A soft tissue's macroscopic behavior is largely determined by its microstructural components (often a collagen fiber network surrounded by a nonfibrillar matrix (NFM)). In the present study, a coupled fiber-matrix model was developed to fully quantify the internal stress field within such a tissue and to explore interactions between the collagen fiber network and nonfibrillar matrix (NFM). Voronoi tessellations (representing collagen networks) were embedded in a continuous three-dimensional NFM. Fibers were represented as one-dimensional nonlinear springs and the NFM, meshed via tetrahedra, was modeled as a compressible neo-Hookean solid. Multidimensional finite element modeling was employed in order to couple the two tissue components and uniaxial tension was applied to the composite representative volume element (RVE). In terms of the overall RVE response (average stress, fiber orientation, and Poisson's ratio), the coupled fiber-matrix model yielded results consistent with those obtained using a previously developed parallel model based upon superposition. The detailed stress field in the composite RVE demonstrated the high degree of inhomogeneity in NFM mechanics, which cannot be addressed by a parallel model. Distributions of maximum/minimum principal stresses in the NFM showed a transition from fiber-dominated to matrix-dominated behavior as the matrix shear modulus increased. The matrix-dominated behavior also included a shift in the fiber kinematics toward the affine limit. We conclude that if only gross averaged parameters are of interest, parallel-type models are suitable. If, however, one is concerned with phenomena, such as individual cell-fiber interactions or tissue failure that could be altered by local variations in the stress field, then the detailed model is necessary in spite of its higher computational cost.

References

References
1.
Fung
,
Y. C.
,
1967
, “
Elasticity of Soft Tissues in Simple Elongation
,”
Am. J. Physiol.
,
213
(
6
), pp.
1532
1544
.
2.
Humphrey
,
J. D.
,
1995
, “
Mechanics of the Arterial Wall: Review and Directions
,”
Crit. Rev. Biomed. Eng.
,
23
(
1–2
), pp.
1
162
.
3.
Horgan
,
C. O.
and
Saccomandi
,
G.
,
2003
, “
A Description of Arterial Wall Mechanics Using Limiting Chain Extensibility Constitutive Models
,”
Biomech. Model. Mechanobiol.
,
1
(
4
), pp.
251
266
.10.1007/s10237-002-0022-z
4.
Lanir
,
Y.
,
1979
, “
A Structural Theory for the Homogeneous Biaxial Stress-Strain Relationships in Flat Collagenous Tissues
,”
J. Biomech.
,
12
(
6
), pp.
423
436
.10.1016/0021-9290(79)90027-7
5.
Decraemer
,
W. F.
,
Maes
,
M. A.
, and
Vanhuyse
,
V. J.
,
1980
, “
An Elastic Stress-Strain Relation for Soft Biological Tissues Based on a Structural Model
,”
J. Biomech.
,
13
(
6
), pp.
463
468
.10.1016/0021-9290(80)90338-3
6.
Kwan
,
M. K.
and
Woo
,
S. L.
,
1989
, “
A Structural Model to Describe the Nonlinear Stress-Strain Behavior for Parallel-Fibered Collagenous Tissues
,”
ASME J. Biomech. Eng.
,
111
(
4
), pp.
361
363
.10.1115/1.3168392
7.
Cortes
,
D. H.
,
Lake
,
S. P.
,
Kadlowec
,
J. A.
,
Soslowsky
,
L. J.
, and
Elliott
,
D. M.
,
2010
, “
Characterizing the Mechanical Contribution of Fiber Angular Distribution in Connective Tissue: Comparison of Two Modeling Approaches
,”
Biomech. Model. Mechanobiol.
,
9
(
5
), pp.
651
658
.10.1007/s10237-010-0194-x
8.
Wagner
,
H. P.
and
Humphrey
,
J. D.
,
2011
, “
Differential Passive and Active Biaxial Mechanical Behaviors of Muscular and Elastic Arteries: Basilar Versus Common Carotid
,”
ASME J. Biomech. Eng.
,
133
(
5
), p.
051009
.10.1115/1.4003873
9.
Hollander
,
Y.
,
Durban
,
D.
,
Lu
,
X.
,
Kassab
,
G. S.
, and
Lanir
,
Y.
