## Abstract

*In vitro*investigations on two carotid bifurcation phantoms, normal and diseased, were conducted, and their relative differences in terms of the flow patterns and WSR distribution were demonstrated. It is shown that high-frame-rate UIV technique can be a non-invasive tool to measure quantitatively the spatio-temporal velocity and WSS distribution.

## Key Words

## Introduction

*in vivo*has not been established (

*in vivo*WSS measurement is limited and relies primarily on computational fluid dynamics (CFD). In this context, numerical simulation is employed to investigate the WSS distribution of a given geometry measured by means of imaging modalities. However, the accuracy of the simulation can be affected by the underlying assumptions on the geometry, wall properties, fluid properties and most importantly the initial and boundary conditions.

*in vivo*, flow velocity information can be assessed using existing non-invasive imaging modalities such as phase contrast magnetic resonance (

*In vitro*studies on two anthropomorphic carotid bifurcation phantoms are also conducted to further highlight the performance of the technique.

## Methods and Materials

### Ultrasound flow simulation

#### Flow model generation

### Poiseuille flow

*r*from the center of the vessel with the velocity as follows:

where

*v*

_{0}is the centreline velocity and R is the radius of the vessel. The WSS, given a fluid with dynamic viscosity

*µ*, is defined as:

_{0}= 20, 50, 80 cm/s) were generated in a 3- mm diameter vessel to investigate the accuracy of flow estimation. The vessel was positioned with a beam-to-flow angle of 60° and the radial velocity errors were calculated. In a 6-mm diameter vessel, steady flow with the centerline velocity of 50 cm/s, Reynolds number (Re) of 1500 and resulting WSR of 333 s

^{−1}was also simulated. The vessel was oriented with a beam-to-flow angle of 90° for the simplicity of the WSS analysis.

### Womersley flow

where $\Delta \text{\hspace{0.05em}}P\mathrm{sin}\left(\omega t+\varphi \right)$ is the pressure difference,

*α*is the Womersley number and $M{\prime}_{10}$ and $\epsilon {\prime}_{10}$ are complex functions of the non-dimensional parameter $\alpha =R\sqrt{\omega /\upsilon}$. From the volume flow rate, it is possible to calculate the velocity profile from a sinusoidal flow (

- Evans D.H.

*Ultrasound Med Biol.*1982; 8: 617-623

where J

_{m}is the m

^{th}order Bessel function of the first kind, $\tau ={j}^{3/2}\alpha $,

*ψn*and

*χn*represent the amplitude and angle of the complex function

*ψn*. Therefore, the velocity profile can be calculated from volume flow rate when the entrance effects are neglected.

Common carotid Diameter = 6.0 mm Heart rate = 60 bpm Viscosity = 0.004 kgm ^{–1} s^{–1} | Common femoral Diameter = 8.4 mm Heart rate = 60 bpm Viscosity = 0.004 kgm ^{–1} s^{–1} | ||||||||
---|---|---|---|---|---|---|---|---|---|

n | f | α | Vn | ϕn | n | f | α | Vn | ϕn |

0 | – | – | 1.00 | – | 0 | – | – | 1.00 | – |

1 | 1.03 | 3.9 | 0.33 | 74 | 1 | 1.03 | – | 1.89 | 32 |

2 | 2.05 | 5.5 | 0.24 | 79 | 2 | 2.05 | 7.7 | 2.49 | 85 |

3 | 3.08 | 6.8 | 0.24 | 121 | 3 | 3.08 | 9.5 | 1.28 | 156 |

4 | 4.10 | 7.8 | 0.12 | 146 | 4 | 4.10 | 10.9 | 0.32 | 193 |

5 | 5.13 | 8.7 | 0.11 | 147 | 5 | 5.13 | 12.2 | 0.27 | 133 |

6 | 6.15 | 9.6 | 0.13 | 197 | 6 | 6.15 | 13.4 | 0.32 | 155 |

7 | 7.18 | 10.3 | 0.06 | 233 | 7 | 7.18 | 14.5 | 0.28 | 195 |

8 | 8.21 | 12.4 | 0.04 | 218 | 8 | 8.21 | 15.5 | 0.01 | 310 |

### Ultrasound simulation

Probe parameter | Imaging parameter | ||
---|---|---|---|

Centre frequency | 8 MHz | Imaging mode | Plane wave |

Number of element | 128 | Transmit frequency | 8 MHz |

Element pitch | 0.2 mm | Excitation pulse | 1 cycle sinusoidal |

Element height | 5 mm | PRF | 10 kHz |

Sampling frequency | 100 MHz | Compounding angle | 5 |

Elevational focus | 20 mm | Angle Range | 20° |

Number of sub-aperture | 4 |

### In vitro flow experiment

#### Ultrafast plane wave imaging system

#### Microbubble contrast agents and contrast imaging mode

^{9}micobubbles/mL with an average size of 1 µm (

^{5}microbubbles/mL. This concentration is clinically relevant and has been used in previous studies (

- Leow C.H.
- Iori F.
- Corbett R.
- Duncan N.
- Caro C.
- Vincent P.
- Tang M.X.

