Abstract

Advances in the field of laser ultrasonics have opened up new possibilities in medical applications. This paper evaluates this technique as a method that would allow for rapid characterization of the elastic properties of soft biological tissue. In doing so, we propose a novel approach that utilizes a low coherence interferometer to detect the laser-induced surface acoustic waves (SAW) from the tissue-mimicking phantoms. A Nd:YAG focused laser line-source is applied to one- and two-layer tissue-mimicking agar-agar phantoms, and the generated SAW signals are detected by a time domain low coherence interferometry system. SAW phase velocity dispersion curves are calculated, from which the elasticity of the specimens is evaluated. We show that the experimental results agree well with those of the theoretical expectations. This study is the first report that a laser-generated SAW phase velocity dispersion technique is applied to soft materials. This technique may open a way for laser ultrasonics to detect the mechanical properties of soft tissues, such as skin.

© 2011 OSA

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2010 (2)

R. K. Wang and A. L. Nuttall, “Phase-sensitive optical coherence tomography imaging of the tissue motion within the organ of Corti at a subnanometer scale: a preliminary study,” J. Biomed. Opt. 15(5), 056005 (2010).
[CrossRef] [PubMed]

T. Z. Pavan, E. L. Madsen, G. R. Frank, A. Adilton O Carneiro, and T. J. Hall, “Nonlinear elastic behavior of phantom materials for elastography,” Phys. Med. Biol. 55(9), 2679–2692 (2010).
[CrossRef] [PubMed]

2009 (3)

2008 (1)

W. Sun, Y. Peng, and J. Xu, “A de-noising method for laser ultrasonic signal based on EMD,” J. Sandong Univ. 38(5), 1–6 (2008).

2006 (2)

Q. J. Huang, Y. Cheng, X. J. Liu, X. D. Xu, and S. Y. Zhang, “Study of the elastic constants in a La0.6Sr0.4MnO3 film by means of laser-generated ultrasonic wave method,” Ultrasonics 44(Suppl 1), e1223–e1227 (2006).
[CrossRef] [PubMed]

A. L’Etang and Z. Y. Huang, “FE simulation of laser generated surface acoustic wave propagation in skin,” Ultrasonics 44(Suppl 1), e1243–e1247 (2006).
[CrossRef] [PubMed]

2005 (1)

A. L’Etang and Z. Y. Huang, “FE simulation of laser ultrasonic surface waves in a biomaterial model,” Appl. Mech. Mater. 3–4, 85–90 (2005).
[CrossRef]

2004 (1)

D. H. Hurley and J. B. Spicer, “Line source representation for laser-generated ultrasound in an elastic transversely isotropic half-space,” J. Acoust. Soc. Am. 116(5), 2914–2922 (2004).
[CrossRef]

2001 (2)

S. Kenderian, B. B. Djordjevic, and R. E. Green., “Point and Line Source Laser Generation of Ultrasound for Inspection of Internal and Surface Flaws in Rail and Structural Materials,” Res. Nondestruct. Eval. 13, 189–200 (2001).

F. Reverdy and B. Audoin, “Ultrasonic measurement of elastic constant of anisotropic materials with laser source and laser receiver focused on the same interface,” J. Appl. Phys. 90(9), 4829–4835 (2001).
[CrossRef]

2000 (3)

T. Nitta, H. Haga, K. Kawabata, K. Abe, and T. Sambongi, “Comparing microscopic with macroscopic elastic properties of polymer gel,” Ultramicroscopy 82(1-4), 223–226 (2000).
[CrossRef] [PubMed]

C. Glorieux, W. Gao, S. E. Kruger, K. Van de Rostyne, W. Lauriks, and J. Thoen, “Surface acoustic wave depth profiling of elastically inhomogeneous materials,” J. Appl. Phys. 88(7), 4394–4400 (2000).
[CrossRef]

J. A. Rogers, A. A. Maznev, M. J. Banet, and K. A. Nelson, “Optical generation and characterization of acoustic waves in thin films: fundamentals and applications,” Annu. Rev. Mater. Sci. 30(1), 117–157 (2000).
[CrossRef]

1998 (1)

D. Schneider, B. Schultrich, H. J. Scheibe, H. Ziegele, and M. Griepentrog, “A laser-acoustic method for testing and classifying hard surface layers,” Thin Solid Films 332(1–2), 157–163 (1998).
[CrossRef]

1997 (1)

D. Schneider and T. A. Schwarz, “Photoacoustic method for characterising thin films,” Surf. Coat. Tech. 91(1-2), 136–146 (1997).
[CrossRef]

1996 (1)

P. A. Doyle and C. M. Scala, “Near-field ultrasonic Rayleigh waves from a laser line source,” Ultrasonics 34(1), 1–8 (1996).
[CrossRef]

1992 (1)

A. Neubrand and P. Hess, “Laser generation and detection of surface acoustic waves: Elastic properties of surface layers,” J. Appl. Phys. 71(1), 227–238 (1992).
[CrossRef]

Abe, K.

