A pulsatile 3D flow relevant to thoracic hemodynamics: CFD - 4D MRI comparison

Application Challenge AC7-04 © copyright ERCOFTAC 2021

Evaluation - Comparison of Test Data and CFD

The comparisons between the post-processed MRI images and downsampled CFD were done on the MRI grid on which both velocity fields are expressed. The streamlines obtained from the CFD simulations before downsampling are provided in Fig. 12 for better assessment of the flow.

Figure 12: CFD streamlines of the phase-averaged velocity magnitude at 4 time instants (peak systole, end diastole and times inbetween). The lines were obtained from two sources localized respectively in the middle of the Ubend and of the collateral.

Qualitative comparison

First a qualitative comparison between the corrected MRI velocity data and the downsampled CFD is shown in several planes of interest. In Fig. 13 the velocity field with its three components as well as the magnitude of the velocity are presented in the coronal (xz)-plane. The velocity in X-direction ( $u$ -component, see Fig. 5 for the definition of the axis) is also shown in a plane along the middle of the collateral (Fig. 14) and in the vertical cross-section in the middle of the bend (Fig. 15). These maps are given for four time instants: peak systole, end diastole and the time instants in between. Similar velocity patterns are seen even in complex flow regions. Both CFD and MRI succeed in capturing main flow features, such as the small separation region in the main branch at peak systole and the recirculation in the aneurysm and the back flow in the bifurcation at end diastole. Through-plane velocity comparison ( $v$ -component) shows larger visual discrepancies, which was expected given the low signal amplitudes collected in the MRI scanner due to the low velocities in this direction. Note that the $v$ -velocity is not zero at the vertical symmetry plane, since the inlet boundary condition comes from the experimental data and it is thus expected that the inlet velocity field is slightly asymmetrical.

Figure 13: Qualitative comparison in the coronal plane

Figure 14: Qualitative comparison across the collateral - Velocity along the X-direction

Figure 15: Qualitative comparison in the cross-section of the bend - Velocity along the X-direction

Flowrates

The flowrates were compared at four planes displayed in Fig. 16, including the inlet, outlet, collateral and bend. They were computed from the post-processed MRI images and downsampled CFD (LR-CFD) velocity field, both expressed on the MRI grid. The results are shown in Fig. 17, where the flowrates obtained from the CFD (ie without downsampling) are also given as reference and prove that the downsampling process does not affect the flowrate computation. Very good agreement is found at the inlet (Fig. 17a) and in the collateral (Fig. 17c). However MRI overestimates the flowrates at the outlet (Fig. 17b) and in the middle of the bend (Fig. 17d). The jet and its associated recirculation at the junction between the collateral and the main descending duct (cf. Fig. 13) could explained this overestimation due to velocity displacement artefacts (Steinman et al., 1997 [18]). These artefacts appear in MR images especially in the presence of oblique and accelerating flows and are caused by the time delays between the different spatial and velocity encodings inherent to the MR process.

Figure 13: Planes on the CFD geometry where flowrates were computed.

Figure 14: Time evolution of flowrates from phase-averaged CFD and MRI

L2-norm error

While the Pearson's correlation $r_{\vert \vert {\textbf {u}}\vert \vert }^{2}$ gives an overview of how similar MRI and downsampled CFD fields are in their entirety, the L2-norm error is a local indicator of how different the velocity fields are. This error is computed at each node position ${\textbf {x}}$ and at each time instant $t$ as:

$L2({\textbf {x}},t)={\frac {\sqrt {(u_{MRI}-u_{LR-CFD})^{2}+(v_{MRI}-v_{LR-CFD})^{2}+(w_{MRI}-w_{LR-CFD})^{2}}}{\overline {\vert \vert {\textbf {u}}_{2D,inlet}\vert \vert }}}$

where ${\textbf {u}}=(u,v,w)$ is the velocity vector associated to the node at the position ${\textbf {x}}$ and ${\overline {\vert \vert {\textbf {u}}_{2D,inlet}\vert \vert }}=0.142m/s$ is the time-averaged bulk velocity at the inlet from the 2D cine PC-MRI acquisitions, which is used as normalization factor of the error.

The L2-norm error averaged over all nodes is shown in Fig. 15. The highest error is found around peak systole, with levels equals to 0.29 for the denoising and unwrapping correction, and as low as 0.14 when the additional no-slip boundary condition is applied at walls. The errors maps corresponding to peak systole are displayed in Fig. 16 and 17. The highest errors occur in the collateral, aneurysm and jet at the outlet after the collateral. These errors are observed in sections of high flow acceleration. As the velocity fields in the MRI are computed by making the assumption that the velocity is approximately constant during data acquisition, this could explain these type of errors. Despite the fact that the flowrate in the collateral is found to be quite accurate (Fig. 14c), high discrepancy between the velocity fields are found in this region.