Jordan A. Chad^{1,2}, J. Jean Chen^{1,2}, and Ofer Pasternak^{3}

Free water mapping using single-shell diffusion MRI has been proposed by regularizing the ill-posed problem, but the effect of this regularization has yet to be systematically investigated. Here we explore the solution space for the two-compartment free water model. Without regularization, the solution space is flat for one-shell data, and while it has a minimum for two-shell data, this minimum is not particularly steep. Regularizing the fit by exploiting smoothness of the tensor creates a steep minimum in both one- and two-shell data, demonstrating the advantage of regularization.

Diffusion MRI provides insight into tissue microstructure by measuring microscopic diffusion through tissue. To separate out the effect of extracellular free diffusion unrestrained by tissue, the conventional diffusion tensor (DTI) model can be extended to the two-compartment free water model^{1}

$$S(f,\mathbf{D}) = S_0 \bigg[ (1-f)e^{-b\hat{\mathbf{q}}^T\mathbf{D}\hat{\mathbf{q}}} + fe^{-bd} \bigg] $$ where $$$f$$$ is the signal fraction of the free water compartment with isotropic $$$d$$$ = 3x10^{-3}mm^{2}/s and $$$\mathbf{D}$$$ is the diffusion tensor for the tissue compartment. This model can be fit to diffusion data $$$\hat{S}$$$ by minimizing $$L_1(f,\mathbf{D}) = \Big| \hat{S} - S(f,\mathbf{D})\Big|^2 $$The functional $$$L_1$$$ contains two free parameters, $$$f$$$ and $$$\mathbf{D}$$$. Typical DTI datasets are acquired with only one b-value, making the minimization ill-posed. It has consequently been proposed to minimize^{1}

$$L_2(f,\mathbf{D}) = \Big| \hat{S} - S(f,\mathbf{D})\Big|^2 + \gamma({\mathbf{D}})$$

where $$$\gamma$$$ is a spatial tensor regularization operator, such as the Laplace-Beltrami operator. Here for simplicity we choose $$$\gamma = || \nabla \mathbf{D} ||$$$ where the spatial gradient $$$\nabla$$$ is taken for each tensor element which are channeled through a Euclidean metric.

In this work, we (a) explore the solution space of $$$L_1$$$ and $$$L_2$$$, and (b) use synthetic data to demonstrate the regularization effect on the fit of standard single-shell diffusion MRI data.

1. Solution Space of $$$L_1$$$

Signal in a single voxel was simulated by synthesizing the two-compartment model with ground truth $$$f=0.4$$$ and $$$\mathbf{D}$$$ with eigenvalues $$$\lambda_1 = 1.5$$$, $$$\lambda_2 = 0.4$$$, $$$\lambda_3 = 0.4$$$. $$$S_0 = 1$$$ for simplicity. Noise was added to the voxel by $$$S =\sqrt{(\tilde{S} + \eta)^2 + \eta^2} $$$ for various levels of $$$\eta$$$. Signal from single-shell data was simulated using b=1000s/mm^{2} with 60 directions and 6 b=0; Multi-shell data was simulated using b=500s/mm^{2} with 60 directions, b=1500s/mm^{2 }with 60 directions, and 12 b=0, based on a protocol previously suggested^{2}. The search space was generated using a brute-force approach, i.e. calculating $$$\mathbf{D}$$$ for each $$$f$$$ in the range 0 to 1, and then evaluating the fit by explicitly calculating $$$L_1$$$ for each obtained $$$f$$$ and $$$\mathbf{D}$$$^{3}.

2. Solution Space of $$$L_2$$$

To obtain realistic synthetic data for regularization, a volunteer was scanned on a 3T MR Siemens Trio with diffusion MRI at 60 directions with b=1000s/mm^{2} + 6 b=0, TR=11.3s, TE=118ms, FOV=25.6cm, 2mm^{3} resolution, 72 slices. Maps of $$$f$$$ and $$$\mathbf{D}$$$ were computed^{1} and in turn plugged into the two-compartment model to synthesize data with varying levels of noise. An initialization scheme for free water imaging^{1} was used on synthetic data with SNR=20 to obtain an initial tensor map for regularization. Then, at each voxel, the search space was generated using the brute-force approach described above. The fitting was evaluated by explicitly calculating $$$L_2$$$, where the initial tensor map was used to compute $$$ \gamma(\mathbf{D})$$$. A voxel with ground-truth $$$f=0.4$$$ was chosen to explore the solution space of $$$L_2$$$. Additionally, synthetic data with SNR=20 was used to explore the regularized fit across a simulated brain slice

As displayed in Figure 1a, the solution space of the fit of single-shell data to the two-compartment model is flat, indicating an ill-posed fit. In the absence of noise, for any chosen $$$f$$$ there is a $$$\mathbf{D}$$$ that results in a perfect fit. Addition of noise worsens the fit, but the fit remains equivalent for each $$$\{f,\mathbf{D}\}$$$ combination. Adding a shell to the data creates a minimum in the solution space, but the minimum is not particularly steep (Figure 1b).

Figure 2 shows that adding regularization ($$$L_2$$$) creates a minimum in the single-shell data solution space and steepens the minimum in two-shell data solution space. The minima of the solution spaces are similar to the ground-truth values even in the presence of noise.

Figure 3 demonstrates that free water maps computed from simulated single-shell data are highly correlated and similar to the simulated ground-truth.

1. Pasternak O, Sochen N, Gur Y, et al. Free water elimination and mapping from diffusion MRI. Magn Reson Med. 2009;62(3):717-730.

2. 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.

3. Bergmann Ø, Westin CF, Pasternak O. Challenges in solving the two-compartment free-water diffusion MRI model. ISMRM 2016: 793