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# Bias of shear wave elasticity measurements in thin layer samples and a simple correction strategy

- Jianqiang Mo†
^{1, 2}, - Hao Xu†
^{1}, - Bo Qiang
^{2}, - Hugo Giambini
^{3, 4}, - Randall Kinnick
^{2}, - Kai-Nan An
^{3}, - Shigao Chen
^{5}and - Zongping Luo
^{1}Email author

**Received: **7 January 2016

**Accepted: **28 July 2016

**Published: **12 August 2016

## Abstract

Shear wave elastography (SWE) is an emerging technique for measuring biological tissue stiffness. However, the application of SWE in thin layer tissues is limited by bias due to the influence of geometry on measured shear wave speed. In this study, we investigated the bias of Young’s modulus measured by SWE in thin layer gelatin–agar phantoms, and compared the result with finite element method and Lamb wave model simulation. The result indicated that the Young’s modulus measured by SWE decreased continuously when the sample thickness decreased, and this effect was more significant for smaller thickness. We proposed a new empirical formula which can conveniently correct the bias without the need of using complicated mathematical modeling. In summary, we confirmed the nonlinear relation between thickness and Young’s modulus measured by SWE in thin layer samples, and offered a simple and practical correction strategy which is convenient for clinicians to use.

## Keywords

## Background

Tissue elastic property, or stiffness, is an important characteristic of biological tissues. It is determined by the cells and extracellular matrix that make up of the tissues. Therefore, pathological changes at cell or molecule level could cause characteristic elastic property changes. It is now well recognized that tissue stiffness is a useful biomarker for evaluation of many diseases. Among the multiple potential clinical applications, tissue stiffness has been extensively investigated as a biomarker for cancer diagnosis and liver fibrosis staging (Cosgrove et al. 2013). Palpation has long been used by physicians to assess tissue stiffness for making diagnosis. Ultrasound elastography initially proposed in 1990’ was a significant advancement (Ophir et al. 1991), which provides objective measurements of tissue elastic property in addition to traditional anatomical images. Since then, this method has been widely studied for diagnoses of liver fibrosis (Bavu et al. 2011; Zhao et al. 2014), breast tumor (Zhang et al. 2015; Stoian et al. 2016), skeleton muscle lesion (Hatta et al. 2015), prostate cancer (Woo et al. 2014), and others.

Two major approaches are currently used for ultrasound elastography: examination of the strain or deformation of a tissue due to an applied force (strain elastography, SE) (Ophir et al. 1991) and analysis of the propagation speed of shear waves generated by mechanical vibration or ultrasound radiation force (shear wave elastography, SWE) (Bercoff et al. 2004). SWE is often preferred over SE, because it can offer quantitative assessment and is less operator-dependent. Therefore, SWE has gained a lot of clinical attention and is currently under active research.

*c*

_{ s }to estimate the Young’s modulus

*E*of tissue through

*ρ*is the density of tissues commonly assumed to be a constant of 1000 kg/m

^{3}. Equation (1) assumes local homogeneousness in an infinite medium. These assumptions are valid for large parenchymal organs such as liver or breast. In organs with thin layer structure, such as myocardium, blood vessel wall, and bladder wall, the shear wave speed is determined by the thickness of the thin layer in addition to the mechanical properties. Therefore, Eq. (1) can no longer give correct estimation of Young’s modulus of tissues within thin layers. Some studies attempted to use a Lamb wave model to describe the propagation of waves in a thin layer, so that the correct shear modulus can be solved from the measured

*c*

_{ s }using the model and the thickness of the thin layer (Brum et al. 2012; Couade et al. 2010; Nguyen et al. 2011; Nenadic et al. 2011). However, this approach involves complex equations and therefore may be difficult for daily use by clinicians.

In this study, we developed a simple empirical formula to correct for the bias induced by SWE measurements in thin layer structures. The effects of the sample thickness and shear wave frequency were investigated. Results of the empirical formula were compared with the Lamb wave model, finite element method (FEM), and experimental data in phantoms of different stiffness and thickness. This method allows easy conversion of measured shear wave speed to Young’s modulus in thin layer structure of any thickness.