T. Nitta, H. Haga, K. Kawabata, K. Abe, and T. Sambongi, “Comparing microscopic with macroscopic elastic properties of polymer gel,” Ultramicroscopy 82(1-4), 223–226 (2000).
[CrossRef] [PubMed]

Adilton O Carneiro, A.

T. Z. Pavan, E. L. Madsen, G. R. Frank, A. Adilton O Carneiro, and T. J. Hall, “Nonlinear elastic behavior of phantom materials for elastography,” Phys. Med. Biol. 55(9), 2679–2692 (2010).
[CrossRef] [PubMed]

An, L.

Audoin, B.

F. Reverdy and B. Audoin, “Ultrasonic measurement of elastic constant of anisotropic materials with laser source and laser receiver focused on the same interface,” J. Appl. Phys. 90(9), 4829–4835 (2001).
[CrossRef]

Banet, M. J.

J. A. Rogers, A. A. Maznev, M. J. Banet, and K. A. Nelson, “Optical generation and characterization of acoustic waves in thin films: fundamentals and applications,” Annu. Rev. Mater. Sci. 30(1), 117–157 (2000).
[CrossRef]

Cheng, Y.

Q. J. Huang, Y. Cheng, X. J. Liu, X. D. Xu, and S. Y. Zhang, “Study of the elastic constants in a La0.6Sr0.4MnO3 film by means of laser-generated ultrasonic wave method,” Ultrasonics 44(Suppl 1), e1223–e1227 (2006).
[CrossRef] [PubMed]

Djordjevic, B. B.

S. Kenderian, B. B. Djordjevic, and R. E. Green., “Point and Line Source Laser Generation of Ultrasound for Inspection of Internal and Surface Flaws in Rail and Structural Materials,” Res. Nondestruct. Eval. 13, 189–200 (2001).

Doyle, P. A.

P. A. Doyle and C. M. Scala, “Near-field ultrasonic Rayleigh waves from a laser line source,” Ultrasonics 34(1), 1–8 (1996).
[CrossRef]

Fleming, S.

Frank, G. R.

T. Z. Pavan, E. L. Madsen, G. R. Frank, A. Adilton O Carneiro, and T. J. Hall, “Nonlinear elastic behavior of phantom materials for elastography,” Phys. Med. Biol. 55(9), 2679–2692 (2010).
[CrossRef] [PubMed]

Gao, W.

C. Glorieux, W. Gao, S. E. Kruger, K. Van de Rostyne, W. Lauriks, and J. Thoen, “Surface acoustic wave depth profiling of elastically inhomogeneous materials,” J. Appl. Phys. 88(7), 4394–4400 (2000).
[CrossRef]

Glorieux, C.

C. Glorieux, W. Gao, S. E. Kruger, K. Van de Rostyne, W. Lauriks, and J. Thoen, “Surface acoustic wave depth profiling of elastically inhomogeneous materials,” J. Appl. Phys. 88(7), 4394–4400 (2000).
[CrossRef]

Green, R. E.

S. Kenderian, B. B. Djordjevic, and R. E. Green., “Point and Line Source Laser Generation of Ultrasound for Inspection of Internal and Surface Flaws in Rail and Structural Materials,” Res. Nondestruct. Eval. 13, 189–200 (2001).

Griepentrog, M.

D. Schneider, B. Schultrich, H. J. Scheibe, H. Ziegele, and M. Griepentrog, “A laser-acoustic method for testing and classifying hard surface layers,” Thin Solid Films 332(1–2), 157–163 (1998).
[CrossRef]

Haga, H.

T. Nitta, H. Haga, K. Kawabata, K. Abe, and T. Sambongi, “Comparing microscopic with macroscopic elastic properties of polymer gel,” Ultramicroscopy 82(1-4), 223–226 (2000).
[CrossRef] [PubMed]

Hall, T. J.

