Jonas Walheim^{1}, Hannes Dillinger^{1}, and Sebastian Kozerke^{1}

We present a 5D Flow MRI approach for mapping the Reynolds stress tensor in the in-vivo aorta within 6 minutes. First and second statistical moments of fluctuating velocities are encoded using six different velocity encoding gradient directions embedded in a Cartesian Golden angle undersampling scheme with data-driven motion detection and locally low-rank imaging reconstruction. It is demonstrated that this approach permits a time-efficient assessment of velocity vector fields, turbulent kinetic energy and Reynolds shear stresses of the aorta in-vivo.

Turbulent
flow encoding
In general flow velocities may be
decomposed into a mean value
$$$\bar{v}$$$ and a fluctuating component $$$v^{'}$$$^{10}:

$$v = \overline{v} + v^{'}.$$

Assuming a Gaussian intra-voxel velocity
distribution of variance
$$$\sigma^2$$$, the MR signal model reads^{10}

$$S(k_v)=S_0 e^{\frac{-\sigma^2 {k_v}^2}{2}}e^{-ik_v \overline{v}}$$

where $$$k_v=\gamma \int_0^TtG(t)dt$$$ denotes the first gradient moment of a bipolar velocity encoding gradient and $$$T$$$ the time of application of gradient $$$G$$$ . The statistical description of velocity fluctuations $$$v^{'}$$$ includes variances and covariances as described by the Reynolds stress tensor (RST)

$$R=\rho \begin{bmatrix}\overline{v_{x}^{'}v_{x}^{'}}&\overline{v_{x}^{'}v_{y}^{'}}&\overline{v_{x}^{'}v_{z}^{'}}\\ \overline{v_{x}^{'}v_{y}^{'}}&\overline{v_{y}^{'}v_{y}^{'}}&\overline{v_{y}^{'}v_{z}^{'}}\\\overline{v_{x}^{'}v_{z}^{'}}&\overline{v_{y}^{'}v_{z}^{'}}&\overline{v_{z}^{'}v_{z}^{'}}\end{bmatrix}$$ with standard deviations $$$\overline{v_{i}^{'}v_{i}^{'}}$$$, covariances
$$$\overline{v_{i}^{'}v_{j}^{'}}$$$ and fluid density
$$$\rho$$$. The magnitude of the measured
signal can be written as^{5}

^{ }

$$|S(\bf{k}_v)|=|S_0|e^{\frac{-1}{2\rho}\bf{k}_v^TR\bf{k}_v}$$

with $$$\bf{k}_v=\begin{bmatrix}k_{vx}&k_{vy}&k_{vz}\end{bmatrix}^T$$$.

For six measurements along six different velocity encodings and by denoting the intra-voxel standard deviations (IVSD) $$$\sigma_{k_v,i}^2=\frac{2}{|\bf{k}_v|^2} ln \frac{|S(\bf{k_v}=\bf{0})|} {|S(\bf{k}_{v,i})|}$$$, the following encoding equation is obtained:

$$\begin{bmatrix}\sigma_{\bf{k}_{v,1}}^2\\...\\\sigma_{\bf{k}_{v,6}}^2\end{bmatrix}=\begin{bmatrix}\left(\begin{array}{c}{k_{vx,1}}^2&{{k}_{vy,1}}^2&{{k}_{vz,1}}^2&{2{k}_{vx,1}}{k}_{vy,1}&{2{k}_{vx,1}}{k}_{vz,1}&{2{k}_{vy,1}}{k}_{vz,1}&\end{array}\right)/ \left |\bf{k_{v,1}} \right| \\...\\\left(\begin{array}{c}{{k}_{vx,6}}^2&{{k}_{vy,6}}^2&{{k}_{vz,6}}^2&{2{k}_{vx,6}}{k}_{vy,6}&{2{k}_{vx,6}}{k}_{vz,6}&{2{k}_{vy,6}}{k}_{vz,6}&\end{array}\right)/ \left|\bf{k_{v,6}} \right|\end{bmatrix}\begin{bmatrix}\overline{v_{x}^{'}v_{x}^{'}}\\\overline{v_{y}^{'}v_{y}^{'}}\\\overline{v_{z}^{'}v_{z}^{'}}\\\overline{v_{x}^{'}v_{y}^{'}}\\\overline{v_{x}^{'}v_{z}^{'}}\\\overline{v_{y}^{'}v_{z}^{'}}\end{bmatrix}=H\begin{bmatrix}\overline{v_{x}^{'}v_{x}^{'}}\\\overline{v_{y}^{'}v_{y}^{'}}\\\overline{v_{z}^{'}v_{z}^{'}}\\\overline{v_{x}^{'}v_{y}^{'}}\\\overline{v_{x}^{'}v_{z}^{'}}\\\overline{v_{y}^{'}v_{z}^{'}}\end{bmatrix}.$$

