# Study of Ultrasonic Guided Wave Propagation in Bone Composite Structures for Revealing Osteoporosis Diagnostic Indicators

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## Abstract

**:**

## 1. Introduction

## 2. Bone Phantoms and Experimental Measurements

## 3. Guided Waves in Bone Phantoms

## 4. H-Function-Based Retrieval of Experimental Dispersion Curves

## 5. MPM-Based Retrieving of GW Parameters

## 6. Diagnostic Indicators

#### 6.1. Effective Material Parameters

#### 6.2. Resonance Response

## 7. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Phantom blanks: plexiglass plates drilled from below for different depths; (

**b**) top view of plates drilled in a checkerboard pattern; (

**c**) experimental setup; and (

**d**) setup with a specimen covered by mammalian tissue.

**Figure 3.**Examples of driving pulses $p\left(t\right)$ (

**top**) and their frequency spectra $\left|P\right(f\left)\right|$ (

**bottom**).

**Figure 4.**Examples of measured signals on the phantoms successively subjected to the three pulses shown in Figure 3: uncoated (

**a**) and coated (

**b**) samples with intact plates; and uncovered (

**c**) and covered (

**d**) 2/3 drilled plates; $h=3$ mm.

**Figure 5.**Examples of time–space waveform profiles measured at the 2/3 drilled phantoms ($h=$ 3 mm, ${h}_{pore}=$ 2 mm; left column) and intact thick-plate phantoms ($h=$ 6 mm, ${h}_{pore}=$ 0 mm; right column); uncoated (

**a**,

**b**) and coated with ${h}_{soft}=$ 2 mm (

**c**,

**d**) and ${h}_{soft}=$ 4 mm (

**e**,

**f**) soft layer; points’ spacing $\Delta x=1$ mm, time discrete $\Delta t=$ 0.03 μs. Straight lines emphasize the propagation of fast and slow wave packets.

**Figure 6.**Slowness dispersion curves for uncoated samples I–III (top, (

**a**–

**c**)) and coated phantoms IV–VI (bottom, (

**d**–

**f**)); $h=3$ mm.

**Figure 9.**Depth dependencies of the ${A}_{0}$ and ${S}_{0}$ fundamental modes excited in uncoated plates I–III (blue and red lines in top subplots (

**a**–

**c**)), and of the first three modes in coated phantoms IV–VI (bottom, (

**d**–

**f**)), green lines are for the additional mode arising in the coated samples, horizontal black lines show interfaces between sublayers); $f=100$ kHz.

**Figure 10.**Lamb wave dispersion curves superimposed on the blurry spots of the H-function were calculated based on experimental data measured on the plexiglass plate of a thickness of 5 mm subjected to two pulses at ${f}_{c}=$ 100 and 300 kHz.

**Figure 11.**Wavenumbers ${\zeta}_{n}\left(f\right)$ extracted from experimental data by the double-sided MPM processing with $\delta =0.1$; blue and red points are for Re ${\zeta}_{n}$ and −Im ${\zeta}_{n}$, respectively.

**Figure 12.**The points from Figure 11 retained after H-filtering with $\epsilon =0.1$; blue and red points are for Re ${\zeta}_{n}$ and −Im ${\zeta}_{n}$, respectively.

**Figure 13.**Restoring the effective material parameters of the hard (plexiglass) layer (

**a**) and its lower drilled part (

**b**) from synthetic data calculated for phantoms I–VI; horizontal lines indicate that the input body wave velocities ${c}_{p}$ and ${c}_{s}$ and markers are for their restored values; sweep driving pulse, Figure 3, right.

**Figure 14.**Amplitudes of frequency spectra ${a}_{n}\left(f\right)$ of the guided waves generated in phantoms I–VI (solid lines) and their total sum (dashed lines); delta pulse, $P=1$.

**Figure 17.**Diagrams of resonance frequencies ${f}_{r}$ depending on the sample structure for the peak patterns in Figure 16, uncovered (

**a**) and covered (

**b**) samples; the same for the h = 6 mm thick plate samples (

**c**,

**d**). Light blue and pink lines are for the resonance frequencies inherent in uncoated phantoms of thickness $h=3$ mm (

**a**), which also appear and keep decreasing in coated samples (

**b**); twice-thicker phantoms yield additional resonance (green line, (

**c**)) that also keeps decreasing in covered samples (

**d**).

Material | ${\mathit{c}}_{\mathit{p}}$ (m/s) | ${\mathit{c}}_{\mathit{s}}$ (m/s) | $\mathit{\rho}$ (kg/m^{3}) | $\mathit{\nu}$ |
---|---|---|---|---|

Soft plastic | 1550 | 369 | 1060 | 0.47 |

Plexiglass (PMMA) | 2700 | 1226 | 1190 | 0.37 |

Its drilled part | 2025 | 920 | 952 | 0.37 |

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## Share and Cite

**MDPI and ACS Style**

Glushkov, E.V.; Glushkova, N.V.; Ermolenko, O.A.; Tatarinov, A.M.
Study of Ultrasonic Guided Wave Propagation in Bone Composite Structures for Revealing Osteoporosis Diagnostic Indicators. *Materials* **2023**, *16*, 6179.
https://doi.org/10.3390/ma16186179

**AMA Style**

Glushkov EV, Glushkova NV, Ermolenko OA, Tatarinov AM.
Study of Ultrasonic Guided Wave Propagation in Bone Composite Structures for Revealing Osteoporosis Diagnostic Indicators. *Materials*. 2023; 16(18):6179.
https://doi.org/10.3390/ma16186179

**Chicago/Turabian Style**

Glushkov, Evgeny V., Natalia V. Glushkova, Olga A. Ermolenko, and Alexey M. Tatarinov.
2023. "Study of Ultrasonic Guided Wave Propagation in Bone Composite Structures for Revealing Osteoporosis Diagnostic Indicators" *Materials* 16, no. 18: 6179.
https://doi.org/10.3390/ma16186179