## Methods

### Experiments

#### Phantom preparation

#### Experimental setup for SWE

#### Shear wave elastography protocol

### Finite element method and the empirical formula

^{3}respectively. The Poisson’s ratio and density of the thin plate were 0.499 and 1050 kg/m

^{3}. The Young’s modulus of the biggest samples (50 mm) of each set was used in our simulation. A vibration stimulus of four cycle of sinusoidal wave with an amplitude of 1 μm and loading frequency of 300, 600 and 900 Hz respectively was applied throughout the entire thickness of the phantoms as a line source. The excitation was applied along the middle axis of the phantoms. A fixed boundary condition was prescribed to the lateral surface of the phantoms. Shear wave velocity was recorded with a 0.5 mm spatial resolution along a line extending away from the source in the middle of the plate. Wave speed at each thickness was calculated from the simulation data and compared with SWE measurements in phantoms of different stiffness and thickness to estimate the center frequency of shear waves in SWE measurements. An empirical formula was then derived from the results of FEM simulation to correct for measurement bias in thin layers. This empirical formula used the thickness of the layer and the “appearant” stiffness measured by SWE in the thin layer to estimate the “true” stiffness of the material. Results were then compared with experimental data in phantoms.

### Anti-symmetric Lamb wave model

*k*

_{ l }=

*ω*/

*c*

_{ L }and \(k_{s} = \omega /\sqrt {{\text{G}}/\rho }\) are the Lamb wave number and shear wave number respectively, \(\beta = \sqrt {k_{l}^{2} - k_{s}^{2} }\), h is the half-thickness of the sample,

*ρ*is density, G is shear modulus,

*c*

_{ L }is the Lamb wave velocity, ω is the angular frequency.

## Results

### Shear wave elastography

### FEM results and empirical model

^{3}, Poisson’s ratio 0.499, thickness of plates 3/6/8/15/30/50 mm. Three different loading frequencies (300/600/900 Hz) were adopted in FEM to investigate the influence of loading frequency on wave speed. Figure 6 shows the comparison of shear wave speed between SWE tests and FEM results with different loading frequency. The shear wave speed from FEM also decreased with the decreasing thickness, and FEM results with frequency of 600 Hz matched best with SWE test results.

*E*

_{0}and

*E*is the corrected and appearant Young’s modulus respectively.

### Validation of the empirical formula

Comparison of appearant and corrected Young’s modulus

Thickness (mm) | Young’s modulus (kPa) | ||||||
---|---|---|---|---|---|---|---|

3 | 6 | 8 | 15 | 30 | 50 | Mean value ± SE | |

2 % agar | |||||||

Appearant | 31.8 | 35.4 | 49.5 | 53.7 | 49.9 | 52.5 | |

Corrected | 59.9 | 42.2 | 56.2 | 54.4 | 49.9 | 52.5 | 52.5 ± 2.5 |

2.5 % agar | |||||||

Appearant | 53.2 | 55.8 | 67.8 | 81.0 | 80.8 | 84.2 | |

Corrected | 132.0 | 75.4 | 82.3 | 83.8 | 80.8 | 84.2 | 89.8 ± 8.5 |

3 % agar | |||||||

Appearant | 78.9 | 93.6 | 117.3 | 168.9 | 181.5 | 182.2 | |

Corrected | 249.2 | 153.4 | 167.8 | 190.8 | 183.1 | 181.2 | 187.6 ± 13.5 |

### Influence of Poisson’s ratio setting on FEM

## Discussion

Our study confirmed that the appearant Young’s modulus measured by SWE is affected by sample thickness. Results from FEM and experiments indicates that shear wave speed decreased with thickness in thin layer samples, and the change was more significant when the thickness was less than 15 mm. When the thickness was larger than 30 mm, the SWE results of the phantoms did not change much. Lamb wave modeling agreed well with SWE experimental data for thin layer samples but had significant error for samples with thickness of 15 mm or above. When we used the empirical formula, the corrected Young’s modulus for the thin phantoms (3, 6, 8 and 15 mm) were much closer to that of the thick phantoms (30 and 50 mm).

