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Research Papers

Mapping the Viscoelastic Behavior of Soft Solids From Time Harmonic Motion

[+] Author and Article Information
Yue Mei

Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77843

Sevan Goenezen

Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77843
e-mail: sevangoenezen@gmail.com

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received October 6, 2017; final manuscript received January 8, 2018; published online February 2, 2018. Assoc. Editor: Yihui Zhang.

J. Appl. Mech 85(4), 041003 (Feb 02, 2018) (11 pages) Paper No: JAM-17-1561; doi: 10.1115/1.4038966 History: Received October 06, 2017; Revised January 08, 2018

We present a nondestructive approach to map the heterogeneous viscoelastic moduli from time harmonic motion via a constrained optimization strategy under the framework of finite element techniques. The adjoint equations are carefully derived to determine the gradient of the objective function with respect to the viscoelastic moduli. The feasibility of this inverse scheme is tested with simulated experiments under various driving frequencies. We observe that the overall strategy results in well-reconstructed moduli. For low frequencies, however, the mapped loss modulus is of inferior quality. To explain this observation, we analyze two simple one-dimensional (1D) models theoretically. The analysis reveals that the known displacement amplitude is less sensitive to the loss modulus value at low frequencies. Thus, we conclude that the inverse method is incapable of finding a well-reconstructed loss modulus distribution for low driving frequencies in the presence of noisy data. Overall, the inverse algorithms presented in this work are highly robust to map the storage and loss modulus with high accuracy given that a proper range of frequencies are utilized.

