Coupled Macroscopic - Microscopic Modeling of Biomaterials
Mathematical modeling of tissues and tissue equivalents (constructs formed
by entrapping cells in a reconstituted biopolymer matrix, typically type
I collagen) is difficult for many reasons, one of which is the lack of
a good fundamental constitutive equation. In part, this lack arises
from the fact that tissues must function on the millimeter to centimeter
length scale, but the events governing the mechanical behavior occur on
the nanometer to micron scale, as can be seen in the scanning
electron micrgraph below. The challenge is to incorporate the
microstructural information in a way that does not render the macroscopic
modeling problem intractable.
Because we believe that the microstructure and the fiber-fiber interactions
are important (in contrast to traditional fiber-reinforced composites,
in which individual fibers do not interact with each other), we have developed
a strategy that consists of a coupled macroscopic-microscopic approach:
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The macroscopic problem is discretized using the finite element method,
just as one would do for any macroscopic mechanical problem.
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Instead of writing a constitutive equation for the stress in each element,
we introduce a microscopic problem (see figure below) based on the local
microstructure of the material. This problem is not meant to account
for all fibers within the space occupied by the finite element, but rather
to be a representation of the underlying microstructure. The solution
to the microscopic problem is project back onto the macroscopic domain,
allowing us to determine the stress.
The coupled approach has two major advantages. First, although the
microscopic problem adds considerable complexity to the overall calculation,
since fibers exist only in a given element, the problem exhibits good scalability.
Second, since we generate different microscopic meshes for each macroscopic
element, we can readily explore how spatial variations in microstructure
(i.e., fiber alignment, thickness, and density, and crosslink density)
affect the macroscopic mechanical behavior.
We are currently collaborating with Bob
Tranquillo on incorporating optically-measured matrix properties into
our model (the model is no good if we can't get data for it!), and we are
also exploring what features of the microstructure are essential in determining
the material properties and constructing a biphasic "wrapper" for our code
to account for interstitial flow in the tissue / tissue equivalent.
This work is supported by the University of Minnesota MRSEC
and by a supercomputing resources grant from the Minnesota
Sypercomputing Institute.
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| Scanning electron micrographs of (a) unaligned and (b) aligned collagen
gels. Scale bar in lower corner of (b) is 1 micron. |