We developed a novel methodology to decompose the WSS vector in WSS and WSS in 3D domains, using 4D flow MRI data. Our method provides a more robust quantification of WSS and WSS in comparison with other reported methods. Magn Reson Med 79:2816-2823, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

Purpose Magnetic resonance elastography (MRE) measures stiffness of soft tissues by analyzing their spatial harmonic response to externally induced shear vibrations. Many MRE methods use inversion‐based reconstruction approaches, which invoke first‐ or second‐order derivatives by finite difference operators (first‐ and second‐FDOs) and thus give rise to a biased frequency dispersion of stiffness estimates. Methods We here demonstrate analytically, numerically, and experimentally that FDO‐based stiffness estimates are affected by (1) noise‐related underestimation of values in the range of high spatial wave support, that is, at lower vibration frequencies, and (2) overestimation of values due to wave discretization at low spatial support, that is, at higher vibration frequencies. Results Our results further demonstrate that second‐FDOs are more susceptible to noise than first‐FDOs and that FDO dispersion depends both on signal‐to‐noise ratio (SNR) and on a lumped parameter A, which is defined as wavelength over pixel size and over a number of pixels per stencil of the FDO. Analytical FDO dispersion functions are derived for optimizing A parameters at a given SNR. As a simple rule of thumb, we show that FDO artifacts are minimized when A/2 is in the range of the square root of 2SNR for the first‐FDO or cubic root of 5SNR for the second‐FDO. Conclusions Taken together, the results of our study provide an analytical solution to a long‐standing, well‐recognized, yet unsolved problem in MRE postprocessing and might thus contribute to the ongoing quest for minimizing inversion artifacts in MRE.

Purpose: We propose a 3D finite-element method for the quantification of vorticity and helicity density from 3D cine phase-contrast (PC) MRI. Methods: By using a 3D finite-element method, we seamlessly estimate velocity gradients in 3D. The robustness and convergence were analyzed using a combined Poiseuille and Lamb-Ossen equation. A computational fluid dynamics simulation was used to compared our method with others available in the literature. Additionally, we computed 3D maps for different 3D cine PC-MRI data sets: phantom without and with coarctation (18 healthy volunteers and 3 patients).Results: We found a good agreement between our method and both the analytical solution of the combined Poiseuille and Lamb-Ossen. The computational fluid dynamics results showed that our method outperforms current approaches to estimate vorticity and helicity values. In the in silico model, we observed that for a tetrahedral element of 2 mm of characteristic length, we underestimated the vorticity in less than 5% with respect to the analytical solution. In patients, we found higher values of helicity density in comparison to healthy volunteers, associated with vortices in the lumen of the vessels. Conclusions: We proposed a novel method that provides entire 3D vorticity and helicity density maps, avoiding the used of reformatted 2D planes from 3D cine PC-MRI. Magn Reson Med 79:541-553,

We study limit cycles of nonlinear oscillators described by the equationẍ + νF (ẋ) + x = 0 with F an odd function. Depending on the nonlinearity, this equation may exhibit different number of limit cycles. We show that limit cycles correspond to relative extrema of a certain functional.Analytical results in the limits ν → 0 and ν → ∞ are in agreement with previously known criteria.For intermediate ν numerical determination of the limit cycles can be obtained.

In this paper we study the Brinkman model as a unified framework to allow the transition between the Darcy and the Stokes problems. We propose an unconditionally stable low-order finite element approach, which is robust with respect to the whole range of physical parameters, and is based on the combination of stabilized equal-order finite elements with a non-symmetric penalty-free Nitsche method for the weak imposition of essential boundary conditions. In particular, we study the properties of the penalty-free Nitsche formulation for the Brinkman setting, extending a recently reported analysis for the case of incompressible elasticity (Boiveau and Burman, IMA J. Numer. Anal. 36 (2016) 770-795). Focusing on the two-dimensional case, we obtain optimal a priori error estimates in a mesh-dependent norm, which, converging to natural norms in the cases of Stokes or Darcy ows, allows to extend the results also to these limits. Moreover, we show that, in order to obtain robust estimates also in the Darcy limit, the formulation shall be equipped with a Grad-Div stabilization and an additional stabilization to control the discontinuities of the normal velocity along the boundary. The conclusions of the analysis are supported by numerical simulations.