Home Mathematics An inverse problem in elastography involving Lamé systems
Article
Licensed
Unlicensed Requires Authentication

An inverse problem in elastography involving Lamé systems

  • Enrique Fernández-Cara ORCID logo and Faustino Maestre ORCID logo EMAIL logo
Published/Copyright: February 14, 2018

Abstract

This paper deals with some inverse problems for the linear elasticity system with origin in elastography: we try to identify the material coefficients from some extra information on (a part of) the boundary. In our main result, we assume that the total variation of the coefficient matrix is a priori bounded. We reformulate the problem as the minimization of a function in an appropriate constraint set. We prove that this extremal problem possesses at least one solution with the help of some regularity results. Two crucial ingredients are a Meyers-like theorem that holds in the context of linear elasticity and a nonlinear interpolation result by Luc Tartar. We also perform some numerical experiments that provide satisfactory results. To this end, we apply the Augmented Lagrangian algorithm, completed with a limited-memory BFGS subalgorithm. Finally, on the basis of these experiments, we illustrate the influence of the starting guess and the errors in the data on the behavior of the iterates.

Funding statement: Partially supported by grants MTM2016-76990-P (DGI-MICINN, Spain) and MTM2014-53309-P (DGI-MINECO, Spain).

References

[1] G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity, Comm. Partial Differential Equations 26 (2001), no. 9–10, 1787–1810. 10.1081/PDE-100107459Search in Google Scholar

[2] H. Ammari, E. Beretta, E. Francini, H. Kang and M. Lim, Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case, J. Math. Pures Appl. (9) 94 (2010), no. 3, 322–339. 10.1016/j.matpur.2010.02.001Search in Google Scholar

[3] P. E. Barbone and N. H. Gokhale, Elastic modulus imaging: on the uniqueness and nonuniqueness of the elastography inverse problem in two dimensions, Inverse Problems 20 (2004), no. 1, 283–296. 10.1088/0266-5611/20/1/017Search in Google Scholar

[4] M. Bellassoued, O. Imanuvilov and M. Yamamoto, Inverse problem of determining the density and two Lamé coefficients by boundary data, SIAM J. Math. Anal. 40 (2008), no. 1, 238–265. 10.1137/070679971Search in Google Scholar

[5] M. Bellassoued, D. Jellali and M. Yamamoto, Lipschitz stability in an inverse problem for a hyperbolic equation with a finite set of boundary data, Appl. Anal. 87 (2008), no. 10–11, 1105–1119. 10.1080/00036810802369231Search in Google Scholar

[6] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, AMS Chelsea Publishing, Providence, 2011. 10.1090/chel/374Search in Google Scholar

[7] J. Bergh and J. Löfström, Interpolations Spaces, Springer, Berlin, 1976. 10.1007/978-3-642-66451-9Search in Google Scholar

[8] E. G. Birgin and J. M. Martínez, Improving ultimate convergence of an augmented Lagrangian method, Optim. Methods Softw. 23 (2008), no. 2, 177–195. 10.1080/10556780701577730Search in Google Scholar

[9] Y. A. Brudnyĭ and N. Y. Krugljak, Interpolation Functors and Interpolation Spaces. Vol. I, North-Holland Math. Libr. 47, North-Holland Publishing, Amsterdam, 1991. Search in Google Scholar

[10] P. Donato and H. Haddadou, Meyers type estimates in elasticity and applications to H-convergence, Adv. Math. Sci. Appl. 16 (2006), no. 2, 537–567. Search in Google Scholar

[11] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren Math. Wiss. 219, Springer, Berlin, 1976. 10.1007/978-3-642-66165-5Search in Google Scholar

[12] L. Escauriaza, Unique continuation for the system of elasticity in the plane, Proc. Amer. Math. Soc. 134 (2006), no. 7, 2015–2018. 10.1090/S0002-9939-05-08413-3Search in Google Scholar

[13] G. Eskin, Uniqueness and nonuniqueness in inverse hyperbolic problems and the black hole phenomenon, Around the Research of Vladimir Maz’ya. III, Int. Math. Ser. (N. Y.) 13, Springer, New York (2010), 77–104. 10.1007/978-1-4419-1345-6_4Search in Google Scholar

[14] E. Fernández-Cara and F. Maestre, On some inverse problems arising in elastography, Inverse Problems 28 (2012), no. 8, Article ID 085001. 10.1088/0266-5611/28/8/085001Search in Google Scholar

[15] M. Ikehata, G. Nakamura and M. Yamamoto, Uniqueness in inverse problems for the isotropic Lamé system, J. Math. Sci. Univ. Tokyo 5 (1998), no. 4, 627–692. Search in Google Scholar

[16] O. Imanuvilov, V. Isakov and M. Yamamoto, An inverse problem for the dynamical Lamé system with two sets of boundary data, Comm. Pure Appl. Math. 56 (2003), no. 9, 1366–1382. 10.1002/cpa.10097Search in Google Scholar

[17] O. Y. Imanuvilov and M. Yamamoto, Carleman estimates for the non-stationary Lamé system and the application to an inverse problem, ESAIM Control Optim. Calc. Var. 11 (2005), no. 1, 1–56. 10.1051/cocv:2004030Search in Google Scholar

[18] O. Y. Imanuvilov and M. Yamamoto, On reconstruction of Lamé coefficients from partial Cauchy data, J. Inverse Ill-Posed Probl. 19 (2011), no. 6, 881–891. 10.1515/jiip.2011.060Search in Google Scholar