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References

Goenezen, S. , Rennie, M. , and Rugonyi, S. , 2012, “ Biomechanics of Early Cardiac Development,” Biomech. Model. Mechanobiol., 11(8), pp. 1187–1204. [CrossRef] [PubMed]
Schmitt, J. , 1998, “ OCT Elastography: Imaging Microscopic Deformation and Strain of Tissue,” Opt. Express, 3(6), pp. 199–211. [CrossRef] [PubMed]
Rogowska, J. , Patel, N. , Plummer, S. , and Brezinski, M. E. , 2006, “ Quantitative Optical Coherence Tomographic Elastography: Method for Assessing Arterial Mechanical Properties,” Br. J. Radiol., 79(945), pp. 707–711. [CrossRef] [PubMed]
Hall, T. , Barbone, P. E. , Oberai, A. A. , Jiang, J. , Dord, J. , Goenezen, S. , and Fisher, T. , 2011, “ Recent Results in Nonlinear Strain and Modulus Imaging,” Curr. Med. Imaging Rev., 7(4), pp. 313–327. [CrossRef] [PubMed]
Ophir, J. , Cespedes, I. , Ponnekanti, H. , Yazdi, Y. , and Li, X. , 1991, “ Elastography: A Quantitative Method for Imaging the Elasticity of Biological Tissues,” Ultrason. Imaging, 13(2), pp. 111–134. [CrossRef] [PubMed]
Ophir, J. , Alam, S. K. , Garra, B. , Kallel, F. , Konofagou, E. , Krouskop, T. , and Varghese, T. , 1999, “ Elastography: Ultrasonic Estimation and Imaging of the Elastic Properties of Tissues,” Proc. Inst. Mech. Eng., Part H: J. Eng. Med., 213(3), pp. 203–33. [CrossRef]
Romano, A. J. , Bucaro, J. A. , Houston, B. H. , Kugel, J. L. , Rossman, P. , Grimm, R. C. , and Ehman, R. , 2004, “ On the Feasibility of Elastic Wave Visualization Within Polymeric Solids Using Magnetic Resonance Elastography,” J. Acoust. Soc. Am., 116(1), pp. 125–132. [CrossRef] [PubMed]
Freimann, F. B. , Muller, S. , Streitberger, K. J. , Guo, J. , Rot, S. , Ghori, A. , Vajkoczy, P. , Reiter, R. , Sack, I. , and Braun, J. , 2013, “ MR Elastography in a Murine Stroke Model Reveals Correlation of Macroscopic Viscoelastic Properties of the Brain With Neuronal Density,” NMR Biomed., 26(11), pp. 1534–1539. [CrossRef] [PubMed]
Streitberger, K. J. , Sack, I. , Krefting, D. , Pfuller, C. , Braun, J. , Paul, F. , and Wuerfel, J. , 2012, “ Brain Viscoelasticity Alteration in Chronic-Progressive Multiple Sclerosis,” PLoS One, 7(1), p. e29888. [CrossRef] [PubMed]
Goenezen, S. , Dord, J. F. , Sink, Z. , Barbone, P. , Jiang, J. , Hall, T. J. , and Oberai, A. A. , 2012, “ Linear and Nonlinear Elastic Modulus Imaging: An Application to Breast Cancer Diagnosis,” Med. Imaging, IEEE Trans., 31(8), pp. 1628–1637. [CrossRef]
Goenezen, S. , Barbone, P. , and Oberai, A. A. , 2011, “ Solution of the Nonlinear Elasticity Imaging Inverse Problem: The Incompressible Case,” Comput. Methods Appl. Mech. Eng., 200(13–16), pp. 1406–1420. [CrossRef] [PubMed]
Mei, Y. , and Goenezen, S. , 2015, “ Spatially Weighted Objective Function to Solve the Inverse Problem in Elasticity for the Elastic Property Distribution,” Computational Biomechanics for Medicine: New Approaches and New Applications, B. J. Doyle , K. Miller , A. Wittek , and P. M. F. Nielson , eds., Springer, New York.
Mei, Y. , Kuznetsov, S. , and Goenezen, S. , 2015, “ Reduced Boundary Sensitivity and Improved Contrast of the Regularized Inverse Problem Solution in Elasticity,” ASME J. Appl. Mech., 83(3), p. 031001. [CrossRef]
Oberai, A. A. , Gokhale, N. H. , and Feijóo, G. R. , 2003, “ Solution of Inverse Problems in Elasticity Imaging Using the Adjoint Method,” Inverse Probl., 19(2), p. 297. [CrossRef]
Guchhait, S. , and Banerjee, B. , 2015, “ Constitutive Error Based Material Parameter Estimation Procedure for Hyperelastic Material,” Comput. Methods Appl. Mech. Eng., 297, pp. 455–475. [CrossRef]
Sridhar, S. L. , Mei, Y. , and Goenezen, S. , 2017, “ Improving the Sensitivity to Map Nonlinear Parameters for Hyperelastic Problems,” Comput. Methods Appl. Mech. Eng., 331, pp. 474–491. [CrossRef]
Mei, Y. , Fulmer, R. , Raja, V. , Wang, S. , and Goenezen, S. , 2016, “ Estimating the Non-Homogeneous Elastic Modulus Distribution From Surface Deformations,” Int. J. Solids Struct., 83, pp. 73–80. [CrossRef]
Mei, Y. , and Goenezen, S. , 2016, “ Non-Destructive Characterization of Heterogeneous Solids From Limited Surface Measurements,” 24th International Congress of Theoretical and Applied Mechanics, Montreal, QC, Canada, Aug. 22–26.
Mei, Y. , Wang, S. , Shen, X. , Rabke, S. , and Goenezen, S. , 2017, “ Mechanics Based Tomography: A Preliminary Feasibility Study,” Sensors, 17(5), p. 1075. [CrossRef]