T. Z. Pavan, E. L. Madsen, G. R. Frank, A. Adilton O Carneiro, and T. J. Hall, “Nonlinear elastic behavior of phantom materials for elastography,” Phys. Med. Biol. 55(9), 2679–2692 (2010).
[CrossRef] [PubMed]

Hess, P.

A. Neubrand and P. Hess, “Laser generation and detection of surface acoustic waves: Elastic properties of surface layers,” J. Appl. Phys. 71(1), 227–238 (1992).
[CrossRef]

Huang, Q. J.

Q. J. Huang, Y. Cheng, X. J. Liu, X. D. Xu, and S. Y. Zhang, “Study of the elastic constants in a La0.6Sr0.4MnO3 film by means of laser-generated ultrasonic wave method,” Ultrasonics 44(Suppl 1), e1223–e1227 (2006).
[CrossRef] [PubMed]

Huang, Z. Y.

A. L’Etang and Z. Y. Huang, “FE simulation of laser generated surface acoustic wave propagation in skin,” Ultrasonics 44(Suppl 1), e1243–e1247 (2006).
[CrossRef] [PubMed]

A. L’Etang and Z. Y. Huang, “FE simulation of laser ultrasonic surface waves in a biomaterial model,” Appl. Mech. Mater. 3–4, 85–90 (2005).
[CrossRef]

Hurley, D. H.

D. H. Hurley and J. B. Spicer, “Line source representation for laser-generated ultrasound in an elastic transversely isotropic half-space,” J. Acoust. Soc. Am. 116(5), 2914–2922 (2004).
[CrossRef]

Kawabata, K.

T. Nitta, H. Haga, K. Kawabata, K. Abe, and T. Sambongi, “Comparing microscopic with macroscopic elastic properties of polymer gel,” Ultramicroscopy 82(1-4), 223–226 (2000).
[CrossRef] [PubMed]

Kenderian, S.

S. Kenderian, B. B. Djordjevic, and R. E. Green., “Point and Line Source Laser Generation of Ultrasound for Inspection of Internal and Surface Flaws in Rail and Structural Materials,” Res. Nondestruct. Eval. 13, 189–200 (2001).

Kruger, S. E.

C. Glorieux, W. Gao, S. E. Kruger, K. Van de Rostyne, W. Lauriks, and J. Thoen, “Surface acoustic wave depth profiling of elastically inhomogeneous materials,” J. Appl. Phys. 88(7), 4394–4400 (2000).
[CrossRef]

L’Etang, A.

A. L’Etang and Z. Y. Huang, “FE simulation of laser generated surface acoustic wave propagation in skin,” Ultrasonics 44(Suppl 1), e1243–e1247 (2006).
[CrossRef] [PubMed]

A. L’Etang and Z. Y. Huang, “FE simulation of laser ultrasonic surface waves in a biomaterial model,” Appl. Mech. Mater. 3–4, 85–90 (2005).
[CrossRef]

Lauriks, W.

C. Glorieux, W. Gao, S. E. Kruger, K. Van de Rostyne, W. Lauriks, and J. Thoen, “Surface acoustic wave depth profiling of elastically inhomogeneous materials,” J. Appl. Phys. 88(7), 4394–4400 (2000).
[CrossRef]

Law, S.

Lee, Y. C.

Liu, X. J.

Q. J. Huang, Y. Cheng, X. J. Liu, X. D. Xu, and S. Y. Zhang, “Study of the elastic constants in a La0.6Sr0.4MnO3 film by means of laser-generated ultrasonic wave method,” Ultrasonics 44(Suppl 1), e1223–e1227 (2006).
[CrossRef] [PubMed]

Madsen, E. L.

T. Z. Pavan, E. L. Madsen, G. R. Frank, A. Adilton O Carneiro, and T. J. Hall, “Nonlinear elastic behavior of phantom materials for elastography,” Phys. Med. Biol. 55(9), 2679–2692 (2010).
[CrossRef] [PubMed]

Maznev, A. A.

J. A. Rogers, A. A. Maznev, M. J. Banet, and K. A. Nelson, “Optical generation and characterization of acoustic waves in thin films: fundamentals and applications,” Annu. Rev. Mater. Sci. 30(1), 117–157 (2000).
[CrossRef]

Nelson, K. A.

