# Geometry and Flow Properties Affect the Phase Shift between Pressure and Shear Stress Waves in Blood Vessels

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## Abstract

**:**

## 1. Introduction

## 2. Physical Model

#### 2.1. Physical Ingredients

**x**over one period [44]:

#### 2.2. Dimensionless Groups

## 3. Numerical Model

## 4. Validation

## 5. Results and Discussion

#### 5.1. Study Cases

- First, we focused on the effect of flow. We performed the simulations of sinusoidal flow through a curved tube without an aneurysm (Figure 1b). We varied the flow properties in two ways:
- (a)
- Varying the Womersley number $\alpha $ while keeping the Reynolds number Re fixed. This was achieved by changing the period T: $\alpha =1.56$, 2.0, and 2.88 for $T=2.72$ s, 1.6 s, and 0.8 s, respectively (see Figure 4). Note that the average flow speed (and thus Re) depends on $\alpha $. Therefore, one also needs to change the pressure drop to maintain a given value of Re;
- (b)
- Varying only Re by altering the flow speed via the amplitude of the pressure gradient: Re = 5, 50, 179, or 250 as a result of changing the average flow velocity to 0.00475 m/s, 0.0475 m/s, 0.17 m/s, or 0.2375 m/s, respectively (see Figure 5);

#### 5.2. Varying Flow Properties in a Curved Tube without Aneurysm

#### 5.3. Curved Tube with Aneurysm

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Time Scales

## References

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**Figure 1.**Three-dimensional geometries studied in this work. The straight circular tube in (

**a**) is used to benchmark the code via comparisons with analytical solutions for pulsatile Womersley flow. (

**b,c**) represent a curved blood vessel without and with a side-wall aneurysm, respectively. The membrane that defines the blood vessel has zero thickness and is represented by a set of vertices and flat triangular facets.

**Figure 2.**Phase shift between pressure, P, and WSS, ${\sigma}_{\mathrm{w}}$. The time delay between pressure and WSS is described by the phase shift $\Delta \phi ={360}^{\circ}\times \Delta t/T$, where $\Delta t=t\left({\sigma}_{\mathrm{w}}^{\mathrm{max}}\right)-t\left({P}^{\mathrm{max}}\right)$. Here, $t\left({\sigma}_{\mathrm{w}}^{\mathrm{max}}\right)/T$ (blue bar) and $t\left({P}^{\mathrm{max}}\right)/T$ (red bar) represent the times of maximum WSS and pressure within one period T, respectively. This kind of behavior is examined at each surface point in the vessel, as each point may have a different time evolution of WSS.

**Figure 3.**Phase shift, $\Delta \phi $, as a function of the Womersley number, $\alpha $, for flow in a straight stiff pipe. The analytical data (black line) are obtained via Equation (5) using a Python script. In simulations using the geometry shown in Figure 1a, we alter the viscosity $\nu $ and/or period T to control $\alpha $. The simulated phase shift is averaged over the entire tube wall (red circles). We observe an excellent agreement, with the maximum numerical error being less than 2%. The inset shows the analytic result for large $\alpha $, suggesting that $\Delta \phi \to {45}^{\circ}$ for $\alpha \to \infty $.

**Figure 4.**Simulation results at a constant Reynolds number (Re = 50) with different Womersley numbers $\alpha $ by varying the period T. (

**a**) $\alpha =1.56$ ($T=2.72$ s), (

**b**) $\alpha =2.0$ ($T=1.6$ s), and (

**c**) $\alpha =2.88$ ($T=0.8$ s). The average flow speed is 0.0475 m/s in all three cases. The flow direction is from left to right. The left, middle, and right columns, respectively, show the phase shift ($\Delta \phi $), time-averaged wall shear stress (TAWSS), and temporal wall shear stress gradient (TWSSG). $\Delta \phi $ ranges from ${14}^{\circ}$ to ${18}^{\circ}$ in (

**a**), from ${18}^{\circ}$ to ${30}^{\circ}$ in (

**b**), and from ${11}^{\circ}$ to ${53}^{\circ}$ in (

**c**). According to Equation (5), the phase shift for flow in a straight tube is ${16}^{\circ}$, ${24}^{\circ}$, and ${35}^{\circ}$ for $\alpha =1.56$, $2.0$, and $2.88$, respectively. The TAWSS and TWSSG are rescaled by their characteristic values 2 Pa and $3.4$ Pa/s, respectively.