Simon, M. , Guo, J. , Papazoglou, S. , Scholand-Engler, H. , Erdmann, C. , Melchert, U. , Bonsanto, M. , Braun, J. , Petersen, D. , Sack, I. , and Wuerfel, J. , 2013, “ Non-Invasive Characterization of Intracranial Tumors by Magnetic Resonance Elastography,” New J. Phys., 15, p. 085024. [CrossRef]
Park, E. , and Maniatty, A. M. , 2006, “ Shear Modulus Reconstruction in Dynamic Elastography: Time Harmonic Case,” Phys. Med. Biol., 51(15), pp. 3697–3721. [CrossRef] [PubMed]
Yin, M. , Rouviere, O. , and Ehman, R. L. , 2005, “ Shear Wave Diffraction Fields Generated by Longitudinal MRE Drivers,” International Society for Magnetic Resonance in Medicine, Miami, FL, May 7–13, p. 2560.
Diaza, M. I. , Aquinoa, W. , and Bonnet, M. , 2015, “ A Modified Error in Constitutive Equation Approach for Frequency-Domain Viscoelasticity Imaging Using Interior Data,” Comput. Methods Appl. Mech. Eng., 296, pp. 129–149. [CrossRef] [PubMed]
Brigham, J. C. , and Aquino, W. , 2009, “ Inverse Viscoelastic Material Characterization Using Pod Reduced-Order Modeling in Acoustic–Structure Interaction,” Comput. Methods Appl. Mech. Eng., 198(9), pp. 893–903. [CrossRef]
Pagnacco, E. , Moreau, A. , and Lemosse, D. , 2007, “ Inverse Strategies for the Identification of Elastic and Viscoelastic Material Parameters Using Full-Field Measurements,” Mater. Sci. Eng.: A, 452–453, pp. 737–745. [CrossRef]
Oliphant, T. E. , Manduca, A. , Ehman, R. L. , and Greenleaf, J. F. , 2001, “ Complex-Valued Stiffness Reconstruction for Magnetic Resonance Elastography by Algebraic Inversion of the Differential Equation,” Magn. Reson. Med., 45(2), pp. 299–310. [CrossRef] [PubMed]
Boulet, T. , Kelso, M. L. , and Othman, S. F. , 2011, “ Microscopic Magnetic Resonance Elastography of Traumatic Brain Injury Model,” J. Neurosci. Methods, 201(2), pp. 296–306. [CrossRef] [PubMed]
Papazoglou, S. , Hamhaber, U. , Braun, J. , and Sack, I. , 2008, “ Algebraic Helmholtz Inversion in Planar Magnetic Resonance Elastography,” Phys. Med. Biol., 53(12), pp. 3147–3158. [CrossRef] [PubMed]
Romano, A. J. , Abraham, P. B. , Rossman, P. J. , Bucaro, J. A. , and Ehman, R. L. , 2005, “ Determination and Analysis of Guided Wave Propagation Using Magnetic Resonance Elastography,” Magn. Reson. Med., 54(4), pp. 893–900. [CrossRef] [PubMed]
Romano, A. , Guo, J. , Prokscha, T. , Meyer, T. , Hirsch, S. , Braun, J. , Sack, I. , and Scheel, M. , 2014, “ In Vivo Waveguide Elastography: Effects of Neurodegeneration in Patients With Amyotrophic Lateral Sclerosis,” Magn. Reson. Med., 72(6), pp. 1755–1761. [CrossRef] [PubMed]
Zhang, Y. X. , Oberai, A. A. , Barbone, P. E. , and Harari, I. , 2012, “ Solution of the Time-Harmonic Viscoelastic Inverse Problem With Interior Data in Two Dimensions,” Int. J. Numer. Methods Eng., 92(13), pp. 1100–1116. [CrossRef]
Goenezen, S. , 2011, Inverse Problems in Finite Elasticity: An Application to Imaging the Nonlinear Elastic Properties of Soft Tissues, Rensselaer Polytechnic Institute, Troy, NY.
Zhu, C. , Byrd, R. H. , Lu, P. , and Nocedal, J. , 1994, “L-BFGS-B: FORTRAN Subroutines for Large-Scale Bound Constrained Optimization,” Northwestern University, Evanston, IL, Technical Report No. NAM-11. http://users.iems.northwestern.edu/~nocedal/PDFfiles/lbfgsb.pdf
Zhu, C. , Byrd, R. H. , Lu, P. , and Nocedal, J. , 1994, “L-BFGS-B: A Limited Memory FORTRAN Code for Solving Bound Constrained Optimization Problems,” Northwestern University, Evanston, IL, Report No. NAM-11.
Dorn, O. , Bertete-Aguirre, H. , Berryman, J. G. , and Papanicolaou, G. C. , 1999, “ A Nonlinear Inversion Method for 3D Electromagnetic Imaging Using Adjoint Fields,” Inverse Probl., 15(6), p. 1523. [CrossRef]
Van Houten, E. E. , Paulsen, K. D. , Miga, M. I. , Kennedy, F. E. , and Weaver, J. B. , 1999, “ An Overlapping Subzone Technique for MR-Based Elastic Property Reconstruction,” Magn. Reson. Med., 42(4), pp. 779–786. [CrossRef] [PubMed]
Green, M. A. , Bilston, L. E. , and Sinkus, R. , 2008, “ In Vivo Brain Viscoelastic Properties Measured by Magnetic Resonance Elastography,” NMR Biomed., 21(7), pp. 755–764. [CrossRef] [PubMed]
Lakes, R. , 2009, Viscoelastic Materials, Cambridge University Press, Cambridge, UK. [CrossRef]
Christensen, R. M. , 2003, Theory of Viscoelasticity, Dover Publications, Mineola, NY.
Aggarwal, A. , 2017, “ An Improved Parameter Estimation and Comparison for Soft Tissue Constitutive Models Containing an Exponential Function,” Biomech. Model. Mechanobiol., 16(4), pp. 1309–1327. [CrossRef] [PubMed]