,
2011
, “
Experimentally Validated Microstructural 3D Constitutive Model of Coronary Arterial Media
,”
ASME J. Biomech. Eng.
,
133
(
3
), p.
031007
.10.1115/1.4003324
10.
Kao
,
P. H.
,
Lammers
,
S. R.
,
Tian
,
L.
,
Hunter
,
K.
,
Stenmark
,
K. R.
,
Shandas
,
R.
, and
Qi
,
H. J.
,
2011
, “
A Microstructurally Driven Model for Pulmonary Artery Tissue
,”
ASME J. Biomech. Eng.
,
133
(
5
), p.
051002
.10.1115/1.4002698
11.
Soares
,
A. L.
,
Stekelenburg
,
M.
, and
Baaijens
,
F. P.
,
2011
, “
Remodeling of the Collagen Fiber Architecture Due to Compaction in Small Vessels Under Tissue Engineered Conditions
,”
ASME J. Biomech. Eng.
,
133
(
7
), p.
071002
.10.1115/1.4003870
12.
Szczesny
,
S. E.
,
Peloquin
,
J. M.
,
Cortes
,
D. H.
,
Kadlowec
,
J. A.
,
Soslowsky
,
L. J.
, and
Elliott
,
D. M.
,
2012
, “
Biaxial Tensile Testing and Constitutive Modeling of Human Supraspinatus Tendon
,”
ASME J. Biomech. Eng.
,
134
(
2
), p.
021004
.10.1115/1.4005852
13.
Holzapfel
,
G. A.
,
Gasser
,
T. C.
, and
Ogden
,
R. W.
,
2000
, “
A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models
,”
J. Elast.
,
61
, pp.
1
48
.10.1023/A:1010835316564
14.
Driessen
,
N. J.
,
Bouten
,
C. V.
, and
Baaijens
,
F. P.
,
2005
, “
A Structural Constitutive Model for Collagenous Cardiovascular Tissues Incorporating the Angular Fiber Distribution
,”
ASME J. Biomech. Eng.
,
127
(
3
), pp.
494
503
.10.1115/1.1894373
15.
Tang
,
H.
,
Buehler
,
M. J.
, and
Moran
,
B.
,
2009
, “
A Constitutive Model of Soft Tissue: From Nanoscale Collagen to Tissue Continuum
,”
Ann. Biomed. Eng.
,
37
(
6
), pp.
1117
1130
.10.1007/s10439-009-9679-0
16.
Nagel
,
T.
and
Kelly
,
D. J.
,
2012
, “
Remodelling of Collagen Fibre Transition Stretch and Angular Distribution in Soft Biological Tissues and Cell-Seeded Hydrogels
,”
Biomech. Model. Mechanobiol.
,
11
(
3–4
), pp.
325
339
.10.1007/s10237-011-0313-3
17.
Quapp
,
K. M.
and
Weiss
,
J. A.
,
1998
, “
Material Characterization of Human Medial Collateral Ligament
,”
ASME J. Biomech. Eng.
,
120
(
6
), pp.
757
763
.10.1115/1.2834890
18.
Guerin
,
H. L.
and
Elliott
,
D. M.
,
2007
, “
Quantifying the Contributions of Structure to Annulus Fibrosus Mechanical Function Using a Nonlinear, Anisotropic, Hyperelastic Model
,”
J. Orthop. Res.
,
25
(
4
), pp.
508
516
.10.1002/jor.20324
19.
Abraham
,
A. C.
,
Moyer
,
J. T.
,
Villegas
,
D. F.
,
Odegard
,
G. M.
, and
Haut Donahue
,
T. L.
,
2011
, “
Hyperelastic Properties of Human Meniscal Attachments
,”
J. Biomech.
,
44
(
3
), pp.
413
418
.10.1016/j.jbiomech.2010.10.001
20.
Cortes
,
D. H.
and
Elliott
,
D. M.
,
2012
, “
Extra-Fibrillar Matrix Mechanics of Annulus Fibrosus in Tension and Compression
,”
Biomech. Model. Mechanobiol.
,
11
(
6
), pp.
781
790
.10.1007/s10237-011-0351-x
21.
