# Comparison of Four-Dimensional Flow Magnetic Resonance Imaging and Particle Image Velocimetry to Quantify Velocity and Turbulence Parameters

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Setup

^{3}and 3.72 × 10

^{−3}kg/m⋅s, respectively. The working fluid was circulated through the flow circuit system at a constant flow rate using a centrifugal pump (EHEIM Universal 3400, Deizisau, Germany). Flow rates of 10 and 20 L/min were set by monitoring with an electromagnetic flowmeter (VN20, Wintech Process, Seoul, Korea). The corresponding Reynolds numbers of the inlet flow at 10 and 20 L/min were 1888 and 3777, respectively. The temperature of the fluid during experimentation was 20 °C. A contrast agent, 30 mL of contrast agent (0.5 mmol/kg, gadofosveset trisodium, VasovistVR, Bayer Schering Pharma AG, Berlin, Germany) was added to 40 L of the working fluid for MRI measurement.

#### 2.2. 4D Flow MRI Measurement

#### 2.3. 4D Flow MRI Turbulence Quantification

_{v}) for a velocity distribution s(v) is expressed by a Fourier transformation as

_{v}represents the level of flow sensitivity, which is related to Venc as k

_{v}= π/Venc. When turbulent flow occurs in the region of interest, the intravoxel velocity variance (IVVV) of the turbulent flow along the i direction, denoted by ${\sigma}_{i}^{2}$, can be estimated from the magnitude ratio between the reference signal without velocity encoding S(0) and the signal with velocity encoding along the i direction S

_{i}(k

_{v}) as

_{ij}can be obtained by measuring six non-orthogonal velocity encodings and finding the least-square solutions of the six-directional phase and magnitude data. R

_{ij}is a six-element symmetric tensor defined as [11]

_{ij}represents the strain rate tensors of the mean velocity field. To compare the 4D flow MRI measurement with the two-dimensional PIV data, the velocity fluctuation is assumed to be two dimensional and the slice-directional velocity was neglected throughout this study.

#### 2.4. Post-Processing of 4D Flow MRI Data

#### 2.5. Setup of PIV

^{3}. The flow field was illuminated with a 0.5-mm-thick thin laser sheet using 532 nm continuous DPSS Laser (Changchun New Industries Optoelectronics Co., Ltd., Changchun, Jilin, China). A high-speed camera (4000 fps, 1280 × 512; VEO 710L, Vision Research lnc., Wanye, NJ, USA) is positioned perpendicular to the flow to measure velocity fields at the center plane of the model. The resolution of the camera was 0.54 $\mathsf{\mu}\mathrm{m}$/pixel with particle sizes of 2–4 pixels in the image. The measurements were taken at three location sections: stenosis apex, upstream, and downstream of the stenosis. These measurement data were combined at the post-processing.

#### 2.6. PIV Analysis

#### 2.7. Statistics

## 3. Results

#### 3.1. Velocity Comparison

^{2}= 0.98 and 0.89 with r

^{2}= 0.98 at the flow rates of 10 and 20 L/min, respectively (Figure 3). The mean bias with 95% limits of agreement was −0.02 ± 0.11 m/s and 0.10 ± 0.24 m/s at the flow rates of 10 and 20 L/min, respectively.

#### 3.2. TKE Comparison

^{2}= 0.68 and 1.66 with r

^{2}= 0.78 at the flow rates of 10 and 20 L/min, respectively (Figure 5). The mean bias with 95% limits of agreement was 13.9 ± 34.2 J/m

^{3}and 59.3 ± 111.3 J/m

^{3}at the flow rates of 10 and 20 L/min, respectively. The TKEplane, which is the cross-sectional sum of TKE at each plane. Consequently, a total TKE of the 4D flow MRI and PIV was 3.8 ± 0.3 mJ and 2.2 ± 0.0 mJ, 13.2 ± 1.1 mJ and 8.7 ± 0.0 mJ at 10 and 20 L/min, respectively (Figure 6).

#### 3.3. Reynolds Stress Tensor (RST) and TP Comparison

^{2}= 0.90 and r

^{2}= 0.87 at the flow rates of 10 and 20 L/min, respectively. The mean bias with 95% limits of agreement was −0.0 ± 12.9 N/m

^{2}and 0.6 ± 67.6 N/m

^{2}at the flow rates of 10 and 20 L/min, respectively (Figure 8).

