Three‐dimensional quantification of vorticity and helicity from 3D cine PC‐MRI using finite‐element interpolations

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Three‐dimensional time‐resolved phase contrast (PC) MRI (3D cine PC‐MRI) has been used for the noninvasive study of hemodynamic parameters of the cardiovascular system 1 and has shown its potential to evaluate different cardiovascular conditions 2. Several methods have been proposed for estimating the quantitative flow parameters from 3D cine PC‐MRI, including the wall shear stress, oscillatory shear index 3, vorticity, and helicity 1. However, there is less knowledge about the importance or effects of secondary flow patterns such as vorticity and helicity within the cardiovascular system. A few studies of vorticity and helicity‐related parameters have shown that these parameters have a direct relationship with geometric changes of the vessels and the reappearance of different diseases 7. Studies based on 3D cine PC‐MRI performed on phantom data have revealed that vortex and helical flow can induce vessel wall injury and vascular remodeling 7. Additionally, other studies have demonstrated that standard surgical procedures induce local and global hemodynamic changes of the treated section, altering the secondary flow patterns, which could lead to the reappearance of the pathology and the need for re‐intervention 12.
Qualitative or semi‐quantitative approaches have been proposed to evaluate the vorticity and helicity from 3D cine PC‐MRI in the context of heart failure and ventricular dysfunction 20. Other semi‐quantitative methods have been developed to assess vortices in the aorta and pulmonary artery, including features such as the vortex size, duration of the vortex, and other scores related to the degree of the rotation of the vortices 8.
Additionally, some other approaches have been reported in the literature to assess vorticity. For instance, the method described in 32 uses an 8‐point circulation approach to study the dynamics of vortex propagation using particle imaging velocimetry data. In 33, the authors propose a fourth‐order central difference approximation, whereas in 34, the authors proposed a fourth‐order Richardson extrapolation method to assess the vorticity in particle imaging velocimetry and 3D PC‐MRI data. In 38, the authors propose the Lambda2 approach, which is an objective method that identifies 3D vortex cores based on their physical fluid dynamics properties. A more sophisticated scheme to measure vortex ring based on the Lagrangian coherent structures (LCS) was proposed in 40. The LCS method is used to define the vortex boundaries, making it possible to measure the vortex volume. The LCS approach has been applied to study vortex ring formation in the left ventricle of the heart using 3D PC‐MRI data 24.
Most of these methods are based on a finite‐differences approach, as in Lorenz et al 41. However, it is well known that finite difference cannot effectively handle complex geometries such as those found in the cardiovascular system, neither can it impose boundary conditions on irregular surfaces in a direct manner and are also sensitive to noise 42. Furthermore, the evaluation of the vorticity and helicity has been typically performed on reformatted 2D planes from 3D cine PC‐MRI data. This approach has been shown to suffer from a loss of accuracy in estimating hemodynamic parameters as a result of the omission of out‐of‐plane velocity information 43. For example, it has recently been shown 6 that 2D analyses of wall shear stress and other related quantities yield poor estimates in vessel geometries with gradients along the central axis, largely undermining the predictiveness of 2D quantification methods. One notable exception is the work of Born et al 44 and Morbiducci et al 45, in which helicity is estimated using 3D velocity data. In particular, they proposed the use of pathline trajectories to assess the helicity in the aorta along the entire cardiac cycle.

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