High-frequency effective dynamical parameters for a viscoelastic layered medium modeling soft tissue
DOI:
https://doi.org/10.31349/RevMexFis.72.051101Keywords:
multiple scales, effective parameters, shear wave elastography, high-frecuency ultrasoundAbstract
Shear wave propagation in a layered medium made of Voigt viscoelastic layers is studied. It is argued that is it possible to obtain closed-form analytical approximate expressions for the effective dynamical parameters of the layered medium in the high-frequency regime. The multiple scales method is applied to obtain the frequency-dependent effective dynamical parameters. Lower order analytical solution approximations for harmonic steady-state was obtained. Numerical comparison against precise numerical solutions via the tranfer matrix method were carried out. This could contribute to the model design of tissue-like phantom materials for ultrasound devices calibration as well as in the processing of signals from shear wave elastography.
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References
R. W. Prager et al., Three-dimensional ultrasound imaging, Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine 224 (2010) 193-223, https://doi.org/10.1243/09544119jeim586
L. Mercier et al., A review of calibration techniques for freehand 3-D ultrasound systems, Ultrasound in Medicine and Biology 31 (2005) 143-165, https://doi.org/10.1016/j.ultrasmedbio.2004.11.001
J. E. Aldrich, Basic physics of ultrasound imaging, Critical Care Medicine 35 (2007) S131-S137, https://doi.org/10.1097/01.ccm.0000260624.99430.22
A. Maccabi et al., Quantitative characterization of viscoelastic behavior in tissue-mimicking phantoms and ex vivo animal tissues, PLOS ONE 13 (2018) e0191919, https://doi.org/10.1371/journal.pone.0191919
M. H. Mozaffari and W.-S. Lee, Freehand 3-D Ultrasound Imaging: A Systematic Review, Ultrasound in Medicine and Biology 43 (2017) 2099-2124, https://doi.org/10.1016/j.ultrasmedbio.2017.06.009
A. Fenster and D. B. Downey, Three-Dimensional Ultrasound Imaging, Annual Review of Biomedical Engineering 2 (2000) 457-475, https://doi.org/10.1146/annurev.bioeng.2.1.457
P.-W. Hsu et al., Freehand 3D Ultrasound Calibration: A Review, p. 47-84 (Springer Berlin Heidelberg, 2008), https://dx.doi.org/10.1007/978-3-540-68993-5_3
S. Wood et al., Design and fabrication of a realistic anthropomorphic heterogeneous head phantom for MR purposes, PLOS ONE 12 (2017) e0183168, https://doi.org/10.1371/journal.pone.0183168
M. Earle, G. D. Portu, and E. DeVos, Agar ultrasound phantoms for low-cost training without refrigeration, African Journal of Emergency Medicine 6 (2016) 18-23, https://doi.org/10.1016/j.afjem.2015.09.003
S. Cournane, A. J. Fagan, and J. E. Browne, Review of ultrasound elastography quality control and training test phantoms, Ultrasound 20 (2011) 16-23, https://doi.org/10.1258/ult.2011.011033
J. Bravo-Castillero and L. F. López Ríos, Variational formulation for fractional hyperbolic problems in the theory of viscoelasticity, Zeitschrift für angewandte Mathematik und Physik 73 (2022), https://doi.org/10.1007/s00033-022-01826-5
K. Manickam, R. R. Machireddy, and S. Seshadri, Characterization of biomechanical properties of agar based tissue mimicking phantoms for ultrasound stiffness imaging techniques, Journal of the Mechanical Behavior of Biomedical Materials 35 (2014) 132-143, https://doi.org/10.1016/j.jmbbm.2014.03.017
T. A. Bigelow and W. D. O’Brien, Scatterer size estimation in pulse-echo ultrasound using focused sources: Calibration measurements and phantom experiments, The Journal of the Acoustical Society of America 116 (2004) 594-602, https://doi.org/10.1121/1.1757453
T. A. Bigelow and W. D. O’Brien, Scatterer size estimation in pulse-echo ultrasound using focused sources: Theoretical approximations and simulation analysis, The Journal of the Acoustical Society of America 116 (2004) 578-593, https://doi.org/10.1121/1.1757452
