We describe a modeling methodology intended as a preliminary step in the identification of appropriate constitutive frameworks for the time-dependent response of biological tissues. The modeling approach comprises a customizable rheological network of viscous and elastic elements governed by user-defined 1D constitutive relationships. The model parameters are identified by iterative nonlinear optimization, minimizing the error between experimental and model-predicted structural (load-displacement) tissue response under a specific mode of deformation. We demonstrate the use of this methodology by determining the minimal rheological arrangement, constitutive relationships, and model parameters for the structural response of various soft tissues, including ex vivo perfused porcine liver in indentation, ex vivo porcine brain cortical tissue in indentation, and ex vivo human cervical tissue in unconfined compression. Our results indicate that the identified rheological configurations provide good agreement with experimental data, including multiple constant strain rate load/unload tests and stress relaxation tests. Our experience suggests that the described modeling framework is an efficient tool for exploring a wide array of constitutive relationships and rheological arrangements, which can subsequently serve as a basis for 3D constitutive model development and finite-element implementations. The proposed approach can also be employed as a self-contained tool to obtain simplified 1D phenomenological models of the structural response of biological tissue to single-axis manipulations for applications in haptic technologies.

1.
Ferry
,
J.
, 1970,
Viscoelastic Properties of Polymers
,
Wiley
,
New York
.
2.
Fung
,
Y. C.
, 1993,
Biomechanics: Mechanical Properties of Living Tissues
,
2nd ed.
,
Springer-Verlag
,
New York
.
3.
Ward
,
I. M.
, 1983,
Mechanical Properties of Solid Polymers
,
Wiley
,
New York
.
4.
Chen
,
E. J.
,
Novakofski
,
J.
,
Jenkins
,
W. K.
, and
O’Brien
,
W. D.
, Jr.
, 1996, “
Young’s Modulus Measurements of Soft Tissues With Application to Elasticity Imaging
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
0885-3010,
43
, pp.
191
194
.
5.
Kalanovic
,
D.
,
Ottensmeyer
,
M. P.
,
Gross
,
J.
,
Buess
,
G.
, and
Dawson
,
S. L.
, 2003, MMVR.
6.
Ottensmeyer
,
M. P.
, 2002, “
In vivo Measurement of Solid Organ Visco-Elastic Properties
,”
Stud. Health Technol. Inform.
0926-9630,
85
, pp.
328
333
.
7.
Bischoff
,
J. E.
,
Arruda
,
E. M.
, and
Grosh
,
K.
, 2000, “
Finite Element Modeling of Human Skin Using an Isotropic, Nonlinear Elastic Constitutive Model
,”
J. Biomech.
0021-9290,
33
, pp.
645
652
.
8.
Brouwer
,
I.
,
Ustin
,
J.
,
Bentley
,
L.
,
Sherman
,
A.
,
Dhruv
,
N.
, and
Tendick
,
F.
, 2001, “
Measuring In Vivo Animal Soft Tissue Properties for Haptic Modeling in Surgical Simulation
,”
Stud. Health Technol. Inform.
0926-9630,
81
, pp.
69
74
.
9.
Brown
,
J. D.
,
Rosen
,
J.
,
Kim
,
Y. S.
,
Chang
,
L.
,
Sinanan
,
M. N.
, and
Hannaford
,
B.
, 2003, “
In-Vivo and In-Situ Compressive Properties of Procine Abdominal Soft Tissues
,”
Stud. Health Technol. Inform.
0926-9630,
94
, pp.
26
32
.
10.
Carter
,
F. J.
,
Frank
,
T. G.
,
Davies
,
P. J.
,
McLean
,
D.
, and
Cuschieri
,
A.
, 2001, “
Measurements and Modelling of the Compliance of Human and Porcine Organs
,”
Med. Image Anal.
1361-8415,
5
, pp.
231
236
.
11.
Davies
,
P. J.
,
Carter
,
F. J.
, and
Cuschieri
,
A.
, 2002, “
Mathematical Modelling for Keyhole Surgery Simulations: A Biomechanical Model for Spleen Tissue
,”
IMA J. Appl. Math.
0272-4960,
67
, pp.
41
67
.
12.
Jordan
,
P.