**Figure 5.**Simulation results at a constant Womersley number ($\alpha =2.88$) with different Reynolds numbers Re by varying the flow velocity. (

**a**) Re = 5, (

**b**) Re = 50, (

**c**) Re = 179, and (

**d**) Re = 250. The flow direction is from left to right. The left, middle, and right columns, respectively, show the phase shift ($\Delta \phi $), time-averaged wall shear stress (TAWSS), and temporal wall shear stress gradient (TWSSG). The corresponding analytical phase shift for flow in a straight tube is ${35}^{\circ}$, according to Equation (5). For a more convenient visual comparison, the same color ranges are applied for all simulations: $\Delta \phi $ ranges from ${32}^{\circ}$ to ${40}^{\circ}$ in (

**a**), from ${11}^{\circ}$ to ${53}^{\circ}$ in (

**b**), from $-{6}^{\circ}$ to ${48}^{\circ}$ in (

**c**), and from $-{28}^{\circ}$ to ${48}^{\circ}$ in (

**d**). The negative phase shift means that the WSS signal arrives before the pressure signal. The TAWSS and TWSSG are rescaled by their characteristic values 2 Pa and $3.4$ Pa/s, respectively.

**Figure 6.**Spatial distributions of the phase shift ($\Delta \phi $), time-averaged wall shear stress (TAWSS), and temporal wall shear stress gradient (TWSSG) for different geometries within one period T. The flow direction is from left to right. In all the cases shown, the average flow velocity, the Reynolds number, and the Womersley value are 0.17 m/s, $\mathrm{Re}=179$, and $\alpha =2.88$, respectively. For this value of the Womersley number, the analytical phase shift for flow in a straight tube is ${35}^{\circ}$. The simulated phase shift ranges from ${33}^{\circ}$ to ${37}^{\circ}$ in (

**a**), from $-{6}^{\circ}$ to ${48}^{\circ}$ in (

**b**), and from $-{30}^{\circ}$ to ${145}^{\circ}$ in (

**c**). For convenient direct visual comparison, the same color ranges are applied for all geometries. The negative phase shift means that the WSS signal arrives before the pressure signal. The TAWSS and TWSSG are rescaled by their characteristic values 2 Pa and $3.4$ Pa/s, respectively.

**Figure 7.**Relation between the phase shift ($\Delta \phi $) and the time-averaged wall shear stress (TAWSS). $\Delta \phi $ and the TAWSS (rescaled by the characteristic value 2 Pa) are computed (

**a**) over the entire curved tube without aneurysm (Figure 6b) and (

**b**) outside and (

**c**) within the aneurysmal region (Figure 6c). In (

**c**), large WSS appears in the flow impingement region. Outside the aneurysm, there is a noticeable positive correlation between $\Delta \phi $ and TAWSS. Within the aneurysmal dome, however, no strong correlation is found.