R. W. Gill, Physics and technology of diagnostic ultrasound: a practitioner’s guide, 2nd ed. (High Frequency Publishing, S.l., 2020), p. 17-24
K. Martin and K. Ramnarine, Physics, In P. R. Hoskins, K. Martin, and A. Thrush, eds., Diagnostic ultrasound: physics and equipment, 3rd ed., p. 7-36 (CRC Press, Boca Raton London New York, 2019), https://doi.org/10.1201/9781138893603
G. A. Holzapfel and R. W. Ogden, Modeling the biomechanical properties of soft biological tissues: Constitutive theories, European Journal of Mechanics-A/Solids 112 (2025) 105634, https://doi.org/10.1016/j.euromechsol.2025.105634
L. A. Taber, Nonlinear theory of elasticity: applications in biomechanics, 1st ed. (World Scientific, River Edge, NJ, 2004), pp. 247-264
C. W. J. Oomens et al., Biomechanics: concepts and computation, Cambridge texts in biomedical engineering, 2nd ed. (Cambridge University Press, Cambridge, United Kingdom; New York, NY, 2018), pp. 270-285
S. Mohammadi, Multiscale biomechanics: theory and applications, 1st ed. (JohnWiley and Sons Ltd, Hoboken, NJ, 2023), pp. 23-74
Y. C. Fung, Biomechanics: mechanical properties of living tissues, 2nd ed. (Springer-Verlag, New York, 1993), pp. 220-314
A. Al-Mayah, ed., Biomechanics of soft tissues: principles and applications, 1st ed. (CRC Press, Boca Raton London New York, 2018), pp. 1-25
V. Andreev, N. Tsybin, and R. Turusov, Layered composite and contact layer. Effective modulus of elasticity, E3S Web of Conferences 97 (2019) 04071, https://doi.org/10.1051/e3sconf/20199704071
J. Willis, Exact effective relations for dynamics of a laminated body, Mechanics of Materials 41 (2009) 385-393, https://doi.org/10.1016/j.mechmat.2009.01.010
J. Bravo-Castillero et al., Homogenization of magnetoelectroelastic multilaminated materials, The Quarterly Journal of Mechanics and Applied Mathematics 61 (2008) 311-332, https://doi.org/10.1093/qjmam/hbn010
M. S. Chaki and J. Bravo-Castillero, Dynamic asymptotic homogenization for wave propagation in magneto-electro-elastic laminated composite periodic structure, Composite Structures 322 (2023) 117410, https://doi.org/10.1016/j.compstruct.2023.117410
A. Srivastava and S. Nemat-Nasser, Overall dynamic properties of three-dimensional periodic elastic composites, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468 (2011) 269-287, https://doi.org/10.1098/rspa.2011.0440
J. H. McElhaney, Dynamic response of bone and muscle tissue, Journal of Applied Physiology 21 (1966) 1231, https://doi.org/10.1152/jappl.1966.21.4.1231
N. W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior (Springer Berlin Heidelberg, 1989), pp. 35-145, https://dx.doi.org/10.1007/978-3-642-73602-5
N. Arnold, J. Scott, and T. R. Bush, A review of the characterizations of soft tissues used in human body modeling: Scope, limitations, and the path forward, Journal of Tissue Viability 32 (2023) 286-304, https://doi.org/10.1016/j.jtv.2023.02.003
S. J. Mostafavi Yazdi and J. Baqersad, Mechanical modeling and characterization of human skin: A review, Journal of Biomechanics 130 (2022) 110864, https://doi.org/10.1016/j.jbiomech.2021.110864
S. Catheline et al., Measurement of viscoelastic properties of homogeneous soft solid using transient elastography: An inverse problem approach, The Journal of the Acoustical Society of America 116 (2004) 3734-3741, https://doi.org/10.1121/1.1815075
B. Zhou and X. Zhang, Comparison of five viscoelastic models for estimating viscoelastic parameters using ultrasound shear wave elastography, Journal of the Mechanical Behavior of Biomedical Materials 85 (2018) 109-116, https://doi.org/10.1016/j.jmbbm.2018.05.041
A. H. Nayfeh, Perturbation Methods (Wiley, 2000), p. 228-302, https://dx.doi.org/10.1002/9783527617609
D. Cioranescu and P. Donato, An Introduction to Homogenization (Oxford University PressOxford, 1999), p. 125-133, https://dx.doi.org/10.1093/oso/9780198565543.001.0001
