# Spectral Decomposition of the Flow and Characterization of the Sound Signals through Stenoses with Different Levels of Severity

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Computational Model

#### 2.2. Physics and Flow Conditions

^{3}and 10

^{−6}Pa.s, respectively. Reference pressure was zero at the outlet, and the no-slip boundary condition was used at the wall boundaries.

^{−5}s was adequate for accurate solution of the turbulent transients and keep the Courant number close to 1. In addition, time step convergence of less than 10

^{−4}was achieved, which is particularly important with SRS models [36]. The governing equations were also discretized using second-order central discretization in space and second-order implicit discretization in time to deliver accurate results.

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

#### 2.4. Fast Fourier Transform (FFT)

#### 2.5. Mesh

^{+}≤ 1 [43,44]. To increase the accuracy of the flow solution, a solution-based mesh refinement was conducted based on turbulent kinetic energy (TKE) as an indication of flow fluctuations and the energy of sound sources. This was accomplished through the following steps: (a) generate an initial coarse mesh on the geometry; (b) solve a steady-state flow and use TKE to threshold and flag the cells that require refining; (c) create a field function to specify the new cell size for the flagged cells for refinement; (d) create a refinement table for the entire domain with the refinement field function as the scalar, and extract the values; (e) add the refinement table to your volume mesher and re-generate 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 reduce computational costs. This semi-automated method reduced computational times by about 30% compared to manual meshing based on different regions (inlet—stenosis—fluctuating zone—reattachment and laminar flow regions). The mesh was determined suitable for the large-eddy simulation (LES) simulation after calculating the ratio of cell size to the minimum Kolmogorov length scale obtained in the fluctuating region (1D to 4D downstream of stenosis). The maximum cell size in this region was set to 0.144 mm to attain the ratio of below 20, which is within the maximum allowable range suggested by [45].

^{+}≤ 1. The evaluation of the mean velocity in flow direction along the pipe indicated that mesh 3 with 1.4 M mesh cells was the most appropriate mesh configuration. In addition, an accurate prediction of pressure drop in the flows with separation depends on resolving the velocity gradients normal to the wall. Hence prism layers were chosen as they allow the solver to resolve near-wall flow more accurately. A 10-layer prism layer mesh with a total thickness of 0.059 mm and layer stretching factor of 1.35 was employed near the boundaries to resolve the velocity gradients normal to the wall.

#### 2.6. Validation

## 3. Results

## 4. Conclusions

- For the least severe cases, the flow solution analysis showed slight disturbances to the flow up to 50% severity. For higher severities, it was observed that the flow velocity increased significantly inside the stenosis, became unstable close to the stenosis wall, and caused significant pressure fluctuations at the plaque surface. It indicates the possibility of higher excitation of the vessel wall in the constricted flow area.
- For the most severe cases (70%, 87%, and 92%), the shear layers around the flow jet became unstable at about x = 1D, breaking into smaller eddies. The fluctuating zone was determined between 1D and 4D downstream of the stenosis, in which as the mean axial velocity decreased, the flow fluctuations increased with distance. This region contained the highest level of flow fluctuations and sound sources.
- While frequency content analysis of RMS of wall pressure fluctuations showed that the severity of 20% did not generate significant turbulent fluctuations compared to the healthy artery, the acoustic energy spectrum increased exponentially with severity levels at the point of maximum excitation at x = 2D for the most severe cases.
- Break frequencies, ranging from 40 to 230 Hz, associated with each specific severity level, were found in this study. An increase in break frequencies was also observed with increased levels of severity. These can suggest a non-invasive approach for predicting the severity of the stenosis.
- As the severity increased over 50%, peak frequencies with higher energies appeared in the acoustic spatial-frequency map of the post-stenotic region. This is another indicator of the progression of the stenosis. Furthermore, additional high-energy frequency ranges of approximately 400–600 Hz, 1000–1400 Hz, and 750–1000 Hz for 70%, 87%, and 92% severities, respectively, were observed, which can help us to estimate the level of severity at late stages.
- Visualization of acoustic pressures filtered at high frequencies of 1000–1400 Hz helped localize the source of the high-frequency fluctuations. As the severity increased, turbulent instabilities were initiated inside the stenosis forming relatively smaller eddies close to the wall and comparable eddies in the shear layers of the flow jet.

