Structure-mechanics relationship in
fibrous biomaterials
Mohd Suhail Rizvi and Sovan Lal Das
Pravartana 2016
12th February 2016
1
Fibrous materials
Articular
cartilage
Aorta
Cellulose fibers
Connective
tissue
Elastic fibers
Collagen fibers
Fibers in natural systems
Engineering fibrous materials
Fibrin fibers
Collagen fibers
Synthetic polymeric
fibers
Plants
Applications
• Scaffolds in tissue engineering
• Drug delivery carriers
• Filteration media and nanosensors
• Wound dressing
• Cosmetic skin masks
1. Anatomy 215 Virtual Microscopy, Indiana University
2. http://histologistics.com
1.
Lisi et al., 2012, PLoS ONE
Cells on substrates
Mechanical properties of the substrates influence
• Cell lineage specification1
• Cell proliferation2 and migration5
• Cytoskeletal organization3
• Cell spreading and stress fiber formation4
• Cell polarization and elongation6
Structural properties of the substrates influence
• Cell morphology7
Random fibers
7,8
• Cell alignment
• Cell differentiation9
• Cell migration10
• Anisotropic cell spreading11
A.J. et al., 2006, Cell
et al, 2009, J. Tissue Eng. Regen. Med.
3Ghosh, K. et al., 2007, Biomaterials
4Yeung, T. et al., 2005, Cell Mot. and Cytoskel.
5Lo, C.M. et al, 2000, Biophys. J.
6Prager-Khoutorsky, M. et al., 2011, Nat. Cell Biol.
1Engler,
2Hadjipanayi, E.
Aligned fibers
7Aubin, H.
et al., 2010, Biomaterials
S.T. et al, 2014, Biomech. Model. Mechanobiol.
9Lu, D. et al. et al., 2014, Biomaterials
10Teixeira, A.I. et al., 2003, J. Cell Sci.
11Saez, A. et al, 2007, Proc. Nat. Acad. Sci.
8Wong,
3
Existing Constitutive Models
• Fibers are considered to be embedded in an isotropic substrate- fibrous composite
m
Energy in terms of invariants of matrix deformation
Response of fiber
Response of fibrous composite
Limitations:
• Applicable to fibrous composites with
straight fibers only
Holzapfel, G.A. et al., 2000, J. Elasticity
Federico, S. et al., 2010, J. Royal Soc. Interface
4
Assumptions
• Fibrous matrix is considered to be planar
• All the fibers are of same diameter
• Fiber diameter is very small (~1μm)
• Fiber bending stiffness is negligible in comparison to the stretching.
• Fibers fused with each other do not detach
Description of fiber structure
θ
R
l
Normalized
fiber length
Description of matrix structure
Fiber length
Fiber orientation
Fiber fusion
(i)
Fiberfiber
fusion
r
(ii)
5
Effective fiber response
• Effective fiber response is defined as the average response of the
fibers passing through two points.
1
λ
Fiber elongation and bending are measured in terms of changes in fiber
length and fiber curvature, respectively.
• The fiber curvature information is not known.
6
Fiber bending
Approximation of
fiber shape
R
l
Root-mean-square (RMS)
curvature of undeformed fiber
RMS curvature
of deformed fiber
Bending energy
of a fiber
• Fiber curvature is approximated in terms of the fiber length
• Fiber response demonstrates
 strain softening for compression
 strain stiffening for elongation
7
Effective fiber response
• For curved fibers the fiber response is bending dominated
 Higher fiber curvature leads to soft fiber response
8
Response of the fibrous matrix
x r+x
r
χ
χ(r)
χ(r+x)
9
Model validation
Matrix elongation
1Experiment
log
Tensile modulus
Change in area
Shear modulus
log
2Experiment
• Model demonstrates qualitative agreement with the experiments.
• Fiber fusion leads to stiffer matrix response
• Aligned fibrous matrix has higher tensile modulus
• Random fibrous matrix has higher shear modulus
1Kumar,
2Kabla,
P., 2010, M. Tech. Thesis, IIT Kanpur
A. et al., 2007, J. R. Soc. Interface
10
Fiber alignment
• The degree of fiber alignment in the fibrous matrix can be defined as
where μθ is the mean fiber orientation.
• Matrix deformation leads to change in
• The probability density function in the deformed configuration is
where
Cell-matrix interaction
Fiber alignment
Displacement field
Our model
Our model
Discrete fiber
model
Experiment
Discrete fiber model
• Displacement due to a polarized cell is highest near the cell extremities.
• Cellular forces lead to remodeling of the fibers in the matrix
Abhilash, A.S. et al., 2014, Biophys. J.
Gjorevski, N. et al., 2012, Biophysi. J.