A. Srivastava and S. Nemat-Nasser, On the limit and applicability of dynamic homogenization, Wave Motion 51 (2014) 1045-1054, https://doi.org/10.1016/j.wavemoti.2014.04.003
R. M. Christensen, Mechanics of Composite Materials (Wiley, 1979), pp. 32-37, https://dx.doi.org/10.1115/1.3153710
N. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media (Springer Netherlands, 1989), p. 12-29, https://dx.doi.org/10.1007/978-94-009-2247-1
R. M. Christensen, Theory of Viscoelasticity, An Introduction, 2nd Edition (ACADEMIC PRESS, 1982), p. 226-226, https://dx.doi.org/10.1115/1.3167591
R. M. Christensen, Theory of Viscoelasticity, An Introduction, 2nd Edition (ACADEMIC PRESS, 1982), p. 21-22, https://dx.doi.org/10.1115/1.3167591
M. Van Dyke and S. Rosenblat, Perturbation Method in Fluid Mechanics, Journal of Applied Mechanics 43 (1976) 189-190, https://doi.org/10.1115/1.3423785
A. Bensoussanet al., Asymptotic Analysis of Periodic Structures, Journal of Applied Mechanics 46 (1979) 477-477, https://doi.org/10.1115/1.3424588
R. V. Craster, J. Kaplunov, and A. V. Pichugin, High-frequency homogenization for periodic media, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466 (2010) 2341-2362, https://doi.org/10.1098/rspa.2009.0612
C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I (Springer New York, 1999), p. 560-567, https://doi.org/10.1007/978-1-4757-3069-2
C. M. Bender and L. M. A. Bettencourt, Multiple-scale analysis of quantum systems, Physical Review D 54 (1996) 7710-7723, https://doi.org/10.1103/physrevd.54.7710
M. J. Lighthill, A Technique for Rendering Approximate Solutions to Physical Problems Uniformly Convergent, Philosophical Magazine 40 (1949) 1179
J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods (Springer New York, 1996), pp. 614- 617, https://dx.doi.org/10.1007/978-1-4612-3968-0
J. V. Uspensky, Theory of equations., McGraw-Hill paperbacks, 1st ed. (New York, 1948), pp. 93-107, OCLC: 525463
M. S. P. M. S. P. Eastham, The spectral theory of periodic differential equations (Edinburgh, Scottish Academic Press [distributed by Chatto and Windus, London], 1973), pp. 1-5, https://archive.org/details/spectraltheoryof0000east
M. Humi and W. Miller, Second Course in Ordinary Differential Equations for Scientists and Engineers, Universitext (Springer US, New York, NY, 1988), pp. 215-221
A. Dell, A. Krynkin, and K. Horoshenkov, The use of the transfer matrix method to predict the effective fluid properties of acoustical systems, Applied Acoustics 182 (2021) 108259, https://doi.org/10.1016/j.apacoust.2021.108259
T. G. Mackay and A. Lakhtakia, The Transfer-Matrix Method in Electromagnetics and Optics (Springer International Publishing, 2020), pp. 25-26, http://dx.doi.org/10.1007/978-3-031-02022-3
E. L. Madsen, H. J. Sathoff, and J. A. Zagzebski, Ultrasonic shear wave properties of soft tissues and tissuelike materials, The Journal of the Acoustical Society of America 74 (1983) 1346-1355, https://doi.org/10.1121/1.390158
Xinmai Yang and C. Church, A simple viscoelastic model for soft tissues the frequency range 6-20 MHz, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 53 (2006) 1404, https://doi.org/10.1109/TUFFC.2006.1665097
X. Feng, G.-Y. Li, and S.-H. Yun, Ultra-wideband optical coherence elastography from acoustic to ultrasonic frequencies, Nature Communications 14 (2023) 4949, https://doi.org/10.1038/s41467-023-40625-y
J. Bravo-Castillero et al., Mathematical Modelling of Composite Phantoms for the Calibration of Ultrasound Devices, Journal of Applied Research and Technology 21 (2023) 154, https://doi.org/10.22201/icat.24486736e.2023.21.2.1535
F. M. Arscott, Periodic Differential Equations: An Introduction to Mathieu, Lame, and Allied Functions (Elsevier, 2014), pp. 26-29
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