Bathe,
K. J.
, 1996, Finite Element Procedures,
Prentice Hall,
Englewood Cliffs, NJ.

Reddy,
J. N.
, 2005, An Introduction to the Finite Element Method,
McGraw-Hill Education,
New York.

Reddy,
J. N.
, 2013, An Introduction to Nonlinear Finite Element Analysis,
Oxford University Press,
New York.

Babuška,
I.
, 1973, “
The Finite Element Method With Lagrangian Multipliers,” Numerische Math.,
20(3), pp. 179–192.

[CrossRef]
Dickinson,
R. J.
, and
Hill,
C. R.
, 1982, “
Measurement of Soft-Tissue Motion Using Correlation Between A-Scans,” Ultrasound Med. Biol.,
8(3), pp. 263–271.

[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,” Current Med. Imaging Rev.,
7(4), pp. 313–327.

[CrossRef]
Hall,
T. J.
,
Bilgen,
M.
,
Insana,
M. F.
, and
Krouskop,
T. A.
, 1997, “
Phantom Materials for Elastography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control,
44(6), pp. 1355–1365.

[CrossRef]
Hall,
T. J.
,
Zhu,
Y.
, and
Spalding,
C. S.
, 2003, “
In Vivo Real-Time Freehand Palpation Imaging,” Ultrasound Med. Biol.,
29(3), pp. 427–435.

[CrossRef] [PubMed]
Jiang,
J.
, and
Hall,
T. J.
, 2009, “
A Generalized Speckle Tracking Algorithm for Ultrasonic Strain Imaging Using Dynamic Programming,” Ultrasound Med. Biol.,
35(11), pp. 1863–1879.

[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. H,
213(3), pp. 203–233.

[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]
Pavan,
T. Z.
,
Madsen,
E. L.
,
Frank,
G. R.
,
Adilton,
O. C. A.
, and
Hall,
T. J.
, 2010, “
Nonlinear Elastic Behavior of Phantom Materials for Elastography,” Phys. Med. Biol.,
55(9), pp. 2679–2692.

[CrossRef] [PubMed]
Regner,
D. M.
,
Hesley,
G. K.
,
Hangiandreou,
N. J.
,
Morton,
M. J.
,
Nordland,
M. R.
,
Meixner,
D. D.
,
Hall,
T. J.
,
Farrell,
M. A.
,
Mandrekar,
J. N.
,
Harmsen,
W. S.
, and
Charboneau,
J. W.
, 2006, “
Breast Lesions: Evaluation With U.S. Strain Imaging–Clinical Experience of Multiple Observers,” Radiology,
238(2), pp. 425–437.

[CrossRef] [PubMed]
Wilson,
L. S.
, and
Robinson,
D. E.
, 1982, “
Ultrasonic Measurement of Small Displacements and Deformations of Tissue,” Ultrason. Imaging,
4(1), pp. 71–82.

[CrossRef] [PubMed]
Zhu,
Y.
, and
Hall,
T. J.
, 2002, “
A Modified Block Matching Method for Real-Time Freehand Strain Imaging,” Ultrason. Imaging,
24(3), pp. 161–176.

[CrossRef] [PubMed]
Atay,
S. M.
,
Kroenke,
C. D.
,
Sabet,
A.
, and
Bayly,
P. V.
, 2008, “
Measurement of the Dynamic Shear Modulus of Mouse Brain Tissue In Vivo by Magnetic Resonance Elastography,” ASME J. Biomech. Eng.,
130(2), p. 021013.

[CrossRef]
Kwon,
O. I.
,
Park,
C.
,
Nam,
H. S.
,
Woo,
E. J.
,
Seo,
J. K.
,
Glaser,
K. J.
,
Manduca,
A.
, and
Ehman,
R. L.
, 2009, “
Shear Modulus Decomposition Algorithm in Magnetic Resonance Elastography,” IEEE Trans. Med. Imaging,
28(10), pp. 1526–1533.

[CrossRef] [PubMed]
Muthupillai,
R.
,
Lomas,
D.
,
Rossman,
P.
,
Greenleaf,
J.
,
Manduca,
A.
, and
Ehman,
R.
, 1995, “
Magnetic Resonance Elastography by Direct Visualization of Propagating Acoustic Strain Waves,” Science,
269(5232), pp. 1854–1857.