^{2}= 0.60 and 1.67 with r

^{2}= 0.78, respectively. The mean bias with 95% limits of agreement was 525.6 ± 2726.5 W/m

^{3}and 4924.1 ± 21,857.5 W/m

^{3}at the flow rates of 10 and 20 L/min, respectively.

#### 3.4. 4D Flow MRI Measurement at the Pulsatile Flow Condition

^{3}, and 64.9 mW/m

^{3}. When the larger Venc was used, the sensitivity of the total TP and TKE were reduced but the results showed the same trend of turbulence (Figures S1–S3 in the Supplementary Materials).

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematics of the experimental setup. (

**A**) the flow circuit. (

**B**) 4D flow MRI. (

**C**) particle image velocimetry.

**Figure 2.**Velocity through the stenosis. (

**A**) Velocity contours. (

**B**) Centerline velocity. Solid line and shading of the plot indicate mean ± standard error of the measurements.

**Figure 3.**Bland–Altman plot of the velocity measurements. (

**A**) Velocity comparison at the flow rate of 10 L/min. (

**B**) Velocity comparison at the flow rate of 20 L/min. Solid red line indicates the slope of the linear regression. Gray dashed line indicates the slope of unity. Black solid line and red dashed line are mean ± 1.96 SD.

**Figure 4.**Turbulence kinetic energy (TKE) through the stenosis. (

**A**) TKE contours. (

**B**) Centerline TKE.

**Figure 5.**Bland–Altman plot of the TKE measurements. (

**A**) TKE comparison at the flow rate of 10 L/min. (

**B**) TKE comparison at the flow rate of 20 L/min. Solid red line indicates the slope of the linear regression. Gray dashed line indicates the slope of unity. Black solid line and red dashed line are mean ± 1.96SD.

**Figure 6.**Planar and total Turbulence kinetic energy (TKE) through the stenosis. (

**A**) Axial directional planar TKE (

**B**) Axial directional accumulative total TKE.

**Figure 7.**Reynolds stress (RST) and turbulence production (TP) through the stenosis. (

**A**) RST contours. (

**B**) TP contours.

**Figure 8.**Bland–Altman plot of the Reynolds stress (RST) and turbulence production (TP) measurements. (

**A**) RST comparison at the flow rate of 10 L/min. (

**B**) RST comparison at the flow rate of 20 L/min. (

**C**) TP comparison at the flow rate of 10 L/min. (

**D**) TP comparison at the flow rate of 20 L/min. Solid red line indicates the slope of the linear regression. Gray dashed line indicates the slope of unity. Black solid line and red dashed line are mean ± 1.96 SD.

**Figure 9.**Planar and total turbulence production (TP) through the stenosis. (

**A**) Axial directional planar TP. (

**B**) Axial directional accumulative total TP.

**Figure 10.**4D flow MRI measurements of velocity and turbulence under pulsatile flow condition. (

**A**) Velocity contours. (

**B**) Turbulence kinetic energy con-tours. (

**C**) Turbulence production contours.

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Kim, D.; Kang, J.; Adeeb, E.; Lee, G.-H.; Yang, D.H.; Ha, H.
Comparison of Four-Dimensional Flow Magnetic Resonance Imaging and Particle Image Velocimetry to Quantify Velocity and Turbulence Parameters. *Fluids* **2021**, *6*, 277.
https://doi.org/10.3390/fluids6080277

**AMA Style**

Kim D, Kang J, Adeeb E, Lee G-H, Yang DH, Ha H.
Comparison of Four-Dimensional Flow Magnetic Resonance Imaging and Particle Image Velocimetry to Quantify Velocity and Turbulence Parameters. *Fluids*. 2021; 6(8):277.
https://doi.org/10.3390/fluids6080277

**Chicago/Turabian Style**

Kim, Doohyeon, Jihun Kang, Ehsan Adeeb, Gyu-Han Lee, Dong Hyun Yang, and Hojin Ha.
2021. "Comparison of Four-Dimensional Flow Magnetic Resonance Imaging and Particle Image Velocimetry to Quantify Velocity and Turbulence Parameters" *Fluids* 6, no. 8: 277.
https://doi.org/10.3390/fluids6080277