J. A. Rogers, A. A. Maznev, M. J. Banet, and K. A. Nelson, “Optical generation and characterization of acoustic waves in thin films: fundamentals and applications,” Annu. Rev. Mater. Sci. 30(1), 117–157 (2000).
[CrossRef]

Neubrand, A.

A. Neubrand and P. Hess, “Laser generation and detection of surface acoustic waves: Elastic properties of surface layers,” J. Appl. Phys. 71(1), 227–238 (1992).
[CrossRef]

Nitta, T.

T. Nitta, H. Haga, K. Kawabata, K. Abe, and T. Sambongi, “Comparing microscopic with macroscopic elastic properties of polymer gel,” Ultramicroscopy 82(1-4), 223–226 (2000).
[CrossRef] [PubMed]

Nuttall, A. L.

R. K. Wang and A. L. Nuttall, “Phase-sensitive optical coherence tomography imaging of the tissue motion within the organ of Corti at a subnanometer scale: a preliminary study,” J. Biomed. Opt. 15(5), 056005 (2010).
[CrossRef] [PubMed]

Pavan, T. Z.

T. Z. Pavan, E. L. Madsen, G. R. Frank, A. Adilton O Carneiro, and T. J. Hall, “Nonlinear elastic behavior of phantom materials for elastography,” Phys. Med. Biol. 55(9), 2679–2692 (2010).
[CrossRef] [PubMed]

Peng, Y.

W. Sun, Y. Peng, and J. Xu, “A de-noising method for laser ultrasonic signal based on EMD,” J. Sandong Univ. 38(5), 1–6 (2008).

Reverdy, F.

F. Reverdy and B. Audoin, “Ultrasonic measurement of elastic constant of anisotropic materials with laser source and laser receiver focused on the same interface,” J. Appl. Phys. 90(9), 4829–4835 (2001).
[CrossRef]

Rogers, J. A.

J. A. Rogers, A. A. Maznev, M. J. Banet, and K. A. Nelson, “Optical generation and characterization of acoustic waves in thin films: fundamentals and applications,” Annu. Rev. Mater. Sci. 30(1), 117–157 (2000).
[CrossRef]

Sambongi, T.

T. Nitta, H. Haga, K. Kawabata, K. Abe, and T. Sambongi, “Comparing microscopic with macroscopic elastic properties of polymer gel,” Ultramicroscopy 82(1-4), 223–226 (2000).
[CrossRef] [PubMed]

Scala, C. M.

P. A. Doyle and C. M. Scala, “Near-field ultrasonic Rayleigh waves from a laser line source,” Ultrasonics 34(1), 1–8 (1996).
[CrossRef]

Scheibe, H. J.

D. Schneider, B. Schultrich, H. J. Scheibe, H. Ziegele, and M. Griepentrog, “A laser-acoustic method for testing and classifying hard surface layers,” Thin Solid Films 332(1–2), 157–163 (1998).
[CrossRef]

Schneider, D.

D. Schneider, B. Schultrich, H. J. Scheibe, H. Ziegele, and M. Griepentrog, “A laser-acoustic method for testing and classifying hard surface layers,” Thin Solid Films 332(1–2), 157–163 (1998).
[CrossRef]

D. Schneider and T. A. Schwarz, “Photoacoustic method for characterising thin films,” Surf. Coat. Tech. 91(1-2), 136–146 (1997).
[CrossRef]

Schultrich, B.

D. Schneider, B. Schultrich, H. J. Scheibe, H. Ziegele, and M. Griepentrog, “A laser-acoustic method for testing and classifying hard surface layers,” Thin Solid Films 332(1–2), 157–163 (1998).
[CrossRef]

Schwarz, T. A.

D. Schneider and T. A. Schwarz, “Photoacoustic method for characterising thin films,” Surf. Coat. Tech. 91(1-2), 136–146 (1997).
[CrossRef]

Spicer, J. B.

D. H. Hurley and J. B. Spicer, “Line source representation for laser-generated ultrasound in an elastic transversely isotropic half-space,” J. Acoust. Soc. Am. 116(5), 2914–2922 (2004).
[CrossRef]

Sun, W.

W. Sun, Y. Peng, and J. Xu, “A de-noising method for laser ultrasonic signal based on EMD,” J. Sandong Univ. 38(5), 1–6 (2008).

Swain, M.

Thoen, J.