This bias in SWE test is due to the slower guided wave propagation speed in thin layers. This guided wave propagation was first studied in Lamb (1917) and has been used for non-destructive testing applications. Isaaz (2002) first adopted Lamb wave equations of motion for estimation of myocardial viscoelasticity. Later on, Nenadic et al. (2011) extended Lamb wave modeling for shear wave experiments on urethane plate and porcine left ventricle free-wall myocardium. Although the Lamb wave model as shown in Eq. (2) can be used to account for geometric effects of the thin layer and solve for the correct material property, the function is relatively complex and the shear modulus is implicitly embedded in the equation. Therefore, it may be difficult for physicians to use this approach in daily clinical practice. In addition, our study demonstrates that Lamb wave model is inaccurate for thick layers. Here we proposed an alternative correction strategy based on a simple empirical formula as shown in Eq. (4), with which the true Young’s modulus can be conveniently calculated from this explicit equation using the sample thickness and the measured shear wave speed. Therefore, the proposed method may be more practical and “user-friendly”. In contrast with the Lamb wave correction approach, this empirical formula can be applied to a wide range of sample thickness.

This simple correction strategy may have significant clinical value in shear wave measurements in thin layer tissues such as myocardium, artery, bladder, and muscle. The thickness of target tissues can be measured by real-time B-mode imaging and used with the measured apparent shear wave speed to calculate the true material properties of the target tissues through the simple empirical formula proposed in this study. Although the study was performed using a linear array transducer, the simple empirical correction approach proposed here can be applied to shear wave measurements obtained with other transducers such as a curve array or a phase array. This correction is useful for reducing the confounding effect of tissue thickness and facilitates better comparison among different patients. The empirical formula is easy to use and can be applied to layer tissues of any thickness, and therefore would be suitable for clinical use.

There are several limitations of this study. First, the agar-gelatin samples prepared from the same solution may have small variations in stiffness across different samples. Large samples cure slower than smaller samples, possibly leading to different final stiffness. Therefore, the assumption that all samples had the same material properties may not necessary be true. This may also explain the discrepancy between experimental results and FEM/empirical-formula predictions. Second, the frequency of shear waves used in the experiments by the Aixplorer scanner was unknown to us, and was obtained by fitting FEM to experimental results. Future studies should investigate if the shear wave center frequency would change for SWE measurements using machines from different vendors. As a first order correction, one can still use the empirical formula and shear wave frequency of this study to provide a corrected measurement that is closer to the “true” answer. Third, the FEM simulation using a line load with four cycle of sinusoidal wave may be somewhat different from the ultrasound push beam used in Aixplorer to generate shear waves. This approximation may cause small discrepancy between FEM and SWE experimental results seen in the study. Fourth, the stiffness of samples used in this study were only calibrated with ultrasound shear wave elastography. An independent validation using mechanical testing would have strengthened the paper. Last, simulations and experiments were performed for samples surrounded by liquid. Therefore, the method may not be adequate for plates surrounded by soft tissues. In addition, the curvature of the plate, which may be important for structures such as vessels, was not considered in this study. Future studies should be conducted to investigate the impact of loading medium and structure curvature for more specific applications.

## Conclusion

In this study, we investigated the confounding effects of sample thickness on ultrasound shear wave elasticity measurements. Phantom studies demonstrated that the measured shear wave speed decreased with the decreasing sample thickness. This observation was confirmed by FEM simulation and Lamb wave analytical model. A simple empirical correction formula was proposed to account for the thickness effects, which was validated by SWE experiments. This simple correction strategy does not require complicated mathematical or engineering expertise, and thus may be useful for clinicians to compare shear wave measurements of thin layer tissue such as myocardium and bladder among different subjects.

## Notes

## Declarations

### Authors’ contributions

Dr. ZL, KA and SC developed the idea and designed the study. JM and HX contributed equally to study design, experiment performing, finite element analysis, data collection and manuscript preparation. BQ, HG, RK participated in experiment performing and data collection. All authors read and approved the final manuscript.

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (81320108018, 31570943 and 31270995), Natural Science Foundation of Jiangsu Province of China (BK20160305), Innovation and Entrepreneurship Program of Jiangsu Province, Jiangsu Provincial Special Program of Medical Science (BL2012004), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), and the National Institute of Arthritis and Musculoskeletal and Skin Diseases of the National Institutes of Health (USA) (R21 AR065550). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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