[19] V. Isakov, An inverse problem for the dynamical Lame system with two sets of local boundary data, Control Theory of Partial Differential Equations, Lect. Notes Pure Appl. Math. 242, Chapman & Hall/CRC, Boca Raton (2005), 101–109. 10.1201/9781420028317.ch7Search in Google Scholar

[20] V. Isakov, J.-N. Wang and M. Yamamoto, An inverse problem for a dynamical Lamé system with residual stress, SIAM J. Math. Anal. 39 (2007/08), no. 4, 1328–1343. 10.1137/060669115Search in Google Scholar

[21] W. Khaled, S. Reichling, O. T. Bruhns and H. Ermert, Ultrasonic strain imaging and reconstructive elastography for biological tissue, Ultrasonics 44 (2006), e199–e202. 10.1016/j.ultras.2006.06.007Search in Google Scholar

[22] E. E. Konofagou, Poroelastography: Imaging the poroelastic properties of tissues, Ultrasound Med. Biol. 27 (2001), 1387–1397. 10.1016/S0301-5629(01)00433-1Search in Google Scholar PubMed

[23] C.-L. Lin, G. Nakamura, G. Uhlmann and J.-N. Wang, Quantitative strong unique continuation for the Lamé system with less regular coefficients, Methods Appl. Anal. 18 (2011), no. 1, 85–92. 10.4310/MAA.2011.v18.n1.a5Search in Google Scholar

[24] C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with Lipschitz coefficients, Math. Ann. 331 (2005), no. 3, 611–629. 10.1007/s00208-004-0597-zSearch in Google Scholar

[25] D. C. Liu and J. Nocedal, On the limited memory BFGS method for large scale optimization, Math. Program. 45 (1989), no. 3, 503–528. 10.1007/BF01589116Search in Google Scholar

[26] A. L. McKnight, J. L. Kugel, P. J. Rossman, A. Manduca, L. C. Hartmann and R. L. Ehman, Elastography of breast cancer: Preliminary results, Amer. J. Roentgenology 178 (2002), 1411–1417. 10.2214/ajr.178.6.1781411Search in Google Scholar PubMed

[27] J. R. McLaughlin and J.-R. Yoon, Unique identifiability of elastic parameters from time-dependent interior displacement measurement, Inverse Problems 20 (2004), no. 1, 25–45. 10.1088/0266-5611/20/1/002Search in Google Scholar

[28] N. G. Meyers, An Lpe-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3) 17 (1963), 189–206. Search in Google Scholar

[29] R. Muthupillai, D. J. Lomas, P. J. Rossman, J. F. Greenleaf, A. Manduca and R. L. Ehman, Magnetic resonance elastography by direct visualization of propagating acoustic strain waves, Science 269 (1995), 1854–1857. 10.1126/science.7569924Search in Google Scholar PubMed

[30] J. Nocedal, Updating quasi-Newton matrices with limited storage, Math. Comp. 35 (1980), no. 151, 773–782. 10.1090/S0025-5718-1980-0572855-7Search in Google Scholar

[31] J. Nocedal and S. J. Wright, Numerical Optimization, Springer Ser. Oper. Res, Springer, New York, 1999. 10.1007/b98874Search in Google Scholar

[32] O. A. Oleînik, A. S. Shamaev and G. A. Yosifian, Mathematical problems in elasticity and homogenization, Stud. Math. Appl. 26, North-Holland Publishing, Amsterdam, 1992. Search in Google Scholar

[33] J. Ophir, I. Cespedes, H. Ponnekanti, Y. Yazdi and X. Li, Elastography: A quantitative method for imaging the elasticity of biological tissues, Ultrasonic Imag. 13 (1991), 111–134. 10.1177/016173469101300201Search in Google Scholar PubMed

[34] P. R. Perriñez, F. E. Kennedy, E. E. W. Van Houten, J. B. Weaver and K. D. Paulsen, Modeling of soft poroelastic tissue in time-harmonic MR elastography, IEEE Trans. Biomed. Eng. 56 (2009), no. 3, 598–608. 10.1109/TBME.2008.2009928Search in Google Scholar PubMed PubMed Central

[35] J. Simon, Compact sets in Lp(0,T;B) spaces, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. 10.1007/BF01762360Search in Google Scholar

[36] R. Sinkus, J. Lorenzen, D. Schrader, M. Lorenzen, D. Dargatz and D. Holz, High-resolution tensor MR elastography for breast tumour detectio, Phys. Med. Biol. 45 (2000), 1649–1664. 10.1088/0031-9155/45/6/317Search in Google Scholar

[37] R. Sinkus, M. Tanter, T. Xydeas, S. Catheline, J. Bercoff and M. Fink, Viscoelastic shear properties of in vivo breast lesions measured by MR elastography, Magn. Resonance Imaging 23 (2005), 159–165. 10.1016/j.mri.2004.11.060Search in Google Scholar

[38] L. Tartar, Interpolation non linéaire et régularité, J. Funct. Anal. 9 (1972), 469–489. 10.1016/0022-1236(72)90022-5Search in Google Scholar

[39] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lect. Notes Unione Mat. Ital. 3, Springer, Berlin, 2007. Search in Google Scholar

Received: 2017-06-21
Revised: 2017-12-09
Accepted: 2018-01-25
Published Online: 2018-02-14
Published in Print: 2018-10-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 10.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jiip-2017-0065/pdf
Scroll to top button