Accordingly, the elements of the RST can be calculated voxel-wise using the pseudoinverse:

$$\begin{bmatrix} \overline{v_{x}^{'}v_{x}^{'}}\\
\overline{v_{y}^{'}v_{y}^{'}}\\\overline{v_{z}^{'}v_{z}^{'}}\\\overline{v_{x}^{'}v_{y}^{'}}\\\overline{v_{x}^{'}v_{z}^{'}}\\\overline{v_{y}^{'}v_{z}^{'}}\end{bmatrix} = (H^T H)^{-1} H \begin{bmatrix}\sigma_{\bf{k}_{v,1}}^2 \\ ... \\ \sigma_{\bf{k}_{v,6}}^2 \end{bmatrix}.$$

*Experiments
*

In a first sub-study, the range of IVSD in the
aorta during systole was retrospectively analyzed in datasets previously
obtained in 9 healthy volunteers and 28 patients with aortic valve stenosis^{1}.

Thereafter, prospective data were collected using a Cartesian Golden angle
undersampling scheme^{11,12}
with data-driven motion detection and locally low-rank image reconstruction
in healthy subjects as illustrated in Figure 1. Scan parameters were: spatial
resolution 2.5x2.5x2.5 mm^{3}, 25 cardiac phases and scan duration of 6
minutes. During image reconstruction, data were sorted into four discrete
respiratory motion bins and respiratory motion resolved datasets were reconstructed
with BART^{13} enforcing
a locally low rank model^{14,15}
along cardiac phases and respiratory motion states. Velocities were encoded
with the Normal+Bisecting (N+B) encoding scheme^{7} as illustrated in Figure 1 using
encoding velocities of 80 cm/s.

*Data analysis
*

* *

Turbulent Kinetic Energy (TKE) was calculated from the main diagonal of the RST as

$$ TKE=\frac{\rho}{2}(\overline{v_{x}^{'}v_{x}^{'}}+\overline{v_{y}^{'}v_{y}^{'}}+\overline{v_{z}^{'}v_{z}^{'}}).$$

Principal stress analysis was performed and the maximum shear stress was calculated from the eigenvalues $$$\delta_1>\delta_2>\delta_3$$$ of the RST as

$$\tau_{max}=0.5(\delta_1-\delta_3)$$ assuming a density of blood of $$$\rho=1060kg/m^3$$$.

1. Binter C, Gotschy A, Sündermann SH, Frank M, Tanner FC, Lüscher TF, Manka R, Kozerke S. Turbulent Kinetic Energy Assessed by Multipoint 4-Dimensional Flow Magnetic Resonance Imaging Provides Additional Information Relative to Echocardiography for the Determination of Aortic Stenosis Severity. Circulation: Cardiovascular Imaging. 2017;10(6):1–8.

2. Haraldsson H, Kefayati S, Ahn S, Dyverfeldt P, Lantz J, Karlsson M, Laub G, Ebbers T, Saloner D. Assessment of Reynolds stress components and turbulent pressure loss using 4D flow MRI with extended motion encoding. Magnetic Resonance in Medicine. 2018;79(4):1962–1971.

3. Casas B, Lantz J, Dyverfeldt P, Ebbers T. 4D Flow MRI-Based Pressure Loss Estimation in Stenotic Flowsâ€Ż: Evaluation Using Numerical Simulations. 2016;1821:1808–1821.

4. Ha H, Kvitting JPE, Dyverfeldt P, Ebbers T. Validation of pressure drop assessment using 4D flow MRI-based turbulence production in various shapes of aortic stenoses. Magnetic Resonance in Medicine. 2018;(February):1–14.

