Recent advances in modulating collagen building blocks enable the design and control of the microstructure and functional properties of collagen matrices for tissue engineering and regenerative medicine. However, this is typically achieved by iterative experimentations and that process can be substantially shortened by computational predictions. Computational efforts to correlate the microstructure of fibrous and/or nonfibrous scaffolds to their functionality such as mechanical or transport properties have been reported, but the predictability is still significantly limited due to the intrinsic complexity of fibrous/nonfibrous networks. In this study, a new computational method is developed to predict two transport properties, permeability and diffusivity, based on a microstructural parameter, the specific number of interfibril branching points (or branching points). This method consists of the reconstruction of a three-dimensional (3D) fibrous matrix structure based on branching points and the computation of fluid velocity and solute displacement to predict permeability and diffusivity. The computational results are compared with experimental measurements of collagen gels. The computed permeability was slightly lower than the measured experimental values, but diffusivity agreed well. The results are further discussed by comparing them with empirical correlations in the literature for the implication for predictive engineering of collagen matrices for tissue engineering applications.

References

References
1.
Crabb
,
R. A. B.
, and
Hubel
,
A.
,
2008
, “
Influence of Matrix Processing on the Optical and Biomechanical Properties of a Corneal Stroma Equivalent
,”
Tissue Eng. Part A
,
14
(
1
), pp.
173
182
.10.1089/ten.a.2007.0139
2.
Roeder
,
B. A.
,
Kokini
,
K.
,
Sturgis
,
J. E.
,
Robinson
,
J. P.
, and
Voytik-Harbin
,
S. L.
,
2002
, “
Tensile Mechanical Properties of Three-Dimensional Type I Collagen Extracellular Matrices With Varied Microstructure
,”
ASME J. Biomech. Eng.
,
124
(
2
), pp.
214
222
.10.1115/1.1449904
3.
Bailey
,
J. L.
,
Critser
,
P. J.
,
Whittington
,
C.
,
Kuske
,
J. L.
,
Yoder
,
M. C.
, and
Voytik-Harbin
,
S. L.
,
2011
, “
Collagen Oligomers Modulate Physical and Biological Properties of Three-Dimensional Self-Assembled Matrices
,”
Biopolymers
,
95
(
2
), pp.
77
93
.10.1002/bip.21537
4.
Chen
,
M. Y.
,
Sun
,
Y. L.
,
Zhao
,
C. F.
,
Zobitz
,
M. E.
,
An
,
K. N.
,
Moran
,
S. L.
, and
Amadio
,
P. C.
,
2007
, “
Factors Related to Contraction and Mechanical Strength of Collagen Gels Seeded With Canine Endotenon Cells
,”
J. Biomed. Mater. Res. Part B
,
82B
(
2
), pp.
519
525
.10.1002/jbm.b.30757
5.
Buehler
,
M. J.
,
2006
, “
Nature Designs Tough Collagen: Explaining the Nanostructure of Collagen Fibrils
,”
Rep. Natl. Acad. Sci. U.S.A.
,
103
(
33
), pp.
12285
12290
.10.1073/pnas.0603216103
6.
Ramanujan
,
S.
,
Pluen
,
A.
,
McKee
,
T. D.
,
Brown
,
E. B.
,
Boucher
,
Y.
, and
Jain
,
R. K.
,
2002
, “
Diffusion and Convection in Collagen Gels: Implications for Transport in the Tumor Interstitium
,”
Biophys. J.
,
83
(
3
), pp.
1650
1660
.10.1016/S0006-3495(02)73933-7
7.
Tierney
,
C. M.
,
Haugh
,
M. G.
,
Liedl
,
J.
,
Mulcahy
,
F.
,
Hayes
,
B.
, and
O′Brien
,
F. J.
,
2009
, “
The Effects of Collagen Concentration and Crosslink Density on the Biological, Structural and Mechanical Properties of Collagen-GAG Scaffolds for Bone Tissue Engineering
,”
J. Mech. Behav. Biomed. Mater.
,
2
(
2
), pp.