Figures

Grahic Jump Location
Fig. 3

The horizontal centerline plot from the left to right for the exact and reconstructed modulus distribution: (a) the storage modulus plot and (b) the loss modulus plot (unit: 100 Pa)

Grahic Jump Location
Fig. 2

The reconstructed viscoelastic modulus distribution for the problem domain in Fig. 1: (a) the reconstructed storage modulus distribution when the driving frequency is 20 Hz; (b) the reconstructed loss modulus distribution when the driving frequency is 20 Hz; (c) the reconstructed storage modulus distribution when the driving frequency is 150 Hz; (d) the reconstructed loss modulus distribution when the driving frequency is 150 Hz; (e) the reconstructed storage modulus distribution when the driving frequency is 300 Hz; and (f) the reconstructed loss modulus distribution when the driving frequency is 300 Hz (unit: 100 Pa)

Grahic Jump Location
Fig. 10

The variation of displacement field along an one-dimensional vibrating string subjected to harmonic motion when the loss modulus is set to 0.5, 1.0, 1.5, 2.0. (a) The real part of the complex-valued displacement field when the driving frequency is 20 Hz; (b) the imaginary part of the complex-valued displacement field when the driving frequency is 20 Hz; (c) the real part of the complex-valued displacement field when the driving frequency is 150 Hz; (d) the imaginary part of the complex-valued displacement field when the driving frequency is 150 Hz; (e) the real part of the complex-valued displacement field when the driving frequency is 300 Hz; and (f) the imaginary part of the complex-valued displacement field when the driving frequency is 300 Hz.

Grahic Jump Location
Fig. 1

The problem domain for a simulated tissue: (a) the target storage modulus distribution and (b) the target loss modulus distribution (unit: 100 Pa)

Grahic Jump Location
Fig. 4

The problem domain for a simulated tissue with a higher loss angle: (a) the target storage modulus distribution and (b) the target loss modulus distribution (unit: 100 Pa)

Grahic Jump Location
Fig. 5

The reconstructed viscoelastic modulus distribution for the problem domain given in Fig. 4. (a) The reconstructed storage modulus distribution for a driving frequency of 20 Hz; (b) the reconstructed loss modulus distribution for a driving frequency of 20 Hz; (c) the reconstructed storage modulus distribution for a driving frequency of 150 Hz; (d) the reconstructed loss modulus distribution for a driving frequency of 150 Hz; (e) the reconstructed storage modulus distribution for a driving frequency of 300 Hz; and (f) the reconstructed loss modulus distribution for a driving frequency of 300 Hz (unit: 100 Pa).

Grahic Jump Location
Fig. 6

The horizontal centerline plot from the left to right for the exact and reconstructed modulus distribution in the case of higher loss angle (a) the storage modulus plot and (b) the loss modulus plot (unit: 100 Pa)

Grahic Jump Location
Fig. 7

The problem domain for a higher loss angle: (a) the target storage modulus distribution and (b) the target loss modulus distribution. In this case, the location of the inclusion is shifted upward closer to the excitation source.

Grahic Jump Location
Fig. 8

The reconstructed viscoelastic modulus distribution for the problem domain where the inclusion is placed upward. (a) The reconstructed storage modulus distribution when the driving frequency is 300 Hz; and (b) the reconstructed loss modulus distribution when the driving frequency is 300 Hz (unit: 100 Pa).

Grahic Jump Location
Fig. 9

The variation of displacement field along an one-dimensional vibrating string subjected to harmonic motion when the loss modulus is set to 0.1, 0.2, 0.3, 0.4. (a) the real part of the complex-valued displacement field when the driving frequency is 20 Hz; (b) the imaginary part of the complex-valued displacement field when the driving frequency is 20 Hz; (c) the real part of the complex-valued displacement field when the driving frequency is 150 Hz; (d) the imaginary part of the complex-valued displacement field when the driving frequency is 150 Hz; (e) the real part of the complex-valued displacement field when the driving frequency is 300 Hz; and (f) the imaginary part of the complex-valued displacement field when the driving frequency is 300 Hz.

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