5. Binter C, Knobloch V, Sigfridsson A, Kozerke S. Direct quantification of turbulent shear stresses by multi-point phase-contrast MRI. Proceedings of the 20th Annual Meeting of ISMRM, Melbourne, Australia.

6. Ha H, Kvitting J-PE, Dyverfeldt P, Ebbers T. 4D Flow MRI quantification of blood flow patterns, turbulence and pressure drop in normal and stenotic prosthetic heart valves. Magnetic Resonance Imaging. 2019;55(July 2018):118–127.

7. Elkins CJ, Alley MT, Saetran L, Eaton JK. Three-dimensional magnetic resonance velocimetry measurements of turbulence quantities in complex flow. Experiments in Fluids. 2009;46(2):285–296.

8. Binter C, Knobloch V, Manka R, Sigfridsson A, Kozerke S. Bayesian multipoint velocity encoding for concurrent flow and turbulence mapping. Magnetic Resonance in Medicine. 2013;69(5):1337–1345.

9. Jonas Walheim and Sebastian Kozerke. 5D Flow MRI – Respiratory Motion Resolved Accelerated 4D Flow Imaging Using Low-Rank + Sparse Reconstruction. In: Proceedings of the 26th Annual Meeting of ISMRM. Presented at the ISMRM. 2018. p. 0032.

10. Dyverfeldt P, Sigfridsson A, Kvitting JPE, Ebbers T. Quantification of intravoxel velocity standard deviation and turbulence intensity by generalizing phase-contrast MRI. Magnetic Resonance in Medicine. 2006;56(4):850–858.

11. Wundrak S, Paul J, Ulrici J, Hell E, Rasche V. A small surrogate for the golden angle in time-resolved radial MRI based on generalized fibonacci sequences. IEEE Transactions on Medical Imaging. 2015;34(6):1262–1269.

12. Cheng JY, Hanneman K, Zhang T, Alley MT, Lai P, Tamir JI, Uecker M, Pauly JM, Lustig M, Vasanawala SS. Comprehensive motion-compensated highly accelerated 4D flow MRI with ferumoxytol enhancement for pediatric congenital heart disease. Journal of Magnetic Resonance Imaging. 2016;43(6):1355–1368.

13. Tamir JI, Ong F, Cheng JY, Uecker M, Lustig M. Generalized Magnetic Resonance Image Reconstruction using The Berkeley Advanced Reconstruction Toolbox. Proceedings of the ISMRM 2016 Data Sampling and Image Reconstruction Workshop. 2016;2486:9660006.

14. Zhang T, Pauly JM, Levesque IR. Accelerating parameter mapping with a locally low rank constraint. Magnetic Resonance in Medicine. 2015;73(2):655–661.

15. Trzasko J, Manduca A, Borisch E. Local versus global low-rank promotion in dynamic MRI series reconstruction. In: Proc. Int. Symp. Magn. Reson. Med. 2011. p. 4371.

Figure 1: Data
acquisition using a Cartesian Golden angle scheme and data-driven
respiratory motion detection. The
data are retrospectively assigned to different motion states. A
Normal+Bisecting (N+B) encoding^{7} with seven different velocity encodings is
played out for the assessment of Reynolds stresses. In addition to encoding
flow velocities in 3 orthogonal directions **k**_{v,1}, k_{v,2}**, k**_{v,3}, the velocity encodings are combined along the
diagonals. Reconstruction is performed using a locally low rank prior^{14,15} which enforces a low rank over cardiac phases
and motion states.

Figure 2: a) Sensitivity of the signal
intensity to different intra-voxel standard deviations (IVSD)
for different encoding velocities
. The dependency of the signal
magnitude on
$$$\sigma$$$ can only be seen for a limited
range. b) Exemplary distributions of IVSD in the ascending aorta of 9 healthy
volunteers show IVSD to be distributed mainly between 0 m/s and 0.4 m/s. For
patients, a wider distribution can be observed with values of IVSD going up to
0.8 m/s.

Figure 3: Exemplary results obtained with 5D
Flow Tensor MRI. Maximum shear stresses derived from the off-diagonal elements
of the RST range from 50 Pa to 100 Pa. TKE
is in the expected range of up to 300 J/m3 in the healthy subject.