Judith Zimmermann^{1,2}, Daniel Demedts^{3}, Michael Markl^{4}, Christian Meierhofer^{2}, Heiko Stern^{2}, and Anja Hennemuth^{3,5}

Wall shear stress (WSS) is a hemodynamic parameter which can be estimated from 4D flow MRI. The aim of this work was to advance the surface inward normal computation for complex (i.e. cone-shaped) vessel geometries and thus to improve the accuracy of wall shear stress estimates. We propose a Gauss gradient field approach to adapt to complex vessel courses and evaluate our method using synthetic flow data and selected patient data. Results show that correct inward normal definition is crucial for reliable WSS estimates, in particular in cases where complex vessel geometries are present.

4D flow MRI allows the image-based estimation of quantitative wall shear stress (WSS) in vascular structures. WSS ($$$\vec{\tau}$$$) at any vessel wall point is defined as $$\vec{\tau}=2\eta\dot{\epsilon}\cdot\vec{n}$$ with $$$\vec{n}$$$ = vessel surface inward normal, $$$\eta$$$ = dynamic viscosity, and $$$\dot{\epsilon}$$$ = velocity deformation tensor. Furthermore, the oscillatory shear index (OSI) is defined as: $$OSI=\frac{1}{2}\left(1-\frac{|\int_{0}^{T}\vec{\tau}\cdot dt| }{\int_{0}^{T}|\vec{\tau}|\cdot dt}\right)$$ Previous works studied the impact of MRI acquisition parameters on WSS estimates^{1} though the burden of inaccurate inward normal definition is unknown. Most commonly, WSS and OSI are estimated by analyzing reformatted 2D planes^{2-4 }and the inward normal at each wall point is only comprised of the vector component lying on the respective plane. However, in case of complex vessel geometries (e.g. stenosis, dilatation) the inward normal may deviate from the analysis plane.The aim of this work was to (1) introduce a method to accurately compute surface inward normals in cone-shaped vessel regions given the typical 4D flow MRI image data, and to (2) study the systematic error of WSS/OSI estimates when only the inplane inward normal component is considered.

**Synthetic data**. Steady flow was simulated through a pipe with
narrowing on a high resolution grid (fig. 1) using Comsol Multiphysics
(resolution = 0.7x0.7x0.7 mm3; min/max vessel diameter = 10.3/33.5
mm; mean velocity at inlet = 3.89 cm/s).

**In-vivo data.** Six 4D flow MRI data sets of patients with stenotic
or dilated vessel geometry were included. Scans were performed on a 1.5T Avanto
(N=3) or 3T Trio (N=3) system with: TE/TR = 2.6/5.1 ms; FOV ~ 400x300x60 mm^{3};
voxel dimension = 1.7-2.5 x 1.7-2.5 x 2.0-2.5 mm^{3}; VENC = 150-300
cm/s; parallel imaging with R = 5; prospective ECG triggering covering the full
cycle; navigator gating. Data pre-processing included phase offset correction
(via polynomial function fitting), background noise masking, and automatic
phase unwrapping if needed.

**WSS
computation. **Vessels were segmented from the generated PC-MRA image using a watershed-based algorithm. For each case, one analysis plane was placed in the
dilated/stenotic vessel region and vessel contours were tracked through all
time frames (fig. 2). The binary vessel mask was used to derive a Gauss
gradient field ($$$\boldsymbol{G}$$$) to advance the computation of the vessel inward
normal at each sampled contour point $$$\boldsymbol{x}$$$:
$$\vec{n}_\boldsymbol{x}=\vec{a}_\boldsymbol{x}+\vec{g}_{\boldsymbol{x},p}$$ where $$$\vec{g}_{\boldsymbol{x},p}=\left(\boldsymbol{G}_\boldsymbol{x}\cdot\vec{p}\right)\vec{p}$$$
is the projection of $$$\boldsymbol{G}_\boldsymbol{x}$$$ (local Gauss
gradient vector) onto the unit plane normal $$$\vec{p}$$$, and $$$\vec{a}_\boldsymbol{x}$$$ is the
conventional inward normal lying on the analysis plane. WSS was then estimated
at each contour point using a B-spline image function based approach^{5} with η = 0.0032 Pa s. Regarding synthetic data, we analyzed single
timepoint WSS (WSS_{single}); regarding in-vivo data, we analyzed WSS at
peak velocity (WSS_{peak}) and OSI. Estimates were binned into twelve
angular segments to compute mean±STD per segment. All image analysis was
implemented using the MevisLab^{6} framework.

**Evaluation.**
WSS_{single},
WSS_{peak}, and OSI were assessed segment-wise regarding absolute and
relative differences of estimates based on conventional vs. adapted inward normal.
Angle deviations between conventional ($$$\vec{a}_\boldsymbol{x}$$$) and adapted ($$$\vec{n}_\boldsymbol{x}$$$)
inward normals were computed.

Close-up
view of **in-vivo data** vessel surface
rendering (grey) with vessel wall inward normal vectors at contour points (red
dots). For each case, white arrows and red arrows represent the normal vectors
defined by the conventional method and by our approach, respectively. Angle
deviations between white and red vectors (computed at each contour point) are
given in median/min/max radian for each case.