Multiscale Modeling of Ligaments and Tendons - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 131

regarding stress and strain at lower scale levels (e.g. within fascicles, fibers and
fibrils) must be obtained. One possible route involves isolating and mechanically
testing individual tissue constituents, such as those described in Sects. 3.3 and 3.4 .
However, such methods have yielded widely variable results, likely due to the
difficulty in consistently isolating substructures without causing tissue damage.
Microscopic imaging studies, such as the confocal studies described in Sect. 3.4 ,
provide considerable promise for use in validation of microscale models. How-
ever, the highly inhomogeneous strain fields and the complex microscale fiber
structure make this a challenging starting point [ 199 , 201 , 202 ].
In order to address these challenges, our lab has developed a surrogate material
for use as a physical model to aid in the development of multiscale modeling and
validation methods. A physical model reduces the number of uncontrolled vari-
ables related to the structural organization of ligaments and tendons. To create the
physical surrogates, dense (*25 % collage/wt), extruded collagen fibers were
embedded within a collagen gel matrix (*0.5 % collagen/wt). Surrogates served
as physical models to emulate features of ligament and tendon tissue in a con-
trolled and reproducible manner. Two different colors of fluorescent beads were
embedded in the fibers and gel matrix (Fig. 14 , left) for use as microscopic fiducial
markers. 3D micromechanical FE models of the surrogates were then constructed
(Fig. 14 , bottom). A constitutive model based on a continuous elliptical fiber
distribution was used to describe the mechanical behavior of the collagen gel and
embedded fibers [ 14 ]. This constitutive model emulated the reorganization of
fibrils with applied strain. The model was curve fit to tensile testing data for
isolated gel and extruded fiber samples and was found to accurately model both the
uniaxial stress-strain behavior and the 2D strain behavior (i.e., the nonlinear
Poisson's function). Micromechanical FE models were subjected to uniaxial strain,
and the macroscale and microscale stress and 2D strain were determined. FEBio
was used for all analysis ( http://www.febio.org )[ 148 ]. To validate the FE models,
the physical surrogates were subjected to tensile loading in a custom testing
apparatus on an inverted confocal microscope. Confocal images were acquired at 6
strain increments at both 2.5X and 10X, while force was measured simultaneously.
Texture correlation was used to measure strain at the macroscale and to measure
strain within the fibers and strain in the inter-fiber matrix at the microscale [ 221 ].
The microscopic 2D strains were inhomogeneous, and the macroscopic 2D
strain was not representative of the microscopic 2D strain (Fig. 14 , right). The
magnitude of the transverse strain in the fibers greatly exceeded the macroscopic
transverse strain, while the magnitude of the transverse matrix strain was signif-
icantly less than the macroscopic strain. The macroscopically measured Poisson's
ratio was 1.72 ± 0.26, which is comparable to experimentally measured values for
tendon and ligament [ 105 , 147 ]. The micromechanical FE model was able to
simultaneously predict the macroscopic stress-strain behavior and the 2D mac-
roscale and microscale strains (Fig. 14 , right). The predicted macroscopic stress
and macroscopic transverse strain closely matched the experimentally measured
values with normalized root mean square (NRMSE) values of 0.015 and 0.085,
respectively.

The predicted microscopic transverse fiber strain was closely