C. Glorieux, W. Gao, S. E. Kruger, K. Van de Rostyne, W. Lauriks, and J. Thoen, “Surface acoustic wave depth profiling of elastically inhomogeneous materials,” J. Appl. Phys. 88(7), 4394–4400 (2000).
[CrossRef]

Van de Rostyne, K.

C. Glorieux, W. Gao, S. E. Kruger, K. Van de Rostyne, W. Lauriks, and J. Thoen, “Surface acoustic wave depth profiling of elastically inhomogeneous materials,” J. Appl. Phys. 88(7), 4394–4400 (2000).
[CrossRef]

Wang, H. C.

Wang, R. K.

R. K. Wang and A. L. Nuttall, “Phase-sensitive optical coherence tomography imaging of the tissue motion within the organ of Corti at a subnanometer scale: a preliminary study,” J. Biomed. Opt. 15(5), 056005 (2010).
[CrossRef] [PubMed]

R. K. Wang and L. An, “Doppler optical micro-angiography for volumetric imaging of vascular perfusion in vivo,” Opt. Express 17(11), 8926–8940 (2009).
[CrossRef] [PubMed]

Xu, J.

W. Sun, Y. Peng, and J. Xu, “A de-noising method for laser ultrasonic signal based on EMD,” J. Sandong Univ. 38(5), 1–6 (2008).

Xu, X. D.

Q. J. Huang, Y. Cheng, X. J. Liu, X. D. Xu, and S. Y. Zhang, “Study of the elastic constants in a La0.6Sr0.4MnO3 film by means of laser-generated ultrasonic wave method,” Ultrasonics 44(Suppl 1), e1223–e1227 (2006).
[CrossRef] [PubMed]

Xue, J.

Zhang, S. Y.

Q. J. Huang, Y. Cheng, X. J. Liu, X. D. Xu, and S. Y. Zhang, “Study of the elastic constants in a La0.6Sr0.4MnO3 film by means of laser-generated ultrasonic wave method,” Ultrasonics 44(Suppl 1), e1223–e1227 (2006).
[CrossRef] [PubMed]

Ziegele, H.

D. Schneider, B. Schultrich, H. J. Scheibe, H. Ziegele, and M. Griepentrog, “A laser-acoustic method for testing and classifying hard surface layers,” Thin Solid Films 332(1–2), 157–163 (1998).
[CrossRef]

Annu. Rev. Mater. Sci. (1)

J. A. Rogers, A. A. Maznev, M. J. Banet, and K. A. Nelson, “Optical generation and characterization of acoustic waves in thin films: fundamentals and applications,” Annu. Rev. Mater. Sci. 30(1), 117–157 (2000).
[CrossRef]

Appl. Mech. Mater. (1)

A. L’Etang and Z. Y. Huang, “FE simulation of laser ultrasonic surface waves in a biomaterial model,” Appl. Mech. Mater. 3–4, 85–90 (2005).
[CrossRef]

Appl. Opt. (1)

J. Acoust. Soc. Am. (1)

D. H. Hurley and J. B. Spicer, “Line source representation for laser-generated ultrasound in an elastic transversely isotropic half-space,” J. Acoust. Soc. Am. 116(5), 2914–2922 (2004).
[CrossRef]

J. Appl. Phys. (3)

F. Reverdy and B. Audoin, “Ultrasonic measurement of elastic constant of anisotropic materials with laser source and laser receiver focused on the same interface,” J. Appl. Phys. 90(9), 4829–4835 (2001).
[CrossRef]

C. Glorieux, W. Gao, S. E. Kruger, K. Van de Rostyne, W. Lauriks, and J. Thoen, “Surface acoustic wave depth profiling of elastically inhomogeneous materials,” J. Appl. Phys. 88(7), 4394–4400 (2000).
[CrossRef]

A. Neubrand and P. Hess, “Laser generation and detection of surface acoustic waves: Elastic properties of surface layers,” J. Appl. Phys. 71(1), 227–238 (1992).
[CrossRef]

J. Biomed. Opt. (1)

R. K. Wang and A. L. Nuttall, “Phase-sensitive optical coherence tomography imaging of the tissue motion within the organ of Corti at a subnanometer scale: a preliminary study,” J. Biomed. Opt. 15(5), 056005 (2010).
[CrossRef] [PubMed]

J. Sandong Univ. (1)

W. Sun, Y. Peng, and J. Xu, “A de-noising method for laser ultrasonic signal based on EMD,” J. Sandong Univ. 38(5), 1–6 (2008).