*Ultrasound Med Biol.*2015; 41: 2926-2937

Probe parameter | Imaging parameter | ||
---|---|---|---|

Probe | L12-3 v | Imaging mode | Plane wave PI |

Centre frequency | 8 MHz | Transmit frequency | 4 MHz |

Number of element | 128 | Excitation pulse | 1 cycle |

Element pitch | 0.2 mm | PRF | 10 kHz |

Element height | 5 mm | Compounding plane wave | 6 (3 positive, 3 negative) |

Sampling frequency | 32 MHz | Angle range | 20° (10° step) |

Elevational focus | 20 mm | Imaging depth | 5 cm |

#### Velocity estimation using a modified-UIV analysis

*in vitro*experiment to evaluate the new method.

#### WSS estimation

where

*µ*is the dynamic viscosity and $\epsilon {\prime}_{12}$ is the tangential component shear rate tensor. To account for any misalignment between the image coordinate and the coordinate along the wall, the shear rate tensor was computed by transforming the original strain rate tensor in the 2-D Cartesian coordinate to the wall-oriented coordinate system as shown in Figure 2 (

where

*εij*is the original strain tensor in the image Cartesian coordinate defined as

*Cij*is the 2-D rotation transformation matrix, defined for each point along the wall in terms of rotation angle

*θ*between the original and wall-oriented coordinate, as follows

### Error quantification

where

*y*and

*yt*are the UIV estimated value, the reference value, ${y}_{t\left(\mathrm{max}\right)}$ is the maximum reference value and n is the number of time sample.

#### Anthropomorphic flow phantoms

## Results

### Ultrasound flow simulation

#### Steady flow

_{0}= 20 cm/s), an incoherent ensemble correlation approach performs as well as the coherent ensemble correlation approach. However, when the flow velocity increases, the accuracy of the coherent approach degrades and significant errors are quantified when the flow is fast (v

_{0}= 80 cm/s). The incoherent ensemble correlation approach, on the other hand, is more robust against the motion artefacts and highly accurate measurement is estimated under all simulated flow conditions.

V0 (cm/s) | Coherent | Incoherent | ||
---|---|---|---|---|

ME (%) | NRMSE (%) | ME (%) | NRMSE (%) | |

20 | –6.08 ± 4.81 | 1.11 ± 0.03 | –6.01 ± 3.24 | 1.10 ± 0.04 |

50 | –13.78 ± 3.75 | 3.35 ± 0.09 | –8.13 ± 2.45 | 2.68 ± 0.06 |

80 | –69.16 ± 27.74 | 12.96 ± 2.09 | –6.57 ± 1.54 | 5.55 ± 0.19 |

#### Pulsatile flow

**.**These results also indicate a robust WSR estimation as the addition noise, or an SNR of 20 dB, does not change the measurement compared with the ideal situation.

Error measures | CCA | CFA | ||
---|---|---|---|---|

Upper wall | Lower wall | Upper wall | Lower wall | |

ME (%) | 6.5 | 4.9 | 7.2 | 10.6 |

NRMSE (%) | 4.7 | 4.6 | 5.0 | 5.2 |

### In vitro results

^{-1}and WSR up to 2000 s

^{-1}can be observed throughout the carotid bifurcation phantom except at the ICA bulb where a relatively low velocity and WSR can be seen. A vortex subsequently emerges near the ICA sinus immediately after the peak systole and slowly dissipates during the post-systolic phase (Fig. 9b), causing a flow reversal and a negative WSR near the carotid bulb. In addition, it is worth noting that at the inner wall of proximal ICA and ECA, where the branching is located, a relatively high WSR can be observed because of the forward streamline flow. When the dicrotic pulse produced a secondary upstroke, the flow moves forward again, as shown in Figure 9c. This is followed by the reappearance of the vortex near the carotid bulb during the post-dicrotic phase, as shown in Figure 9d. However, during the diastole phase, the flow moves relatively slowly and therefore lower WSR is expected.

## Discussion

*in vitro*experiment. However, the simulation of the contrast pulse sequence and response are possible and can be achieved by adaptation of the Creanius simulator (

## Conclusions

*in vitro*experiment demonstrated the potential of this technique for vascular applications

*in vivo*.

## Acknowledgments

## Supplementary Data

- Video S1
Spatially and temporally resolved quantification of flow velocity vectors and wall shear rate within a computer simulated common carotid artery. Flow velocity vectors are indicated as arrows within the vessel while wall shear rate indicated as coloured straight lines segmenting the vessel wall and the lumen. The images were generated at 1k fps but played back at 100 fps.

- Video S2
Spatially and temporally resolved quantification of flow velocity vectors and wall shear rate within a computer simulated common femoral artery. Flow velocity vectors are indicated as arrows within the vessel while wall shear rate indicated as coloured straight lines segmenting the vessel wall and the lumen. The images were generated at 1k fps but played back at 100 fps.

- Video S3
Spatially and temporally resolved quantification of flow velocity vectors and wall shear rate within a flow phantom mimicking a healthy carotid bifurcation under pulsatile flow condition. Flow velocity vectors are indicated as arrows within the vessel while wall shear rate indicated as coloured curved lines segmenting the vessel wall and the lumen. Images were acquired at 1.67 k fps and played back at 100 fps.

- Video S4
Spatially and temporally resolved quantification of a highly dynamic flow and wall shear rate within a phantom mimicking a diseased carotid bifurcation with a stenosis under pulsatile flow condition. Flow velocity vectors are indicated as arrows within the vessel while wall shear rate indicated as coloured curved lines segmenting the vessel wall and the lumen. Images were acquired at 1.67 k fps and played back at 100 fps.

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