**Figure 8.**(

**a**) Flow field at a cross-section of the curved tube midway between the inlet and outlet of the tube for $\mathrm{Re}=179$ and $\alpha =2.88$. Velocity is rescaled by the characteristic value $0.17$ m/s. The color scale gives the primary (axial) flow speed. The highest velocity appears below the geometric center of the cross-section. Streamlines serve to visualize the secondary (cross-sectional) flow field. (

**b**) Relation of the phase shift between pressure and velocity signals $\Delta \phi $ and peak velocity (rescaled by $U=0.17$ m/s). $\Delta \phi $ and the peak velocity are computed along the vertical line shown in (

**a**). The red solid line in (

**b**) represents the analytical solution of oscillatory flow through a straight pipe [53]. Farther away from the site of highest velocity, the phase shift is smaller in the straight tube; this also applies to the curved pipe in addition to a small region above the site of highest velocity. Note that the point of highest velocity is defined as the point on the cross-section that has the largest velocity averaged over a full period. The peak velocity at any given point indicates the maximum velocity within a full period at that point.

**Table 1.**Simulation parameters for the benchmark tests in simulation units. The simulation box is $({N}_{x},\phantom{\rule{3.33333pt}{0ex}}{N}_{y},\phantom{\rule{3.33333pt}{0ex}}{N}_{z})=(256,\phantom{\rule{3.33333pt}{0ex}}57,\phantom{\rule{3.33333pt}{0ex}}57)$. The fluid density is $\rho =1$, and the tube radius is $R=22.9$ (tube length $L=12R$). We apply stiff springs on all Lagrangian nodes to mimic nearly rigid walls. The elastic shear modulus ${\kappa}_{\mathrm{s}}$, area dilation modulus ${\kappa}_{\alpha}$, and spring modulus ${\kappa}_{\mathrm{sp}}$ have the same magnitude of 0.3. The bending modulus is ${\kappa}_{\mathrm{b}}=0.03$. All other relevant parameters are listed below. ${P}^{\prime}\left(t\right)$: pulsatile pressure gradient (note Equation (4) with ${P}_{0}^{\prime}$ indicating the amplitude); T: period; $\eta $: dynamic viscosity. For blood flow in major brain arteries, the Womersley number, $\alpha $, is around 3. In validation tests, the Womersley number varies from 1.02 to 2.88 via changing the period and/or viscosity. The Reynolds number is either 5 or 179.

$\mathit{\alpha}$ | ${\mathit{P}}_{0}^{\prime}$ | T | $\mathit{\eta}$ |
---|---|---|---|

1.02 | $2.4\xb7{10}^{-7}$ | 262,492 | 0.012 |

1.52 | $2.4\xb7{10}^{-7}$ | 119,314 | 0.012 |

2.06 | $2.4\xb7{10}^{-7}$ | 64,494 | 0.012 |

2.54 | $6.5\xb7{10}^{-7}$ | 153,393 | 0.0033 |

2.88 | $6.5\xb7{10}^{-7}$ | 119,409 | 0.0033 |

Physical | Simulation | ||
---|---|---|---|

Variable | Units | Units | |

Radius | R | 2.0 mm | 22.9 |

Viscosity | $\eta $ | 4 mPa s | 0.0033 |

Density | $\rho $ | 1.055 g/cm^{3} | 1.0 |

Period | T | 0.8 s | 119,409 |

Pressure gradient | ${P}_{0}^{\prime}$ | 1357 Pa/m | $6.5\xb7{10}^{-7}$ |

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**MDPI and ACS Style**

Wang, H.; Krüger, T.; Varnik, F. Geometry and Flow Properties Affect the Phase Shift between Pressure and Shear Stress Waves in Blood Vessels. *Fluids* **2021**, *6*, 378.
https://doi.org/10.3390/fluids6110378

**AMA Style**

Wang H, Krüger T, Varnik F. Geometry and Flow Properties Affect the Phase Shift between Pressure and Shear Stress Waves in Blood Vessels. *Fluids*. 2021; 6(11):378.
https://doi.org/10.3390/fluids6110378

**Chicago/Turabian Style**

Wang, Haifeng, Timm Krüger, and Fathollah Varnik. 2021. "Geometry and Flow Properties Affect the Phase Shift between Pressure and Shear Stress Waves in Blood Vessels" *Fluids* 6, no. 11: 378.
https://doi.org/10.3390/fluids6110378