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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.