Opt. Express (2)

Phys. Med. Biol. (1)

T. Z. Pavan, E. L. Madsen, G. R. Frank, A. Adilton O Carneiro, and T. J. Hall, “Nonlinear elastic behavior of phantom materials for elastography,” Phys. Med. Biol. 55(9), 2679–2692 (2010).
[CrossRef] [PubMed]

Res. Nondestruct. Eval. (1)

S. Kenderian, B. B. Djordjevic, and R. E. Green., “Point and Line Source Laser Generation of Ultrasound for Inspection of Internal and Surface Flaws in Rail and Structural Materials,” Res. Nondestruct. Eval. 13, 189–200 (2001).

Surf. Coat. Tech. (1)

D. Schneider and T. A. Schwarz, “Photoacoustic method for characterising thin films,” Surf. Coat. Tech. 91(1-2), 136–146 (1997).
[CrossRef]

Thin Solid Films (1)

D. Schneider, B. Schultrich, H. J. Scheibe, H. Ziegele, and M. Griepentrog, “A laser-acoustic method for testing and classifying hard surface layers,” Thin Solid Films 332(1–2), 157–163 (1998).
[CrossRef]

Ultramicroscopy (1)

T. Nitta, H. Haga, K. Kawabata, K. Abe, and T. Sambongi, “Comparing microscopic with macroscopic elastic properties of polymer gel,” Ultramicroscopy 82(1-4), 223–226 (2000).
[CrossRef] [PubMed]

Ultrasonics (3)

Q. J. Huang, Y. Cheng, X. J. Liu, X. D. Xu, and S. Y. Zhang, “Study of the elastic constants in a La0.6Sr0.4MnO3 film by means of laser-generated ultrasonic wave method,” Ultrasonics 44(Suppl 1), e1223–e1227 (2006).
[CrossRef] [PubMed]

A. L’Etang and Z. Y. Huang, “FE simulation of laser generated surface acoustic wave propagation in skin,” Ultrasonics 44(Suppl 1), e1243–e1247 (2006).
[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

In homogeneous material (left), the phase velocity is constant while in layered material (right), the wave dispersion occurs because the phase velocity depends on the frequency [3].

Fig. 2
Fig. 2

System set up of SAW generation and detection.

Fig. 4
Fig. 4

SAW signal of one layer 3.5% agar-agar phantom with the distance of 0.5 mm (top) to 3 mm (bottom) to laser pulse, with 0.5 mm/step. Each SAW signal is purposely shifted vertically by equal distance in order to better illustrate the results captured from different positions. The same also applies to Fig. 7.

Fig. 5
Fig. 5

Autocorrelation spectrum of the detected SAW of one layer 3.5% agar-agar phantom.

Fig. 6
Fig. 6

Comparison of phase velocity dispersion between 3.5% agar-agar phantom and 2% agar-agar phantom.

Fig. 7
Fig. 7

SAW signal of double layer 2% agar on 3.5% agar phantom with the distance of 1 mm to 6 mm to laser pulse, with 1 mm/step.

Fig. 8
Fig. 8

Autocorrelation spectrum of the detected SAW of double layers 2% on 3.5% agar-agar phantom.

Fig. 9
Fig. 9

Comparison of phase velocity dispersion between one layer 3.5% agar-agar phantom and two layer phantoms 2% on 3.5% agar-agar phantom and 5% on 3.5% agar-agar phantoms.

Fig. 10
Fig. 10

The relationship between agar concentration and phase velocity (a) and the relationship between agar concentration and Young’s modulus where the values pointed by the black arrows indicate the values extracted from the prior publications (b).

Tables (1)

Tables Icon

Table 1 The relationship between concentration of agar-agar phantom and the phase velocity

Equations (5)

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c = 0.87 + 1.12 v 1 + v ( E 2 ρ ( 1 + v ) ) 1 2 ,
I ( k ) = 2 I 1 I 2 cos ( φ + 2 k Δ z + φ n o i s e ) ,
Y 12 ( f ) = Y 1 ( f ) Y 2 ( f ) ¯ = A 1 A 2 e i ( φ 2 φ 1 ) ,
Δ φ / 2 π = ( x 1 x 2 ) / λ .
V = ( x 1 x 2 ) 2 π f / Δ φ .

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