## 5. Future Works

- Sound analysis of flow-induced acoustics in patient-specific models derived from medical imaging, with more realistic flow properties, which may lead to specific alterations in the generated sounds compared to simplified models.
- Additional severities to derive a general correlation between the emerged signals and severity levels, which can assist to develop an algorithm for early detection of the stenosis.
- Expanding the current methodology to combine computational fluid dynamics, finite element analysis, and sound analysis techniques to conduct an in-depth investigation on the propagation of flow-induced sound waves through artery wall and the surrounding tissue.
- Combination of POD and frequency-based flow decomposition methods to study possible characteristic frequencies of flow structures for aortic aneurysm.
- Considering Pulsatile flow to account for pressure fluctuations, especially in the accelerating and decelerating phases.
- The proposed approach was performed on a few levels of severity. This needs to be tested on several cases, cross-validated by experimental sound analysis, and expanded by signal processing techniques.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Toth, P.P. Subclinical atherosclerosis: What it is, what it means and what we can do about it. Int. J. Clin. Pract.
**2008**, 62, 1246–1254. [Google Scholar] [CrossRef] [PubMed][Green Version] - Go, A.S.; Mozaffarian, D.; Roger, V.L.; Benjamin, E.J.; Berry, J.D.; Borden, W.B.; Bravata, D.M.; Dai, S.; Ford, E.S.; Fox, C.S.; et al. Heart disease and stroke statistics-2013 update: A Report from the American Heart Association. Circulation
**2013**, 127, e6–e245. [Google Scholar] [CrossRef] [PubMed] - Rosamond, W.; Flegal, K.; Friday, G.; Furie, K.; Go, A.; Greenlund, K.; Haase, N.; Ho, M.; Howard, V.; Kissela, B.; et al. Heart disease and stroke statistics—2007 Update: A report from the American Heart Association Statistics Committee and Stroke Statistics Subcommittee. Circulation
**2007**, 115, e69–e171. [Google Scholar] [CrossRef] [PubMed] - Fryar, C.D.; Chen, T.C.; Li, X. Prevalence of uncontrolled risk factors for cardiovascular disease: United States, 1999–2010. In NCHS Data Brief; US Department of Health and Human Services, Centers for Disease Control and Prevention, National Center for Health Statistics: Hyattsville, MD, USA, 2012; pp. 1–8. [Google Scholar] [PubMed]
- Doriot, P.A.; Dorsaz, P.A.; Dorsaz, L.; Rutishauser, W. Overestimation of stenosis severity by single plane geometric measurements. In Proceedings of the Computers in Cardiology 1996, Indianapolis, IN, USA, 8–11 September 1996. [Google Scholar] [CrossRef]
- Alshuhri, A.A.; Holsgrove, T.P.; Miles, A.W.; Cunningham, J.L. Development of a non-invasive diagnostic technique for acetabular component loosening in total hip replacements. Med. Eng. Phys.
**2015**, 37, 739–745. [Google Scholar] [CrossRef][Green Version] - Irie, S.; Inoue, K.; Yoshida, K.; Mamou, J.; Kobayashi, K.; Maruyama, H.; Yamaguchi, T. Speed of sound in diseased liver observed by scanning acoustic microscopy with 80 MHz and 250 MHz. J. Acoust. Soc. Am.
**2016**, 139, 512–519. [Google Scholar] [CrossRef] [PubMed] - Fredberg, J.J. Origin and character of vascular murmurs: Model studies. J. Acoust. Soc. Am.
**1977**, 61, 1077–1085. [Google Scholar] [CrossRef] - Mittal, R.; Simmons, S.P.; Najjar, F. Numerical study of pulsatile flow in a constricted channel. J. Fluid Mech.