[CrossRef] [PubMed]
Neu,
C. P.
,
Arastu,
H. F.
,
Curtiss,
S.
, and
Reddi,
A. H.
, 2009, “
Characterization of Engineered Tissue Construct Mechanical Function by Magnetic Resonance Imaging,” J. Tissue Eng. Regener. Med.,
3(6), pp. 477–485.

[CrossRef]
Othman,
S. F.
,
Curtis,
E. T.
,
Plautz,
S. A.
,
Pannier,
A. K.
,
Butler,
S. D.
, and
Xu,
H.
, 2012, “
MR Elastography Monitoring of Tissue-Engineered Constructs,” NMR Biomed.,
25(3), pp. 452–463.

[CrossRef] [PubMed]
Othman,
S. F.
,
Xu,
H.
,
Royston,
T. J.
, and
Magin,
R. L.
, 2005, “
Microscopic Magnetic Resonance Elastography (

*μ*MRE),” Magn. Reson. Med.,
54(3), pp. 605–615.

[CrossRef] [PubMed]
Sack,
I.
,
Buntkowsky,
G.
,
Bernarding,
J.
, and
Braun,
J.
, 2001, “
Magnetic Resonance Elastography: A Method for the Noninvasive and Spatially Resolved Observation of Phase Transitions in Gels,” J. Am. Chem. Soc.,
123(44), pp. 11087–11088.

[CrossRef] [PubMed]
Shah,
N. S.
,
Kruse,
S. A.
,
Lager,
D. J.
,
Farell-Baril,
G.
,
Lieske,
J. C.
,
King,
B. F.
, and
Ehman,
R.
, 2004, “
Evaluation of Renal Parenchymal Disease in a Rat Model With Magnetic Resonance Elastography,” Magn. Reson. Med.,
52(1), pp. 56–64.

[CrossRef] [PubMed]
Peng,
L.
,
Xin,
Y.
,
Liang,
S.
,
Aiping,
L.
,
Rugonyi,
S.
, and
Wang,
R. K.
, 2011, “
Measurement of Strain and Strain Rate in Embryonic Chick Heart In Vivo Using Spectral Domain Optical Coherence Tomography,” IEEE Trans. Biomed. Eng.,
58(8), pp. 2333–2338.

[CrossRef]
Schmitt,
J.
, 1998, “
OCT Elastography: Imaging Microscopic Deformation and Strain of Tissue,” Opt. Express,
3(6), pp. 199–211.

[CrossRef] [PubMed]
Falzon,
G.
,
Pearson,
S.
, and
Murison,
R.
, 2008, “
Analysis of Collagen Fibre Shape Changes in Breast Cancer,” Phys. Med. Biol.,
53(23), pp. 6641–6652.

[CrossRef] [PubMed]
Burnside,
E. S.
,
Hall,
T. J.
,
Sommer,
A. M.
,
Hesley,
G. K.
,
Sisney,
G. A.
,
Svensson,
W. E.
, and
Hangiandreou,
N. J.
, 2007, “
Ultrasound Strain Imaging to Improve the Decision to Biopsy Solid Breast Masses,” Radiology,
245(2), pp. 401–410.

[CrossRef] [PubMed]
Garra,
B. S.
,
Cespedes,
E. I.
,
Ophir,
J.
,
Spratt,
S. R.
,
Zuurbier,
R. A.
,
Magnant,
C. M.
, and
Pennanen,
M. F.
, 1997, “
Elastography of Breast Lesions: Initial Clinical Results,” Radiology,
202(1), pp. 79–86.

[CrossRef] [PubMed]
Hiltawsky,
K. M.
,
Kruger,
M.
,
Starke,
C.
,
Heuser,
L.
,
Ermert,
H.
, and
Jensen,
A.
, 2001, “
Freehand Ultrasound Elastography of Breast Lesions: Clinical Results,” Ultrasound Med. Biol.,
27(11), pp. 1461–1469.

