Skip to main content
SpringerLink
Log in

Mapped Chebyshev pseudo-spectral method for simulating the shear wave propagation in the plane of symmetry of a transversely isotropic viscoelastic medium

  • Original Article
  • Published:
Medical & Biological Engineering & Computing Aims and scope Submit manuscript

Abstract

Shear wave elastography is a versatile technique that is being applied to many organs. However, in tissues that exhibit anisotropic material properties, special care must be taken to estimate shear wave propagation accurately and efficiently. A two-dimensional simulation method is implemented to simulate the shear wave propagation in the plane of symmetry in transversely isotropic viscoelastic media. The method uses a mapped Chebyshev pseudo-spectral method to calculate the spatial derivatives and an Adams–Bashforth–Moulton integrator with variable step sizes for time marching. The boundaries of the two-dimensional domain are surrounded by perfectly matched layers to approximate an infinite domain and minimize reflection errors. In an earlier work, we proposed a solution for estimating the apparent shear wave elasticity and viscosity of the spatial group velocity as a function of rotation angle through a low-frequency approximation by a Taylor expansion. With the solver implemented in MATLAB, the simulated results in this paper match well with the theory. Compared to the finite element method simulations we used before, the pseudo-spectral solver consumes less memory and is faster and achieves better accuracy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Alexandrescu A, Bueno-Orovio A, Salgueiro JR, Perez-Garcia VM (2009) Mapped Chebyshev pseudospectral method for the study of multiple scale phenomena. Comput Phys Commun 180:912–919. doi:10.1016/j.cpc.2008.12.018

    Article  CAS  Google Scholar 

  2. Amador C, Urban MW, Chen S, Greenleaf JF (2011) Shearwave dispersion ultrasound vibrometry (SDUV) on swine kidney. IEEE Trans Ultrason Ferroelectr Freq Control 58:2608–2619

    Article  PubMed  PubMed Central  Google Scholar 

  3. Aristizabal S, Amador C, Qiang B, Kinnick RR, Nenadic IZ, Greenleaf JF, Urban MW (2014) Shear wave vibrometry evaluation in transverse isotropic tissue mimicking phantoms and skeletal muscle. Phys Med Biol 59:7735–7752. doi:10.1088/0031-9155/59/24/7735

    Article  PubMed  PubMed Central  Google Scholar 

  4. Bammer R, Acar B, Moseley ME (2003) In vivo MR tractography using diffusion imaging. Eur J Radiol 45:223–234

    Article  PubMed  Google Scholar 

  5. Basser PJ, Mattiello J, LeBihan D (1994) MR diffusion tensor spectroscopy and imaging. Biophys J 66:259–267. doi:10.1016/S0006-3495(94)80775-1

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  6. Basser PJ, Pierpaoli C (1996) Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. J Magn Reson, Ser B 111:209–219. doi:10.1006/jmrb.1996.0086

    Article  CAS  Google Scholar 

  7. Bercoff J, Tanter M, Fink M (2004) Supersonic shear imaging: a new technique for soft tissue elasticity mapping. IEEE Trans Ultrason Ferroelectr Freq Control 51:396–409

    Article  PubMed  Google Scholar 

  8. Boyd JP (2001) Chebyshev and Fourier spectral methods, 2nd edn. Dover Publications, Mineola

    Google Scholar 

  9. Brum J, Bernal M, Gennisson JL, Tanter M (2014) In vivo evaluation of the elastic anisotropy of the human Achilles tendon using shear wave dispersion analysis. Phys Med Biol 59:505–523

    Article  CAS  PubMed  Google Scholar 

  10. Budzik JF, Balbi V, Verclytte S, Pansini V, Le Thuc V, Cotten A (2014) Diffusion tensor imaging in musculoskeletal disorders. Radiographics 34:E56–E72. doi:10.1148/rg.343125062

    Article  PubMed  Google Scholar 

  11. Carcione JM (1993) Seismic modeling in viscoelastic media. Geophysics 58:110–120. doi:10.1190/1.1443340

    Article  Google Scholar 

  12. Carcione JM (2007) Wave fields in real media: wave propagation in anisotropic, anelastic, porous and electromagnetic media. Handbook of geophysical exploration Seismic exploration, vol 38, 2nd edn. Elsevier, Amsterdam

    Google Scholar 

  13. Carcione JM, Kosloff D, Kosloff R (1988) Wave propagation simulation in a linear viscoelastic medium. Geophys J 95:597–611