**2003**, 485, 337–378. [Google Scholar] [CrossRef][Green Version] - Chang, Y.; Kim, N.; Stenfelt, S. The development of a whole-head human finite-element model for simulation of the transmission of bone-conducted sound. J. Acoust. Soc. Am.
**2016**, 140, 1635–1651. [Google Scholar] [CrossRef][Green Version] - Khalili, F.; Gamage, P.P.T.; Meguid, I.A.; Mansy, H.A. A coupled CFD-FEA study of sound generated in a stenosed artery and transmitted through tissue layers. In Proceedings of the SoutheastCon 2018, St. Petersburg, FL, USA, 19–22 April 2018. [Google Scholar]
- Lees, R.S.; Dewey, C.F. Phonoangiography: A new noninvasive diagnostic method for studying arterial disease. Proc. Natl. Acad. Sci. USA
**1970**, 67, 935–942. [Google Scholar] [CrossRef][Green Version] - Lee, S.E.; Lee, S.W.; Fischer, P.F.; Bassiouny, H.S.; Loth, F. Direct numerical simulation of transitional flow in a stenosed carotid bifurcation. J. Biomech.
**2008**, 41, 2551–2561. [Google Scholar] [CrossRef][Green Version] - Salman, H.E.; Yazicioglu, Y. Flow-induced vibration analysis of constricted artery models with surrounding soft tissue. J. Acoust. Soc. Am.
**2017**, 142, 1913–1925. [Google Scholar] [CrossRef] - Salman, H.E.; Yazicioglu, Y. Experimental and numerical investigation on soft tissue dynamic response due to turbulence-induced arterial vibration. Med. Biol. Eng. Comput.
**2019**, 57, 1737–1752. [Google Scholar] [CrossRef] - Tobin, R.J.; Chang, I.D. Wall pressure spectra scaling downstream of stenoses in steady tube flow. J. Biomech.
**1976**, 9, 633–640. [Google Scholar] [CrossRef] - Borisyuk, A.O. Model study of noise field in the human chest due to turbulent flow in a larger blood vessel. J. Fluids Struct.
**2003**, 17, 1095–1110. [Google Scholar] [CrossRef] - Seo, J.H.; Mittal, R. A coupled flow-acoustic computational study of bruits from a modeled stenosed artery. Med. Biol. Eng. Comput.
**2012**, 50, 1025–1035. [Google Scholar] [CrossRef] - Gamage, P.T. Modeling of Flow Generated Sound in a Constricted Duct at Low Mach Number Flow. Master’s Thesis, University of Central Florida, Orlando, FL, USA, 2017; p. 5668. Available online: https://stars.library.ucf.edu/etd/5668 (accessed on 30 December 2020).
- Gamage, P.T.; Khalili, F.; Mansy, H.A. Aero--acoustics in constricted pipe flow at low mach number. J. Appl. Biotechnol. Bioeng.
**2018**, 5, 306–309. [Google Scholar] - Bakhshinejad, A.; Baghaie, A.; Vali, A.; Saloner, D.; Rayz, V.L.; D’Souza, R.M. Merging computational fluid dynamics and 4D Flow MRI using proper orthogonal decomposition and ridge regression. J. Biomech.
**2017**, 58, 162–173. [Google Scholar] [CrossRef] - Janiga, G. Quantitative assessment of 4D hemodynamics in cerebral aneurysms using proper orthogonal decomposition. J. Biomech.
**2019**, 82, 80–86. [Google Scholar] [CrossRef] - Kefayati, S.; Poepping, T.L. Transitional flow analysis in the carotid artery bifurcation by proper orthogonal decomposition and particle image velocimetry. Med. Eng. Phys.
**2013**, 35, 898–909. [Google Scholar] [CrossRef] - Grinberg, L.; Yakhot, A.; Karniadakis, G.E. Analyzing transient turbulence in a stenosed carotid artery by proper orthogonal decomposition. Ann. Biomed. Eng.
**2009**, 37, 2200–2217. [Google Scholar] [CrossRef] [PubMed] - Natarajan, T.; MacDonald, D.E.; Najafi, M.; Coppin, P.W.; Steinman, D.A. Spectral decomposition and illustration-inspired visualisation of highly disturbed cerebrovascular blood flow dynamics. Comput. Method Biomech. Biomed. Eng. Imaging Vis.
