{1}{2}\sum_{i=1}^m\omega_i^2\left( y_{i}-\sum_{j=1}^{22}A_{ij}\gamma_{j}\right)^2$$$. (3)

The weights $$$\omega_i$$$ are set equal to the measured signal $$$s_i$$$, $$$A$$$ is the $$$\left(m\times22\right)$$$ diffusion encoding matrix, and $$$\gamma$$$ is the solution parameter matrix estimated by

$$$\gamma=\left( A^T S^2 A\right)^{-1}A^TS^2y$$$, (4)

where $$$S$$$ is a diagonal matrix of the measured diffusion signal, and $$$y$$$ is the natural log of the free water adjusted signal given by

$$$y_{ik}=ln\left\{\frac{s_i-s_0f_k\,exp\left(-b\,D_{iso}\right)}{\left(1-f_k\right)}\right\}$$$, (5)

with $$$k=1,…,n$$$ and $$$n$$$ is the number of $$$f$$$-values fitted simultaneously. Our method is implemented in Python using the Dipy libraries. The initial WLS solution is found by searching for the solution over intervals of $$$f=\left\{0,0.1,0.2,…1\right\}$$$. A second and third iteration are used to improve the solution over smaller intervals sizes of 0.01 and 0.001 respectively. The solution is further refined during NLS fitting using a modified Levenberg-Marquard algorithm to minimize the NLS objective function

$$$F_{NLS}=\frac{1}{2}\sum_{i=1}^m\left[s_i-S_0\,f\,exp\left(-b_i\,D_{iso}\right)-\left(1-f\right)exp\left(\sum_{j=1}^{22}A_{ij}\gamma_{j}\right)\right]^2$$$

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2. Lyall AE, et al. Greater extracellular free-water in first-episode psychosis predicts better neurocognitive functioning. Mol Psychiatry 2018;23(3):701–707.

3. Pasternak O, et al. Estimation of extracellular volume from regularized multi-Shell diffusion MRI. Med Image Comput Comput Assist Interv 2012;15(Pt 2):305-312

4. Hoy AR, et al. Optimization of a free water elimination two-compartment model for diffusion tensor imaging. NeuroImage 2014;103:323-333.

5. Collier, et al. Diffusion Kurtosis Imaging With Free Water Elimination: A Bayesian Estimation Approach. Magn Resn Med 2018;80(2):802-813.

6. Jensen JH, et al. Diffusional kurtosis imaging: the quantification of non-gaussian water diffusion by means of magnetic resonance imaging. Magn Reson Med 2005;53(6):1432-1440.

7. Jensen JH and Helpern JA (2010). MRI quantification of
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8. Henriques RN, et al. bioRxiv 2017. doi: http://dx.doi.org/10.1101/108795.

Figure 1: An
axial slice used to illustrate the parametric maps produced using our FWE-DKI method.
A b0 structural image is shown for anatomical reference (a). Our FWE-DKI method
can output the free water volume fraction (b), diffusivity measures such as
fractional anisotropy (FA) and mean diffusivity (MD) in (c) and (d), and
kurtosis maps mean kurtosis (MK), axial kurtosis (AK), and radial kurtosis
(RK)) in (e), (f) and (g) respectively.