# Spectral Decomposition and Sound Source Localization of Highly Disturbed Flow through a Severe Arterial Stenosis

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Computational Model

#### 2.2. Physics and Flow Conditions

^{3}and, 10

^{−6}Pa·s, respectively.

^{−5}s was evaluated to be sufficient, through a time independence study, to ensure the accuracy of the turbulent transients and a Courant number close to 1. In addition, the time step convergence was less than 10

^{−4}, which is particularly important with SRS models [41]. The initial velocity was set to a value close to the inlet velocity to reduce the initial residual errors. The governing equations were discretized using second-order central discretization in space and second-order implicit discretization in time.

#### 2.3. Proper Orthogonal Decomposition (POD) Analysis

#### 2.4. Mesh

^{+}≤ 1 [10,45]. Mesh should be finer in those areas where high physical gradients are present. Therefore, to increase the accuracy of the flow solution and capture the high flow fluctuations, a solution-based mesh refinement method was used based on a flow parameter. In this study, turbulent kinetic energy (TKE) was selected, since it is an indication of flow fluctuations and energy of sound sources. This mesh refinement was accomplished through the following steps: (1) generate an initial coarse mesh on the geometry, (2) solve a steady-state flow and use TKE to threshold and identify the cells that require refining, (3) create a field function to set a new cell size for the flagged cells for refinement, (4) create a refinement table for the entire domain with the refinement field function as the scalar and extract the values, and (5) add the refinement table to your volume mesher and regenerate the volume mesh. This way, the mesh was optimally refined based on an important flow parameter in the region of interest to avoid unnecessary mesh cells throughout the domain and to reduce the computational costs. This semiautomated refinement method reduced the computational times by about 30% compared to a manual meshing method based on the different predefined regions (inlet, stenosis, fluctuating zone, reattachment, and laminar flow regions).

^{+}≤ 1. After an evaluation of the mean velocity in flow direction along the pipe, the mesh with the maximum cell size of 0.144 mm and total number of mesh cells about 1.4 M was selected. This was also determined suitable for the large eddy simulation (LES) simulations after calculating the ratio of the cell size to the minimum Kolmogorov length scale obtained in the fluctuating region (1D to 4D downstream of stenosis). A ratio of below 20 was attained, which is within the maximum allowable range suggested by [46].

#### 2.5. Validation

## 3. Results and Discussions

^{4}s

^{−1}to clearly present the flow fluctuations in the post-stenotic region. The LES model could accurately predict the fluctuations and capture smaller eddies in this region. Helicity is proportional to the flow velocity and vorticity, and it indicates the potential for the development of a helical flow. Figure 6c suggests that intense vortical structures started to appear inside the stenosis where the flow was mostly unstable close to the stenosis wall (plaque surface).

## 4. Conclusions

- The analysis of the flow solution showed that the flow velocity increased significantly inside the stenosis and became unstable, leading to significant pressure fluctuations at the plaque surface. It indicates the possibility of increased fluid-solid interactions and subsequent excitation of the vessel wall;
- As the flow jet entered into the expansion region, flow separation occurred at x = 1D, where large eddies started to cascade into smaller eddies with higher rotational energies, up to 4D downstream of the stenosis. These eddies were the origin of flow-induced acoustic radiation, which was mostly concentrated around x = 11.5 mm, as the point of maximum excitation. It is important to avoid this region and move the measurement probe further downstream (i.e., x > 4D) of the vessel during coronary catheterization measurements such as the fractional flow reserve (FFR) for accurate readings;
- The analysis of the spectra of the recorded pressures at the wall also showed that the most energetic POD mode of the flow appeared in the same regions, which complimented the results from the fluid dynamics analysis;
- The spectral decomposition of the pressure fluctuations showed broadband acoustic sources distributed in the same region (1D to 4D) generated from turbulence.
- Low-frequency (i.e., <40 Hz) acoustic fluctuations were observed mostly around the flow jet and the separation regions, which were correlated with larger eddies. The break frequency, as a characteristic of the sound transmitted through the vessel wall and surrounding tissue, was considered in the temporal filtering of the acoustic pressure;
- At higher frequency ranges between 80 Hz to 220 Hz, the fluctuations related to smaller eddies appeared at the entrance of the stenosis and, in the middle of the fluctuating region, extended up to 4D downstream of the stenosis;
- The results also showed organized ring-like isosurfaces of fluctuations inside the stenosis at high frequencies over 220 Hz.