**2020**, 8, 182–193. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Beach, T.G.; Maarouf, C.L.; Brooks, R.G.; Shirohi, S.; Daugs, I.D.; Sue, L.I.; Sabbagh, M.N.; Walker, D.G.; Lue, L.; Roher, A.E. Reduced clinical and postmortem measures of cardiac pathology in subjects with advanced Alzheimer’s disease. BMC Geriatr.
**2011**, 11, 3. [Google Scholar] [CrossRef][Green Version] - Khalili, F.; Gamage, P.P.T.; Mansy, H.A. Verification of Turbulence Models for Flow in a Constricted Pipe at Low Reynolds Number. In Proceedings of the 3rd Thermal and Fluids Engineering Conference (TFEC), Fort Lauderdale, FL, USA, 4–7 March 2018; pp. 1–10. [Google Scholar]
- Yazicioglu, Y.; Royston, T.J.; Spohnholtz, T.; Martin, B.; Loth, F.; Bassiouny, H.S. Acoustic radiation from a fluid-filled, subsurface vascular tube with internal turbulent flow due to a constriction. J. Acoust. Soc. Am.
**2005**, 118, 1193–1209. [Google Scholar] [CrossRef] - Sandgren, T.; Sonesson, B.; Ahlgren, A.R.; Lanne, T. The diameter of the common femoral artery in healthy human: Influence of sex, age, and body size. J. Vasc. Surg.
**1999**, 29, 503–510. [Google Scholar] [CrossRef][Green Version] - Chami, H.A.; Keyes, M.J.; Vita, J.A.; Mitchell, G.F.; Larson, M.G.; Fan, S.; Vasan, R.S.; O’Connor, G.T.; Benjamin, E.J.; Gottlieb, D.J. Brachial artery diameter, blood flow and flow-mediated dilation in sleep-disordered breathing. Vasc. Med.
**2009**, 14, 351–360. [Google Scholar] [CrossRef] - Gayathri, K.; Shailendhra, K. Pulsatile blood flow in large arteries: Comparative study of Burton’s and McDonald’s models. Appl. Math. Mech. Engl. Ed.
**2014**, 35, 575–590. [Google Scholar] [CrossRef] - Ozden, K.; Sert, C.; Yazicioglu, Y. Numerical investigation of wall pressure fluctuations downstream of concentric and eccentric blunt stenosis models. Proc. Inst. Mech. Eng. Part H J. Eng. Med.
**2020**, 234, 48–60. [Google Scholar] [CrossRef] [PubMed] - Tan, F.P.P.; Wood, N.B.; Tabor, G.; Xu, X.Y. Comparison of les of steady transitional flow in an idealized stenosed axisymmetric artery model with a RANS transitional model. J. Biomech. Eng.
**2011**, 133, 051001. [Google Scholar] [CrossRef] - Paul, M.C.; Molla, M.M. Investigation of physiological pulsatile flow in a model arterial stenosis using large-eddy and direct numerical simulations. Appl. Math. Model.
**2012**, 36, 4393–4413. [Google Scholar] [CrossRef] - RHIE, C.M.; CHOW, W.L. Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J.
**1983**, 21, 1525–1532. [Google Scholar] [CrossRef] - Fathi, M.F.; Bakhshinejad, A.; Baghaie, A.; Saloner, D.; Sacho, R.H.; Rayz, V.L.; D’Souza, R.M. Denoising and spatial resolution enhancement of 4D flow MRI using proper orthogonal decomposition and lasso regularization. Comput. Med. Imaging Graph.
**2018**, 70, 165–172. [Google Scholar] [CrossRef] [PubMed] - Le, T.B.; Akerkouch, L. On the Modal Analysis of Blood Flow Dynamics in Brain Aneurysms. In Proceedings of the 2020 Design of Medical Devices Conference, 2020 Design of Medical Devices Conference, Minneapolis, MN, USA, 6–9 April 2020. [Google Scholar]
- Darwish, A.; Di Labbio, G.; Saleh, W.; Kadem, L. Proper Orthogonal Decomposition Analysis of the Flow Downstream of a Dysfunctional Bileaflet Mechanical Aortic Valve. Cardiovasc. Eng. Technol.
