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 flow rates 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 flow rates obtained from the CFD (ie without downsampling) are also given as reference and prove that the downsampling process does not affect the flow rate computation. Very good agreement is found at the inlet (Fig. 17a) and in the collateral (Fig. 17c). However MRI overestimates the flow rates 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 16: Planes on the CFD geometry where flow rates were computed.

Figure 17: Time evolution of flow rates from phase-averaged CFD and MRI

L2-norm error

The L2-norm, also known as Euclidean distance, is used to assess the point-wise similarity between the velocity fields obtained from the two methods, namely the experimental MR acquisitions and the downsampled in silico CFD simulations. This metric can be seen as a local indicator of how different the velocity fields are. It 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. 18. It appears that the higher the flow rate at the inlet, the higher the L2-norm error. Increased errors are found around peak systole, whereas the lowest ones are found at the diastole.

Error maps are given in Fig. 19 and 20 at peak systole and end diastole, respectively in the coronal (xz)-plane and in the collateral transverse (xy)-plane. The highest errors occur in the collateral, aneurysm and jet at the junction of the collateral with the main descending pipe. 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 flow rate in the collateral is found to be quite accurate (Fig. 17c), high discrepancy between the velocity fields are found in this region. Indeed, as it appears in Fig. 20, the velocity field in the center of the collateral duct obtained by MR acquisitions overestimates the values from the CFD simulation. High L2-norm values are also observed in particular at the collateral walls. An explanation for this trend could come from the small number of voxels along the collateral diameter and partial volume artifacts. That is to say the voxels straddling the phantom wall in the MRI acquisition include signal coming from both inside and outside the fluid volume, where the latter leads to random noise on the reconstructed velocity.