    Article  Google Scholar 

  14. Carcione JM, Poletto F, Gei D (2004) 3-D wave simulation in anelastic media using the Kelvin–Voigt constitutive equation. J Comput Phys 196:282–297. doi:10.1016/j.jcp.2003.10.024

    Article  Google Scholar 

  15. Catheline S, Gennisson JL, Delon G, Fink M, Sinkus R, Abouelkaram S, Culioli J (2004) Measurement of viscoelastic properties of homogeneous soft solid using transient elastography: an inverse problem approach. J Acoust Soc Am 116:3734–3741

    Article  CAS  PubMed  Google Scholar 

  16. Chatelin S, Bernal M, Deffieux T, Papadacci C, Flaud P, Nahas A, Boccara C, Gennisson JL, Tanter M, Pernot M (2014) Anisotropic polyvinyl alcohol hydrogel phantom for shear wave elastography in fibrous biological soft tissue: a multimodality characterization. Phys Med Biol 59:6923–6940. doi:10.1088/0031-9155/59/22/6923

    Article  PubMed  Google Scholar 

  17. Chui C, Kobayashi E, Chen X, Hisada T, Sakuma I (2007) Transversely isotropic properties of porcine liver tissue: experiments and constitutive modelling. Med Biol Eng Comput 45:99–106

    Article  CAS  PubMed  Google Scholar 

  18. Corovic S, Zupanic A, Kranjc S, Al Sakere B, Leroy-Willig A, Mir LM, Miklavcic D (2010) The influence of skeletal muscle anisotropy on electroporation: in vivo study and numerical modeling. Med Biol Eng Comput 48:637–648

    Article  PubMed  PubMed Central  Google Scholar 

  19. Eby SF, Song P, Chen S, Chen Q, Greenleaf JF, An KN (2013) Validation of shear wave elastography in skeletal muscle. J Biomech 46:2381–2387. doi:10.1016/j.jbiomech.2013.07.033

    Article  PubMed  Google Scholar 

  20. Galban CJ, Maderwald S, Stock F, Ladd ME (2007) Age-related changes in skeletal muscle as detected by diffusion tensor magnetic resonance Imaging. J Gerontol Ser A Biol Sci Med Sci 62:453–458

    Article  Google Scholar 

  21. Garmirian LP, Chin AB, Rutkove SB (2009) Discriminating neurogenic from myopathic disease via measurement of muscle anisotropy. Muscle Nerve 39:16–24. doi:10.1002/mus.21115

    Article  PubMed  PubMed Central  Google Scholar 

  22. Gates F, McCammond D, Zingg W, Kunov H (1980) In vivo stiffness properties of the canine diaphragm muscle. Med Biol Eng Comput 18:625–632

    Article  CAS  PubMed  Google Scholar 

  23. Gennisson JL, Catheline S, Chaffai S, Fink M (2003) Transient elastography in anisotropic medium: application to the measurement of slow and fast shear wave speeds in muscles. J Acoust Soc Am 114:536–541

    Article  PubMed  Google Scholar 

  24. Gennisson JL, Deffieux T, Mace E, Montaldo G, Fink M, Tanter M (2010) Viscoelastic and anisotropic mechanical properties of in vivo muscle tissue assessed by supersonic shear imaging. Ultrasound Med Biol 36:789–801. doi:10.1016/j.ultrasmedbio.2010.02.013

    Article  PubMed  Google Scholar 

  25. Gennisson JL, Grenier N, Combe C, Tanter M (2012) Supersonic shear wave elastography of in vivo pig kidney: influence of blood pressure, urinary pressure and tissue anisotropy. Ultrasound Med Biol 38:1559–1567. doi:10.1016/j.ultrasmedbio.2012.04.013

    Article  PubMed  Google Scholar 

  26. Guillard H, Male JM, Peyret R (1992) Adaptive spectral methods with application to mixing layer computations. J Comput Phys 102:114–127. doi:10.1016/S0021-9991(05)80010-5

    Article  Google Scholar 

  27. Khalil C, Budzik JF, Kermarrec E, Balbi V, Le Thuc V, Cotten A (2010) Tractography of peripheral nerves and skeletal muscles. Eur J Radiol 76:391–397. doi:10.1016/j.ejrad.2010.03.012

    Article  CAS  PubMed  Google Scholar 

  28. Komatitsch D, Martin R (2007) An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation. Geophysics Sm72:Sm155–Sm167. doi:10.1190/1.2757586

    Article  Google Scholar 

  29. Kosloff D, Talezer H (1993) A modified Chebyshev pseudospectral method with an O(N-1) time step restriction. J Comput Phys 104:457–469. doi:10.1006/jcph.1993.1044

