Introduction

Non-Gaussian intravoxel incoherent motion (NG-IVIM)¹ has been proposed to simultaneously quantify the perfusion and non-Gaussian diffusion properties in tissues. However, accurate parameter estimation for NG-IVIM is usually challenged by noise. The noncentral χ-distribution noise would introduce bias in the estimated NG-IVIM parameters. In addition, severe noise easily causes the estimated parameter values have large variance. To improve the accuracy and precision of parameter estimation for NG-IVIM, we propose to use an unbiased vector non-local means (UVNLM) filter to denoise and correct the noise bias before NG-IVIM model fitting.

Methods

UVNLM subtracts the noise bias from the vector NLM (VNLM)² filtered image, which can be expressed as:$$UVNLM(m(x_{i}))=\sqrt{(VNLM(m(x_{i})))^{2}-2L\sigma^{2}}$$(1)
where L is the number of coil channels, σ is the standard deviation (SD) of noncentral χ-distribution noise, m(x_i) is a vector of size Ndwi indicating the signal profile at position x_i in image m, NDWI is the total number of non-diffusion and diffusion images. VNLM(m(x_i)) is a spatial domain filter² that replaces each signal vector m(x_i) in the image with a weighted average of every neighboring signal vector m(x_j) in its “search region” V_i:$$VNLM(m(x_{i}))=\sum_{x_{j}\in V_{i}}w(x_{i},x_{j})m(x_{j})$$(2)
w(x_i,x_j) is the weight that controls the contribution of neighboring pixel x_j within search window to the target pixel x_i and is defined as:$$w(x_{i},x_{j})=\frac{1}{Z(x_{i})}exp(-\frac{G_{a}||P(x_{i})-P(x_{j})||_2^2}{h^{2}}),\forall x_{j}\in V_{i} and x_{j} \neq x_{i}$$(3)
with$${Z(x_{i})}=\sum_{x_{j}\in V_{i}}exp(-\frac{G_{a}||P(x_{i})-P(x_{j})||_2^2}{h^{2}})$$(4)
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(x_i) and P(x_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 N_DWI. To avoid the over-weighting due to the self-similarity when x_j = x_i, the weight w(x_i, x_i) is calculated as w(x_i, x_i) = max{w(x_i, x_j), $$$\forall$$$x_j≠x_i}. With the UVNLM, the parameters of NG-IVIM model can be estimated by using the following objective function:$$\min_{\theta}\sum_{n=1}^{N_{DWI}}\|UVNLM(S_{m}(b_{n}))-S(b_{n};{\theta})||_{2}^{2}$$(5)
where S_m is the measured signal profile of a single pixel, b_n the diffusion weighting strength of nth diffusion weighted image, θ = {f, D*, Dapp, Kapp, S₀} the NG-IVIM model parameter, S the noise-free diffusion signal which can be written as follows:$$S(b;\theta)=S_{0}(fexp(-bD^{*})+(1-f)exp(-bD_{app}+\frac{(bD_{app})^{2}K_{app}}{6}))$$(6)
where S₀ is the signal without diffusion gradient (b = 0), f the perfusion fraction, D* the pseudodiffusion coefficient related to the blood flow velocity, Dapp the apparent diffusion coefficient, Kapp the apparent kurtosis.

Simulations were performed based on a synthetic liver NG-IVIM data. The liver parenchyma parameters of f = 0.15, D* = 50 ×10-3 mm²/s, Dapp = 1.5×10-3 mm²/s, and Kapp = 0.8 were used to simulate the reference NG-IVIM data with 18 b-values of 0, 10, 20, 30, 40, 60, 80, 100, 150, 200, 300, 400, 500, 600, 800, 1000, 1500, 2000 s/mm². The reference images were then used to generate noncentral Chi (L = 8) distributed serial images with three SNRs of 10, 30, 50.

We compared the following three methods: 1) conventional nonlinear least squares (NLS) without denoising and bias correction; 2) NLS after principal component analysis denoising (referred as PCA-NLS); 3) NLS after UVNLM (referred as UVNLM-NLS).

Results

Fig. 1 presents the RMSEs of NG-IVIM maps estimated by the different methods. The proposed UVNLM-NLS method outperformed the conventional NLS and PCA-NLS for D*, Dapp, Kapp at all three SNRs. In Figs. 2-4, a visual comparison of the estimated NG-IVIM maps and their corresponding error maps is presented for conventional NLS (Fig. 2), PCA-NLS (Fig. 3), and the proposed UVNLM-NLS (Fig. 4). It can be seen that the parameter maps from the proposed method were visually closest to the reference parameter maps among all compared methods under three SNRs of 10, 30, 50.

Discussion and conclusion

In order to reduce the noise effect on the NG-IVIM parameter estimation, we adopted a UVNLM scheme to reduce both the noise variance and bias on the image domain before the model fitting. Simulation results demonstrated that the UVNLM scheme can significantly improve parameter estimation for NG-IVIM. Our future work would be to validate the usefulness of UVNLM scheme for improving parameter estimation of NG-IVIM on clinical in vivo data.

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

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