Abstract
We briefly review the iterative solution of the inverse problem in elasticity, which is posed as a constrained optimization method. The objective function minimizes the discrepancy between a measured and a computed displacement field in the L-2 norm and is subject to the static equations of equilibrium in elasticity. We realize that this inverse formulation is sensitive to Dirichlet and Neumann boundary conditions, i.e., sensitive to varying spatial deformations in the region of interest. This problem arises in particular, when solving the inverse problem for more than one inclusion in a homogeneous background, where the inclusions represent diseased tissues such as cysts, benign tumors, malignant tumors, etc. In order to address this issue, we propose to introduce a new formulation of the objective function, where the displacement correlation term is spatially weighted. We refer to this new formulation as the spatially weighted objective function and show that it improves the uniqueness of the inverse problem solution.
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References
Barbone, P.E., Oberai, A.A., Bamber, J.C., Berry, G.P., Dord, J.F., Ferreira, E.R., Goenezen, S., Hall, T.J.: Nonlinear and poroelastic biomechanical imaging: elastography beyond Youngs modulus. In: Neu, C., Genin, G. (eds.) CRC Handbook of Imaging in Biological Mechanics. CRC Press and Taylor & Francis, New York (2014)
Fung, Y.C.: Biomechanics: Mechanical Properties of Living Tissues, p. 433. Springer, New York (1981)
Martins, P.A.L.S., Natal Jorge, R.M., Ferreira, A.J.M.: A comparative study of several material models for prediction of hyperelastic properties: application to silicone-rubber and soft tissues. Strain 42(3), 135–147 (2006)
Garra, B.S., Cespedes, E.I., Ophir, J., Spratt, S.R., Zuurbier, R.A., Magnant, C.M., Pennanen, M.F.: Elastography of breast lesions: initial clinical results. Radiology 202(1), 79–86 (1997)
Goenezen, S., Dord, J.F., Sink, Z., Barbone, P., Jiang, J., Hall, T.J., Oberai, A.A.: Linear and nonlinear elastic modulus imaging: an application to breast cancer diagnosis. IEEE Trans. Med. Imaging 31(8), 1628–1637 (2012)
de Korte, C.L., van der Steen, A.F., Cespedes, E.I., Pasterkamp, G.: Intravascular ultrasound elastography in human arteries: initial experience in vitro. Ultrasound Med. Biol. 24(3), 401–408 (1998)
Ophir, J., Alam, S.K., Garra, B., Kallel, F., Konofagou, E., Krouskop, T., Varghese, T.: Elastography: ultrasonic estimation and imaging of the elastic properties of tissues. Proc. Inst. Mech. Eng. H 213(3), 203–233 (1999)
Ophir, J., Cespedes, I., Ponnekanti, H., Yazdi, Y., Li, X.: Elastography: a quantitative method for imaging the elasticity of biological tissues. Ultrason. Imaging 13(2), 111–134 (1991)
Skovoroda, A.R., Lubinski, L.A., Emelianov, S.Y., O’Donnell, M.: Reconstructive elasticity imaging for large deformations. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46(3), 523–535 (1999)
Prusa, V., Rajagopal, K.R., Saravanan, U.: Fidelity of the estimation of the deformation gradient from data deduced from the motion of markers placed on a body that is subject to an inhomogeneous deformation field. J. Biomech. Eng. 135(8), 081004 (2013)
Doyley, M.M., Meaney, P.M., Bamber, J.C.: Evaluation of an iterative reconstruction method for quantitative elastography. Phys. Med. Biol. 45(6), 1521–1540 (2000)
Goenezen, S., Barbone, P., Oberai, A.A.: Solution of the nonlinear elasticity imaging inverse problem: the incompressible case. Comput. Methods Appl. Mech. Eng. 200(13–16), 1406–1420 (2011)
Goenezen, S., Oberai, A.A., Dord, J., Sink, Z., Barbone, P.: In: Nonlinear Elasticity Imaging Bioengineering Conference (NEBEC), 2011 IEEE 37th Annual Northeast, pp. 1–2 (2011)
Hall, T., Barbone, P.E., Oberai, A.A., Jiang, J., Dord, J., Goenezen, S., Fisher, T.: Recent results in nonlinear strain and modulus imaging. Curr. Med. Imaging Rev. 7(4), 313–327 (2011)
Kallel, F., Bertrand, M.: Tissue elasticity reconstruction using linear perturbation method. IEEE Trans. Med. Imaging 15(3), 299–313 (1996)
Oberai, A.A., Gokhale, N.H., Feijóo, G.R.: Solution of inverse problems in elasticity imaging using the adjoint method. Inverse Prob. 19(2), 297 (2003)
Maniatty, A.M., Liu, Y., Klaas, O., Shephard, M.S.: Higher order stabilized finite element method for hyperelastic finite deformation. Comput. Methods Appl. Mech. Eng. 191(13–14), 1491–1503 (2002)
Vogel, C.R.: Computational Methods for Inverse Problems. Society for Industrial and Applied Mathematics, Philadelphia (2002)
Zhu, C., Byrd, R.H., Lu, P., Nocedal, J.: L-BFGS-B: a limited memory FORTRAN code for solving bound constrained optimization problems. Tech. Report, NAM-11, EECS Department, Northwestern University (1994)
Goenezen, S.: Inverse problems in finite elasticity: an application to imaging the nonlinear elastic properties of soft tissues. Ph.D. Dissertation (2011)
Tyagi, M., Goenezen, S., Barbone, P.E., Oberai, A.A.: Algorithms for quantitative quasi-static elasticity imaging using force data. Int. J. Numer. Methods Biomed. Eng. 30(12) (2014)
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Mei, Y., Goenezen, S. (2015). Spatially Weighted Objective Function to Solve the Inverse Elasticity Problem for the Elastic Modulus. In: Doyle, B., Miller, K., Wittek, A., Nielsen, P. (eds) Computational Biomechanics for Medicine. Springer, Cham. https://doi.org/10.1007/978-3-319-15503-6_5
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DOI: https://doi.org/10.1007/978-3-319-15503-6_5
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