Purpose

To propose a unified framework that integrates multiple prior information including nonlocal structural self-similarity (NSS), local spatial smoothness (LSS), physical relevance (PR) of DKI model, and noise characteristic of magnitude diffusion MRI (dMRI) images for improved DKI tensor estimation.

Methods

The diffusion tensor (DT) and kurtosis tensor (KT) of all pixels across the image (namely DT and KT fields) are simultaneously estimated by our unified estimation framework. Suppose the size of a 2D image is M×N, the unified estimation framework for DT field D_f$$$\in$$$R^M×N×3×3 and KT field W_f$$$\in$$$R^{M×N×3×3×3×3} can be formulated as:$$\min_{ {\bf \theta} }F({\bf \theta})=\min_{ {\bf \theta} }\{\sum_{ x_{i}\in Ω}\sum_{ x_{j}\in V_{i}}w(x_{i},x_{j})\sum_{n=1}^{N_{DWI}}\|S_{m}(b_{n};{g_{n}};x_{j})-f(b_{n};{g_{n}};{\bf \theta}(x_{i}))||_{2}^{2}+\lambda\sum_{q=1}^{Q}TV({\bf \theta}_{q})+R({\bf \theta}_{D_{f}},{\bf \theta}_{W_{f}})\}$$(1)
Here, the first term is a weighted nonlinear least squares term which incorporates the nonlocal structural self-similarity (NSS) prior information to reduce the noise effect on the model fitting. θ$$$\in$$$R^M×N×Q denotes all the parameter maps where Q is the total number of DKI model parameters for each pixel. S_m is the measured signal with the diffusion weighting strength bn along the gradient direction g_n, f(⁕) the first-moment noise-corrected model (M¹NCM) as a function of θ and can be formulated as:$$f({\bf \theta})=\sigma\sqrt{\frac{\pi}{2}}\frac{(2C-1)!!}{2^{C-1}(C-1)!!}F_{1}^{1}(-0.5;C;(\frac{S(b;g;{\bf \theta})}{\sqrt{2}\sigma})^{2})$$(2)
where σ the standard deviation (SD) of complex Gaussian noise, C the number of coils, !! the double factorial, $$$F_{1}^{1}$$$the confluent hypergeometric function, S the underlying noise-free signal. w(x_i, x_j) is the weight that controls the contribution of neighboring pixel x_j within search window V_i to the tensor estimation of target pixel x_i within image domain Ω and is defined as:$$w(x_{i},x_{j})=\frac{1}{Z(x_{i})}exp(-\frac{_{G_{a}\|P_{i}-P_{j}||_{2}^{2}}}{h^{2}}),\forall x_{j}\in V_{i} and x_{j}\neq x_{i}$$(3)
where Z(x_i) is the normalized constant, G_a a normalized Gaussian kernel (the SD of a), h controls the degree of smoothing and determined as h = βσ, where β is a scalar. P_i and P_j denote 3D patch of p×p×N_DWI, where p is the patch size in the spatial domain, and the patch size along the diffusion dimension is set to the total number of non-DW and DW images (N_DWI).
The second term is a total variation (TV) regularization term which constraints the local spatial smoothness (LSS) on each parameter map θ_q$$$\in$$$R^M×N.
The third term ensures that the DKI model is physically relevant^1,2. For each diffusion encoding direction g, we use the following function to construct the R(θ_Df, θ_Wf):$$R({\bf \theta}_{D_{f}},{\bf \theta}_{W_{f}})=\sum_{ x_{i}\in Ω}\sum_{n=1}^{N_{DWI}}(exp(\frac{{K_{f}(g_{n},x_{i})-3/b_{max}/}D_{f}(g_{n},x_{i})}{c_{1}})+exp(\frac{{K_{f}(g_{n},x_{i})}}{c_{2}}))$$(4)
Here, c₁ and c₂ are small values to ensure R(θ_Df, θ_Wf) approaches infinity (or a big number) when the physical relevance is not satisfied. We solved the unconstrained nonlinear optimization problem (Eq. (1)) by using the limited-memory BFGS Quasi-Newton method (L-BFGS).
To quantitatively evaluate the performance of the proposed M¹NCM-NSS-LSS-PR algorithm, simulations were performed based on one subject (“100307”) from HCP diffusion data³. First, the reference DKI tensors were estimated using constrained weighted least square estimator⁴ after preprocessing of denoising and bias correction. Second, the estimated DKI tensors were used to simulate the reference data with one non-DW image and 45 b = 1000 and 45 b = 2000 s/mm² DW images. Third, the reference serial images were used to generate noncentral Chi (C = 8) distributed serial images with stationary noise levels of 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09 and unstationary noise levels of 0.02 and 0.03. A modulation map with factors 1-3 was multiplied with the unstationary noise level to generate the spatially unstationary noise. The Gaussian noise can be computed in advance by using background area for stationary noise case and be treated as an unknown parameter for unstationary noise case.
To validate the effect of introducing different prior information on the estimation of DKI parameters, we compared the following M1NCM-based methods: 1) M¹NCM; 2) M¹NCM-NSS; 3) M¹NCM-LSS; 4) M¹NCM-NSS-LSS; 5) M¹NCM-NSS-LSS-PR. We also compared the proposed M¹NCM-NSS-LSS-PR algorithm with two widely used methods: nonlinear least squares (NLS) and constrained weighted linear least squares (CWLLS)⁴. For the sake of fairness, we adopted an unbiased vector nonlocal means (UVNLM) filter⁵ to denoise and correct the noise bias prior to the NLS or CWLLS estimation (referred as UVNLM-NLS and UVNLM-CWLLS, respectively).