202
209
.10.1016/j.jmbbm.2008.08.007
8.
Johnson
,
E. M.
,
Berk
,
D. A.
,
Jain
,
R. K.
, and
Deen
,
W. M.
,
1996
, “
Hindered Diffusion in Agarose Gels: Test of Effective Medium Model
,”
Biophys. J.
,
70
(
2
), pp.
1017
1023
.10.1016/S0006-3495(96)79645-5
9.
Kosto
,
K. B.
, and
Deen
,
W. M.
,
2004
, “
Diffusivities of Macromolecules in Composite Hydrogels
,”
AIChE J.
,
50
(
11
), pp.
2648
2658
.10.1002/aic.10216
10.
Erikson
,
A.
,
Andersen
,
H. N.
,
Naess
,
S. N.
,
Sikorski
,
P.
, and
Davies
,
C. D.
,
2008
, “
Physical and Chemical Modifications of Collagen Gels: Impact on Diffusion
,”
Biopolymers
,
89
(
2
), pp.
135
143
.10.1002/bip.20874
11.
O′Brien
,
F. J.
,
Harley
,
B. A.
,
Waller
,
M. A.
,
Yannas
,
I. V.
,
Gibson
,
L. J.
, and
Prendergast
,
P. J.
,
2007
, “
The Effect of Pore Size on Permeability and Cell Attachment in Collagen Scaffolds for Tissue Engineering
,”
Technol. Health Care
,
15
(
1
), pp.
3
17
.
12.
Karande
,
T. S.
,
Ong
,
J. L.
, and
Agrawal
,
C. M.
,
2004
, “
Diffusion in Musculoskeletal Tissue Engineering Scaffolds: Design Issues Related to Porosity, Permeability, Architecture, and Nutrient Mixing
,”
Ann. Biomed. Eng.
,
32
(
12
), pp.
1728
1743
.10.1007/s10439-004-7825-2
13.
Al-Munajjed
,
A. A.
,
Hien
,
M.
,
Kujat
,
R.
,
Gleeson
,
J. P.
, and
Hammer
,
J.
,
2008
, “
Influence of Pore Size on Tensile Strength, Permeability and Porosity of Hyaluronan-Collagen Scaffolds
,”
J. Mater. Sci.-Mater. Med.
,
19
(
8
), pp.
2859
2864
.10.1007/s10856-008-3422-5
14.
Hollister
,
S. J.
,
2005
, “
Porous Scaffold Design for Tissue Engineering
,”
Nat. Mater.
,
4
(
7
), pp.
518
524
.10.1038/nmat1421
15.
Drummond
,
J. E.
, and
Tahir
,
M. I.
,
1984
, “
Laminar Viscous-Flow Through Regular Arrays of Parallel Solid Cylinders
,”
Int. J. Multiphase Flow
,
10
(
5
), pp.
515
540
.10.1016/0301-9322(84)90079-X
16.
Bechtold
,
G.
, and
Ye
,
L.
,
2003
, “
Influence of Fibre Distribution on the Transverse Flow Permeability in Fibre Bundles
,”
Compos. Sci. Technol.
,
63
(
14
), pp.
2069
2079
.10.1016/S0266-3538(03)00112-X
17.
Phillips
,
R. J.
,
2000
, “
A Hydrodynamic Model for Hindered Diffusion of Proteins and Micelles in Hydrogels
,”
Biophys. J.
,
79
(
6
), pp.
3350
3353
.10.1016/S0006-3495(00)76566-0
18.
Tomadakis
,
M. M.
, and
Robertson
,
T. J.
,
2005
, “
Viscous Permeability of Random Fiber Structures: Comparison of Electrical and Diffusional Estimates With Experimental and Analytical Results
,”
J. Compos. Mater.
,
39
(
2
), pp.
163
188
.10.1177/0021998305046438
19.
Masoud
,
H.
, and
Alexeev
,
A.
,
2010
, “
Permeability and Diffusion Through Mechanically Deformed Random Polymer Networks
,”
Macromolecules
,
43
(
23
), pp.