**2021**. [Google Scholar] [CrossRef] - Habibi, M.; Dawson, S.T.M.; Arzani, A. Data-Driven pulsatile blood flow physics with dynamic mode decomposition. Fluids
**2020**, 5, 111. [Google Scholar] [CrossRef] - Janiga, G. Novel feature-based visualization of the unsteady blood flow in intracranial aneurysms with the help of proper orthogonal decomposition (POD). Comput. Med. Imaging Graph.
**2019**, 73, 30–38. [Google Scholar] [CrossRef] [PubMed] - Ballarin, F.; Faggiano, E.; Ippolito, S.; Manzoni, A.; Quarteroni, A.; Rozza, G.; Scrofani, R. Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization. J. Comput. Phys.
**2016**, 315, 609–628. [Google Scholar] [CrossRef] - Ariff, M.; Salim, S.M.; Cheah, S.C. Wall Y + Approach for Dealing With Turbulent Flow Over a Surface Mounted Cube: Part 1–Low Reynolds Number. In Proceedings of the Seventh International Conference on CFD in the Minerals and Process Industries, Melbourne, Australia, 9–11 December 2009; pp. 1–6. [Google Scholar] [CrossRef]
- Salim, S.M.; Cheah, S.C. Wall y + Strategy for Dealing with Wall-bounded Turbulent Flows. In Proceedings of the he International MultiConference of Engineers and Computer Scientists, Hong Kong, China, 18–20 March 2009; Volume II, pp. 1–6. [Google Scholar]
- Celik, I.; Klein, M.; Janicka, J. Assessment measures for engineering LES applications. J. Fluids Eng. Trans. ASME
**2009**, 131, 031102. [Google Scholar] [CrossRef] - Lu, P.C.; Gross, D.R.; Hwang, N.H.C. Intravascular pressure and velocity fluctuations in pulmonic arterial stenosis. J. Biomech.
**1980**, 13, 291–300. [Google Scholar] [CrossRef] - Borisyuk, A.O. Modeling of noise generation by a vascular stenosis. Int. J. Fluid Mech. Res.
**2002**, 29, 24. [Google Scholar] [CrossRef] - Gamage, P.T.; Azad, M.K.; Taebi, A.; Sandler, R.H.; Mansy, H.A. Clustering of SCG Events Using Unsupervised Machine Learning. In Signal Processing in Medicine and Biology; Springer: Berlin/Heidelberg, Germany, 2020; pp. 205–233. [Google Scholar]
- Borisyuk, A.O. Experimental study of wall pressure fluctuations in rigid and elastic pipes behind an axisymmetric narrowing. J. Fluids Struct.
**2010**, 26, 658–674. [Google Scholar] [CrossRef] - Mamun, K.; Rahman, M.M.; Akhter, M.N.; Ali, M. Physiological non-Newtonian blood flow through single stenosed artery. In Proceedings of the AIP Conference Proceedings, Dhaka, Bangladesh, 18–20 December 2015. [Google Scholar]
- Jabir, E.; Lal, S.A. Numerical analysis of blood flow through an elliptic stenosis using large eddy simulation. Proc. Inst. Mech. Eng. Part H J. Eng. Med.
**2016**, 230, 709–726. [Google Scholar] [CrossRef] [PubMed] - Mancini, V.; Bergersen, A.W.; Vierendeels, J.; Segers, P.; Valen-Sendstad, K. High-Frequency Fluctuations in Post-stenotic Patient Specific Carotid Stenosis Fluid Dynamics: A Computational Fluid Dynamics Strategy Study. Cardiovasc. Eng. Technol.
**2019**, 10, 277–298. [Google Scholar] [CrossRef][Green Version] - Valen-Sendstad, K.; Steinman, D.A. Mind the gap: Impact of computational fluid dynamics solution strategy on prediction of intracranial aneurysm hemodynamics and rupture status indicators. Am. J. Neuroradiol.
**2014**, 35, 536–543. [Google Scholar] [CrossRef] [PubMed][Green Version] - Raschi, M.; Mut, F.; Byrne, G.; Putman, C.M.; Tateshima, S.; Viñuela, F.; Tanoue, T.; Tanishita, K.; Cebral, J.R. CFD and PIV analysis of hemodynamics in a growing intracranialaneurysm. Int. J. Numer. Method. Biomed. Eng.
**2012**, 28, 214–228. [Google Scholar] [CrossRef] [PubMed][Green Version]