[CrossRef] [PubMed]
Itoh,
A.
,
Ueno,
E.
,
Tohno,
E.
,
Kamma,
H.
,
Takahashi,
H.
,
Shiina,
T.
,
Yamakawa,
M.
, and
Matsumura,
T.
, 2006, “
Breast Disease: Clinical Application of U.S. Elastography for Diagnosis,” Radiol. Soc. North Am.,
239(2), pp. 341–350.

de Korte,
C. L.
,
van der Steen,
A. F.
,
Cespedes,
E. I.
, and
Pasterkamp,
G.
, 1998, “
Intravascular Ultrasound Elastography in Human Arteries: Initial Experience In Vitro,” Ultrasound Med. Biol.,
24(3), pp. 401–408.

[CrossRef] [PubMed]
Schaar,
J. A.
,
de Korte,
C. L.
,
Mastik,
F.
,
Strijder,
C.
,
Pasterkamp,
G.
,
Boersma,
E.
,
Serruys,
P. W.
, and
van der Steen,
A. F. W.
, 2003, “
Characterizing Vulnerable Plaque Features With Intravascular Elastography,” Circulation,
108(21), pp. 2636–2641.

[CrossRef] [PubMed]
Shiina,
T.
,
Nitta,
N.
, and
Yamagishi,
M.
, 2002, “
Coronary Artery Characterization Based on Tissue Elasticity Imaging—In Vivo Assessment,” IEEE Ultrasonics Symposium, Vol.
1852, pp. 1855–1858.

Skovoroda,
A. R.
,
Lubinski,
L. A.
,
Emelianov,
S. Y.
, and
O'Donnell,
M.
, 1999, “
Reconstructive Elasticity Imaging for Large Deformations,” IEEE Trans. Ultrason. Ferroelectr Freq Control,
46(3), pp. 523–535.

[CrossRef] [PubMed]
Zhu,
Y.
,
Hall,
T.
, and
Jiang,
J.
, 2003, “
A Finite-Element Approach for Young's Modulus Reconstruction,” IEEE Trans. Med. Imaging,
22(7), pp. 890–901.

[CrossRef] [PubMed]
Prusa,
V.
,
Rajagopal,
K. R.
, and
Saravanan,
U.
, 2013, “
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,” ASME J. Biomech. Eng.,
135(8), p. 081004.

[CrossRef]
Doyley,
M. M.
,
Meaney,
P. M.
, and
Bamber,
J. C.
, 2000, “
Evaluation of an Iterative Reconstruction Method for Quantitative Elastography,” Phys. Med. Biol.,
45(6), pp. 1521–1540.

[CrossRef] [PubMed]
Kallel,
F.
, and
Bertrand,
M.
, 1996, “
Tissue Elasticity Reconstruction Using Linear Perturbation Method,” IEEE Trans. Med. Imaging,
15(3), pp. 299–313.

[CrossRef] [PubMed]
Barbone,
P.
,
Oberai,
A. A.
,
Bamber,
J. C.
,
Berry,
G. P.
,
Dord,
J.
,
Ferreira,
E. R.
,
Goenezen,
S.
, and
Hall,
T.
, 2014, “
Biomechanical Imaging: Elastography Beyond Young's Modulus,” CRC Handbook of Imaging in Biological Mechanics,
CRC Press,
Boca Raton, FL.

Goenezen,
S.
, 2011, “
Inverse Problems in Finite Elasticity: An Application to Imaging the Nonlinear Elastic Properties of Soft Tissues,” Ph.D. dissertation,
Rensselaer Polytechnic Institute,
Troy, NY.

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.

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,” IEEE Trans. Med. Imaging,
31(8), pp. 1628–1637.

[CrossRef] [PubMed]
Goenezen,
S.
,
Oberai,
A. A.
,
Dord,
J.
,
Sink,
Z.
, and
Barbone,
P.
, 2011, “
Nonlinear Elasticity Imaging,” IEEE 37th Annual Northeast Bioengineering Conference (NEBEC), Troy, NY, Apr. 1–3.

Gokhale,
N. H.
,
Barbone,
P.
, and
Oberai,
A. A.
, 2008, “
Solution of the Nonlinear Elasticity Imaging Inverse Problem: The Compressible Case,” Inverse Probl.,
24(4), pp. 1406–1420.

[CrossRef]
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,
Cham, Switzerland.