10117
10122
.10.1021/ma102052m
20.
Chandran
,
P. L.
,
Stylianopoulos
,
T.
, and
Barocas
,
V. H.
,
2008
, “
Microstructure-Based, Multiscale Modeling for the Mechanical Behavior of Hydrated Fiber Networks
,”
Multiscale Model. Simul.
,
7
(
1
), pp.
22
43
.10.1137/070689504
21.
Stylianopoulos
,
T.
, and
Barocas
,
V. H.
,
2007
, “
Volume-Averaging Theory for the Study of the Mechanics of Collagen Networks
,”
Comput. Methods Appl. Mech. Eng.
,
196
(
31–32
), pp.
2981
2990
.10.1016/j.cma.2006.06.019
22.
Stylianopoulos
,
T.
,
Yeckel
,
A.
,
Derby
,
J. J.
,
Luo
,
X. J.
,
Shephard
,
M. S.
,
Sander
,
E. A.
, and
Barocas
,
V. H.
,
2008
, “
Permeability Calculations in Three-Dimensional Isotropic and Oriented Fiber Networks
,”
Phys. Fluids
,
20
(
12
), p.
123601
.10.1063/1.3021477
23.
Stylianopoulos
,
T.
,
Diop-Frimpong
,
B.
,
Munn
,
L. L.
, and
Jain
,
R. K.
,
2010
, “
Diffusion Anisotropy in Collagen Gels and Tumors: The Effect of Fiber Network Orientation
,”
Biophys. J.
,
99
(
10
), pp.
3119
3128
.10.1016/j.bpj.2010.08.065
24.
Whittington
,
C. F.
,
Brandner
,
E.
,
Teo
,
K. Y.
,
Han
,
B.
,
Nauman
,
E.
, and
Voytik-Harbin
,
S. L.
,
2013
, “
Oligomers Modulate Interfibril Branching and Mass Transport Properties of Collagen Matrices
,”
Microsc. Microanal.
,
19
(
5
), pp.
1323
1333
.10.1017/S1431927613001931
25.
Whittington
,
C. F.
,
Yoder
,
M. C.
, and
Voytik-Harbin
,
S. L.
,
2013
, “
Collagen-Polymer Guidance of Vessel Network Formation and Stabilization by Endothelial Colony Forming Cells In Vitro
,”
Macromol. Biosci.
,
13
(
9
), pp.
1135
1149
.10.1002/mabi.201300128
26.
Pedersen
,
J. A.
,
Boschetti
,
F.
, and
Swartz
,
M. A.
,
2007
, “
Effects of Extracellular Fiber Architecture on Cell Membrane Shear Stress in a 3D Fibrous Matrix
,”
J. Biomech.
,
40
(
7
), pp.
1484
1492
.10.1016/j.jbiomech.2006.06.023
27.
Pedersen
,
J. A.
,
Lichter
,
S.
, and
Swartz
,
M. A.
,
2010
, “
Cells in 3D Matrices Under Interstitial Flow: Effects of Extracellular Matrix Alignment on Cell Shear Stress and Drag Forces
,”
J. Biomech.
,
43
(
5
), pp.
900
905
.10.1016/j.jbiomech.2009.11.007
28.
Swartz
,
M. A.
, and
Fleury
,
M. E.
,
2007
, “
Interstitial Flow and Its Effects in Soft Tissues
,”
Annu. Rev. Biomed. Eng.
,
9
, pp.
229
256
.10.1146/annurev.bioeng.9.060906.151850
29.
Snyder
,
M. A.
, and
Vlachos
,
D. G.
,
2007
, “
Nano-Patterned Standards for Improving the Quantitative Capability of Laser Scanning Confocal Microscopy for Materials Characterization
,”
Microporous Mesoporous Mater.
,
102
(
1–3
), pp.
101
110
.10.1016/j.micromeso.2006.12.028
30.
Park
,
S.
,
Seawright
,
A.
,
Park
,
S.
,
Craig Dutton
,
J.
,
Grinnell
,
F.