Results

Fig. 1 presents the RMSEs of FA, MD, MK maps estimated by the different algorithms, using the simulated data with stationary noise levels of 0.02-0.09. Table 1 shows the RMSE values of the FA, MD, MK maps as well as the noise maps estimated by the proposed M¹NCM-based algorithms, using the simulated data with unstationary noise levels of 0.02 and 0.03. In terms of RMSE for each parameter map, M¹NCM-NSS-LSS and M¹NCM-NSS-LSS-PR outperformed other compared algorithms at all the noise levels. In Figs. 2 and 3, a visual comparison of the estimated FA, MD, MK maps and their corresponding error maps is presented. It can be seen that the parameter maps from the proposed M¹NCM-NSS-LSS or M¹NCM-NSS-LSS-PR methods had the minimum errors and were visually closest to the reference parameter maps among all compared methods.

Discussion and conclusion

The proposed M¹NCM-NSS-LSS-PR outperformed the compared methods on the simulated data with both the spatially stationary and unstationary noise distributions. The proposed method can be easily extended to a 3D version for the volume diffusion data, which will further improve the accuracy of DKI tensor estimation.

Acknowledgements

This study was funded by China Postdoctoral Science Foundation (NO.2020M672526), Guangdong Basic and Applied Basic Research Foundation (NO.2019A1515110976), National Natural Science Foundation of China (NO.61971214, 81601564), Natural Science Foundation of Guangdong Province (NO.2019A1515011513), Guangdong-Hong Kong-Macao Greater Bay Area Center for Brain Science and Brain-Inspired Intelligence Fund (NO.2019022).

References

[1] Veraart, J., et al. “Constrained maximum likelihood estimation of the diffusion kurtosis tensor using a rician noise model”. Magn Reson Med 66, 678-686. (2011). [2] Tabesh, A., et al. “Estimation of tensors and tensor-derived measures in diffusional kurtosis imaging”. Magn Reson Med 65, 823-836. (2011). [3] https://www.humanconnectome.org/ [4] Veraart, J., et al. “Weighted linear least squares estimation of diffusion mri parameters: Strengths, limitations, and pitfalls”. Neuroimage 81, 335-346. (2013). [5] Zhou, M.X., et al. “Evaluation of non-local means based denoising filters for diffusion kurtosis imaging using a new phantom”. PLoS One 10, e0116986. (2015).