    Article  Google Scholar 

  30. Le Bihan D, Mangin JF, Poupon C, Clark CA, Pappata S, Molko N, Chabriat H (2001) Diffusion tensor imaging: concepts and applications. J Magn Reson Imaging 13:534–546

    Article  PubMed  Google Scholar 

  31. Lee W-N, Pernot M, Couade M, Messas E, Bruneval P, Bel A, Hagege AA, Fink M, Tanter M (2012) Mapping myocardial fiber orientation using echocardiography-based shear wave imaging. IEEE Trans Med Imaging 31:554–562

    Article  PubMed  Google Scholar 

  32. Lee WN, Larrat B, Pernot M, Tanter M (2012) Ultrasound elastic tensor imaging: comparison with MR diffusion tensor imaging in the myocardium. Phys Med Biol 57:5075–5095. doi:10.1088/0031-9155/57/16/5075

    Article  PubMed  Google Scholar 

  33. Li Y, Bou Matar O (2010) Convolutional perfectly matched layer for elastic second-order wave equation. J Acoust Soc Am 127:1318–1327. doi:10.1121/1.3290999

    Article  PubMed  Google Scholar 

  34. Martin R, Komatitsch D (2009) An unsplit convolutional perfectly matched layer technique improved at grazing incidence for the viscoelastic wave equation. Geophys J Int 179:333–344. doi:10.1111/j.1365-246X.2009.04278.x

    Article  Google Scholar 

  35. Martin R, Komatitsch D, Gedney SD, Bruthiaux E (2010) A high-order time and space formulation of the unsplit perfectly matched layer for the seismic wave equation using auxiliary differential equations (ADE-PML). Comp Model Eng 56:17–41

    Google Scholar 

  36. Masutani Y, Aoki S, Abe O, Hayashi N, Otomo K (2003) MR diffusion tensor imaging: recent advance and new techniques for diffusion tensor visualization. Eur J Radiol 46:53–66

    Article  PubMed  Google Scholar 

  37. MATLAB manual (2013). Mathworks Inc.

  38. McGarry MDJ, Van Houten EEW (2008) Use of a Rayleigh damping model in elastography. Med Biol Eng Comput 46:759–766

    Article  PubMed  Google Scholar 

  39. Moseley ME, Cohen Y, Kucharczyk J, Mintorovitch J, Asgari HS, Wendland MF, Tsuruda J, Norman D (1990) Diffusion-weighted MR imaging of anisotropic water diffusion in cat central nervous system. Radiology 176:439–445. doi:10.1148/radiology.176.2.2367658

    Article  CAS  PubMed  Google Scholar 

  40. Muthupillai R, Ehman RL (1996) Magnetic resonance elastography. Nat Med 2:601–603

    Article  CAS  PubMed  Google Scholar 

  41. Ng KT, Yan R (2003) Three-dimensional pseudospectral modelling of cardiac propagation in an inhomogeneous anisotropic tissue. Med Biol Eng Comput 41:618–624

    Article  CAS  PubMed  Google Scholar 

  42. Palmeri ML, Sharma AC, Bouchard RR, Nightingale RW, Nightingale KR (2005) A finite-element method model of soft tissue response to impulsive acoustic radiation force. IEEE Trans Ultrason Ferroelectr Freq Control 52:1699–1712

    Article  PubMed  PubMed Central  Google Scholar 

  43. Palmeri ML, Wang MH, Dahl JJ, Frinkley KD, Nightingale KR (2008) Quantifying hepatic shear modulus in vivo using acoustic radiation force. Ultrasound Med Biol 34:546–558. doi:10.1016/j.ultrasmedbio.2007.10.009

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  44. Pierpaoli C, Basser PJ (1996) Toward a quantitative assessment of diffusion anisotropy. Magn Reson Med 36:893–906. doi:10.1002/mrm.1910360612

    Article  CAS  PubMed  Google Scholar 

  45. Qi J, Olsen NJ, Price RR, Winston JA, Park JH (2008) Diffusion-weighted imaging of inflammatory myopathies: polymyositis and dermatomyositis. J Magn Reson Imaging 27:212–217. doi:10.1002/jmri.21209

    Article  PubMed  Google Scholar 

  46. Qiang B, Brigham JC, Aristizabal S, Greenleaf JF, Zhang X, Urban MW (2015) Modeling transversely isotropic, viscoelastic, incompressible tissue-like materials with application in ultrasound shear wave elastography. Phys Med Biol 60:1289–1306. doi:10.1088/0031-9155/60/3/1289