Wagner
,
D. R.
and
Lotz
,
J. C.
,
2004
, “
Theoretical Model and Experimental Results for the Nonlinear Elastic Behavior of Human Annulus Fibrosus
,”
J. Orthop. Res.
,
22
(
4
), pp.
901
909
.10.1016/j.orthres.2003.12.012
22.
Peng
,
X. Q.
,
Guo
,
Z. Y.
, and
Moran
,
B.
,
2006
, “
An Anisotropic Hyperelastic Constitutive Model With Fiber-Matrix Shear Interaction for the Human Annulus Fibrosus
,”
ASME J. Appl. Mech.
,
73
, pp.
815
824
.10.1115/1.2069987
23.
O'Connell
,
G. D.
,
Guerin
,
H. L.
, and
Elliott
,
D. M.
,
2009
, “
Theoretical and Uniaxial Experimental Evaluation of Human Annulus Fibrosus Degeneration
,”
ASME J. Biomech. Eng.
,
131
(
11
), p.
111007
.10.1115/1.3212104
24.
Lake
,
S. P.
,
Hadi
,
M. F.
,
Lai
,
V. K.
, and
Barocas
,
V. H.
,
2012
, “
Mechanics of a Fiber Network Within a Non-Fibrillar Matrix: Model and Comparison With Collagen-Agarose Co-gels
,”
Ann. Biomed. Eng.
,
40
(
10
), pp.
2111
2121
.10.1007/s10439-012-0584-6
25.
Lake
,
S. P.
and
Barocas
,
V. H.
,
2011
, “
Mechanical and Structural Contribution of Non-Fibrillar Matrix in Uniaxial Tension: A Collagen-Agarose Co-Gel Model
,”
Ann. Biomed. Eng.
,
39
(
7
), pp.
1891
1903
.10.1007/s10439-011-0298-1
26.
Chandran
,
P. L.
and
Barocas
,
V. H.
,
2006
, “
Affine Versus Non-Affine Fibril Kinematics in Collagen Networks: Theoretical Studies of Network Behavior
,”
ASME J. Biomech. Eng.
,
128
(
2
), pp.
259
270
.10.1115/1.2165699
27.
Stylianopoulos
,
T.
and
Barocas
,
V. H.
,
2007
, “
Multiscale, Structure-Based Modeling for the Elastic Mechanical Behavior of Arterial Walls
,”
ASME J. Biomech. Eng.
,
129
(
4
), pp.
611
618
.10.1115/1.2746387
28.
Sander
,
E. A.
,
Stylianopoulos
,
T.
,
Tranquillo
,
R. T.
, and
Barocas
,
V. H.
,
2009
, “
Image-Based Multiscale Modeling Predicts Tissue-Level and Network-Level Fiber Reorganization in Stretched Cell-Compacted Collagen Gels
,”
Proc. Natl. Acad. Sci. U.S.A.
,
106
(
42
), pp.
17675
17680
.10.1073/pnas.0903716106
29.
Kang
,
J.
,
Steward
,
R. L.
,
Kim
,
Y.
,
Schwartz
,
R. S.
,
LeDuc
,
P. R.
, and
Puskar
,
K. M.
,
2011
, “
Response of an Actin Filament Network Model Under Cyclic Stretching Through a Coarse Grained Monte Carlo Approach
,”
J. Theor. Biol.
,
274
(
1
), pp.
109
119
.10.1016/j.jtbi.2011.01.011
30.
Nachtrab
,
S.
,
Kapfer
,
S. C.
,
Arns
,
C. H.
,
Madadi
,
M.
,
Mecke
,
K.
, and
Schroder-Turk
,
G. E.
,
2011
, “
Morphology and Linear-Elastic Moduli of Random Network Solids
,”
Adv. Mater.
,
23
(
22–23
), pp.
2633
2637
.10.1002/adma.201004094
31.
Stein
,
A. M.
,
Vader
,
D. A.
,
Weitz
,
D. A.
, and
Sander
,
L. M.
,
2011
, “
The Micromechanics of Three-Dimensional Collagen-I Gels
,”
Complexity
,
16
(
4
), pp.
22
28
.10.1002/cplx.20332
32.
