Chaehyuk Im^{1}, Seungbin Ko^{1}, Jeesoo Lee^{1,2}, Jee-Hyun Cho^{3}, Doosang Kim^{4}, Sang Hyung Lee^{5}, and Simon Song^{1,2}

Flow data measured by 4D flow MRI often result in inaccurate wall shear stress estimation due to near-wall noise in velocity measurements. We propose wall-bounded divergence-free smoothing (WB-DFS) to denoise the flow data. This method minimizes a residual error under the divergence-free condition for a wall-bounded flow and simultaneously performs data smoothing. The denoising performance of WB-DFS was found to be the best among methods reported in

** **The DFS method proposed by Wang et al. ^{4} is based on the combination of a penalized least squares technique and the divergence corrective scheme method. Mathematically, a solution is obtained when the objective function (Eq.1) is minimized while the constraint condition (Eq.2) is satisfied.

$$ F(U_c)=(U_c-U_{exp})^T(U_c-U_{exp})+sR(U_c) \tag{1}$$

$$ subject. to. \nabla \cdot U_c = 0 \tag{2}$$

To incorporate wall-boundary information into the DFS method, the segmentation mask of a flow geometry is applied into Eq. 1 and Eq. 2, and the final solution $$$U_c$$$ can be obtained by linear algebraic calculation as follows.

$$ U_c = \Phi_{wb}(I + s\Sigma)^{-1} \Phi_{wb}^TU_{exp} \tag{3}$$

The subscript $$$c$$$ and $$$exp$$$ denote the corrected data and raw experimental data, respectively. Also, $$$\Phi_{wb}$$$, $$$\Sigma$$$, and $$$s$$$ denote the bases of WB-DFS, singular value matrix, and the smoothing parameter. To compare the denoising performance of WB-DFS and other divergence reduction methods (FDM ^{5}, DFWs ^{5}, RBF ^{6}, and DFS), we used the CFD results for a stenosed pipe flow by Ong ^{5} as a control, and the CFD results to which 10 % Gaussian noise and near-wall outliers were added to mimic experimental data with uncertainty.

- Nayak KS, Nielsen J-F, Bernstein MA, et al. Cardiovascular magnetic resonance phase contrast imaging. Journal of Cardiovascular Magnetic Resonance 2015;17(1):71.
- Lustig M, Pauly JM. SPIRiT: Iterative self‐consistent parallel imaging reconstruction from arbitrary k‐space. Magnetic resonance in medicine 2010;64(2):457-471.
- Markl M, Bammer R, Alley M, et al. Generalized reconstruction of phase contrast MRI: analysis and correction of the effect of gradient field distortions. Magnetic resonance in medicine 2003;50(4):791-801.
- Wang C, Gao Q, Wang H, Wei R, Li T, Wang J. Divergence-free smoothing for volumetric PIV data. Experiments in Fluids 2016;57(1):15.
- Ong F, Uecker M, Tariq U, et al. Robust 4D flow denoising using divergence‐free wavelet transform. Magnetic resonance in medicine 2015;73(2):828-842.
- Busch J, Giese D, Wissmann L, Kozerke S. Reconstruction of divergence‐free velocity fields from cine 3D phase‐contrast flow measurements. Magnetic resonance in medicine 2013;69(1):200-210.
- Craven P, Wahba G. Smoothing noisy data with spline functions. Numerische mathematik 1978;31(4):377-403.

Figure. 1. Velocity
magnitude and velocity magnitude error. From left to right: Clean CFD data,
noisy data, and denoised results of FDM, DFWs, RBF, DFS, and WB-DFS,
respectively. The noisy data contains a random 10% Gaussian noise over the
entire flow field and outliers near the wall. The red arrow indicates the
streamwise direction of the flow. The inlet of the stenosis is magnified at the
top and the near-wall region at the outlet of the stenosis is magnified at the
bottom. The velocity magnitude error is magnified by 10 times in the
rectangular insets.

Figure. 2. Divergence
magnitude of each data. From left to right: Original CFD data, noisy CFD data,
and results of applying FDM, DFWs, RBF, DFS, and WB-DFS to noisy CFD,
respectively. The noisy data contains 10% Gaussian noise and the outliers are
in the range of ±5 standard deviation at a random position near the vessel
boundary.

Figure. 3. Performance evaluation of WB-DFS. Top left: Flow-rate variation during a cardiac cycle and a patient-specific 3D model of a carotid CT scan after carotid endarterectomy. $$$t_f$$$ denotes time fraction during the cardiac cycle and the white arrow indicates the flow direction. Top right: Divergence magnitude map on a slice of the carotid phantom flow at two time steps. Bottom: Divergence magnitude distribution at each time step during the cardiac cycle.

Figure. 4. Velocity vector visualization at a slice of the carotid artery phantom flow at $$$t_f=0.3$$$.