    Article  PubMed  PubMed Central  Google Scholar 

  47. Rudenko OV, Sarvazyan AP (2014) Wave anisotropy of shear viscosity and elasticity. Acoust Phys 60:710–718. doi:10.1134/s1063771014060141

    Article  Google Scholar 

  48. Sansalone M, Carino NJ, Hsu NN (1987) A finite-element study of transient wave-propagation in plates. J Res Nat Bur Stand 92:267–278. doi:10.6028/Jres.092.025

    Article  CAS  Google Scholar 

  49. Saotome T, Sekino M, Eto F, Ueno S (2006) Evaluation of diffusional anisotropy and microscopic structure in skeletal muscles using magnetic resonance. Magn Reson Imaging 24:19–25. doi:10.1016/j.mri.2005.09.009

    Article  PubMed  Google Scholar 

  50. Sarvazyan AP, Rudenko OV, Swanson SD, Fowlkes JB, Emelianov SY (1998) Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics. Ultrasound Med Biol 24:1419–1435. doi:10.1016/S0301-5629(98)00110-0

    Article  CAS  PubMed  Google Scholar 

  51. Trefethen LN (2000) Spectral methods in MATLAB. Software, environments, tools. Society for Industrial and Applied Mathematics, Philadelphia, PA

  52. Urban MW, Chen S, Fatemi M (2012) A review of shearwave dispersion ultrasound vibrometry (SDUV) and its applications. Curr Med Imaging Rev 8:27–36

    Article  PubMed  PubMed Central  Google Scholar 

  53. Virieux J, Calandra H, Plessix RE (2011) A review of the spectral, pseudo-spectral, finite-difference and finite-element modelling techniques for geophysical imaging. Geophys Prospect 59:794–813. doi:10.1111/j.1365-2478.2011.00967.x

    Article  Google Scholar 

  54. Wang M, Byram B, Palmeri M, Rouze N, Nightingale K (2013) Imaging transverse isotropic properties of muscle by monitoring acoustic radiation force induced shear waves using a 2-D matrix ultrasound array. IEEE Trans Med Imaging 32:1671–1684. doi:10.1109/TMI.2013.2262948

    Article  PubMed  PubMed Central  Google Scholar 

  55. Zhan Z, Ng KT (2000) Two-dimensional Chebyshev pseudospectral modelling of cardiac propagation. Med Biol Eng Comput 38:311–318

    Article  CAS  PubMed  Google Scholar 

  56. Zhang J, Zhang G, Morrison B, Mori S, Sheikh KA (2008) Magnetic resonance imaging of mouse skeletal muscle to measure denervation atrophy. Exp Neurol 212:448–457. doi:10.1016/j.expneurol.2008.04.033

    Article  PubMed  PubMed Central  Google Scholar 

  57. Zhao H, Song P, Urban MW, Greenleaf JF, Chen S (2012) Shear wave speed measurement using an unfocused ultrasound beam. Ultrasound Med Biol 38:1646–1655

    Article  PubMed  PubMed Central  Google Scholar 

Download references

Acknowledgments

This work was supported by Grant R01DK092255 from the National Institute of Diabetes and digestive and Kidney Diseases (NIDDK) and National Institutes of Health (NIH). The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIDDK and NIH.

Author information

Authors and Affiliations

  1. Department of Physiology and Biomedical Engineering, Mayo Clinic College of Medicine, Rochester, MN, 55905, USA

    Bo Qiang, James F. Greenleaf & Matthew W. Urban

  2. Department of Civil and Environmental Engineering, Department of Bioengineering, Swanson School of Engineering, University of Pittsburgh, Pittsburgh, PA, 15261, USA

    John C. Brigham

  3. Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI, 48824, USA

    Robert J. McGough

  4. Department of Radiology, Mayo Clinic College of Medicine, Rochester, MN, 55905, USA

    Matthew W. Urban

  5. The Nielsen Company, Oldsmar, FL, 34677, USA

    Bo Qiang

  6. School of Engineering and Computing Sciences, Durham University, South Road, Durham, DH1 3LE, UK

    John C. Brigham

Authors
  1. Bo Qiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John C. Brigham
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Robert J. McGough
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. James F. Greenleaf
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Matthew W. Urban
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bo Qiang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiang, B., Brigham, J.C., McGough, R.J. et al. Mapped Chebyshev pseudo-spectral method for simulating the shear wave propagation in the plane of symmetry of a transversely isotropic viscoelastic medium. Med Biol Eng Comput 55, 389–401 (2017). https://doi.org/10.1007/s11517-016-1522-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11517-016-1522-9

Keywords

Search

Navigation