Billiar
,
K. L.
and
Sacks
,
M. S.
,
2000
, “
Biaxial Mechanical Properties of the Native and Glutaraldehyde-Treated Aortic Valve Cusp—Part II: A Structural Constitutive Model
,”
ASME J. Biomech. Eng.
,
122
(
4
), pp.
327
335
.10.1115/1.1287158
33.
Holzapfel
,
G. A.
,
2000
,
Nonlinear Solid Mechanics: A Contiuum Approach for Engineering
,
John Wiley and Sons
,
New York
.
34.
Nagel
,
T.
and
Kelly
,
D. J.
,
2010
, “
The Influence of Fiber Orientation on the Equilibrium Properties of Neutral and Charged Biphasic Tissues
,”
ASME J. Biomech. Eng.
,
132
(
11
), p.
114506
.10.1115/1.4002589
35.
Benkherourou
,
M.
,
Rochas
,
C.
,
Tracqui
,
P.
,
Tranqui
,
L.
, and
Gumery
,
P. Y.
,
1999
, “
Standardization of a Method for Characterizing Low-Concentration Biogels: Elastic Properties of Low-Concentration Agarose Gels
,”
ASME J. Biomech. Eng.
,
121
(
2
), pp.
184
187
.10.1115/1.2835102
36.
Lake
,
S. P.
,
Hald
,
E. S.
, and
Barocas
,
V. H.
,
2011
, “
Collagen-Agarose Co-Gels as a Model for Collagen-Matrix Interaction in Soft Tissues Subjected to Indentation
,”
J. Biomed. Mater. Res. Part A
,
99
(
4
), pp.
507
515
.10.1002/jbm.a.33183
37.
Chandran
,
P. L.
and
Barocas
,
V. H.
,
2007
, “
Deterministic Material-Based Averaging Theory Model of Collagen Gel Micromechanics
,”
ASME J. Biomech. Eng.
,
129
(
2
), pp.
137
147
.10.1115/1.2472369
38.
Stylianopoulos
,
T.
and
Barocas
,
V. H.
,
2007
, “
Volume-Averaging Theory for the Study of the Mechanics of Collagen Networks
,”
Comput. Methods Biomech. Biomed. Eng.
,
196
(
31–32
), pp.
2981
2990
.10.1016/j.cma.2006.06.019
39.
Weiler
,
K. J.
1988
, “
The Radial-Edge Structure: A Topological Representation for Non-Manifold Geometric Boundary Representations
,”
Geometric Modeling for CAD Applications
,
M. J.
Wozny
,
H. W.
McLaughlin
, and
J. L.
Encarnacao
, eds.,
North Holland
,
Amsterdam
, pp.
3
36
.
41.
Simmetrix web page
, http://www.simmetrix.com
42.
Shephard
,
M. S.
,
2000
, “
Meshing Environment for Geometry-Based Analysis
,”
Int. J. Numer. Methods Eng.
,
47
(
1–3
), pp.
169
190
.10.1002/(SICI)1097-0207(20000110/30)47:1/3<169::AID-NME766>3.0.CO;2-A
43.
Franck
,
C.
,
Maskarinec
,
S. A.
,
Tirrell
,
D. A.
, and
Ravichandran
,
G.
,
2011
, “
Three-Dimensional Traction Force Microscopy: A New Tool for Quantifying Cell-Matrix Interactions
,”
PLoS ONE
,
6
(
3
), p.
e17833
.10.1371/journal.pone.0017833
44.
Hadi
,
M. F.
,
Sander
,
E. A.
, and
Barocas
,
V. H.
,
2012
, “
Multiscale Model Predicts Tissue-Level Failure From Collagen Fiber-Level Damage
,”
ASME J. Biomech. Eng.
,
134
(
9
), p.
091005
.10.1115/1.4007097
45.
Hadi
,
M. F.
,
Sander
,
E. A.
,
Ruberti
,
J. W.
, and
Barocas
,
V. H.
,
2012
, “
Simulated Remodeling of Loaded Collagen Networks via Strain-Dependent Enzymatic Degradation and Constant-Rate Fiber Growth
,”
Mech. Mater.
,
44
, pp.
72
82
.10.1016/j.mechmat.2011.07.003
46.