, and
Han
,
B.
,
2015
, “
Preservation of Tissue Microstructure and Functionality During Freezing by Modulation of Cytoskeletal Structure
,”
J. Mech. Behav. Biomed. Mater.
,
45C
, pp.
32
44
.10.1016/j.jmbbm.2015.01.014
31.
Johnson
,
E. M.
, and
Deen
,
W. M.
,
1996
, “
Hydraulic Permeability of Agarose Gels
,”
AIChE J.
,
42
(
5
), pp.
1220
1224
.10.1002/aic.690420504
32.
Valentine
,
M. T.
,
Perlman
,
Z. E.
,
Gardel
,
M. L.
,
Shin
,
J. H.
,
Matsudaira
,
P.
,
Mitchison
,
T. J.
, and
Weitz
,
D. A.
,
2004
, “
Colloid Surface Chemistry Critically Affects Multiple Particle Tracking Measurements of Biomaterials
,”
Biophys. J.
,
86
(
6
), pp.
4004
4014
.10.1529/biophysj.103.037812
33.
Abouali
,
O.
,
Nikbakht
,
A.
,
Ahmadi
,
G.
, and
Saadabadi
,
S.
,
2009
, “
Three-Dimensional Simulation of Brownian Motion of Nano-Particles in Aerodynamic Lenses
,”
Aerosol Sci. Technol.
,
43
(
3
), pp.
205
215
.10.1080/02786820802587888
34.
Roache
,
P. J.
,
1997
, “
Quantification of Uncertainty in Computational Fluid Dynamics
,”
Annu. Rev. Fluid Mech.
,
29
, pp.
123
160
.10.1146/annurev.fluid.29.1.123
35.
Barth
,
T. J.
, and
Jesperson
,
D. C.
,
1989
, “
The Design and Application of Upwind Schemes on Unstructured Meshes
,”
AIAA
Paper No. 89-0366.10.2514/6.1989-366
36.
Kreger
,
S. T.
,
Bell
,
B. J.
,
Bailey
,
J.
,
Stites
,
E.
,
Kuske
,
J.
,
Waisner
,
B.
, and
Voytik-Harbin
,
S. L.
,
2010
, “
Polymerization and Matrix Physical Properties as Important Design Considerations for Soluble Collagen Formulations
,”
Biopolymers
,
93
(
8
), pp.
690
707
.10.1002/bip.21431
37.
Nicholson
,
C.
, and
Tao
,
L.
,
1993
, “
Hindered Diffusion of High-Molecular-Weight Compounds in Brain Extracellular Microenvironment Measured With Integrative Optical Imaging
,”
Biophys. J.
,
65
(
6
), pp.
2277
2290
.10.1016/S0006-3495(93)81324-9
38.
Thorne
,
R. G.
, and
Nicholson
,
C.
,
2006
, “
In Vivo Diffusion Analysis With Quantum Dots and Dextrans Predicts the Width of Brain Extracellular Space
,”
Proc. Natl. Acad. Sci. U. S. A.
,
103
(
14
), pp.
5567
5572
.10.1073/pnas.0509425103
39.
Marelli
,
B.
,
Ghezzi
,
C. E.
,
Barralet
,
J. E.
, and
Nazhat
,
S. N.
,
2011
, “
Collagen Gel Fibrillar Density Dictates the Extent of Mineralization In Vitro
,”
Soft Matter
,
7
(
21
), pp.
9898
9907
.10.1039/c1sm06027a
40.
Salami
,
S.
,
Rondeau-Mouro
,
C.
,
Barhoum
,
M.
,
van Duynhoven
,
J.
, and
Mariette
,
F.
,
2014
, “
Translational and Rotational Diffusion of Flexible PEG and Rigid Dendrimer Probes in Sodium Caseinate Dispersions and Acid Gels
,”
Biopolymers
,
101
(
9
), pp.
959
965
.10.1002/bip.22492
41.
Hosseini
,
S. A.
, and
Tafreshi
,
H. V.
,
2010
, “
Modeling Permeability of 3-D Nanofiber Media in Slip Flow Regime
,”
Chem. Eng. Sci.