Byrd,
R. H.
,
Lu,
P.
,
Nocedal,
J.
, and
Zhu,
C.
, 1995, “
A Limited Memory Algorithm for Bound Constrained Optimization,” SIAM J. Sci. Comput.,
16(5), pp. 1190–1208.

[CrossRef]
Zhu,
C.
,
Byrd,
R. H.
,
Lu,
P.
, and
Nocedal,
J.
, 1994, “
L-BFGS-B: FORTRAN Subroutines for Large Scale Bound Constrained Optimization',” EECS Department, Northwestern University, Evanston, IL, Technical Report No. NAM-11.

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,” EECS Department, Northwestern University, Evanston, IL, Technical 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), pp. 1523–1558.

Tyagi,
M.
,
Goenezen,
S.
,
Barbone,
P.
, and
Oberai,
A. A.
, 2014, “
Algorithms for Quantitative Quasi-Static Elasticity Imaging Using Force Data,” Int. J. Numer. Methods Biomed. Eng.,
30(12), pp. 1421–1436.

[CrossRef]
Hughes,
T. J. R.
,
Franca,
L. P.
, and
Balestra,
M.
, 1986, “
A New Finite Element Formulation for Computational Fluid Dynamics: V. Circumventing the Babuska-Brezzi Condition: A Stable Petrov-Galerkin Formulation of the Stokes Problem Accommodating Equal-Order Interpolations,” Comput. Methods Appl. Mech. Eng.,
59(1), pp. 85–99.

[CrossRef]
Maniatty,
A. M.
,
Liu,
Y.
,
Klaas,
O.
, and
Shephard,
M. S.
, 2002, “
Higher Order Stabilized Finite Element Method for Hyperelastic Finite Deformation,” Comput. Methods Appl. Mech. Eng.,
191(13–14), pp. 1491–1503.

[CrossRef]
Oberai,
A. A.
,
Gokhale,
N. H.
,
Goenezen,
S.
,
Barbone,
P.
,
Hall,
T.
,
Sommer,
A. M.
, and
Jiang,
J.
, 2009, “
Linear and Nonlinear Elasticity Imaging of Tissue In-Vivo: Demonstration of Feasibility,” Phys. Med. Biol.,
54(5), pp. 1191–1207.

[CrossRef] [PubMed]
Calvetti,
D.
,
Morigi,
S.
,
Reichel,
L.
, and
Sgallari,
F.
, 2000, “
Tikhonov Regularization and the L-Curve for Large Discrete Ill-Posed Problems,” J. Comput. Appl. Math.,
123(2), pp. 423–446.

[CrossRef]
Chvetsov,
A. V.
, 2005, “
L-Curve Analysis of Radiotherapy Optimization Problems,” Med. Phys.,
32(8), pp. 2598–2605.

[CrossRef] [PubMed]
Vogel,
C. R.
, 1996, “
Non-Convergence of the L-Curve Regularization Parameter Selection Method,” Inverse Probl.,
12(4), pp. 535–547.

[CrossRef]
Vogel,
C. R.
, 2002, Computational Methods for Inverse Problems,
Society for Industrial and Applied Mathematics,
Philadelphia, PA.

Anzengruber,
S.
, and
Ramlau,
R.
, 2010, “
Morozov's Discrepancy Principle for Tikhonov-Type Functionals With Nonlinear Operators,” Inverse Probl.,
26(2), p. 025001.

Bonesky,
T.
, 2009, “
Morozov's Discrepancy Principle and Tikhonov-Type Functionals,” Inverse Probl.,
25(1), p. 015015.

Frick,
K.
,
Lorenz,
D.
, and
Resmerita,
E.
, 2011, “
Morozov's Principle for the Augmented Lagrangian Method Applied to Linear Inverse Problems,” Multiscale Model. Simul.,
9(4), pp. 1528–1548.

[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), pp. 297–313.

[CrossRef]
Richards,
M. S.
, and
Doyley,
M. M.
, 2011, “
Investigating the Impact of Spatial Priors on the Performance of Model-Based IVUS Elastography,” Phys. Med. Biol.,
56(22), pp. 7223–7246.

[CrossRef] [PubMed]