Pence
,
T. J.
,
Monroe
,
R. J.
, and
Wright
,
N. T.
,
2008
, “
On the Computation of Stress in Affine Versus Nonaffine Fibril Kinematics Within Planar Collagen Network Models
,”
ASME J. Biomech. Eng.
,
130
(
4
), p.
041009
.10.1115/1.2917432
47.
Sopakayang
,
R.
,
De Vita
,
R.
,
Kwansa
,
A.
, and
Freeman
,
J. W.
,
2012
, “
Elastic and Viscoelastic Properties of a Type I Collagen Fiber
,”
J. Theor. Biol.
,
293
, pp.
197
205
.10.1016/j.jtbi.2011.10.018
48.
Silver
,
F. H.
,
Ebrahimi
,
A.
, and
Snowhill
,
P. B.
,
2002
, “
Viscoelastic Properties of Self-Assembled Type I Collagen Fibers: Molecular Basis of Elastic and Viscous Behaviors
,”
Connect. Tissue Res.
,
43
(
4
), pp.
569
580
.10.1080/03008200290001302
49.
Broedersz
,
C. P.
,
Sheinman
,
M.
, and
MacKintosh
,
F. C.
,
2012
, “
Filament-Length-Controlled Elasticity in 3D Fiber Networks
,”
Phys. Rev. Lett.
,
108
(
6
), p.
078102
.10.1103/PhysRevLett.108.078102
50.
Head
,
D. A.
,
Levine
,
A. J.
, and
MacKintosh
,
F. C.
,
2003
, “
Distinct Regimes of Elastic Response and Deformation Modes of Cross-Linked Cytoskeletal and Semiflexible Polymer Networks
,”
Phys. Rev. E
,
68
(
6
), p.
061907
.
51.
Chandran
,
P. L.
and
Barocas
,
V. H.
,
2004
, “
Microstructural Mechanics of Collagen Gels in Confined Compression: Poroelasticity, Viscoelasticity, and Collapse
,”
ASME J. Biomech. Eng.
,
126
(
2
), pp.
152
166
.10.1115/1.1688774
52.
Knapp
,
D.
,
Barocas
,
V. H.
,
Moon
,
A.
,
Yoo
,
K.
,
Petzold
,
L.
, and
Tranquillo
,
R. T.
,
1997
, “
Rheology of Reconstituted Type I Collagen Gel in Confined Compression
,”
J. Rheol.
,
41
, p.
971
.10.1122/1.550817
53.
Hsu
,
S.
,
Jamieson
,
A. M.
, and
Blackwell
,
J.
,
1994
, “
Viscoelastic Studies of Extracellular Matrix Interactions in a Model Native Collagen Gel System
,”
Biorheology
,
31
(
1
), pp.
21
36
.
54.
Meghezi
,
S.
,
Couet
,
F.
,
Chevallier
,
P.
, and
Mantovani
,
D.
,
2012
, “
Effects of a Pseudophysiological Environment on the Elastic and Viscoelastic Properties of Collagen Gels
,”
Int. J. Biomater.
,
2012
, p.
319290
.
55.
Krishnan
,
L.
,
Weiss
,
J. A.
,
Wessman
,
M. D.
, and
Hoying
,
J. B.
,
2004
, “
Design and Application of a Test System for Viscoelastic Characterization of Collagen Gels
,”
Tissue Eng.
,
10
(
1–2
), pp.
241
252
.10.1089/107632704322791880
56.
Pryse
,
K. M.
,
Nekouzadeh
,
A.
,
Genin
,
G. M.
,
Elson
,
E. L.
, and
Zahalak
,
G. I.
,
2003
, “
Incremental Mechanics of Collagen Gels: New Experiments and a New Viscoelastic Model
,”
Ann. Biomed. Eng.
,
31
(
10
), pp.
1287
1296
.10.1114/1.1615571
57.
Wagenseil
,
J. E.
,
Wakatsuki
,
T.
,
Okamoto
,
R. J.
,
Zahalak
,
G. I.
, and
Elson
,
E. L.
,
2003
, “
One-Dimensional Viscoelastic Behavior of Fibroblast Populated Collagen Matrices
,”
ASME J. Biomech. Eng.
,
125
(
5
), pp.
719
725
.10.1115/1.1614818
You do not currently have access to this content.