,
Socrate
,
S.
,
Zickler
,
T. ,
and
Howe
,
R.
, 2009, “
Constitutive Modeling of Porcine Liver in Indentation Using 3D Ultrasound Imaging
,”
J. Mech. Behav. Biomed. Mater.
1751-6161,
2
, pp.
192
201
.
13.
Kauer
,
M.
,
Vuskovic
,
V.
,
Dual
,
J.
,
Szekely
,
G.
, and
Bajka
,
M.
, 2002, “
Inverse Finite Element Characterization of Soft Tissues
,”
Med. Image Anal.
1361-8415,
6
, pp.
275
287
.
14.
Kerdok
,
A. E.
,
Ottensmeyer
,
M. P.
, and
Howe
,
R. D.
, 2006, “
Effects of Perfusion on the Viscoelastic Characteristics of Liver
,”
J. Biomech.
0021-9290,
39
, pp.
2221
2231
.
15.
Kerdok
,
A. E.
, 2006, “
Characterizing the Nonlinear Mechanical Response of Liver to Surgical Manipulation
,” Ph.D. thesis, Harvard University, Cambridge, MA.
16.
Kim
,
J.
,
Tay
,
B.
,
Stylopoulos
,
N.
,
Rattner
,
D.
, and
Srinivasan
,
M.
, 2003,
Medical Image Computing and Computer Assisted Intervention
,
Springer
,
Berlin/Heidelberg
.
17.
Miller
,
K.
, 2000, “
Constitutive Modelling of Abdominal Organs
,”
J. Biomech.
0021-9290,
33
, pp.
367
373
.
18.
Nava
,
A.
,
Mazza
,
E.
,
Furrer
,
M.
,
Villiger
,
P.
, and
Reinhart
,
W. H.
, 2008, “
In Vivo Mechanical Characterization of Human Liver
,”
Med. Image Anal.
1361-8415,
12
, pp.
203
216
.
19.
Rubin
,
M. B.
, and
Bodner
,
S. R.
, 2002, “
A Three-Dimensional Nonlinear Model for Dissipative Response of Soft Tissue
,”
Int. J. Solids Struct.
0020-7683,
39
, pp.
5081
5099
.
20.
Socrate
,
S.
, and
Boyce
,
M. C.
, 2001,
Proceedings of the ASME 2001 Bioengineering Conference
, Vol.
50
, pp.
597
598
.
21.
Suh
,
J. -K.
, and
Spilker
,
R. L.
, 1994, “
Indentation Analysis of Biphasic Articular Cartilage: Nonlinear Phenomena Under Finite Deformation
,”
ASME J. Biomech. Eng.
0148-0731,
116
, pp.
1
9
.
22.
Febvay
,
S.
,
Socrate
,
S.
, and
House
,
M.
, 2003,
ASME International Mechanical Engineering Congress and Exposition (IMECE)
, Washington, DC.
23.
Hu
,
T.
, and
Desai
,
J. P.
, 2004,
International Symposium on Medical Simulation
, Cambridge, MA, p.
294
.
24.
Miller
,
K.
, 2001, “
How to Test Very Soft Biological Tissues in Extension?
,”
J. Biomech.
0021-9290,
34
, pp.
651
657
.
25.
Myers
,
K. M.
,
Paskaleva
,
A. P.
,
House
,
M.
, and
Socrate
,
S.
, 2008, “
Mechanical and Biochemical Properties of Human Cervical Tissue
,”
Acta Biomater.
1742-7061,
4
, pp.
104
116
.
26.
Dokos
,
S.
,
LeGrice
,
I. J.
,
Smaill
,
B. H.
,
Kar
,
J.
, and
Young
,
A. A.
, 2000, “
A Triaxial-Measurement Shear-Test Device for Soft Biological Tissues
,”
ASME J. Biomech. Eng.
0148-0731,
122
, pp.
471
478
.
27.
Liu
,
Z.
, and
Bilston
,
L. E.
, 2002, “
Large Deformation Shear Properties of Liver Tissue
,”
Biorheology
0006-355X,
39
, pp.
735
742
.
28.
Valtorta
,
D.
, and
Mazza
,
E.
