Li Guo^{1,2,3}, Xinyuan Zhang^{2,3}, Changqing Wang^{4}, Jian Lyu^{2,3}, Yingjie Mei^{5}, Ruiliang Lu^{1}, Mingyong Gao^{1}, and Yanqiu Feng^{2,3}

^{1}Department of MRI, The First People’s Hospital of Foshan (Affiliated Foshan Hospital of Sun Yat-sen University), Foshan, China, ^{2}School of Biomedical Engineering, Southern Medical University, Guangzhou, China, ^{3}Guangdong Provincial Key Laboratory of Medical Image Processing, Southern Medical University, Guangzhou, China, ^{4}School of Biomedical Engineering, Anhui Medical University, Hefei, China, ^{5}Philips Healthcare, Guangzhou, China

The noncentral Chi noise in magnitude image may significantly
affect the reliability of quantitative analysis in diffusion-weighted (DW)
magnetic resonance imaging (MRI), especially at high b-value and/or higher order
modeling of diffusion signal such as diffusion kurtosis imaging (DKI). We
developed a novel first-moment noise-corrected curve fitting model with
adaptive neighborhood regularization (MN^{1}CM-ANR) algorithm for DKI. By
fitting the signal to its first-moment (i.e. the expectation of the signal), MN^{1}CM-ANR can effectively compensate
the bias due to the noncentral Chi noise. In addition, by exploiting the
neighboring pixels to regularize the curve fitting, MN^{1}CM-ANR can reduce
the measurement variance.

$$E(S_{M})=\sigma\sqrt{\frac{\pi}{2}}\frac{(2L-1)!!}{2^{L-1}(L-1)!}1F1(-\frac{1}{2};L;-(\frac{S^{DKI}}{\sqrt{2}\sigma})^{2}),{~~~~~~}(1)$$

where

To further reduce the variance in DKI parameter estimates, an adaptive neighborhood regularization (ANR) was incorporated into the MN

$$\min_{\theta}||S_{x_{i}}-f(\theta)||_2^2+\sum_{x_{j}\in_{W_{i}},j\neq{i}}\alpha(x_{i},x_{j})||S_{x_{i}}-f(\theta)||_2^2,\forall{x_{i}}\in{I},{~~~~~~}(2)$$

where the first term is the data fidelity, and the second one is the regularization.

$$\alpha(x_{i},x_{j})=exp(-\frac{||S_{x_{i}}-S_{x_{j}}||_2^2}{12h^{2}}),\forall{x_{j}}\in{W_{i}}{~~}and {~~}x_{j}\neq{x_{j}},{~~~~~~}(3)$$

where

To evaluate the performance of the proposed MN

We compared the following methods using the simulation: 1) NLS, curve fitting without denoising and noise-correction; 2) NLM+NLS, curve fitting after non-local means (NLM) denoising

Figure
1. FA, colored FA, and the corresponding error maps of the NLS, NLM+NLS, MN^{1}CM,
NLM+MN^{1}CM, MN^{1}CM-ANR. The reference maps are presented in
the first column. The error images show the absolute difference between the
reference map and the estimated map. The color is based on the orientation of
the primary eigenvector of diffusion tensors: red for right-left, green for
anterior-posterior, and blue for inferior-superior. Numbers in bottom right
represent RMSE.

Figure
2. MD, MK, and the corresponding error maps of the NLS, NLM+NLS, MN^{1}CM,
NLM+MN^{1}CM, MN^{1}CM-ANR. The reference maps are presented in
the first column. The error images show the absolute difference between the
reference map and the estimated map. Numbers in bottom right represent RMSE.

Figure
3. Quantitative comparison of FA, MD, and MK estimation. (a) FA quantification
errors. (b) MD quantification errors. (c) MK quantifications. The bias (solid
line) and the standard variation (dash line) are also shown.