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Regularization stabilizes the fit of the two-compartment free water diffusion MRI model
Jordan A. Chad1,2, J. Jean Chen1,2, and Ofer Pasternak3

1Rotman Research Institute, Baycrest Health Sciences, Toronto, ON, Canada, 2Medical Biophysics, University of Toronto, Toronto, ON, Canada, 3Brigham and Women's Hospital, Harvard Medical School, Boston, MA, United States

### Synopsis

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.

### Introduction

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 model1

$$S(f,\mathbf{D}) = S_0 \bigg[ (1-f)e^{-b\hat{\mathbf{q}}^T\mathbf{D}\hat{\mathbf{q}}} + fe^{-bd} \bigg]$$ where $f$ is the signal fraction of the free water compartment with isotropic $d$ = 3x10-3mm2/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 minimize1

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

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.

### Methods

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/mm2 with 60 directions and 6 b=0; Multi-shell data was simulated using b=500s/mm2 with 60 directions, b=1500s/mm2 with 60 directions, and 12 b=0, based on a protocol previously suggested2. 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/mm2 + 6 b=0, TR=11.3s, TE=118ms, FOV=25.6cm, 2mm3 resolution, 72 slices. Maps of $f$ and $\mathbf{D}$ were computed1 and in turn plugged into the two-compartment model to synthesize data with varying levels of noise. An initialization scheme for free water imaging1 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

### Results

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.

### Discussion and Conclusion

While the fit of single-shell data to the two-compartment free water model is naturally ill-posed, this work demonstrates that regularizing the fit by exploiting the expected smoothness of the tensor introduces clear minima that are in good agreement with simulated ground truth. Regularization also improves the steepness of the solution space with multi-shell data. In addition to regularization, it is required to have a proper initialization and application on data where smoothly-varying diffusivity may be expected. Future work will investigate these aspects to further explore the potential for free water mapping in widely-available single-shell DTI data.

### Acknowledgements

This work was funded in part by the Jack and Rita Catherall Fund.

### References

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

### Figures

Figure 1. Non-regularized solution space (L1) in a simulated voxel of one- and two-shell data with ground-truth f=0.4. The solution space is flat for single-shell data and minima exist in two-shell data. Standard deviation of noise is a percentage of S0, as indicated in the legend.

Figure 2. Regularized solution space (L2) in a voxel of simulated full-brain one- and two-shell data with ground-truth f=0.4. Regularization introduces steep minima in comparison with Figure 1. Standard deviation of added noise is a percentage of S0, as indicated in the legend. Note that the solution space never quite hits 0, as a regularized model fit of 0 would imply spatial constancy of the tensor.

Figure 3. Regularized fit of free water maps from single-shell data. Computed free water values are highly correlated and in good agreement with simulated ground truth values across the slice.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)
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