Acta Mechanica Solida Sinica

, Volume 25, Issue 5, pp 510–519 | Cite as

Modified Szabo’s Wave Equation for Arbitrarily Frequency-Dependent Viscous Dissipation in Soft Matter with Applications to 3D Ultrasonic Imaging

  • Xiaodi Zhang
  • Wen Chen
  • Chuanzeng Zhang


Soft matters are observed anomalous viscosity behaviors often characterized by a power law frequency-dependent attenuation in acoustic wave propagation. Recent decades have witnessed a fast growing research on developing various models for such anomalous viscosity behaviors, among which one of the present authors proposed the modified Szabo’s wave equation via the positive fractional derivative. The purpose of this study is to apply the modified Szabo’s wave equation to simulate a recent ultrasonic imaging technique called the clinical amplitude-velocity reconstruction imaging (CARI) of breast tumors which are of typical soft tissue matters. Investigations have been made on the effects of the size and position of tumors on the quality of ultrasonic medical imaging. It is observed from numerical results that the sound pressure along the reflecting line, which indicates the detection results, varies obviously with sizes and lateral positions of tumors, but remains almost the same for different axial positions.

Key words

soft matter viscosity frequency-dependent dissipation modified Szabo’s wave equation positive fractional derivative ultrasonic imaging 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Lu, K. and Liu, J., Soft matter physics—a new field of physics. Physics, 2009, 38(7): 453–461 (in Chinese).Google Scholar
  2. [2]
    Jones, R.A.L., Soft Condensed Matter. Beijing: Science Press, 2008.Google Scholar
  3. [3]
    Daoud, M. and Willams, C.E., Soft Matter Physics. Springer, 1999.Google Scholar
  4. [4]
    Gaul, L., The influence of damping on waves and vibrations. Mechanical Systems and Signal Processing, 1999, 13(1): 1–30.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Szabo, T.L. and Wu, J., A model for longitudinal and shear wave propagation in viscoelastic media. Journal of the Acoustical Society of America, 2000, 107(5): 2437–2446.CrossRefGoogle Scholar
  6. [6]
    Szabo, T.L., Time domain wave equations for lossy media obeying a frequency power law. Journal of the Acoustical Society of America, 1994, 96(1): 491–500.CrossRefGoogle Scholar
  7. [7]
    Chen, W. and Holm, S., Modified Szabo’s wave equation models for lossy media obeying frequency power law. Journal of the Acoustical Society of America, 2003, 114(5): 2570–2574.CrossRefGoogle Scholar
  8. [8]
    Carcione, J.M., Cavallini, F., Mainardi, F., et al., Time-domain modeling of constant-Q seismic waves using fractional derivatives. Pure and Applied Geophysics, 2002, 159: 1719–1736.CrossRefGoogle Scholar
  9. [9]
    D’astrous, F.T. and Foster, F.S., Frequency dependence of ultrasound attenuation and backscatter in breast tissue. Ultrasound in Medicine & Biology, 1986, 12(10): 795–808.CrossRefGoogle Scholar
  10. [10]
    Enelund, M. and Olsson, P., Damping described by fading memory-analysis and application to fractional derivative models. International Journal of Solids and Structures, 1999, 36: 939–970.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Podlubny, I., Fractional Differential Equations. San Diego: Academic Press, 1999zbMATHGoogle Scholar
  12. [12]
    Wismer, M.G. and Ludwig, R., An explicit numerical time domain formulation to simulate pulsed pressure waves in viscous fluids exhibiting arbitrary frequency power law attenuation. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 1995, 42(6): 1040–1049.CrossRefGoogle Scholar
  13. [13]
    Pritz, T., Frequency power law of material damping. Applied Acoustics, 2004, 65: 1027–1036.CrossRefGoogle Scholar
  14. [14]
    Waters, K.R., Hughes, M.S. and Mobley, J., et al., On the applicability of Kramers-Kronig relations for ultrasonic attenuation obeying a frequency power law. Journal of the Acoustical Society of America, 2000, 108(2): 556–563.CrossRefGoogle Scholar
  15. [15]
    Waters, K.R., Hughes, M.S. and Mobley, J., et al., Differential forms of the Kramers-Kronig dispersion relations. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 2003, 50(1): 68–76.CrossRefGoogle Scholar
