VIGRE seminar, spring 2000:
Soft Tissue Modeling
- Instructors
-
Jay Walton,
David Dobson,
Jeff Morgan,
Patrick Wilber
- Students enrolled
-
Heather Gollmer,
Vincent Lemoine,
Autumn Moody,
Nicholas Neumann,
Matthew Riddle,
Melinda Williams
(undergraduate mathematics students);
Nandini Duraiswamy,
Yuan Xu
(graduate bio-engineering students);
Guido David,
Sami Hamid,
Damali Moore,
John Ryan,
Ediz Tufekcioglu,
David Zeigler
(graduate mathematics students)
- Description
-
The general topic involved soft-tissue
modeling. Specifically, we worked with Jay Humphrey
from the Bioengineering Department. The problem
concerned modeling the mechanical response of
arteries. The clinical issue involves the important
phenomenon of mechanically induced chemical changes in
living cells. For example, it is conjectured that
mechanical stresses and strains play an important role
in the initial formation and subsequent growth of
atherosclerotic plaques on the inner surfaces of
arteries. Humphrey's students are conducting mechanical
tests on arteries trying to model their mechanical
response to inflation, elongation and torsion. They had
been trying to fit certain standard models due to
Y. C. Fung from nonlinear elasticity to their data
with very limited success. They had difficulty in
matching their experimental data and also in trying to
solve boundary value problems numerically. One of
Humphrey's students who was actually collecting data on
arteries was a student in the class. So we had access
to the best data available. The mathematics was a very
heavy go for the undergraduates, but we did begin to
develop some new models. Pat Wilber and Jay Walton
decided to check the standard models the engineers were
using for certain mathematical properties, namely
strong ellipticity and other ellipticity or convexity
properties, that have a strong bearing on the
qualitative predictions they make and their numerical
stability. These investigations led to two papers by
Walton and Wilber in which they show that the standard
models fail to satisfy strong ellipticity for ANY
choice of parameters (to say the least, this surprised
the experienced practitioners) and developed some new
classes of models for which they gave convenient tests
for when strong ellipticity holds and when it doesn't.
One of the main challenges for the modeling and
analysis of arterial tissue is that it is highly
nonhomogeneous and anisotropic The students found that
taking isotropic models or models with spatially
averaged fibre orientation and stiffnesses do not do a
very satisfactory job of modeling the experimental
data. The class was divided into groups who worked with
different modeling approaches on the same data. They
then wrote reports and gave presentations on their
results comparing how each approach fared. The new
approaches got reasonable agreement with some of the
test data, but not so good with the other. Some of this
work carried over to future VIGRE courses.
- Impact
-
Two papers were started by Walton and Wilber.
One of them has now been accepted by the
International Journal of Non-Linear Mechanics.