**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 Laser Doppler Anemometry (LDA) axial velocity measurements for a constricted pipe representing arterial stenosis.

**Figure 3.**Validation of computational fluid dynamics (CFD) results of 92% stenosis at Re = 1600 with the LDA measurements.

**Figure 5.**Mean axial velocity at six different locations (1D to 6D) downstream of stenosis for all severity cases.

**Figure 6.**Pressure fluctuations on the arterial wall and root-mean-square (RMS) of pressure fluctuations on the middle cross-section of the flow domain showing the concentration and high-energy fluctuation through and downstream of the stenosis.

**Figure 7.**Point of maximum excitation in the post-stenotic region by analyzing (

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

**b**) turbulent kinetic energy (TKE) on the stenosis centerline.

**Figure 8.**For all severity cases: (

**a**) Sound pressure level (SPL) variation at different frequencies of acoustic pressure and (

**b**) exponential increase in RMS of pressure fluctuations and mean axial velocity with an increase in stenosis severity.

**Figure 10.**For 87% stenosis: (

**a**) frequency content of the flow, (

**b**) isosurfaces of proper orthogonal decomposition (POD) mode 1 of RMS of acoustic pressure, (

**c**) snapshot of high-energy fluctuations for different frequency ranges and the bandwidth of 1000–1400 Hz.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Khalili, F.; Gamage, P.T.; Taebi, A.; Johnson, M.E.; Roberts, R.B.; Mitchell, J.
Spectral Decomposition of the Flow and Characterization of the Sound Signals through Stenoses with Different Levels of Severity. *Bioengineering* **2021**, *8*, 41.
https://doi.org/10.3390/bioengineering8030041

**AMA Style**

Khalili F, Gamage PT, Taebi A, Johnson ME, Roberts RB, Mitchell J.
Spectral Decomposition of the Flow and Characterization of the Sound Signals through Stenoses with Different Levels of Severity. *Bioengineering*. 2021; 8(3):41.
https://doi.org/10.3390/bioengineering8030041

**Chicago/Turabian Style**

Khalili, Fardin, Peshala T. Gamage, Amirtahà Taebi, Mark E. Johnson, Randal B. Roberts, and John Mitchell.
2021. "Spectral Decomposition of the Flow and Characterization of the Sound Signals through Stenoses with Different Levels of Severity" *Bioengineering* 8, no. 3: 41.
https://doi.org/10.3390/bioengineering8030041