,
65
(
6
), pp.
2249
2254
.10.1016/j.ces.2009.12.002
42.
Myllyharju
,
J.
, and
Kivirikko
,
K. I.
,
2004
, “
Collagens, Modifying Enzymes and Their Mutations in Humans, Flies and Worms
,”
Trends Genet.
,
20
(
1
), pp.
33
43
.10.1016/j.tig.2003.11.004
43.
Israelowitz
,
M.
,
Rizvi
,
S. W. H.
,
Kramer
,
J.
, and
von Schroeder
,
H. P.
,
2005
, “
Computational Modeling of Type I Collagen Fibers to Determine the Extracellular Matrix Structure of Connective Tissues
,”
Protein Eng. Des. Sel.
,
18
(
7
), pp.
329
335
.10.1093/protein/gzi037
44.
Sanjeevi
,
R.
,
Ramanthan
,
N.
, and
Viswanathan
,
B.
,
1976
, “
Pore-Size Distribution in Collagen Fiber Using Water-Vapor Adsorption Studies
,”
J. Colloid Interface Sci.
,
57
(
2
), pp.
207
211
.10.1016/0021-9797(76)90194-6
45.
Boki
,
K.
,
Kawasaki
,
N.
,
Minami
,
K.
, and
Takahashi
,
H.
,
1993
, “
Structural-Analysis of Collagen-Fibers by Nitrogen Adsorption Method
,”
J. Colloid Interface Sci.
,
157
(
1
), pp.
55
59
.10.1006/jcis.1993.1157
46.
Giesa
,
T.
, and
Buehler
,
M. J.
,
2013
, “
Nanoconfinement and the Strength of Biopolymers
,”
Annu. Rev. Biophys.
,
42
, pp.
651
673
.10.1146/annurev-biophys-083012-130345
47.
Hambli
,
R.
, and
Barkaoui
,
A.
,
2012
, “
Physically Based 3D Finite Element Model of a Single Mineralized Collagen Microfibril
,”
J. Theor. Biol
,
301
, pp.
28
41
.10.1016/j.jtbi.2012.02.007
48.
Barkaoui
,
A.
,
Chamekh
,
A.
,
Merzouki
,
T.
,
Hambli
,
R.
, and
Mkaddem
,
A.
,
2014
, “
Multiscale Approach Including Microfibril Scale to Assess Elastic Constants of Cortical Bone Based on Neural Network Computation and Homogenization Method
,”
Int. J. Numer. Methods Biomed. Eng.
,
30
(
3
), pp.
318
338
.10.1002/cnm.2604
49.
Barkaoui
,
A.
, and
Hambli
,
R.
,
2011
, “
Finite Element 3D Modeling of Mechanical Behavior of Mineralized Collagen Microfibrils
,”
J. Appl. Biomater. Biomech.
,
9
(
3
), pp.
199
206
.
50.
Davies
,
C. N.
,
1952
, “
The Separation of Airborne Dust and Particles
,”
Proc. Inst. Mech. Eng.
,
1
(
4
), pp.
185
213
.
51.
Spielman
,
L.
, and
Goren
,
S. L.
,
1968
, “
Model for Predicting Pressure Drop and Filtration Efficiency in Fibrous Media
,”
Environ. Sci. Technol.
,
2
(
4
), pp.
279
287
.10.1021/es60016a003
52.
Jackson
,
G. W.
, and
James
,
D. F.
,
1986
, “
The Permeability of Fibrous Porous-Media
,”
Can. J. Chem. Eng.
,
64
(
3
), pp.
364
374
.10.1002/cjce.5450640302
53.
Yazdchi
,
K.
,
Srivastava
,
S.
, and
Luding
,
S.
,
2011
, “
Microstructural Effects on the Permeability of Periodic Fibrous Porous Media
,”
Int. J. Multiphase Flow
,
37
(
8
), pp.
956
966
.10.1016/j.ijmultiphaseflow.2011.05.003
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