In situ Mechanical Characterization of Hydrogel ... - Cornell University

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Oct 29, 2013 (3 years and 5 months ago)

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In situ
Mechanical Characterization of Hydrogel
Extracellular Matrices using Ferromagnetic Beads

Matthew S. Hall

PhD Student

Department of Biological and Environmental Engineering

Cornell University

Motivation: Cellular Mechanosensing


Substrate stiffness affects
cellular behaviors
including cell growth,
adhesion, differentiation,
and migration

Engler et al.
Cell
. 2006.

Cell Seeded 3D Collagen Hydrogel

Red= Collagen Fibers

Green= Fluorescent Beads

Blue= Breast Cancer
C
ell


Cells actively remodel the
collagen hydrogel changing
both local and bulk
mechanical properties

In situ
Material Characterization Challenges


Sterile conditions


μ
N force regime


(E= 50
-
5000 [Pa])


3D
μ
m scale
g
eometry


Nonlinear material behavior



Solution: Embed Ferromagnetic Beads


Apply a constant magnetic
field to produce F
mag



Ferromagnetic bead is
carboxylated to covalently
bond to the collagen fibers




no slip boundary
condition


Ferromagnetic


Bead

F
mag

Axisymmetric Geometry

0 Natural BC

r: 0 Natural BC

z: 0 Essential BC

r: 0 Natural BC

z: 0 Essential BC

z: Essential BC

(satisfy F
mag
)

z

r

z: 0 Natural BC

r
: 0 Essential BC

r
: 0 Essential BC

z: 0 Natural BC

r
: 0 Natural BC

Collagen

Hydrogel

Strain Hardening Material

Hall
-
Abstract MAE 5700




Roeder AE et al.
J
Biomech

Eng
. 2002.

Uniaxial Tension Test for 2.0 [mg/ml] Collagen I

Inverse Finite Element for Material
Characterization

Experiment:


Apply a known force (apply magnetic field to ferromagnetic
bead)


Measure axisymmetric discrete displacement field as a function
of time (track green fluorescent beads)

Finite Element Analysis:


Implement axisymmetric geometry and nonlinear neo
-
hookean

material model in Finite Element simulation


Refine material constants until the Finite Element simulation
converges to the experimental displacement field

Neo
-
hookean

Material Model

Weak Form

Linearization

Newton
-
Raphson

Solution Process


Initialize for 0 displacement and 0 strain


Iterate over 2 nested loops:


Outer loop: incrementally apply bead displacement




and stop when F
mag

is satisfied


Inner loop: solve and update nodal positions until solution



converge

Inner Loop

Inverse FEA for Material Characterization

Loop over entire solution varying material properties
μ
1

and K

1

until the simulation displacement field converges to the
experimental displacement field.


References

1.
Long R, Hall MS, Wu M, Hui CY. Effects of gel thickness on microscopic
indentation measurements of gel modulus.
Biophysics Journal
. 108, 5614
-
5619
(2011).

2.
Hall MS, Long R, Hui CY, Wu M. Mapping 3D stress and strain fields within a soft
hydrogel using a fluorescence microscope.
Biophysical Journal
. 102, 2241
-
2250.
(2012).

3.
Engler et al. Matrix Elasticity Directs Stem Cell Lineage Specification.
Cell
. 126,
677

689.
2006.

4.
Roeder BA et al. Tensile Mechanical Properties of Three
-
Dimensional Type I
Collagen Extracellular Matrices with Varied Microstructure. J Biomech Eng. 124.
2002.

5.
Wriggers, Peter. Nonlinear Finite Element Methods. Springer
-
Verlag Berlin
Heidelberg. 2008.

6.
Hinton E. NAFEMS Introduction to Nonlinear Finite Element Analysis. Bell and
Bain Glasgow.1992.

7.
Bonet, Javier and Wood, Richard. Nonlinear Continuum Mechanics for Finite
Element Analysis. Cambridge Press. 1997
.

8.
Bower, AF. Applied Mechanics of Solids. CRC Press. 2010.