Comparing the performances of different divergence-free approaches to increase the accuracy of 4D flow MRI velocity data

Cardiovascular diseases (CVDs) are the leading cause of global death, accounting for 30% of all fatalities. The understanding of the hemodynamics of cardiovascular flows in vivo is important to evaluate diseases progression and the efficacy of surgical treatments. A new promising diagnostic technique for cardiovascular flow assessment is "4D flow MRI”. It overcomes the limitations associated with traditional methods, like Doppler echocardiography and 2D time-resolved phase-contrast (PC) MRI and it is capable to provide information on the temporal and spatial evolution of 3D blood flow, with complete volumetric coverage. However, noise-like errors significantly affect the accuracy of these measurements. In this context, the aim of this study is to enhance the quality of 4D flow MRI data by exploiting the physical law of the incompressibility of blood, which results in the imposition of a divergence-free condition on the flow field. In particular, three divergence-free approaches - Finite Difference Method (FDM), Divergence-Free Wavelets (DFW) and Radial Basis Functions (RBF) based methods – have been considered in this study to reduce the divergence of the velocity field provided by 4D flow MRI, reducing the deviation between the filtered flow field and the measured one. In detail, the Finite Difference Method (FDM) projects the velocity data into a divergence-free space using Helmholtz-Hodge decomposition, solved by employing first-order finite differences with periodic boundary conditions. Radial Basis Functions (RBF)-based method used normalized convolution to locally approximate the acquired velocity field into divergence-free radial basis functions (RBFs), used as convolution kernels. The normalized convolution operation is equivalent to solving a weighted linear least squares problem, the result of which is a vector of coefficients that is used to reconstruct the divergence-free flow field. Divergence-Free Wavelets (DFW)-based method applies the Discrete Wavelet Transform, realized through a bank of filters, separately, to each velocity component. The resulting divergence-free and non–divergence-free coefficients, linearly combined, provide a sparse representation of flow data, which were subjected to the denoising procedure using a soft-thresholding technique, called SureShrink. Furthermore, a second-generation denoising method, named “cycle spinning”, was considered to reduce blocking artifacts. Comparative analysis of these methods, applied on synthetic velocity data obtained from a CFD simulation of a realistic aorta at the systolic peak and with Gaussian noise superimposed, demonstrated, both qualitatively and quantitatively, the effective restoration of the original flow field and a reduction in the divergence of noisy data. Quantitative results showed that the DFW-based method performed better than FDM and RBF-based method. The no-divergence constraint in the DFW-based method is more relaxed, making it less sensitive to segmentation errors. Moreover, the DFW-based method with partial cycle spinning further enhanced its performance. In conclusion, the finding of this study can be significant because the integration of these denoising techniques into 4D flow MRI data could improve the accuracy of hemodynamic assessments, providing a significant advance in the diagnostic and therapeutic management of cardiovascular disease.