# A robust comparison approach of velocity data between MRI and CFD based on divergence-free space projection

## Summary (2 min read)

### 1. INTRODUCTION

- Recently, 4D flow MRI combining 3D spatial encoding with three-directional velocity-encoding has revealed great potential [1].
- Such measurements are limited by many factors such as acquisition times, signal-to-noise ratio and resolution depending on the set-up and region of interest.
- Numerical phantoms play an important role in the assessment and validation of hemodynamics.
- A reasonable comparison method has not yet been developed.
- This work was supported by the Swiss National Science Foundation grant 320030-149567.

### 2. EXPERIMENTAL SETTING

- A silicon replica of a healthy human aortic arch (Elastrat, Switzerland) (B) was connected to a centrifugal pump (A) (BG-GP 636, Einhell Germany AG, Germany, maximum pressure 3.9 bar) via PVC tubing with 19 mm inner diameter with a total length of 20 m.
- Inlet and outlet were connected to a reservoir (D) resulting in an open circuit.
- The flow rate was monitored in 10 min intervals by PhaseContrast MRI.
- For the acquisition of the flow field a 3D spoiled Gradient-Echo sequence with flow encoding gradients was used.
- Linear Phase Correction was applied to compensate for the eddy-current induced background phase.

### 3. PRE-PROCESSINGS OF THE NUMERICAL PHANTOM

- Flow reconstruction yields both the proton density images and three-directional velocity data, denoted by uMRI.
- The phantom aorta was segmented semi-automatically with snake evolution methods using ITK-SNAP [11], see Fig. 2 (left).
- The surface mesh was smoothed with VMTK [12].
- The mesh was built using snappyHexMesh, an OpenFOAM utility for creating hexahedral meshes.

### 3.1. Denoising of MRI data

- The three-dimensional velocity data was composed with the segmentation binary mask, such that only the original values were kept within the surface of the segmented aorta.
- Values outside the surface were set to zero, such that they are not considered by the latter interpolation of measurements onto the mesh domain.
- After these processes, one is still left with the noise within the aorta, for which fast, accurate and automatic post-processings are required to get an appropriate input for numerical computations.
- This problem has been extensively studied in the existing literature [13].
- In order to use the MRI acquisitions as initial conditions for the numerical simulations, the denoised data were projected onto the mesh domain using the linear interpolation provided by ITK [14].

### 3.2. Projection onto the divergence-free space

- In what follows, the authors look for the divergence-free component of the denoised velocity, simply denoted by ũ⋆ = ũ+ u⋆.
- The authors use no-slip boundary condition at the aortic wall Γ5 to ensure the well-posedness of the problem, while homogeneous Neumann boundary condition is considered on the remaining boundaries.
- Let n denote the outward unit normal vector on the boundary.

### 4.2. Multiscale coupling with zero-dimensional model

- Reduced order modeling represent a useful formalism that can provide partial but accurate information about the arterial hemodynamics.
- During the past decades, there has been significant developments using several techniques of multiscale modeling, where reduced order models, also referred to as lumped-parameter models, are coupled with multidimensional description of the cardiovascular system [16].
- Such reduced order models provide boundary conditions to be coupled with the detailed three-dimensional model.
- Using the flow rate QΓ provided by the fluid solver, i.e.
- In the presented work, the coupled problem is solved until the steady state, where a threshold tolerance ǫ is reached.

### 5. NUMERICAL EXAMPLES AND VALIDATION

- The numerical phantom was implemented using icoFoam, an OpenFOAM [18] solver for incompressible, laminar Navier-Stokes equations using the PISO algorithm.
- The spatial accuracy is studied by computing the error in L2, denoted by ‖.‖0,2,Ω, and L ∞, denoted by ‖.‖0,∞,Ω, norms with respect to the exact solution, respectively for several finite element polynomial approximations.
- Notice that div ũ⋆ < 0.05s−1. In the second experiment, the authors performed numerical computations under laminar flow conditions with helical flow patterns using the denoised data as depicted in the graph in Section 4.2.
- Comparisons in Fig. 6 show that the velocity magnitudes turned out to be almost identical in the defined planes P1, P2 and P3 across the aorta, see Fig.
- HFI measures the alignment between the local velocity u and the vorticity w vectors, and it is given by the normalized helicity density HFI = u ·w/(|u| |w|).

### 6. CONCLUSION

- A novel approach has been proposed to perform reliable comparisons between CFD and MRI.
- Phantom experiments of laminar flow under helical patterns have been performed.
- Numerical computations were compared against MRI acquisitions, and results show good agreement.
- Stability and the numerical issues will be provided in a forthcoming work.

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##### References

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