## 5. Application Feasibility

## 6. Future Works

- The modeling of flow-induced acoustics in patient-specific models derived from medical imaging. Although the simplified concentric stenosis geometry can help derive qualitative conclusions, the patient-specific irregular stenosis profiles can lead to specific alterations in the generated sounds;
- An acoustic analysis of the progression of stenosis at different levels of severity. The understanding of sound signatures of a stenosis at different stages of the disease can assist to develop an algorithm for the early detection of the stenosis;
- A pulsatile patient-specific flow. The steady flow assumption in this study represented the peak systole of the pulsatile flow. However, the pulsatile flow, with the turbulent diffusion during the diastole with lower flow rates, generates more homogeneous spectra;
- Modeling of elastic wall structural response. Although we verified the use of a rigid wall for this study according to the literature, we should agree that, for the studies focused more on the correlation of hemodynamic parameters with the gradual development of stenosis size and the interactions between the flow and the artery wall, especially with different stiffness of the stenosis, artery, and surrounding tissue, the modeling assumption of an elastic wall becomes more relevant and acceptable.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A sectional view of the flow domain. Flow is from left to right. The figure is out of scale, and the dimensions are in mm.

**Figure 2.**Experimental setup of the Laser Doppler Anemometry (LDA) axial velocity measurements for a constricted pipe representing arterial stenosis.

**Figure 3.**Validation of the computational results of 70% stenosis at Re = 1600 compared with the LDA measurements.

**Figure 4.**Validation of the CFD results of 92% stenosis at Re = 1600 compared with the LDA measurements.

**Figure 5.**Mean pressure on the arterial wall in the post-stenotic region compared to [26].

**Figure 6.**Instantaneous flow parameters: (

**a**) axial velocity; (

**b**) vorticity; (

**c**) helicity; (

**d**) the turbulent kinetic energy (TKE) on the isosurfaces of Q-criterion; (

**e**) wall pressure fluctuations, and (

**f**) root mean square (RMS) of the pressure fluctuations at the middle cross-section of the flow domain.

**Figure 7.**A cross-sectional view of the pressure fluctuations and time series data of the wall pressure fluctuations on the sample points at different locations downstream of the stenosis. Note that only a part of the flow domain close to the stenosis is shown in this figure.

**Figure 8.**Point of maximum excitation in the post-stenotic region by analyzing (

**a**) the RMS of the pressure fluctuations on the wall and (

**b**) TKE on the stenosis centerline.

**Figure 9.**Energy % distribution of the first 50 proper orthogonal decomposition (POD) modes of flow through the stenosis.

**Figure 10.**Visualization of POD mode 1 with the highest energy in the stenosis and post-stenotic region.

**Figure 11.**(

**a**) Acoustic spatial-frequency map of the post-stenotic region and (

**b**) fast Fourier-transform (FFT) of the wall pressure fluctuations at the point of maximum excitation (x = 11.5 mm) in the post-stenotic region.

**Figure 12.**FFT of the wall pressure fluctuations at 41 nodes along the wall in the post-stenotic region. Break frequencies and ranges of locations are included. SPL: sound pressure level.

**Figure 13.**Snapshots of high-energy fluctuations are shown with isosurfaces of the RMS of acoustic pressure for different bandwidths.

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**MDPI and ACS Style**

Khalili, F.; Gamage, P.T.; Taebi, A.; Johnson, M.E.; Roberts, R.B.; Mitchel, J.
Spectral Decomposition and Sound Source Localization of Highly Disturbed Flow through a Severe Arterial Stenosis. *Bioengineering* **2021**, *8*, 34.
https://doi.org/10.3390/bioengineering8030034

**AMA Style**

Khalili F, Gamage PT, Taebi A, Johnson ME, Roberts RB, Mitchel J.
Spectral Decomposition and Sound Source Localization of Highly Disturbed Flow through a Severe Arterial Stenosis. *Bioengineering*. 2021; 8(3):34.
https://doi.org/10.3390/bioengineering8030034

**Chicago/Turabian Style**

Khalili, Fardin, Peshala T. Gamage, Amirtahà Taebi, Mark E. Johnson, Randal B. Roberts, and John Mitchel.
2021. "Spectral Decomposition and Sound Source Localization of Highly Disturbed Flow through a Severe Arterial Stenosis" *Bioengineering* 8, no. 3: 34.
https://doi.org/10.3390/bioengineering8030034