  16. [16]
    Waters, K.R., Mobley, J. and Miller, J.G., Causality-imposed (Kramers-Kronig) relationships between attenuation and dispersion. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 2005, 52(5): 822–833.CrossRefGoogle Scholar
  17. [17]
    Szabo, T.L., Diagnostic Ultrasound Imaging. Burlington: Elsevier, 2004.Google Scholar
  18. [18]
    Nachman, A.I., Smith, J.F. and Waag, R.C., An equation for acoustic propagation in inhomogeneous media with relaxation losses. Journal of the Acoustical Society of America, 1990, 88(3): 1584–1595.CrossRefGoogle Scholar
  19. [19]
    Yuan, X., Borup, D. and Wiskin, J., et al., Simulation of acoustic wave prop dispersive media with relaxation using FDTD method with PML absorbing boundary condition. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 1999, 46(1): 14–23.CrossRefGoogle Scholar
  20. [20]
    Caputo, M. and Mainardi, F., A new dissipation model based on memory mechanism. Pure and Applied Geophysics, 1971, 91(1): 134–147.CrossRefGoogle Scholar
  21. [21]
    Wismer, M.G., Finite element analysis of broadband acoustic pulses through inhomogenous media with power law attenuation. Journal of the Acoustical Society of America, 2006, 120(6): 3493–3502.CrossRefGoogle Scholar
  22. [22]
    Holm, S. and Sinkus, R., A unifying fractional wave equation for compressional and shear waves. Journal of the Acoustical Society of America, 2010, 127(1): 542–548.CrossRefGoogle Scholar
  23. [23]
    Holm, S. and Naholm, S.P., A causal and fractional all-frequency wave equation for lossy media. Journal of the Acoustical Society of America, 2011, 130(4): 2195–2202.CrossRefGoogle Scholar
  24. [24]
    WFUMB. World Federation for Ultrasound in Medicine and Biology News. vol.4, no.2, Ultrasound in Medicine & Biology, 1997, 23.Google Scholar
  25. [25]
    Wells, P.N.T., Current status and future technical advances of ultrasonic imaging. Engineering in Medicine and Biology, 2000, 19(5): 14–20.CrossRefGoogle Scholar
  26. [26]
    Chen, W., Zhang, X. and Cai, X., A study on modified Szabo’s wave equation modeling of frequency-dependent dissipation in ultrasonic medical imaging. Physica Scripta, 2009, T136(014014): 1–5.Google Scholar
  27. [27]
    Richter, K., Clinical amplitude/velocity reconstructive imaging (CARI)—a new sonographic method for detecting breast lesions. The British Journal of Radiology, 1995, 68: 375–384.CrossRefGoogle Scholar
  28. [28]
    Richter, K. and Heywang-Köbrunner, S.H., Quantitative parameters measured by a new sonographic method for differentiation of benign and malignant breast disease. Investigative Radiology, 1995, 30(7): 401–411.CrossRefGoogle Scholar
  29. [29]
    Szabo, T.L., Causal theories and data for acoustic attenuation obeying a frequency power law. Journal of the Acoustical Society of America, 1995, 97(1): 14–24.MathSciNetCrossRefGoogle Scholar
  30. [30]
    Lin, Y. and Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation. Journal of Computational Physics, 2007, 225: 1533–1552.MathSciNetCrossRefGoogle Scholar
  31. [31]
    Bounaïm, A., Holm, S. and Chen, W. et al., Sensitivity of the ultrasonic CARI technique for breast tumor detection using a FETD scheme. Ultrasonics, 2004, 42: 919–925.CrossRefGoogle Scholar
  32. [32]
    Bounaïm, A., Holm, S. and Chen, W., et al., Detectability of breast lesions with CARI ultrasonography using a bioacoustic computational approach. Computers & Mathematics with Applications, 2007, 54(1): 96–106.MathSciNetCrossRefGoogle Scholar
  33. [33]
    Bounaïm, A., Holm, S. and Chen, W., et al., Quantification of the CARI breast imaging sensitivity by 2D/3D numerical time-domain ultrasound wave propagation. Mathematics and Computers in Simulation, 2004, 65(4–5): 521–534.MathSciNetCrossRefGoogle Scholar
  34. [34]
    Weiwad, W., Heinig, A. and Goetz, L., et al., Direct measurement of sound velocity in various specimens of breast tissue. Investigative Radiology, 2000, 35(12): 721–726.CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2012

Authors and Affiliations

  1. 1.Institute of Soft Matter Mechanics, Department of Engineering MechanicsHohai UniversityNanjingChina
  2. 2.Department of Civil EngineeringUniversity of SiegenSiegenGermany

Personalised recommendations