12
Cell-cell interaction
Displacement field
Low
Displacement
High
Fiber alignment
Low
Alignment
parameter
Discrete fiber model
High
•Fiber remodeling is observed between two cells- formation of tracks
Cell-cell interaction energy
Cell-cell separation
Strong interaction
Low
Interaction energy
Weak interaction
High
• Cell-cell interaction depends on cell orientations and their separation
Abhilash, A.S. et al., 2014, Biophys. J.
13
Generalized Model
• For fibrous composites with single family of straight fibers, the energy is
expressed in terms of 5 invariants.
• For two family fibrous composites, energy is written in terms of 9 invariants
Isotropic part
Family-1 fibers
Family-2 fibers
Fiber-fiber interaction
Holzapfel, G., Nonlinear Solid Mechanics
n
m
Generalized Model
• For fibrous composites with single family of straight fibers
• For fibrous composites with multiple independent families of straight fibers
• For curved fibers without fiber fusion
No bending stiffness
Finite bending stiffness
Here u is the effective fiber response.
Generalized Model
• Inclusion of fiber breakage- Mullins effect
• Assumption- a fiber breaks if its stretch exceeds a threshold value λb
• Fiber breakage or damage criterion
• In general
• For fibrous matrix
• These relations describe the mechanical response of the fibrous composite including
• Effect of curved fibers
• Diverse fiber orientations
• Fiber breakage
Future directions
• Validation of model predictions using experiments
Improvement in the model
• Incorporation of structural details in the model
• Study of matrix response against biaxial loading
• Implementation of the fibrous composite model
• Microscopic mechanical response of matrix against dynamic loading
Study of cell-matrix interactions
• Incorporation of detailed cell geometry
• Study of the fibrous matrix response against forces applied by multiple
cells
• Incorporation of cytoskeletal remodeling in response to matrix elasticity
17
Thank You
18
Microscopic elasticity
Random fibers
Aligned fibers
All boundaries fixed
Two boundaries fixed
• Elasticity experienced by the cells is dependent on the
 location of cell on the matrix
 direction
 applied boundary conditions
21
Cell-matrix interaction: location
• Cell experiences substrate stiffness in of terms of work done for substrate
deformation
• This work done is also known as “self-energy of the cell”
Random
fibrous matrix
a
b
c
c
α
a
b
c
b
b
a
c
b
b
a
• Near the boundaries of the random fibrous matrix- anisotropic elasticity
• For clamped-free-clamped-free boundary condition – whole matrix is anisotropic
Anisotropic substrate
Bischofs, I. B. et al, 2003, PNAS; Bischofs, I. B. et al, 2004, PRE
Saez, A. et al., 2007, PNAS.
Isotropic substrate
22
Effect of fiber alignment
Aligned fibers
α
• Fiber alignment leads to anisotropic elasticity of fibrous matrix
Experiments
Random fibers
Saha, S. et al., 2012, Langmuir
Teh, T.K.H. et al., 2012, Tissue Engg. Part A
Aligned fibers
Effect of boundary conditions
Free edge
Fixed edge
• Self energy of a cell is-
• high near a fixed edge of random fibrous matrix
• low near a free edge of random fibrous matrix
Near free edge
Near fixed edge
• Boundary conditions of fibrous matrix can influence the cell behavior
Grinnell, F., 2000, Trends in Cell Biol.
Summary
1. Model
• Fiber bending is not negligible for cell-matrix interaction
• Fiber bending demonstrates strain stiffening and strain softening in fiber
elongation and compression, respectively
• Fiber fusion leads to stiffer matrix response
2. Cell-matrix interaction
• Matrix stiffness experienced by the cell is dependent on the cell location,
orientation, and matrix boundary conditions
• Cellular forces cause fiber remodeling in the fibrous matrix
• Fiber alignment leads to anisotropic mechanical properties of the matrix
3. Cell-cell interaction
• Fibers align in the region between two cells
• Cell-cell interaction depends on the cell orientations and the separation
between them
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Model parameters
• Fiber diameter (a)
• All fibers are not of same diameter
• But the diameter of a fiber does not change with its length
• aeffective ~ 1μm
• Constitutive relation for fiber material (τ(γf ))
• During mechanical testing large strains are applied
• Large strains lead to stretching of fibers
• Bending/Stretching ~ a2 ~ 10 -12
• Therefore, fiber bending can be ignored
0
Curved fiber
τ
• linear viscoelastic for dynamic loading
τfr
• Fiber breaks when stress in the fiber exceeds a threshold value
Straight load
• Before breaking, fiber response is
bearing fiber
• linear elastic for quasi-static loading
Fiber
breakage
ε0
εfr
Strain of the fibrous matrix
Cell-matrix interactions
• Cells apply localized forces to the substrate via focal adhesions.
Focal
adhesion
Stress
fiber
Substrate
• The modeling framework presented till now is not applicable
• The understanding of behavior of the fibrous matrices against localized
forces is needed
Burridge, K. et al., 2013, J. Cell. Biol.
Balaban, N. Q. et al., 2001, Nature Cell Biology
Gjorevski, N. et al., 2012, Biophys. J.
27