, 2005, “
Dynamic Measurement of Soft Tissue Viscoelastic Properties With a Torsional Resonator Device
,”
Med. Image Anal.
1361-8415,
9
, pp.
481
490
.
29.
Nava
,
A.
,
Mazza
,
E.
,
Kleinermann
,
F.
,
Avis
,
N. J.
,
McClure
,
J.
, and
Bajka
,
M.
, 2004, “
Evaluation of the Mechanical Properties of Human Liver and Kidney Through Aspiration Experiments
,”
Technol. Health Care
0928-7329,
12
, pp.
269
280
.
30.
Criscione
,
J. C.
, 2003, “
Rivlin’s Representation Formula Is Ill-Conceived for the Determination of Response Functions via Biaxial Testing
,”
J. Elast.
0374-3535,
70
, pp.
129
147
.
31.
Ogden
,
R. W.
,
Saccomandi
,
G.
, and
Sgura
,
I.
, 2004, “
Fitting Hyperelastic Models to Experimental Data
,”
Comput. Mech.
0178-7675,
34
, pp.
484
502
.
32.
Johnson
,
T. P. M.
,
Socrate
,
S.
, and
Boyce
,
M. C.
, 2010, “
A Viscoelastic, Viscoplastic Model of Cortical Bone Valid at Low and High Strain Rates
,”
Acta Biomater.
1742-7061,
6
, pp.
4073
4080
.
33.
Prevost
,
T. P.
,
Balakrishnan
,
A.
,
Suresh
,
S.
, and
Socrate
,
S.
, 2011, “
Biomechanics of Brain Tissue
,”
Acta Biomater.
1742-7061,
7
, pp.
83
95
.
34.
Mazza
,
E.
,
Nava
,
A.
,
Hahnloser
,
D.
,
Jochum
,
W.
, and
Bajka
,
M.
, 2007, “
The Mechanical Response of Human Liver and Its Relation to Histology: An In Vivo Study
,”
Med. Image Anal.
1361-8415,
11
, pp.
663
672
.
35.
Shampine
,
L. F.
, 1994,
Numerical Solution of Ordinary Differential Equations
,
Chapman and Hall
,
New York
.
36.
Gasser
,
T. C.
,
Ogden
,
R. W.
, and
Holzapfel
,
G. A.
, 2006, “
Hyperelastic Modelling of Arterial Layers With Distributed Collagen Fibre Orientations
,”
J. R. Soc., Interface
1742-5689,
3
, pp.
15
35
.
37.
Bischoff
,
J. E.
,
Arruda
,
E. M.
, and
Grosh
,
K.
, 2004, “
A Rheological Network Model for the Continuum Anisotropic and Viscoelastic Behavior of Soft Tissue
,”
Biomech. Model. Mechanobiol.
1617-7959,
3
, pp.
56
65
.
38.
Arruda
,
E. M.
, and
Boyce
,
M. C.
, 1993, “
A Three-Dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials
,”
J. Mech. Phys. Solids
0022-5096,
41
, pp.
389
412
.
39.
Bergström
,
J. S.
, and
Boyce
,
M. C.
, 2001, “
Constitutive Modeling of the Time-Dependent and Cyclic Loading of Elastomers and Application to Soft Biological Tissues
,”
Mech. Mater.
0167-6636,
33
, pp.
523
530
.
40.
Lagarias
,
J. C.
,
Reeds
,
J. A.
,
Wright
,
M. H.
, and
Wright
,
P. E.
, 1998, “
Convergence Properties of the Nelder–Mead Simplex Method in Low Dimensions
,”
SIAM J. Optim.
1052-6234,
9
, pp.
112
147
.
41.
Jordan
,
P.
, 2008, “
Image-Based Mechanical Characterization of Soft Tissue Using Three Dimensional Ultrasound
,” Ph.D. thesis, Harvard University, Cambridge, MA.
42.
Balakrishnan
,
A.
, 2007, “
Development of Novel Dynamic Indentation Techniques for Soft Tissue Applications
,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
43.
Myers
,
K. M.
,
Socrate
,
S.
,
Paskaleva
,
A.
, and
House
,
M.
, 2010,
ASME J. Biomech. Eng.
0148-0731,
132
, p.
021003
.
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