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.

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. 9 the velocity field along its three components as well as the magnitude of the velocity are presented in the coronal plane. The velocity field along the 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. 10) and in the cross-section in the middle of the bend (Fig. 11). These maps are given for four time instants: peak systole, peak 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 peak 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.

Figure 9: Qualitative comparison in the coronal plane

Figure 10: Qualitative comparison across the collateral - Velocity along the X-direction

Figure 11: Qualitative comparison in the cross-section of the bend - Velocity along the X-direction

Pearson's correlation

The Pearson's product moment correlation was computed over the cardiac cycle to assess the linearity between MRI and downsampled CFD (LR-CFD) results. Indeed this indicator is widely used for statistics within the medical community. It is defined as:

$r_{\vert \vert {\textbf {u}}\vert \vert }^{2}={\frac {(\sum _{i=1}^{N}(\vert \vert {\textbf {u}}_{i,LR-CFD}\vert \vert -\vert \vert {\overline {{\textbf {u}}_{LR-CFD}}}\vert \vert )(\vert \vert {\textbf {u}}_{i,MRI}\vert \vert -\vert \vert {\overline {{\textbf {u}}_{MRI}}}\vert \vert ))^{2}}{\sum _{i=1}^{N}(\vert \vert {\textbf {u}}_{i,LR-CFD}\vert \vert -\vert \vert {\overline {{\textbf {u}}_{LR-CFD}}}\vert \vert )^{2}\sum _{i=1}^{N}((\vert \vert {\textbf {u}}_{i,MRI}\vert \vert -\vert \vert {\overline {{\textbf {u}}_{MRI}}}\vert \vert )^{2}}}$

where $\vert \vert {\textbf {u}}_{i}\vert \vert$ is the magnitude of the velocity at the $i^{th}$ node and $\vert \vert {\overline {\textbf {u}}}\vert \vert$ is the averaged magnitude velocity. The correlation over the cardiac cycle is shown on Fig. 12. At peak systole, a $r_{\vert \vert {\textbf {u}}\vert \vert }^{2}$ value of 0.84 is found when the MRI image is denoised and unwrapped (phase offset correction) and 0.97 when the additional no-slip boundary condition is applied at walls. This is in agreement with the results reported in Puiseux et al. [2] (cf. table 3).

Figure 12: Pearson's correlation $r_{\vert \vert {\textbf {u}}\vert \vert }^{2}$ between post-processed MRI and downsampled CFD

Flowrates

The flowrates were compared at four planes displayed in Fig. 13, 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. 14, where the flowrates obtained from the CFD are also given as reference and prove that the downsampling process do not affect the flowrate computation. Very good agreement is found at the inlet (Fig. 1a) and in the collateral (Fig. 14c). However MRI overestimates the flowrates at the outlet (Fig. 14b) and in the middle of the bend (Fig. 14d). The recirculation after the collateral and just upstream of the outlet could explained this overestimation, due to velocity displacement artefacts (Steinman et al., 1997 [18]).

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

U_PAVG corresponds to the CFD phase-averaged velocity field

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.