Purpose

Diffusion-weighted (DW) magnetic resonance imaging (MRI) remains challenging as a result of the low signal-to-noise (SNR). Multichannel array coils are usually applied, and the magnitude reconstruction results in non-central Chi noise distribution. This noise may introduce significant bias and variance in the estimated diffusion parameters, especially at high b-values and/or higher order modeling of diffusion signal such as diffusion kurtosis imaging (DKI) ¹. The goal of this work is to propose a novel curve fitting model with adaptive neighborhood regularization for DKI, which can improve the accuracy and precision of DKI parameters.

Methods

To reduce the bias in DKI parameters, a first-moment noise-corrected (MN¹CM) curve fitting model was adopted: fitting the DKI signals at each pixel to its first-moment (i.e. the expectation of the signals). The first-moment of the measured signal S_M can be formulated as ²:
$$E(S_{M})=\sigma\sqrt{\frac{\pi}{2}}\frac{(2L-1)!!}{2^{L-1}(L-1)!}1F1(-\frac{1}{2};L;-(\frac{S^{DKI}}{\sqrt{2}\sigma})^{2}),{~~~~~~}(1)$$
where σ is the standard deviation of noise; L is the number of receiver coils; !! is the double factorial: n!! = n(n-2)(n-4)…; ₁F₁ is the confluent hyper-geometric function; S^DKI represents the theoretical DKI model signal.
To further reduce the variance in DKI parameter estimates, an adaptive neighborhood regularization (ANR) was incorporated into the MN¹CM curve fitting model: neighboring pixels with similar DKI signals were simultaneously fitted to reduce the effect of noise and the weighting of each neighboring pixel was adaptively determined based on the interpixel signal similarity. The MN¹CM curve fitting with ANR (named as MN¹CM-ANR) algorithm can be given by minimizing the following objective function:
$$\min_{\theta}||S_{x_{i}}-f(\theta)||_2^2+\sum_{x_{j}\in_{W_{i}},j\neq{i}}\alpha(x_{i},x_{j})||S_{x_{i}}-f(\theta)||_2^2,\forall{x_{i}}\in{I},{~~~~~~}(2)$$
where the first term is the data fidelity, and the second one is the regularization. θ denotes the parameter vector containing the non-DW signal S₀, diffusion tensor D, and kurtosis tensor K; $$$S_{x_{i}}$$$ is the vector representing the measured DKI signals at target pixel x_i in image domain I; f(·) is the MN¹CM curve fitting model (right-hand in Eq. (1)); x_j is the neighboring pixel in search window W_i around target pixel x_i; α(x_i,x_j) is the regularization parameter and was adaptively calculated as follows ^3,4:
$$\alpha(x_{i},x_{j})=exp(-\frac{||S_{x_{i}}-S_{x_{j}}||_2^2}{12h^{2}}),\forall{x_{j}}\in{W_{i}}{~~}and {~~}x_{j}\neq{x_{j}},{~~~~~~}(3)$$
where h is the smoothing parameter and is related to the noise level (h = βσ, where β is a tuning parameter). Nonlinear optimization was implemented to solve Eq. (2). A rapid look-up table method was used to accelerate the calculation of the confluent hypergeometric function ⁵.
To evaluate the performance of the proposed MN¹CM-ANR algorithm, simulation was performed based on the MASSIVE database ⁶. First, the reference D, K and tensor-derived parameters were estimated using nonlinear least square (NLS) estimator. Second, the estimated D, K were used to simulate the reference data with 10 non-DW signal and DW signal on four shells with b-values of 500 s/mm², 1000 s/mm², 2000 s/mm², and 3000 s/mm². The number of diffusion gradient directions per shell is 36, 50, 50, and 50. Third, the reference serial images were used to generate noncentral Chi (L = 4) distributed serial images with SNR of 15.
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 ⁷; 3) MN¹CM, noise-correction curve fitting without denoising; 4) NLM+MN¹CM, noise-correction curve fitting after denoising; 5) MN¹CM-ANR, simultaneous noise-correction curve fitting and denoising. In this study, the search window W_i was set to 9×9 and the smoothing parameter h was set to 1.5σ for MN¹CM-ANR. With regard to the NLM+NLS, NLM+MN¹CM, the optimal h was selected by visual inspection of the denoised image.

Results

Figures 1 and 2 show the fitted fractional anisotropy (FA) maps, colored FA maps, mean diffusivity (MD) maps, mean kurtosis (MK) maps and their corresponding error maps. Numbers in bottom right are root-mean-square-error (RMSE). The NLS and MN¹CM methods produced noisy maps. Although the NLM+NLS, and NLM+MN¹CM methods produced maps that are less affected by the noise, measurement errors were obvious. The MN¹CM-ANR algorithm effectively reduced the effect of noise on maps and reduced the measurement errors. Figure 3 presents the FA, MD and MK quantification errors for all the algorithms. The FA and MK errors of MN¹CM-ANR are more concentrated around zeros with smaller bias (solid line) and narrower variance (dashed line).

Discussion

We developed a novel first-moment noise-corrected curve fitting model with adaptive neighborhood regularization (MN¹CM-ANR) algorithm for DKI. MN¹CM-ANR can simultaneously correcting the noise bias and exploiting information redundancy across spatial domain and along diffusion encoding directions. Simulation results demonstrated that the MN¹CM-ANR method can successfully mitigate the effect of noise on DKI parameters and reduced the bias and variance of the estimated diffusion parameters. Note that NLM+MN¹CM also used both the noise-corrected curve fitting and neighborhood pixels to reduce the effect of noise but in a two-stage pattern, which may induce the potential error propagation from the denoising stage to the curve fitting stage. We expect to further improve MN¹CM-ANR by incorporating outlier detection scheme ^8,9 to deal with the physiological noise such as artifacts caused by motion or system instabilities in the future.

Conclusion

The MN¹CM-ANR can reduce the effect of noise on DKI parameters with improved accuracy and precision.

Acknowledgements

No acknowledgement found.

References

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